Exotic phase transition and superconductivity in layered titanium-oxypnictides implied by a computational phonon analysis

TL;DR

This study uses ab initio phonon analysis to investigate phase transitions and superconductivity in layered titanium-oxypnictides, revealing a systematic dependence on the pnictogen element that influences the transition mechanism.

Contribution

It provides the first computational phonon analysis explaining the structural phase transition in titanium-oxypnictides and highlights the pnictogen-dependent shift from conventional to exotic mechanisms.

Findings

Explains experimental structure of Na2Ti2Pn2O with lighter pnictogen using electron-phonon theory.

Identifies discrepancies with heavier pnictogen, suggesting alternative mechanisms.

Shows systematic dependence of phase transition on pnictogen element.

Abstract

We applied $ab$ $initio$ phonon analysis to layered titanium-oxypnictides, Na$_2$Ti$_2Pn_2$O ($Pn$ = As, Sb), and found a clear contrast between the cases with lighter/heavier pnictogen in comparisons with experiments. The result completely explains the experimental structure, $C$2/$m$ for $Pn$ = As, within the conventional electron-phonon framework, while there arise discrepancies when the pnictogen gets heavier, being in the same trend for the BaTi$_2Pn_2$O ($Pn$ = As, Sb, Bi) case. The fact implies a systematic dependence on pnictogen to tune the mechanism of the phase transition and superconductivity from conventional to exotic.

Tables

Table 1. Table 1: The results of geometry optimization for undistorted Na 2 Ti P 2 ​ n 2 subscript 𝑃 2 subscript 𝑛 2 {}_{2}Pn_{2} O structure ( P ​ n 𝑃 𝑛 Pn = As, Sb). The relaxed conventional lattice parameters are a 𝑎 a = 4.071128 Å, c 𝑐 c = 15.440517 Å, ( a 𝑎 a = 4.13957 Å, c 𝑐 c = 16.97766 Å) for P ​ n 𝑃 𝑛 Pn = As (Sb). The final enthalpy is -980.0514 Ry (-1323.3103 Ry) for P ​ n 𝑃 𝑛 Pn = As (Sb) per formula unit [Na 2 Ti P 2 ​ n 2 subscript 𝑃 2 subscript 𝑛 2 {}_{2}Pn_{2} O].

	$Pn$ = As ($I$4/$mmm$)
	label	$x$	$y$	$z$	wyckoff
	Na	0.50000	0.50000	0.18146	4$e$
	Ti	0.50000	0.00000	0.00000	4$c$
	As	0.00000	0.00000	0.11752	4$e$
	O	0.50000	0.50000	0.00000	2$b$
	$Pn$ = Sb ($I$4/$mmm$)
	Na	0.50000	0.50000	0.18369	4$e$
	Ti	0.50000	0.00000	0.00000	4$c$
	Sb	0.00000	0.00000	0.12015	4$e$
	O	0.50000	0.50000	0.00000	2$b$

Table 2. Table 2: The results of geometry optimization for Na 2 Ti P 2 ​ n 2 subscript 𝑃 2 subscript 𝑛 2 {}_{2}Pn_{2} O superlattice ( P ​ n 𝑃 𝑛 Pn = As, Sb) obtained by phonon calculations. The conventional lattice parameters after relaxation are a ′ superscript 𝑎 ′ a^{\prime} =11.51310 Å, b ′ superscript 𝑏 ′ b^{\prime} = 11.51244 Å, c ′ superscript 𝑐 ′ c^{\prime} = 8.23431 Å, β 𝛽 \beta = 110.45774 ∘ ( a ′ superscript 𝑎 ′ a^{\prime} = 17.04343 Å, b ′ superscript 𝑏 ′ b^{\prime} = 5.83920 Å, c ′ superscript 𝑐 ′ c^{\prime} = 5.83039 Å) for P ​ n 𝑃 𝑛 Pn = As (Sb). The final enthalpy is -980.0517 Ry (-1323.3120 Ry) for P ​ n 𝑃 𝑛 Pn = As (Sb) per formula unit [Na 2 Ti P 2 ​ n 2 subscript 𝑃 2 subscript 𝑛 2 {}_{2}Pn_{2} O].

