Parameterization of Stillinger-Weber Potential for Two-Dimensional Atomic Crystals

TL;DR

This paper develops parameter sets for the Stillinger-Weber potential tailored to 156 two-dimensional atomic crystals, enabling efficient nonlinear simulations of their physical and mechanical properties.

Contribution

It introduces a systematic method to derive Stillinger-Weber potential parameters from the valence force field model for a wide range of 2D materials.

Findings

Parameters for 156 2D crystals are provided.

Resources include code and files for various simulations.

Potential enables accurate nonlinear physical modeling.

Abstract

We parametrize the Stillinger-Weber potential for 156 two-dimensional atomic crystals. Parameters for the Stillinger-Weber potential are obtained from the valence force field model following the analytic approach (Nanotechnology 26, 315706 (2015)), in which the valence force constants are determined by the phonon spectrum. The Stillinger-Weber potential is an efficient nonlinear interaction, and is applicable for numerical simulations of nonlinear physical or mechanical processes. The supplemental resources for all simulations in the present work are available online in Ref. 1, including a fortran code to generate crystals' structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the Stillinger-Weber potential using GULP, and files for phonon calculations with the valence force field model using GULP.

Tables

Table 1. Table 1: 1H-MX 2 , with M as the transition metal and X as oxygen or dichalcogenide. The structure is shown in Fig.  1 .

1H-ScO2	1H-ScS2	1H-ScSe2	1H-ScTe2	1H-TiTe2	1H-VO2	1H-VS2	1H-VSe2	1H-VTe2
1H-CrO2	1H-CrS2	1H-CrSe2	1H-CrTe2	1H-MnO2	1H-FeO2	1H-FeS2	1H-FeSe2	1H-FeTe2
1H-CoTe2	1H-NiS2	1H-NiSe2	1H-NiTe2	1H-NbS2	1H-NbSe2	1H-MoO2	1H-MoS2	1H-MoSe2
1H-MoTe2	1H-TaS2	1H-TaSe2	1H-WO2	1H-WS2	1H-WSe2	1H-WTe2

Table 2. Table 2: 1T-MX 2 , with M as the transition metal and X as oxygen or dichalcogenide. The structure is shown in Fig.  71 .

1T-ScO2	1T-ScS2	1T-ScSe2	1T-ScTe2	1T-TiS2	1T-TiSe2	1T-TiTe2	1T-VS2	1T-VSe2
1T-VTe2	1T-MnO2	1T-MnS2	1T-MnSe2	1T-MnTe2	1T-CoTe2	1T-NiO2	1T-NiS2	1T-NiSe2
1T-NiTe2	1T-ZrS2	1T-ZrSe2	1T-ZrTe2	1T-NbS2	1T-NbSe2	1T-NbTe2	1T-MoS2	1T-MoSe2
1T-MoTe2	1T-TcS2	1T-TcSe2	1T-TcTe2	1T-RhTe2	1T-PdS2	1T-PdSe2	1T-PdTe2	1T-SnS2
1T-SnSe2	1T-HfS2	1T-HfSe2	1T-HfTe2	1T-TaS2	1T-TaSe2	1T-TaTe2	1T-WS2	1T-WSe2
1T-WTe2	1T-ReS2	1T-ReSe2	1T-ReTe2	1T-IrTe2	1T-PtS2	1T-PtSe2	1T-PtTe2

Table 3. Table 3: Puckered (p-) M, with M from group V. The structure is shown in Fig.  178 or Fig.  183 .

black phosphorus

p-arsenene

p-antimonene

p-bismuthene

Table 4. Table 4: Puckered MX, with M from group IV and X from group VI. The structure is shown in Fig.  189 , and particularly Fig.  191 for p-MX with X=O.

	p-SiO	p-GeO	p-SnO
p-CS	p-SiS	p-GeS	p-SnS
p-CSe	p-SiSe	p-GeSe	p-SnSe
p-CTe	p-SiTe	p-GeTe	p-SnTe

Table 5. Table 5: Buckled (b-) M, with M from group IV or V. The structure is shown in Fig.  222 .

silicene	germanene	stanene	indiene
blue phosphorus	b-arsenene	b-antimonene	b-bismuthene

Table 6. Table 6: Buckled MX, with M from group IV and X from group VI. The structure is shown in Fig.  239 .

b-CO	b-SiO	b-GeO	b-SnO
b-CS	b-SiS	b-GeS	b-SnS
b-CSe	b-SiSe	b-GeSe	b-SnSe
b-CTe	b-SiTe	b-GeTe	b-SnTe

Table 7. Table 7: Buckled MX, with both M and X from group IV, or M from group III and X from group V. The structure is shown in Fig.  239 .

b-SnGe

b-SiGe

b-SnSi

b-InP

b-InAs

b-InSb

b-GaAs

b-GaP

b-AlSb

Table 8. Table 8: Bi-buckled MX, with M from group III and X from group VI. The structure is shown in Fig.  290 .

BO	AlO	GaO	InO
BS	AlS	GaS	InS
BSe	AlSe	GaSe	InSe
BTe	AlTe	GaTe	InTe

Table 9. Table 9: Others.

borophene

Table 10. Table 10: The VFF model for single-layer 1H-ScO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	9.417	4.825	4.825	4.825
$r_0$ or ${\theta }_0$	2.090	98.222	58.398	98.222

Table 11. Table 11: Two-body SW potential parameters for single-layer 1H-ScO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-O	7.506	1.380	9.540	0.0	2.939

Table 12. Table 12: Three-body SW potential parameters for single-layer 1H-ScO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-O-O}$	63.576	98.222	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460
${\theta }_{\mathrm{Sc}-O-O^{\prime }}$	85.850	58.398	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460
${\theta }_{O-\mathrm{Sc}-\mathrm{Sc}}$	63.576	98.222	1.380	1.380	0.0	2.939	0.0	2.939	0.0	3.460

Table 13. Table 13: SW potential parameters for single-layer 1H-ScO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc1-O1-O1	1.000	1.380	2.129	0.000	1.000	0.000	7.506	2.627	4	0	0.0
Sc1-O1-O3	1.000	0.000	0.000	63.576	1.000	-0.143	0.000	0.000	4	0	0.0
Sc1-O1-O2	1.000	0.000	0.000	85.850	1.000	0.524	0.000	0.000	4	0	0.0
O1-Sc1-Sc3	1.000	0.000	0.000	63.576	1.000	-0.143	0.000	0.000	4	0	0.0

Table 14. Table 14: The VFF model for single-layer 1H-ScS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	5.192	2.027	2.027	2.027
$r_0$ or ${\theta }_0$	2.520	94.467	64.076	94.467

Table 15. Table 15: Two-body SW potential parameters for single-layer 1H-ScS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-S	5.505	1.519	20.164	0.0	3.498

Table 16. Table 16: Three-body SW potential parameters for single-layer 1H-ScS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-S-S}$	22.768	94.467	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132
${\theta }_{\mathrm{Sc}-S-S^{\prime }}$	27.977	64.076	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132
${\theta }_{S-\mathrm{Sc}-\mathrm{Sc}}$	22.768	94.467	1.519	1.519	0.0	3.498	0.0	3.498	0.0	4.132

Table 17. Table 17: SW potential parameters for single-layer 1H-ScS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc1-S1-S1	1.000	1.519	2.303	0.000	1.000	0.000	5.505	3.784	4	0	0.0
Sc1-S1-S3	1.000	0.000	0.000	22.768	1.000	-0.078	0.000	0.000	4	0	0.0
Sc1-S1-S2	1.000	0.000	0.000	27.977	1.000	0.437	0.000	0.000	4	0	0.0
S1-Sc1-Sc3	1.000	0.000	0.000	22.768	1.000	-0.078	0.000	0.000	4	0	0.0

Table 18. Table 18: The VFF model for single-layer 1H-ScSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	5.192	2.027	2.027	2.027
$r_0$ or ${\theta }_0$	2.650	92.859	66.432	92.859

Table 19. Table 19: Two-body SW potential parameters for single-layer 1H-ScSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-Se	5.853	1.533	24.658	0.0	3.658

Table 20. Table 20: Three-body SW potential parameters for single-layer 1H-ScSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-\mathrm{Se}-\mathrm{Se}}$	21.292	92.859	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327
${\theta }_{\mathrm{Sc}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	25.280	66.432	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327
${\theta }_{\mathrm{Se}-\mathrm{Sc}-\mathrm{Sc}}$	21.292	92.859	1.533	1.533	0.0	3.658	0.0	3.658	0.0	4.327

Table 21. Table 21: SW potential parameters for single-layer 1H-ScSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc1-Se1-Se1	1.000	1.533	2.386	0.000	1.000	0.000	5.853	4.464	4	0	0.0
Sc1-Se1-Se3	1.000	0.000	0.000	21.292	1.000	-0.050	0.000	0.000	4	0	0.0
Sc1-Se1-Se2	1.000	0.000	0.000	25.280	1.000	0.400	0.000	0.000	4	0	0.0
Se1-Sc1-Sc3	1.000	0.000	0.000	21.292	1.000	-0.050	0.000	0.000	4	0	0.0

Table 22. Table 22: The VFF model for single-layer 1H-ScTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	5.192	2.027	2.027	2.027
$r_0$ or ${\theta }_0$	2.890	77.555	87.364	87.364

Table 23. Table 23: Two-body SW potential parameters for single-layer 1H-ScTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-Te	4.630	1.050	34.879	0.0	3.761

Table 24. Table 24: Three-body SW potential parameters for single-layer 1H-ScTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-\mathrm{Te}-\mathrm{Te}}$	11.848	77.555	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504
${\theta }_{\mathrm{Sc}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	11.322	87.364	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504
${\theta }_{\mathrm{Te}-\mathrm{Sc}-\mathrm{Sc}}$	11.848	77.555	1.050	1.050	0.0	3.761	0.0	3.761	0.0	4.504

Table 25. Table 25: SW potential parameters for single-layer 1H-ScTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc1-Te1-Te1	1.000	1.050	3.581	0.000	1.000	0.000	4.630	28.679	4	0	0.0
Sc1-Te1-Te3	1.000	0.000	0.000	11.848	1.000	0.216	0.000	0.000	4	0	0.0
Sc1-Te1-Te2	1.000	0.000	0.000	11.322	1.000	0.046	0.000	0.000	4	0	0.0
Te1-Sc1-Sc3	1.000	0.000	0.000	11.848	1.000	0.216	0.000	0.000	4	0	0.0

Table 26. Table 26: The VFF model for single-layer 1H-TiTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ti}-\mathrm{Ti}}{\left(\Delta \theta \right)}^2$
parameter	4.782	3.216	3.216	3.216
$r_0$ or ${\theta }_0$	2.750	82.323	81.071	82.323

Table 27. Table 27: Two-body SW potential parameters for single-layer 1H-TiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ti-Te	4.414	1.173	28.596	0.0	3.648

Table 28. Table 28: Three-body SW potential parameters for single-layer 1H-TiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ti}-\mathrm{Te}-\mathrm{Te}}$	22.321	82.323	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354
${\theta }_{\mathrm{Ti}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	22.463	81.071	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354
${\theta }_{\mathrm{Te}-\mathrm{Ti}-\mathrm{Ti}}$	11.321	82.323	1.173	1.173	0.0	3.648	0.0	3.648	0.0	4.354

Table 29. Table 29: SW potential parameters for single-layer 1H-TiTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Ti and X=Te).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ti1-Te1-Te1	1.000	1.173	3.110	0.000	1.000	0.000	4.414	15.100	4	0	0.0
Ti1-Te1-Te3	1.000	0.000	0.000	22.321	1.000	0.134	0.000	0.000	4	0	0.0
Ti1-Te1-Te2	1.000	0.000	0.000	22.463	1.000	0.155	0.000	0.000	4	0	0.0
Te1-Ti1-Ti3	1.000	0.000	0.000	22.321	1.000	0.134	0.000	0.000	4	0	0.0

Table 30. Table 30: The VFF model for single-layer 1H-VO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{V-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-V-V}{\left(\Delta \theta \right)}^2$
parameter	9.417	4.825	4.825	4.825
$r_0$ or ${\theta }_0$	1.920	89.356	71.436	89.356

Table 31. Table 31: Two-body SW potential parameters for single-layer 1H-VO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-O	5.105	1.011	6.795	0.0	2.617

Table 32. Table 32: Three-body SW potential parameters for single-layer 1H-VO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-O-O}$	43.951	89.356	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105
${\theta }_{V-O-O^{\prime }}$	48.902	71.436	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105
${\theta }_{O-V-V}$	43.951	89.356	1.011	1.011	0.0	2.617	0.0	2.617	0.0	3.105

Table 33. Table 33: SW potential parameters for single-layer 1H-VO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V1-O1-O1	1.000	1.011	2.589	0.000	1.000	0.000	5.105	6.509	4	0	0.0
V1-O1-O3	1.000	0.000	0.000	43.951	1.000	0.011	0.000	0.000	4	0	0.0
V1-O1-O2	1.000	0.000	0.000	48.902	1.000	0.318	0.000	0.000	4	0	0.0
O1-V1-V3	1.000	0.000	0.000	43.951	1.000	0.011	0.000	0.000	4	0	0.0

Table 34. Table 34: The VFF model for single-layer 1H-VS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{V-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-V-V}{\left(\Delta \theta \right)}^2$
parameter	8.392	4.074	4.074	4.074
$r_0$ or ${\theta }_0$	2.310	83.954	78.878	83.954

Table 35. Table 35: Two-body SW potential parameters for single-layer 1H-VS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-S	5.714	1.037	14.237	0.0	3.084

Table 36. Table 36: Three-body SW potential parameters for single-layer 1H-VS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-S-S}$	30.059	83.954	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676
${\theta }_{V-S-S^{\prime }}$	30.874	78.878	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676
${\theta }_{S-V-V}$	30.059	83.954	1.037	1.037	0.0	3.084	0.0	3.084	0.0	3.676

Table 37. Table 37: SW potential parameters for single-layer 1H-VS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=V and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V1-S1-S1	1.000	1.037	2.973	0.000	1.000	0.000	5.714	12.294	4	0	0.0
V1-S1-S3	1.000	0.000	0.000	30.059	1.000	0.105	0.000	0.000	4	0	0.0
V1-S1-S2	1.000	0.000	0.000	30.874	1.000	0.193	0.000	0.000	4	0	0.0
S1-V1-V3	1.000	0.000	0.000	30.059	1.000	0.105	0.000	0.000	4	0	0.0

Table 38. Table 38: The VFF model for single-layer 1H-VSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-V-V}{\left(\Delta \theta \right)}^2$
parameter	6.492	4.716	4.716	4.716
$r_0$ or ${\theta }_0$	2.450	82.787	80.450	82.787

Table 39. Table 39: Two-body SW potential parameters for single-layer 1H-VSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-Se	4.817	1.061	18.015	0.0	3.256

Table 40. Table 40: Three-body SW potential parameters for single-layer 1H-VSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-\mathrm{Se}-\mathrm{Se}}$	33.299	82.787	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884
${\theta }_{V-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	33.702	80.450	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884
${\theta }_{\mathrm{Se}-V-V}$	33.299	82.787	1.061	1.061	0.0	3.256	0.0	3.256	0.0	3.884

Table 41. Table 41: SW potential parameters for single-layer 1H-VSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=V and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V1-Se1-Se1	1.000	1.061	3.070	0.000	1.000	0.000	4.817	14.236	4	0	0.0
V1-Se1-Se3	1.000	0.000	0.000	33.299	1.000	0.126	0.000	0.000	4	0	0.0
V1-Se1-Se2	1.000	0.000	0.000	33.702	1.000	0.166	0.000	0.000	4	0	0.0
Se1-V1-V3	1.000	0.000	0.000	33.299	1.000	0.126	0.000	0.000	4	0	0.0

Table 42. Table 42: The VFF model for single-layer 1H-VTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-V-V}{\left(\Delta \theta \right)}^2$
parameter	6.371	4.384	4.384	4.384
$r_0$ or ${\theta }_0$	2.660	81.708	81.891	81.708

Table 43. Table 43: Two-body SW potential parameters for single-layer 1H-VTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-Te	5.410	1.112	25.032	0.0	3.520

Table 44. Table 44: Three-body SW potential parameters for single-layer 1H-VTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-\mathrm{Te}-\mathrm{Te}}$	29.743	81.708	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203
${\theta }_{V-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	29.716	81.891	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203
${\theta }_{\mathrm{Te}-V-V}$	29.743	81.708	1.112	1.112	0.0	3.520	0.0	3.520	0.0	4.203

Table 45. Table 45: SW potential parameters for single-layer 1H-VTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=V and X=Te).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V1-Te1-Te1	1.000	1.112	3.164	0.000	1.000	0.000	5.410	16.345	4	0	0.0
V1-Te1-Te3	1.000	0.000	0.000	29.743	1.000	0.144	0.000	0.000	4	0	0.0
V1-Te1-Te2	1.000	0.000	0.000	29.716	1.000	0.141	0.000	0.000	4	0	0.0
Te1-V1-V3	1.000	0.000	0.000	29.743	1.000	0.144	0.000	0.000	4	0	0.0

Table 46. Table 46: The VFF model for single-layer 1H-CrO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Cr}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Cr}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Cr}-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Cr}-\mathrm{Cr}}{\left(\Delta \theta \right)}^2$
parameter	12.881	8.039	8.039	8.039
$r_0$ or ${\theta }_0$	1.880	86.655	75.194	86.655

Table 47. Table 47: Two-body SW potential parameters for single-layer 1H-CrO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Cr-O	6.343	0.916	6.246	0.0	2.536

Table 48. Table 48: Three-body SW potential parameters for single-layer 1H-CrO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Cr}-O-O}$	65.805	86.655	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016
${\theta }_{\mathrm{Cr}-O-O^{\prime }}$	70.163	75.194	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016
${\theta }_{O-\mathrm{Cr}-\mathrm{Cr}}$	65.805	86.655	0.916	0.916	0.0	2.536	0.0	2.536	0.0	3.016

Table 49. Table 49: SW potential parameters for single-layer 1H-CrO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Cr1-O1-O1	1.000	0.916	2.769	0.000	1.000	0.000	6.242	8.871	4	0	0.0
Cr1-O1-O3	1.000	0.000	0.000	65.805	1.000	0.058	0.000	0.000	4	0	0.0
Cr1-O1-O2	1.000	0.000	0.000	70.163	1.000	0.256	0.000	0.000	4	0	0.0
O1-Cr1-Cr3	1.000	0.000	0.000	65.805	1.000	0.058	0.000	0.000	4	0	0.0

Table 50. Table 50: The VFF model for single-layer 1H-CrS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Cr-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Cr-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Cr-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-Cr-Cr}{\left(\Delta \theta \right)}^2$
parameter	8.752	4.614	4.614	4.614
$r_0$ or ${\theta }_0$	2.254	83.099	80.031	83.099

Table 51. Table 51: Two-body SW potential parameters for single-layer 1HCrS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Cr-S	5.544	0.985	12.906	0.0	2.999

Table 52. Table 52: Three-body SW potential parameters for single-layer 1H-CrS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Cr}-S-S}$	32.963	83.099	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577
${\theta }_{\mathrm{Cr}-S-S^{\prime }}$	33.491	80.031	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577
${\theta }_{S-\mathrm{Cr}-\mathrm{Cr}}$	32.963	83.099	0.985	0.985	0.0	2.999	0.0	2.999	0.0	3.577

Table 53. Table 53: SW potential parameters for single-layer 1H-CrS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Cr and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Cr1-S1-S1	1.000	0.985	3.043	0.000	1.000	0.000	5.544	13.683	4	0	0.0
Cr1-S1-S3	1.000	0.000	0.000	32.963	1.000	0.120	0.000	0.000	4	0	0.0
Cr1-S1-S2	1.000	0.000	0.000	33.491	1.000	0.173	0.000	0.000	4	0	0.0
S1-Cr1-Cr3	1.000	0.000	0.000	32.963	1.000	0.120	0.000	0.000	4	0	0.0

Table 54. Table 54: The VFF model for single-layer 1H-CrSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Cr-Se}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Cr-Se-Se}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Cr-Se-S{e}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Se-Cr-Cr}{\left(\Delta \theta \right)}^2$
parameter	9.542	4.465	4.465	4.465
$r_0$ or ${\theta }_0$	2.380	82.229	81.197	82.229

Table 55. Table 55: Two-body SW potential parameters for single-layer 1H-CrSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Cr-Se	6.581	1.012	16.043	0.0	3.156

Table 56. Table 56: Three-body SW potential parameters for single-layer 1H-CrSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Cr}-\mathrm{Se}-\mathrm{Se}}$	30.881	82.229	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767
${\theta }_{\mathrm{Cr}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	31.044	81.197	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767
${\theta }_{\mathrm{Se}-\mathrm{Cr}-\mathrm{Cr}}$	30.881	82.229	1.012	1.012	0.0	3.156	0.0	3.156	0.0	3.767

Table 57. Table 57: SW potential parameters for single-layer 1H-CrSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Cr and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Cr1-Se1-Se1	1.000	1.012	3.118	0.000	1.000	0.000	6.581	15.284	4	0	0.0
Cr1-Se1-Se3	1.000	0.000	0.000	30.881	1.000	0.135	0.000	0.000	4	0	0.0
Cr1-Se1-Se2	1.000	0.000	0.000	31.044	1.000	0.153	0.000	0.000	4	0	0.0
Se1-Cr1-Cr3	1.000	0.000	0.000	30.881	1.000	0.135	0.000	0.000	4	0	0.0

Table 58. Table 58: The VFF model for single-layer 1H-CrTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Cr-Te}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Cr-Te-Te}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Cr-Te-T{e}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Te-Cr-Cr}{\left(\Delta \theta \right)}^2$
parameter	8.197	4.543	4.543	4.543
$r_0$ or ${\theta }_0$	2.580	82.139	81.316	82.139

Table 59. Table 59: Two-body SW potential parameters for single-layer 1H-CrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Cr-Te	6.627	1.094	22.154	0.0	3.420

Table 60. Table 60: Three-body SW potential parameters for single-layer 1H-CrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Cr}-\mathrm{Te}-\mathrm{Te}}$	31.316	82.139	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082
${\theta }_{\mathrm{Cr}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	31.447	81.316	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082
${\theta }_{\mathrm{Te}-\mathrm{Cr}-\mathrm{Cr}}$	31.316	82.139	1.094	1.094	0.0	3.420	0.0	3.420	0.0	4.082

Table 61. Table 61: SW potential parameters for single-layer 1H-CrTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Cr and X=Te).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Cr1-Te1-Te1	1.000	1.094	3.126	0.000	1.000	0.000	6.627	15.461	4	0	0.0
Cr1-Te1-Te3	1.000	0.000	0.000	31.316	1.000	0.137	0.000	0.000	4	0	0.0
Cr1-Te1-Te2	1.000	0.000	0.000	31.447	1.000	0.151	0.000	0.000	4	0	0.0
Te1-Cr1-Cr3	1.000	0.000	0.000	31.316	1.000	0.137	0.000	0.000	4	0	0.0

Table 62. Table 62: The VFF model for single-layer 1H-MnO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mn}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Mn}-\mathrm{Mn}}{\left(\Delta \theta \right)}^2$
parameter	9.382	6.253	6.253	6.253
$r_0$ or ${\theta }_0$	1.870	88.511	72.621	88.511

Table 63. Table 63: Two-body SW potential parameters for single-layer 1H-MnO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mn-O	4.721	0.961	6.114	0.0	2.540

Table 64. Table 64: Three-body SW potential parameters for single-layer 1H-MnO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mn}-O-O}$	55.070	88.511	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016
${\theta }_{\mathrm{Mn}-O-O^{\prime }}$	60.424	72.621	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016
${\theta }_{O-\mathrm{Mn}-\mathrm{Mn}}$	55.070	88.511	0.961	0.961	0.0	2.540	0.0	2.540	0.0	3.016

Table 65. Table 65: SW potential parameters for single-layer 1H-MnO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mn1-O1-O1	1.000	0.961	2.643	0.000	1.000	0.000	4.721	7.158	4	0	0.0
Mn1-O1-O3	1.000	0.000	0.000	55.070	1.000	0.026	0.000	0.000	4	0	0.0
Mn1-O1-O2	1.000	0.000	0.000	60.424	1.000	0.299	0.000	0.000	4	0	0.0
O1-Mn1-Mn3	1.000	0.000	0.000	55.070	1.000	0.026	0.000	0.000	4	0	0.0

Table 66. Table 66: The VFF model for single-layer 1H-FeO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Fe}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Fe}-\mathrm{Fe}}{\left(\Delta \theta \right)}^2$
parameter	8.377	3.213	3.213	3.213
$r_0$ or ${\theta }_0$	1.880	88.343	72.856	88.343

Table 67. Table 67: Two-body SW potential parameters for single-layer 1H-FeO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Fe-O	4.242	0.962	6.246	0.0	2.552

Table 68. Table 68: Three-body SW potential parameters for single-layer 1H-FeO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Fe}-O-O}$	28.105	88.343	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031
${\theta }_{\mathrm{Fe}-O-O^{\prime }}$	30.754	72.856	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031
${\theta }_{O-\mathrm{Fe}-\mathrm{Fe}}$	28.105	88.343	0.962	0.962	0.0	2.552	0.0	2.552	0.0	3.031

Table 69. Table 69: SW potential parameters for single-layer 1H-FeO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Fe1-O1-O1	1.000	0.962	2.654	0.000	1.000	0.000	4.242	7.298	4	0	0.0
Fe1-O1-O3	1.000	0.000	0.000	28.105	1.000	0.029	0.000	0.000	4	0	0.0
Fe1-O1-O2	1.000	0.000	0.000	30.754	1.000	0.295	0.000	0.000	4	0	0.0
O1-Fe1-Fe3	1.000	0.000	0.000	28.105	1.000	0.029	0.000	0.000	4	0	0.0

Table 70. Table 70: The VFF model for single-layer 1H-FeS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Fe}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Fe}-\mathrm{Fe}}{\left(\Delta \theta \right)}^2$
parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.220	87.132	74.537	87.132

Table 71. Table 71: Two-body SW potential parameters for single-layer 1H-FeS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Fe-S	4.337	1.097	12.145	0.0	3.000

Table 72. Table 72: Three-body SW potential parameters for single-layer 1H-FeS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Fe}-S-S}$	33.060	87.132	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567
${\theta }_{\mathrm{Fe}-S-S^{\prime }}$	35.501	74.537	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567
${\theta }_{S-\mathrm{Fe}-\mathrm{Fe}}$	33.060	87.132	1.097	1.097	0.0	3.000	0.0	3.000	0.0	3.567

Table 73. Table 73: SW potential parameters for single-layer 1H-FeS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Fe1-S1-S1	1.000	1.097	2.735	0.000	1.000	0.000	4.337	8.338	4	0	0.0
Fe1-S1-S3	1.000	0.000	0.000	33.060	1.000	0.050	0.000	0.000	4	0	0.0
Fe1-S1-S2	1.000	0.000	0.000	35.501	1.000	0.267	0.000	0.000	4	0	0.0
S1-Fe1-Fe3	1.000	0.000	0.000	33.060	1.000	0.050	0.000	0.000	4	0	0.0

Table 74. Table 74: The VFF model for single-layer 1H-FeSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Fe}-\mathrm{Fe}}{\left(\Delta \theta \right)}^2$
parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.350	86.488	75.424	86.488

Table 75. Table 75: Two-body SW potential parameters for single-layer 1H-FeSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Fe-Se	4.778	1.139	15.249	0.0	3.168

Table 76. Table 76: Three-body SW potential parameters for single-layer 1H-FeSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Fe}-\mathrm{Se}-\mathrm{Se}}$	32.235	86.488	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768
${\theta }_{\mathrm{Fe}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	34.286	75.424	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768
${\theta }_{\mathrm{Se}-\mathrm{Fe}-\mathrm{Fe}}$	32.235	86.488	1.139	1.139	0.0	3.168	0.0	3.168	0.0	3.768

Table 77. Table 77: SW potential parameters for single-layer 1H-FeSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Fe1-Se1-Se1	1.000	1.139	2.781	0.000	1.000	0.000	4.778	9.049	4	0	0.0
Fe1-Se1-Se3	1.000	0.000	0.000	32.235	1.000	0.061	0.000	0.000	4	0	0.0
Fe1-Se1-Se2	1.000	0.000	0.000	34.286	1.000	0.252	0.000	0.000	4	0	0.0
Se1-Fe1-Fe3	1.000	0.000	0.000	32.235	1.000	0.061	0.000	0.000	4	0	0.0

Table 78. Table 78: The VFF model for single-layer 1H-FeTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Fe}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Fe}-\mathrm{Fe}}{\left(\Delta \theta \right)}^2$
parameter	6.338	3.964	3.964	3.964
$r_0$ or ${\theta }_0$	2.530	86.904	74.851	86.904

Table 79. Table 79: Two-body SW potential parameters for single-layer 1H-FeTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Fe-Te	5.599	1.242	20.486	0.0	3.416

Table 80. Table 80: Three-body SW potential parameters for single-layer 1H-FeTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Fe}-\mathrm{Te}-\mathrm{Te}}$	32.766	86.904	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062
${\theta }_{\mathrm{Fe}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	35.065	74.851	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062
${\theta }_{\mathrm{Te}-\mathrm{Fe}-\mathrm{Fe}}$	32.766	86.904	1.242	1.242	0.0	3.416	0.0	3.416	0.0	4.062

Table 81. Table 81: SW potential parameters for single-layer 1H-FeTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Fe1-Te1-Te1	1.000	1.242	2.751	0.000	1.000	0.000	5.599	8.615	4	0	0.0
Fe1-Te1-Te3	1.000	0.000	0.000	32.766	1.000	0.054	0.000	0.000	4	0	0.0
Fe1-Te1-Te2	1.000	0.000	0.000	35.065	1.000	0.261	0.000	0.000	4	0	0.0
Te1-Fe1-Fe3	1.000	0.000	0.000	32.766	1.000	0.054	0.000	0.000	4	0	0.0

Table 82. Table 82: The VFF model for single-layer 1H-CoTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Co}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Co}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Co}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Co}-\mathrm{Co}}{\left(\Delta \theta \right)}^2$
parameter	6.712	2.656	2.656	2.656
$r_0$ or ${\theta }_0$	2.510	89.046	71.873	89.046

Table 83. Table 83: Two-body SW potential parameters for single-layer 1H-CoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Co-Te	6.169	1.310	19.846	0.0	3.417

Table 84. Table 84: Three-body SW potential parameters for single-layer 1H-CoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Co}-\mathrm{Te}-\mathrm{Te}}$	23.895	89.046	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055
${\theta }_{\mathrm{Co}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	26.449	71.873	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055
${\theta }_{\mathrm{Te}-\mathrm{Co}-\mathrm{Co}}$	23.895	89.046	1.310	1.310	0.0	3.417	0.0	3.417	0.0	4.055

Table 85. Table 85: SW potential parameters for single-layer 1H-CoTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Co1-Te1-Te1	1.000	1.310	2.608	0.000	1.000	0.000	6.169	6.739	4	0	0.0
Co1-Te1-Te3	1.000	0.000	0.000	23.895	1.000	0.017	0.000	0.000	4	0	0.0
Co1-Te1-Te2	1.000	0.000	0.000	26.449	1.000	0.311	0.000	0.000	4	0	0.0
Te1-Co1-Co3	1.000	0.000	0.000	23.895	1.000	0.017	0.000	0.000	4	0	0.0

Table 86. Table 86: The VFF model for single-layer 1H-NiS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	6.933	3.418	3.418	3.418
$r_0$ or ${\theta }_0$	2.240	98.740	57.593	98.740

Table 87. Table 87: Two-body SW potential parameters for single-layer 1H-NiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-S	6.425	1.498	12.588	0.0	3.156

Table 88. Table 88: Three-body SW potential parameters for single-layer 1H-NiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-S-S}$	46.062	98.740	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713
${\theta }_{\mathrm{Ni}-S-S^{\prime }}$	63.130	57.593	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713
${\theta }_{S-\mathrm{Ni}-\mathrm{Ni}}$	46.062	98.740	1.498	1.498	0.0	3.156	0.0	3.156	0.0	3.713

Table 89. Table 89: SW potential parameters for single-layer 1H-NiS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni1-S1-S1	1.000	1.498	2.107	0.000	1.000	0.000	6.425	2.502	4	0	0.0
Ni1-S1-S3	1.000	0.000	0.000	46.062	1.000	-0.152	0.000	0.000	4	0	0.0
Ni1-S1-S2	1.000	0.000	0.000	63.130	1.000	0.536	0.000	0.000	4	0	0.0
S1-Ni1-Ni3	1.000	0.000	0.000	46.062	1.000	-0.152	0.000	0.000	4	0	0.0

Table 90. Table 90: The VFF model for single-layer 1H-NiSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	4.823	2.171	2.171	2.171
$r_0$ or ${\theta }_0$	2.350	90.228	70.206	90.228

Table 91. Table 91: Two-body SW potential parameters for single-layer 1H-NiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-Se	4.004	1.267	15.249	0.0	3.213

Table 92. Table 92: Three-body SW potential parameters for single-layer 1H-NiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-\mathrm{Se}-\mathrm{Se}}$	20.479	90.228	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809
${\theta }_{\mathrm{Ni}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	23.132	70.206	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809
${\theta }_{\mathrm{Se}-\mathrm{Ni}-\mathrm{Ni}}$	20.479	90.228	1.267	1.267	0.0	3.213	0.0	3.213	0.0	3.809

Table 93. Table 93: SW potential parameters for single-layer 1H-NiSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni1-Se1-Se1	1.000	1.267	2.535	0.000	1.000	0.000	4.004	5.913	4	0	0.0
Ni1-Se1-Se3	1.000	0.000	0.000	20.479	1.000	-0.004	0.000	0.000	4	0	0.0
Ni1-Se1-Se2	1.000	0.000	0.000	23.132	1.000	0.339	0.000	0.000	4	0	0.0
Se1-Ni1-Ni3	1.000	0.000	0.000	20.479	1.000	-0.004	0.000	0.000	4	0	0.0

Table 94. Table 94: The VFF model for single-layer 1H-NiTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	6.712	2.656	2.656	2.656
$r_0$ or ${\theta }_0$	2.540	89.933	70.624	89.933

Table 95. Table 95: Two-body SW potential parameters for single-layer 1H-NiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-Te	6.461	1.359	20.812	0.0	3.469

Table 96. Table 96: Three-body SW potential parameters for single-layer 1H-NiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-\mathrm{Te}-\mathrm{Te}}$	24.759	89.933	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114
${\theta }_{\mathrm{Ni}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	27.821	70.624	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114
${\theta }_{\mathrm{Te}-\mathrm{Ni}-\mathrm{Ni}}$	24.759	89.933	1.359	1.359	0.0	3.469	0.0	3.469	0.0	4.114

Table 97. Table 97: SW potential parameters for single-layer 1H-NiTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni1-Te1-Te1	1.000	1.359	2.553	0.000	1.000	0.000	6.461	6.107	4	0	0.0
Ni1-Te1-Te3	1.000	0.000	0.000	24.759	1.000	0.001	0.000	0.000	4	0	0.0
Ni1-Te1-Te2	1.000	0.000	0.000	27.821	1.000	0.332	0.000	0.000	4	0	0.0
Te1-Ni1-Ni3	1.000	0.000	0.000	24.759	1.000	0.001	0.000	0.000	4	0	0.0

Table 98. Table 98: The VFF model for single-layer 1H-NbS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Nb-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Nb-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Nb-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-Nb-Nb}{\left(\Delta \theta \right)}^2$
parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.470	84.140	78.626	84.140

Table 99. Table 99: Two-body SW potential parameters for single-layer 1H-NbS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Nb-S	6.439	1.116	18.610	0.0	3.300

Table 100. Table 100: Three-body SW potential parameters for single-layer 1H-NbS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Nb}-S-S}$	35.748	84.140	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933
${\theta }_{\mathrm{Nb}-S-S^{\prime }}$	36.807	78.626	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933
${\theta }_{S-\mathrm{Nb}-\mathrm{Nb}}$	35.748	84.140	1.116	1.116	0.0	3.300	0.0	3.300	0.0	3.933

Table 101. Table 101: SW potential parameters for single-layer 1H-NbS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Nb and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Nb1-S1-S1	1.000	1.116	2.958	0.000	1.000	0.000	6.439	12.014	4	0	0.0
Nb1-S1-S3	1.000	0.000	0.000	35.748	1.000	0.102	0.000	0.000	4	0	0.0
Nb1-S1-S2	1.000	0.000	0.000	36.807	1.000	0.197	0.000	0.000	4	0	0.0
S1-Nb1-Nb3	1.000	0.000	0.000	35.748	1.000	0.102	0.000	0.000	4	0	0.0

Table 102. Table 102: The VFF model for single-layer 1H-NbSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Nb}-\mathrm{Nb}}{\left(\Delta \theta \right)}^2$
parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.600	83.129	79.990	83.129

Table 103. Table 103: Two-body SW potential parameters for single-layer 1H-NbSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Nb-Se	6.942	1.138	22.849	0.0	3.460

Table 104. Table 104: Three-body SW potential parameters for single-layer 1H-NbSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Nb}-\mathrm{Se}-\mathrm{Se}}$	34.409	83.129	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127
${\theta }_{\mathrm{Nb}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	34.973	79.990	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127
${\theta }_{\mathrm{Se}-\mathrm{Nb}-\mathrm{Nb}}$	34.409	83.129	1.138	1.138	0.0	3.460	0.0	3.460	0.0	4.127

Table 105. Table 105: SW potential parameters for single-layer 1H-NbSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Nb and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Nb1-Se1-Se1	1.000	1.138	3.041	0.000	1.000	0.000	6.942	13.631	4	0	0.0
Nb1-Se1-Se3	1.000	0.000	0.000	34.409	1.000	0.120	0.000	0.000	4	0	0.0
Nb1-Se1-Se2	1.000	0.000	0.000	34.973	1.000	0.174	0.000	0.000	4	0	0.0
Se1-Nb1-Nb3	1.000	0.000	0.000	34.409	1.000	0.120	0.000	0.000	4	0	0.0

Table 106. Table 106: The VFF model for single-layer 1H-MoO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mo}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Mo}-\mathrm{Mo}}{\left(\Delta \theta \right)}^2$
parameter	14.622	8.410	8.410	8.410
$r_0$ or ${\theta }_0$	2.000	88.054	73.258	88.054

Table 107. Table 107: Two-body SW potential parameters for single-layer 1H-MoO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-O	8.317	1.015	8.000	0.0	2.712

Table 108. Table 108: Three-body SW potential parameters for single-layer 1H-MoO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-O-O}$	72.735	88.054	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222
${\theta }_{\mathrm{Mo}-O-O^{\prime }}$	79.226	73.258	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222
${\theta }_{O-\mathrm{Mo}-\mathrm{Mo}}$	72.735	88.054	1.015	1.015	0.0	2.712	0.0	2.712	0.0	3.222

Table 109. Table 109: SW potential parameters for single-layer 1H-MoO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo1-O1-O1	1.000	1.015	2.673	0.000	1.000	0.000	8.317	7.541	4	0	0.0
Mo1-O1-O3	1.000	0.000	0.000	72.735	1.000	0.034	0.000	0.000	4	0	0.0
Mo1-O1-O2	1.000	0.000	0.000	79.226	1.000	0.288	0.000	0.000	4	0	0.0
O1-Mo1-Mo3	1.000	0.000	0.000	72.735	1.000	0.034	0.000	0.000	4	0	0.0

Table 110. Table 110: The VFF model parameters for single-layer 1H-MoS 2 from Ref.  Wakabayashi, Smith, and Nicklow, 1975 . The second line gives the expression for each VFF term. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Mo-S}{\left(\Delta r_{Mo-S}\right)}^2$	$\frac{1}{2}{K}_{MoSS}{\left(\Delta {\theta }_{MoSS}\right)}^2$	$\frac{1}{2}{K}_{SMoMo}{\left(\Delta {\theta }_{SMoMo}\right)}^2$
parameter	8.640	5.316	4.891
$r_0$ or ${\theta }_0$	2.382	80.581	80.581

Table 111. Table 111: Two-body SW potential parameters for single-layer 1H-MoS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{\mathrm{Mo}-S}$	6.918	1.252	17.771	0.0	3.16

Table 112. Table 112: Three-body SW potential parameters for single-layer 1H-MoS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{MoSS}}$	67.883	81.788	1.252	1.252	0.0	3.16	0.0	3.16	0.0	3.78
${\theta }_{\mathrm{SMoMo}}$	62.449	81.788	1.252	1.252	0.0	3.16	0.0	3.16	0.0	4.27

Table 113. Table 113: SW potential parameters for single-layer 1H-MoS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Mo and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo1-S1-S1	1.000	1.252	2.523	0.000	1.000	0.000	6.918	7.223	4	0	0.0
Mo1-S1-S3	1.000	0.000	0.000	67.883	1.000	0.143	0.000	0.000	4	0	0.0
S1-Mo1-Mo3	1.000	0.000	0.000	62.449	1.000	0.143	0.000	0.000	4	0	0.0

Table 114. Table 114: The VFF model for single-layer 1H-MoSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{Mo-Se}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Mo-Se-Se}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Mo-Se-S{e}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{Se-Mo-Mo}{\left(\Delta \theta \right)}^2$
parameter	7.928	6.945	6.945	5.782
$r_0$ or ${\theta }_0$	2.528	82.119	81.343	82.119

Table 115. Table 115: Two-body SW potential parameters for single-layer 1H-MoSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-Se	5.737	0.913	18.787	0.0	3.351

Table 116. Table 116: Three-body SW potential parameters for single-layer 1H-MoSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-\mathrm{Se}-\mathrm{Se}}$	32.526	82.119	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000
${\theta }_{\mathrm{Mo}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	32.654	81.343	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000
${\theta }_{\mathrm{Se}-\mathrm{Mo}-\mathrm{Mo}}$	27.079	82.119	0.913	0.913	0.0	3.351	0.0	3.351	0.0	4.000

Table 117. Table 117: SW potential parameters for single-layer 1H-MoSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Mo and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo1-Se1-Se1	1.000	0.913	3.672	0.000	1.000	0.000	5.737	27.084	4	0	0.0
Mo1-Se1-Se3	1.000	0.000	0.000	32.526	1.000	0.137	0.000	0.000	4	0	0.0
Mo1-Se1-Se2	1.000	0.000	0.000	32.654	1.000	0.151	0.000	0.000	4	0	0.0
Se1-Mo1-Mo3	1.000	0.000	0.000	27.079	1.000	0.137	0.000	0.000	4	0	0.0

Table 118. Table 118: The VFF model for single-layer 1H-MoTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Mo}-\mathrm{Mo}}{\left(\Delta \theta \right)}^2$
parameter	6.317	6.184	6.184	5.225
$r_0$ or ${\theta }_0$	2.730	81.111	82.686	81.111

Table 119. Table 119: Two-body SW potential parameters for single-layer 1H-MoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-Te	5.086	0.880	24.440	0.0	3.604

Table 120. Table 120: Three-body SW potential parameters for single-layer 1H-MoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-\mathrm{Te}-\mathrm{Te}}$	23.705	81.111	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\mathrm{Mo}-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	23.520	82.686	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\mathrm{Te}-\mathrm{Mo}-\mathrm{Mo}}$	20.029	81.111	0.880	0.880	0.0	3.604	0.0	3.604	0.0	4.305

Table 121. Table 121: SW potential parameters for single-layer 1H-MoTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Mo and X=Te).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo1-Te1-Te1	1.000	0.900	4.016	0.000	1.000	0.000	5.169	37.250	4	0	0.0
Mo1-Te1-Te3	1.000	0.000	0.000	24.163	1.000	0.143	0.000	0.000	4	0	0.0
Te1-Mo1-Mo3	1.000	0.000	0.000	20.416	1.000	0.143	0.000	0.000	4	0	0.0

Table 122. Table 122: The VFF model for single-layer 1H-TaS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ta}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Ta}-\mathrm{Ta}}{\left(\Delta \theta \right)}^2$
parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.480	83.879	78.979	83.879

Table 123. Table 123: Two-body SW potential parameters for single-layer 1H-TaS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ta-S	6.446	1.111	18.914	0.0	3.310

Table 124. Table 124: Three-body SW potential parameters for single-layer 1H-TaS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ta}-S-S}$	35.396	83.879	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945
${\theta }_{\mathrm{Ta}-S-S^{\prime }}$	36.321	78.979	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945
${\theta }_{S-\mathrm{Ta}-\mathrm{Ta}}$	35.396	83.879	1.111	1.111	0.0	3.310	0.0	3.310	0.0	3.945

Table 125. Table 125: SW potential parameters for single-layer 1H-TaS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Ta and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ta1-S1-S1	1.000	1.111	2.979	0.000	1.000	0.000	6.446	12.408	4	0	0.0
Ta1-S1-S3	1.000	0.000	0.000	35.396	1.000	0.107	0.000	0.000	4	0	0.0
Ta1-S1-S2	1.000	0.000	0.000	36.321	1.000	0.191	0.000	0.000	4	0	0.0
S1-Ta1-Ta3	1.000	0.000	0.000	35.396	1.000	0.107	0.000	0.000	4	0	0.0

Table 126. Table 126: The VFF model for single-layer 1H-TaSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Ta}-\mathrm{Ta}}{\left(\Delta \theta \right)}^2$
parameter	8.230	4.811	4.811	4.811
$r_0$ or ${\theta }_0$	2.590	83.107	80.019	83.107

Table 127. Table 127: Two-body SW potential parameters for single-layer 1H-TaSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ta-Se	6.885	1.133	22.499	0.0	3.446

Table 128. Table 128: Three-body SW potential parameters for single-layer 1H-TaSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ta}-\mathrm{Se}-\mathrm{Se}}$	34.381	83.107	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111
${\theta }_{\mathrm{Ta}-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	34.936	80.019	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111
${\theta }_{\mathrm{Se}-\mathrm{Ta}-\mathrm{Ta}}$	34.381	83.107	1.133	1.133	0.0	3.446	0.0	3.446	0.0	4.111

Table 129. Table 129: SW potential parameters for single-layer 1H-TaSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=Ta and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ta1-Se1-Se1	1.000	1.133	3.043	0.000	1.000	0.000	6.885	13.668	4	0	0.0
Ta1-Se1-Se3	1.000	0.000	0.000	34.381	1.000	0.120	0.000	0.000	4	0	0.0
Ta1-Se1-Se2	1.000	0.000	0.000	34.936	1.000	0.173	0.000	0.000	4	0	0.0
Se1-Ta1-Ta3	1.000	0.000	0.000	34.381	1.000	0.120	0.000	0.000	4	0	0.0

Table 130. Table 130: The VFF model for single-layer 1H-WO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{W-O-O^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-W-W}{\left(\Delta \theta \right)}^2$
parameter	15.318	10.276	10.276	10.276
$r_0$ or ${\theta }_0$	2.030	87.206	74.435	87.206

Table 131. Table 131: Two-body SW potential parameters for single-layer 1H-WO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-O	8.781	1.005	8.491	0.0	2.744

Table 132. Table 132: Three-body SW potential parameters for single-layer 1H-WO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-O-O}$	85.955	87.206	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262
${\theta }_{W-O-O^{\prime }}$	92.404	74.435	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262
${\theta }_{O-W-W}$	85.955	87.206	1.005	1.005	0.0	2.744	0.0	2.744	0.0	3.262

Table 133. Table 133: SW potential parameters for single-layer 1H-WO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W1-O1-O1	1.000	1.005	2.730	0.000	1.000	0.000	8.781	8.316	4	0	0.0
W1-O1-O3	1.000	0.000	0.000	85.955	1.000	0.049	0.000	0.000	4	0	0.0
W1-O1-O2	1.000	0.000	0.000	92.404	1.000	0.268	0.000	0.000	4	0	0.0
O1-W1-W3	1.000	0.000	0.000	85.955	1.000	0.049	0.000	0.000	4	0	0.0

Table 134. Table 134: The VFF model for single-layer 1H-WS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{W-S-S^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-W-W}{\left(\Delta \theta \right)}^2$
parameter	8.701	7.421	7.421	6.607
$r_0$ or ${\theta }_0$	2.390	81.811	81.755	81.811