	$Pn$ = As ($C$2/$m$)
	label	$x$	$y$	$z$	wyckoff
	Na	0.90907	0.00000	0.63699	4$i$
	Na	0.40943	0.00000	0.63694	4$i$
	Na	0.15932	0.24981	0.63725	8$j$
	Ti	0.13031	0.11979	0.00000	8$j$
	Ti	0.88019	0.36969	0.00000	8$j$
	As	0.80877	0.00000	0.23625	4$i$
	As	0.30933	0.00000	0.23625	4$i$
	As	0.05905	0.24969	0.23623	8$j$
	O	0.00000	0.00000	0.00000	2$a$
	O	0.25000	0.25000	0.00000	4$e$
	O	0.00000	0.50000	0.00000	2$b$
	$Pn$ = Sb ($Cmce$)
	Na	0.18411	0.00000	0.00000	8$d$
	Ti	0.00000	0.23266	0.26812	8$f$
	Sb	0.37883	0.00000	0.00000	8$d$
	O	0.00000	0.00000	0.00000	4$a$

Table 3. Table 3: The results of geometry optimization for experimentally observed Na 2 Ti 2 Sb 2 O superlattice. The conventional lattice parameters after relaxation are a ′ superscript 𝑎 ′ a^{\prime} = 11.73782 Å, b ′ superscript 𝑏 ′ b^{\prime} = 11.73541 Å, and c ′ superscript 𝑐 ′ c^{\prime} = 16.82525 Å. The final enthalpy is -1323.3109 Ry per formula unit [Na 2 Ti 2 Sb 2 O].

	$Pn$ = Sb ($Cmcm$)
	label	$x$	$y$	$z$	wyckoff
	Na	0.00000	0.37498	0.93293	8$f$
	Na	0.00000	0.12503	0.43294	8$f$
	Na	0.74999	0.12500	0.93275	16$h$
	Ti	0.37417	0.00083	0.25000	8$g$
	Ti	0.62584	0.24918	0.25000	8$g$
	Ti	0.87581	-0.00085	0.25000	8$g$
	Ti	0.12418	0.25082	0.25000	8$g$
	Sb	0.00000	0.37533	0.12909	8$f$
	Sb	0.00000	0.12532	0.62908	8$f$
	Sb	0.75033	0.12499	0.12907	16$h$
	O	0.00000	0.62500	0.25000	4$c$
	O	0.00000	0.12500	0.25000	4$c$
	O	0.75000	0.37500	0.25000	8$g$

Full text

PACS numbers: 71.45.Lr, 74.25.Kc, 74.70.-b

I Introduction

Recently discovered layered titanium-oxypnictides, $A$Ti${}_{2}Pn_{2}$O [$A=$ Na2, Ba, (SrF)2, (SmO)2; $Pn=$ As, Sb, Bi]  III et al. (1997); Ozawa et al. (2001); Liu et al. (2009); Wang et al. (2010); Doan et al. (2012); Yajima et al. (2012, 2013a, 2013b); Zhai et al. (2013); Nakano et al. (2013); Pachmayr and Johrendt (2014); von Rohr et al. (2014), is a new superconducting family, and their superconducting mechanisms have attracted intensive attentions because their crystal and electronic structures are similar to the exotic superconductors such as cuprates Bednorz and Müller (1986) or iron arsenides Kamihara et al. (2008). The mechanism has been expected to be conventional, namely electron-phonon driven Subedi (2013); von Rohr et al. (2013); Kitagawa et al. (2013); Nozaki et al. (2013), and accompanying singularities in resistivity and susceptibility at low temperature are regarded due to the conventional charge density wave (CDW) Kitagawa et al. (2013); von Rohr et al. (2013); Nozaki et al. (2013); Gooch et al. (2013); Tan et al. (2015). Recent works are, however, pointing out the possibility of exotic mechanism for these compounds: For BaTi${}_{2}Pn_{2}$O ($Pn$ = As, Sb), Frandsen et al. reported the breaking four-fold symmetry at low temperature without any superlattice peaks by Neutron diffraction and TEM Frandsen et al. (2014), and suggested the possibility of the intra-unit-cell nematic CDW Lawler et al. (2010); Fujita et al. (2014) to make account for the breaking. A theoretical suggestion was made that such an intra-unit-cell nematic CDW could be realized by the orbital ordering mediated by spin-fluctuations Nakaoka et al. (2016). The Uemura classification scheme Uemura et al. (1991) also supports exotic superconducting mechanisms in these compounds Kamusella et al. (2014). Though our latest phonon evaluations Nakano et al. (2016) partly refuted that the breaking could be explained within the conventional electron-phonon mechanism at least for $Pn$ = As, it is still unknown if this is the case for $Pn$ = Sb and Bi, leaving debates about whether the mechanism is conventional or exotic.