Table 135. Table 135: Two-body SW potential parameters for single-layer 1H-WS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-S	5.664	0.889	15.335	0.0	3.164

Table 136. Table 136: Three-body SW potential parameters for single-layer 1H-WS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-S-S}$	37.687	81.811	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778
${\theta }_{W-S-S^{\prime }}$	37.697	81.755	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778
${\theta }_{S-W-W}$	33.553	81.811	0.889	0.889	0.0	3.164	0.0	3.164	0.0	3.778

Table 137. Table 137: SW potential parameters for single-layer 1H-WS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=W and X=S).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W1-S1-S1	1.000	0.889	3.558	0.000	1.000	0.000	5.664	24.525	4	0	0.0
W1-S1-S3	1.000	0.000	0.000	37.687	1.000	0.142	0.000	0.000	4	0	0.0
W1-S1-S2	1.000	0.000	0.000	37.697	1.000	0.143	0.000	0.000	4	0	0.0
S1-W1-W3	1.000	0.000	0.000	33.553	1.000	0.142	0.000	0.000	4	0	0.0

Table 138. Table 138: The VFF model for single-layer 1H-WSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Se}-{\mathrm{Se}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-W-W}{\left(\Delta \theta \right)}^2$
parameter	8.286	8.513	8.513	7.719
$r_0$ or ${\theta }_0$	2.510	80.693	83.140	80.693

Table 139. Table 139: Two-body SW potential parameters for single-layer 1H-WSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-Se	5.476	0.706	16.273	0.0	3.308

Table 140. Table 140: Three-body SW potential parameters for single-layer 1H-WSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-\mathrm{Se}-\mathrm{Se}}$	25.607	80.693	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953
${\theta }_{W-\mathrm{Se}-{\mathrm{Se}}^{\prime }}$	25.287	83.240	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953
${\theta }_{\mathrm{Se}-W-W}$	23.218	80.693	0.706	0.706	0.0	3.308	0.0	3.308	0.0	3.953

Table 141. Table 141: SW potential parameters for single-layer 1H-WSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=W and X=Se).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W1-Se1-Se1	1.000	0.706	4.689	0.000	1.000	0.000	5.476	65.662	4	0	0.0
W1-Se1-Se3	1.000	0.000	0.000	25.607	1.000	0.162	0.000	0.000	4	0	0.0
W1-Se1-Se2	1.000	0.000	0.000	25.287	1.000	0.118	0.000	0.000	4	0	0.0
Se1-W1-W3	1.000	0.000	0.000	23.218	1.000	0.162	0.000	0.000	4	0	0.0

Table 142. Table 142: The VFF model for single-layer 1H-WTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Te}-{\mathrm{Te}}^{\prime }}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-W-W}{\left(\Delta \theta \right)}^2$
parameter	5.483	7.016	7.016	5.718
$r_0$ or ${\theta }_0$	2.730	81.111	82.686	81.111

Table 143. Table 143: Two-body SW potential parameters for single-layer 1H-WTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-Te	4.326	0.778	22.774	0.0	3.604

Table 144. Table 144: Three-body SW potential parameters for single-layer 1H-WTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-\mathrm{Te}-\mathrm{Te}}$	21.313	81.111	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{W-\mathrm{Te}-{\mathrm{Te}}^{\prime }}$	21.147	82.686	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305
${\theta }_{\mathrm{Te}-W-W}$	17.370	81.111	0.778	0.778	0.0	3.604	0.0	3.604	0.0	4.305

Table 145. Table 145: SW potential parameters for single-layer 1H-WTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  2 (with M=W and X=Te).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W1-Te1-Te1	1.000	0.778	4.632	0.000	1.000	0.000	4.326	62.148	4	0	0.0
W1-Te1-Te3	1.000	0.000	0.000	21.313	1.000	0.155	0.000	0.000	4	0	0.0
W1-Te1-Te2	1.000	0.000	0.000	21.147	1.000	0.127	0.000	0.000	4	0	0.0
Te1-W1-W3	1.000	0.000	0.000	17.370	1.000	0.155	0.000	0.000	4	0	0.0

Table 146. Table 146: The VFF model for single-layer 1T-ScO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	11.926	3.258	3.258
$r_0$ or ${\theta }_0$	2.07	102.115	102.115

Table 147. Table 147: Two-body SW potential parameters for single-layer 1T-ScO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-O	10.187	1.493	9.180	0.0	2.949

Table 148. Table 148: Three-body SW potential parameters for single-layer 1T-ScO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-O-O}$	50.913	102.115	1.493	1.493	0.0	2.949	0.0	2.949	0.0	4.399
${\theta }_{O-\mathrm{Sc}-\mathrm{Sc}}$	50.913	102.115	1.493	1.493	0.0	2.949	0.0	2.949	0.0	4.399

Table 149. Table 149: SW potential parameters for single-layer 1T-ScO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc-O1-O1	1.000	1.493	1.975	50.913	1.000	-0.210	10.187	1.847	4	0	0.0

Table 150. Table 150: The VFF model for single-layer 1T-ScS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	3.512	1.593	1.593
$r_0$ or ${\theta }_0$	2.50	92.771	92.771

Table 151. Table 151: Two-body SW potential parameters for single-layer 1T-ScS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-S	3.516	1.443	19.531	0.0	3.450

Table 152. Table 152: Three-body SW potential parameters for single-layer 1T-ScS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-S-S}$	16.674	92.771	1.443	1.443	0.0	3.450	0.0	3.450	0.0	4.945
${\theta }_{S-\mathrm{Sc}-\mathrm{Sc}}$	16.674	92.771	1.443	1.443	0.0	3.450	0.0	3.450	0.0	4.945

Table 153. Table 153: SW potential parameters for single-layer 1T-ScS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc-S1-S1	1.000	1.443	2.390	16.674	1.000	-0.048	3.516	4.504	4	0	0.0

Table 154. Table 154: The VFF model for single-layer 1T-ScSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.64	83.621	83.621

Table 155. Table 155: Two-body SW potential parameters for single-layer 1T-ScSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-Se	3.884	1.173	24.288	0.0	3.520

Table 156. Table 156: Three-body SW potential parameters for single-layer 1T-ScSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-\mathrm{Se}-\mathrm{Se}}$	17.479	83.621	1.173	1.173	0.0	3.520	0.0	3.520	0.0	4.808
${\theta }_{\mathrm{Se}-\mathrm{Sc}-\mathrm{Sc}}$	17.479	83.621	1.173	1.173	0.0	3.520	0.0	3.520	0.0	4.808

Table 157. Table 157: SW potential parameters for single-layer 1T-ScSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc-Se1-Se1	1.000	1.173	3.000	17.479	1.000	0.111	3.884	12.814	4	0	0.0

Table 158. Table 158: The VFF model for single-layer 1T-ScTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sc}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Sc}-\mathrm{Sc}}{\left(\Delta \theta \right)}^2$
parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.85	81.481	81.481

Table 159. Table 159: Two-body SW potential parameters for single-layer 1T-ScTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sc-Te	4.269	1.183	32.988	0.0	3.768

Table 160. Table 160: Three-body SW potential parameters for single-layer 1T-ScTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sc}-\mathrm{Te}-\mathrm{Te}}$	16.139	81.481	1.183	1.183	0.0	3.768	0.0	3.768	0.0	5.082
${\theta }_{\mathrm{Te}-\mathrm{Sc}-\mathrm{Sc}}$	16.139	81.481	1.183	1.183	0.0	3.768	0.0	3.768	0.0	5.082

Table 161. Table 161: SW potential parameters for single-layer 1T-ScTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sc-Te1-Te1	1.000	1.183	3.185	16.139	1.000	0.148	4.269	16.841	4	0	0.0

Table 162. Table 162: The VFF model for single-layer 1T-TiS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ti}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ti}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Ti}-\mathrm{Ti}}{\left(\Delta \theta \right)}^2$
parameter	9.815	3.754	3.754
$r_0$ or ${\theta }_0$	2.390	87.984	87.984

Table 163. Table 163: Two-body SW potential parameters for single-layer 1T-TiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ti-S	7.958	1.210	16.314	0.0	3.240

Table 164. Table 164: Three-body SW potential parameters for single-layer 1T-TiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ti}-S-S}$	32.377	87.984	1.210	1.210	0.0	3.240	0.0	3.240	0.0	4.535
${\theta }_{S-\mathrm{Ti}-\mathrm{Ti}}$	32.377	87.984	1.210	1.210	0.0	3.240	0.0	3.240	0.0	4.535

Table 165. Table 165: SW potential parameters for single-layer 1T-TiS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ti-S-S	1.000	1.210	2.677	32.377	1.000	0.035	7.958	7.602	4	0	0.0

Table 166. Table 166: The VFF model for single-layer 1T-TiSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Ti}-\mathrm{Ti}}{\left(\Delta \theta \right)}^2$
parameter	7.712	3.363	3.363
$r_0$ or ${\theta }_0$	2.510	86.199	86.199

Table 167. Table 167: Two-body SW potential parameters for single-layer 1T-TiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ti-Se	6.582	1.207	19.846	0.0	3.380

Table 168. Table 168: Three-body SW potential parameters for single-layer 1T-TiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ti}-\mathrm{Se}-\mathrm{Se}}$	27.044	86.199	1.207	1.207	0.0	3.380	0.0	3.380	0.0	4.685
${\theta }_{\mathrm{Se}-\mathrm{Ti}-\mathrm{Ti}}$	27.044	86.199	1.207	1.207	0.0	3.380	0.0	3.380	0.0	4.685

Table 169. Table 169: SW potential parameters for single-layer 1T-TiSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ti-Se-Se	1.000	1.207	2.801	27.044	1.000	0.066	6.582	9.362	4	0	0.0

Table 170. Table 170: The VFF model for single-layer 1T-TiTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ti}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ti}-\mathrm{Ti}}{\left(\Delta \theta \right)}^2$
parameter	3.758	3.217	3.217
$r_0$ or ${\theta }_0$	2.730	83.621	83.621

Table 171. Table 171: Two-body SW potential parameters for single-layer 1T-TiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ti-Te	3.542	1.213	27.773	0.0	3.640

Table 172. Table 172: Three-body SW potential parameters for single-layer 1T-TiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ti}-\mathrm{Te}-\mathrm{Te}}$	23.439	83.621	1.213	1.213	0.0	3.640	0.0	3.640	0.0	4.972
${\theta }_{\mathrm{Te}-\mathrm{Ti}-\mathrm{Ti}}$	23.439	83.621	1.213	1.213	0.0	3.640	0.0	3.640	0.0	4.972

Table 173. Table 173: SW potential parameters for single-layer 1T-TiTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ti-Te-Te	1.000	1.213	3.000	23.439	1.000	0.111	3.542	12.814	4	0	0.0

Table 174. Table 174: The VFF model for single-layer 1T-VS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-V-V}{\left(\Delta \theta \right)}^2$
parameter	11.562	4.237	4.237
$r_0$ or ${\theta }_0$	2.310	84.288	84.288

Table 175. Table 175: Two-body SW potential parameters for single-layer 1T-VS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-S	7.943	1.048	14.237	0.0	3.088

Table 176. Table 176: Three-body SW potential parameters for single-layer 1T-VS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-S-S}$	31.659	84.288	1.048	1.048	0.0	3.088	0.0	3.088	0.0	4.235
${\theta }_{S-V-V}$	31.659	84.288	1.048	1.048	0.0	3.088	0.0	3.088	0.0	4.235

Table 177. Table 177: SW potential parameters for single-layer 1T-VS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V-S1-S1	1.000	1.048	2.946	31.659	1.000	0.100	7.943	11.797	4	0	0.0

Table 178. Table 178: The VFF model for single-layer 1T-VSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-V-V}{\left(\Delta \theta \right)}^2$
parameter	11.562	4.237	4.237
$r_0$ or ${\theta }_0$	2.440	83.201	83.201

Table 179. Table 179: Two-body SW potential parameters for single-layer 1T-VSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-Se	8.606	1.070	17.723	0.0	3.248

Table 180. Table 180: Three-body SW potential parameters for single-layer 1T-VSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-\mathrm{Se}-\mathrm{Se}}$	30.387	83.201	1.070	1.070	0.0	3.248	0.0	3.248	0.0	4.426
${\theta }_{\mathrm{Se}-V-V}$	30.387	83.201	1.070	1.070	0.0	3.248	0.0	3.248	0.0	4.426

Table 181. Table 181: SW potential parameters for single-layer 1T-VSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V-Se1-Se1	1.000	1.070	3.035	30.387	1.000	0.118	8.606	13.507	4	0	0.0

Table 182. Table 182: The VFF model for single-layer 1T-VTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{V-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{V-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-V-V}{\left(\Delta \theta \right)}^2$
parameter	10.476	3.814	3.814
$r_0$ or ${\theta }_0$	2.640	81.885	81.885

Table 183. Table 183: Two-body SW potential parameters for single-layer 1T-VTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
V-Te	8.805	1.110	24.288	0.0	3.496

Table 184. Table 184: Three-body SW potential parameters for single-layer 1T-VTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{V-\mathrm{Te}-\mathrm{Te}}$	26.043	81.885	1.110	1.110	0.0	3.496	0.0	3.496	0.0	4.726
${\theta }_{\mathrm{Te}-V-V}$	26.043	81.885	1.110	1.110	0.0	3.496	0.0	3.496	0.0	4.726

Table 185. Table 185: SW potential parameters for single-layer 1T-VTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
V-Te1-Te1	1.000	1.110	3.149	26.043	1.000	0.141	8.805	15.980	4	0	0.0

Table 186. Table 186: The VFF model for single-layer 1T-MnO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mn}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Mn}-\mathrm{Mn}}{\left(\Delta \theta \right)}^2$
parameter	15.371	4.822	4.822
$r_0$ or ${\theta }_0$	1.88	97.181	97.181

Table 187. Table 187: Two-body SW potential parameters for single-layer 1T-MnO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mn-O	9.675	1.212	6.246	0.0	2.635

Table 188. Table 188: Three-body SW potential parameters for single-layer 1T-MnO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mn}-O-O}$	60.755	97.181	1.212	1.212	0.0	2.635	0.0	2.635	0.0	3.852
${\theta }_{O-\mathrm{Mn}-\mathrm{Mn}}$	60.755	97.181	1.212	1.212	0.0	2.635	0.0	2.635	0.0	3.852

Table 189. Table 189: SW potential parameters for single-layer 1T-MnO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mn-O1-O1	1.000	1.212	2.175	60.755	1.000	-0.125	9.675	2.899	4	0	0.0

Table 190. Table 190: The VFF model for single-layer 1T-MnS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mn}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Mn}-\mathrm{Mn}}{\left(\Delta \theta \right)}^2$
parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.27	86.822	86.822

Table 191. Table 191: Two-body SW potential parameters for single-layer 1T-MnS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mn-S	3.127	1.111	13.276	0.0	3.064

Table 192. Table 192: Three-body SW potential parameters for single-layer 1T-MnS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mn}-S-S}$	19.765	86.822	1.111	1.111	0.0	3.064	0.0	3.064	0.0	4.262
${\theta }_{S-\mathrm{Mn}-\mathrm{Mn}}$	19.765	86.822	1.111	1.111	0.0	3.064	0.0	3.064	0.0	4.262

Table 193. Table 193: SW potential parameters for single-layer 1T-MnS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mn-S1-S1	1.000	1.111	2.757	19.765	1.000	0.055	3.127	8.700	4	0	0.0

Table 194. Table 194: The VFF model for single-layer 1T-MnSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mn}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Mn}-\mathrm{Mn}}{\left(\Delta \theta \right)}^2$
parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.39	86.330	86.330

Table 195. Table 195: Two-body SW potential parameters for single-layer 1T-MnSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mn-Se	3.422	1.153	16.314	0.0	3.220

Table 196. Table 196: Three-body SW potential parameters for single-layer 1T-MnSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mn}-\mathrm{Se}-\mathrm{Se}}$	19.390	86.330	1.153	1.153	0.0	3.220	0.0	3.220	0.0	4.467
${\theta }_{\mathrm{Se}-\mathrm{Mn}-\mathrm{Mn}}$	19.390	86.330	1.153	1.153	0.0	3.220	0.0	3.220	0.0	4.467

Table 197. Table 197: SW potential parameters for single-layer 1T-MnSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mn-Se1-Se1	1.000	1.153	2.792	19.390	1.000	0.064	3.422	9.219	4	0	0.0

Table 198. Table 198: The VFF model for single-layer 1T-MnTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mn}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mn}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Mn}-\mathrm{Mn}}{\left(\Delta \theta \right)}^2$
parameter	4.407	2.399	2.399
$r_0$ or ${\theta }_0$	2.59	86.219	86.219

Table 199. Table 199: Two-body SW potential parameters for single-layer 1T-MnTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mn-Te	4.007	1.246	22.499	0.0	3.488

Table 200. Table 200: Three-body SW potential parameters for single-layer 1T-MnTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mn}-\mathrm{Te}-\mathrm{Te}}$	19.307	86.219	1.246	1.246	0.0	3.488	0.0	3.488	0.0	4.836
${\theta }_{\mathrm{Te}-\mathrm{Mn}-\mathrm{Mn}}$	19.307	86.219	1.246	1.246	0.0	3.488	0.0	3.488	0.0	4.836

Table 201. Table 201: SW potential parameters for single-layer 1T-MnTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mn-Te1-Te1	1.000	1.246	2.800	19.307	1.000	0.066	4.007	9.340	4	0	0.0

Table 202. Table 202: The VFF model for single-layer 1T-CoTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Co}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Co}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Co}-\mathrm{Co}}{\left(\Delta \theta \right)}^2$
parameter	4.726	3.035	3.035
$r_0$ or ${\theta }_0$	2.512	91.501	91.501

Table 203. Table 203: Two-body SW potential parameters for single-layer 1T-CoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Co-Te	4.628	1.402	19.899	0.0	3.450

Table 204. Table 204: Three-body SW potential parameters for single-layer 1T-CoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Co}-\mathrm{Te}-\mathrm{Te}}$	30.149	91.501	1.402	1.402	0.0	3.450	0.0	3.450	0.0	4.915
${\theta }_{\mathrm{Te}-\mathrm{Co}-\mathrm{Co}}$	30.149	91.501	1.402	1.402	0.0	3.450	0.0	3.450	0.0	4.915

Table 205. Table 205: SW potential parameters for single-layer 1T-CoTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Co-Te1-Te1	1.000	1.402	2.461	30.149	1.000	-0.026	4.628	5.151	4	0	0.0

Table 206. Table 206: The VFF model for single-layer 1T-NiO 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-O-O}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{O-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	15.925	4.847	4.847
$r_0$ or ${\theta }_0$		97.653	97.653

Table 207. Table 207: Two-body SW potential parameters for single-layer 1T-NiO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-O	9.709	1.199	5.731	0.0	2.583

Table 208. Table 208: Three-body SW potential parameters for single-layer 1T-NiO 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-O-O}$	62.317	97.653	1.199	1.199	0.0	2.583	0.0	2.583	0.0	3.784
${\theta }_{O-\mathrm{Ni}-\mathrm{Ni}}$	62.317	97.653	1.199	1.199	0.0	2.583	0.0	2.583	0.0	3.784

Table 209. Table 209: SW potential parameters for single-layer 1T-NiO 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni-O1-O1	1.000	1.199	2.154	62.317	1.000	-0.133	9.709	2.772	4	0	0.0

Table 210. Table 210: The VFF model for single-layer 1T-NiS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	9.385	2.952	2.952
$r_0$ or ${\theta }_0$	2.232	96.000	96.000

Table 211. Table 211: Two-body SW potential parameters for single-layer 1T-NiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-S	8.098	1.398	12.409	0.0	3.115

Table 212. Table 212: Three-body SW potential parameters for single-layer 1T-NiS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-S-S}$	35.372	96.000	1.398	1.398	0.0	3.115	0.0	3.115	0.0	4.532
${\theta }_{S-\mathrm{Ni}-\mathrm{Ni}}$	35.372	96.000	1.398	1.398	0.0	3.115	0.0	3.115	0.0	4.532

Table 213. Table 213: SW potential parameters for single-layer 1T-NiS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni-S1-S1	1.000	1.398	2.228	35.372	1.000	-0.105	8.098	3.249	4	0	0.0

Table 214. Table 214: The VFF model for single-layer 1T-NiSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	8.814	3.149	3.149
$r_0$ or ${\theta }_0$	2.339	95.798	95.798

Table 215. Table 215: Two-body SW potential parameters for single-layer 1T-NiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-Se	8.313	1.458	14.971	0.0	3.263

Table 216. Table 216: Three-body SW potential parameters for single-layer 1T-NiSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-\mathrm{Se}-\mathrm{Se}}$	37.407	95.798	1.458	1.458	0.0	3.263	0.0	3.263	0.0	4.742
${\theta }_{\mathrm{Se}-\mathrm{Ni}-\mathrm{Ni}}$	37.407	95.798	1.458	1.458	0.0	3.263	0.0	3.263	0.0	4.742

Table 217. Table 217: SW potential parameters for single-layer 1T-NiSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni-Se1-Se1	1.000	1.458	2.238	37.407	1.000	-0.101	8.313	3.315	4	0	0.0

Table 218. Table 218: The VFF model for single-layer 1T-NiTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ni}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ni}-\mathrm{Ni}}{\left(\Delta \theta \right)}^2$
parameter	4.230	2.429	2.429
$r_0$ or ${\theta }_0$	2.532	94.702	94.702
or ${\theta }_0$	2.635	95.999	95.999

Table 219. Table 219: Two-body SW potential parameters for single-layer 1T-NiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ni-Te	4.554	1.536	20.554	0.0	3.518

Table 220. Table 220: Three-body SW potential parameters for single-layer 1T-NiTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ni}-\mathrm{Te}-\mathrm{Te}}$	27.553	94.702	1.536	1.536	0.0	3.518	0.0	3.518	0.0	5.088
${\theta }_{\mathrm{Te}-\mathrm{Ni}-\mathrm{Ni}}$	27.553	94.702	1.536	1.536	0.0	3.518	0.0	3.518	0.0	5.088

Table 221. Table 221: SW potential parameters for single-layer 1T-NiTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ni-Te1-Te1	1.000	1.536	2.291	27.553	1.000	-0.082	4.554	3.696	4	0	0.0

Table 222. Table 222: The VFF model for single-layer 1T-ZrS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Zr}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Zr}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Zr}-\mathrm{Zr}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.580	91.305	91.305

Table 223. Table 223: Two-body SW potential parameters for single-layer 1T-ZrS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Zr-S	8.149	1.432	22.154	0.0	3.541

Table 224. Table 224: Three-body SW potential parameters for single-layer 1T-ZrS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Zr}-S-S}$	42.170	91.305	1.432	1.432	0.0	3.541	0.0	3.541	0.0	5.041
${\theta }_{S-\mathrm{Zr}-\mathrm{Zr}}$	42.170	91.305	1.432	1.432	0.0	3.541	0.0	3.541	0.0	5.041

Table 225. Table 225: SW potential parameters for single-layer 1T-ZrS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Zr-S1-S1	1.000	1.432	2.473	42.177	1.000	-0.023	8.149	5.268	4	0	0.0
S1-Zr-Zr	1.000	1.432	2.473	42.177	1.000	-0.023	8.149	5.268	4	0	0.0

Table 226. Table 226: The VFF model for single-layer 1T-ZrSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Zr}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Zr}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Zr}-\mathrm{Zr}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.667	88.058	88.058

Table 227. Table 227: Two-body SW potential parameters for single-layer 1T-ZrSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Zr-Se	8.022	1.354	25.297	0.0	3.617

Table 228. Table 228: Three-body SW potential parameters for single-layer 1T-ZrSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Zr}-\mathrm{Se}-\mathrm{Se}}$	37.051	88.058	1.354	1.354	0.0	3.617	0.0	3.617	0.0	5.064
${\theta }_{\mathrm{Se}-\mathrm{Zr}-\mathrm{Zr}}$	37.051	88.058	1.354	1.354	0.0	3.617	0.0	3.617	0.0	5.064

Table 229. Table 229: SW potential parameters for single-layer 1T-ZrSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Zr-Se-Se	1.000	1.354	2.671	37.051	1.000	0.034	8.022	7.527	4	0	0.0
Se-Zr-Zr	1.000	1.354	2.671	37.051	1.000	0.034	8.022	7.527	4	0	0.0

Table 230. Table 230: The VFF model for single-layer 1T-ZrTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Zr}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Zr}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Zr}-\mathrm{Zr}}{\left(\Delta \theta \right)}^2$
parameter	2.974	3.681	3.681
$r_0$ or ${\theta }_0$	2.902	87.301	87.301

Table 231. Table 231: Two-body SW potential parameters for single-layer 1T-ZrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Zr-Te	3.493	1.441	35.467	0.0	3.925

Table 232. Table 232: Three-body SW potential parameters for single-layer 1T-ZrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Zr}-\mathrm{Te}-\mathrm{Te}}$	30.905	87.301	1.441	1.441	0.0	3.925	0.0	3.925	0.0	5.473
${\theta }_{\mathrm{Te}-\mathrm{Zr}-\mathrm{Zr}}$	30.905	87.301	1.441	1.441	0.0	3.925	0.0	3.925	0.0	5.473

Table 233. Table 233: SW potential parameters for single-layer 1T-ZrTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Zr-Te1-Te1	1.000	1.441	2.723	30.905	1.000	0.047	3.493	8.225	4	0	0.0

Table 234. Table 234: The VFF model for single-layer 1T-NbS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Nb}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Nb}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Nb}-\mathrm{Nb}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.450	84.671	84.671

Table 235. Table 235: Two-body SW potential parameters for single-layer 1T-NbS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Nb-S	6.192	1.125	18.015	0.0	3.280

Table 236. Table 236: Three-body SW potential parameters for single-layer 1T-NbS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Nb}-S-S}$	32.472	84.671	1.125	1.125	0.0	3.280	0.0	3.280	0.0	4.508
${\theta }_{S-\mathrm{Nb}-\mathrm{Nb}}$	32.472	84.671	1.125	1.125	0.0	3.280	0.0	3.280	0.0	4.508

Table 237. Table 237: SW potential parameters for single-layer 1T-NbS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Nb-S1-S1	1.000	1.125	2.916	32.472	1.000	0.093	6.192	11.247	4	0	0.0
S1-Nb-Nb	1.000	1.125	2.916	32.472	1.000	0.093	6.192	11.247	4	0	0.0

Table 238. Table 238: The VFF model for single-layer 1T-NbSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Nb}-\mathrm{Nb}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.570	82.529	82.529

Table 239. Table 239: Two-body SW potential parameters for single-layer 1T-NbSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Nb-Se	6.430	1.104	21.812	0.0	3.412

Table 240. Table 240: Three-body SW potential parameters for single-layer 1T-NbSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Nb}-\mathrm{Se}-\mathrm{Se}}$	29.956	82.528	1.104	1.104	0.0	3.412	0.0	3.412	0.0	4.631
${\theta }_{\mathrm{Se}-\mathrm{Nb}-\mathrm{Nb}}$	29.956	82.528	1.104	1.104	0.0	3.412	0.0	3.412	0.0	4.631

Table 241. Table 241: SW potential parameters for single-layer 1T-NbSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Nb-Se1-Se1	1.000	1.104	3.092	29.956	1.000	0.130	6.430	14.706	4	0	0.0

Table 242. Table 242: The VFF model for single-layer 1T-NbTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Nb}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Nb}-\mathrm{Nb}}{\left(\Delta \theta \right)}^2$
parameter	3.559	4.863	4.863
$r_0$ or ${\theta }_0$	2.770	79.972	79.972

Table 243. Table 243: Two-body SW potential parameters for single-layer 1T-NbTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Nb-Te	3.123	1.094	29.437	0.0	3.640

Table 244. Table 244: Three-body SW potential parameters for single-layer 1T-NbTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Nb}-\mathrm{Te}-\mathrm{Te}}$	30.968	79.972	1.094	1.094	0.0	3.640	0.0	3.640	0.0	4.863
${\theta }_{\mathrm{Te}-\mathrm{Nb}-\mathrm{Nb}}$	30.968	79.972	1.094	1.094	0.0	3.640	0.0	3.640	0.0	4.863

Table 245. Table 245: SW potential parameters for single-layer 1T-NbTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Nb-Te1-Te1	1.000	1.094	3.328	30.968	1.000	0.174	3.123	20.560	4	0	0.0

Table 246. Table 246: The VFF model for single-layer 1T-MoS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mo}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Mo}-\mathrm{Mo}}{\left(\Delta \theta \right)}^2$
parameter	3.523	10.394	10.394
$r_0$ or ${\theta }_0$	2.419	82.799	82.799

Table 247. Table 247: Two-body SW potential parameters for single-layer 1T-MoS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-S	2.550	1.048	17.129	0.0	3.215

Table 248. Table 248: Three-body SW potential parameters for single-layer 1T-MoS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-S-S}$	73.436	82.799	1.048	1.048	0.0	3.215	0.0	3.215	0.0	4.371
${\theta }_{S-\mathrm{Mo}-\mathrm{Mo}}$	73.436	82.799	1.048	1.048	0.0	3.215	0.0	3.215	0.0	4.371

Table 249. Table 249: SW potential parameters for single-layer 1T-MoS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo-S1-S1	1.000	1.048	3.069	73.436	1.000	0.125	2.550	14.207	4	0	0.0

Table 250. Table 250: The VFF model for single-layer 1T-MoSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Mo}-\mathrm{Mo}}{\left(\Delta \theta \right)}^2$
parameter	2.964	14.753	14.753
$r_0$ or ${\theta }_0$	2.529	80.501	80.501

Table 251. Table 251: Two-body SW potential parameters for single-layer 1T-MoSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-Se	2.201	1.017	20.463	0.0	3.331

Table 252. Table 252: Three-body SW potential parameters for single-layer 1T-MoSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-\mathrm{Se}-\mathrm{Se}}$	95.770	80.501	1.017	1.017	0.0	3.331	0.0	3.331	0.0	4.465
${\theta }_{\mathrm{Se}-\mathrm{Mo}-\mathrm{Mo}}$	95.770	80.501	1.017	1.017	0.0	3.331	0.0	3.331	0.0	4.465

Table 253. Table 253: SW potential parameters for single-layer 1T-MoSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo-Se1-Se1	1.000	1.017	3.276	95.770	1.000	0.165	2.201	19.152	4	0	0.0

Table 254. Table 254: The VFF model for single-layer 1T-MoTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Mo}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Mo}-\mathrm{Mo}}{\left(\Delta \theta \right)}^2$
parameter	3.074	12.516	12.516
$r_0$ or ${\theta }_0$	2.729	79.700	79.700

Table 255. Table 255: Two-body SW potential parameters for single-layer 1T-MoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Mo-Te	2.597	1.068	27.720	0.0	3.582

Table 256. Table 256: Three-body SW potential parameters for single-layer 1T-MoTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Mo}-\mathrm{Te}-\mathrm{Te}}$	78.925	79.700	1.068	1.068	0.0	3.582	0.0	3.582	0.0	4.777
${\theta }_{\mathrm{Te}-\mathrm{Mo}-\mathrm{Mo}}$	78.925	79.700	1.068	1.068	0.0	3.582	0.0	3.582	0.0	4.777

Table 257. Table 257: SW potential parameters for single-layer 1T-MoTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Mo-Te1-Te1	1.000	1.068	3.355	78.925	1.000	0.179	2.597	21.328	4	0	0.0

Table 258. Table 258: The VFF model for single-layer 1T-TcS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Tc}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Tc}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Tc}-\mathrm{Tc}}{\left(\Delta \theta \right)}^2$
parameter	2.986	11.141	11.141
$r_0$ or ${\theta }_0$	2.392	79.800	79.800

Table 259. Table 259: Two-body SW potential parameters for single-layer 1T-TcS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Tc-S	1.945	0.939	16.380	0.0	3.142

Table 260. Table 260: Three-body SW potential parameters for single-layer 1T-TcS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Tc}-S-S}$	70.512	79.800	0.939	0.939	0.0	3.142	0.0	3.142	0.0	4.193
${\theta }_{S-\mathrm{Tc}-\mathrm{Tc}}$	70.512	79.800	0.939	0.939	0.0	3.142	0.0	3.142	0.0	4.193

Table 261. Table 261: SW potential parameters for single-layer 1T-TcS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Tc-S1-S1	1.000	0.939	3.345	70.512	1.000	0.177	1.945	21.038	4	0	0.0

Table 262. Table 262: The VFF model for single-layer 1T-TcSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Tc}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Tc}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Tc}-\mathrm{Tc}}{\left(\Delta \theta \right)}^2$
parameter	3.467	10.636	10.636
$r_0$ or ${\theta }_0$	2.506	78.001	78.001

Table 263. Table 263: Two-body SW potential parameters for single-layer 1T-TcSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Tc-Se	2.355	0.925	19.723	0.0	3.267

Table 264. Table 264: Three-body SW potential parameters for single-layer 1T-TcSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Tc}-\mathrm{Se}-\mathrm{Se}}$	63.150	78.001	0.925	0.925	0.0	3.267	0.0	3.267	0.0	4.309
${\theta }_{\mathrm{Se}-\mathrm{Tc}-\mathrm{Tc}}$	63.150	78.001	0.925	0.925	0.0	3.267	0.0	3.267	0.0	4.309

Table 265. Table 265: SW potential parameters for single-layer 1T-TcSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Tc-Se1-Se1	1.000	0.925	3.532	63.150	1.000	0.208	2.355	26.932	4	0	0.0

Table 266. Table 266: The VFF model for single-layer 1T-TcTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Tc}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Tc}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Tc}-\mathrm{Tc}}{\left(\Delta \theta \right)}^2$
parameter	0.785	8.894	8.894
$r_0$ or ${\theta }_0$	2.690	78.801	78.801

Table 267. Table 267: Two-body SW potential parameters for single-layer 1T-TcTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Tc-Te	0.628	1.021	26.181	0.0	3.519

Table 268. Table 268: Three-body SW potential parameters for single-layer 1T-TcTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Tc}-\mathrm{Te}-\mathrm{Te}}$	54.313	78.801	1.021	1.021	0.0	3.519	0.0	3.519	0.0	4.665
${\theta }_{\mathrm{Te}-\mathrm{Tc}-\mathrm{Tc}}$	54.313	78.801	1.021	1.021	0.0	3.519	0.0	3.519	0.0	4.665

Table 269. Table 269: SW potential parameters for single-layer 1T-TcTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Tc-Te1-Te1	1.000	1.021	3.447	54.313	1.000	0.194	0.628	24.110	4	0	0.0

Table 270. Table 270: The VFF model for single-layer 1T-RhTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Rh}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Rh}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Rh}-\mathrm{Rh}}{\left(\Delta \theta \right)}^2$
parameter	4.366	1.869	1.869
$r_0$ or ${\theta }_0$	2.633	91.001	91.001

Table 271. Table 271: Two-body SW potential parameters for single-layer 1T-RhTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Rh-Te	4.640	1.450	24.038	0.0	3.610

Table 272. Table 272: Three-body SW potential parameters for single-layer 1T-RhTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Rh}-\mathrm{Te}-\mathrm{Te}}$	18.192	91.001	1.450	1.450	0.0	3.610	0.0	3.610	0.0	5.131
${\theta }_{\mathrm{Te}-\mathrm{Rh}-\mathrm{Rh}}$	18.192	91.001	1.450	1.450	0.0	3.610	0.0	3.610	0.0	5.131

Table 273. Table 273: SW potential parameters for single-layer 1T-RhTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Rh-Te1-Te1	1.000	1.450	2.490	18.192	1.000	-0.017	4.640	5.436	4	0	0.0

Table 274. Table 274: The VFF model for single-layer 1T-PdS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pd}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pd}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Pd}-\mathrm{Pd}}{\left(\Delta \theta \right)}^2$
parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.401	94.998	94.998

Table 275. Table 275: Two-body SW potential parameters for single-layer 1T-PdS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pd-S	10.116	1.467	16.625	0.0	3.340

Table 276. Table 276: Three-body SW potential parameters for single-layer 1T-PdS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pd}-S-S}$	35.859	94.998	1.467	1.467	0.0	3.340	0.0	3.340	0.0	4.837
${\theta }_{S-\mathrm{Pd}-\mathrm{Pd}}$	35.859	94.998	1.467	1.467	0.0	3.340	0.0	3.340	0.0	4.837

Table 277. Table 277: SW potential parameters for single-layer 1T-PdS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pd-S1-S1	1.000	1.467	2.276	35.859	1.000	-0.087	10.116	3.588	4	0	0.0

Table 278. Table 278: The VFF model for single-layer 1T-PdSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pd}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pd}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Pd}-\mathrm{Pd}}{\left(\Delta \theta \right)}^2$
parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.493	94.999	94.999

Table 279. Table 279: Two-body SW potential parameters for single-layer 1T-PdSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pd-Se	10.902	1.523	19.310	0.0	3.467

Table 280. Table 280: Three-body SW potential parameters for single-layer 1T-PdSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pd}-\mathrm{Se}-\mathrm{Se}}$	35.859	94.999	1.523	1.523	0.0	3.467	0.0	3.467	0.0	5.021
${\theta }_{\mathrm{Se}-\mathrm{Pd}-\mathrm{Pd}}$	35.859	94.999	1.523	1.523	0.0	3.467	0.0	3.467	0.0	5.021

Table 281. Table 281: SW potential parameters for single-layer 1T-PdSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pd-Se1-Se1	1.000	1.523	2.276	35.859	1.000	-0.087	10.902	3.588	4	0	0.0

Table 282. Table 282: The VFF model for single-layer 1T-PdTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pd}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pd}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Pd}-\mathrm{Pd}}{\left(\Delta \theta \right)}^2$
parameter	10.374	3.122	3.122
$r_0$ or ${\theta }_0$	2.635	95.999	95.999

Table 283. Table 283: Two-body SW potential parameters for single-layer 1T-PdTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pd-Te	12.474	1.650	24.101	0.0	3.678

Table 284. Table 284: Three-body SW potential parameters for single-layer 1T-PdTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pd}-\mathrm{Te}-\mathrm{Te}}$	37.406	95.999	1.650	1.650	0.0	3.678	0.0	3.678	0.0	5.350
${\theta }_{\mathrm{Te}-\mathrm{Pd}-\mathrm{Pd}}$	37.406	95.999	1.650	1.650	0.0	3.678	0.0	3.678	0.0	5.350

Table 285. Table 285: SW potential parameters for single-layer 1T-PdTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pd-Te1-Te1	1.000	1.650	2.229	37.406	1.000	-0.105	12.474	3.250	4	0	0.0

Table 286. Table 286: The VFF model for single-layer 1T-SnS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sn}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sn}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Sn}-\mathrm{Sn}}{\left(\Delta \theta \right)}^2$
parameter	7.872	5.817	5.817
$r_0$ or ${\theta }_0$	2.570	90.173	90.173

Table 287. Table 287: Two-body SW potential parameters for single-layer 1T-SnS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-S	7.805	1.384	21.812	0.0	3.513

Table 288. Table 288: Three-body SW potential parameters for single-layer 1T-SnS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sn}-S-S}$	54.748	90.173	1.384	1.384	0.0	3.513	0.0	3.513	0.0	4.972
${\theta }_{S-\mathrm{Sn}-\mathrm{Sn}}$	54.748	90.173	1.384	1.384	0.0	3.513	0.0	3.513	0.0	4.972

Table 289. Table 289: SW potential parameters for single-layer 1T-SnS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-S1-S1	1.000	1.384	2.539	54.748	1.000	-0.003	7.805	5.949	4	0	0.0

Table 290. Table 290: The VFF model for single-layer 1T-SnSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Sn}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Sn}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Sn}-\mathrm{Sn}}{\left(\Delta \theta \right)}^2$
parameter	7.872	5.817	5.817
$r_0$ or ${\theta }_0$	2.704	89.044	89.044

Table 291. Table 291: Two-body SW potential parameters for single-layer 1T-SnSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-Se	8.395	1.411	26.730	0.0	3.681

Table 292. Table 292: Three-body SW potential parameters for single-layer 1T-SnSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Sn}-\mathrm{Se}-\mathrm{Se}}$	52.322	89.044	1.411	1.411	0.0	3.681	0.0	3.681	0.0	5.180
${\theta }_{\mathrm{Se}-\mathrm{Sn}-\mathrm{Sn}}$	52.322	89.044	1.411	1.411	0.0	3.681	0.0	3.681	0.0	5.180

Table 293. Table 293: SW potential parameters for single-layer 1T-SnSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Se1-Se1	1.000	1.411	2.609	52.322	1.000	0.017	8.395	6.743	4	0	0.0

Table 294. Table 294: The VFF model for single-layer 1T-HfS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Hf}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Hf}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Hf}-\mathrm{Hf}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.550	91.078	91.078

Table 295. Table 295: Two-body SW potential parameters for single-layer 1T-HfS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Hf-S	7.917	1.407	21.141	0.0	3.497

Table 296. Table 296: Three-body SW potential parameters for single-layer 1T-HfS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Hf}-S-S}$	41.798	91.078	1.407	1.407	0.0	3.497	0.0	3.497	0.0	4.973
${\theta }_{S-\mathrm{Hf}-\mathrm{Hf}}$	41.798	91.078	1.407	1.407	0.0	3.497	0.0	3.497	0.0	4.973

Table 297. Table 297: SW potential parameters for single-layer 1T-HfS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Hf-S-S	1.000	1.407	2.485	41.798	1.000	-0.019	7.917	5.394	4	0	0.0
S-Hf-Hf	1.000	1.407	2.485	41.798	1.000	-0.019	7.917	5.394	4	0	0.0

Table 298. Table 298: The VFF model for single-layer 1T-HfSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Hf}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Hf}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Hf}-\mathrm{Hf}}{\left(\Delta \theta \right)}^2$
parameter	7.930	4.283	4.283
$r_0$ or ${\theta }_0$	2.642	88.093	88.093

Table 299. Table 299: Two-body SW potential parameters for single-layer 1T-HfSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Hf-Se	7.871	1.341	24.361	0.0	3.583

Table 300. Table 300: Three-body SW potential parameters for single-layer 1T-HfSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Hf}-\mathrm{Se}-\mathrm{Se}}$	37.039	88.093	1.341	1.341	0.0	3.583	0.0	3.583	0.0	5.018
${\theta }_{\mathrm{Se}-\mathrm{Hf}-\mathrm{Hf}}$	37.039	88.093	1.341	1.341	0.0	3.583	0.0	3.583	0.0	5.018

Table 301. Table 301: SW potential parameters for single-layer 1T-HfSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Hf-Se-Se	1.000	1.341	2.672	37.039	1.000	0.033	7.871	7.533	4	0	0.0
Se-Hf-Hf	1.000	1.341	2.672	37.039	1.000	0.033	7.871	7.533	4	0	0.0

Table 302. Table 302: The VFF model for single-layer 1T-HfTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Hf}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Hf}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Hf}-\mathrm{Hf}}{\left(\Delta \theta \right)}^2$
parameter	3.328	3.877	3.877
$r_0$ or ${\theta }_0$	2.856	87.801	87.801

Table 303. Table 303: Two-body SW potential parameters for single-layer 1T-HfTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Hf-Te	3.835	1.439	33.262	0.0	3.869

Table 304. Table 304: Three-body SW potential parameters for single-layer 1T-HfTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Hf}-\mathrm{Te}-\mathrm{Te}}$	33.196	87.801	1.439	1.439	0.0	3.869	0.0	3.869	0.0	5.410
${\theta }_{\mathrm{Te}-\mathrm{Hf}-\mathrm{Hf}}$	33.196	87.801	1.439	1.439	0.0	3.869	0.0	3.869	0.0	5.410

Table 305. Table 305: SW potential parameters for single-layer 1T-HfTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Hf-Te1-Te1	1.000	1.439	2.690	33.196	1.000	0.038	3.835	7.764	4	0	0.0

Table 306. Table 306: The VFF model for single-layer 1T-TaS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ta}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Ta}-\mathrm{Ta}}{\left(\Delta \theta \right)}^2$
parameter	11.192	4.774	4.774
$r_0$ or ${\theta }_0$	2.458	85.999	85.999

Table 307. Table 307: Two-body SW potential parameters for single-layer 1T-TaS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ta-S	9.110	1.174	18.246	0.0	3.307

Table 308. Table 308: Three-body SW potential parameters for single-layer 1T-TaS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ta}-S-S}$	38.092	85.999	1.174	1.174	0.0	3.307	0.0	3.307	0.0	4.579
${\theta }_{S-\mathrm{Ta}-\mathrm{Ta}}$	38.092	85.999	1.174	1.174	0.0	3.307	0.0	3.307	0.0	4.579

Table 309. Table 309: SW potential parameters for single-layer 1T-TaS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ta-S1-S1	1.000	1.174	2.816	38.092	1.000	0.070	9.110	9.589	4	0	0.0

Table 310. Table 310: The VFF model for single-layer 1T-TaSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Ta}-\mathrm{Ta}}{\left(\Delta \theta \right)}^2$
parameter	9.348	4.535	4.535
$r_0$ or ${\theta }_0$	2.561	84.999	84.999

Table 311. Table 311: Two-body SW potential parameters for single-layer 1T-TaSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ta-Se	8.045	1.188	21.505	0.0	3.433

Table 312. Table 312: Three-body SW potential parameters for single-layer 1T-TaSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ta}-\mathrm{Se}-\mathrm{Se}}$	34.820	84.999	1.188	1.188	0.0	3.433	0.0	3.3433	0.0	4.727
${\theta }_{\mathrm{Se}-\mathrm{Ta}-\mathrm{Ta}}$	34.820	84.999	1.188	1.188	0.0	3.433	0.0	3.3433	0.0	4.727

Table 313. Table 313: SW potential parameters for single-layer 1T-TaSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ta-Se1-Se1	1.000	1.188	2.891	34.820	1.000	0.087	8.045	10.813	4	0	0.0

Table 314. Table 314: The VFF model for single-layer 1T-TaTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ta}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ta}-\mathrm{Ta}}{\left(\Delta \theta \right)}^2$
parameter	3.442	4.516	4.516
$r_0$ or ${\theta }_0$	2.770	82.999	82.999

Table 315. Table 315: Two-body SW potential parameters for single-layer 1T-TaTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ta-Te	3.283	1.207	29.415	0.0	3.684

Table 316. Table 316: Three-body SW potential parameters for single-layer 1T-TaTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ta}-\mathrm{Te}-\mathrm{Te}}$	32.144	82.999	1.207	1.207	0.0	3.684	0.0	3.684	0.0	5.014
${\theta }_{\mathrm{Te}-\mathrm{Ta}-\mathrm{Ta}}$	32.144	82.999	1.207	1.207	0.0	3.684	0.0	3.684	0.0	5.014

Table 317. Table 317: SW potential parameters for single-layer 1T-TaTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ta-Se1-Se1	1.000	1.188	2.891	34.820	1.000	0.087	8.045	10.813	4	0	0.0

Table 318. Table 318: The VFF model for single-layer 1T-WS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-W-W}{\left(\Delta \theta \right)}^2$
parameter	4.395	10.087	10.087
$r_0$ or ${\theta }_0$	2.413	82.799	82.799

Table 319. Table 319: Two-body SW potential parameters for single-layer 1T-WS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-S	3.163	1.045	16.937	0.0	3.206

Table 320. Table 320: Three-body SW potential parameters for single-layer 1T-WS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-S-S}$	71.264	82.799	1.045	1.045	0.0	3.206	0.0	3.206	0.0	4.359
${\theta }_{S-W-W}$	71.264	82.799	1.045	1.045	0.0	3.206	0.0	3.206	0.0	4.359

Table 321. Table 321: SW potential parameters for single-layer 1T-WS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W-S1-S1	1.000	1.045	3.069	71.264	1.000	0.125	3.163	14.209	4	0	0.0

Table 322. Table 322: The VFF model for single-layer 1T-WSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-W-W}{\left(\Delta \theta \right)}^2$
parameter	2.556	15.375	15.375
$r_0$ or ${\theta }_0$	2.521	80.501	80.501