Examining whether the observed superlattices are explained by the conventional ab initio framework or not can provide a critical clue for the questions. For experimental side, Davies et al. Davies et al. (2016) have just synthesized large single crystals of Na2Ti${}_{2}Pn_{2}$O ($Pn$ = As, Sb) (Fig. 1) in order to achieve reliable diffaction measurements, and observed clear superlattice peaks of $2\times 2\times 2$ and $2\times 2\times 1$ for $Pn$ = As and Sb, respectively. For $Pn$ = As, a recent phonon calculationChen et al. (2016) reported possible superlattices but they could not explain the observed peaks. In this paper, we report that our ab initio phonon calculations can explain the observed $C$2/$m$ (monoclinic No.12) structure for $Pn$ = As, implying that the conventional electron-phonon driven mechanism is likely for this system. For $Pn$ = Sb, on the other hand, we obtained $Cmce$ (Orthorhombic No.64) superlattice, being inconsistent with the observed one [$Cmcm$ (Orthorhombic No.63)]. The discrepancy would support exotic mechanisms likely for $Pn$ = Sb in contrast. Based on the results combined with preceding studiesKuroki et al. (2009); Miyake et al. (2010); Singh (2012); Subedi (2013); Yan and Lu (2013); Nakano et al. (2016), we propose a new possibility that a heavier $Pn$ causes stronger electron correlations and induces exotic phase transition and superconductivity that cannot be explained by the conventional framework.

II Method

All the calculations were done within DFT using GGA-PBE exchange-correlation functionals Perdew et al. (1996), implemented in Quantum Espresso package. Giannozzi et al. (2009) We adopted PAW Blöchl (1994) pseudo potentials. The present PAW implementation takes into account the scalar-relativistic effects upon a careful comparison with all-electron calculations. Jollet et al. (2014); Kucukbenli et al. (2014) We restricted ourselves to spin unpolarized calculations. Lattice instabilities were detected by the negative (imaginary) phonon dispersions evaluated for undistorted and distorted structures. Taking each of the negative phonon modes, the structural relaxations along the mode were evaluated by the BFGS optimization scheme with the structural symmetries fixed to $C$2/$m$ for Na2Ti2As2O, $Cmce$ for Na2Ti2Sb2O. For phonon calculations, we used the linear response theory implemented in Quantum Espresso package. Baroni et al. (2001) Crystal structures and Fermi surfaces were depicted by using VESTA Momma and Izumi (2011) and XCrySDen Kokalj (1999), respectively.

To deal with all the compounds systematically, we checked the convergence of plane-wave cutoff energies ($E_{\rm cut}$), $k$-meshes, $q$-meshes, and smearing parameters. The most strict condition among the compounds was taken to achieve the convergence within $\pm 1.0$ mRy per formula unit in the ground state energy, resulting in $E_{\rm cut}^{(\rm WF)}=90~{}$ Ry for wavefunction and $E_{\rm cut}^{(\rho)}=800~{}$ Ry for charge density. For undistorted Na2Ti2As2O and Na2Ti2Sb2O, ($6\times 6\times 6$) $k$-meshes were used for the Brillouin-zone integration. Phonon dispersions were calculated on ($6\times 6\times 6$) $q$-meshes. For distorted Na2Ti2As2O ($C$2/$m$) and Na2Ti2Sb2O ($Cmce$) superlattices, ($4\times 4\times 4$) $k$-meshes were used. For the distorted Na2Ti2Sb2O superlattice, ($4\times 4\times 4$) $q$-meshes were used to calculate phonon dispersions. We also calculated a ground state energy of the experimentally observed Na2Ti2Sb2O ($Cmcm$) superlattice Davies et al. (2016) with ($6\times 6\times 3$) $k$-meshes. The Marzari-Vanderbilt cold smearing scheme Marzari et al. (1999) with a broadening width of 0.01 Ry was applied to all the compounds. All the calculations were performed with the primitive lattices.