Table 323. Table 323: Two-body SW potential parameters for single-layer 1T-WSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-Se	1.885	1.013	20.186	0.0	3.320

Table 324. Table 324: Three-body SW potential parameters for single-layer 1T-WSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-\mathrm{Se}-\mathrm{Se}}$	99.800	80.501	1.013	1.013	0.0	3.320	0.0	3.320	0.0	4.450
${\theta }_{\mathrm{Se}-W-W}$	99.800	80.501	1.013	1.013	0.0	3.320	0.0	3.320	0.0	4.450

Table 325. Table 325: SW potential parameters for single-layer 1T-WSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W-Se1-Se1	1.000	1.013	3.277	99.800	1.000	0.165	1.885	19.156	4	0	0.0

Table 326. Table 326: The VFF model for single-layer 1T-WTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{W-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{W-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-W-W}{\left(\Delta \theta \right)}^2$
parameter	2.272	19.437	19.437
$r_0$ or ${\theta }_0$	2.720	79.999	79.999

Table 327. Table 327: Two-body SW potential parameters for single-layer 1T-WTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
W-Te	1.924	1.075	27.376	0.0	3.575

Table 328. Table 328: Three-body SW potential parameters for single-layer 1T-WTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{W-\mathrm{Te}-\mathrm{Te}}$	123.899	79.999	1.075	1.075	0.0	3.575	0.0	3.575	0.0	4.777
${\theta }_{\mathrm{Te}-W-W}$	123.899	79.999	1.075	1.075	0.0	3.575	0.0	3.575	0.0	4.777

Table 329. Table 329: SW potential parameters for single-layer 1T-WTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
W-Te1-Te1	1.000	1.075	3.325	123.899	1.000	0.174	1.924	20.483	4	0	0.0

Table 330. Table 330: The VFF model for single-layer 1T-ReS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Re}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Re}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Re}-\mathrm{Re}}{\left(\Delta \theta \right)}^2$
parameter	2.684	10.829	10.829
$r_0$ or ${\theta }_0$	2.405	79.498	79.498

Table 331. Table 331: Two-body SW potential parameters for single-layer 1T-ReS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Re-S	1.751	0.934	16.714	0.0	3.154

Table 332. Table 332: Three-body SW potential parameters for single-layer 1T-ReS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Re}-S-S}$	67.797	79.498	0.934	0.934	0.0	3.154	0.0	3.154	0.0	4.201
${\theta }_{S-\mathrm{Re}-\mathrm{Re}}$	67.797	79.498	0.934	0.934	0.0	3.154	0.0	3.154	0.0	4.201

Table 333. Table 333: SW potential parameters for single-layer 1T-ReS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Re-S1-S1	1.000	0.934	3.375	67.797	1.000	0.182	1.751	21.916	4	0	0.0

Table 334. Table 334: The VFF model for single-layer 1T-ReSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Re}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Re}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Re}-\mathrm{Re}}{\left(\Delta \theta \right)}^2$
parameter	4.313	12.674	12.674
$r_0$ or ${\theta }_0$	2.515	76.999	76.999

Table 335. Table 335: Two-body SW potential parameters for single-layer 1T-ReSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Re-Se	2.866	0.896	20.001	0.0	3.265

Table 336. Table 336: Three-body SW potential parameters for single-layer 1T-ReSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Re}-\mathrm{Se}-\mathrm{Se}}$	72.666	76.999	0.896	0.896	0.0	3.265	0.0	3.265	0.0	4.277
${\theta }_{\mathrm{Se}-\mathrm{Re}-\mathrm{Re}}$	72.666	76.999	0.896	0.896	0.0	3.265	0.0	3.265	0.0	4.277

Table 337. Table 337: SW potential parameters for single-layer 1T-ReSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Re-Se1-Se1	1.000	0.896	3.645	72.666	1.000	0.225	2.866	31.036	4	0	0.0

Table 338. Table 338: The VFF model for single-layer 1T-ReTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Re}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Re}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Re}-\mathrm{Re}}{\left(\Delta \theta \right)}^2$
parameter	1.724	14.812	14.812
$r_0$ or ${\theta }_0$	2.703	77.501	77.501

Table 339. Table 339: Two-body SW potential parameters for single-layer 1T-ReTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Re-Te	1.343	0.980	26.678	0.0	3.517

Table 340. Table 340: Three-body SW potential parameters for single-layer 1T-ReTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Re}-\mathrm{Te}-\mathrm{Te}}$	86.424	77.501	0.980	0.980	0.0	3.517	0.0	3.517	0.0	4.622
${\theta }_{\mathrm{Te}-\mathrm{Re}-\mathrm{Re}}$	86.424	77.501	0.980	0.980	0.0	3.517	0.0	3.517	0.0	4.622

Table 341. Table 341: SW potential parameters for single-layer 1T-ReTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Re-Te1-Te1	1.000	0.980	3.587	86.424	1.000	0.216	1.343	28.891	4	0	0.0

Table 342. Table 342: The VFF model for single-layer 1T-IrTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ir}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ir}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Ir}-\mathrm{Ir}}{\left(\Delta \theta \right)}^2$
parameter	5.334	2.182	2.182
$r_0$ or ${\theta }_0$	2.649	93.002	93.002

Table 343. Table 343: Two-body SW potential parameters for single-layer 1T-IrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ir-Te	6.030	1.538	24.621	0.0	3.658

Table 344. Table 344: Three-body SW potential parameters for single-layer 1T-IrTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Ir}-\mathrm{Te}-\mathrm{Te}}$	23.056	93.002	1.538	1.538	0.0	3.658	0.0	3.658	0.0	5.250
${\theta }_{\mathrm{Te}-\mathrm{Ir}-\mathrm{Ir}}$	23.056	93.002	1.538	1.538	0.0	3.658	0.0	3.658	0.0	5.250

Table 345. Table 345: SW potential parameters for single-layer 1T-IrTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ir-Te1-Te1	1.000	1.538	2.370	23.056	1.000	-0.052	6.030	4.398	4	0	0.0

Table 346. Table 346: The VFF model for single-layer 1T-PtS 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pt}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pt}-S-S}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{S-\mathrm{Pt}-\mathrm{Pt}}{\left(\Delta \theta \right)}^2$
parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.371	96.00	96.00

Table 347. Table 347: Two-body SW potential parameters for single-layer 1T-PtS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pt-S	11.806	1.485	15.796	0.0	3.309

Table 348. Table 348: Three-body SW potential parameters for single-layer 1T-PtS 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pt}-S-S}$	59.607	96.00	1.485	1.485	0.0	3.309	0.0	3.309	0.0	4.813
${\theta }_{S-\mathrm{Pt}-\mathrm{Pt}}$	59.607	96.00	1.485	1.485	0.0	3.309	0.0	3.309	0.0	4.813

Table 349. Table 349: SW potential parameters for single-layer 1T-PtS 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pt-S1-S1	1.000	1.485	2.229	59.607	1.000	-0.105	11.806	3.250	4	0	0.0

Table 350. Table 350: The VFF model for single-layer 1T-PtSe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pt}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pt}-\mathrm{Se}-\mathrm{Se}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Se}-\mathrm{Pt}-\mathrm{Pt}}{\left(\Delta \theta \right)}^2$
parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.467	95.999	95.999

Table 351. Table 351: Two-body SW potential parameters for single-layer 1T-PtSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pt-Se	12.781	1.545	18.511	0.0	3.443

Table 352. Table 352: Three-body SW potential parameters for single-layer 1T-PtSe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pt}-\mathrm{Se}-\mathrm{Se}}$	59.608	95.999	1.545	1.545	0.0	3.443	0.0	3.443	0.0	5.008
${\theta }_{\mathrm{Se}-\mathrm{Pt}-\mathrm{Pt}}$	59.608	95.999	1.545	1.545	0.0	3.443	0.0	3.443	0.0	5.008

Table 353. Table 353: SW potential parameters for single-layer 1T-PtSe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pt-Se1-Se1	1.000	1.545	2.229	59.608	1.000	-0.105	12.781	3.250	4	0	0.0

Table 354. Table 354: The VFF model for single-layer 1T-PtTe 2 . The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Pt}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Pt}-\mathrm{Te}-\mathrm{Te}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{Te}-\mathrm{Pt}-\mathrm{Pt}}{\left(\Delta \theta \right)}^2$
parameter	12.128	4.975	4.975
$r_0$ or ${\theta }_0$	2.661	95.998	95.998

Table 355. Table 355: Two-body SW potential parameters for single-layer 1T-PtTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Pt-Te	14.877	1.667	25.081	0.0	3.714

Table 356. Table 356: Three-body SW potential parameters for single-layer 1T-PtTe 2 used by GULP Gale ( 1997 ) as expressed in Eq. ( 4 ). The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} in the first line indicates the bending energy for the angle with atom i as the apex.

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{\mathrm{Pt}-\mathrm{Te}-\mathrm{Te}}$	59.607	95.998	1.667	1.667	0.0	3.714	0.0	3.714	0.0	5.403
${\theta }_{\mathrm{Te}-\mathrm{Pt}-\mathrm{Pt}}$	59.607	95.998	1.667	1.667	0.0	3.714	0.0	3.714	0.0	5.403

Table 357. Table 357: SW potential parameters for single-layer 1T-PtTe 2 used by LAMMPS lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Pt-Te1-Te1	1.000	1.667	2.229	59.607	1.000	-0.104	14.877	3.250	4	0	0.0

Table 358. Table 358: The VFF model for black phosphorus. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  178  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$
parameter	10.542	10.542	7.048	7.048
$r_0$ or ${\theta }_0$	2.238	2.260	96.581	102.307

Table 359. Table 359: Two-body SW potential parameters for black phosphorus used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  178  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	4.172	0.551	12.543	0.0	2.793
$r_{14}$	4.976	0.685	13.044	0.0	2.882

Table 360. Table 360: Three-body SW potential parameters for black phosphorus used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  178  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	25.965	96.581	0.551	0.551	0.0	2.793	0.0	2.793	2.793	3.365
${\theta }_{134}$	29.932	102.307	0.551	0.685	0.0	2.793	0.0	2.882	2.882	3.772

Table 361. Table 361: SW potential parameters for black phosphorus used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types (P 1 and P 2 ) in the first column are displayed in Fig.  178 .

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
P1-P1-P1	1.000	0.551	5.069	25.965	1.000	-0.115	4.172	136.080	4	0	0.0
P1-P2-P2	1.000	0.685	4.207	0.000	1.000	0.000	4.976	59.245	4	0	0.0
P1-P1-P2	1.000	0.000	0.000	29.932	1.000	-0.213	0.000	0.000	4	0	0.0

Table 362. Table 362: The VFF model for p-arsenene. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  178  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$
parameter	7.936	7.936	7.456	7.456
$r_0$ or ${\theta }_0$	2.508	2.495	94.400	100.692

Table 363. Table 363: Two-body SW potential parameters for p-arsenene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  178  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	3.180	0.455	19.782	0.0	3.042
$r_{14}$	4.477	0.737	19.375	0.0	3.173

Table 364. Table 364: Three-body SW potential parameters for p-arsenene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  178  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	20.597	94.400	0.455	0.455	0.0	3.042	0.0	3.042	3.628	4.225
${\theta }_{134}$	26.831	100.692	0.455	0.737	0.0	3.042	0.0	3.173	3.173	4.149

Table 365. Table 365: SW potential parameters for p-arsenene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  178  (b), with element symbol P substituted by As.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
As1-As1-As1	1.000	0.455	6.686	20.597	1.000	-0.077	3.180	461.556	4	0	0.0
As1-As2-As2	1.000	0.737	4.305	0.000	1.000	0.000	4.477	65.671	4	0	0.0
As1-As1-As2	1.000	0.000	0.000	26.831	1.000	-0.186	0.000	0.000	4	0	0.0

Table 366. Table 366: The VFF model for p-antimonene. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  183  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.675	7.675	6.534	12.068	12.068
$r_0$ or ${\theta }_0$	2.950	2.870	95.380	88.300	102.800

Table 367. Table 367: Two-body SW potential parameters for p-antimonene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  183  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	1.750	0.122	37.867	0.0	3.250
$r_{14}$	11.221	1.843	33.923	0.0	4.020

Table 368. Table 368: Three-body SW potential parameters for p-antimonene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  183  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	7.435	95.380	0.122	0.122	0.0	3.250	0.0	3.250	0.0	4.545
${\theta }_{134}$	45.054	88.380	1.843	0.122	0.0	4.020	0.0	3.250	0.0	5.715
${\theta }_{415}$	47.338	102.800	1.843	0.122	0.0	4.020	0.0	3.250	0.0	6.105

Table 369. Table 369: SW potential parameters for p-antimonene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  184 .

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sb5-Sb4-Sb4	1.000	5.103	0.958	635.059	1.000	-0.094	38.498	0.056	4	0	0.0
Sb1-Sb2-Sb2	1.000	1.924	2.102	0.000	1.000	0.000	11.708	2.476	4	0	0.0
Sb5-Sb4-Sb6	1.000	0.000	0.000	431.139	1.000	0.030	0.000	0.000	4	0	0.0
Sb2-Sb3-Sb1	1.000	0.000	0.000	452.994	1.000	-0.222	0.000	0.000	4	0	0.0

Table 370. Table 370: The VFF model for p-bismuthene. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  183  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.675	7.675	2.267	8.347	8.347
$r_0$ or ${\theta }_0$	3.110	3.097	94.018	86.486	103.491

Table 371. Table 371: Two-body SW potential parameters for p-bismuthene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  183  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	1.777	0.109	46.775	0.0	3.401
$r_{14}$	12.322	1.872	45.998	0.0	4.301

Table 372. Table 372: Three-body SW potential parameters for p-bismuthene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  183  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	2.408	94.018	0.109	0.109	0.0	3.401	0.0	3.401	0.0	4.745
${\theta }_{134}$	28.842	86.486	1.872	0.109	0.0	4.301	0.0	3.401	0.0	5.982
${\theta }_{415}$	30.388	103.491	1.872	0.109	0.0	4.301	0.0	3.401	0.0	6.473

Table 373. Table 373: SW potential parameters for p-bismuthene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  184 with elemental symbol Sb substituted by Bi.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Bi1-Bi8-Bi8	1.000	2.737	1.571	112.813	1.000	-0.070	18.974	1.000	4	0	0.0
Bi1-Bi2-Bi2	1.000	2.808	1.532	0.000	1.000	0.000	19.577	0.888	4	0	0.0
Bi1-Bi8-Bi2	1.000	0.000	0.000	429.598	1.000	0.061	0.000	0.000	4	0	0.0
Bi2-Bi3-Bi1	1.000	0.000	0.000	452.618	1.000	-0.233	0.000	0.000	4	0	0.0

Table 374. Table 374: The VFF model for the single-layer p-SiO. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	12.191	12.191	4.817	3.123	3.123
$r_0$ or ${\theta }_0$	1.843	1.859	95.989	96.000	132.005

Table 375. Table 375: Two-body SW potential parameters for the single-layer p-SiO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	19.127	2.720	5.769	0.0	2.962
$r_{14}$	7.105	1.133	5.972	0.0	2.585

Table 376. Table 376: Three-body SW potential parameters for the single-layer p-SiO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	314.008	95.989	2.720	2.720	0.0	2.962	0.0	2.962	0.0	3.720
${\theta }_{134}$	85.406	96.000	1.133	2.720	0.0	2.585	0.0	2.962	0.0	3.875
${\theta }_{415}$	152.982	132.005	1.133	2.720	0.0	2.585	0.0	2.962	0.0	4.194

Table 377. Table 377: SW potential parameters for p-SiO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Si and X=O.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si1-O2-O2	1.000	2.720	1.089	314.008	1.000	-0.104	19.127	0.105	4	0	0.0
Si1-O1-O1	1.000	1.133	2.282	0.000	1.000	0.000	7.105	3.630	4	0	0.0
Si1-O1-O2	1.000	0.000	0.000	85.406	1.000	-0.105	0.000	0.000	4	0	0.0
O1-Si1-Si2	1.000	0.000	0.000	152.982	1.000	-0.669	0.000	0.000	4	0	0.0

Table 378. Table 378: The VFF model for the single-layer p-GeO. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	12.191	12.191	4.817	3.123	3.123
$r_0$ or ${\theta }_0$	1.956	1.986	102.692	93.300	128.213

Table 379. Table 379: Two-body SW potential parameters for the single-layer p-GeO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	21.562	2.889	7.319	0.0	3.144
$r_{14}$	9.258	1.384	7.778	0.0	2.815

Table 380. Table 380: Three-body SW potential parameters for the single-layer p-GeO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	326.824	102.692	2.889	2.889	0.0	3.144	0.0	3.144	0.0	3.928
${\theta }_{134}$	94.550	93.300	1.384	2.889	0.0	2.815	0.0	3.144	0.0	3.933
${\theta }_{415}$	152.646	128.213	1.384	2.889	0.0	2.815	0.0	3.144	0.0	4.284

Table 381. Table 381: SW potential parameters for p-GeO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Ge and X=O.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge1-O2-O2	1.000	2.889	1.089	326.824	1.000	-0.220	21.562	0.105	4	0	0.0
Ge1-O1-O1	1.000	1.384	2.034	0.000	1.000	0.000	9.258	2.119	4	0	0.0
Ge1-O1-O2	1.000	0.000	0.000	94.550	1.000	-0.058	0.000	0.000	4	0	0.0
O1-Ge1-Ge2	1.000	0.000	0.000	152.646	1.000	-0.619	0.000	0.000	4	0	0.0

Table 382. Table 382: The VFF model for the single-layer p-SnO. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	9.208	9.208	2.835	3.023	3.023
$r_0$ or ${\theta }_0$	2.127	2.163	106.117	90.000	126.496

Table 383. Table 383: Two-body SW potential parameters for the single-layer p-SnO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	11.711	2.107	10.234	0.0	3.185
$r_{14}$	8.879	1.612	10.945	0.0	3.096

Table 384. Table 384: Three-body SW potential parameters for the single-layer p-SnO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	82.293	106.117	2.107	2.107	0.0	3.185	0.0	3.185	0.0	4.082
${\theta }_{134}$	62.178	90.000	1.612	2.107	0.0	3.096	0.0	3.185	0.0	4.017
${\theta }_{415}$	96.214	126.496	1.612	2.107	0.0	3.096	0.0	3.185	0.0	4.426

Table 385. Table 385: SW potential parameters for p-SnO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Sn and X=O.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn1-O2-O2	1.000	2.107	1.512	82.293	1.000	-0.278	11.711	0.519	4	0	0.0
Sn1-O1-O1	1.000	1.612	1.921	0.000	1.000	0.000	8.879	1.623	4	0	0.0
Sn1-O1-O2	1.000	0.000	0.000	62.178	1.000	0.000	0.000	0.000	4	0	0.0
O1-Sn1-Sn2	1.000	0.000	0.000	96.214	1.000	-0.595	0.000	0.000	4	0	0.0

Table 386. Table 386: The VFF model for the single-layer p-CS. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	9.291	9.291	3.933	3.075	3.075
$r_0$ or ${\theta }_0$	1.757	1.849	105.384	118.100	104.288

Table 387. Table 387: Two-body SW potential parameters for the single-layer p-CS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	8.898	1.894	4.765	0.0	2.669
$r_{14}$	5.791	1.220	5.844	0.0	2.600

Table 388. Table 388: Three-body SW potential parameters for the single-layer p-CS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	134.527	105.384	1.894	1.894	0.0	2.669	0.0	2.669	0.0	3.559
${\theta }_{134}$	79.992	118.100	1.220	1.894	0.0	2.600	0.0	2.669	0.0	4.046
${\theta }_{415}$	66.283	104.288	1.220	1.894	0.0	2.600	0.0	2.669	0.0	3.921

Table 389. Table 389: SW potential parameters for p-CS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=C and X=S.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C1-S2-S2	1.000	1.894	1.410	134.527	1.000	-0.265	8.898	0.371	4	0	0.0
C1-S1-S1	1.000	1.220	2.131	0.000	1.000	0.000	5.791	2.637	4	0	0.0
C1-S1-S2	1.000	0.000	0.000	79.992	1.000	-0.471	0.000	0.000	4	0	0.0
S1-C1-C2	1.000	0.000	0.000	66.283	1.000	-0.247	0.000	0.000	4	0	0.0

Table 390. Table 390: The VFF model for the single-layer p-SiS. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.135	7.135	2.512	2.922	2.922
$r_0$ or ${\theta }_0$	2.300	2.344	93.554	96.500	111.710

Table 391. Table 391: Two-body SW potential parameters for the single-layer p-SiS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	3.878	0.797	13.992	0.0	2.977
$r_{14}$	6.051	1.301	15.094	0.0	3.217

Table 392. Table 392: Three-body SW potential parameters for the single-layer p-SiS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	13.284	93.554	0.797	0.797	0.0	2.977	0.0	2.977	0.0	4.063
${\theta }_{134}$	21.310	96.500	1.301	0.797	0.0	3.217	0.0	2.977	0.0	4.232
${\theta }_{415}$	24.372	111.710	1.301	0.797	0.0	3.217	0.0	2.977	0.0	4.423

Table 393. Table 393: SW potential parameters for p-SiS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Si and X=S.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si1-S2-S2	1.000	0.797	3.735	13.284	1.000	-0.062	3.878	34.661	4	0	0.0
Si1-S1-S1	1.000	1.301	2.474	0.000	1.000	0.000	6.051	5.276	4	0	0.0
Si1-S1-S2	1.000	0.000	0.000	21.310	1.000	-0.113	0.000	0.000	4	0	0.0
S1-Si1-Si2	1.000	0.000	0.000	24.372	1.000	-0.370	0.000	0.000	4	0	0.0

Table 394. Table 394: The VFF model for the single-layer p-GeS. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	6.364	6.364	2.153	3.896	3.896
$r_0$ or ${\theta }_0$	2.462	2.423	95.402	94.400	104.837

Table 395. Table 395: Two-body SW potential parameters for the single-layer p-GeS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	2.096	0.351	18.371	0.0	2.926
$r_{14}$	6.694	1.571	17.234	0.0	3.398

Table 396. Table 396: Three-body SW potential parameters for the single-layer p-GeS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	4.905	95.402	0.350	0.350	0.0	2.926	0.0	2.926	0.0	4.067
${\theta }_{134}$	20.842	94.400	1.571	0.350	0.0	3.398	0.0	2.926	0.0	4.292
${\theta }_{415}$	22.173	104.837	1.571	0.350	0.0	3.398	0.0	2.296	0.0	4.438

Table 397. Table 397: SW potential parameters for p-GeS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Ge and X=S.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge1-S4-S4	1.000	0.351	8.332	4.905	1.000	-0.094	2.096	1227.130	4	0	0.0
Ge1-S1-S1	1.000	1.571	2.163	0.000	1.000	0.000	6.694	2.830	4	0	0.0
Ge1-S1-S4	1.000	0.000	0.000	20.842	1.000	-0.077	0.000	0.000	4	0	0.0
S1-Ge1-Ge2	1.000	0.000	0.000	22.173	1.000	-0.256	0.000	0.000	4	0	0.0

Table 398. Table 398: The VFF model for the single-layer p-SnS. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	4.163	4.163	1.776	5.841	5.841
$r_0$ or ${\theta }_0$	2.728	2.595	95.762	89.000	101.887

Table 399. Table 399: Two-body SW potential parameters for the single-layer p-SnS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	0.782	0.106	27.692	0.0	2.997
$r_{14}$	5.636	1.887	22.674	0.0	3.702

Table 400. Table 400: Three-body SW potential parameters for the single-layer p-SnS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	1.968	95.762	0.106	0.106	0.0	2.997	0.0	2.997	0.0	4.197
${\theta }_{134}$	23.839	89.000	1.887	0.106	0.0	3.702	0.0	2.997	0.0	4.366
${\theta }_{415}$	24.888	101.887	1.887	0.106	0.0	3.702	0.0	2.997	0.0	4.566

Table 401. Table 401: SW potential parameters for p-SnS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Sn and X=S.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn1-S4-S4	1.000	0.106	28.347	1.971	1.000	-0.100	0.783	221784.222	4	0	0.0
Sn1-S1-S1	1.000	1.887	1.961	0.000	1.000	0.000	5.636	1.787	4	0	0.0
Sn1-S1-S4	1.000	0.000	0.000	23.839	1.000	0.017	0.000	0.000	4	0	0.0
S1-Sn1-Sn2	1.000	0.000	0.000	24.888	1.000	-0.206	0.000	0.000	4	0	0.0

Table 402. Table 402: The VFF model for the single-layer p-CSe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	10.120	10.120	4.505	3.910	3.910
$r_0$ or ${\theta }_0$	1.961	2.014	101.354	113.000	100.563

Table 403. Table 403: Two-body SW potential parameters for the single-layer p-CSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	6.141	1.114	7.394	0.0	2.700
$r_{14}$	7.411	1.316	8.226	0.0	2.828

Table 404. Table 404: Three-body SW potential parameters for the single-layer p-CSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	47.768	101.354	1.114	1.114	0.0	2.700	0.0	2.700	0.0	3.667
${\theta }_{134}$	52.464	113.000	1.316	1.114	0.0	2.828	0.0	2.700	0.0	4.157
${\theta }_{415}$	46.000	100.563	1.316	1.114	0.0	2.828	0.0	2.700	0.0	4.032

Table 405. Table 405: SW potential parameters for p-CSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=C and X=Se.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C1-Se4-Se4	1.000	1.114	2.424	47.768	1.000	-0.197	6.141	4.802	4	0	0.0
C1-Se1-Se1	1.000	1.316	2.149	0.000	1.000	0.000	7.411	2.743	4	0	0.0
C1-Se1-Se4	1.000	0.000	0.000	52.464	1.000	-0.391	0.000	0.000	4	0	0.0
Se1-C1-C2	1.000	0.000	0.000	46.000	1.000	-0.183	0.000	0.000	4	0	0.0

Table 406. Table 406: The VFF model for the single-layer p-SiSe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.192	7.192	3.222	5.890	5.890
$r_0$ or ${\theta }_0$	2.524	2.448	95.513	98.200	97.686

Table 407. Table 407: Two-body SW potential parameters for the single-layer p-SiSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	1.883	0.230	20.292	0.0	2.905
$r_{14}$	8.098	1.665	17.956	0.0	3.457

Table 408. Table 408: Three-body SW potential parameters for the single-layer p-SiSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	5.440	95.513	0.230	0.230	0.0	2.905	0.0	2.905	0.0	4.069
${\theta }_{134}$	28.616	98.200	1.665	0.230	0.0	3.457	0.0	2.905	0.0	4.379
${\theta }_{415}$	28.545	97.686	1.665	0.230	0.0	3.457	0.0	2.905	0.0	4.369

Table 409. Table 409: SW potential parameters for p-SiSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Si and X=Se.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si1-Se4-Se4	1.000	0.230	12.628	5.440	1.000	-0.096	1.883	7245.111	4	0	0.0
Si1-Se1-Se1	1.000	1.665	2.076	0.000	1.000	0.000	8.098	2.335	4	0	0.0
Si1-Se1-Se4	1.000	0.000	0.000	28.616	1.000	-0.143	0.000	0.000	4	0	0.0
Se1-Si1-Si2	1.000	0.000	0.000	28.545	1.000	-0.134	0.000	0.000	4	0	0.0

Table 410. Table 410: The VFF model for the single-layer p-GeSe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	5.063	5.063	2.249	5.927	5.927
$r_0$ or ${\theta }_0$	2.661	2.544	96.322	97.000	93.964

Table 411. Table 411: Two-body SW potential parameters for the single-layer p-GeSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	0.962	0.115	25.070	0.0	2.938
$r_{14}$	6.572	1.846	20.943	0.0	3.628

Table 412. Table 412: Three-body SW potential parameters for the single-layer p-GeSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	2.614	96.322	0.115	0.115	0.0	2.938	0.0	2.938	0.0	4.133
${\theta }_{134}$	25.087	97.400	1.846	0.115	0.0	3.628	0.0	2.938	0.0	4.455
${\theta }_{415}$	24.789	93.964	1.846	0.155	0.0	3.628	0.0	2.938	0.0	4.404

Table 413. Table 413: SW potential parameters for p-GeSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Ge and X=Se.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge1-Se4-Se4	1.000	0.115	25.561	2.614	1.000	-0.110	0.962	143723.555	4	0	0.0
Ge1-Se1-Se1	1.000	1.846	1.965	0.000	1.000	0.000	6.572	1.804	4	0	0.0
Ge1-Se1-Se4	1.000	0.000	0.000	25.087	1.000	-0.129	0.000	0.000	4	0	0.0
Se1-Ge1-Ge2	1.000	0.000	0.000	24.789	1.000	-0.069	0.000	0.000	4	0	0.0

Table 414. Table 414: The VFF model for the single-layer p-SnSe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	3.872	3.872	3.157	7.674	7.674
$r_0$ or ${\theta }_0$	2.887	2.730	95.087	92.500	95.411

Table 415. Table 415: Two-body SW potential parameters for the single-layer p-SnSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	0.565	0.056	34.734	0.0	3.088
$r_{14}$	5.811	1.989	27.773	0.0	3.895

Table 416. Table 416: Three-body SW potential parameters for the single-layer p-SnSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	2.777	95.087	0.056	0.056	0.0	3.088	0.0	3.088	0.0	4.357
${\theta }_{134}$	27.996	92.500	1.989	0.056	0.0	3.895	0.0	3.088	0.0	4.530
${\theta }_{415}$	28.193	95.411	1.989	0.056	0.0	3.895	0.0	3.088	0.0	4.576

Table 417. Table 417: SW potential parameters for p-SnSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Sn and X=Se.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn1-Se4-Se4	1.000	0.056	55.166	2.777	1.000	-0.089	0.565	3537820.961	4	0	0.0
Sn1-Se1-Se1	1.000	1.989	1.959	0.000	1.000	0.000	5.811	1.776	4	0	0.0
Sn1-Se1-Se4	1.000	0.000	0.000	27.996	1.000	-0.044	0.000	0.000	4	0	0.0
Se1-Sn1-Sn2	1.000	0.000	0.000	28.193	1.000	-0.094	0.000	0.000	4	0	0.0

Table 418. Table 418: The VFF model for the single-layer p-CTe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	13.235	13.235	4.210	3.901	3.901
$r_0$ or ${\theta }_0$	2.164	2.181	103.122	110.000	90.854

Table 419. Table 419: Two-body SW potential parameters for the single-layer p-CTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	4.165	0.424	10.965	0.0	2.643
$r_{14}$	12.519	1.569	11.313	0.0	3.106

Table 420. Table 420: Three-body SW potential parameters for the single-layer p-CTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	13.040	103.122	0.424	0.424	0.0	2.643	0.0	2.643	0.0	3.639
${\theta }_{134}$	29.206	110.000	1.569	0.424	0.0	3.106	0.0	2.643	0.0	4.280
${\theta }_{415}$	25.795	90.854	1.569	0.424	0.0	3.106	0.0	2.643	0.0	4.043

Table 421. Table 421: SW potential parameters for p-CTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=C and X=Te.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C1-Te4-Te4	1.000	0.424	6.232	13.040	1.000	-0.227	4.165	338.925	4	0	0.0
C1-Te1-Te1	1.000	1.569	1.979	0.000	1.000	0.000	12.519	1.866	4	0	0.0
C1-Te1-Te4	1.000	0.000	0.000	29.206	1.000	-0.342	0.000	0.000	4	0	0.0
Te1-C1-C2	1.000	0.000	0.000	25.795	1.000	-0.015	0.000	0.000	4	0	0.0

Table 422. Table 422: The VFF model for the single-layer p-SiTe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	9.479	2.892	3.145	10.111	10.111
$r_0$ or ${\theta }_0$	2.641	2.772	102.142	100.200	92.760

Table 423. Table 423: Two-body SW potential parameters for the single-layer p-SiTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	2.394	0.194	24.324	0.0	2.999
$r_{14}$	3.818	1.722	29.522	0.0	3.864

Table 424. Table 424: Three-body SW potential parameters for the single-layer p-SiTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	4.868	102.142	0.194	0.194	0.0	2.999	0.0	2.999	0.0	4.204
${\theta }_{134}$	43.419	100.200	1.722	0.194	0.0	3.864	0.0	2.999	0.0	4.577
${\theta }_{415}$	42.155	92.760	1.722	0.194	0.0	3.864	0.0	2.999	0.0	4.289

Table 425. Table 425: SW potential parameters for p-SiTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Si and X=Te.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si1-Te4-Te4	1.000	0.194	15.442	4.868	1.000	-0.210	2.394	17093.960	4	0	0.0
Si1-Te1-Te1	1.000	1.722	2.245	0.000	1.000	0.000	3.818	3.360	4	0	0.0
Si1-Te1-Te4	1.000	0.000	0.000	43.419	1.000	-0.177	0.000	0.000	4	0	0.0
Te1-Si1-Si2	1.000	0.000	0.000	42.155	1.000	-0.048	0.000	0.000	4	0	0.0

Table 426. Table 426: The VFF model for the single-layer p-GeTe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.074	7.074	2.611	5.876	5.876
$r_0$ or ${\theta }_0$	2.736	2.883	101.517	100.400	91.402

Table 427. Table 427: Two-body SW potential parameters for the single-layer p-GeTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	1.708	0.165	28.018	0.0	3.072
$r_{14}$	9.854	1.745	34.542	0.0	4.005

Table 428. Table 428: Three-body SW potential parameters for the single-layer p-GeTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	3.626	101.517	0.165	0.165	0.0	3.072	0.0	3.072	0.0	4.307
${\theta }_{134}$	23.509	100.400	1.745	0.165	0.0	4.005	0.0	3.072	0.0	4.659
${\theta }_{415}$	22.756	91.402	1.745	0.165	0.0	4.005	0.0	3.072	0.0	4.434

Table 429. Table 429: SW potential parameters for p-GeTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Ge and X=Te.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge1-Te4-Te4	1.000	0.165	18.665	3.626	1.000	-0.200	1.708	38204.858	4	0	0.0
Ge1-Te1-Te1	1.000	1.745	2.295	0.000	1.000	0.000	9.854	3.725	4	0	0.0
Ge1-Te1-Te4	1.000	0.000	0.000	23.509	1.000	-0.181	0.000	0.000	4	0	0.0
Te1-Ge1-Ge2	1.000	0.000	0.000	22.756	1.000	-0.024	0.000	0.000	4	0	0.0

Table 430. Table 430: The VFF model for the single-layer p-SnTe. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  189  (c). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{123}{\left(\Delta {\theta }_{123}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{415}{\left(\Delta {\theta }_{415}\right)}^2$
parameter	7.074	7.074	2.611	2.611	5.876
$r_0$ or ${\theta }_0$	2.931	3.164	101.578	96.000	94.045

Table 431. Table 431: Two-body SW potential parameters for the single-layer p-SnTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  189  (c).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{12}$	1.662	0.131	36.901	0.0	3.241
$r_{14}$	11.054	1.777	50.109	0.0	4.349

Table 432. Table 432: Three-body SW potential parameters for the single-layer p-SnTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  189  (c).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{123}$	3.170	101.578	0.131	0.131	0.0	3.241	0.0	3.241	0.0	4.562
${\theta }_{134}$	20.298	96.000	1.777	0.131	0.0	4.349	0.0	3.241	0.0	4.766
${\theta }_{415}$	20.176	94.045	1.777	0.131	0.0	4.349	0.0	3.241	0.0	4.595

Table 433. Table 433: SW potential parameters for p-SnTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  190 , with M=Sn and X=Te.

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn1-Te4-Te4	1.000	0.131	24.712	3.170	1.000	-0.201	1.662	124727.735	4	0	0.0
Sn1-Te1-Te1	1.000	1.777	2.448	0.000	1.000	0.000	11.054	5.028	4	0	0.0
Sn1-Te1-Te4	1.000	0.000	0.000	20.298	1.000	-0.105	0.000	0.000	4	0	0.0
Te1-Sn1-Sn2	1.000	0.000	0.000	20.176	1.000	-0.071	0.000	0.000	4	0	0.0

Table 434. Table 434: The VFF model for silicene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	18.387	3.465
$r_0$ or ${\theta }_0$	2.279	116.218

Table 435. Table 435: Two-body SW potential parameters for silicene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-Si	19.343	1.668	16.186	0.0	3.075

Table 436. Table 436: Three-body SW potential parameters for silicene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-Si-Si	142.310	116.218	1.668	1.668	0.0	3.075	0.0	3.075	3.075	4.181

Table 437. Table 437: SW potential parameters for silicene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-Si-Si	1.000	1.668	1.844	142.310	1.000	-0.442	19.343	2.091	4	0	0.0

Table 438. Table 438: The VFF model for germanene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	18.387	3.465
$r_0$ or ${\theta }_0$	2.443	112.358

Table 439. Table 439: Two-body SW potential parameters for germanene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ge-Ge	19.570	1.607	21.372	0.0	3.252

Table 440. Table 440: Three-body SW potential parameters for germanene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ge-Ge-Ge	107.735	112.358	1.607	1.607	0.0	3.252	0.0	3.252	3.252	4.4

Table 441. Table 441: SW potential parameters for germanene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge-Ge-Ge	1.000	1.607	2.024	107.735	1.000	-0.380	19.570	3.205	4	0	0.0

Table 442. Table 442: The VFF model for stanene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	10.489	1.372
$r_0$ or ${\theta }_0$	2.836	111.224

Table 443. Table 443: Two-body SW potential parameters for stanene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-Sn	19.542	2.227	42.047	0.0	3.758

Table 444. Table 444: Three-body SW potential parameters for stanene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-Sn-Sn	98.863	111.224	2.227	2.227	0.0	3.758	0.0	3.758	3.758	5.076

Table 445. Table 445: SW potential parameters for stanene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Sn-Sn	1.000	2.227	1.687	98.863	1.000	-0.362	19.542	1.709	4	0	0.0

Table 446. Table 446: The VFF model for indiene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	2.128	1.175
$r_0$ or ${\theta }_0$	2.890	94.372

Table 447. Table 447: Two-body SW potential parameters for indiene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In-In	1.537	0.946	41.855	0.0	3.565

Table 448. Table 448: Three-body SW potential parameters for indiene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In-In-In	9.745	94.372	0.946	0.946	0.0	3.565	0.0	4.565	3.565	4.686

Table 449. Table 449: SW potential parameters for indiene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In-In-In	1.000	0.946	3.768	9.745	1.000	-0.076	1.537	52.262	4	0	0.0

Table 450. Table 450: The VFF model for blue phosphorus. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.270	94.209

Table 451. Table 451: Two-body SW potential parameters for blue phosphorus used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
P-P	5.706	0.491	13.276	0.0	2.798

Table 452. Table 452: Three-body SW potential parameters for blue phosphorus used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
P-P-P	16.605	94.209	0.491	0.491	0.0	2.798	0.0	2.798	2.798	3.677

Table 453. Table 453: SW potential parameters for blue phosphorus used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
P-P-P	1.000	0.491	5.699	16.605	1.000	-0.073	5.706	228.424	4	0	0.0

Table 454. Table 454: The VFF model for b-arsenene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.510	91.964

Table 455. Table 455: Two-body SW potential parameters for b-arsenene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
As-As	6.418	0.482	19.846	0.0	3.060

Table 456. Table 456: Three-body SW potential parameters for b-arsenene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
As-As-As	14.845	91.964	0.482	0.482	0.0	3.060	0.0	3.060	3.060	4.004

Table 457. Table 457: SW potential parameters for b-arsenene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
As-As-As	1.000	0.482	6.349	14.845	1.000	-0.034	6.418	367.693	4	0	0.0

Table 458. Table 458: The VFF model for b-antimonene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	15.372	5.138
$r_0$ or ${\theta }_0$	2.890	90.927

Table 459. Table 459: Two-body SW potential parameters for b-antimonene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sb-Sb	8.173	0.523	34.879	0.0	3.505

Table 460. Table 460: Three-body SW potential parameters for b-antimonene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sb-Sb-Sb	14.100	90.927	0.523	0.523	0.0	3.505	0.0	3.503	3.505	4.577

Table 461. Table 461: SW potential parameters for b-antimonene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sb-Sb-Sb	1.000	0.523	6.702	14.100	1.000	-0.016	8.173	466.184	4	0	0.0

Table 462. Table 462: The VFF model for b-bismuthene. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	11.529	3.853
$r_0$ or ${\theta }_0$	3.045	90.901

Table 463. Table 463: Two-body SW potential parameters for b-bismuthene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Bi-Bi	6.805	0.552	42.985	0.0	3.693

Table 464. Table 464: Three-body SW potential parameters for b-bismuthene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Bi-Bi-Bi	10.574	90.901	0.552	0.552	0.0	3.693	0.0	3.693	3.693	4.821

Table 465. Table 465: SW potential parameters for b-bismuthene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Bi-Bi-Bi	1.000	0.552	6.690	10.574	1.000	-0.016	6.805	462.978	4	0	0.0

Table 466. Table 466: The VFF model for b-CO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	16.063	5.221
$r_0$ or ${\theta }_0$	1.636	97.181

Table 467. Table 467: Two-body SW potential parameters for b-CO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
C-O	7.656	1.054	3.582	0.0	2.293

Table 468. Table 468: Three-body SW potential parameters for b-CO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
C-O-O	65.778	97.181	1.054	1.054	0.0	2.293	0.0	2.293	0.0	3.352
O-C-C	65.778	97.181	1.054	1.054	0.0	2.293	0.0	2.293	0.0	3.352

Table 469. Table 469: SW potential parameters for b-CO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C-O-O	1.000	1.054	2.175	65.778	1.000	-0.125	7.812	2.900	4	0	0.0
O-C-C	1.000	1.054	2.175	65.778	1.000	-0.125	7.812	2.900	4	0	0.0

Table 470. Table 470: The VFF model for b-CS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	9.390	4.722
$r_0$ or ${\theta }_0$	1.880	97.921

Table 471. Table 471: Two-body SW potential parameters for b-CS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
C-S	6.014	1.233	6.246	0.0	2.641

Table 472. Table 472: Three-body SW potential parameters for b-CS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
C-S-S	61.413	97.921	1.233	1.233	0.0	2.641	0.0	2.641	0.0	3.874
S-C-C	61.413	97.921	1.233	1.233	0.0	2.641	0.0	2.641	0.0	3.874

Table 473. Table 473: SW potential parameters for b-CS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C-S-S	1.000	1.233	2.142	61.413	1.000	-0.138	6.014	2.703	4	0	0.0
S-C-C	1.000	1.233	2.142	61.413	1.000	-0.138	6.014	2.703	4	0	0.0

Table 474. Table 474: The VFF model for b-CSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	10.425	5.031
$r_0$ or ${\theta }_0$	2.055	96.362

Table 475. Table 475: Two-body SW potential parameters for b-CSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
C-Se	7.691	1.298	8.917	0.0	2.872

Table 476. Table 476: Three-body SW potential parameters for b-CSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
C-Se-Se	61.215	96.362	1.298	1.298	0.0	2.872	0.0	2.872	0.0	4.184
Se-C-C	61.215	96.362	1.298	1.298	0.0	2.872	0.0	2.872	0.0	4.184

Table 477. Table 477: SW potential parameters for b-CSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C-Se-Se	1.000	1.298	2.212	61.215	1.000	-0.111	7.691	3.137	4	0	0.0
Se-C-C	1.000	1.298	2.212	61.215	1.000	-0.111	7.691	3.137	4	0	0.0

Table 478. Table 478: The VFF model for b-CTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	9.367	4.311
$r_0$ or ${\theta }_0$	2.231	97.239

Table 479. Table 479: Two-body SW potential parameters for b-CTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
C-Te	8.314	1.440	12.387	0.0	3.127

Table 480. Table 480: Three-body SW potential parameters for b-CTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
C-Te-Te	54.451	97.239	1.440	1.440	0.0	3.127	0.0	3.127	0.0	4.573
Te-C-C	54.451	97.239	1.440	1.440	0.0	3.127	0.0	3.127	0.0	4.573

Table 481. Table 481: SW potential parameters for b-CTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
C-Te-Te	1.000	1.440	2.172	54.451	1.000	-0.126	8.314	2.883	4	0	0.0
Te-C-C	1.000	1.440	2.172	54.451	1.000	-0.126	8.314	2.883	4	0	0.0

Table 482. Table 482: The VFF model for b-SiO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	9.315	3.300
$r_0$ or ${\theta }_0$	1.884	96.676

Table 483. Table 483: Two-body SW potential parameters for b-SiO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-O	5.819	1.200	6.299	0.0	2.636

Table 484. Table 484: Three-body SW potential parameters for b-SiO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-O-O	40.695	96.676	1.200	1.200	0.0	2.636	0.0	2.636	0.0	3.845
O-Si-Si	40.695	96.676	1.200	1.200	0.0	2.636	0.0	2.636	0.0	3.845

Table 485. Table 485: SW potential parameters for b-SiO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-O-O	1.000	1.200	2.197	40.695	1.000	-0.116	5.819	3.043	4	0	0.0
O-Si-Si	1.000	1.200	2.197	40.695	1.000	-0.116	5.819	3.043	4	0	0.0

Table 486. Table 486: The VFF model for b-SiS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.441	4.802
$r_0$ or ${\theta }_0$	2.321	90.581

Table 487. Table 487: Two-body SW potential parameters for b-SiS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-S	6.897	1.264	14.510	0.0	3.177

Table 488. Table 488: Three-body SW potential parameters for b-SiS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-S-S	45.954	90.581	1.264	1.264	0.0	3.177	0.0	3.177	0.0	4.506
S-Si-Si	45.954	90.581	1.264	1.264	0.0	3.177	0.0	3.177	0.0	4.506

Table 489. Table 489: SW potential parameters for b-SiS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-S-S	1.000	1.264	2.514	45.954	1.000	-0.010	6.897	5.687	4	0	0.0
S-Si-Si	1.000	1.264	2.514	45.954	1.000	-0.010	6.897	5.687	4	0	0.0

Table 490. Table 490: The VFF model for b-SiSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.441	4.802
$r_0$ or ${\theta }_0$	2.477	90.590

Table 491. Table 491: Two-body SW potential parameters for b-SiSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-Se	7.857	1.349	18.822	0.0	3.391

Table 492. Table 492: Three-body SW potential parameters for b-SiSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-Se-Se	45.968	90.590	1.349	1.349	0.0	3.391	0.0	3.391	0.0	4.810
Se-Si-Si	45.968	90.590	1.349	1.349	0.0	3.391	0.0	3.391	0.0	4.810

Table 493. Table 493: SW potential parameters for b-SiSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-Se-Se	1.000	1.349	2.514	45.968	1.000	-0.010	7.857	5.6683	4	0	0.0
Se-Si-Si	1.000	1.349	2.514	45.968	1.000	-0.010	7.857	5.6683	4	0	0.0

Table 494. Table 494: The VFF model for b-SiTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.418	4.349
$r_0$ or ${\theta }_0$	2.690	90.779

Table 495. Table 495: Two-body SW potential parameters for b-SiTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-Te	9.285	1.473	26.181	0.0	3.685

Table 496. Table 496: Three-body SW potential parameters for b-SiTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-Te-Te	41.952	90.779	1.473	1.473	0.0	3.685	0.0	3.685	0.0	5.232
Te-Si-Si	41.952	90.779	1.473	1.473	0.0	3.685	0.0	3.685	0.0	5.232

Table 497. Table 497: SW potential parameters for b-SiTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-Te-Te	1.000	0.725	4.497	27.653	1.000	-0.014	4.428	113.714	4	0	0.0
Te-Si-Si	1.000	0.725	4.497	27.653	1.000	-0.014	4.428	113.714	4	0	0.0

Table 498. Table 498: The VFF model for b-GeO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	9.315	3.300
$r_0$ or ${\theta }_0$	2.032	100.475

Table 499. Table 499: Two-body SW potential parameters for b-GeO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ge-O	7.390	1.413	8.524	0.0	2.879

Table 500. Table 500: Three-body SW potential parameters for b-GeO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ge-O-O	47.962	100.475	1.413	1.413	0.0	2.879	0.0	2.879	0.0	4.267
O-Ge-Ge	47.962	100.475	1.413	1.413	0.0	2.879	0.0	2.879	0.0	4.267

Table 501. Table 501: SW potential parameters for b-GeO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge-O-O	1.000	1.413	2.037	47.962	1.000	-0.182	7.390	2.136	4	0	0.0
O-Ge-Ge	1.000	1.413	2.037	47.962	1.000	-0.182	7.390	2.136	4	0	0.0

Table 502. Table 502: The VFF model for b-GeS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.428	91.725

Table 503. Table 503: Two-body SW potential parameters for b-GeS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ge-S	7.657	1.363	17.377	0.0	3.338

Table 504. Table 504: Three-body SW potential parameters for b-GeS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ge-S-S	35.249	91.725	1.363	1.363	0.0	3.338	0.0	3.338	0.0	4.761
S-Ge-Ge	35.249	91.725	1.363	1.363	0.0	3.338	0.0	3.338	0.0	4.761

Table 505. Table 505: SW potential parameters for b-GeS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge-S-S	1.000	1.363	2.448	35.249	1.000	-0.030	7.657	5.030	4	0	0.0
S-Ge-Ge	1.000	1.363	2.448	35.249	1.000	-0.030	7.657	5.030	4	0	0.0

Table 506. Table 506: The VFF model for b-GeSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.568	91.406

Table 507. Table 507: Two-body SW potential parameters for b-GeSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ge-Se	8.498	1.430	21.745	0.0	3.526

Table 508. Table 508: Three-body SW potential parameters for b-GeSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ge-Se-Se	34.791	91.406	1.430	1.430	0.0	3.526	0.0	3.526	0.0	5.021
Se-Ge-Ge	34.791	91.406	1.430	1.430	0.0	3.526	0.0	3.526	0.0	5.021

Table 509. Table 509: SW potential parameters for b-GeSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge-Se-Se	1.000	1.430	2.466	34.791	1.000	-0.025	8.498	5.205	4	0	0.0
Se-Ge-Ge	1.000	1.430	2.466	34.791	1.000	-0.025	8.498	5.205	4	0	0.0

Table 510. Table 510: The VFF model for b-GeTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.322	3.516
$r_0$ or ${\theta }_0$	2.768	90.718

Table 511. Table 511: Two-body SW potential parameters for b-GeTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ge-Te	9.704	1.513	29.352	0.0	3.791

Table 512. Table 512: Three-body SW potential parameters for b-GeTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ge-Te-Te	33.832	90.718	1.513	1.513	0.0	3.791	0.0	3.791	0.0	5.381
Te-Ge-Ge	33.832	90.718	1.513	1.513	0.0	3.791	0.0	3.791	0.0	5.381

Table 513. Table 513: SW potential parameters for b-GeTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ge-Te-Te	1.000	1.513	2.506	33.832	1.000	-0.013	9.704	5.605	4	0	0.0
Te-Ge-Ge	1.000	1.513	2.506	33.832	1.000	-0.013	9.704	5.605	4	0	0.0

Table 514. Table 514: The VFF model for b-SnO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	9.315	3.300
$r_0$ or ${\theta }_0$	2.204	102.677

Table 515. Table 515: Two-body SW potential parameters for b-SnO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-O	9.133	1.609	11.798	0.0	3.146

Table 516. Table 516: Three-body SW potential parameters for b-SnO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-O-O	52.875	102.677	1.609	1.609	0.0	3.146	0.0	3.146	0.0	4.702
O-Sn-Sn	52.875	102.677	1.609	1.609	0.0	3.146	0.0	3.146	0.0	4.702

Table 517. Table 517: SW potential parameters for b-SnO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-O-O	1.000	1.609	1.955	52.875	1.000	-0.219	9.133	1.760	4	0	0.0
O-Sn-Sn	1.000	1.609	1.955	52.875	1.000	-0.219	9.133	1.760	4	0	0.0

Table 518. Table 518: The VFF model for b-SnS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.616	91.793

Table 519. Table 519: Two-body SW potential parameters for b-SnS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-S	7.392	1.472	23.416	0.0	3.597

Table 520. Table 520: Three-body SW potential parameters for b-SnS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-S-S	27.243	91.793	1.472	1.472	0.0	3.597	0.0	3.597	0.0	5.132
S-Sn-Sn	27.243	91.793	1.472	1.472	0.0	3.597	0.0	3.597	0.0	5.132

Table 521. Table 521: SW potential parameters for b-SnS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-S-S	1.000	1.472	2.444	27.243	1.000	-0.031	7.392	4.994	4	0	0.0
S-Sn-Sn	1.000	1.472	2.444	27.243	1.000	-0.031	7.392	4.994	4	0	0.0

Table 522. Table 522: The VFF model for b-SnSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.747	90.923

Table 523. Table 523: Two-body SW potential parameters for b-SnSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-Se	7.976	1.510	28.471	0.0	3.765

Table 524. Table 524: Three-body SW potential parameters for b-SnSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-Se-Se	26.294	90.923	1.510	1.510	0.0	3.765	0.0	3.765	0.0	5.349
Se-Sn-Sn	26.294	90.923	1.510	1.510	0.0	3.765	0.0	3.765	0.0	5.349

Table 525. Table 525: SW potential parameters for b-SnSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Se-Se	1.000	1.510	2.494	26.294	1.000	-0.016	7.976	5.482	4	0	0.0
Se-Sn-Sn	1.000	1.510	2.494	26.294	1.000	-0.016	7.976	5.482	4	0	0.0

Table 526. Table 526: The VFF model for b-SnTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	6.909	2.710
$r_0$ or ${\theta }_0$	2.947	89.542

Table 527. Table 527: Two-body SW potential parameters for b-SnTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-Te	8.864	1.559	37.713	0.0	4.019

Table 528. Table 528: Three-body SW potential parameters for b-SnTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-Te-Te	24.867	89.542	1.559	1.559	0.0	4.019	0.0	4.019	0.0	5.670
Te-Sn-Sn	24.867	89.542	1.559	1.559	0.0	4.019	0.0	4.019	0.0	5.670

Table 529. Table 529: SW potential parameters for b-SnTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Te-Te	1.000	1.559	2.577	24.867	1.000	0.008	8.864	6.378	4	0	0.0
Te-Sn-Sn	1.000	1.559	2.577	24.867	1.000	0.008	8.864	6.378	4	0	0.0

Table 530. Table 530: The VFF model for b-SnGe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.390	3.112
$r_0$ or ${\theta }_0$	2.570	112.350

Table 531. Table 531: Two-body SW potential parameters for b-SnGe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-Ge	13.674	2.267	21.812	0.0	3.777

Table 532. Table 532: Three-body SW potential parameters for b-SnGe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-Ge-Ge	77.881	112.350	2.267	2.267	0.0	3.777	0.0	3.777	0.0	5.833
Ge-Sn-Sn	77.881	112.350	2.267	2.267	0.0	3.777	0.0	3.777	0.0	5.833

Table 533. Table 533: SW potential parameters for b-SnGe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Ge-Ge	1.000	2.267	1.666	77.881	1.000	-0.380	13.674	0.826	4	0	0.0
Ge-Sn-Sn	1.000	2.267	1.666	77.881	1.000	-0.380	13.674	0.826	4	0	0.0

Table 534. Table 534: The VFF model for b-SiGe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	16.390	3.112
$r_0$ or ${\theta }_0$	2.310	114.702

Table 535. Table 535: Two-body SW potential parameters for b-SiGe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Si-Ge	22.576	2.122	14.237	0.0	3.417

Table 536. Table 536: Three-body SW potential parameters for b-SiGe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Si-Ge-Ge	87.197	114.702	2.122	2.122	0.0	3.417	0.0	3.417	0.0	5.314
Ge-Si-Si	87.197	114.702	2.122	2.122	0.0	3.417	0.0	3.417	0.0	5.314

Table 537. Table 537: SW potential parameters for b-SiGe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Si-Ge-Ge	1.000	2.122	1.610	87.197	1.000	-0.418	22.576	0.702	4	0	0.0
Ge-Si-Si	1.000	2.122	1.610	87.197	1.000	-0.418	22.576	0.702	4	0	0.0

Table 538. Table 538: The VFF model for b-SnSi. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	10.315	2.880
$r_0$ or ${\theta }_0$	2.520	113.298

Table 539. Table 539: Two-body SW potential parameters for b-SnSi used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Sn-Si	16.463	2.260	20.164	0.0	3.713

Table 540. Table 540: Three-body SW potential parameters for b-SnSi used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Sn-Si-Si	75.415	113.298	2.260	2.260	0.0	3.713	0.0	3.713	0.0	5.751
Si-Sn-Sn	75.415	113.298	2.260	2.260	0.0	3.713	0.0	3.713	0.0	5.751

Table 541. Table 541: SW potential parameters for b-SnSi used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Sn-Si-Si	1.000	2.260	1.643	75.415	1.000	-0.396	16.463	0.773	4	0	0.0
Si-Sn-Sn	1.000	2.260	1.643	75.415	1.000	-0.396	16.463	0.773	4	0	0.0

Table 542. Table 542: The VFF model for b-InP. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	12.903	2.384
$r_0$ or ${\theta }_0$	2.460	115.895

Table 543. Table 543: Two-body SW potential parameters for b-InP used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In-P	20.610	2.306	18.311	0.0	3.651

Table 544. Table 544: Three-body SW potential parameters for b-InP used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In-P-P	70.782	115.895	2.306	2.306	0.0	3.651	0.0	3.651	0.0	5.696
P-In-In	70.782	115.895	2.306	2.306	0.0	3.651	0.0	3.651	0.0	5.696

Table 545. Table 545: SW potential parameters for b-InP used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In-P-P	1.000	2.306	1.583	70.782	1.000	-0.437	20.610	0.648	4	0	0.0
P-In-In	1.000	2.306	1.583	70.782	1.000	-0.437	20.610	0.648	4	0	0.0

Table 546. Table 546: The VFF model for b-InAs. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	10.903	2.384
$r_0$ or ${\theta }_0$	2.550	114.115

Table 547. Table 547: Two-body SW potential parameters for b-InAs used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In-As	18.099	2.320	21.141	0.0	3.766

Table 548. Table 548: Three-body SW potential parameters for b-InAs used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In-As-As	64.931	114.115	2.320	2.320	0.0	3.766	0.0	3.766	0.0	5.847
As-In-In	64.931	114.115	2.320	2.320	0.0	3.766	0.0	3.766	0.0	5.847

Table 549. Table 549: SW potential parameters for b-InAs used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In-As-As	1.000	2.320	1.624	64.931	1.000	-0.409	18.099	0.730	4	0	0.0
As-In-In	1.000	2.320	1.624	64.931	1.000	-0.409	18.099	0.730	4	0	0.0

Table 550. Table 550: The VFF model for b-InSb. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	8.903	2.384
$r_0$ or ${\theta }_0$	2.740	113.012

Table 551. Table 551: Two-body SW potential parameters for b-InSb used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In-Sb	16.706	2.445	28.182	0.0	4.034

Table 552. Table 552: Three-body SW potential parameters for b-InSb used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In-Sb-Sb	61.578	113.012	2.445	2.445	0.0	4.034	0.0	4.034	0.0	6.243
Sb-In-In	61.578	113.012	2.445	2.445	0.0	4.034	0.0	4.034	0.0	6.243

Table 553. Table 553: SW potential parameters for b-InSb used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In-Sb-Sb	1.000	2.445	1.650	61.578	1.000	-0.391	16.706	0.788	4	0	0.0
Sb-In-In	1.000	2.445	1.650	61.578	1.000	-0.391	16.706	0.788	4	0	0.0

Table 554. Table 554: The VFF model for b-GaAs. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	12.903	3.284
$r_0$ or ${\theta }_0$	2.360	114.513

Table 555. Table 555: Two-body SW potential parameters for b-GaAs used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga-As	18.485	2.161	15.510	0.0	3.489

Table 556. Table 556: Three-body SW potential parameters for b-GaAs used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga-As-As	91.177	114.513	2.161	2.161	0.0	3.489	0.0	3.489	0.0	5.423
As-Ga-Ga	91.177	114.513	2.161	2.161	0.0	3.489	0.0	3.489	0.0	5.423

Table 557. Table 557: SW potential parameters for b-GaAs used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga-As-As	1.000	2.161	1.614	91.177	1.000	-0.415	18.485	0.711	4	0	0.0
As-Ga-Ga	1.000	2.161	1.614	91.177	1.000	-0.415	18.485	0.711	4	0	0.0

Table 558. Table 558: The VFF model for b-GaP. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	16.050	3.022
$r_0$ or ${\theta }_0$	2.250	117.152

Table 559. Table 559: Two-body SW potential parameters for b-GaP used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga-P	21.948	2.152	12.814	0.0	3.350

Table 560. Table 560: Three-body SW potential parameters for b-GaP used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga-P-P	95.438	117.152	2.152	2.152	0.0	3.350	0.0	3.350	0.0	5.246
P-Ga-Ga	95.438	117.152	2.152	2.152	0.0	3.350	0.0	3.350	0.0	5.246

Table 561. Table 561: SW potential parameters for b-GaP used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga-P-P	1.000	2.152	1.557	95.438	1.000	-0.456	21.948	0.597	4	0	0.0
P-Ga-Ga	1.000	2.152	1.557	95.438	1.000	-0.456	21.948	0.597	4	0	0.0

Table 562. Table 562: The VFF model for b-AlSb. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching	angle bending
expression	$\frac{1}{2}{K}_r{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\theta }{\left(\Delta \theta \right)}^2$
parameter	12.050	3.022
$r_0$ or ${\theta }_0$	2.570	114.791

Table 563. Table 563: Two-body SW potential parameters for b-AlSb used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Al-Sb	20.580	2.365	21.812	0.0	3.803

Table 564. Table 564: Three-body SW potential parameters for b-AlSb used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Al-Sb-Sb	85.046	114.791	2.365	2.365	0.0	3.803	0.0	3.803	0.0	5.915
Sb-Al-Al	85.046	114.791	2.365	2.365	0.0	3.803	0.0	3.803	0.0	5.915

Table 565. Table 565: SW potential parameters for b-AlSb used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Al-Sb-Sb	1.000	2.365	1.608	85.046	1.000	-0.419	20.580	0.697	4	0	0.0
Sb-Al-Al	1.000	2.365	1.608	85.046	1.000	-0.419	20.580	0.697	4	0	0.0

Table 566. Table 566: The VFF model for BO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{B-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{B-B}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{BOO}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{BBO}}{\left(\Delta \theta \right)}^2$
parameter	23.030	15.512	5.577	6.209
$r_0$ or ${\theta }_0$	1.520	1.770	106.764	112.059

Table 567. Table 567: Two-body SW potential parameters for BO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
B1-O1	11.725	1.207	2.669	0.0	2.197
B1-B2	6.749	0.875	4.908	0.0	2.392

Table 568. Table 568: Three-body SW potential parameters for BO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
B1-O1-O1	107.486	106.764	1.207	1.207	0.0	2.197	0.0	2.197	0.0	3.333
B1-B2-O1	87.662	112.059	0.875	1.207	0.0	2.392	0.0	2.197	0.0	3.198

Table 569. Table 569: SW potential parameters for BO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
B1-O1-O1	1.000	1.207	1.820	107.486	1.000	-0.288	11.725	1.256	4	0	0.0
B1-B2-B2	1.000	0.875	2.734	0.000	1.000	0.000	6.749	8.377	4	0	0.0
B1-B2-O1	1.000	0.000	0.000	87.662	1.000	-0.376	0.000	0.000	4	0	0.0

Table 570. Table 570: The VFF model for AlO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Al}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Al}-\mathrm{Al}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlOO}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlAlO}}{\left(\Delta \theta \right)}^2$
parameter	18.189	6.410	3.182	1.318
$r_0$ or ${\theta }_0$	1.830	2.620	107.947	110.956

Table 571. Table 571: Two-body SW potential parameters for AlO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Al1-O1	13.758	1.488	5.608	0.0	2.655
Al1-Al2	3.609	0.678	23.560	0.0	3.287

Table 572. Table 572: Three-body SW potential parameters for AlO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Al1-O1-O1	64.759	107.947	1.488	1.488	0.0	2.655	0.0	2.655	0.0	4.043
Al1-O1-Al2	12.688	110.956	1.488	0.678	0.0	2.655	0.0	3.287	0.0	4.213

Table 573. Table 573: SW potential parameters for AlO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Al1-O1-O1	1.000	1.488	1.785	64.759	1.000	-0.308	13.758	1.145	4	0	0.0
Al1-Al2-Al2	1.000	0.678	4.846	0.000	1.000	0.000	3.609	111.363	4	0	0.0
Al1-Al2-O1	1.000	0.000	0.000	12.688	1.000	-0.358	0.000	0.000	4	0	0.0

Table 574. Table 574: The VFF model for GaO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ga}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Ga}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaOO}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaGaO}}{\left(\Delta \theta \right)}^2$
parameter	18.189	6.410	3.182	1.628
$r_0$ or ${\theta }_0$	1.940	2.510	107.051	111.794

Table 575. Table 575: Two-body SW potential parameters for GaO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga1-O1	15.178	1.550	7.082	0.0	2.807
Ga1-Ga2	4.225	0.890	19.846	0.0	3.257

Table 576. Table 576: Three-body SW potential parameters for GaO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga1-O1-O1	62.149	107.051	1.550	1.550	0.0	2.807	0.0	2.807	0.0	4.262
Ga1-O1-Ga2	18.443	111.794	1.550	0.890	0.0	2.807	0.0	3.257	0.0	4.269

Table 577. Table 577: SW potential parameters for GaO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga1-O1-O1	1.000	1.550	1.811	62.149	1.000	-0.293	15.178	1.227	4	0	0.0
Ga1-Ga2-Ga2	1.000	0.890	3.661	0.000	1.000	0.000	4.225	31.685	4	0	0.0
Ga1-Ga2-O1	1.000	0.000	0.000	18.443	1.000	-0.371	0.000	0.000	4	0	0.0

Table 578. Table 578: The VFF model for InO. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{In}-O}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{In}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{InOO}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{InInO}}{\left(\Delta \theta \right)}^2$
parameter	16.916	4.250	2.171	1.138
$r_0$ or ${\theta }_0$	2.160	2.860	107.328	111.538

Table 579. Table 579: Two-body SW potential parameters for InO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In1-O1	17.600	1.735	10.884	0.0	3.128
In1-In2	3.440	0.945	33.453	0.0	3.682

Table 580. Table 580: Three-body SW potential parameters for InO used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In1-O1-O1	42.946	107.328	1.735	1.735	0.0	3.128	0.0	3.128	0.0	4.754
In1-O1-In2	12.470	111.538	1.735	0.945	0.0	3.128	0.0	3.682	0.0	4.800

Table 581. Table 581: SW potential parameters for InO used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In1-O1-O1	1.000	1.735	1.803	42.946	1.000	-0.298	17.600	1.201	4	0	0.0
In1-In2-In2	1.000	0.945	3.895	0.000	1.000	0.000	3.440	41.864	4	0	0.0
In1-In2-O1	1.000	0.000	0.000	12.470	1.000	-0.367	0.000	0.000	4	0	0.0

Table 582. Table 582: The VFF model for BS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{B-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{B-B}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{BSS}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{BBS}}{\left(\Delta \theta \right)}^2$
parameter	17.138	16.385	5.144	3.861
$r_0$ or ${\theta }_0$	1.940	1.720	102.691	115.613

Table 583. Table 583: Two-body SW potential parameters for BS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
B1-S1	13.021	1.417	7.082	0.0	2.769
B1-B2	14.613	1.809	4.376	0.0	2.602

Table 584. Table 584: Three-body SW potential parameters for BS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
B1-S1-S1	82.459	102.691	1.417	1.417	0.0	2.769	0.0	2.769	0.0	4.139
B1-B2-S1	102.002	115.613	1.809	1.417	0.0	2.602	0.0	2.769	0.0	3.717

Table 585. Table 585: SW potential parameters for BS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
B1-S1-S1	1.000	1.417	1.955	82.459	1.000	-0.220	13.021	1.758	4	0	0.0
B1-B2-B2	1.000	1.809	1.438	0.000	1.000	0.000	14.613	0.408	4	0	0.0
B1-B2-S1	1.000	0.000	0.000	102.002	1.000	-0.432	0.000	0.000	4	0	0.0

Table 586. Table 586: The VFF model for AlS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Al}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Al}-\mathrm{Al}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlSS}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlAlS}}{\left(\Delta \theta \right)}^2$
parameter	11.065	4.912	3.210	1.900
$r_0$ or ${\theta }_0$	2.320	2.590	100.600	117.324

Table 587. Table 587: Two-body SW potential parameters for AlS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Al1-S1	11.476	1.618	14.485	0.0	3.289
Al1-Al2	4.575	1.280	22.499	0.0	3.500

Table 588. Table 588: Three-body SW potential parameters for AlS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Al1-S1-S1	46.910	100.600	1.618	1.618	0.0	3.289	0.0	3.289	0.0	4.877
Al1-S1-Al2	26.090	117.324	1.280	1.618	0.0	3.500	0.0	3.289	0.0	4.853

Table 589. Table 589: SW potential parameters for AlS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Al1-S1-S1	1.000	1.618	2.032	46.910	1.000	-0.184	11.476	2.112	4	0	0.0
Al1-Al2-Al2	1.000	1.280	2.735	0.000	1.000	0.000	4.575	8.388	4	0	0.0
Al1-Al2-S1	1.000	0.000	0.000	26.090	1.000	-0.459	0.000	0.000	4	0	0.0

Table 590. Table 590: The VFF model for GaS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ga}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Ga}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaSS}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaGaS}}{\left(\Delta \theta \right)}^2$
parameter	10.014	6.133	2.925	1.900
$r_0$ or ${\theta }_0$	2.360	2.470	100.921	117.065

Table 591. Table 591: Two-body SW potential parameters for GaS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga1-S1	10.825	1.658	15.510	0.0	3.349
Ga1-Ga2	6.316	1.506	18.610	0.0	3.434

Table 592. Table 592: Three-body SW potential parameters for GaS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga1-S1-S1	43.355	100.921	1.658	1.658	0.0	3.349	0.0	3.349	0.0	4.972
Ga1-S1-Ga2	30.536	117.065	1.658	1.506	0.0	3.349	0.0	3.434	0.0	4.809

Table 593. Table 593: SW potential parameters for GaS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga1-S1-S1	1.000	1.658	2.020	43.355	1.000	-0.189	10.825	2.052	4	0	0.0
Ga1-Ga2-Ga2	1.000	1.506	2.280	0.000	1.000	0.000	6.316	3.615	4	0	0.0
Ga1-Ga2-S1	1.000	0.000	0.000	30.536	1.000	-0.455	0.000	0.000	4	0	0.0

Table 594. Table 594: The VFF model for InS. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{In}-S}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{In}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{InSS}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{InInS}}{\left(\Delta \theta \right)}^2$
parameter	10.014	4.533	2.179	1.412
$r_0$ or ${\theta }_0$	2.560	2.820	100.624	117.305

Table 595. Table 595: Two-body SW potential parameters for InS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In1-S1	12.652	1.787	21.475	0.0	3.629
In1-In2	5.202	1.454	31.620	0.0	3.833

Table 596. Table 596: Three-body SW potential parameters for InS used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In1-S1-S1	31.876	100.624	1.787	1.787	0.0	3.629	0.0	3.629	0.0	5.382
In1-S1-In2	19.900	117.305	1.787	1.454	0.0	3.629	0.0	3.833	0.0	5.325

Table 597. Table 597: SW potential parameters for InS used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In1-S1-S1	1.000	1.787	2.031	31.876	1.000	-0.184	12.652	2.108	4	0	0.0
In1-In2-In2	1.000	1.454	2.635	0.000	1.000	0.000	5.202	7.067	4	0	0.0
In1-In2-S1	1.000	0.000	0.000	19.990	1.000	-0.459	0.000	0.000	4	0	0.0

Table 598. Table 598: The VFF model for BSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{B-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{B-B}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{BSeSe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{BBSe}}{\left(\Delta \theta \right)}^2$
parameter	17.138	15.227	5.144	3.113
$r_0$ or ${\theta }_0$	2.100	1.710	101.394	116.681

Table 599. Table 599: Two-body SW potential parameters for BSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
B1-Se1	14.825	1.491	9.724	0.0	2.985
B1-B2	17.700	2.252	4.275	0.0	2.691

Table 600. Table 600: Three-body SW potential parameters for BSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
B1-S1-Se1	77.850	101.394	1.491	1.491	0.0	2.985	0.0	2.985	0.0	4.440
B1-B2-Se1	104.372	116.681	2.252	1.491	0.0	2.691	0.0	2.985	0.0	3.923

Table 601. Table 601: SW potential parameters for BSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
B1-Se1-Se1	1.000	1.491	2.002	77.850	1.000	-0.198	14.825	1.968	4	0	0.0
B1-B2-B2	1.000	2.252	1.195	0.000	1.000	0.000	17.700	0.166	4	0	0.0
B1-B2-Se1	1.000	0.000	0.000	104.372	1.000	-0.449	0.000	0.000	4	0	0.0

Table 602. Table 602: The VFF model for AlSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{Al-Se}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{Al-Al}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{AlSeSe}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{AlAlSe}{\left(\Delta \theta \right)}^2$
parameter	9.831	4.487	2.916	1.659
$r_0$ or ${\theta }_0$	2.470	2.570	99.846	117.926

Table 603. Table 603: Two-body SW potential parameters for AlSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Al1-Se1	11.362	1.694	18.610	0.0	3.493
Al1-Al2	4.974	1.558	21.812	0.0	3.570

Table 604. Table 604: Three-body SW potential parameters for AlSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Al1-Se1-Se1	41.235	99.846	1.694	1.694	0.0	3.493	0.0	3.493	0.0	5.164
Al1-Se1-Al2	26.418	117.926	1.558	1.694	0.0	3.570	0.0	3.493	0.0	5.029

Table 605. Table 605: SW potential parameters for AlSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Al1-Se1-Se1	1.000	1.694	2.062	41.235	1.000	-0.171	11.362	2.260	4	0	0.0
Al1-Al2-Al2	1.000	1.558	2.292	0.000	1.000	0.000	4.974	3.704	4	0	0.0
Al1-Al2-Se1	1.000	0.000	0.000	26.418	1.000	-0.468	0.000	0.000	4	0	0.0

Table 606. Table 606: The VFF model for GaSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Ga}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaSeSe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaGaSe}}{\left(\Delta \theta \right)}^2$
parameter	10.014	5.400	2.925	1.701
$r_0$ or ${\theta }_0$	2.500	2.460	99.636	118.092

Table 607. Table 607: Two-body SW potential parameters for GaSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga1-Se1	11.798	1.706	19.531	0.0	3.533
Ga1-Ga2	6.479	1.765	18.311	0.0	3.502

Table 608. Table 608: Three-body SW potential parameters for GaSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga1-Se1-Se1	40.978	99.636	1.706	1.706	0.0	3.533	0.0	3.533	0.0	5.218
Ga1-Se1-Ga2	31.031	118.092	1.765	1.706	0.0	3.502	0.0	3.533	0.0	4.985

Table 609. Table 609: SW potential parameters for GaSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga1-Se1-Se1	1.000	1.706	2.070	40.978	1.000	-0.167	11.798	2.305	4	0	0.0
Ga1-Ga2-Ga2	1.000	1.765	1.984	0.000	1.000	0.000	6.479	1.888	4	0	0.0
Ga1-Ga2-Se1	1.000	0.000	0.000	31.031	1.000	-0.471	0.000	0.000	4	0	0.0

Table 610. Table 610: The VFF model for InSe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{Se}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{In}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{InSeSe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{InInSe}}{\left(\Delta \theta \right)}^2$
parameter	9.812	4.185	2.090	1.227
$r_0$ or ${\theta }_0$	2.690	2.810	99.296	118.361

Table 611. Table 611: Two-body SW potential parameters for InSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In1-Se1	13.281	1.822	26.181	0.0	3.797
In1-In2	5.414	1.661	31.174	0.0	2.890

Table 612. Table 612: Three-body SW potential parameters for InSe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In1-Se1-Se1	28.853	99.296	1.822	1.822	0.0	3.797	0.0	3.797	0.0	5.601
In1-Se1-In2	19.120	118.361	1.822	1.661	0.0	3.797	0.0	3.890	0.0	5.489

Table 613. Table 613: SW potential parameters for InSe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In1-Se1-Se1	1.000	1.822	2.084	28.853	1.000	-0.162	13.281	2.377	4	0	0.0
In1-In2-In2	1.000	1.661	2.342	0.000	1.000	0.000	5.414	4.094	4	0	0.0
In1-In2-Se1	1.000	0.000	0.000	19.120	1.000	-0.475	0.000	0.000	4	0	0.0

Table 614. Table 614: The VFF model for BTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{B-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{B-B}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{BTeTe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{BBTe}}{\left(\Delta \theta \right)}^2$
parameter	13.287	14.502	5.466	2.515
$r_0$ or ${\theta }_0$	2.310	1.710	100.809	117.156

Table 615. Table 615: Two-body SW potential parameters for BTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
B1-Te1	13.727	1.619	14.237	0.0	3.277
B1-B2	24.140	2.933	4.275	0.0	2.830

Table 616. Table 616: Three-body SW potential parameters for BTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
B1-Te1-Te1	80.622	100.809	1.619	1.619	0.0	3.277	0.0	3.277	0.0	4.863
B1-B2-Te1	116.3301	117.156	2.933	1.619	0.0	2.830	0.0	3.277	0.0	4.199

Table 617. Table 617: SW potential parameters for BTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
B1-Te1-Te1	1.000	1.619	2.024	80.622	1.000	-0.188	13.727	2.073	4	0	0.0
B1-B2-B2	1.000	2.933	0.965	0.000	1.000	0.000	24.140	0.058	4	0	0.0
B1-B2-Te1	1.000	0.000	0.000	116.301	1.000	-0.456	0.000	0.000	4	0	0.0

Table 618. Table 618: The VFF model for AlTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Al}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Al}-\mathrm{Al}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlTeTe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{AlAlTe}}{\left(\Delta \theta \right)}^2$
parameter	8.077	3.859	2.820	1.518
$r_0$ or ${\theta }_0$	2.700	2.580	99.124	118.495

Table 619. Table 619: Two-body SW potential parameters for AlTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Al1-Te1	10.971	1.821	26.572	0.0	3.809
Al1-Al2	5.523	2.002	22.154	0.0	3.716

Table 620. Table 620: Three-body SW potential parameters for AlTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Al1-Te1-Te1	38.638	99.124	1.821	1.821	0.0	3.809	0.0	3.809	0.0	5.614
Al1-Te1-Al2	29.574	118.495	2.002	1.821	0.0	3.716	0.0	3.809	0.0	5.330

Table 621. Table 621: SW potential parameters for AlTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Al1-Te1-Te1	1.000	1.821	2.091	38.638	1.000	-0.159	10.971	2.415	4	0	0.0
Al1-Al2-Al2	1.000	2.002	1.856	0.000	1.000	0.000	5.523	1.379	4	0	0.0
Al1-Al2-Te1	1.000	0.000	0.000	29.574	1.000	-0.477	0.000	0.000	4	0	0.0

Table 622. Table 622: The VFF model for GaTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{Ga}-\mathrm{Ga}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaTeTe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{GaGaTe}}{\left(\Delta \theta \right)}^2$
parameter	7.382	4.366	2.841	1.519
$r_0$ or ${\theta }_0$	2.700	2.460	99.781	117.978

Table 623. Table 623: Two-body SW potential parameters for GaTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
Ga1-Te1	10.179	1.849	26.572	0.0	3.817
Ga1-Ga2	6.750	2.239	18.311	0.0	3.634

Table 624. Table 624: Three-body SW potential parameters for GaTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
Ga1-Te1-Te1	40.060	99.781	1.849	1.849	0.0	3.817	0.0	3.817	0.0	5.642
Ga1-Te1-Ga2	34.354	117.978	2.239	1.849	0.0	3.634	0.0	3.817	0.0	5.238

Table 625. Table 625: SW potential parameters for GaTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
Ga1-Te1-Te1	1.000	1.849	2.065	40.060	1.000	-0.170	10.179	2.274	4	0	0.0
Ga1-Ga2-Ga2	1.000	2.239	1.623	0.000	1.000	0.000	6.750	0.728	4	0	0.0
Ga1-Ga2-Te1	1.000	0.000	0.000	34.354	1.000	-0.469	0.000	0.000	4	0	0.0

Table 626. Table 626: The VFF model for InTe. The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{Te}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{In}-\mathrm{In}}{\left(\Delta r\right)}^2$	$\frac{1}{2}{K}_{\mathrm{InTeTe}}{\left(\Delta \theta \right)}^2$	$\frac{1}{2}{K}_{\mathrm{InInTe}}{\left(\Delta \theta \right)}^2$
parameter	5.592	3.928	2.419	1.227
$r_0$ or ${\theta }_0$	2.890	2.810	99.148	118.477

Table 627. Table 627: Two-body SW potential parameters for InTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
In1-Te1	8.707	1.950	34.879	0.0	4.077
In1-In2	6.312	2.068	31.174	0.0	4.015

Table 628. Table 628: Three-body SW potential parameters for InTe used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
In1-Te1-Te1	33.178	99.148	1.950	1.950	0.0	4.077	0.0	4.077	0.0	6.011
In1-Te1-In2	22.833	118.477	2.068	1.950	0.0	4.015	0.0	4.077	0.0	5.741

Table 629. Table 629: SW potential parameters for InTe used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ).

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
In1-Te1-Te1	1.000	1.950	2.090	33.178	1.000	-0.159	8.707	2.410	4	0	0.0
In1-In2-In2	1.000	2.068	1.942	0.000	1.000	0.000	6.312	1.704	4	0	0.0
In1-In2-Te1	1.000	0.000	0.000	22.833	1.000	-0.477	0.000	0.000	4	0	0.0

Table 630. Table 630: The VFF model for borophene. The second line gives an explicit expression for each VFF term, where atom indexes are from Fig.  323  (a). The third line is the force constant parameters. Parameters are in the unit of e ​ V Å 2 𝑒 𝑉 superscript italic-Å 2 \frac{eV}{\AA^{2}} for the bond stretching interactions, and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in unit of Å italic-Å \AA ) for the bond stretching interaction and the initial angle (in unit of degrees) for the angle bending interaction. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has atom i as the apex.

VFF type	bond stretching		angle bending
expression	$\frac{1}{2}{K}_{14}{\left(\Delta r_{14}\right)}^2$	$\frac{1}{2}{K}_{12}{\left(\Delta r_{12}\right)}^2$	$\frac{1}{2}{K}_{134}{\left(\Delta {\theta }_{134}\right)}^2$	$\frac{1}{2}{K}_{135}{\left(\Delta {\theta }_{135}\right)}^2$
parameter	20.673	6.025	3.523	4.651
$r_0$ or ${\theta }_0$	1.614	1.880	64.581	99.318

Table 631. Table 631: Two-body SW potential parameters for borophene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 3 ). The quantity ( r i ​ j subscript 𝑟 𝑖 𝑗 r_{ij} ) in the first line lists one representative term for the two-body SW potential between atoms i and j. Atom indexes are from Fig.  323  (a).

	$A$ (eV)	$\rho$ (Å)	$B$ (Å4)	$r_{\min }\left(\AA \right)$	$r_{\max }$ (Å)
$r_{14}$	8.974	0.971	3.393	0.0	2.240
$r_{12}$	2.098	0.618	6.246	0.0	2.419

Table 632. Table 632: Three-body SW potential parameters for borophene used by GULP, Gale ( 1997 ) as expressed in Eq. ( 4 ). The first line ( θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} ) presents one representative term for the three-body SW potential. The angle θ i ​ j ​ k subscript 𝜃 𝑖 𝑗 𝑘 \theta_{ijk} has the atom i as the apex. Atom indexes are from Fig.  323  (a).

	$K$ (eV)	${\theta }_0$ (degree)	${\rho }_1$ (Å)	${\rho }_2$ (Å)	$r_{\mathrm{min12}}$ (Å)	$r_{\mathrm{max12}}$ (Å)	$r_{\mathrm{min13}}$ (Å)	$r_{\mathrm{max13}}$ (Å)	$r_{\mathrm{min23}}$ (Å)	$r_{\mathrm{max23}}$ (Å)
${\theta }_{134}$	32.074	64.581	0.971	0.618	0.0	2.240	0.0	2.419	0.0	2.419
${\theta }_{135}$	23.668	99.318	0.618	0.618	0.0	2.419	0.0	2.410	2.240	3.047

Table 633. Table 633: SW potential parameters for borophene used by LAMMPS, lammps ( 2012 ) as expressed in Eqs. ( 9 ) and ( 10 ). Atom types in the first column are displayed in Fig.  325 .

	$ϵ$ (eV)	$\sigma$ (Å)	$a$	$\lambda$	$\gamma$	$\cos {\theta }_0$	$A_L$	$B_L$	$p$	$q$	tol
B1-B5-B5	1.000	0.971	2.307	0.000	1.000	0.000	8.974	3.817	4	0	0.0
B1-B2-B2	1.000	0.618	3.914	0.000	1.000	0.000	2.098	42.820	4	0	0.0
B1-B2-B5	1.000	0.000	0.000	32.074	1.000	0.429	0.000	0.000	4	0	0.0
B1-B2-B4	1.000	0.000	0.000	23.668	1.000	-0.162	0.000	0.000	4	0	0.0

Equations

$$V_r$$

$$V_{\theta }$$

$$V_2\left(r_{ij}\right)$$

$$V_3\left({\theta }_{ijk}\right)$$

$$\rho$$

$$A$$

$$K$$

$$\alpha$$

$$V_2\left(r_{ij}\right)$$

$$V_3\left({\theta }_{ijk}\right)$$

$$A_L$$

$$\sigma$$

$$B_L$$

$$a$$

$$\lambda$$

Full text

Layered Crystal, Stillinger-Weber Potential, Molecular Dynamics Simulation, Empirical Potential

I Introduction

The atomic interaction is of essential importance in the numerical investigation of most physical or mechanical processes. The present work provides parameters for the Stillinger-Weber (SW) empirical potential for 156 two-dimensional atomic crystal (TDACs). In practical applications, these layered materials are usually played as lego on atomic scale to construct the van der Waals heterostructures with comprehensive properties.Geim and Grigorieva (2013) The computational cost of ab initio for the heterostructure will be substantially increased as compared with one individual atomic layer, because the unit cell for the heterostructure is typically very large resulting from the mismatch of the lattice constants of different layered components. The empirical potential will be a competitive alternative to help out this difficult situation, considering their high efficiency.

In the early stage before 1980s, the computation ability of the scientific community was quite limited. At that time, the valence force field (VFF) model was one popular empirical potential for the description of the atomic interaction, since the VFF model is linear and can be applied in the analytic derivation of most elastic quantities.Yu (2010) In this model, each VFF term corresponds to a particular motion style in the crystal. Hence, each parameter in the VFF model usually has clear physical essence, which is beneficial for the parameterization of this model. For instance, the bond stretching term in the VFF model is directly related to the frequency of the longitudinal optical phonon modes, so the force constant of the bond stretching term can be determined from the frequencies of the longitudinal optical phonon modes. The VFF model can thus serve as the starting point for developing atomic empirical potentials for different crystals.

While the VFF model is beneficial for the fastest numerical simulation, its strong limitation is the absence of nonlinear effect. Due to this limitation, the VFF model is not applicable to nonlinear phenomena, for which other potential models with nonlinear components are required. Some representative potential models are (in the order of their simulation costs) SW potential,Stillinger and Weber (1985) Tersoff potential,Tersoff (1986) Brenner potential,Brenner et al. (2002) ab initio approaches, and etc. The SW potential is one of the simplest potential forms with nonlinear effects included. An advanced feature for the SW potential is that it includes the nonlinear effect, and keeps the numerical simulation at a very fast level.

Considering its distinct advantages, the present article aims at providing the SW potential for 156 TDACs. We will determine parameters for the SW potential from the VFF model, following the analytic approach proposed by one of the present author (J.W.J).Jiang (2015a) The VFF constants are fitted to the phonon spectrum or the elastic properties in the TDACs.

In this paper, we parametrize the SW potential for 156 TDACs. All structures discussed in the present work are listed in Tables 1,  2,  3,  4,  5,  6,  7,  8, and  9. The supplemental materials are freely available online in Ref. Jia, , including a fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the SW potential using GULP, and files for phonon calculations with the valence force field model using GULP.

II VFF model and SW potential

II.1 VFF model

The VFF model is one of the most widely used linear model for the description of atomic interactions.Yu (2010) The bond stretching and the angle bending are two typical motion styles for most covalent bonding materials. The bond stretching describes the energy variation for a bond due to a bond variation $\Delta r=r-r_{0}$, with $r_{0}$ as the initial bond length. The angle bending gives the energy increment for an angle resulting from an angle variation $\Delta\theta=\theta-\theta_{0}$, with $\theta_{0}$ as the initial angle. In the VFF model, the energy variations for the bond stretching and the angle bending are described by the following quadratic forms,

[TABLE]

where $K_{r}$ and $K_{\theta}$ are two force constant parameters. These two potential expressions in Eqs. (1) and  (2) are directly related to the optical phonon modes in the crystal. Hence, their force constant parameters $K_{r}$ and $K_{\theta}$ are usually determined by fitting to the phonon dispersion.

II.2 SW potential

In the SW potential, energy increments for the bond stretching and angle bending are described by the following two-body and three-body forms,

[TABLE]

where $V_{2}$ corresponds to the bond stretching and $V_{3}$ associates with the angle bending. The cut-offs $r_{\rm max}$, $r_{\rm max12}$ and $r_{\rm max13}$ are geometrically determined by the material’s structure. There are five unknown geometrical parameters, i.e., $\rho$ and $B$ in the two-body $V_{2}$ term and $\rho_{1}$, $\rho_{2}$, and $\theta_{0}$ in the three-body $V_{3}$ term, and two energy parameters $A$ and $K$. There is a constraint among these parameters due to the equilibrium condition,Jiang (2015a)

[TABLE]

where $d$ is the equilibrium bond length from experiments. Eq. (5) ensures that the bond has an equilibrium length of $d$ and the $V_{2}$ interaction for this bond is at the energy minimum state at the equilibrium configuration.

The energy parameters $A$ and $K$ in the SW potential can be analytically derived from the VFF model as follows,

[TABLE]

where the coefficient $\alpha$ in Eq. (6) is,

[TABLE]

In some situations, the SW potential is also written into the following form,

[TABLE]

The parameters here can be determined by comparing the SW potential forms in Eqs. (9) and (10) with Eqs. (3) and (4). It is obvious that $p=4$ and $q=0$. Eqs. (9) and (10) have two more parameters than Eqs. (3) and (4), so we can set $\epsilon=1.0$ eV and $\gamma=1.0$. The other parameters in Eqs. (9) and (10) are related to these parameters in Eqs. (3) and (4) by the following equations

[TABLE]

The SW potential is implemented in GULP using Eqs. (3) and (4). The SW potential is implemented in LAMMPS using Eqs. (9) and (10).

In the rest of this article, we will develop the VFF model and the SW potential for layered crystals. The VFF model will be developed by fitting to the phonon dispersion from experiments or first-principles calculations. The SW potential will be developed following the above analytic parameterization approach. In this work, GULPGale (1997) is used for the calculation of phonon dispersion and the fitting process, while LAMMPSlammps (2012) is used for molecular dynamics simulations. The OVITOStukowski (2010) and XCRYSDENKokalj (2003) packages are used for visualization. All simulation scripts for GULP and LAMMPS are available online in Ref. Jia, .

III 1H-ScO2

Most existing theoretical studies on the single-layer 1H-ScO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScO2.

The structure for the single-layer 1H-ScO2 is shown in Fig. 1 (with M=Sc and X=O). Each Sc atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.16$ Å, and the bond length $d_{\rm Sc-O}=2.09$ Å. The resultant angles are $\theta_{\rm ScOO}=\theta_{\rm OScSc}=98.222^{\circ}$ and $\theta_{\rm ScOO^{\prime}}=58.398^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 10 shows four VFF terms for the single-layer 1H-ScO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along $\Gamma$M as shown in Fig. 3 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 3 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 11. The parameters for the three-body SW potential used by GULP are shown in Tab. 12. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 13. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-ScO2 using LAMMPS, because the angles around atom Sc in Fig. 1 (with M=Sc and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Sc and X=O) shows that, for 1H-ScO2, we can differentiate these angles around the Sc atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 4 shows the stress-strain curve for the tension of a single-layer 1H-ScO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 126.3 N/m and 125.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$.

There is no available value for nonlinear quantities in the single-layer 1H-ScO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -652.8 N/m and -683.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.2 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.7 Nm*-1* at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

IV 1H-ScS2

Most existing theoretical studies on the single-layer 1H-ScS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScS2.

The structure for the single-layer 1H-ScS2 is shown in Fig. 1 (with M=Sc and X=S). Each Sc atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.70$ Å, and the bond length $d_{\rm Sc-S}=2.52$ Å. The resultant angles are $\theta_{\rm ScSS}=\theta_{\rm SScSc}=94.467^{\circ}$ and $\theta_{\rm ScSS^{\prime}}=64.076^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 14 shows four VFF terms for the single-layer 1H-ScS2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 5 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 5 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 15. The parameters for the three-body SW potential used by GULP are shown in Tab. 16. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 17. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-ScS2 using LAMMPS, because the angles around atom Sc in Fig. 1 (with M=Sc and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Sc and X=S) shows that, for 1H-ScS2, we can differentiate these angles around the Sc atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 6 shows the stress-strain curve for the tension of a single-layer 1H-ScS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 43.8 N/m and 43.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.30$.

There is no available value for nonlinear quantities in the single-layer 1H-ScS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -146.9 N/m and -159.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

V 1H-ScSe2

Most existing theoretical studies on the single-layer 1H-ScSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScSe2.

The structure for the single-layer 1H-ScSe2 is shown in Fig. 1 (with M=Sc and X=Se). Each Sc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.84$ Å, and the bond length $d_{\rm Sc-Se}=2.65$ Å. The resultant angles are $\theta_{\rm ScSeSe}=\theta_{\rm SeScSc}=92.859^{\circ}$ and $\theta_{\rm ScSeSe^{\prime}}=66.432^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 18 shows four VFF terms for the single-layer 1H-ScSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 7 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 7 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 19. The parameters for the three-body SW potential used by GULP are shown in Tab. 20. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 21. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-ScSe2 using LAMMPS, because the angles around atom Sc in Fig. 1 (with M=Sc and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Sc and X=Se) shows that, for 1H-ScSe2, we can differentiate these angles around the Sc atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 8 shows the stress-strain curve for the tension of a single-layer 1H-ScSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 39.4 N/m and 39.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.32$.

There is no available value for nonlinear quantities in the single-layer 1H-ScSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -115.7 N/m and -135.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.35 in the zigzag direction at the low temperature of 1 K.

VI 1H-ScTe2

Most existing theoretical studies on the single-layer 1H-ScTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScTe2.

The structure for the single-layer 1H-ScTe2 is shown in Fig. 1 (with M=Sc and X=Te). Each Sc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.62$ Å, and the bond length $d_{\rm Sc-Te}=2.89$ Å. The resultant angles are $\theta_{\rm ScTeTe}=\theta_{\rm TeScSc}=77.555^{\circ}$ and $\theta_{\rm ScTeTe^{\prime}}=87.364^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 22 shows four VFF terms for the single-layer 1H-ScTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 9 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. There is only one (longitudinal) acoustic branch available. We find that the VFF parameters can be chosen to be the same as that of the 1H-ScSe2, from which the longitudinal acoustic branch agrees with the ab initio results as shown in Fig. 9 (a). It has also been shown that the VFF parameters can be the same for TaSe2 and NbSe2 of similar structure.Feldman (1982) Fig. 9 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 23. The parameters for the three-body SW potential used by GULP are shown in Tab. 24. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 25. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-ScTe2 using LAMMPS, because the angles around atom Sc in Fig. 1 (with M=Sc and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Sc and X=Te) shows that, for 1H-ScTe2, we can differentiate these angles around the Sc atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 10 shows the stress-strain curve for the tension of a single-layer 1H-ScTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 29.3 N/m and 28.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.38$.