III Results and Discussion

Figure 2 (a) shows the phonon dispersions for $Pn$ = As, giving imaginary frequencies appearing around $N$ = $(2\pi/a)\cdot$(1/2, 0, $a/c$) and $\Sigma^{\prime}$ = $(2\pi/a)\cdot$(1/2, 0, 0), where ${a}$ and ${c}$ are conventional lattice constants for the undistorted structure. The compatible modes with the experimental observation Davies et al. (2016) that the twice larger periodicity along $c$-axis realized are $N$ = $(2\pi/a)\cdot$(1/2, 0, $a/c$) and $N^{\prime}$ = $(2\pi/a)\cdot$(0, 1/2, $a/c$). We note that the previous phonon calculation Chen et al. (2016) could not reproduce this doubled periodicity. We therefore took these for further lattice relaxations from the original $I$4/$mmm$ structure along the mode displacements to get $C$2/$m$ (monoclinic, No.12) superlattice (Fig. 4 (a) and Fig. 5 (a)), being consistent with the experiments Davies et al. (2016). The resultant optimized geometry after the relaxation gives fairly good agreement with the experiments Davies et al. (2016) within deviations at most 2.5% (See Appendix C). We could not confirm whether or not the negative modes disappear in the $C$2/$m$ superlattice just because of intractable computational costs for enlarged reciprocal space by the lowered symmetry.

For $Pn$ = Sb, we obtained phonon dispersions as shown in Fig. 2 (b). Imaginary frequencies appear widely around $X$ = $(2\pi/a)\cdot$(1/2, 1/2, 0) and $P$ = $(2\pi/a)\cdot$(1/2, 1/2, $a/2c$). The experiments Davies et al. (2016) reported that the twice larger super periodicity is realized only in $ab$-plane, not along $c$-axis. It is therefore $X$ mode being likely to realize the instability toward the superlattice. By optimizing the geometry along the mode displacement, we obtained $Cmce$ (orthorhombic, No.64) (Fig. 4 (b) and Fig. 5 (b)), being inconsistent with the observed $Cmcm$ (orthorhombic, No.63) structure (Fig. 4 (c)) in the experimentDavies et al. (2016). To confirm the present conclusion further, we examined the stability by seeing if the imaginary frequencies disappear at the relaxed structure, as shown in Fig. 3. In addition to confirming the stability, we compared enthalpies between our $Cmce$ and the observed $Cmcm$. The optimized geometry with $Cmce$ is found to be more stable by 11 mRy per formula unit than that with $Cmcm$. These results mean that the experimentally observed $Cmcm$ is energetically unfavorable within DFT framework.

Putting the discrepancy for $Pn$ = Sb aside for a while, we can provide a plausible explanation for the lattice instabilities as follows: The lattice instability mainly derives from in-plane Ti vibration. To make account for the inter-atomic forces to restore the lattice displacements, we can ignore (at least for the discussion of the instability) the contribution from Na because there are no bonds between Na and Ti (Fig. 1). Starting with $X$ mode for $Pn$ = Sb (Fig. 4 (b)), it generates a displacement such that a rectangular formed by Ti rotates to vibrate within $ab$-plane. A possible restoring force would come from inter-ion interactions between Ti3+ and nearest O2-, but the force is orthogonal to the displacement and hence we can ignore it. Primary force would therefore come from Ti-$Pn$ interaction for Ti approaching to $Pn$ when the rectangular rotates. The trend in the electronegativity (As = 2.18, Sb = 2.05) implies a weaker Ti-$Pn$ interaction for $Pn$ = Sb. The fact that we get the instability only for $Pn$ = Sb implies that the threshold would be around 2.1 in terms of the electronegativity against the instability. For $N$ mode (Fig. 4 (a)), the imaginary frequency appears commonly for $Pn$ = As and Sb. This is also accountable along the above discussion. In this mode, the Ti-$Pn$ interaction has litte effect on the restoring forces because Ti and $Pn$ move in the same direction. The possible restoring force would originate from nearest Oxygen and again this is orthogonal to the displacement, giving too weak contribution leading to the instability. This is the reason why the imaginary frequencies appear at $N$ mode for both $Pn$.