There is no available value for nonlinear quantities in the single-layer 1H-ScTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -43.2 N/m and -59.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.7 Nm*-1* at the ultimate strain of 0.33 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.7 Nm*-1* at the ultimate strain of 0.45 in the zigzag direction at the low temperature of 1 K.

VII 1H-TiTe2

Most existing theoretical studies on the single-layer 1H-TiTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-TiTe2.

The structure for the single-layer 1H-TiTe2 is shown in Fig. 1 (with M=Ti and X=Se). Each Ti atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ti atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.62$ Å, and the bond length $d_{\rm Ti-Te}=2.75$ Å. The resultant angles are $\theta_{\rm TiTeTe}=\theta_{\rm TeTiTi}=82.323^{\circ}$ and $\theta_{\rm TiTeTe^{\prime}}=81.071^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 26 shows the VFF terms for the 1H-TiTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 11 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 11 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 27. The parameters for the three-body SW potential used by GULP are shown in Tab. 28. Parameters for the SW potential used by LAMMPS are listed in Tab. 29. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-TiTe2 using LAMMPS, because the angles around atom Ti in Fig. 1 (with M=Ti and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ti and X=Te) shows that, for 1H-TiTe2, we can differentiate these angles around the Ti atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ti atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TiTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 12 shows the stress-strain curve for the tension of a single-layer 1H-TiTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 47.9 N/m and 47.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.29$.

There is no available value for the nonlinear quantities in the single-layer 1H-TiTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -158.6 N/m and -176.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.6 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.3 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

VIII 1H-VO2

Most existing theoretical studies on the single-layer 1H-VO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-VO2.

The structure for the single-layer 1H-VO2 is shown in Fig. 1 (with M=V and X=O). Each V atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three V atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.70$ Å, and the bond length $d_{\rm V-O}=1.92$ Å. The resultant angles are $\theta_{\rm VOO}=\theta_{\rm OVV}=89.356^{\circ}$ and $\theta_{\rm VOO^{\prime}}=71.436^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 30 shows four VFF terms for the single-layer 1H-VO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 13 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 13 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 31. The parameters for the three-body SW potential used by GULP are shown in Tab. 32. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 33. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-VO2 using LAMMPS, because the angles around atom V in Fig. 1 (with M=V and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=V and X=O) shows that, for 1H-VO2, we can differentiate these angles around the V atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 14 shows the stress-strain curve for the tension of a single-layer 1H-VO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 133.0 N/m and 132.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1H-VO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -652.3 N/m and -705.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.3 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.7 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

IX 1H-VS2

Most existing theoretical studies on the single-layer 1H-VS2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VS2.

The structure for the single-layer 1H-VS2 is shown in Fig. 1 (with M=V and X=S). Each V atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three V atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.09$ Å, and the bond length $d_{\rm V-S}=2.31$ Å. The resultant angles are $\theta_{\rm VSS}=\theta_{\rm SVV}=83.954^{\circ}$ and $\theta_{\rm VSS^{\prime}}=78.878^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 34 shows the VFF terms for the 1H-VS2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 15 (a). The ab initio calculations for the phonon dispersion are from Ref. Isaacs and Marianetti, 2016. The phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012) Fig. 15 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 35. The parameters for the three-body SW potential used by GULP are shown in Tab. 36. Parameters for the SW potential used by LAMMPS are listed in Tab. 37. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-VS2 using LAMMPS, because the angles around atom V in Fig. 1 (with M=V and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=V and X=S) shows that, for 1H-VS2, we can differentiate these angles around the V atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 16 shows the stress-strain curve for the tension of a single-layer 1H-VS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 86.5 N/m and 85.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.28$.

There is no available value for the nonlinear quantities in the single-layer 1H-VS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -302.0 N/m and -334.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.5 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

X 1H-VSe2

Most existing theoretical studies on the single-layer 1H-VSe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VSe2.

The structure for the single-layer 1H-VSe2 is shown in Fig. 1 (with M=V and X=Se). Each V atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three V atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.24$ Å, and the bond length $d_{\rm V-Se}=2.45$ Å. The resultant angles are $\theta_{\rm VSeSe}=\theta_{\rm SeVV}=82.787^{\circ}$ and $\theta_{\rm VSeSe^{\prime}}=80.450^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 38 shows the VFF terms for the 1H-VSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 17 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 17 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 39. The parameters for the three-body SW potential used by GULP are shown in Tab. 40. Parameters for the SW potential used by LAMMPS are listed in Tab. 41. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-VSe2 using LAMMPS, because the angles around atom V in Fig. 1 (with M=V and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=V and X=Se) shows that, for 1H-VSe2, we can differentiate these angles around the V atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 18 shows the stress-strain curve for the tension of a single-layer 1H-VSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 81.7 N/m and 80.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for the nonlinear quantities in the single-layer 1H-VSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -335.2 N/m and -363.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

XI 1H-VTe2

Most existing theoretical studies on the single-layer 1H-VTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VTe2.

The structure for the single-layer 1H-VTe2 is shown in Fig. 1 (with M=V and X=Te). Each V atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three V atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.48$ Å, and the bond length $d_{\rm V-Te}=2.66$ Å. The resultant angles are $\theta_{\rm VTeTe}=\theta_{\rm TeVV}=81.708^{\circ}$ and $\theta_{\rm VTeTe^{\prime}}=81.891^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 42 shows the VFF terms for the 1H-VTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 19 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 19 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 43. The parameters for the three-body SW potential used by GULP are shown in Tab. 44. Parameters for the SW potential used by LAMMPS are listed in Tab. 45. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-VTe2 using LAMMPS, because the angles around atom V in Fig. 1 (with M=V and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=V and X=Te) shows that, for 1H-VTe2, we can differentiate these angles around the V atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 20 shows the stress-strain curve for the tension of a single-layer 1H-VTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 68.1 N/m and 66.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.28$.

There is no available value for the nonlinear quantities in the single-layer 1H-VTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -237.4 N/m and -260.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

XII 1H-CrO2

Most existing theoretical studies on the single-layer 1H-CrO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CrO2.

The structure for the single-layer 1H-CrO2 is shown in Fig. 1 (with M=Cr and X=O). Each Cr atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Cr atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.58$ Å, and the bond length $d_{\rm Cr-O}=1.88$ Å. The resultant angles are $\theta_{\rm CrOO}=\theta_{\rm OCrCr}=86.655^{\circ}$ and $\theta_{\rm CrOO^{\prime}}=75.194^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 46 shows four VFF terms for the single-layer 1H-CrO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 21 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 21 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 47. The parameters for the three-body SW potential used by GULP are shown in Tab. 48. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 49. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-CrO2 using LAMMPS, because the angles around atom Cr in Fig. 1 (with M=Cr and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Cr and X=O) shows that, for 1H-CrO2, we can differentiate these angles around the Cr atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 22 shows the stress-strain curve for the tension of a single-layer 1H-CrO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 210.6 N/m and 209.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.13$.

There is no available value for nonlinear quantities in the single-layer 1H-CrO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -1127.7 N/m and -1185.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.4 Nm*-1* at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 18.7 Nm*-1* at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

XIII 1H-CrS2

Most existing theoretical studies on the single-layer 1H-CrS2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrS2.

The structure for the single-layer 1H-CrS2 is shown in Fig. 1 (with M=Cr and X=S). Each Cr atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Cr atoms. The structural parameters are from Ref. Zhuang et al., 2014, including the lattice constant $a=2.99$ Å, and the bond length $d_{\rm Cr-S}=2.254$ Å. The resultant angles are $\theta_{\rm CrSS}=\theta_{\rm SCrCr}=83.099^{\circ}$ and $\theta_{\rm CrSS^{\prime}}=80.031^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 50 shows four VFF terms for the 1H-CrS2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 23 (a). The ab initio calculations for the phonon dispersion are from Ref. Zhuang et al., 2014. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012) Fig. 23 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 51. The parameters for the three-body SW potential used by GULP are shown in Tab. 52. Parameters for the SW potential used by LAMMPS are listed in Tab. 53. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-CrS2 using LAMMPS, because the angles around atom Cr in Fig. 1 (with M=Cr and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Cr and X=S) shows that, for 1H-CrS2, we can differentiate these angles around the Cr atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 24 shows the stress-strain curve for the tension of a single-layer 1H-CrS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 98.4 N/m and 97.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 112.0 N/m from Refs Cakir, Peeters, and Sevik, 2014, or 111.9 N/m from Ref. Alyoruk et al., 2015. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.26$, which agrees with the ab initio value of 0.27.Cakir, Peeters, and Sevik (2014); Alyoruk et al. (2015)

There is no available value for the nonlinear quantities in the single-layer 1H-CrS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -364.8 N/m and -409.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.4 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.8 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

XIV 1H-CrSe2

Most existing theoretical studies on the single-layer 1H-CrSe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrSe2.

The structure for the single-layer 1H-CrSe2 is shown in Fig. 1 (with M=Cr and X=Se). Each Cr atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Cr atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.13$ Å, and the bond length $d_{\rm Cr-Se}=2.38$ Å. The resultant angles are $\theta_{\rm CrSeSe}=\theta_{\rm SeCrCr}=82.229^{\circ}$ and $\theta_{\rm CrSeSe^{\prime}}=81.197^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 54 shows four VFF terms for the 1H-CrSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 25 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 25 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 55. The parameters for the three-body SW potential used by GULP are shown in Tab. 56. Parameters for the SW potential used by LAMMPS are listed in Tab. 57. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-CrSe2 using LAMMPS, because the angles around atom Cr in Fig. 1 (with M=Cr and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Cr and X=Se) shows that, for 1H-CrSe2, we can differentiate these angles around the Cr atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 26 shows the stress-strain curve for the tension of a single-layer 1H-CrSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 90.0 N/m and 89.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 88.0 N/m from Refs Cakir, Peeters, and Sevik, 2014, or 87.9 N/m from Ref. Alyoruk et al., 2015. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.30$, which agrees with the ab initio value of 0.30.Cakir, Peeters, and Sevik (2014); Alyoruk et al. (2015)

There is no available value for the nonlinear quantities in the single-layer 1H-CrSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most two-dimensional atomic layered materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -279.6 N/m and -318.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.0 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.4 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XV 1H-CrTe2

Most existing theoretical studies on the single-layer 1H-CrTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrTe2.

The structure for the single-layer 1H-CrTe2 is shown in Fig. 1 (with M=Cr and X=Te). Each Cr atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Cr atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.39$ Å, and the bond length $d_{\rm Cr-Te}=2.58$ Å. The resultant angles are $\theta_{\rm CrTeTe}=\theta_{\rm TeCrCr}=82.139^{\circ}$ and $\theta_{\rm CrTeTe^{\prime}}=81.316^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 58 shows three VFF terms for the 1H-CrTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 27 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 27 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 59. The parameters for the three-body SW potential used by GULP are shown in Tab. 60. Parameters for the SW potential used by LAMMPS are listed in Tab. 61. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-CrTe2 using LAMMPS, because the angles around atom Cr in Fig. 1 (with M=Cr and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Cr and X=Te) shows that, for 1H-CrTe2, we can differentiate these angles around the Cr atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 28 shows the stress-strain curve for the tension of a single-layer 1H-CrTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 77.2 N/m and 76.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 63.9 N/m from Refs Cakir, Peeters, and Sevik, 2014 and  Alyoruk et al., 2015. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.30$, which agrees with the ab initio value of 0.30.Cakir, Peeters, and Sevik (2014); Alyoruk et al. (2015)

There is no available value for the nonlinear quantities in the single-layer 1H-CrTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most two-dimensional atomic layered materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -237.1 N/m and -280.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XVI 1H-MnO2

Most existing theoretical studies on the single-layer 1H-MnO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MnO2.

The structure for the single-layer 1H-MnO2 is shown in Fig. 1 (with M=Mn and X=O). Each Mn atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.61$ Å, and the bond length $d_{\rm Mn-O}=1.87$ Å. The resultant angles are $\theta_{\rm MnOO}=\theta_{\rm OMnMn}=88.511^{\circ}$ and $\theta_{\rm MnOO^{\prime}}=72.621^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 62 shows four VFF terms for the single-layer 1H-MnO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 29 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Typically, the transverse acoustic branch has a linear dispersion, so is higher than the flexural branch. However, it can be seen that the transverse acoustic branch is close to the flexural branch, which should be due to the underestimation from the ab initio calculations. Fig. 29 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 63. The parameters for the three-body SW potential used by GULP are shown in Tab. 64. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 65. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-MnO2 using LAMMPS, because the angles around atom Mn in Fig. 1 (with M=Mn and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Mn and X=O) shows that, for 1H-MnO2, we can differentiate these angles around the Mn atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Mn atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MnO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 30 shows the stress-strain curve for the tension of a single-layer 1H-MnO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MnO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MnO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 161.1 N/m and 160.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.10$.

There is no available value for nonlinear quantities in the single-layer 1H-MnO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -915.9 N/m and -957.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.1 Nm*-1* at the ultimate strain of 0.17 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.7 Nm*-1* at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

XVII 1H-FeO2

Most existing theoretical studies on the single-layer 1H-FeO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeO2.

The structure for the single-layer 1H-FeO2 is shown in Fig. 1 (with M=Fe and X=O). Each Fe atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.62$ Å, and the bond length $d_{\rm Fe-O}=1.88$ Å. The resultant angles are $\theta_{\rm FeOO}=\theta_{\rm OFeFe}=88.343^{\circ}$ and $\theta_{\rm FeOO^{\prime}}=72.856^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 66 shows four VFF terms for the single-layer 1H-FeO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 31 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 31 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 67. The parameters for the three-body SW potential used by GULP are shown in Tab. 68. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 69. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-FeO2 using LAMMPS, because the angles around atom Fe in Fig. 1 (with M=Fe and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Fe and X=O) shows that, for 1H-FeO2, we can differentiate these angles around the Fe atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 32 shows the stress-strain curve for the tension of a single-layer 1H-FeO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 100.2 N/m and 99.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer 1H-FeO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -423.4 N/m and -460.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XVIII 1H-FeS2

Most existing theoretical studies on the single-layer 1H-FeS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeS2.

The structure for the single-layer 1H-FeS2 is shown in Fig. 1 (with M=Fe and X=S). Each Fe atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.06$ Å, and the bond length $d_{\rm Fe-S}=2.22$ Å. The resultant angles are $\theta_{\rm FeSS}=\theta_{\rm SFeFe}=87.132^{\circ}$ and $\theta_{\rm FeSS^{\prime}}=74.537^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 70 shows four VFF terms for the single-layer 1H-FeS2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 33 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 33 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 71. The parameters for the three-body SW potential used by GULP are shown in Tab. 72. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 73. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-FeS2 using LAMMPS, because the angles around atom Fe in Fig. 1 (with M=Fe and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Fe and X=S) shows that, for 1H-FeS2, we can differentiate these angles around the Fe atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 34 shows the stress-strain curve for the tension of a single-layer 1H-FeS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 83.6 N/m and 83.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$.

There is no available value for nonlinear quantities in the single-layer 1H-FeS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -377.5 N/m and -412.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

XIX 1H-FeSe2

Most existing theoretical studies on the single-layer 1H-FeSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeSe2.

The structure for the single-layer 1H-FeSe2 is shown in Fig. 1 (with M=Fe and X=Se). Each Fe atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.22$ Å, and the bond length $d_{\rm Fe-Se}=2.35$ Å. The resultant angles are $\theta_{\rm FeSeSe}=\theta_{\rm SeFeFe}=86.488^{\circ}$ and $\theta_{\rm FeSeSe^{\prime}}=75.424^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 74 shows four VFF terms for the single-layer 1H-FeSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 35 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 35 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 75. The parameters for the three-body SW potential used by GULP are shown in Tab. 76. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 77. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-FeSe2 using LAMMPS, because the angles around atom Fe in Fig. 1 (with M=Fe and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Fe and X=Se) shows that, for 1H-FeSe2, we can differentiate these angles around the Fe atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 36 shows the stress-strain curve for the tension of a single-layer 1H-FeSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 77.3 N/m and 77.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer 1H-FeSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -323.8 N/m and -360.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.8 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.4 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XX 1H-FeTe2

Most existing theoretical studies on the single-layer 1H-FeTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeTe2.

The structure for the single-layer 1H-FeTe2 is shown in Fig. 1 (with M=Fe and X=Te). Each Fe atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.48$ Å, and the bond length $d_{\rm Fe-Te}=2.53$ Å. The resultant angles are $\theta_{\rm FeTeTe}=\theta_{\rm TeFeFe}=86.904^{\circ}$ and $\theta_{\rm FeTeTe^{\prime}}=74.851^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 78 shows four VFF terms for the single-layer 1H-FeTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 37 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 37 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 79. The parameters for the three-body SW potential used by GULP are shown in Tab. 80. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 81. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-FeTe2 using LAMMPS, because the angles around atom Fe in Fig. 1 (with M=Fe and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Fe and X=Te) shows that, for 1H-FeTe2, we can differentiate these angles around the Fe atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 38 shows the stress-strain curve for the tension of a single-layer 1H-FeTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 69.6 N/m and 69.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.25$.

There is no available value for nonlinear quantities in the single-layer 1H-FeTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -267.5 N/m and -302.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.1 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

XXI 1H-CoTe2

Most existing theoretical studies on the single-layer 1H-CoTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CoTe2.

The structure for the single-layer 1H-CoTe2 is shown in Fig. 1 (with M=Co and X=Te). Each Co atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Co atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.52$ Å, and the bond length $d_{\rm Co-Te}=2.51$ Å. The resultant angles are $\theta_{\rm CoTeTe}=\theta_{\rm TeCoCo}=89.046^{\circ}$ and $\theta_{\rm CoTeTe^{\prime}}=71.873^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 82 shows four VFF terms for the single-layer 1H-CoTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 39 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 39 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 83. The parameters for the three-body SW potential used by GULP are shown in Tab. 84. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 85. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-CoTe2 using LAMMPS, because the angles around atom Co in Fig. 1 (with M=Co and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Co and X=Te) shows that, for 1H-CoTe2, we can differentiate these angles around the Co atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Co atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CoTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 40 shows the stress-strain curve for the tension of a single-layer 1H-CoTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 53.7 N/m and 54.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.32$.

There is no available value for nonlinear quantities in the single-layer 1H-CoTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -157.2 N/m and -187.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.2 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.9 Nm*-1* at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

XXII 1H-NiS2

Most existing theoretical studies on the single-layer 1H-NiS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiS2.

The structure for the single-layer 1H-NiS2 is shown in Fig. 1 (with M=Ni and X=S). Each Ni atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.40$ Å, and the bond length $d_{\rm Ni-S}=2.24$ Å. The resultant angles are $\theta_{\rm NiSS}=\theta_{\rm SNiNi}=98.740^{\circ}$ and $\theta_{\rm NiSS^{\prime}}=57.593^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 86 shows four VFF terms for the single-layer 1H-NiS2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 41 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 41 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 87. The parameters for the three-body SW potential used by GULP are shown in Tab. 88. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 89. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-NiS2 using LAMMPS, because the angles around atom Ni in Fig. 1 (with M=Ni and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ni and X=S) shows that, for 1H-NiS2, we can differentiate these angles around the Ni atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ni atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 42 shows the stress-strain curve for the tension of a single-layer 1H-NiS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 84.0 N/m and 82.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer 1H-NiS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -403.2 N/m and -414.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.7 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

XXIII 1H-NiSe2

Most existing theoretical studies on the single-layer 1H-NiSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiSe2.

The structure for the single-layer 1H-NiSe2 is shown in Fig. 1 (with M=Ni and X=Se). Each Ni atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.33$ Å, and the bond length $d_{\rm Ni-Se}=2.35$ Å. The resultant angles are $\theta_{\rm NiSeSe}=\theta_{\rm SeNiNi}=90.228^{\circ}$ and $\theta_{\rm NiSeSe^{\prime}}=70.206^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 90 shows four VFF terms for the single-layer 1H-NiSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 43 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 43 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 91. The parameters for the three-body SW potential used by GULP are shown in Tab. 92. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 93. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-NiSe2 using LAMMPS, because the angles around atom Ni in Fig. 1 (with M=Ni and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ni and X=Se) shows that, for 1H-NiSe2, we can differentiate these angles around the Ni atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ni atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 44 shows the stress-strain curve for the tension of a single-layer 1H-NiSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 47.6 N/m and 47.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.27$.

There is no available value for nonlinear quantities in the single-layer 1H-NiSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -173.9 N/m and -197.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

XXIV 1H-NiTe2

Most existing theoretical studies on the single-layer 1H-NiTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiTe2.

The structure for the single-layer 1H-NiTe2 is shown in Fig. 1 (with M=Ni and X=Te). Each Ni atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.59$ Å, and the bond length $d_{\rm Ni-Te}=2.54$ Å. The resultant angles are $\theta_{\rm NiTeTe}=\theta_{\rm TeNiNi}=89.933^{\circ}$ and $\theta_{\rm NiTeTe^{\prime}}=70.624^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 94 shows four VFF terms for the single-layer 1H-NiTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 45 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) The transverse acoustic branch is very close to the longitudinal acoustic branch in the ab initio calculations. Fig. 45 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 95. The parameters for the three-body SW potential used by GULP are shown in Tab. 96. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 97. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-NiTe2 using LAMMPS, because the angles around atom Ni in Fig. 1 (with M=Ni and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ni and X=Te) shows that, for 1H-NiTe2, we can differentiate these angles around the Ni atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ni atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 46 shows the stress-strain curve for the tension of a single-layer 1H-NiTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 53.2 N/m and 53.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.32$.

There is no available value for nonlinear quantities in the single-layer 1H-NiTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -156.6 N/m and -184.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.1 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 Nm*-1* at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

XXV 1H-NbS2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbS2.FlcMullan (1983) In this section, we will develop the SW potential for the single-layer 1H-NbS2.

The structure for the single-layer 1H-NbS2 is shown in Fig. 1 (with M=Nb and X=S). Each Nb atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Nb atoms. The structural parameters are from Ref. FlcMullan, 1983, including the lattice constant $a=3.31$ Å, and the bond length $d_{\rm Nb-S}=2.47$ Å. The resultant angles are $\theta_{\rm NbSS}=\theta_{\rm SNbNb}=84.140^{\circ}$ and $\theta_{\rm NbSS^{\prime}}=78.626^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 98 shows four VFF terms for the 1H-NbS2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 47 (a). The theoretical phonon frequencies (gray pentagons) are from Ref. FlcMullan, 1983, which are the phonon dispersion of bulk 2H-NbS2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbS2, as the inter-layer interaction in the bulk 2H-NbS2 only induces weak effects on the two inplane acoustic branches. The inter-layer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbS2 (gray pentagons). Fig. 47 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 99. The parameters for the three-body SW potential used by GULP are shown in Tab. 100. Parameters for the SW potential used by LAMMPS are listed in Tab. 101. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-NbS2 using LAMMPS, because the angles around atom Nb in Fig. 1 (with M=Nb and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Nb and X=S) shows that, for 1H-NbS2, we can differentiate these angles around the Nb atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Nb atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 48 shows the stress-strain curve for the tension of a single-layer 1H-NbS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 87.7 N/m and 87.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.27$.

There is no available value for the nonlinear quantities in the single-layer 1H-NbS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -315.3 N/m and -355.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4  Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

XXVI 1H-NbSe2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbSe2.Feldman (1982); FlcMullan (1983) In this section, we will develop the SW potential for the single-layer 1H-NbSe2.

The structure for the single-layer 1H-NbSe2 is shown in Fig. 1 (with M=Nb and X=Se). Each Nb atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Nb atoms. The structural parameters are from Ref. FlcMullan, 1983, including the lattice constant $a=3.45$ Å, and the bond length $d_{\rm Nb-Se}=2.60$ Å. The resultant angles are $\theta_{\rm NbSeSe}=\theta_{\rm SNbNb}=83.129^{\circ}$ and $\theta_{\rm NbSeSe^{\prime}}=79.990^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 102 shows four VFF terms for the 1H-NbSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 49 (a). The theoretical phonon frequencies (gray pentagons) are from Ref. FlcMullan, 1983, which are the phonon dispersion of bulk 2H-NbSe2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbSe2, as the inter-layer interaction in the bulk 2H-NbSe2 only induces weak effects on the two inplane acoustic branches. The inter-layer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbSe2 (gray pentagons). It turns out that the VFF parameters for the single-layer 1H-NbSe2 are the same as the single-layer NbS2. The phonon dispersion for single-layer 1H-NbSe2 was also shown in Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 49 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 103. The parameters for the three-body SW potential used by GULP are shown in Tab. 104. Parameters for the SW potential used by LAMMPS are listed in Tab. 105. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-NbSe2 using LAMMPS, because the angles around atom Nb in Fig. 1 (with M=Nb and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Nb and X=Se) shows that, for 1H-NbSe2, we can differentiate these angles around the Nb atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Nb atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 50 shows the stress-strain curve for the tension of a single-layer 1H-NbSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 80.2 N/m and 80.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.29$.

There is no available value for the nonlinear quantities in the single-layer 1H-NbSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -258.8 N/m and -306.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

XXVII 1H-MoO2

Most existing theoretical studies on the single-layer 1H-MoO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MoO2.

The structure for the single-layer 1H-MoO2 is shown in Fig. 1 (with M=Mo and X=O). Each Mo atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.78$ Å, and the bond length $d_{\rm Mo-O}=2.00$ Å. The resultant angles are $\theta_{\rm MoOO}=\theta_{\rm OMoMo}=88.054^{\circ}$ and $\theta_{\rm MoOO^{\prime}}=73.258^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 106 shows four VFF terms for the single-layer 1H-MoO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 51 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 51 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 107. The parameters for the three-body SW potential used by GULP are shown in Tab. 108. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 109. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-MoO2 using LAMMPS, because the angles around atom Mo in Fig. 1 (with M=Mo and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Mo and X=O) shows that, for 1H-MoO2, we can differentiate these angles around the Mo atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Mo atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 52 shows the stress-strain curve for the tension of a single-layer 1H-MoO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 210.0 N/m and 209.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1H-MoO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -1027.8 N/m and -1106.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 21.0 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.1 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

XXVIII 1H-MoS2

Several potentials have been proposed to describe the interaction for the single-layer 1H-MoS2. In 1975, Wakabayashi et al. developed a VFF model to calculate the phonon spectrum of the bulk 2H-MoS2.Wakabayashi, Smith, and Nicklow (1975) In 2009, Liang et al. parameterized a bond-order potential for 1H-MoS2,Liang, Phillpot, and Sinnott (2009) which is based on the bond order concept underlying the Brenner potential.Brenner et al. (2002) A separate force field model was parameterized in 2010 for MD simulations of bulk 2H-MoS2.Varshney et al. (2010) The present author (J.W.J.) and his collaborators parameterized the SW potential for 1H-MoS2 in 2013,Jiang, Park, and Rabczuk (2013) which was improved by one of the present author (J.W.J.) in 2015.Jiang (2015a) Recently, another set of parameters for the SW potential were proposed for the single-layer 1H-MoS2.Kandemir et al. (2016)

We show the VFF model and the SW potential for single-layer 1H-MoS2 in this section. These potentials have been developed in previous works. The VFF model presented here is from Ref. Wakabayashi, Smith, and Nicklow, 1975, while the SW potential presented in this section is from Ref. Jiang, 2015a.

The structural parameters for the single-layer 1H-MoS2 are from the first-principles calculations as shown in Fig. 1 (with M=Mo and X=S).Molina-Sánchez and Wirtz (2011) The Mo atom layer in the single-layer 1H-MoS2 is sandwiched by two S atom layers. Accordingly, each Mo atom is surrounded by six S atoms, while each S atom is connected to three Mo atoms. The bond length between neighboring Mo and S atoms is $d=2.382$ Å, and the angles are $\theta_{\rm MoSS}=80.581^{\circ}$ and $\theta_{\rm SMoMo}=80.581^{\circ}$.

The VFF model for single-layer 1H-MoS2 is from Ref. Wakabayashi, Smith, and Nicklow, 1975, which is able to describe the phonon spectrum and the sound velocity accurately. We have listed the first three leading force constants for single-layer 1H-MoS2 in Tab. 110, neglecting other weak interaction terms. The SW potential parameters for single-layer 1H-MoS2 used by GULP are listed in Tabs. 111 and  112. The SW potential parameters for single-layer 1H-MoS2 used by LAMMPSlammps (2012) are listed in Tab. 113. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-MoS2 using LAMMPS, because the angles around atom Mo in Fig. 1 (with M=Mo and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Mo and X=S) shows that, for 1H-MoS2, we can differentiate these angles around the Mo atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Mo atom.

We use GULP to compute the phonon dispersion for the single-layer 1H-MoS2 as shown in Fig. 53. The results from the VFF model are quite comparable with the experiment data. The phonon dispersion from the SW potential is the same as that from the VFF model, which indicates that the SW potential has fully inherited the linear portion of the interaction from the VFF model.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 54 shows the stress-strain curve during the tension of a single-layer 1H-MoS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 97 N/m and 96 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, eg. $120\pm 30$ N/m from Refs Cooper et al., 2013a, b, or $180\pm 60$ N/m from Ref. Bertolazzi, Brivio, and Kis, 2011. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -418 N/m and -461 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.27$.

XXIX 1H-MoSe2

There is a recent parameter set for the SW potential in the single-layer 1H-MoSe2.Kandemir et al. (2016) In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoSe2.

The structure for the single-layer 1H-MoSe2 is shown in Fig. 1 (with M=Mo and X=Se). Each Mo atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Mo atoms. The structural parameters are from Ref. Horzum et al., 2013, including the lattice constant $a=3.321$ Å, and the bond length $d_{\rm Mo-Se}=2.528$ Å. The resultant angles are $\theta_{\rm MoSeSe}=\theta_{\rm SeMoMo}=82.119^{\circ}$ and $\theta_{\rm MoSeSe^{\prime}}=81.343^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 114 shows four VFF terms for the 1H-MoSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 55 (a). The ab initio calculations for the phonon dispersion are from Ref. Horzum et al., 2013. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012); Huang, Da, and Liang (2013); Sevik (2014); Kumar and Schwingenschlogl (2015); Huang, Zhang, and Zhang (2016) Fig. 55 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 115. The parameters for the three-body SW potential used by GULP are shown in Tab. 116. Parameters for the SW potential used by LAMMPS are listed in Tab. 117. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-MoSe2 using LAMMPS, because the angles around atom Mo in Fig. 1 (with M=Mo and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Mo and X=Se) shows that, for 1H-MoSe2, we can differentiate these angles around the Mo atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Mo atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 56 shows the stress-strain curve during the tension of a single-layer 1H-MoSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 103.0 N/m and 101.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, eg. 103.9 N/m from Refs Cakir, Peeters, and Sevik, 2014, or 113.9 N/m from Ref. Li, Medhekar, and Shenoy, 2013. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.24$, which agrees quite well with the ab initio value of 0.23.Cakir, Peeters, and Sevik (2014)

We have determined the nonlinear parameter to be $B=0.46d^{4}$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations.Li, Medhekar, and Shenoy (2013) We have extracted the value of $D=-383.7$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -365.4 N/m and -402.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.6 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.0 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

XXX 1H-MoTe2

Most existing theoretical studies on the single-layer 1H-MoTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoTe2.

The structure for the single-layer 1H-MoTe2 is shown in Fig. 1 (with M=Mo and X=Te). Each Mo atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Mo atoms. The structural parameters are from Ref. Guo et al., 2015, including the lattice constant $a=3.55$ Å, and the bond length $d_{\rm Mo-Te}=2.73$ Å. The resultant angles are $\theta_{\rm MoTeTe}=\theta_{\rm TeMoMo}=81.111^{\circ}$ and $\theta_{\rm MoTeTe^{\prime}}=82.686^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 118 shows four VFF terms for the 1H-MoTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 57 (a). The ab initio calculations for the phonon dispersion are from Ref. Guo et al., 2015. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012); Kan et al. (2015); Huang, Zhang, and Zhang (2016) Fig. 57 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 119. The parameters for the three-body SW potential used by GULP are shown in Tab. 120. Parameters for the SW potential used by LAMMPS are listed in Tab. 121. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-MoTe2 using LAMMPS, because the angles around atom Mo in Fig. 1 (with M=Mo and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Mo and X=Te) shows that, for 1H-MoTe2, we can differentiate these angles around the Mo atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Mo atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 58 shows the stress-strain curve for the tension of a single-layer 1H-MoTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 79.8 N/m and 78.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, eg. 79.4 N/m from Refs Cakir, Peeters, and Sevik, 2014, or 87.0 N/m from Ref. Li, Medhekar, and Shenoy, 2013. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.25$, which agrees with the ab initio value of 0.24.Cakir, Peeters, and Sevik (2014)

We have determined the nonlinear parameter to be $B=0.44d^{4}$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations.Li, Medhekar, and Shenoy (2013) We have extracted the value of $D=-278.2$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -250.5 N/m and -276.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.7 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.1 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XXXI 1H-TaS2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-TaS2.FlcMullan (1983) In this section, we will develop the SW potential for the single-layer 1H-TaS2.

The structure for the single-layer 1H-TaS2 is shown in Fig. 1 (with M=Ta and X=S). Each Ta atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Ta atoms. The structural parameters are from Ref. FlcMullan, 1983, including the lattice constant $a=3.315$ Å, and the bond length $d_{\rm Ta-S}=2.48$ Å. The resultant angles are $\theta_{\rm TaSS}=\theta_{\rm STaTa}=83.879^{\circ}$ and $\theta_{\rm TaSS^{\prime}}=78.979^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 122 shows the VFF terms for the 1H-TaS2, one of which is the bond stretching interaction shown by Eq. (1) while the others are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 59 (a). The theoretical phonon frequencies (gray pentagons) are from Ref. FlcMullan, 1983, which are the phonon dispersion of bulk 2H-TaS2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaS2, as the inter-layer interaction in the bulk 2H-TaS2 only induces weak effects on the two inplane acoustic branches. The inter-layer coupling will strengthen the out-of-plane acoustic (flexural) branch, so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-TaS2 (gray pentagons). Fig. 59 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 123. The parameters for the three-body SW potential used by GULP are shown in Tab. 124. Parameters for the SW potential used by LAMMPS are listed in Tab. 125. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-TaS2 using LAMMPS, because the angles around atom Ta in Fig. 1 (with M=Ta and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ta and X=S) shows that, for 1H-TaS2, we can differentiate these angles around the Ta atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 60 shows the stress-strain curve for the tension of a single-layer 1H-TaS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TaS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TaS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.4 N/m and 86.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.27$.

There is no available value for the nonlinear quantities in the single-layer 1H-TaS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -313.0 N/m and -349.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

XXXII 1H-TaSe2

The VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-TaSe2.Feldman (1982); FlcMullan (1983) In this section, we will develop the SW potential for the single-layer 1H-TaSe2.

The structure for the single-layer 1H-TaSe2 is shown in Fig. 1 (with M=Ta and X=Se). Each Ta atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Ta atoms. The structural parameters are from Ref. FlcMullan, 1983, including the lattice constant $a=3.436$ Å, and the bond length $d_{\rm Ta-Se}=2.59$ Å. The resultant angles are $\theta_{\rm TaSeSe}=\theta_{\rm SeTaTa}=83.107^{\circ}$ and $\theta_{\rm TaSeSe^{\prime}}=80.019^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 126 shows the VFF terms for the 1H-TaSe2, one of which is the bond stretching interaction shown by Eq. (1) while the others are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 61 (a). The theoretical phonon frequencies (gray pentagons) are from Ref. FlcMullan, 1983, which are the phonon dispersion of bulk 2H-TaSe2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaSe2, as the inter-layer interaction in the bulk 2H-TaSe2 only induces weak effects on the two in-plane acoustic branches. The inter-layer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-TaSe2 (gray pentagons). Fig. 61 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 127. The parameters for the three-body SW potential used by GULP are shown in Tab. 128. Parameters for the SW potential used by LAMMPS are listed in Tab. 129. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-TaSe2 using LAMMPS, because the angles around atom Ta in Fig. 1 (with M=Ta and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=Ta and X=Se) shows that, for 1H-TaSe2, we can differentiate these angles around the Ta atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 62 shows the stress-strain curve for the tension of a single-layer 1H-TaSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TaSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TaSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 80.8 N/m and 81.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.29$.

There is no available value for the nonlinear quantities in the single-layer 1H-TaSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -263.3 N/m and -308.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.3 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

XXXIII 1H-WO2

Most existing theoretical studies on the single-layer 1H-WO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-WO2.

The structure for the single-layer 1H-WO2 is shown in Fig. 1 (with M=W and X=O). Each W atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three W atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.80$ Å, and the bond length $d_{\rm W-O}=2.03$ Å. The resultant angles are $\theta_{\rm WOO}=\theta_{\rm OWW}=87.206^{\circ}$ and $\theta_{\rm WOO^{\prime}}=74.435^{\circ}$, in which atoms O and O’ are from different (top or bottom) group.

Table 130 shows four VFF terms for the single-layer 1H-WO2, one of which is the bond stretching interaction shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 63 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 63 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 131. The parameters for the three-body SW potential used by GULP are shown in Tab. 132. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 133. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-WO2 using LAMMPS, because the angles around atom W in Fig. 1 (with M=W and X=O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=W and X=O) shows that, for 1H-WO2, we can differentiate these angles around the W atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 64 shows the stress-strain curve for the tension of a single-layer 1H-WO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 237.1 N/m and 237.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer 1H-WO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -1218.0 N/m and -1312.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 22.6 Nm*-1* at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 21.8 Nm*-1* at the ultimate strain of 0.21 in the zigzag direction at the low temperature of 1 K.

XXXIV 1H-WS2

Most existing theoretical studies on the single-layer 1H-WS2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WS2.

The structure for the single-layer 1H-WS2 is shown in Fig. 1 (with M=W and X=S). Each W atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three W atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.13$ Å, and the bond length $d_{\rm W-S}=2.39$ Å. The resultant angles are $\theta_{\rm WSS}=\theta_{\rm SWW}=81.811^{\circ}$ and $\theta_{\rm WSS^{\prime}}=81.755^{\circ}$, in which atoms S and S’ are from different (top or bottom) group.

Table 134 shows the VFF terms for the 1H-WS2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 65 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Da, and Liang, 2013. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012); Molina-Sánchez and Wirtz (2011); Gu and Yang (2014); Huang et al. (2014); Huang, Zhang, and Zhang (2016) Fig. 65 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 135. The parameters for the three-body SW potential used by GULP are shown in Tab. 136. Parameters for the SW potential used by LAMMPS are listed in Tab. 137. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-WS2 using LAMMPS, because the angles around atom W in Fig. 1 (with M=W and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=W and X=S) shows that, for 1H-WS2, we can differentiate these angles around the W atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 66 shows the stress-strain curve for the tension of a single-layer 1H-WS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 121.5 N/m along both armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 139.6 N/m from Ref. Cakir, Peeters, and Sevik, 2014, or 148.5 N/m from Ref. Li, Medhekar, and Shenoy, 2013. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.21$, which agrees with the ab initio value of 0.22.Cakir, Peeters, and Sevik (2014)

We have determined the nonlinear parameter to be $B=0.47d^{4}$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations.Li, Medhekar, and Shenoy (2013) We have extracted the value of $D=-502.9$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -472.8 N/m and -529.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.7 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 14.1 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XXXV 1H-WSe2

Most existing theoretical studies on the single-layer 1H-WSe2 are based on the first-principles calculations. Norouzzadeh and Singh provided one set of parameters for the SW potential for the single-layer 1H-WSe2.Norouzzadeh and Singh (2017) In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WSe2.

The structure for the single-layer 1H-WSe2 is shown in Fig. 1 (with M=W and X=Se). Each W atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three W atoms. The structural parameters are from Ref. Ataca, Sahin, and Ciraci, 2012, including the lattice constant $a=3.25$ Å, and the bond length $d_{\rm W-Se}=2.51$ Å. The resultant angles are $\theta_{\rm WSeSe}=\theta_{\rm SeWW}=80.693^{\circ}$ and $\theta_{\rm WSeSe^{\prime}}=83.240^{\circ}$, in which atoms Se and Se’ are from different (top or bottom) group.

Table 138 shows three VFF terms for the 1H-WSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 67 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Da, and Liang, 2013. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012); Huang et al. (2014); Zhou and Chen (2015); Kumar and Schwingenschlogl (2015); Huang, Zhang, and Zhang (2016) Fig. 67 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 139. The parameters for the three-body SW potential used by GULP are shown in Tab. 140. Parameters for the SW potential used by LAMMPS are listed in Tab. 141. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-WSe2 using LAMMPS, because the angles around atom W in Fig. 1 (with M=W and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=W and X=Se) shows that, for 1H-WSe2, we can differentiate these angles around the W atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 68 shows the stress-strain curve for the tension of a single-layer 1H-WSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 124.1 N/m and 123.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 116.0 N/m from Ref. Cakir, Peeters, and Sevik, 2014, or 126.2 N/m from Ref. Li, Medhekar, and Shenoy, 2013. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$, which agrees with the ab initio value of 0.19.Cakir, Peeters, and Sevik (2014)

We have determined the nonlinear parameter to be $B=0.41d^{4}$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations.Li, Medhekar, and Shenoy (2013) We have extracted the value of $D=-413.1$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -400.4 N/m and -444.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 17.8 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 17.1 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XXXVI 1H-WTe2

Most existing theoretical studies on the single-layer 1H-WTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WTe2.

The bulk WTe2 has the trigonally coordinated H phase structure.Mar, Jobic, and Ibers (1992) However, it has been predicted that the structure of the single-layer WTe2 can be either the trigonally coordinated H phaseAtaca, Sahin, and Ciraci (2012) or the octahedrally coordinated $T_{d}$ phase,Dawson and Bullett (1987); Brown (1996); Jana et al. (2015); Jiang, Gao, and Wang (2016) with $T_{d}$ phase as the more stable structure.Torun et al. (2016) We will thus consider both phases in the present paper. This section is devoted to the H phase for the WTe2 (1H-WTe2), while the SW potential for the $T_{d}$-WTe2 (1T-WTe2) is presented in another section.

The structure for the single-layer 1H-WTe2 is shown in Fig. 1 (with M=W and X=Te). Each W atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three W atoms. The structural parameters are from Ref. Torun et al., 2016, including the lattice constant $a=3.55$ Å, and the bond length $d_{\rm W-Te}=2.73$ Å. The resultant angles are $\theta_{\rm WTeTe}=\theta_{\rm TeWW}=81.111^{\circ}$ and $\theta_{\rm WTeTe^{\prime}}=82.686^{\circ}$, in which atoms Te and Te’ are from different (top or bottom) group.

Table 142 shows the VFF terms for the 1H-WTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 69 (a). The ab initio calculations for the phonon dispersion are from Ref. Torun et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Ataca, Sahin, and Ciraci (2012) Fig. 69 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 143. The parameters for the three-body SW potential used by GULP are shown in Tab. 144. Parameters for the SW potential used by LAMMPS are listed in Tab. 145. We note that twelve atom types have been introduced for the simulation of the single-layer 1H-WTe2 using LAMMPS, because the angles around atom W in Fig. 1 (with M=W and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS.Jiang, Park, and Rabczuk (2013); Zhou and Jiang (2017) According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Fig. 2 (with M=W and X=Te) shows that, for 1H-WTe2, we can differentiate these angles around the W atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 70 shows the stress-strain curve for the tension of a single-layer 1H-WTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-WTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-WTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 82.7 N/m and 81.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, eg. 86.4 N/m from Ref. Cakir, Peeters, and Sevik, 2014, or 93.9 N/m from Ref. Li, Medhekar, and Shenoy, 2013. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$, which agrees with the ab initio value of 0.18.Cakir, Peeters, and Sevik (2014)

We have determined the nonlinear parameter to be $B=0.41d^{4}$ in Eq. (5) by fitting to the third-order nonlinear elastic constant $D$ from the ab initio calculations.Li, Medhekar, and Shenoy (2013) We have extracted the value of $D=-280.3$ N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -269.4 N/m and -297.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.8 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.3 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XXXVII 1T-ScO2

Most existing theoretical studies on the single-layer 1T-ScO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScO2.

The structure for the single-layer 1T-ScO2 is shown in Fig. 71 (with M=Sc and X=O). Each Ni atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.22$ Å and the bond length $d_{\rm Sc-O}=2.07$ Å. The resultant angles are $\theta_{\rm ScOO}=102.115^{\circ}$ with O atoms from the same (top or bottom) group, and $\theta_{\rm OScSc}=102.115^{\circ}$.

Table 146 shows three VFF terms for the single-layer 1T-ScO2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sc-O-O}$ is for the angle $\theta_{\rm Sc-O-O}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 72 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 72 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 147. The parameters for the three-body SW potential used by GULP are shown in Tab. 148. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 149.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 73 shows the stress-strain curve for the tension of a single-layer 1T-ScO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 100.9 N/m and 100.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer 1T-ScO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -422.4 N/m and -453.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.7 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.3 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XXXVIII 1T-ScS2

Most existing theoretical studies on the single-layer 1T-ScS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScS2.

The structure for the single-layer 1T-ScS2 is shown in Fig. 71 (with M=Sc and X=S). Each Sc atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.62$ Å, and the bond length $d_{\rm Sc-S}=2.50$ Å. The resultant angle is $\theta_{\rm SScSc}=92.771^{\circ}$ and $\theta_{\rm ScSS}=92.771^{\circ}$ with S atoms from the same (top or bottom) group.

Table 150 shows three VFF terms for the single-layer 1T-ScS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sc-S-S}$ is for the angle $\theta_{\rm Sc-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 74 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 74 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 151. The parameters for the three-body SW potential used by GULP are shown in Tab. 152. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 153.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 75 shows the stress-strain curve for the tension of a single-layer 1T-ScS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 30.0 N/m and 29.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1T-ScS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -113.7 N/m and -124.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.8 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.6 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

XXXIX 1T-ScSe2

Most existing theoretical studies on the single-layer 1T-ScSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScSe2.

The structure for the single-layer 1T-ScSe2 is shown in Fig. 71 (with M=Sc and X=Se). Each Sc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.52$ Å, and the bond length $d_{\rm Sc-Se}=2.64$ Å. The resultant angle is $\theta_{\rm SeScSc}=83.621^{\circ}$ and $\theta_{\rm ScSeSe}=83.621^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 154 shows three VFF terms for the single-layer 1T-ScSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sc-Se-Se}$ is for the angle $\theta_{\rm Sc-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 76 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 76 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 155. The parameters for the three-body SW potential used by GULP are shown in Tab. 156. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 157.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 77 shows the stress-strain curve for the tension of a single-layer 1T-ScSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 36.4 N/m and 36.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$.

There is no available value for nonlinear quantities in the single-layer 1T-ScSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -113.7 N/m and -130.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.3 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XL 1T-ScTe2

Most existing theoretical studies on the single-layer 1T-ScTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScTe2.

The structure for the single-layer 1T-ScTe2 is shown in Fig. 71 (with M=Sc and X=Te). Each Sc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.72$ Å, and the bond length $d_{\rm Sc-Te}=2.85$ Å. The resultant angle is $\theta_{\rm TeScSc}=81.481^{\circ}$ and $\theta_{\rm ScTeTe}=81.481^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 158 shows three VFF terms for the single-layer 1T-ScTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sc-Te-Te}$ is for the angle $\theta_{\rm Sc-Te-Te}$ with both Te atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 78 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 78 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 159. The parameters for the three-body SW potential used by GULP are shown in Tab. 160. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 161.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 79 shows the stress-strain curve for the tension of a single-layer 1T-ScTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ScTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ScTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 31.4 N/m and 31.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer 1T-ScTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -81.2 N/m and -96.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

XLI 1T-TiS2

Most existing theoretical studies on the single-layer 1T-TiS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiS2.

The structure for the single-layer 1T-TiS2 is shown in Fig. 71 (with M=Ti and X=S). Each Ti atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.32$ Å and the bond length $d_{\rm Ti-S}=2.39$ Å. The resultant angles are $\theta_{\rm TiSS}=87.984^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm STiTi}=87.984^{\circ}$.

Table 162 shows three VFF terms for the single-layer 1T-TiS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ti-S-S}$ is for the angle $\theta_{\rm Ti-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 85 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 163. The parameters for the three-body SW potential used by GULP are shown in Tab. 164. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 165.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 80 shows the stress-strain curve for the tension of a single-layer 1T-TiS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 75.0 N/m and 74.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 85 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TiS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -220.8 N/m and -264.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.4 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 81 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

XLII 1T-TiSe2

Most existing theoretical studies on the single-layer 1T-TiSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiSe2.

The structure for the single-layer 1T-TiSe2 is shown in Fig. 71 (with M=Ti and X=Se). Each Ti atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.43$ Å and the bond length $d_{\rm Ti-Se}=2.51$ Å. The resultant angles are $\theta_{\rm TiSeSe}=86.199^{\circ}$ with Se atoms from the same (top or bottom) group, and $\theta_{\rm SeTiTi}=86.199^{\circ}$.

Table 166 shows three VFF terms for the single-layer 1T-TiSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ti-Se-Se}$ is for the angle $\theta_{\rm Ti-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 70 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 167. The parameters for the three-body SW potential used by GULP are shown in Tab. 168. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 169.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 82 shows the stress-strain curve for the tension of a single-layer 1T-TiSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 59.2 N/m and 58.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 70 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TiSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -166.5 N/m and -201.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.7 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 83 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

XLIII 1T-TiTe2

Most existing theoretical studies on the single-layer 1T-TiTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TiTe2.

The structure for the single-layer 1T-TiTe2 is shown in Fig. 71 (with M=Ti and X=Te). Each Ti atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ti atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.64$ Å and the bond length $d_{\rm Ti-Te}=2.73$ Å. The resultant angles are $\theta_{\rm TiTeTe}=83.621^{\circ}$ with Te atoms from the same (top or bottom) group, and $\theta_{\rm TeTiTi}=83.621^{\circ}$.

Table 170 shows three VFF terms for the single-layer 1T-TiTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ti-Te-Te}$ is for the angle $\theta_{\rm Ti-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 46 N/m and the Poisson’s ratio as 0.15.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 171. The parameters for the three-body SW potential used by GULP are shown in Tab. 172. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 173.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TiTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 84 shows the stress-strain curve for the tension of a single-layer 1T-TiTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 41.4 N/m and 41.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 46 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TiTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -161.3 N/m and -181.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.7 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Fig. 85 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

XLIV 1T-VS2

Most existing theoretical studies on the single-layer 1T-VS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VS2.

The structure for the single-layer 1T-VS2 is shown in Fig. 71 (with M=V and X=S). Each V atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three V atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.10$ Å and the bond length $d_{\rm V-S}=2.31$ Å. The resultant angles are $\theta_{\rm VSS}=84.288^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SVV}=84.288^{\circ}$.

Table 174 shows three VFF terms for the single-layer 1T-VS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm V-S-S}$ is for the angle $\theta_{\rm V-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 86 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. The lowest acoustic branch (flexural mode) is linear and very close to the inplane transverse acoustic branch in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 86 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 175. The parameters for the three-body SW potential used by GULP are shown in Tab. 176. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 177.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 87 shows the stress-strain curve for the tension of a single-layer 1T-VS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 87.1 N/m and 86.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.21$.

There is no available value for nonlinear quantities in the single-layer 1T-VS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -230.5 N/m and -283.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.3 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.7 Nm*-1* at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

XLV 1T-VSe2

Most existing theoretical studies on the single-layer 1T-VSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VSe2.

The structure for the single-layer 1T-VSe2 is shown in Fig. 71 (with M=V and X=Se). Each V atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three V atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.24$ Å and the bond length $d_{\rm V-Se}=2.44$ Å. The resultant angles are $\theta_{\rm VSeSe}=83.201^{\circ}$ with Se atoms from the same (top or bottom) group, and $\theta_{\rm SeVV}=83.201^{\circ}$.

Table 178 shows three VFF terms for the single-layer 1T-VSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm V-Se-Se}$ is for the angle $\theta_{\rm V-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 88 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 88 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 179. The parameters for the three-body SW potential used by GULP are shown in Tab. 180. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 181.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 89 shows the stress-strain curve for the tension of a single-layer 1T-VSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 78.4 N/m and 78.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer 1T-VSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -168.5 N/m and -218.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.1 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.5 Nm*-1* at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

XLVI 1T-VTe2

Most existing theoretical studies on the single-layer 1T-VTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-VTe2.

The structure for the single-layer 1T-VTe2 is shown in Fig. 71 (with M=V and X=Te). Each V atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three V atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.46$ Å and the bond length $d_{\rm V-Te}=2.64$ Å. The resultant angles are $\theta_{\rm VTeTe}=81.885^{\circ}$ with Te atoms from the same (top or bottom) group, and $\theta_{\rm TeVV}=81.885^{\circ}$.

Table 182 shows three VFF terms for the single-layer 1T-VTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm V-Te-Te}$ is for the angle $\theta_{\rm V-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 67 N/m and the Poisson’s ratio as 0.24.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 183. The parameters for the three-body SW potential used by GULP are shown in Tab. 184. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 185.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-VTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 90 shows the stress-strain curve for the tension of a single-layer 1T-VTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-VTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-VTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 61.2 N/m and 61.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.24$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 67 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-VTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -95.8 N/m and -135.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.5 Nm*-1* at the ultimate strain of 0.30 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.0 Nm*-1* at the ultimate strain of 0.34 in the zigzag direction at the low temperature of 1 K.

Fig. 91 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

XLVII 1T-MnO2

Most existing theoretical studies on the single-layer 1T-MnO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnO2.

The structure for the single-layer 1T-MnO2 is shown in Fig. 71 (with M=Mn and X=O). Each Mn atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.82$ Å and the bond length $d_{\rm Mn-O}=1.88$ Å. The resultant angles are $\theta_{\rm MnOO}=97.181^{\circ}$ with O atoms from the same (top or bottom) group, and $\theta_{\rm OMnMn}=97.181^{\circ}$.

Table 186 shows three VFF terms for the single-layer 1T-MnO2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mn-O-O}$ is for the angle $\theta_{\rm Mn-O-O}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 92 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 92 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 187. The parameters for the three-body SW potential used by GULP are shown in Tab. 188. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 189.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 93 shows the stress-strain curve for the tension of a single-layer 1T-MnO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 156.3 N/m and 155.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.12$.

There is no available value for nonlinear quantities in the single-layer 1T-MnO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -711.7 N/m and -756.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 16.8 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 16.2 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

XLVIII 1T-MnS2

Most existing theoretical studies on the single-layer 1T-MnS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnS2.

The structure for the single-layer 1T-MnS2 is shown in Fig. 71 (with M=Mn and X=S). Each Mn atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.12$ Å and the bond length $d_{\rm Mn-S}=2.27$ Å. The resultant angles are $\theta_{\rm MnSS}=86.822^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SMnMn}=86.822^{\circ}$.

Table 190 shows three VFF terms for the single-layer 1T-MnS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mn-S-S}$ is for the angle $\theta_{\rm Mn-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 94 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 94 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 191. The parameters for the three-body SW potential used by GULP are shown in Tab. 192. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 193.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 95 shows the stress-strain curve for the tension of a single-layer 1T-MnS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 47.1 N/m and 46.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer 1T-MnS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -193.8 N/m and -210.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XLIX 1T-MnSe2

Most existing theoretical studies on the single-layer 1T-MnSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnSe2.

The structure for the single-layer 1T-MnSe2 is shown in Fig. 71 (with M=Mn and X=Se). Each Mn atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.27$ Å and the bond length $d_{\rm Mn-Se}=2.39$ Å. The resultant angles are $\theta_{\rm MnSeSe}=86.330^{\circ}$ with Se atoms from the same (top or bottom) group, and $\theta_{\rm SeMnMn}=86.330^{\circ}$.

Table 194 shows three VFF terms for the single-layer 1T-MnSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mn-Se-Se}$ is for the angle $\theta_{\rm Mn-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 96 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 96 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 195. The parameters for the three-body SW potential used by GULP are shown in Tab. 196. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 197.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 97 shows the stress-strain curve for the tension of a single-layer 1T-MnSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 43.2 N/m and 42.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1T-MnSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -163.4 N/m and -179.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

L 1T-MnTe2

Most existing theoretical studies on the single-layer 1T-MnTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MnTe2.

The structure for the single-layer 1T-MnTe2 is shown in Fig. 71 (with M=Mn and X=Te). Each Mn atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.54$ Å and the bond length $d_{\rm Mn-Te}=2.59$ Å. The resultant angles are $\theta_{\rm MnTeTe}=86.219^{\circ}$ with Te atoms from the same (top or bottom) group, and $\theta_{\rm TeMnMn}=86.219^{\circ}$.

Table 198 shows three VFF terms for the single-layer 1T-MnTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mn-Te-Te}$ is for the angle $\theta_{\rm Mn-Te-Te}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 98 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 98 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 199. The parameters for the three-body SW potential used by GULP are shown in Tab. 200. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 201.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MnTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 99 shows the stress-strain curve for the tension of a single-layer 1T-MnTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MnTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MnTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 38.5 N/m and 38.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer 1T-MnTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -133.5 N/m and -149.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

LI 1T-CoTe2

Most existing theoretical studies on the single-layer 1T-CoTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-CoTe2.

The structure for the single-layer 1T-CoTe2 is shown in Fig. 71 (with M=Co and X=Te). Each Co atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Co atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.5983$ Å, and the bond length $d_{\rm Co-Te}=2.5117$ Å, which is derived from the angle $\theta_{\rm TeCoCo}=91.5^{\circ}$. The other angle is $\theta_{\rm CoTeTe}=91.5^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 202 shows three VFF terms for the single-layer 1T-CoTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Co-Te-Te}$ is for the angle $\theta_{\rm Co-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 59 N/m and the Poisson’s ratio as 0.14.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 203. The parameters for the three-body SW potential used by GULP are shown in Tab. 204. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 205.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-CoTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 100 shows the stress-strain curve for the tension of a single-layer 1T-CoTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-CoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-CoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 50.5 N/m and 50.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.13$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 59 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-CoTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -221.5 N/m and -238.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.6 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.4 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Fig. 101 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LII 1T-NiO2

Most existing theoretical studies on the single-layer 1T-NiO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiO2.

The structure for the single-layer 1T-NiO2 is shown in Fig. 71 (with M=Ni and X=O). Each Ni atom is surrounded by six O atoms. These O atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each O atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=2.77$ Å and the bond length $d_{\rm Ni-O}=1.84$ Å. The resultant angles are $\theta_{\rm NiOO}=97.653^{\circ}$ with O atoms from the same (top or bottom) group, and $\theta_{\rm ONiNi}=97.653^{\circ}$.

Table 206 shows three VFF terms for the single-layer 1T-NiO2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ni-O-O}$ is for the angle $\theta_{\rm Ni-O-O}$ with both O atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 102 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 102 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 207. The parameters for the three-body SW potential used by GULP are shown in Tab. 208. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 209.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiO2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 103 shows the stress-strain curve for the tension of a single-layer 1T-NiO2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 163.3 N/m and 162.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.12$.

There is no available value for nonlinear quantities in the single-layer 1T-NiO2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -748.7 N/m and -796.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 17.4 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 16.8 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

LIII 1T-NiS2

Most existing theoretical studies on the single-layer 1T-NiS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiS2.

The structure for the single-layer 1T-NiS2 is shown in Fig. 71 (with M=Ni and X=S). Each Ni atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.3174$ Å, and the bond length $d_{\rm Ni-S}=2.2320$ Å, which is derived from the angle $\theta_{\rm SNiNi}=96^{\circ}$. The other angle is $\theta_{\rm NiSS}=96^{\circ}$ with S atoms from the same (top or bottom) group.

Table 210 shows three VFF terms for the single-layer 1T-NiS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ni-S-S}$ is for the angle $\theta_{\rm Ni-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 104 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 104 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 211. The parameters for the three-body SW potential used by GULP are shown in Tab. 212. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 213.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 105 shows the stress-strain curve for the tension of a single-layer 1T-NiS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 74.2 N/m and 73.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1T-NiS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -274.5 N/m and -301.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

LIV 1T-NiSe2

Most existing theoretical studies on the single-layer 1T-NiSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiSe2.

The structure for the single-layer 1T-NiSe2 is shown in Fig. 71 (with M=Ni and X=Se). Each Ni atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.4712$ Å, and the bond length $d_{\rm Ni-Se}=2.3392$ Å, which is derived from the angle $\theta_{\rm SeNiNi}=95.8^{\circ}$. The other angle is $\theta_{\rm NiSeSe}=95.8^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 214 shows three VFF terms for the single-layer 1T-NiSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ni-Se-Se}$ is for the angle $\theta_{\rm Ni-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 106 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. Fig. 106 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 215. The parameters for the three-body SW potential used by GULP are shown in Tab. 216. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 217.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 107 shows the stress-strain curve for the tension of a single-layer 1T-NiSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 70.9 N/m and 70.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1T-NiSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -263.7 N/m and -289.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.7 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

LV 1T-NiTe2

Most existing theoretical studies on the single-layer 1T-NiTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NiTe2.

The structure for the single-layer 1T-NiTe2 is shown in Fig. 71 (with M=Ni and X=Te). Each Ni atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.7248$ Å, and the bond length $d_{\rm Ni-Te}=2.5321$ Å, which is derived from the angle $\theta_{\rm TeNiNi}=94.7^{\circ}$. The other angle is $\theta_{\rm NiTeTe}=94.7^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 218 shows three VFF terms for the single-layer 1T-NiTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ni-Te-Te}$ is for the angle $\theta_{\rm Ni-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 44 N/m and the Poisson’s ratio as 0.14.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 219. The parameters for the three-body SW potential used by GULP are shown in Tab. 220. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 221.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NiTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 108 shows the stress-strain curve for the tension of a single-layer 1T-NiTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 42.6 N/m and 42.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.14$.

There is no available value for nonlinear quantities in the single-layer 1T-NiTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -187.6 N/m and -200.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.7 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.6 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Fig. 109 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LVI 1T-ZrS2

Most existing theoretical studies on the single-layer 1T-ZrS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrS2.

The structure for the single-layer 1T-ZrS2 is shown in Fig. 71 (with M=Zr and X=S). Each Zr atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations,Li, Kang, and Li (2014) including the lattice constant $a=3.690$ Å and the bond length $d_{\rm Zr-S}=2.58$ Å. The resultant angles are $\theta_{\rm ZrSS}=91.305^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SZrZr}=91.305^{\circ}$.

Table 222 shows three VFF terms for the single-layer 1T-ZrS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Zr-S-S}$ is for the angle $\theta_{\rm Zr-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 110 (a). The ab initio calculations for the phonon dispersion are from Ref. Gu and Yang, 2014. Similar phonon dispersion can also be found in other ab initio calculations.Huang, Zhang, and Zhang (2016) Fig. 110 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 223. The parameters for the three-body SW potential used by GULP are shown in Tab. 224. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 225.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 111 shows the stress-strain curve for the tension of a single-layer 1T-ZrS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 71.8 N/m and 71.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are close to the ab initio results at 0 K temperature, eg. 75.74 Nm*-1* in Ref. Li, Kang, and Li, 2014. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$, which are comparable with the ab initio resultLi, Kang, and Li (2014) of 0.22.

There is no available value for nonlinear quantities in the single-layer 1T-ZrS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -268.9 N/m and -305.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.9 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.5 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

LVII 1T-ZrSe2

Most existing theoretical studies on the single-layer 1T-ZrSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrSe2.

The structure for the single-layer 1T-ZrSe2 is shown in Fig. 71 (with M=Zr and X=Se). Each Zr atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations,Zhang et al. (2014a) including the lattice constant $a=3.707$ Å, and the position of the Se atom with respective to the Zr atomic plane $h=1.591$ Å. The resultant angles are $\theta_{\rm ZrSeSe}=88.058^{\circ}$ with Se atoms from the same (top or bottom) group, and $\theta_{\rm SeZrZr}=88.058^{\circ}$.

Table 226 shows three VFF terms for the single-layer 1T-ZrSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Zr-Se-Se}$ is for the angle $\theta_{\rm Zr-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 112 (a). The ab initio calculations for the phonon dispersion are from Ref. Ding et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Huang, Zhang, and Zhang (2016) Fig. 112 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 227. The parameters for the three-body SW potential used by GULP are shown in Tab. 228. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 229.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 113 shows the stress-strain curve for the tension of a single-layer 1T-ZrSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 66.7 N/m and 66.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer 1T-ZrSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -219.6 N/m and -256.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

LVIII 1T-ZrTe2

Most existing theoretical studies on the single-layer 1T-ZrTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ZrTe2.

The structure for the single-layer 1T-ZrTe2 is shown in Fig. 71 (with M=Zr and X=Te). Each Zr atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Zr atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=4.0064$ Å, and the bond length $d_{\rm Zr-Te}=2.9021$ Å, which is derived from the angle $\theta_{\rm TeZrZr}=87.3^{\circ}$. The other angle is $\theta_{\rm ZrTeTe}=87.3^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 230 shows three VFF terms for the single-layer 1T-ZrTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Zr-Te-Te}$ is for the angle $\theta_{\rm Zr-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 44 N/m and the Poisson’s ratio as 0.13.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 231. The parameters for the three-body SW potential used by GULP are shown in Tab. 232. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 233.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ZrTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 114 shows the stress-strain curve for the tension of a single-layer 1T-ZrTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ZrTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ZrTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 39.2 N/m and 39.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.10$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 44 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ZrTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -187.2 N/m and -201.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.0 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Fig. 115 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LIX 1T-NbS2

Most existing theoretical studies on the single-layer 1T-NbS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbS2.

The structure for the single-layer 1T-NbS2 is shown in Fig. 71 (with M=Nb and X=S). Each Nb atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.30$ Å and the bond length $d_{\rm Nb-S}=2.45$ Å. The resultant angles are $\theta_{\rm NbSS}=84.671^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SNbNb}=84.671^{\circ}$.

Table 234 shows three VFF terms for the single-layer 1T-NbS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Nb-S-S}$ is for the angle $\theta_{\rm Nb-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 116 (a). The ab initio calculations for the phonon dispersion are from Ref. Ataca, Sahin, and Ciraci, 2012. The lowest acoustic branch (flexural mode) is linear and very close to the inplane transverse acoustic branch in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 116 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 235. The parameters for the three-body SW potential used by GULP are shown in Tab. 236. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 237.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 117 shows the stress-strain curve for the tension of a single-layer 1T-NbS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 73.8 N/m and 73.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.18$.

There is no available value for nonlinear quantities in the single-layer 1T-NbS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -250.5 N/m and -290.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.7 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.4 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

LX 1T-NbSe2

Most existing theoretical studies on the single-layer 1T-NbSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbSe2.

The structure for the single-layer 1T-NbSe2 is shown in Fig. 71 (with M=Nb and X=Se). Each Nb atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.39$ Å and the bond length $d_{\rm Nb-Se}=2.57$ Å. The resultant angles are $\theta_{\rm NbSeSe}=82.529^{\circ}$ with Se atoms from the same (top or bottom) group, and $\theta_{\rm SeNbNb}=82.529^{\circ}$.

Table 238 shows three VFF terms for the single-layer 1T-NbSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Nb-Se-Se}$ is for the angle $\theta_{\rm Nb-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 73 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 239. The parameters for the three-body SW potential used by GULP are shown in Tab. 240. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 241.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 118 shows the stress-strain curve for the tension of a single-layer 1T-NbSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 67.1 N/m and 66.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 73 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically lead to about 10% underestimation for the Young’s modulus value.

There is no available value for nonlinear quantities in the single-layer 1T-NbSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -193.5 N/m and -233.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.7 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.3 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 119 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXI 1T-NbTe2

Most existing theoretical studies on the single-layer 1T-NbTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-NbTe2.

The structure for the single-layer 1T-NbTe2 is shown in Fig. 71 (with M=Nb and X=Te). Each Nb atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Nb atoms. The structural parameters are from the first-principles calculations,Ataca, Sahin, and Ciraci (2012) including the lattice constant $a=3.56$ Å and the bond length $d_{\rm Nb-Te}=2.77$ Å. The resultant angles are $\theta_{\rm NbTeTe}=79.972^{\circ}$ with Te atoms from the same (top or bottom) group, and $\theta_{\rm TeNbNb}=79.972^{\circ}$.

Table 242 shows three VFF terms for the single-layer 1T-NbTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Nb-Te-Te}$ is for the angle $\theta_{\rm Nb-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 56 N/m and the Poisson’s ratio as 0.11.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 243. The parameters for the three-body SW potential used by GULP are shown in Tab. 244. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 245.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-NbTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 120 shows the stress-strain curve for the tension of a single-layer 1T-NbTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-NbTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-NbTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 52.2 N/m and 51.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.11$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 56 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-NbTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -237.7 N/m and -265.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Fig. 121 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXII 1T-MoS2

Most existing theoretical studies on the single-layer 1T-MoS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoS2.

The structure for the single-layer 1T-MoS2 is shown in Fig. 71 (with M=Mo and X=S). Each Mo atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.1998$ Å, and the bond length $d_{\rm Mo-S}=2.4193$ Å, which is derived from the angle $\theta_{\rm SMoMo}=82.8^{\circ}$. The other angle is $\theta_{\rm MoSS}=82.8^{\circ}$ with S atoms from the same (top or bottom) group.

Table 246 shows three VFF terms for the single-layer 1T-MoS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mo-S-S}$ is for the angle $\theta_{\rm Mo-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 103 N/m and the Poisson’s ratio as -0.07.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoS2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 247. The parameters for the three-body SW potential used by GULP are shown in Tab. 248. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 249.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 122 shows the stress-strain curve for the tension of a single-layer 1T-MoS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 88.7 N/m and 88.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.07$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 103 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-MoS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -595.2 N/m and -624.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.5 Nm*-1* at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 Nm*-1* at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Fig. 123 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXIII 1T-MoSe2

Most existing theoretical studies on the single-layer 1T-MoSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoSe2.

The structure for the single-layer 1T-MoSe2 is shown in Fig. 71 (with M=Mo and X=Se). Each Mo atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.2685$ Å, and the bond length $d_{\rm Mo-Se}=2.5293$ Å, which is derived from the angle $\theta_{\rm SeMoMo}=80.5^{\circ}$. The other angle is $\theta_{\rm MoSeSe}=80.5^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 250 shows three VFF terms for the single-layer 1T-MoSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mo-Se-Se}$ is for the angle $\theta_{\rm Mo-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 104 N/m and the Poisson’s ratio as -0.13.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoSe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 251. The parameters for the three-body SW potential used by GULP are shown in Tab. 252. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 253.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 124 shows the stress-strain curve for the tension of a single-layer 1T-MoSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 88.2 N/m and 87.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.13$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 104 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-MoSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -632.6 N/m and -629.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.2 Nm*-1* at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Fig. 125 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXIV 1T-MoTe2

Most existing theoretical studies on the single-layer 1T-MoTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-MoTe2.

The structure for the single-layer 1T-MoTe2 is shown in Fig. 71 (with M=Mo and X=Te). Each Mo atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.4970$ Å, and the bond length $d_{\rm Mo-Te}=2.7287$ Å, which is derived from the angle $\theta_{\rm TeMoMo}=79.7^{\circ}$. The other angle is $\theta_{\rm MoTeTe}=79.7^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 254 shows three VFF terms for the single-layer 1T-MoTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Mo-Te-Te}$ is for the angle $\theta_{\rm Mo-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 92 N/m and the Poisson’s ratio as -0.07.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-MoTe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 255. The parameters for the three-body SW potential used by GULP are shown in Tab. 256. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 257.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-MoTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 126 shows the stress-strain curve for the tension of a single-layer 1T-MoTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-MoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-MoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 81.6 N/m and 81.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.07$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 92 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-MoTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -543.1 N/m and -558.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.0 Nm*-1* at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.0 Nm*-1* at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Fig. 127 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXV 1T-TcS2

Most existing theoretical studies on the single-layer 1T-TcS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcS2.

The structure for the single-layer 1T-TcS2 is shown in Fig. 71 (with M=Tc and X=S). Each Tc atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.0692$ Å, and the bond length $d_{\rm Tc-S}=2.3924$ Å, which is derived from the angle $\theta_{\rm STcTc}=79.8^{\circ}$. The other angle is $\theta_{\rm TcSS}=79.8^{\circ}$ with S atoms from the same (top or bottom) group.

Table 258 shows three VFF terms for the single-layer 1T-TcS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Tc-S-S}$ is for the angle $\theta_{\rm Tc-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 94 N/m and the Poisson’s ratio as -0.10.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcS2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 259. The parameters for the three-body SW potential used by GULP are shown in Tab. 260. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 261.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 128 shows the stress-strain curve for the tension of a single-layer 1T-TcS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 84.3 N/m and 84.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.10$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 94 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TcS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -572.0 N/m and -588.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.0 Nm*-1* at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.16 in the zigzag direction at the low temperature of 1 K.

Fig. 129 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXVI 1T-TcSe2

Most existing theoretical studies on the single-layer 1T-TcSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcSe2.

The structure for the single-layer 1T-TcSe2 is shown in Fig. 71 (with M=Tc and X=Se). Each Tc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.1543$ Å, and the bond length $d_{\rm Tc-Se}=2.5061$ Å, which is derived from the angle $\theta_{\rm SeTcTc}=78^{\circ}$. The other angle is $\theta_{\rm TcSeSe}=78^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 262 shows three VFF terms for the single-layer 1T-TcSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Tc-Se-Se}$ is for the angle $\theta_{\rm Tc-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 104 N/m and the Poisson’s ratio as -0.04.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcSe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 263. The parameters for the three-body SW potential used by GULP are shown in Tab. 264. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 265.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 130 shows the stress-strain curve for the tension of a single-layer 1T-TcSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 88.8 N/m and 88.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.04$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 104 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TcSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -565.7 N/m and -587.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.8 Nm*-1* at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.8 Nm*-1* at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Fig. 131 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXVII 1T-TcTe2

Most existing theoretical studies on the single-layer 1T-TcTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TcTe2.

The structure for the single-layer 1T-TcTe2 is shown in Fig. 71 (with M=Tc and X=Te). Each Tc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Tc atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.4149$ Å, and the bond length $d_{\rm Tc-Te}=2.6900$ Å, which is derived from the angle $\theta_{\rm TeTcTc}=78.8^{\circ}$. The other angle is $\theta_{\rm TcTeTe}=78.8^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 266 shows three VFF terms for the single-layer 1T-TcTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Tc-Te-Te}$ is for the angle $\theta_{\rm Tc-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 34 N/m and the Poisson’s ratio as -0.36.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-TcTe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 267. The parameters for the three-body SW potential used by GULP are shown in Tab. 268. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 269.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TcTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 132 shows the stress-strain curve for the tension of a single-layer 1T-TcTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TcTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TcTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 28.6 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.21$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 34 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TcTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -207.8 N/m and -208.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 1.9 Nm*-1* at the ultimate strain of 0.11 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 2.0 Nm*-1* at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K. The ultimate strain decreases to be about 0.01 at 300 K, so the single-layer 1T-TcTe2 is not very stable at higher temperature. It is because this material is very soft and the Poisson’s ratio is very small (negative value).

Fig. 133 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXVIII 1T-RhTe2

Most existing theoretical studies on the single-layer 1T-RhTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-RhTe2.

The structure for the single-layer 1T-RhTe2 is shown in Fig. 71 (with M=Rh and X=Te). Each Rh atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Rh atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.7563$ Å, and the bond length $d_{\rm Rh-Te}=2.6332$ Å, which is derived from the angle $\theta_{\rm TeRhRh}=91^{\circ}$. The other angle is $\theta_{\rm RhTeTe}=91^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 270 shows three VFF terms for the single-layer 1T-RhTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Rh-Te-Te}$ is for the angle $\theta_{\rm Rh-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 37 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 271. The parameters for the three-body SW potential used by GULP are shown in Tab. 272. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 273.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-RhTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 134 shows the stress-strain curve for the tension of a single-layer 1T-RhTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-RhTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-RhTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 32.1 N/m and 32.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 37 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-RhTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -103.1 N/m and -116.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.6 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.4 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 135 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXIX 1T-PdS2

Most existing theoretical studies on the single-layer 1T-PdS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdS2.

The structure for the single-layer 1T-PdS2 is shown in Fig. 71 (with M=Pd and X=S). Each Pd atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.5408$ Å, and the bond length $d_{\rm Pd-S}=2.4013$ Å, which is derived from the angle $\theta_{\rm SPdPd}=95^{\circ}$. The other angle is $\theta_{\rm PdSS}=95^{\circ}$ with S atoms from the same (top or bottom) group.

Table 274 shows three VFF terms for the single-layer 1T-PdS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pd-S-S}$ is for the angle $\theta_{\rm Pd-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 77 N/m and the Poisson’s ratio as 0.53.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 275. The parameters for the three-body SW potential used by GULP are shown in Tab. 276. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 277.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 136 shows the stress-strain curve for the tension of a single-layer 1T-PdS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 69.9 N/m and 69.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 77 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-PdS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -222.0 N/m and -248.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.1 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.7 Nm*-1* at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

Fig. 137 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXX 1T-PdSe2

Most existing theoretical studies on the single-layer 1T-PdSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdSe2.

The structure for the single-layer 1T-PdSe2 is shown in Fig. 71 (with M=Pd and X=Se). Each Pd atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.6759$ Å, and the bond length $d_{\rm Pd-Se}=2.4929$ Å, which is derived from the angle $\theta_{\rm SePdPd}=95^{\circ}$. The other angle is $\theta_{\rm PdSeSe}=95^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 278 shows three VFF terms for the single-layer 1T-PdSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pd-Se-Se}$ is for the angle $\theta_{\rm Pd-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 66 N/m and the Poisson’s ratio as 0.45.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 279. The parameters for the three-body SW potential used by GULP are shown in Tab. 280. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 281.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 138 shows the stress-strain curve for the tension of a single-layer 1T-PdSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 65.5 N/m and 65.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.21$.

There is no available value for nonlinear quantities in the single-layer 1T-PdSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -194.7 N/m and -222.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.9 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.5 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

Fig. 139 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXI 1T-PdTe2

Most existing theoretical studies on the single-layer 1T-PdTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PdTe2.

The structure for the single-layer 1T-PdTe2 is shown in Fig. 71 (with M=Pd and X=Te). Each Pd atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Pd atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.9162$ Å, and the bond length $d_{\rm Pd-Te}=2.6349$ Å, which is derived from the angle $\theta_{\rm TePdPd}=96^{\circ}$. The other angle is $\theta_{\rm PdTeTe}=96^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 282 shows three VFF terms for the single-layer 1T-PdTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pd-Te-Te}$ is for the angle $\theta_{\rm Pd-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 63 N/m and the Poisson’s ratio as 0.35.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 283. The parameters for the three-body SW potential used by GULP are shown in Tab. 284. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 285.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PdTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 140 shows the stress-strain curve for the tension of a single-layer 1T-PdTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PdTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PdTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 61.6 N/m and 61.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer 1T-PdTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -178.8 N/m and -203.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.6 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 Nm*-1* at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

Fig. 141 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXII 1T-SnS2

Most existing theoretical studies on the single-layer 1T-SnS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-SnS2.

The structure for the single-layer 1T-SnS2 is shown in Fig. 71 (with M=Sn and X=S). Each Sn atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Sn atoms. The structural parameters are from the first-principles calculations,Huang, Zhang, and Zhang (2016) including the lattice constant $a=3.640$ Å, and the bond length $d_{\rm Sn-S}=2.570$ Å. The resultant angles are $\theta_{\rm SSnSn}=90.173^{\circ}$, and $\theta_{\rm SnSS}=90.173^{\circ}$ with S atoms from the same (top or bottom) group.

Table 286 shows three VFF terms for the single-layer 1T-SnS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sn-S-S}$ is for the angle $\theta_{\rm Sn-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 142 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Zhang, and Zhang, 2016. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 142 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 287. The parameters for the three-body SW potential used by GULP are shown in Tab. 288. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 289.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-SnS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 143 shows the stress-strain curve for the tension of a single-layer 1T-SnS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-SnS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-SnS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 88.4 N/m and 87.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.13$.

There is no available value for nonlinear quantities in the single-layer 1T-SnS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -392.8 N/m and -421.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.6 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.3 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

LXXIII 1T-SnSe2

Most existing theoretical studies on the single-layer 1T-SnSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-SnSe2.

The structure for the single-layer 1T-SnSe2 is shown in Fig. 71 (with M=Sn and X=Se). Each Sn atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Sn atoms. The structural parameters are from the first-principles calculations,Huang, Zhang, and Zhang (2016) including the lattice constant $a=3.792$ Å, and the bond length $d_{\rm Sn-Se}=2.704$ Å. The resultant angles are $\theta_{\rm SeSnSn}=89.044^{\circ}$, and $\theta_{\rm SnSeSe}=89.044^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 290 shows three VFF terms for the single-layer 1T-SnSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Sn-Se-Se}$ is for the angle $\theta_{\rm Sn-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 144 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Zhang, and Zhang, 2016. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 144 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 291. The parameters for the three-body SW potential used by GULP are shown in Tab. 292. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 293.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-SnSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 145 shows the stress-strain curve for the tension of a single-layer 1T-SnSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-SnSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-SnSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 82.0 N/m and 81.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer 1T-SnSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -339.2 N/m and -368.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.1 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

LXXIV 1T-HfS2

Most existing theoretical studies on the single-layer 1T-HfS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfS2.

The structure for the single-layer 1T-HfS2 is shown in Fig. 71 (with M=Hf and X=S). Each Hf atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations,Kang, Sahin, and Peeters (2015) including the lattice constant $a=3.64$ Å and the bond length $d_{\rm Hf-S}=2.55$ Å. The resultant angles are $\theta_{\rm HfSS}=91.078^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SHfHf}=91.078^{\circ}$.

Table 294 shows three VFF terms for the single-layer 1T-HfS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Hf-S-S}$ is for the angle $\theta_{\rm Hf-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 146 (a). The ab initio calculations for the phonon dispersion are from Ref. Gu and Yang, 2014. Similar phonon dispersion can also be found in other ab initio calculations.Chen (2016); Huang, Zhang, and Zhang (2016) Fig. 146 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 295. The parameters for the three-body SW potential used by GULP are shown in Tab. 296. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 297.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 147 shows the stress-strain curve for the tension of a single-layer 1T-HfS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 73.3 N/m and 72.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are close to the ab initio results at 0 K temperature, eg. 79.86 Nm*-1* in Ref. Kang, Sahin, and Peeters, 2015. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$, which agrees reasonably with the ab initio resultKang, Sahin, and Peeters (2015) of 0.19.

There is no available value for nonlinear quantities in the single-layer 1T-HfS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -280.9 N/m and -317.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.9 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

LXXV 1T-HfSe2

Most existing theoretical studies on the single-layer 1T-HfSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfSe2.

The structure for the single-layer 1T-HfSe2 is shown in Fig. 71 (with M=Hf and X=Se). Each Hf atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations,Zhang et al. (2014a) including the lattice constant $a=3.673$ Å, and the position of the Se atom with respective to the Hf atomic plane $h=1.575$ Å. The resultant angles are $\theta_{\rm HfSeSe}=88.093^{\circ}$ with S atoms from the same (top or bottom) group, and $\theta_{\rm SeHfHf}=88.093^{\circ}$.

Table 298 shows three VFF terms for the single-layer 1T-HfSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Hf-Se-Se}$ is for the angle $\theta_{\rm Hf-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 148 (a). The ab initio calculations for the phonon dispersion are from Ref. Ding et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Huang, Zhang, and Zhang (2016) Fig. 148 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 299. The parameters for the three-body SW potential used by GULP are shown in Tab. 300. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 301.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 149 shows the stress-strain curve for the tension of a single-layer 1T-HfSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 67.3 N/m and 67.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.18$.

There is no available value for nonlinear quantities in the single-layer 1T-HfSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -221.5 N/m and -258.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.7 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

LXXVI 1T-HfTe2

Most existing theoretical studies on the single-layer 1T-HfTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-HfTe2.

The structure for the single-layer 1T-HfTe2 is shown in Fig. 71 (with M=Hf and X=Te). Each Hf atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Hf atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.9606$ Å, and the bond length $d_{\rm Hf-Te}=2.8559$ Å, which is derived from the angle $\theta_{\rm TeHfHf}=87.8^{\circ}$. The other angle is $\theta_{\rm HfTeTe}=87.8^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 302 shows three VFF terms for the single-layer 1T-HfTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Hf-Te-Te}$ is for the angle $\theta_{\rm Hf-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 50 N/m and the Poisson’s ratio as 0.10.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 303. The parameters for the three-body SW potential used by GULP are shown in Tab. 304. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 305.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-HfTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 150 shows the stress-strain curve for the tension of a single-layer 1T-HfTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-HfTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-HfTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.1 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.10$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 50 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-HfTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -204.3 N/m and -220.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.4 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.3 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Fig. 151 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXVII 1T-TaS2

Most existing theoretical studies on the single-layer 1T-TaS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaS2.

The structure for the single-layer 1T-TaS2 is shown in Fig. 71 (with M=Ta and X=S). Each Ta atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.3524$ Å, and the bond length $d_{\rm Ta-S}=2.4578$ Å, which is derived from the angle $\theta_{\rm STaTa}=86^{\circ}$. The other angle is $\theta_{\rm TaSS}=86^{\circ}$ with S atoms from the same (top or bottom) group.

Table 306 shows three VFF terms for the single-layer 1T-TaS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ta-S-S}$ is for the angle $\theta_{\rm Ta-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 101 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 307. The parameters for the three-body SW potential used by GULP are shown in Tab. 308. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 309.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 152 shows the stress-strain curve for the tension of a single-layer 1T-TaS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 87.8 N/m and 87.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 101 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TaS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -276.3 N/m and -313.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.7 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.2 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 153 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXVIII 1T-TaSe2

Most existing theoretical studies on the single-layer 1T-TaSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaSe2.

The structure for the single-layer 1T-TaSe2 is shown in Fig. 71 (with M=Ta and X=Se). Each Ta atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.4602$ Å, and the bond length $d_{\rm Ta-Se}=2.5609$ Å, which is derived from the angle $\theta_{\rm SeTaTa}=85^{\circ}$. The other angle is $\theta_{\rm TaSeSe}=85^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 310 shows three VFF terms for the single-layer 1T-TaSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ta-Se-Se}$ is for the angle $\theta_{\rm Ta-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 85 N/m and the Poisson’s ratio as 0.20.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 311. The parameters for the three-body SW potential used by GULP are shown in Tab. 312. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 313.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 154 shows the stress-strain curve for the tension of a single-layer 1T-TaSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 74.6 N/m and 74.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 85 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TaSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -231.7 N/m and -265.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.4 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 155 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXIX 1T-TaTe2

Most existing theoretical studies on the single-layer 1T-TaTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-TaTe2.

The structure for the single-layer 1T-TaTe2 is shown in Fig. 71 (with M=Ta and X=Te). Each Ta atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ta atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.6702$ Å, and the bond length $d_{\rm Ta-Te}=2.7695$ Å, which is derived from the angle $\theta_{\rm TeTaTa}=83^{\circ}$. The other angle is $\theta_{\rm TaTeTe}=83^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 314 shows three VFF terms for the single-layer 1T-TaTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ta-Te-Te}$ is for the angle $\theta_{\rm Ta-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 57 N/m and the Poisson’s ratio as 0.10.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 315. The parameters for the three-body SW potential used by GULP are shown in Tab. 316. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 317.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-TaTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 156 shows the stress-strain curve for the tension of a single-layer 1T-TaTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-TaTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-TaTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 50.3 N/m and 50.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.10$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 57 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-TaTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -247.1 N/m and -262.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

Fig. 157 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXX 1T-WS2

Most existing theoretical studies on the single-layer 1T-WS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WS2.

The structure for the single-layer 1T-WS2 is shown in Fig. 71 (with M=W and X=S). Each W atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three W atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.1908$ Å, and the bond length $d_{\rm W-S}=2.4125$ Å, which is derived from the angle $\theta_{\rm SWW}=82.8^{\circ}$. The other angle is $\theta_{\rm WSS}=82.8^{\circ}$ with S atoms from the same (top or bottom) group.

Table 318 shows three VFF terms for the single-layer 1T-WS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm W-S-S}$ is for the angle $\theta_{\rm W-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 113 N/m and the Poisson’s ratio as -0.03.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WS2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 319. The parameters for the three-body SW potential used by GULP are shown in Tab. 320. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 321.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 158 shows the stress-strain curve for the tension of a single-layer 1T-WS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 100.2 N/m and 99.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.03$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 113 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-WS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -666.6 N/m and -660.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.7 Nm*-1* at the ultimate strain of 0.15 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.7 Nm*-1* at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Fig. 159 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXI 1T-WSe2

Most existing theoretical studies on the single-layer 1T-WSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WSe2.

The structure for the single-layer 1T-WSe2 is shown in Fig. 71 (with M=W and X=Se). Each W atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three W atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.2574$ Å, and the bond length $d_{\rm W-Se}=2.5207$ Å, which is derived from the angle $\theta_{\rm SeWW}=80.5^{\circ}$. The other angle is $\theta_{\rm WSeSe}=80.5^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 322 shows three VFF terms for the single-layer 1T-WSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm W-Se-Se}$ is for the angle $\theta_{\rm W-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 94 N/m and the Poisson’s ratio as -0.15.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WSe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 323. The parameters for the three-body SW potential used by GULP are shown in Tab. 324. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 325.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 160 shows the stress-strain curve for the tension of a single-layer 1T-WSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 80.5 N/m and 80.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.15$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 94 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-WSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -666.1 N/m and -580.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 Nm*-1* at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.6 Nm*-1* at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Fig. 161 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXII 1T-WTe2

Most existing theoretical studies on the single-layer 1T-WTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-WTe2.

The structure for the single-layer 1T-WTe2 is shown in Fig. 71 (with M=W and X=Te). Each W atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three W atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.4970$ Å, and the bond length $d_{\rm W-Te}=2.7202$ Å, which is derived from the angle $\theta_{\rm TeWW}=80.0^{\circ}$. The other angle is $\theta_{\rm WTeTe}=80.0^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 326 shows three VFF terms for the single-layer 1T-WTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm W-Te-Te}$ is for the angle $\theta_{\rm W-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 88 N/m and the Poisson’s ratio as -0.18.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-WTe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 327. The parameters for the three-body SW potential used by GULP are shown in Tab. 328. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 329.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-WTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 162 shows the stress-strain curve for the tension of a single-layer 1T-WTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-WTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-WTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 75.9 N/m and 75.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.18$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 88 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-WTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -546.0 N/m and -551.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.1 Nm*-1* at the ultimate strain of 0.12 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K.

Fig. 163 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXIII 1T-ReS2

Most existing theoretical studies on the single-layer 1T-ReS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReS2.

The structure for the single-layer 1T-ReS2 is shown in Fig. 71 (with M=Re and X=S). Each Re atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Re atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.0750$ Å, and the bond length $d_{\rm Re-S}=2.4045$ Å, which is derived from the angle $\theta_{\rm SReRe}=79.5^{\circ}$. The other angle is $\theta_{\rm ReSS}=79.5^{\circ}$ with S atoms from the same (top or bottom) group.

Table 330 shows three VFF terms for the single-layer 1T-ReS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Re-S-S}$ is for the angle $\theta_{\rm Re-S-S}$ with both S atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 90 N/m and the Poisson’s ratio as -0.11.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReS2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 331. The parameters for the three-body SW potential used by GULP are shown in Tab. 332. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 333.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 164 shows the stress-strain curve for the tension of a single-layer 1T-ReS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 78.1 N/m and 77.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.11$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 90 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ReS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -537.1 N/m and -550.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.5 Nm*-1* at the ultimate strain of 0.13 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.6 Nm*-1* at the ultimate strain of 0.15 in the zigzag direction at the low temperature of 1 K.

Fig. 165 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXIV 1T-ReSe2

Most existing theoretical studies on the single-layer 1T-ReSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReSe2.

The structure for the single-layer 1T-ReSe2 is shown in Fig. 71 (with M=Re and X=Se). Each Re atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Re atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.1311$ Å, and the bond length $d_{\rm Re-Se}=2.5149$ Å, which is derived from the angle $\theta_{\rm SeReRe}=77^{\circ}$. The other angle is $\theta_{\rm ReSeSe}=77^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 334 shows three VFF terms for the single-layer 1T-ReSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Re-Se-Se}$ is for the angle $\theta_{\rm Re-Se-Se}$ with both Se atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 123 N/m and the Poisson’s ratio as -0.03.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReSe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 335. The parameters for the three-body SW potential used by GULP are shown in Tab. 336. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 337.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 166 shows the stress-strain curve for the tension of a single-layer 1T-ReSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 108.2 N/m and 107.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.03$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 123 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ReSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -669.3 N/m and -699.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.5 Nm*-1* at the ultimate strain of 0.14 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.4 Nm*-1* at the ultimate strain of 0.17 in the zigzag direction at the low temperature of 1 K.

Fig. 167 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXV 1T-ReTe2

Most existing theoretical studies on the single-layer 1T-ReTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ReTe2.

The structure for the single-layer 1T-ReTe2 is shown in Fig. 71 (with M=Re and X=Te). Each Re atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Re atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.3834$ Å, and the bond length $d_{\rm Re-Te}=2.7027$ Å, which is derived from the angle $\theta_{\rm TeReRe}=77.5^{\circ}$. The other angle is $\theta_{\rm ReTeTe}=77.5^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 338 shows three VFF terms for the single-layer 1T-ReTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Re-Te-Te}$ is for the angle $\theta_{\rm Re-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 71 N/m and the Poisson’s ratio as -0.22.Yu, Yan, and Ruzsinszky (2017) The ab initio calculations have predicted a negative Poisson’s ratio in the 1T-ReTe2, which was attributed to the orbital coupling in this material. The orbital coupling enhances the angle bending interaction in the VFF model. As a result, the value of the angle bending parameter is much larger than the bond stretching force constant parameter, which is typical in auxetic materials with negative Poisson’s ratio.Jiang et al. (2016)

The parameters for the two-body SW potential used by GULP are shown in Tab. 339. The parameters for the three-body SW potential used by GULP are shown in Tab. 340. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 341.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ReTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 168 shows the stress-strain curve for the tension of a single-layer 1T-ReTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-ReTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-ReTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 59.4 N/m and 59.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=-0.17$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 71 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-ReTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -416.1 N/m and -425.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.0 Nm*-1* at the ultimate strain of 0.12 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.1 Nm*-1* at the ultimate strain of 0.14 in the zigzag direction at the low temperature of 1 K.

Fig. 169 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXVI 1T-IrTe2

Most existing theoretical studies on the single-layer 1T-IrTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-IrTe2.

The structure for the single-layer 1T-IrTe2 is shown in Fig. 71 (with M=Ir and X=Te). Each Ir atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Ir atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.8431$ Å, and the bond length $d_{\rm Ir-Te}=2.6490$ Å, which is derived from the angle $\theta_{\rm TeIrIr}=93^{\circ}$. The other angle is $\theta_{\rm IrTeTe}=93^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 342 shows three VFF terms for the single-layer 1T-IrTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Ir-Te-Te}$ is for the angle $\theta_{\rm Ir-Te-Te}$ with both Te atoms from the same (top or bottom) group. We find that there are actually only two parameters in the VFF model, so we can determine their value by fitting to the Young’s modulus and the Poisson’s ratio of the system. The ab initio calculations have predicted the Young’s modulus to be 45 N/m and the Poisson’s ratio as 0.22.Yu, Yan, and Ruzsinszky (2017)

The parameters for the two-body SW potential used by GULP are shown in Tab. 343. The parameters for the three-body SW potential used by GULP are shown in Tab. 344. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 345.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-IrTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 170 shows the stress-strain curve for the tension of a single-layer 1T-IrTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-IrTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-IrTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 38.6 N/m and 38.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$. The fitted Young’s modulus value is about 10% smaller than the ab initio result of 45 N/m,Yu, Yan, and Ruzsinszky (2017) as only short-range interactions are considered in the present work. The long-range interactions are ignored, which typically leads to about 10% underestimation for the value of the Young’s modulus.

There is no available value for nonlinear quantities in the single-layer 1T-IrTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -127.7 N/m and -142.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.4 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Fig. 171 shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

LXXXVII 1T-PtS2

Most existing theoretical studies on the single-layer 1T-PtS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtS2.

The structure for the single-layer 1T-PtS2 is shown in Fig. 71 (with M=Pt and X=S). Each Pt atom is surrounded by six S atoms. These S atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each S atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.5237$ Å, and the bond length $d_{\rm Pt-S}=2.3708$ Å, which is derived from the angle $\theta_{\rm SPtPt}=96^{\circ}$. The other angle is $\theta_{\rm PtSS}=96^{\circ}$ with S atoms from the same (top or bottom) group.

Table 346 shows three VFF terms for the single-layer 1T-PtS2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pt-S-S}$ is for the angle $\theta_{\rm Pt-S-S}$ with both S atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 172 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Zhang, and Zhang, 2016. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 172 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 347. The parameters for the three-body SW potential used by GULP are shown in Tab. 348. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 349.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtS2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 173 shows the stress-strain curve for the tension of a single-layer 1T-PtS2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 105.9 N/m and 105.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$.

There is no available value for nonlinear quantities in the single-layer 1T-PtS2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -420.6 N/m and -457.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.8 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.3 Nm*-1* at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

LXXXVIII 1T-PtSe2

Most existing theoretical studies on the single-layer 1T-PtSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtSe2.

The structure for the single-layer 1T-PtSe2 is shown in Fig. 71 (with M=Pt and X=Se). Each Pt atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Se atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.6662$ Å, and the bond length $d_{\rm Pt-Se}=2.4667$ Å, which is derived from the angle $\theta_{\rm SePtPt}=96^{\circ}$. The other angle is $\theta_{\rm PtSeSe}=96^{\circ}$ with Se atoms from the same (top or bottom) group.

Table 350 shows three VFF terms for the single-layer 1T-PtSe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pt-Se-Se}$ is for the angle $\theta_{\rm Pt-Se-Se}$ with both Se atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 174 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Zhang, and Zhang, 2016. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may due to the violation of the rigid rotational invariance.Jiang et al. (2015) Fig. 174 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 351. The parameters for the three-body SW potential used by GULP are shown in Tab. 352. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 353.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtSe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 175 shows the stress-strain curve for the tension of a single-layer 1T-PtSe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 101.1 N/m and 100.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer 1T-PtSe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -391.4 N/m and -424.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.5 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.1 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

LXXXIX 1T-PtTe2

Most existing theoretical studies on the single-layer 1T-PtTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-PtTe2.

The structure for the single-layer 1T-PtTe2 is shown in Fig. 71 (with M=Pt and X=Te). Each Pt atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (eg. atoms 1, 3, and 5) and bottom group (eg. atoms 2, 4, and 6). Each Te atom is connected to three Pt atoms. The structural parameters are from the first-principles calculations,Yu, Yan, and Ruzsinszky (2017) including the lattice constant $a=3.9554$ Å, and the bond length $d_{\rm Pt-Te}=2.6613$ Å, which is derived from the angle $\theta_{\rm TePtPt}=96^{\circ}$. The other angle is $\theta_{\rm PtTeTe}=96^{\circ}$ with Te atoms from the same (top or bottom) group.

Table 354 shows three VFF terms for the single-layer 1T-PtTe2, one of which is the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). We note that the angle bending term $K_{\rm Pt-Te-Te}$ is for the angle $\theta_{\rm Pt-Te-Te}$ with both Te atoms from the same (top or bottom) group. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 176 (a). The ab initio calculations for the phonon dispersion are from Ref. Huang, Zhang, and Zhang, 2016. Fig. 176 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 355. The parameters for the three-body SW potential used by GULP are shown in Tab. 356. Some representative parameters for the SW potential used by LAMMPS are listed in Tab. 357.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-PtTe2 under uniaxial tension at 1.0 K and 300.0 K. Fig. 177 shows the stress-strain curve for the tension of a single-layer 1T-PtTe2 of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1T-PtTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1T-PtTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 89.1 N/m and 88.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer 1T-PtTe2. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -306.8 N/m and -340.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.1 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.6 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

XC Black phosphorus

The black phosphorus is also named the $\alpha$ phosphorus. There are several empirical potentials available for the atomic interaction in the black phosphorus. A VFF model was proposed for the single-layer black phosphorus in 1982.Kaneta, Katayama-Yoshida, and Morita (1982) One of the present author (J.W.J.) simplified this VFF model by ignoring some angle-angle crossing terms, and use the simplified VFF model to develop the SW potential for the black phosphorus.Jiang (2015a) However, the mechanical properties from this SW potential are smaller than first-principles calculations, as some angle-angle crossing VFF terms can not be implemented in the SW potential. We will thus propose a new set of SW potential for the single-layer black phosphorus in this section.

The structure of the single-layer black phosphorus is shown in Fig. 178, with structural parameters from the ab initio calculations.Du et al. (2010) The black phosphorus has a puckered configuration as shown in Fig. 178 (b), where the pucker is perpendicular to the x-direction. The bases for the rectangular unit cell are $a_{1}=4.422$ Å and $a_{2}=3.348$ Å. For bulk black phosphorus, the basis lattice vector in the third direction is $a_{3}=10.587$ Å. There are four phosphorus atoms in the basic unit cell, and their relative coordinates are $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v)$, and $(0.5+u,0.5,-v)$ with $u=0.0821$ and $v=0.1011$. Atoms are categorized into the top and bottom groups. Atoms in the top group are denoted by P1, while atoms in the bottom group are denoted by P2.

Table 358 shows four VFF terms for the single-layer black phosphorus, two of which are the bond stretching interactions shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$, as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms $\theta_{123}$ and $\theta_{134}$, which have very similar chemical environment. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 179 (a). The ab initio calculations for the phonon dispersion are from Ref. Zhu and Tomanek, 2014. Similar phonon dispersion can also be found in other ab initio calculations.Qin et al. (2014); Elahi et al. (2014); Ong et al. (2014); Aierken et al. (2015); Jiang (2015b); Jain and McGaughey (2015); Zhang et al. (2016a) Fig. 179 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 359. The parameters for the three-body SW potential used by GULP are shown in Tab. 360. Parameters for the SW potential used by LAMMPS are listed in Tab. 361.

Fig. 180 shows the stress strain relations for the black phosphorus of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 24.3 Nm*-1* and 90.5 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. These values agree quite well with previously reported ab initio results, eg. 28.9 Nm*-1* and 101.6 Nm*-1* from Ref. Qiao et al., 2014, or 24.4 Nm*-1* and 92.1 Nm*-1* from Ref. Wei and Peng, 2014, or 24.3 Nm*-1* and 80.2 Nm*-1* from Ref. Qin et al., 2014. The ultimate stress is about 4.27 Nm*-1* at the critical strain of 0.33 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.0 Nm*-1* at the critical strain of 0.19 in the zigzag direction at the low temperature of 1 K. These values agree quite well with the ab initio results at 0 K.Wei and Peng (2014)

It should be noted that the Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.058$ and $\nu_{yx}=0.22$. These values are obviously smaller than first-principles calculations results, eg. 0.4 and 0.93 from Ref. Jiang and Park, 2014, or 0.17 and 0.62 from Ref. Qin et al., 2014, or 0.24 and 0.81 from Ref. Elahi et al., 2014. The Poisson’s ratio can not be obtained correctly by the VFF model proposed in 1982Kaneta, Katayama-Yoshida, and Morita (1982) and the SW potentialJiang (2015a) either.Midtvedt and Croy (2016) These failures are due to the missing of one angle-angle crossing term,Jiang (2016) which has not been implemented in the package LAMMPS and is not included in the present work.

XCI p-arsenene

Present studies on the puckered (p-) arsenene, also named $\alpha$ arsenene, are based on first-principles calculations, and no empirical potential has been proposed for the p-arsenene. We will thus parametrize a set of VFF model for the single-layer p-arsenene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-arsenene.

The structure of the single-layer p-arsenene is exactly the same as that of the black phosphorus as shown in Fig. 178. Structural parameters for p-arsenene are from the ab initio calculations.Xu et al. (2016) The pucker of the p-arsenene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_{1}=4.77$ Å and $a_{2}=3.68$ Å. There are four As atoms in the basic unit cell, and their relative coordinates are $(-u,0,-v)$, $(u,,v)$, $(0.5-u,0.5,v)$, and $(0.5+u,0.5,-v)$ with $u=0.0714$ and $v=0.108$. The value of the dimensionless parameter $u$ is extracted from the geometrical parameters provided in Ref. Xu et al., 2016. The other dimensionless parameter $v$ is a ratio based on the lattice constant in the out-of-plane z-direction, so the other basis $a_{3}=11.11$ Å from Ref. Zhang et al., 2015 is also adopted in extracting the value of $v$. We note that the main purpose of the usage of $u$ and $v$ in representing atomic coordinates is to follow the same convention for all puckered structures. The resultant atomic coordinates are the same as that in Ref. Xu et al., 2016.

Table 362 shows four VFF terms for the single-layer p-arsenene, two of which are the bond stretching interactions shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$, as these two bonds have very close bond length value. The force constant parameters happen to be the same for the two angle bending terms $\theta_{123}$ and $\theta_{134}$. These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 181 (a). The ab initio calculations for the phonon dispersion are from Ref. Xu et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Kamal and Ezawa (2015); Zeraati et al. (2015); Yang and Liu (2016); Zhang et al. (2016a) We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 181 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 363. The parameters for the three-body SW potential used by GULP are shown in Tab. 364. Parameters for the SW potential used by LAMMPS are listed in Tab. 365.

Fig. 182 shows the stress strain relations for the p-arsenene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 20.7 Nm*-1* and 73.0 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -56.4 N/m and -415.5 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 Nm*-1* at the critical strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 Nm*-1* at the critical strain of 0.18 in the zigzag direction at the low temperature of 1 K.

XCII p-antimonene

Present studies on the puckered (p-) antimonene, also named $\alpha$ antimonene, are based on first-principles calculations, and no empirical potential has been proposed for the p-antimonene. We will thus parametrize a set of VFF model for the single-layer p-antimonene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-antimonene.

The structure of the single-layer p-antimonene is shown in Fig. 183, which is similar as that of the black phosphorus as shown in Fig. 178. Structural parameters for p-antimonene are from the ab initio calculations.Xu et al. (2016) The pucker of the p-antimonene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_{1}=4.73$ Å and $a_{2}=4.36$ Å. There are four Sb atoms in the basic unit cell, and their relative coordinates are $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.044$, $v=0.128$ and $w=0.0338$. The value of the dimensionless parameter $u$ is extracted from the geometrical parameters (bond lengths and bond angles) provided in Ref. Xu et al., 2016. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, so an arbitrary value of $a_{3}=11.11$ Å is adopted in extracting the values of $v$ and $w$. The value of $a_{3}$ has no effect on the actual position of each Sb atom. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention of black phosphorus. The resultant atomic coordinates are the same as that in Ref. Xu et al., 2016.

As shown in Fig. 183 (b), a specific feature in the puckered configuration of the p-antimonene is that Sb atoms in the top/bottom group are further divided into two subgroups with different z-coordinates. Specifically, in Fig. 183 (c), there is a difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. As a result of the nonzero value of $w$, there are two different inter-group angles, i.e., $\theta_{134}=88.3^{\circ}$ and $\theta_{415}=102.8^{\circ}$. We have $w=0$ for the ideal puckered configuration of the black phosphorus.

Table 366 shows five VFF terms for the single-layer p-antimonene, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$, as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. As a result, there are only three force constant parameters, i.e., $K_{12}=K_{14}$, $K_{123}$, and $K_{134}=K_{415}$. These three force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 185 (a). The ab initio calculations for the phonon dispersion are from Ref. Xu et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Wang, Pandey, and Karna (2015); Zheng et al. (2016); Zhang et al. (2016a) We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 185 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 367. The parameters for the three-body SW potential used by GULP are shown in Tab. 368. Parameters for the SW potential used by LAMMPS are listed in Tab. 369. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 184, which technically increases the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 183 (c).

Fig. 186 shows the stress strain relations for the p-antimonene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 18.3 Nm*-1* and 65.2 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.08$ and $\nu_{yx}=0.29$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -22.1 N/m and -354.1 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.7 Nm*-1* at the critical strain of 0.37 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.4 Nm*-1* at the critical strain of 0.17 in the zigzag direction at the low temperature of 1 K.

XCIII p-bismuthene

Present studies on the puckered (p-) bismuthene, which is also named $\alpha$ bismuthene, are based on first-principles calculations, and no empirical potential has been proposed for the p-bismuthene. We will thus parametrize a set of VFF model for the single-layer p-bismuthene in this section. We will also derive the SW potential based on the VFF model for the single-layer p-bismuthene.

The structure of the single-layer p-bismuthene is the same as p-antimonene as shown in Fig. 183. Structural parameters for p-bismuthene are from the ab initio calculations.Akturk, Akturk, and Ciraci (2016) The pucker of the p-bismuthene is perpendicular to the x (armchair)-direction. The bases for the rectangular unit cell are $a_{1}=4.94$ Å and $a_{2}=4.55$ Å. There are four Bi atoms in the basic unit cell, and their relative coordinates are $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0405$, $v=0.130$ and $w=0.0391$. The value of the dimensionless parameter $u$ is extracted from the geometrical parameters provided in Ref. Akturk, Akturk, and Ciraci, 2016. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, so an arbitrary value of $a_{3}=11.81$ Å is adopted in extracting the values of $v$ and $w$. The value of $a_{3}$ has no effect on the actual position of each Bi atom. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention of black phosphorus. The resultant atomic coordinates are the same as that in Ref. Akturk, Akturk, and Ciraci, 2016.

As shown in Fig. 183 (b), a specific feature in the puckered configuration of the p-bismuthene is that Bi atoms in the top/bottom group are further divided into two subgroups with different z-coordinates. Specifically, in Fig. 183 (c), there is a difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. As a result of the nonzero value of $w$, there are two different inter-group angles, i.e., $\theta_{134}=86.486^{\circ}$ and $\theta_{415}=103.491^{\circ}$. We have $w=0$ for the ideal puckered configuration of the black phosphorus.

Table 370 shows five VFF terms for the single-layer p-bismuthene, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are reasonably chosen to be the same for the two bond stretching terms denoted by $r_{12}$ and $r_{14}$, as these two bonds have very close bond length value. The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. As a result, there are only three force constant parameters, i.e., $K_{12}=K_{14}$, $K_{123}$, and $K_{134}=K_{415}$. These three force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 187 (a). The ab initio calculations for the phonon dispersion are from Ref. Akturk, Akturk, and Ciraci, 2016. Similar phonon dispersion can also be found in other ab initio calculations.Zhang et al. (2016a) Fig. 187 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 371. The parameters for the three-body SW potential used by GULP are shown in Tab. 372. Parameters for the SW potential used by LAMMPS are listed in Tab. 373. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 184, which helps to increase the cutoff for the bond stretching interaction between atoms like 1 and 2 in Fig. 183 (c).

Fig. 188 shows the stress strain relations for the p-bismuthene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 10.2 Nm*-1* and 26.2 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.24$ and $\nu_{yx}=0.61$. These values are very close to the ab initio calculations, eg. $\nu_{xy}=0.261$ and $\nu_{yx}=0.648$ in Ref. Akturk, Akturk, and Ciraci, 2016. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -12.4 N/m and -86.4 N/m at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 Nm*-1* at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 Nm*-1* at the critical strain of 0.29 in the zigzag direction at the low temperature of 1 K.

XCIV p-SiO

Present studies on the puckered (p-) SiO are based on first-principles calculations, and no empirical potential has been proposed for the p-SiO. We will thus parametrize the SW potential for the single-layer p-SiO in this section.

The structure of the single-layer p-SiO is shown in Fig. 189, with M=Si and X=O. Structural parameters for p-SiO are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.1501$, $v=0.0605$ and $w=0.0800$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.701$ Å and $a_{2}=2.739$ Å, bond lengths $d_{12}=1.843$ Å and $d_{14}=2.859$ Å, and the angle $\theta_{145}=96.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SiO is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$. The p-SiO has a zigzag configuration as shown in Fig. 191, which is a specific case of the puckered structure shown in Fig. 189.

Table 374 shows five VFF terms for the single-layer p-SiO, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 192 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 192 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 375. The parameters for the three-body SW potential used by GULP are shown in Tab. 376. Parameters for the SW potential used by LAMMPS are listed in Tab. 377.

Fig. 193 shows the stress strain relations for the single-layer p-SiO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The structure of p-SiO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 81.3 Nm*-1* in the zigzag direction at 1 K, which is obtained by linear fitting of the stress strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The value of $D$ is -432.4 Nm*-1* at 1 K along the zigzag direction. The ultimate stress is about 5.3 Nm*-1* at the critical strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 Nm*-1* at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

XCV p-GeO

Present studies on the puckered (p-) GeO are based on first-principles calculations, and no empirical potential has been proposed for the p-GeO. We will thus parametrize the SW potential for the single-layer p-GeO in this section.

The structure of the single-layer p-GeO is shown in Fig. 189, with M=Ge and X=O. Structural parameters for p-GeO are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.1622$, $v=0.0616$ and $w=0.0884$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.801$ Å and $a_{2}=3.055$ Å, bond lengths $d_{12}=1.956$ Å and $d_{14}=1.986$ Å, and the angle $\theta_{145}=93.3^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-GeO is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$. The p-GeO has a zigzag configuration as shown in Fig. 191, which is a specific case of the puckered structure shown in Fig. 189.

Table 378 shows five VFF terms for the single-layer p-GeO, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 194 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 194 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 379. The parameters for the three-body SW potential used by GULP are shown in Tab. 380. Parameters for the SW potential used by LAMMPS are listed in Tab. 381.

Fig. 195 shows the stress strain relations for the single-layer p-GeO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The structure of p-GeO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 14.5 Nm*-1* and 78.9 Nm*-1* in the armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.09$ and $\nu_{yx}=0.65$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are 22.0 Nm*-1* and -383.3 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 4.8 Nm*-1* at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.2 Nm*-1* at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

XCVI p-SnO

Present studies on the puckered (p-) SnO are based on first-principles calculations, and no empirical potential has been proposed for the p-SnO. We will thus parametrize the SW potential for the single-layer p-SnO in this section.

The structure of the single-layer p-SnO is shown in Fig. 189, with M=Sn and X=O. Structural parameters for p-SnO are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.1485$, $v=0.0818$ and $w=0.0836$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.764$ Å and $a_{2}=3.400$ Å, bond lengths $d_{12}=2.127$ Å and $d_{14}=2.163$ Å, and the angle $\theta_{145}=90.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SnO is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$. The p-SnO has a zigzag configuration as shown in Fig. 191, which is a specific case of the puckered structure shown in Fig. 189.

Table 382 shows five VFF terms for the single-layer p-SnO, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 196 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 196 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 383. The parameters for the three-body SW potential used by GULP are shown in Tab. 384. Parameters for the SW potential used by LAMMPS are listed in Tab. 385.

Fig. 197 shows the stress strain relations for the single-layer p-SnO of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The structure of p-SnO is so soft along the armchair direction that the Young’s modulus is almost zero in the armchair direction. The Young’s modulus is 52.8 Nm*-1* in the zigzag direction at 1 K, which is obtained by linear fitting of the stress strain relations in [0, 0.01]. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The value of $D$ is -204.5 Nm*-1* at 1 K along the zigzag direction. The ultimate stress is about 3.8 Nm*-1* at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.8 Nm*-1* at the critical strain of 0.26 in the zigzag direction at the low temperature of 1 K.

XCVII p-CS

Present studies on the puckered (p-) CS are based on first-principles calculations, and no empirical potential has been proposed for the p-CS. We will thus parametrize the SW potential for the single-layer p-CS in this section.

The structure of the single-layer p-CS is shown in Fig. 189, with M=C and X=S. Structural parameters for p-CS are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.1302$, $v=0.0733$ and $w=-0.0248$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.323$ Å and $a_{2}=2.795$ Å, bond lengths $d_{12}=1.757$ Å and $d_{14}=1.849$ Å, and the angle $\theta_{145}=118.1^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-CS is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 386 shows five VFF terms for the single-layer p-CS, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 198 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital bacs set is adopted. Fig. 198 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 387. The parameters for the three-body SW potential used by GULP are shown in Tab. 388. Parameters for the SW potential used by LAMMPS are listed in Tab. 389.

Fig. 199 shows the stress strain relations for the single-layer p-CS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. There is a structural transition around 0.16 at 1 K, where the C atom is twisted. The Young’s modulus is 16.2 Nm*-1* and 70.5 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=-0.06$ and $\nu_{yx}=-0.27$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -27.3 Nm*-1* and -447.2 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 4.3 Nm*-1* at the critical strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.2 Nm*-1* at the critical strain of 0.22 in the zigzag direction at the low temperature of 1 K.

XCVIII p-SiS

Present studies on the puckered (p-) SiS are based on first-principles calculations, and no empirical potential has been proposed for the p-SiS. We will thus parametrize the SW potential for the single-layer p-SiS in this section.

The structure of the single-layer p-SiS is shown in Fig. 189, with M=Si and X=S. Structural parameters for p-SiS are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0884$, $v=0.1093$ and $w=0.0316$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.774$ Å and $a_{2}=3.352$ Å, bond lengths $d_{12}=2.300$ Å and $d_{14}=2.344$ Å, and the angle $\theta_{145}=96.5^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SiS is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 390 shows five VFF terms for the single-layer p-SiS, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 200 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 200 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 391. The parameters for the three-body SW potential used by GULP are shown in Tab. 392. Parameters for the SW potential used by LAMMPS are listed in Tab. 393.

Fig. 201 shows the stress strain relations for the single-layer p-SiS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 10.9 Nm*-1* and 34.8 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.04$ and $\nu_{yx}=0.12$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -24.1 Nm*-1* and -145.2 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 Nm*-1* at the critical strain of 0.39 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 Nm*-1* at the critical strain of 0.25 in the zigzag direction at the low temperature of 1 K.

XCIX p-GeS

Present studies on the puckered (p-) GeS are based on first-principles calculations, and no empirical potential has been proposed for the p-GeS. We will thus parametrize the SW potential for the single-layer p-GeS in this section.

The structure of the single-layer p-GeS is shown in Fig. 189, with M=Ge and X=S. Structural parameters for p-GeS are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0673$, $v=0.1173$ and $w=0.0228$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.492$ Å and $a_{2}=3.642$ Å, bond lengths $d_{12}=2.462$ Å and $d_{14}=2.423$ Å, and the angle $\theta_{145}=94.4^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-GeS is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 394 shows five VFF terms for the single-layer p-GeS, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 202 (a). The ab initio calculations are from Ref. Qin et al., 2016. Fig. 202 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 395. The parameters for the three-body SW potential used by GULP are shown in Tab. 396. Parameters for the SW potential used by LAMMPS are listed in Tab. 397. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Ge and X=S, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 203 shows the stress strain relations for the single-layer p-GeS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 10.6 Nm*-1* and 32.1 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.10$ and $\nu_{yx}=0.29$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -20.4 Nm*-1* and -118.8 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.4 Nm*-1* at the critical strain of 0.39 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 Nm*-1* at the critical strain of 0.24 in the zigzag direction at the low temperature of 1 K.

C p-SnS

Present studies on the puckered (p-) SnS are based on first-principles calculations, and no empirical potential has been proposed for the p-SnS. We will thus parametrize the SW potential for the single-layer p-SnS in this section.

The structure of the single-layer p-SnS is shown in Fig. 189, with M=Sn and X=S. Structural parameters for p-SnS are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0426$, $v=0.1284$ and $w=0.0308$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.347$ Å and $a_{2}=4.047$ Å, bond lengths $d_{12}=2.728$ Å and $d_{14}=2.595$ Å, and the angle $\theta_{145}=89.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SnS is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 398 shows five VFF terms for the single-layer p-SnS, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 204 (a). The ab initio calculations are from Ref. Qin et al., 2016. Fig. 204 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 399. The parameters for the three-body SW potential used by GULP are shown in Tab. 400. Parameters for the SW potential used by LAMMPS are listed in Tab. 401. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Sn and X=S, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 205 shows the stress strain relations for the single-layer p-SnS of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 9.6 Nm*-1* and 24.5 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.18$ and $\nu_{yx}=0.47$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -14.7 Nm*-1* and -80.3 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 Nm*-1* at the critical strain of 0.36 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.1 Nm*-1* at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

CI p-CSe

Present studies on the puckered (p-) CSe are based on first-principles calculations, and no empirical potential has been proposed for the p-CSe. We will thus parametrize the SW potential for the single-layer p-CSe in this section.

The structure of the single-layer p-CSe is shown in Fig. 189, with M=C and X=Se. Structural parameters for p-CSe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.1079$, $v=0.0894$ and $w=-0.0229$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.299$ Å and $a_{2}=3.034$ Å, bond lengths $d_{12}=1.961$ Å and $d_{14}=2.014$ Å, and the angle $\theta_{145}=113.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-CSe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 402 shows five VFF terms for the single-layer p-CSe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 206 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 206 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 403. The parameters for the three-body SW potential used by GULP are shown in Tab. 404. Parameters for the SW potential used by LAMMPS are listed in Tab. 405. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=C and X=Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 207 shows the stress strain relations for the single-layer p-CSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 17.2 Nm*-1* and 75.4 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=-0.02$ and $\nu_{yx}=-0.11$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -46.3 Nm*-1* and -442.0 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.0 Nm*-1* at the critical strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.5 Nm*-1* at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

CII p-SiSe

Present studies on the puckered (p-) SiSe are based on first-principles calculations, and no empirical potential has been proposed for the p-SiSe. We will thus parametrize the SW potential for the single-layer p-SiSe in this section.

The structure of the single-layer p-SiSe is shown in Fig. 189, with M=Si and X=Se. Structural parameters for p-SiSe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0572$, $v=0.1198$ and $w=-0.0011$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.400$ Å and $a_{2}=3.737$ Å, bond lengths $d_{12}=2.524$ Å and $d_{14}=2.448$ Å, and the angle $\theta_{145}=98.2^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SiSe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 406 shows five VFF terms for the single-layer p-SiSe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 208 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 208 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 407. The parameters for the three-body SW potential used by GULP are shown in Tab. 408. Parameters for the SW potential used by LAMMPS are listed in Tab. 409. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Si and X=Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 209 shows the stress strain relations for the single-layer p-SiSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 14.4 Nm*-1* and 44.6 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.09$ and $\nu_{yx}=0.30$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -28.8 Nm*-1* and -176.6 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.1 Nm*-1* at the critical strain of 0.37 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 Nm*-1* at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

CIII p-GeSe

Present studies on the puckered (p-) GeSe are based on first-principles calculations, and no empirical potential has been proposed for the p-GeSe. We will thus parametrize the SW potential for the single-layer p-GeSe in this section.

The structure of the single-layer p-GeSe is shown in Fig. 189, with M=Ge and X=Se. Structural parameters for p-GeSe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0439$, $v=0.1258$ and $w=-0.0080$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.302$ Å and $a_{2}=3.965$ Å, bond lengths $d_{12}=2.661$ Å and $d_{14}=2.544$ Å, and the angle $\theta_{145}=97.4^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-GeSe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 410 shows five VFF terms for the single-layer p-GeSe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 210 (a). The ab initio calculations are from Ref. Qin et al., 2016. Fig. 210 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 411. The parameters for the three-body SW potential used by GULP are shown in Tab. 412. Parameters for the SW potential used by LAMMPS are listed in Tab. 413. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Ge and X=Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 211 shows the stress strain relations for the single-layer p-GeSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 11.1 Nm*-1* and 32.0 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.14$ and $\nu_{yx}=0.42$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -19.3 Nm*-1* and -114.7 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 Nm*-1* at the critical strain of 0.36 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.9 Nm*-1* at the critical strain of 0.20 in the zigzag direction at the low temperature of 1 K.

CIV p-SnSe

Present studies on the puckered (p-) SnSe are based on first-principles calculations, and no empirical potential has been proposed for the p-SnSe. We will thus parametrize the SW potential for the single-layer p-SnSe in this section.

The structure of the single-layer p-SnSe is shown in Fig. 189, with M=Sn and X=Se. Structural parameters for p-SnSe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0313$, $v=0.1358$ and $w=0.0074$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.453$ Å and $a_{2}=4.260$ Å, bond lengths $d_{12}=2.887$ Å and $d_{14}=2.730$ Å, and the angle $\theta_{145}=92.5^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SnSe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 414 shows five VFF terms for the single-layer p-SnSe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 212 (a). The ab initio calculations are from Ref. Zhang et al., 2016b. Fig. 212 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 415. The parameters for the three-body SW potential used by GULP are shown in Tab. 416. Parameters for the SW potential used by LAMMPS are listed in Tab. 417. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Sn and X=Se, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 213 shows the stress strain relations for the single-layer p-SnSe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 11.4 Nm*-1* and 34.1 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.11$ and $\nu_{yx}=0.33$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -22.0 Nm*-1* and -128.8 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.3 Nm*-1* at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 Nm*-1* at the critical strain of 0.15 in the zigzag direction at the low temperature of 1 K.

CV p-CTe

Present studies on the puckered (p-) CTe are based on first-principles calculations, and no empirical potential has been proposed for the p-CTe. We will thus parametrize the SW potential for the single-layer p-CTe in this section.

The structure of the single-layer p-CTe is shown in Fig. 189, with M=C and X=Te. Structural parameters for p-CTe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0837$, $v=0.1041$ and $w=-0.0371$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=3.889$ Å and $a_{2}=3.390$ Å, bond lengths $d_{12}=2.164$ Å and $d_{14}=2.181$ Å, and the angle $\theta_{145}=110.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-CTe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 418 shows five VFF terms for the single-layer p-CTe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 214 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 214 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 419. The parameters for the three-body SW potential used by GULP are shown in Tab. 420. Parameters for the SW potential used by LAMMPS are listed in Tab. 421. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=C and X=Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 215 shows the stress strain relations for the single-layer p-CTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 10.8 Nm*-1* and 89.1 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.02$ and $\nu_{yx}=0.20$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -15.3 Nm*-1* and -419.6 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.0 Nm*-1* at the critical strain of 0.43 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.4 Nm*-1* at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

CVI p-SiTe

Present studies on the puckered (p-) SiTe are based on first-principles calculations, and no empirical potential has been proposed for the p-SiTe. We will thus parametrize the SW potential for the single-layer p-SiTe in this section.

The structure of the single-layer p-SiTe is shown in Fig. 189, with M=Si and X=Te. Structural parameters for p-SiTe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0581$, $v=0.1363$ and $w=-0.0173$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.300$ Å and $a_{2}=4.109$ Å, bond lengths $d_{12}=2.641$ Å and $d_{14}=2.772$ Å, and the angle $\theta_{145}=100.200^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SiTe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 422 shows five VFF terms for the single-layer p-SiTe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 216 (a). The ab initio calculations are from Ref. Chen, Sun, and Jena, 2016. Fig. 216 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 423. The parameters for the three-body SW potential used by GULP are shown in Tab. 424. Parameters for the SW potential used by LAMMPS are listed in Tab. 425. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Si and X=Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 217 shows the stress strain relations for the single-layer p-SiTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 14.0 Nm*-1* and 53.6 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.12$ and $\nu_{yx}=0.47$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -32.9 Nm*-1* and -183.2 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.2 Nm*-1* at the critical strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.0 Nm*-1* at the critical strain of 0.22 in the zigzag direction at the low temperature of 1 K.

CVII p-GeTe

Present studies on the puckered (p-) GeTe are based on first-principles calculations, and no empirical potential has been proposed for the p-GeTe. We will thus parametrize the SW potential for the single-layer p-GeTe in this section.

The structure of the single-layer p-GeTe is shown in Fig. 189, with M=Ge and X=Te. Structural parameters for p-GeTe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0538$, $v=0.1422$ and $w=-0.0216$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.376$ Å and $a_{2}=4.238$ Å, bond lengths $d_{12}=2.736$ Å and $d_{14}=2.883$ Å, and the angle $\theta_{145}=100.4^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-GeTe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 426 shows five VFF terms for the single-layer p-GeTe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 218 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 218 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 427. The parameters for the three-body SW potential used by GULP are shown in Tab. 428. Parameters for the SW potential used by LAMMPS are listed in Tab. 429. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Ge and X=Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 219 shows the stress strain relations for the single-layer p-GeTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 8.1 Nm*-1* and 41.6 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.09$ and $\nu_{yx}=0.49$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -10.5 Nm*-1* and -143.7 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.6 Nm*-1* at the critical strain of 0.53 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 Nm*-1* at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

CVIII p-SnTe

Present studies on the puckered (p-) SnTe are based on first-principles calculations, and no empirical potential has been proposed for the p-SnTe. We will thus parametrize the SW potential for the single-layer p-SnTe in this section.

The structure of the single-layer p-SnTe is shown in Fig. 189, with M=Sn and X=Te. Structural parameters for p-SnTe are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) There are four atoms in the unit cell with relative coordinates as $(-u,0,-v)$, $(u,0,v)$, $(0.5-u,0.5,v+w)$, and $(0.5+u,0.5,-v+w)$ with $u=0.0478$, $v=0.1567$ and $w=-0.0050$. The value of these dimensionless parameters are extracted from the geometrical parameters provided in Ref. Kamal, Chakrabarti, and Ezawa, 2016, including lattice constants $a_{1}=4.581$ Å and $a_{2}=4.542$ Å, bond lengths $d_{12}=2.931$ Å and $d_{14}=3.164$ Å, and the angle $\theta_{145}=96.0^{\circ}$. The dimensionless parameters $v$ and $w$ are ratios based on the lattice constant in the out-of-plane z-direction, which is arbitrarily chosen as $a_{3}=10.0$ Å. We note that the main purpose of the usage of $u$, $v$, and $w$ in representing atomic coordinates is to follow the same convention for all puckered structures in the present work. The resultant atomic coordinates are the same as that in Ref. Kamal, Chakrabarti, and Ezawa, 2016.

As shown in Fig. 189, a specific feature in the puckered configuration of the p-SnTe is that there is a small difference of $wa_{3}$ between the z-coordinate of atom 1 and the z-coordinates of atoms 2 and 3. Similarly, atom 4 is higher than atoms 5 and 6 for $wa_{3}$ along the z-direction. The sign of $w$ determines which types of atoms take the out-most positions, e.g., atoms 1, 5, and 6 are the out-most atoms if $w>0$ in Fig. 189 (c), while atoms 2, 3, and 4 will take the out-most positions for $w<0$.

Table 430 shows five VFF terms for the single-layer p-SnTe, two of which are the bond stretching interactions shown by Eq. (1) while the other three terms are the angle bending interaction shown by Eq. (2). The force constant parameters are the same for the two angle bending terms $\theta_{134}$ and $\theta_{415}$, which have the same arm lengths. All force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 220 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 220 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion.

The parameters for the two-body SW potential used by GULP are shown in Tab. 431. The parameters for the three-body SW potential used by GULP are shown in Tab. 432. Parameters for the SW potential used by LAMMPS are listed in Tab. 433. Eight atom types have been introduced for writing the SW potential script used by LAMMPS as shown in Fig. 190 with M=Sn and X=Te, which helps to increase the cutoff for the bond stretching interaction between atom 1 and atom 2 in Fig. 189 (c).

Fig. 221 shows the stress strain relations for the single-layer p-SnTe of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 6.6 Nm*-1* and 38.5 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=0.10$ and $\nu_{yx}=0.57$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -7.2 Nm*-1* and -114.5 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 2.4 Nm*-1* at the critical strain of 0.61 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.3 Nm*-1* at the critical strain of 0.21 in the zigzag direction at the low temperature of 1 K.

CIX Silicene

There have been several empirical potentials available for the silicene. A many-body potential based on the Lennard-Jones and Axilrod-Teller functions was used to describe the interaction within the single-layer silicene.Ince and Erkoc (2011) The modified embedded-atom potentialBaskes (1992) was used by Pei et al to simulate the thermal transport in the single-layer silicnene in 2013.Pei et al. (2013) The environment-dependent interatomic potentialJusto et al. (1998) was also used to simulate the silicene.Chavez-Castillo, Rodriguez-Meza, and Meza-Montes (2015) In particular, the original set of SW parametersStillinger and Weber (1985) for the silicon were found to be not suitable for the planar silicene, so two sets of optimized parameters for the SW potential were proposed to simulate the thermal conductivity in the single-layer silicene in 2014.Zhang et al. (2014b) We will develop a new SW potential to describe the interaction within the silicene in this section, with specific focus on the mechanical properties of the silicene.

The structure of the silicene is shown in Fig. 222, with structural parameters from the ab initio calculations.Ge, Yao, and Lu (2016) The silicene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=0.45$ Å and the lattice constant is 3.87 Å, which results in a bond length of 2.279 Å.

Table 434 shows the VFF model for the silicene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 223 (a). The ab initio calculations for the phonon dispersion are from Ref. Ge, Yao, and Lu, 2016. Similar phonon dispersion can also be found in other ab initio calculations.Li et al. (2013); Scalise et al. (2013); Roome and Carey (2014); Yang et al. (2014); Wang et al. (2014); Xie, Hu, and Bao (2014); Gu and Yang (2015); Huang, Gong, and Zeng (2015); Wang, Feng, and Ruan (2015); Ge, Yao, and Lu (2016); Xie et al. (2016); Kuang et al. (2016); Peng et al. (2016a) Fig. 223 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

We note that the present SW potential is fitted perfectly to the three acoustic phonon branches, so it can give a nice description for the elastic deformation of the silicene. As a trade off, the optical phonons are overestimated by the present SW potential. Hence, the present SW potential is in particular suitable for the simulation of mechanical or thermal processes which are dominated by acoustic phonons, while the present SW potential may cause a systematic error for the optical absorption process which mainly involves the optical phonons. One can introduce the long-range interactions (eg. the second-nearest-neighboring interaction) to give a good description for both acoustic and optical phonon branches, see one such example for borophene in Ref. Zhou and Jiang, 2017. It is because the long-range interaction mainly contributes to the acoustic phonon branches, while it makes only neglectable contribution to the optical phonon branches. As another solution, the SW potential can give reasonable descriptions for the optical phonon branches by reducing its accuracy in describing acoustic phonon branches as done in Ref. Zhang et al., 2014b.

The parameters for the two-body SW potential used by GULP are shown in Tab. 435. The parameters for the three-body SW potential used by GULP are shown in Tab. 436. Parameters for the SW potential used by LAMMPS are listed in Tab. 437.

Fig. 224 shows the stress strain relations for the silicene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 63.3 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the silicene. The value of the Young’s modulus agrees with the value of 63.8 Nm*-1* from the ab initio calculations.Peng, Wen, and De (2013) The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.15$, which are smaller but comparable with the ab initio results of 0.325.Peng, Wen, and De (2013) The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -212.5 Nm*-1* and -267.5 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 8.6 Nm*-1* at the critical strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 Nm*-1* at the critical strain of 0.23 in the zigzag direction at the low temperature of 1 K.

The stress-strain curves shown in Fig. 224 disclose a structural transition at the strain around 0.076 for the silicene at the low temperature of 1 K. The buckled configuration of the silicene is flattened during this structural transition, which can be seen from these two insets in Fig. 224. This structural transition was also observed in the ab initio calculations,Wang et al. (2014) where the critical strain for the structural transition is 0.2. At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations; i.e., the buckled configuration of the silicene can be strongly disturbed by the thermal vibration at higher temperatures.

CX Germanene

In a recent work, the Tersoff potential was applied to simulate the thermal conductivity of the germanene nanoribbon.Balatero et al. (2015) We will provide the SW potential to describe the interaction within the germanene in this section.

The structure of the germanene is shown in Fig. 222, with structural parameters from the ab initio calculations.Ge, Yao, and Lu (2016) The germanene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=0.69$ Å and the lattice constant is 4.06 Å, which results in a bond length of 2.443 Å.

Table 438 shows the VFF model for the germanene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 225 (a). The ab initio calculations for the phonon dispersion are from Ref. Ge, Yao, and Lu, 2016. Similar phonon dispersion can also be found in other ab initio calculations.Scalise et al. (2013); Roome and Carey (2014); Huang, Gong, and Zeng (2015); Ge, Yao, and Lu (2016); Kuang et al. (2016); Zaveh, Roknabadi, and T. Morshedloo (2016); Peng et al. (2016a) Fig. 225 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 439. The parameters for the three-body SW potential used by GULP are shown in Tab. 440. Parameters for the SW potential used by LAMMPS are listed in Tab. 441.

Fig. 226 shows the stress strain relations for the germanene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 53.2 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the germanene. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.19$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -229.2 Nm*-1* and -278.2 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.5 Nm*-1* at the critical strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.3 Nm*-1* at the critical strain of 0.27 in the zigzag direction at the low temperature of 1 K.

The stress-strain curves shown in Fig. 226 disclose a structural transition for the germanene at the low temperature of 1 K. The critical strains for the structural transition are 0.15 and 0.16 along the armchair and zigzag directions, respectively. The buckled configuration of the germanene is flattened during this structural transition, which can be seen from these two insets in Fig. 226. At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations; i.e., the buckled configuration of the germanene can be strongly disturbed by the thermal vibration at higher temperatures.

CXI Stanene

There are several available empirical potentials for the description of the interaction within the stanene. The modified embedded atom method potential was applied to simulate mechanical properties for the stanene.Mojumder, Amin, and Islam (2015) A VFF model was fitted for the stanene in 2015.Modarresi et al. (2015) The Tersoff potential was parameterized to describe the interaction for stanene.Cherukara et al. (2016) In this section, we will develop the SW potential for the stanene.

The structure of the stanene is shown in Fig. 222, with structural parameters from the ab initio calculations.Zaveh, Roknabadi, and T. Morshedloo (2016) The stanene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=0.86$ Å and the lattice constant is 4.68 Å, which results in a bond length of 2.836 Å.

Table 442 shows the VFF model for the stanene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 227 (a). The ab initio calculations for the phonon dispersion are from Ref. Zaveh, Roknabadi, and T. Morshedloo, 2016 with the spin-orbit coupling effect. Similar phonon dispersion can also be found in other ab initio calculations.van den Broek et al. (2014); Kuang et al. (2016); Zaveh, Roknabadi, and T. Morshedloo (2016); Peng et al. (2016b); Zhou et al. (2016); Peng et al. (2016a) Fig. 227 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 443. The parameters for the three-body SW potential used by GULP are shown in Tab. 444. Parameters for the SW potential used by LAMMPS are listed in Tab. 445.

Fig. 228 shows the stress strain relations for the stanene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 17.0 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the stanene. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.29$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -37.2 Nm*-1* and -69.4 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 Nm*-1* at the critical strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.6 Nm*-1* at the critical strain of 0.29 in the zigzag direction at the low temperature of 1 K.

The stress-strain curves shown in Fig. 228 disclose a structural transition for the stanene at the low temperature of 1 K. The critical strain for the structural transition is about 0.15 along the armchair and zigzag directions. The buckled configuration of the stanene is flattened during this structural transition, which can be seen from these two insets in Fig. 228. At temperatures above 300 K, this structural transition is blurred by stronger thermal vibrations; i.e., the buckled configuration of the stanene can be strongly disturbed by the thermal vibration at higher temperatures.

CXII Indiene

In this section, we will develop the SW potential for the indiene, i.e., the single layer of Indium atoms. The structure of the indiene is shown in Fig. 222, with structural parameters from the ab initio calculations.Singh et al. (2016) The indiene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The lattice constant is 4.24 Å and the bond length is 2.89 Å, which results in the buckling height of $h=1.536$ Å.

Table 446 shows the VFF model for the indiene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 229 (a). The ab initio calculations for the phonon dispersion are from Ref. Singh et al., 2016. We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 229 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 447. The parameters for the three-body SW potential used by GULP are shown in Tab. 448. Parameters for the SW potential used by LAMMPS are listed in Tab. 449.

Fig. 230 shows the stress strain relations for the indiene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K. The Young’s modulus is 8.4 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus of the indiene is very small; i.e., the indiene is very soft. As a result, we find that the structure becomes unstable at room temperature. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.18$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -42.0 Nm*-1* and -50.2 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 0.77 Nm*-1* at the critical strain of 0.16 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 0.73 Nm*-1* at the critical strain of 0.19 in the zigzag direction at the low temperature of 1 K.

CXIII Blue phosphorus

The blue phosphorus is also named $\beta$-phosphorus. Present studies on the blue phosphorus are based on first-principles calculations, and no empirical potential has been proposed for the blue phosphorus. We will thus parametrize a set of VFF model for the single-layer blue phosphorus in this section. We will also derive the SW potential based on the VFF model for the single-layer blue phosphorus.

The structure of the single-layer blue phosphorus is shown in Fig. 222, with structural parameters from the ab initio calculations.Zhu and Tomanek (2014) The blue phosphorus has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=1.211$ Å. The lattice constant is 3.326 Å, and the bond length is 2.270 Å.

Table 450 shows the VFF model for the single-layer blue phosphorus. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 231 (a). The ab initio calculations for the phonon dispersion are from Ref. Zhu and Tomanek, 2014. Similar phonon dispersion can also be found in other ab initio calculations.Aierken et al. (2015); Ge, Yao, and Lu (2016); Zhang et al. (2016a) Fig. 231 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 451. The parameters for the three-body SW potential used by GULP are shown in Tab. 452. Parameters for the SW potential used by LAMMPS are listed in Tab. 453.

Fig. 232 shows the stress strain relations for the single-layer blue phosphorus of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 60.5 Nm*-1* and 60.6 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the blue phosphorus. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.18$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -195.3 Nm*-1* and -237.0 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 8.0 Nm*-1* at the critical strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.6 Nm*-1* at the critical strain of 0.25 in the zigzag direction at the low temperature of 1 K.

CXIV b-arsenene

Present studies on the buckled (b-) arsenene, whch is also named $\beta$ arsenene, are based on first-principles calculations, and no empirical potential has been proposed for the b-arsenene. We will thus parametrize a set of VFF model for the single-layer b-arsenene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-arsenene.

The structure of the single-layer b-arsenene is shown in Fig. 222, with structural parameters from the ab initio calculations.Xu et al. (2016) The b-arsenene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=1.40$ Å. The lattice constant is 3.61 Å, and the bond length is 2.51 Å.

Table 454 shows the VFF model for the single-layer b-arsenene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 233 (a). The ab initio calculations for the phonon dispersion are from Ref. Xu et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Kamal and Ezawa (2015); Zeraati et al. (2015); Zhang et al. (2016a) We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 233 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 455. The parameters for the three-body SW potential used by GULP are shown in Tab. 456. Parameters for the SW potential used by LAMMPS are listed in Tab. 457.

Fig. 234 shows the stress strain relations for the single-layer b-arsenene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 50.8 Nm*-1* and 49.9 Nm*-1* in the armchair and zigzag directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-arsenene. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.21$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -127.6 Nm*-1* and -153.6 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.6 Nm*-1* at the critical strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.2 Nm*-1* at the critical strain of 0.28 in the zigzag direction at the low temperature of 1 K.

CXV b-antimonene

The buckled (b-) antimonene is a Sb allotrope, which is also named $\beta$ antimonene. Present studies on the b-antimonene are based on first-principles calculations, and no empirical potential has been proposed for the b-antimonene. We will thus parametrize a set of VFF model for the single-layer b-antimonene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-antimonene.

The structure of the single-layer b-antimonene is shown in Fig. 222, with structural parameters from the ab initio calculations.Xu et al. (2016) The b-antimonene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=1.65$ Å. The lattice constant is 4.12 Å, and the bond length is 2.89 Å.

Table 458 shows the VFF model for the single-layer b-antimonene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 235 (a). The ab initio calculations for the phonon dispersion are from Ref. Xu et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Kamal and Ezawa (2015); Zeraati et al. (2015); Zhang et al. (2016a) Fig. 235 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 459. The parameters for the three-body SW potential used by GULP are shown in Tab. 460. Parameters for the SW potential used by LAMMPS are listed in Tab. 461.

Fig. 236 shows the stress strain relations for the single-layer b-antimonene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 39.6 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-antimonene. The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.24$. The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -62.6 Nm*-1* and -91.5 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 7.1 Nm*-1* at the critical strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.7 Nm*-1* at the critical strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXVI b-bismuthene

The buckled (b-) bismuthene is a Bi allotrope, which is also named $beta$ bismuthene. Most studies on the b-bismuthene are based on first-principles calculations, while a modified Morse potential was proposed for the b-bismuthene in 2013.Cheng et al. (2013) We will parametrize a set of VFF model for the single-layer b-bismuthene in this section. We will also derive the SW potential based on the VFF model for the single-layer b-bismuthene.

The structure of the single-layer b-bismuthene is shown in Fig. 222, with structural parameters from the ab initio calculations.Xu et al. (2016) The b-bismuthene has a buckled configuration as shown in Fig. 222 (b), where the buckle is along the zigzag direction. The height of the buckle is $h=1.73$ Å. The lattice constant is 4.34 Å, and the bond length is 3.045 Å.

Table 462 shows the VFF model for the single-layer b-bismuthene. The force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 237 (a). The ab initio calculations for the phonon dispersion are from Ref. Zhang et al., 2016a. Similar phonon dispersion can also be found in other ab initio calculations.Akturk, Akturk, and Ciraci (2016) We note that the lowest-frequency branch aroung the $\Gamma$ point from the VFF model is lower than the ab initio results. This branch is the flexural branch, which should be a quadratic dispersion. However, the ab initio calculations give a linear dispersion for the flexural branch due to the violation of the rigid rotational invariance in the first-principles package,Jiang et al. (2015) so ab initio calculations typically overestimate the frequency of this branch. Fig. 237 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 463. The parameters for the three-body SW potential used by GULP are shown in Tab. 464. Parameters for the SW potential used by LAMMPS are listed in Tab. 465.

Fig. 238 shows the stress strain relations for the single-layer b-bismuthene of size $100\times 100$ Å. The structure is uniaxially stretched in the armchair or zigzag directions at 1 K and 300 K. The Young’s modulus is 27.0 Nm*-1* in both armchair and zigzag directions at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. The Young’s modulus is isotropic for the b-bismuthene. The value of the Young’s modulus is close to the value of 23.9 Nm*-1* from the ab initio calculations.Akturk, Akturk, and Ciraci (2016) The Poisson’s ratios from the VFF model and the SW potential are $\nu_{xy}=\nu_{yx}=0.25$, which are comparable with the ab initio results of 0.327.Akturk, Akturk, and Ciraci (2016) The third-order nonlinear elastic constant $D$ can be obtained by fitting the stress-strain relation to $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with E as the Young’s modulus. The values of $D$ are -34.3 Nm*-1* and -54.5 Nm*-1* at 1 K along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 Nm*-1* at the critical strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 Nm*-1* at the critical strain of 0.33 in the zigzag direction at the low temperature of 1 K.

CXVII b-CO

Present studies on the buckled (b-) CO are based on first-principles calculations, and no empirical potential has been proposed for the b-CO. We will thus parametrize a set of SW potential for the single-layer b-CO in this section.

The structure of the single-layer b-CO is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-CO has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.454 Å and the bond length 1.636 Å.

Table 466 shows the VFF model for the single-layer b-CO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 240 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 240 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 467. The parameters for the three-body SW potential used by GULP are shown in Tab. 468. Parameters for the SW potential used by LAMMPS are listed in Tab. 469.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CO under uniaxial tension at 1.0 K and 300.0 K. Fig. 241 shows the stress-strain curve for the tension of a single-layer b-CO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 99.1 N/m and 98.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.08$.

There is no available value for nonlinear quantities in the single-layer b-CO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -513.8 N/m and -542.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.4 Nm*-1* at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.2 Nm*-1* at the ultimate strain of 0.21 in the zigzag direction at the low temperature of 1 K.

CXVIII b-CS

Present studies on the buckled (b-) CS are based on first-principles calculations, and no empirical potential has been proposed for the b-CS. We will thus parametrize a set of SW potential for the single-layer b-CS in this section.

The structure of the single-layer b-CS is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-CS has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.836 Å and the bond length 1.880 Å.

Table 470 shows the VFF model for the single-layer b-CS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 242 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 242 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 471. The parameters for the three-body SW potential used by GULP are shown in Tab. 472. Parameters for the SW potential used by LAMMPS are listed in Tab. 473.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CS under uniaxial tension at 1.0 K and 300.0 K. Fig. 243 shows the stress-strain curve for the tension of a single-layer b-CS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 63.5 N/m and 63.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.05$.

There is no available value for nonlinear quantities in the single-layer b-CS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -352.5 N/m and -372.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.7 Nm*-1* at the ultimate strain of 0.17 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.5 Nm*-1* at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

CXIX b-CSe

Present studies on the buckled (b-) CSe are based on first-principles calculations, and no empirical potential has been proposed for the b-CSe. We will thus parametrize a set of SW potential for the single-layer b-CSe in this section.

The structure of the single-layer b-CSe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-CSe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.063 Å and the bond length 2.055 Å.

Table 474 shows the VFF model for the single-layer b-CSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 244 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 244 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 475. The parameters for the three-body SW potential used by GULP are shown in Tab. 476. Parameters for the SW potential used by LAMMPS are listed in Tab. 477.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 245 shows the stress-strain curve for the tension of a single-layer b-CSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 61.6 N/m and 61.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.09$.

There is no available value for nonlinear quantities in the single-layer b-CSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -306.6 N/m and -324.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

CXX b-CTe

Present studies on the buckled (b-) CTe are based on first-principles calculations, and no empirical potential has been proposed for the b-CTe. We will thus parametrize a set of SW potential for the single-layer b-CTe in this section.

The structure of the single-layer b-CTe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-CTe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.348 Å and the bond length 2.231 Å.

Table 478 shows the VFF model for the single-layer b-CTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 246 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 246 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 479. The parameters for the three-body SW potential used by GULP are shown in Tab. 480. Parameters for the SW potential used by LAMMPS are listed in Tab. 481.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-CTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 247 shows the stress-strain curve for the tension of a single-layer b-CTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-CTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-CTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 48.8 N/m and 48.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.12$.

There is no available value for nonlinear quantities in the single-layer b-CTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -306.6 N/m and -324.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

CXXI b-SiO

Present studies on the buckled (b-) SiO are based on first-principles calculations, and no empirical potential has been proposed for the b-SiO. We will thus parametrize a set of SW potential for the single-layer b-SiO in this section.

The structure of the single-layer b-SiO is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SiO has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 2.815 Å and the bond length 1.884 Å.

Table 482 shows the VFF model for the single-layer b-SiO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 248 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 248 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 483. The parameters for the three-body SW potential used by GULP are shown in Tab. 484. Parameters for the SW potential used by LAMMPS are listed in Tab. 485.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiO under uniaxial tension at 1.0 K and 300.0 K. Fig. 249 shows the stress-strain curve for the tension of a single-layer b-SiO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 51.3 N/m and 50.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.11$.

There is no available value for nonlinear quantities in the single-layer b-SiO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -247.8 N/m and -253.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.2 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.1 Nm*-1* at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

CXXII b-SiS

Present studies on the buckled (b-) SiS are based on first-principles calculations, and no empirical potential has been proposed for the b-SiS. We will thus parametrize a set of SW potential for the single-layer b-SiS in this section.

The structure of the single-layer b-SiS is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SiS has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.299 Å and the bond length 2.321 Å.

Table 486 shows the VFF model for the single-layer b-SiS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 250 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 250 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 487. The parameters for the three-body SW potential used by GULP are shown in Tab. 488. Parameters for the SW potential used by LAMMPS are listed in Tab. 489.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiS under uniaxial tension at 1.0 K and 300.0 K. Fig. 251 shows the stress-strain curve for the tension of a single-layer b-SiS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 45.5 N/m and 45.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.13$.

There is no available value for nonlinear quantities in the single-layer b-SiS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -196.4 N/m and -217.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.1 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

CXXIII b-SiSe

Present studies on the buckled (b-) SiSe are based on first-principles calculations, and no empirical potential has been proposed for the b-SiSe. We will thus parametrize a set of SW potential for the single-layer b-SiSe in this section.

The structure of the single-layer b-SiSe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SiSe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.521 Å and the bond length 2.477 Å.

Table 490 shows the VFF model for the single-layer b-SiSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 252 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 252 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 491. The parameters for the three-body SW potential used by GULP are shown in Tab. 492. Parameters for the SW potential used by LAMMPS are listed in Tab. 493.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 253 shows the stress-strain curve for the tension of a single-layer b-SiSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 41.8 N/m and 41.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer b-SiSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -169.9 N/m and -188.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.7 Nm*-1* at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

CXXIV b-SiTe

Present studies on the buckled (b-) SiTe are based on first-principles calculations, and no empirical potential has been proposed for the b-SiTe. We will thus parametrize a set of SW potential for the single-layer b-SiTe in this section.

The structure of the single-layer b-SiTe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Chen, Sun, and Jena (2016) The b-SiTe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 3.83 Å and the bond length 2.689 Å. The resultant height of the buckle is $h=1.53$ Å.

Table 494 shows the VFF model for the single-layer b-SiTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 254 (a). The ab initio calculations for the phonon dispersion are from Ref. Chen, Sun, and Jena, 2016. Fig. 254 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 495. The parameters for the three-body SW potential used by GULP are shown in Tab. 496. Parameters for the SW potential used by LAMMPS are listed in Tab. 497.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 255 shows the stress-strain curve for the tension of a single-layer b-SiTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 34.3 N/m and 34.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values agrees with the ab initio result at 0 K temperature, eg. 34.1 Nm*-1* in Ref. Chen, Sun, and Jena, 2016. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.18$, which agrees with the ab initio resultChen, Sun, and Jena (2016) of 0.18.

There is no available value for nonlinear quantities in the single-layer b-SiTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -119.3 N/m and -137.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.6 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.4 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

CXXV b-GeO

Present studies on the buckled (b-) GeO are based on first-principles calculations, and no empirical potential has been proposed for the b-GeO. We will thus parametrize a set of SW potential for the single-layer b-GeO in this section.

The structure of the single-layer b-GeO is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-GeO has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.124 Å and the bond length 2.032 Å.

Table 498 shows the VFF model for the single-layer b-GeO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 256 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 256 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 499. The parameters for the three-body SW potential used by GULP are shown in Tab. 500. Parameters for the SW potential used by LAMMPS are listed in Tab. 501.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeO under uniaxial tension at 1.0 K and 300.0 K. Fig. 257 shows the stress-strain curve for the tension of a single-layer b-GeO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 47.5 N/m and 46.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.11$.

There is no available value for nonlinear quantities in the single-layer b-GeO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -224.6 N/m and -232.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.8 Nm*-1* at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

CXXVI b-GeS

Present studies on the buckled (b-) GeS are based on first-principles calculations, and no empirical potential has been proposed for the b-GeS. We will thus parametrize a set of SW potential for the single-layer b-GeS in this section.

The structure of the single-layer b-GeS is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-GeS has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.485 Å and the bond length 2.428 Å.

Table 502 shows the VFF model for the single-layer b-GeS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 258 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 258 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 503. The parameters for the three-body SW potential used by GULP are shown in Tab. 504. Parameters for the SW potential used by LAMMPS are listed in Tab. 505.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeS under uniaxial tension at 1.0 K and 300.0 K. Fig. 259 shows the stress-strain curve for the tension of a single-layer b-GeS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 34.9 N/m and 34.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.18$.

There is no available value for nonlinear quantities in the single-layer b-GeS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -128.1 N/m and -135.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.5 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.3 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

CXXVII b-GeSe

Present studies on the buckled (b-) GeSe are based on first-principles calculations, and no empirical potential has been proposed for the b-GeSe. We will thus parametrize a set of SW potential for the single-layer b-GeSe in this section.

The structure of the single-layer b-GeSe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-GeSe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.676 Å and the bond length 2.568 Å.

Table 506 shows the VFF model for the single-layer b-GeSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 260 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 260 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 507. The parameters for the three-body SW potential used by GULP are shown in Tab. 508. Parameters for the SW potential used by LAMMPS are listed in Tab. 509.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 261 shows the stress-strain curve for the tension of a single-layer b-GeSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 31.6 N/m and 31.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer b-GeSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -105.2 N/m and -118.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.4 Nm*-1* at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.2 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

CXXVIII b-GeTe

Present studies on the buckled (b-) GeTe are based on first-principles calculations, and no empirical potential has been proposed for the b-GeTe. We will thus parametrize a set of SW potential for the single-layer b-GeTe in this section.

The structure of the single-layer b-GeTe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-GeTe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.939 Å and the bond length 2.768 Å.

Table 510 shows the VFF model for the single-layer b-GeTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 262 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 262 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 511. The parameters for the three-body SW potential used by GULP are shown in Tab. 512. Parameters for the SW potential used by LAMMPS are listed in Tab. 513.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GeTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 263 shows the stress-strain curve for the tension of a single-layer b-GeTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GeTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GeTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 27.7 N/m and 28.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.21$.

There is no available value for nonlinear quantities in the single-layer b-GeTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -80.4 N/m and -95.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.3 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.1 Nm*-1* at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

CXXIX b-SnO

Present studies on the buckled (b-) SnO are based on first-principles calculations, and no empirical potential has been proposed for the b-SnO. We will thus parametrize a set of SW potential for the single-layer b-SnO in this section.

The structure of the single-layer b-SnO is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SnO has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.442 Å and the bond length 2.204 Å.

Table 514 shows the VFF model for the single-layer b-SnO. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 264 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 264 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 515. The parameters for the three-body SW potential used by GULP are shown in Tab. 516. Parameters for the SW potential used by LAMMPS are listed in Tab. 517.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnO under uniaxial tension at 1.0 K and 300.0 K. Fig. 265 shows the stress-strain curve for the tension of a single-layer b-SnO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 43.7 N/m and 43.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.12$.

There is no available value for nonlinear quantities in the single-layer b-SnO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -199.9 N/m and -215.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.7 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 4.5 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

CXXX b-SnS

Present studies on the buckled (b-) SnS are based on first-principles calculations, and no empirical potential has been proposed for the b-SnS. We will thus parametrize a set of SW potential for the single-layer b-SnS in this section.

The structure of the single-layer b-SnS is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SnS has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.757 Å and the bond length 2.616 Å.

Table 518 shows the VFF model for the single-layer b-SnS. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 266 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 266 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 519. The parameters for the three-body SW potential used by GULP are shown in Tab. 520. Parameters for the SW potential used by LAMMPS are listed in Tab. 521.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnS under uniaxial tension at 1.0 K and 300.0 K. Fig. 267 shows the stress-strain curve for the tension of a single-layer b-SnS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 23.8 N/m and 24.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.20$.

There is no available value for nonlinear quantities in the single-layer b-SnS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -71.8 N/m and -88.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.4 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

CXXXI b-SnSe

Present studies on the buckled (b-) SnSe are based on first-principles calculations, and no empirical potential has been proposed for the b-SnSe. We will thus parametrize a set of SW potential for the single-layer b-SnSe in this section.

The structure of the single-layer b-SnSe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SnSe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 3.916 Å and the bond length 2.747 Å.

Table 522 shows the VFF model for the single-layer b-SnSe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 268 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 268 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 523. The parameters for the three-body SW potential used by GULP are shown in Tab. 524. Parameters for the SW potential used by LAMMPS are listed in Tab. 525.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 269 shows the stress-strain curve for the tension of a single-layer b-SnSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 22.0 N/m and 22.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer b-SnSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -61.6 N/m and -73.69 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.5 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXXXII b-SnTe

Present studies on the buckled (b-) SnTe are based on first-principles calculations, and no empirical potential has been proposed for the b-SnTe. We will thus parametrize a set of SW potential for the single-layer b-SnTe in this section.

The structure of the single-layer b-SnTe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Kamal, Chakrabarti, and Ezawa (2016) The b-SnTe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, including the lattice constant 4.151 Å and the bond length 2.947 Å.

Table 526 shows the VFF model for the single-layer b-SnTe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 270 (a). The ab initio calculations for the phonon dispersion are calculated from the SIESTA package.Soler et al. (2002) The generalized gradients approximation is applied to account for the exchange-correlation function with Perdew, Burke, and Ernzerhof parameterization,Perdew, Burke, and Ernzerhof (1996) and the double-$\zeta$ orbital basis set is adopted. Fig. 270 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 527. The parameters for the three-body SW potential used by GULP are shown in Tab. 528. Parameters for the SW potential used by LAMMPS are listed in Tab. 529.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 271 shows the stress-strain curve for the tension of a single-layer b-SnTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus are 19.6 N/m and 19.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer b-SnTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -48.7 N/m and -54.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 3.4 Nm*-1* at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 3.3 Nm*-1* at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

CXXXIII b-SnGe

Present studies on the buckled SnGe (b-SnGe) are based on first-principles calculations, and no empirical potential has been proposed for the b-SnGe. We will thus parametrize a set of SW potential for the single-layer b-SnGe in this section.

The structure of the single-layer b-SnGe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-SnGe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.27 Å and the bond length 2.57 Å. The resultant height of the buckle is $h=0.73$ Å.

Table 530 shows the VFF model for the single-layer b-SnGe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 272 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 272 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 531. The parameters for the three-body SW potential used by GULP are shown in Tab. 532. Parameters for the SW potential used by LAMMPS are listed in Tab. 533.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnGe under uniaxial tension at 1.0 K and 300.0 K. Fig. 273 shows the stress-strain curve for the tension of a single-layer b-SnGe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnGe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnGe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 36.8 N/m along both armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.11$.

There is no available value for nonlinear quantities in the single-layer b-SnGe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -171.6 N/m and -197.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 4.9 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.0 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXXXIV b-SiGe

Present studies on the buckled SiGe (b-SiGe) are based on first-principles calculations, and no empirical potential has been proposed for the b-SiGe. We will thus parametrize a set of SW potential for the single-layer b-SiGe in this section.

The structure of the single-layer b-SiGe is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-SiGe has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 3.89 Å and the bond length 2.31 Å. The resultant height of the buckle is $h=0.55$ Å.

Table 534 shows the VFF model for the single-layer b-SiGe. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 274 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 274 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 535. The parameters for the three-body SW potential used by GULP are shown in Tab. 536. Parameters for the SW potential used by LAMMPS are listed in Tab. 537.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SiGe under uniaxial tension at 1.0 K and 300.0 K. Fig. 275 shows the stress-strain curve for the tension of a single-layer b-SiGe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SiGe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SiGe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 54.6 N/m and 54.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$.

There is no available value for nonlinear quantities in the single-layer b-SiGe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -186.7 N/m and -233.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.0 Nm*-1* at the ultimate strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.2 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXXXV b-SnSi

Present studies on the buckled SnSi (b-SnSi) are based on first-principles calculations, and no empirical potential has been proposed for the b-SnSi. We will thus parametrize a set of SW potential for the single-layer b-SnSi in this section.

The structure of the single-layer b-SnSi is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-SnSi has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.21 Å and the bond length 2.52 Å. The resultant height of the buckle is $h=0.67$ Å.

Table 538 shows the VFF model for the single-layer b-SnSi. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 276 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 276 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 539. The parameters for the three-body SW potential used by GULP are shown in Tab. 540. Parameters for the SW potential used by LAMMPS are listed in Tab. 541.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-SnSi under uniaxial tension at 1.0 K and 300.0 K. Fig. 277 shows the stress-strain curve for the tension of a single-layer b-SnSi of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-SnSi is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-SnSi. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.0 N/m and 38.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.14$.

There is no available value for nonlinear quantities in the single-layer b-SnSi. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -150.5 N/m and -174.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.5 Nm*-1* at the ultimate strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.6 Nm*-1* at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

CXXXVI b-InP

Present studies on the buckled InP (b-InP) are based on first-principles calculations, and no empirical potential has been proposed for the b-InP. We will thus parametrize a set of SW potential for the single-layer b-InP in this section.

The structure of the single-layer b-InP is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-InP has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.17 Å and the bond length 2.46 Å. The resultant height of the buckle is $h=0.51$ Å.

Table 542 shows the VFF model for the single-layer b-InP. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 278 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 278 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 543. The parameters for the three-body SW potential used by GULP are shown in Tab. 544. Parameters for the SW potential used by LAMMPS are listed in Tab. 545.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InP under uniaxial tension at 1.0 K and 300.0 K. Fig. 279 shows the stress-strain curve for the tension of a single-layer b-InP of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InP is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InP. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.3 N/m and 38.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer b-InP. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -119.3 N/m and -132.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.2 Nm*-1* at the ultimate strain of 0.35 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.1 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXXXVII b-InAs

Present studies on the buckled InAs (b-InAs) are based on first-principles calculations, and no empirical potential has been proposed for the b-InAs. We will thus parametrize a set of SW potential for the single-layer b-InAs in this section.

The structure of the single-layer b-InAs is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-InAs has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.28 Å and the bond length 2.55 Å. The resultant height of the buckle is $h=0.62$ Å.

Table 546 shows the VFF model for the single-layer b-InAs. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 280 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 280 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 547. The parameters for the three-body SW potential used by GULP are shown in Tab. 548. Parameters for the SW potential used by LAMMPS are listed in Tab. 549.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InAs under uniaxial tension at 1.0 K and 300.0 K. Fig. 281 shows the stress-strain curve for the tension of a single-layer b-InAs of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InAs is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InAs. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 33.9 N/m and 34.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer b-InAs. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -85.0 N/m and -130.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.3 Nm*-1* at the ultimate strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.5 Nm*-1* at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

CXXXVIII b-InSb

Present studies on the buckled InSb (b-InSb) are based on first-principles calculations, and no empirical potential has been proposed for the b-InSb. We will thus parametrize a set of SW potential for the single-layer b-InSb in this section.

The structure of the single-layer b-InSb is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-InSb has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.57 Å and the bond length 2.74 Å. The resultant height of the buckle is $h=0.73$ Å.

Table 550 shows the VFF model for the single-layer b-InSb. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 282 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 282 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 551. The parameters for the three-body SW potential used by GULP are shown in Tab. 552. Parameters for the SW potential used by LAMMPS are listed in Tab. 553.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-InSb under uniaxial tension at 1.0 K and 300.0 K. Fig. 283 shows the stress-strain curve for the tension of a single-layer b-InSb of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-InSb is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-InSb. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 28.6 N/m and 28.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.17$.

There is no available value for nonlinear quantities in the single-layer b-InSb. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -85.4 N/m and -121.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.8 Nm*-1* at the ultimate strain of 0.31 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.0 Nm*-1* at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

CXXXIX b-GaAs

Present studies on the buckled GaAs (b-GaAs) are based on first-principles calculations, and no empirical potential has been proposed for the b-GaAs. We will thus parametrize a set of SW potential for the single-layer b-GaAs in this section.

The structure of the single-layer b-GaAs is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-GaAs has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 3.97 Å and the bond length 2.36 Å. The resultant height of the buckle is $h=0.55$ Å.

Table 554 shows the VFF model for the single-layer b-GaAs. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 284 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 284 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 555. The parameters for the three-body SW potential used by GULP are shown in Tab. 556. Parameters for the SW potential used by LAMMPS are listed in Tab. 557.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GaAs under uniaxial tension at 1.0 K and 300.0 K. Fig. 285 shows the stress-strain curve for the tension of a single-layer b-GaAs of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GaAs is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GaAs. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 50.5 N/m and 50.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.13$.

There is no available value for nonlinear quantities in the single-layer b-GaAs. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -199.5 N/m and -258.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.3 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 Nm*-1* at the ultimate strain of 0.30 in the zigzag direction at the low temperature of 1 K.

CXL b-GaP

Present studies on the buckled GaP (b-GaP) are based on first-principles calculations, and no empirical potential has been proposed for the b-GaP. We will thus parametrize a set of SW potential for the single-layer b-GaP in this section.

The structure of the single-layer b-GaP is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-GaP has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 3.84 Å and the bond length 2.25 Å. The resultant height of the buckle is $h=0.40$ Å.

Table 558 shows the VFF model for the single-layer b-GaP. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 286 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 286 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 559. The parameters for the three-body SW potential used by GULP are shown in Tab. 560. Parameters for the SW potential used by LAMMPS are listed in Tab. 561.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-GaP under uniaxial tension at 1.0 K and 300.0 K. Fig. 287 shows the stress-strain curve for the tension of a single-layer b-GaP of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-GaP is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-GaP. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 57.2 N/m and 57.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.14$.

There is no available value for nonlinear quantities in the single-layer b-GaP. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -186.4 N/m and -261.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.3 Nm*-1* at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 Nm*-1* at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

CXLI b-AlSb

Present studies on the buckled AlSb (b-AlSb) are based on first-principles calculations, and no empirical potential has been proposed for the b-AlSb. We will thus parametrize a set of SW potential for the single-layer b-AlSb in this section.

The structure of the single-layer b-AlSb is shown in Fig. 239. The structural parameters are from the ab initio calculations.Sahin et al. (2009) The b-AlSb has a buckled configuration as shown in Fig. 239 (b), where the buckle is along the zigzag direction. This structure can be determined by two independent geometrical parameters, eg. the lattice constant 4.33 Å and the bond length 2.57 Å. The resultant height of the buckle is $h=0.60$ Å.

Table 562 shows the VFF model for the single-layer b-AlSb. The force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 288 (a). The ab initio calculations for the phonon dispersion are from Ref. Sahin et al., 2009. Fig. 288 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 563. The parameters for the three-body SW potential used by GULP are shown in Tab. 564. Parameters for the SW potential used by LAMMPS are listed in Tab. 565.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer b-AlSb under uniaxial tension at 1.0 K and 300.0 K. Fig. 289 shows the stress-strain curve for the tension of a single-layer b-AlSb of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer b-AlSb is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer b-AlSb. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 41.7 N/m and 42.0 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.15$.

There is no available value for nonlinear quantities in the single-layer b-AlSb. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -142.4 N/m and -190.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.1 Nm*-1* at the ultimate strain of 0.29 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.1 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CXLII BO

Present studies on the BO are based on first-principles calculations, and no empirical potential has been proposed for the BO. We will thus parametrize a set of SW potential for the single-layer BO in this section.

The structure of the single-layer BO is shown in Fig. 290 with M=B and X=O. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The BO has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 2.44 Å, the bond length $d_{\rm B-O}=1.52$ Å, and the bond length $d_{\rm B-B}=1.77$ Å.

Table 566 shows the VFF model for the single-layer BO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 291 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 291 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 567. The parameters for the three-body SW potential used by GULP are shown in Tab. 568. Parameters for the SW potential used by LAMMPS are listed in Tab. 569.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BO under uniaxial tension at 1.0 K and 300.0 K. Fig. 292 shows the stress-strain curve for the tension of a single-layer BO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 299.6 N/m and 297.7 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.11$.

There is no available value for nonlinear quantities in the single-layer BO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -1554.7 N/m and -1585.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 28.9 Nm*-1* at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 28.2 Nm*-1* at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

CXLIII AlO

Present studies on the AlO are based on first-principles calculations, and no empirical potential has been proposed for the AlO. We will thus parametrize a set of SW potential for the single-layer AlO in this section.

The structure of the single-layer AlO is shown in Fig. 290 with M=Al and X=O. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The AlO has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 2.96 Å, the bond length $d_{\rm Al-O}=1.83$ Å, and the bond length $d_{\rm Al-Al}=2.62$ Å.

Table 570 shows the VFF model for the single-layer AlO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 293 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 293 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 571. The parameters for the three-body SW potential used by GULP are shown in Tab. 572. Parameters for the SW potential used by LAMMPS are listed in Tab. 573.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlO under uniaxial tension at 1.0 K and 300.0 K. Fig. 294 shows the stress-strain curve for the tension of a single-layer AlO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 149.3 N/m and 148.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer AlO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -563.9 N/m and -565.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.6 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.4 Nm*-1* at the ultimate strain of 0.34 in the zigzag direction at the low temperature of 1 K.

CXLIV GaO

Present studies on the GaO are based on first-principles calculations, and no empirical potential has been proposed for the GaO. We will thus parametrize a set of SW potential for the single-layer GaO in this section.

The structure of the single-layer GaO is shown in Fig. 290 with M=Ga and X=O. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The GaO has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.12 Å, the bond length $d_{\rm Ga-O}=1.94$ Å, and the bond length $d_{\rm Ga-Ga}=2.51$ Å.

Table 574 shows the VFF model for the single-layer GaO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 295 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 295 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 575. The parameters for the three-body SW potential used by GULP are shown in Tab. 576. Parameters for the SW potential used by LAMMPS are listed in Tab. 577.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaO under uniaxial tension at 1.0 K and 300.0 K. Fig. 296 shows the stress-strain curve for the tension of a single-layer GaO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 137.2 N/m and 136.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer GaO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -467.5 N/m and -529.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.6 Nm*-1* at the ultimate strain of 0.28 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.8 Nm*-1* at the ultimate strain of 0.35 in the zigzag direction at the low temperature of 1 K.

CXLV InO

Present studies on the InO are based on first-principles calculations, and no empirical potential has been proposed for the InO. We will thus parametrize a set of SW potential for the single-layer InO in this section.

The structure of the single-layer InO is shown in Fig. 290 with M=In and X=O. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The InO has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.48 Å, the bond length $d_{\rm In-O}=2.16$ Å, and the bond length $d_{\rm In-In}=2.86$ Å.

Table 578 shows the VFF model for the single-layer InO. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 297 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 297 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 579. The parameters for the three-body SW potential used by GULP are shown in Tab. 580. Parameters for the SW potential used by LAMMPS are listed in Tab. 581.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InO under uniaxial tension at 1.0 K and 300.0 K. Fig. 298 shows the stress-strain curve for the tension of a single-layer InO of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InO is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InO. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 85.7 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.29$.

There is no available value for nonlinear quantities in the single-layer InO. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -157.3 N/m and -210.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.9 Nm*-1* at the ultimate strain of 0.38 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 23.6 Nm*-1* at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

CXLVI BS

Present studies on the BS are based on first-principles calculations, and no empirical potential has been proposed for the BS. We will thus parametrize a set of SW potential for the single-layer BS in this section.

The structure of the single-layer BS is shown in Fig. 290 with M=B and X=S. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The BS has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.03 Å, the bond length $d_{\rm B-S}=1.94$ Å, and the bond length $d_{\rm B-B}=1.72$ Å.

Table 582 shows the VFF model for the single-layer BS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 299 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 299 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 583. The parameters for the three-body SW potential used by GULP are shown in Tab. 584. Parameters for the SW potential used by LAMMPS are listed in Tab. 585.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BS under uniaxial tension at 1.0 K and 300.0 K. Fig. 300 shows the stress-strain curve for the tension of a single-layer BS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 179.4 N/m and 178.5 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$.

There is no available value for nonlinear quantities in the single-layer BS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -793.2 N/m and -823.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 20.0 Nm*-1* at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 19.5 Nm*-1* at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

CXLVII AlS

Present studies on the AlS are based on first-principles calculations, and no empirical potential has been proposed for the AlS. We will thus parametrize a set of SW potential for the single-layer AlS in this section.

The structure of the single-layer AlS is shown in Fig. 290 with M=Al and X=S. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The AlS has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.57 Å, the bond length $d_{\rm Al-S}=2.32$ Å, and the bond length $d_{\rm Al-Al}=2.59$ Å.

Table 586 shows the VFF model for the single-layer AlS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 301 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 301 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 587. The parameters for the three-body SW potential used by GULP are shown in Tab. 588. Parameters for the SW potential used by LAMMPS are listed in Tab. 589.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlS under uniaxial tension at 1.0 K and 300.0 K. Fig. 302 shows the stress-strain curve for the tension of a single-layer AlS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 85.2 N/m and 84.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.22$.

There is no available value for nonlinear quantities in the single-layer AlS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -289.7 N/m and -302.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.9 Nm*-1* at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.9 Nm*-1* at the ultimate strain of 0.36 in the zigzag direction at the low temperature of 1 K.

CXLVIII GaS

Present studies on the GaS are based on first-principles calculations, and no empirical potential has been proposed for the GaS. We will thus parametrize a set of SW potential for the single-layer GaS in this section.

The structure of the single-layer GaS is shown in Fig. 290 with M=Ga and X=S. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The GaS has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.64 Å, the bond length $d_{\rm Ga-S}=2.36$ Å, and the bond length $d_{\rm Ga-Ga}=2.47$ Å.

Table 590 shows the VFF model for the single-layer GaS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 303 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 303 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 591. The parameters for the three-body SW potential used by GULP are shown in Tab. 592. Parameters for the SW potential used by LAMMPS are listed in Tab. 593.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaS under uniaxial tension at 1.0 K and 300.0 K. Fig. 304 shows the stress-strain curve for the tension of a single-layer GaS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 76.2 N/m along the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer GaS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materigas. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -254.5 N/m and -269.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.0 Nm*-1* at the ultimate strain of 0.36 in the zigzag direction at the low temperature of 1 K.

CXLIX InS

Present studies on the InS are based on first-principles calculations, and no empirical potential has been proposed for the InS. We will thus parametrize a set of SW potential for the single-layer InS in this section.

The structure of the single-layer InS is shown in Fig. 290 with M=In and X=S. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The InS has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.94 Å, the bond length $d_{\rm In-S}=2.56$ Å, and the bond length $d_{\rm In-In}=2.82$ Å.

Table 594 shows the VFF model for the single-layer InS. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 305 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 305 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 595. The parameters for the three-body SW potential used by GULP are shown in Tab. 596. Parameters for the SW potential used by LAMMPS are listed in Tab. 597.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InS under uniaxial tension at 1.0 K and 300.0 K. Fig. 306 shows the stress-strain curve for the tension of a single-layer InS of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InS is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InS. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 52.9 N/m and 53.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.29$.

There is no available value for nonlinear quantities in the single-layer InS. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -115.9 N/m and -141.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.2 Nm*-1* at the ultimate strain of 0.32 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.8 Nm*-1* at the ultimate strain of 0.42 in the zigzag direction at the low temperature of 1 K.

CL BSe

Present studies on the BSe are based on first-principles calculations, and no empirical potential has been proposed for the BSe. We will thus parametrize a set of SW potential for the single-layer BSe in this section.

The structure of the single-layer BSe is shown in Fig. 290 with M=B and X=Se. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The BSe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.25 Å, the bond length $d_{\rm B-Se}=2.10$ Å, and the bond length $d_{\rm B-B}=1.71$ Å.

Table 598 shows the VFF model for the single-layer BSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 307 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 307 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 599. The parameters for the three-body SW potential used by GULP are shown in Tab. 600. Parameters for the SW potential used by LAMMPS are listed in Tab. 601.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 308 shows the stress-strain curve for the tension of a single-layer BSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 157.3 N/m and 156.4 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.19$.

There is no available value for nonlinear quantities in the single-layer BSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -627.0 N/m and -655.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.2 Nm*-1* at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 18.7 Nm*-1* at the ultimate strain of 0.31 in the zigzag direction at the low temperature of 1 K.

CLI AlSe

Present studies on the AlSe are based on first-principles calculations, and no empirical potential has been proposed for the AlSe. We will thus parametrize a set of SW potential for the single-layer AlSe in this section.

The structure of the single-layer AlSe is shown in Fig. 290 with M=Al and X=Se. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The AlSe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.78 Å, the bond length $d_{\rm Al-Se}=2.47$ Å, and the bond length $d_{\rm Al-Al}=2.57$ Å.

Table 602 shows the VFF model for the single-layer AlSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 309 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 309 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 603. The parameters for the three-body SW potential used by GULP are shown in Tab. 604. Parameters for the SW potential used by LAMMPS are listed in Tab. 605.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 310 shows the stress-strain curve for the tension of a single-layer AlSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 69.4 N/m and 69.2 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.24$.

There is no available value for nonlinear quantities in the single-layer AlSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -217.3 N/m and -231.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.4 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.5 Nm*-1* at the ultimate strain of 0.38 in the zigzag direction at the low temperature of 1 K.

CLII GaSe

Present studies on the GaSe are based on first-principles calculations, and no empirical potential has been proposed for the GaSe. We will thus parametrize a set of SW potential for the single-layer GaSe in this section.

The structure of the single-layer GaSe is shown in Fig. 290 with M=Ga and X=Se. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The GaSe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.82 Å, the bond length $d_{\rm Ga-Se}=2.50$ Å, and the bond length $d_{\rm Ga-Ga}=2.46$ Å.

Table 606 shows the VFF model for the single-layer GaSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 311 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 311 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 607. The parameters for the three-body SW potential used by GULP are shown in Tab. 608. Parameters for the SW potential used by LAMMPS are listed in Tab. 609.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 312 shows the stress-strain curve for the tension of a single-layer GaSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 68.3 N/m and 67.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.25$.

There is no available value for nonlinear quantities in the single-layer GaSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -206.1 N/m and -219.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 10.5 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 Nm*-1* at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

CLIII InSe

Present studies on the InSe are based on first-principles calculations, and no empirical potential has been proposed for the InSe. We will thus parametrize a set of SW potential for the single-layer InSe in this section.

The structure of the single-layer InSe is shown in Fig. 290 with M=In and X=Se. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The InSe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 4.10 Å, the bond length $d_{\rm In-Se}=2.69$ Å, and the bond length $d_{\rm In-In}=2.81$ Å.

Table 610 shows the VFF model for the single-layer InSe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 313 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 313 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 611. The parameters for the three-body SW potential used by GULP are shown in Tab. 612. Parameters for the SW potential used by LAMMPS are listed in Tab. 613.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InSe under uniaxial tension at 1.0 K and 300.0 K. Fig. 314 shows the stress-strain curve for the tension of a single-layer InSe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InSe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InSe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 45.7 N/m and 45.8 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.30$.

There is no available value for nonlinear quantities in the single-layer InSe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -81.6 N/m and -103.5 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.8 Nm*-1* at the ultimate strain of 0.35 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.9 Nm*-1* at the ultimate strain of 0.44 in the zigzag direction at the low temperature of 1 K.

CLIV BTe

Present studies on the BTe are based on first-principles calculations, and no empirical potential has been proposed for the BTe. We will thus parametrize a set of SW potential for the single-layer BTe in this section.

The structure of the single-layer BTe is shown in Fig. 290 with M=B and X=Te. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The BTe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior B-B bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 3.56 Å, the bond length $d_{\rm B-Te}=2.31$ Å, and the bond length $d_{\rm B-B}=1.71$ Å.

Table 614 shows the VFF model for the single-layer BTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 315 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 315 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 615. The parameters for the three-body SW potential used by GULP are shown in Tab. 616. Parameters for the SW potential used by LAMMPS are listed in Tab. 617.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer BTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 316 shows the stress-strain curve for the tension of a single-layer BTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer BTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer BTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 130.6 N/m and 129.7 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.16$.

There is no available value for nonlinear quantities in the single-layer BTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -560.4 N/m and -588.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.9 Nm*-1* at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 14.4 Nm*-1* at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

CLV AlTe

Present studies on the AlTe are based on first-principles calculations, and no empirical potential has been proposed for the AlTe. We will thus parametrize a set of SW potential for the single-layer AlTe in this section.

The structure of the single-layer AlTe is shown in Fig. 290 with M=Al and X=Te. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The AlTe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Al-Al bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 4.11 Å, the bond length $d_{\rm Al-Te}=2.70$ Å, and the bond length $d_{\rm Al-Al}=2.58$ Å.

Table 618 shows the VFF model for the single-layer AlTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 317 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 317 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 619. The parameters for the three-body SW potential used by GULP are shown in Tab. 620. Parameters for the SW potential used by LAMMPS are listed in Tab. 621.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer AlTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 318 shows the stress-strain curve for the tension of a single-layer AlTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer AlTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer AlTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 55.8 N/m and 54.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.24$.

There is no available value for nonlinear quantities in the single-layer AlTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -171.4 N/m and -179.0 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.4 Nm*-1* at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 Nm*-1* at the ultimate strain of 0.39 in the zigzag direction at the low temperature of 1 K.

CLVI GaTe

Present studies on the GaTe are based on first-principles calculations, and no empirical potential has been proposed for the GaTe. We will thus parametrize a set of SW potential for the single-layer GaTe in this section.

The structure of the single-layer GaTe is shown in Fig. 290 with M=Ga and X=Te. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The GaTe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior Ga-Ga bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 4.13 Å, the bond length $d_{\rm Ga-Te}=2.70$ Å, and the bond length $d_{\rm Ga-Ga}=2.46$ Å.

Table 622 shows the VFF model for the single-layer GaTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 319 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 319 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 623. The parameters for the three-body SW potential used by GULP are shown in Tab. 624. Parameters for the SW potential used by LAMMPS are listed in Tab. 625.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer GaTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 320 shows the stress-strain curve for the tension of a single-layer GaTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer GaTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer GaTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 55.2 N/m and 55.3 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer GaTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -183.2 N/m and -195.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 7.8 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 Nm*-1* at the ultimate strain of 0.37 in the zigzag direction at the low temperature of 1 K.

CLVII InTe

Present studies on the InTe are based on first-principles calculations, and no empirical potential has been proposed for the InTe. We will thus parametrize a set of SW potential for the single-layer InTe in this section.

The structure of the single-layer InTe is shown in Fig. 290 with M=In and X=Te. The structural parameters are from the ab initio calculations.Demirci et al. (2017) The InTe has a bi-buckled configuration as shown in Fig. 290 (b), where the buckle is along the zigzag direction. Two buckling layers are symmetrically integrated through the interior In-In bonds, forming a bi-buckled configuration. This structure can be determined by three independent geometrical parameters, eg. the lattice constant 4.40 Å, the bond length $d_{\rm In-Te}=2.89$ Å, and the bond length $d_{\rm In-In}=2.81$ Å.

Table 626 shows the VFF model for the single-layer InTe. The force constant parameters are determined by fitting to the six low-frequency branches in the phonon dispersion along the $\Gamma$M as shown in Fig. 321 (a). The ab initio calculations for the phonon dispersion are from Ref. Demirci et al., 2017. Fig. 321 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 627. The parameters for the three-body SW potential used by GULP are shown in Tab. 628. Parameters for the SW potential used by LAMMPS are listed in Tab. 629.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer InTe under uniaxial tension at 1.0 K and 300.0 K. Fig. 322 shows the stress-strain curve for the tension of a single-layer InTe of dimension $100\times 100$ Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer InTe is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer InTe. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 40.6 N/m and 40.9 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is $\nu_{xy}=\nu_{yx}=0.23$.

There is no available value for nonlinear quantities in the single-layer InTe. We have thus used the nonlinear parameter $B=0.5d^{4}$ in Eq. (5), which is close to the value of $B$ in most materials. The value of the third order nonlinear elasticity $D$ can be extracted by fitting the stress-strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$ with $E$ as the Young’s modulus. The values of $D$ from the present SW potential are -130.4 N/m and -142.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 Nm*-1* at the ultimate strain of 0.38 in the zigzag direction at the low temperature of 1 K.

CLVIII Borophene

Most existing theoretical studies on the monolayer of boron atoms (borophene) are based on the first-principles calculations. The ReaxFF force field model was developed for the borophene recently.Le, Mortazavi, and Rabczuk (2016) The present authors have provided the VFF model and the SW potential to describe the atomic interaction within the borophene,Zhou and Jiang (2017) which includes the second-nearest-neighboring interactions. In the present work, we present a more efficient SW potential with only the first-nearest-neighboring interactions.

The structure of the borophene is shown in Fig. 323, with structural parameters from the ab initio calculations.Wang et al. (2016) Borophene has a puckered configuration as shown in Fig. 323 (b), where the pucker is perpendicular to the x-direction. The height of the pucker is $h=0.911$ Å, which is the distance between the top chain and the bottom chain along the out-of-plane z-direction. The two lattice bases are $a_{1}=2.866$ Å and $a_{2}=1.614$ Å for the inplane rectangular unit cell. There are two inequivalent boron atoms in the unit cell. Boron atoms are categorized into the top chain and the bottom chain. The top chain includes atoms like 1, 4, and 7. The bottom chain includes atoms like 2, 3, 5, and 6.

Table 630 shows four VFF terms for the borophene, two of which are the bond stretching interaction shown by Eq. (1) while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the $\Gamma$X as shown in Fig. 324 (a). The ab initio calculations for the phonon dispersion are from Ref. Wang et al., 2016. Similar phonon dispersion can also be found in other ab initio calculations.Pang et al. (2016) Fig. 324 (b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Tab. 631. The parameters for the three-body SW potential used by GULP are shown in Tab. 632. Parameters for the SW potential used by LAMMPS are listed in Tab. 633. We note that twelve atom types have been introduced for the simulation of borophene using LAMMPS, because the angles around atom 1 in Fig. 323 (a) are not distinguishable in LAMMPS. We thus need to differentiate these angles by assigning these six neighboring atoms (2, 3, 4, 5, 6, 7) with different atom types. Fig. 325 shows that twelve atom types are necessary for the purpose of differentiating all six neighbors around one B atom.

Fig. 326 shows the stress strain relations for the borophene of size $100\times 100$ Å. The structure is uniaxially stretched in the x or y directions at 1 K and 300 K. The Young’s modulus is 162.7 Nm*-1* and 385.0 Nm*-1* in the x and y directions respectively at 1 K, which are obtained by linear fitting of the stress strain relations in [0, 0.01]. These values are in good agreement with the ab initio results at 0 K temperature, eg. 170 Nm*-1* and 398 Nm*-1* in Ref. Mannix et al., 2015, or 166 Nm*-1* and 389 Nm*-1* in Ref. Wang et al., 2016, or 163 Nm*-1* and 399 Nm*-1* in Ref. Zhang et al., 2017. Previous ab initio calculations obtained negative Poisson’s ratio for the uniaxial stretching of the borophene in the x and y directions, e.g. -0.02 and -0.04 in Refs Mannix et al., 2015 and Wang et al., 2016. The Poisson’s ratio from the present SW potential are -0.03 and -0.07 along the x and y directions respectively, which are quite comparable with the ab initio results.

The third-order nonlinear constant ($D$) can be obtained by fitting the stress strain relation to the function $\sigma=E\epsilon+\frac{1}{2}D\epsilon^{2}$, with $E$ as the Young’s modulus. The obtained values of $D$ are -1100.1 Nm*-1* and -2173.6 Nm*-1* in the x and y directions, respectively. The ultimate stress is about 12.3 Nm*-1* at the critical strain of 0.17 in the x-direction at the low temperature of 1 K, which agree quite well with the ab initio results at 0 K.Wang et al. (2016); Pang et al. (2016); Zhang et al. (2017) The ultimate stress is about 32.9 Nm*-1* at the critical strain of 0.16 in the y-direction at the low temperature of 1 K, which are quite comparable with ab initio results at 0 K.Wang et al. (2016); Pang et al. (2016); Zhang et al. (2017)

CLIX Conclusion Remarks

As a final remark, we note some major advantages and deficiencies for the SW potential parameters provided in the present work. On the one hand, the key feature of the SW potential is its high efficiency, which is maintained by using minimum potential parameters in the present work, so the interaction range is limited to the first-nearest-neighboring atoms. As a result, the present SW potential parameters are of high computational efficiency. On the other hand, since the interaction is limited to short-range, the optical branches in the phonon spectrum are typically overestimated by the present SW potential. It is because we have ignored the long-range interactions, which contribute mostly to the acoustic phonon branches while have neglectable contribution to the optical phonon branches. The short-range interaction has thus been strengthened to give an accurate description for the acoustic phonon branches and the elastic properties, which leads to the overestimation of the optical phonon branches as a trade off. Hence, there will be systematic overestimation for simulating optical processes using the present SW parameters.

We also note that the mathematical form of the SW potential is not suitable for the atomic-thick planar structures, such as graphene and b-BN, because the SW potential are not able to resist the bending motion of these real planar crystals.Arroyo and Belytschko (2004); Jiang et al. (2013)

In conclusion, we have provided the SW potential parameters for 156 layered crystals. The supplemental resources for all simulations in the present work are available online in Ref. Jia, , including a fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculation with the SW potential using GULP, and files for phonon calculations with the valence force field model using GULP.

Acknowledgements The work is supported by the Recruitment Program of Global Youth Experts of China, the National Natural Science Foundation of China (NSFC) under Grant No. 11504225 and the start-up funding from Shanghai University.