Getting back to the discrepancy, the present results confront the sharp contrast between $Pn$ = As and Sb in terms of whether the conventional electron-phonon treatment for the phase transition works or not. Since it is clearly reported Davies et al. (2016) that the measured XRD results cannot be identified by the $\sqrt{2}\times\sqrt{2}\times 1$ superlattice (Fig. 4 (b)), the discrepancy would be intrinsic so that it should be accounted for by those mechanism beyond the conventional ones, such as strong electronic correlations. As discussed above, the present predictions about the instabilities even including $Pn$ = Sb can be explained to some extent by a plausible physical picture, but another mechanism could dominate over this to critically decide which mode is chosen as the phonon condensation mode when the temperature decreases.

One of the possible mechanisms would be the spatial anisotropy introduced via spin-orbit couplings Kim and Kee (2015) under the enhanced polarizations by the electronic correlation: Kuroki et al. (2009); Miyake et al. (2010) In layered titanium-oxypnictides, the hybridization between Ti-3$d$ and $p$-orbitals of $Pn$ Singh (2012); Subedi (2013); Yan and Lu (2013); Nakano et al. (2016) is one of the critical factor for the transport property about how much the valence electrons are localized. For iron arsenide superconductors, such tendency toward the localization is well captured by the trend of $h$, the vertical distances between Fe layer and $Pn$ or $Ch$ Kuroki et al. (2009); Miyake et al. (2010) (Fig. 1). When $h$ gets larger, the covalency gets weaker to make Fe-3$d$ more localized and then the spin/orbital polarizations get enhanced as one of the electronic correlation effect. Kuroki et al. (2009); Miyake et al. (2010) If we apply the similar analysis to the present case, we get $h_{Pn=Bi}>h_{Pn=Sb}>h_{Pn=As}$ Yajima et al. (2013a, 2012); Doan et al. (2012); Ozawa et al. (2000); Liu et al. (2010); Wang et al. (2010) for the experimental geometry. This trend would support the correlation effect gets more enhanced for Sb than As.

The trend in $h$ above is again consistent with our previous study for BaTi${}_{2}Pn_{2}$O ($Pn$ = As, Sb, Bi), where we showed that the conventional electron-phonon framework could explain experiments only for $Pn$ = As, but not for the other heavier $Pn$. Nakano et al. (2016) It might be a general tendency also applicable to the layered titanium-oxypnictides that the larger $h$ enhances the electronic correlations as the origin of the exotic mechanism. Such mechanisms have actually been proposed by several authors Kim and Kee (2015); Nakaoka et al. (2016) such as the phase transition driven not by CDW but by the orbital ordering Kim and Kee (2015), or the orbital ordering induced by spin-fluctuation Nakaoka et al. (2016). We note that there are some papers reporting ’weak correlations’ in these compounds Yan and Lu (2013); Huang et al. (2013); Kim and Kee (2015); Tan et al. (2015). It apparently seems contrary but we have to be careful that some reports support ’weaker than Fe-based compounds’ Huang et al. (2013) while other ’weaker correlation in XC potentials’. Tan et al. (2015); Yan and Lu (2013) These are therefore not necessarily contrary to the present statement that ’the correlation gets stronger to overwhelm the conventional electron-phonon mechansm’. The above exotic orbital orderings Kim and Kee (2015); Nakaoka et al. (2016) are reported to be possible even when ’the correlation is weak’, but again we wonder if this would be such a situation as $E_{\rm corr.}^{\rm Fe}>E_{\rm corr.}^{\rm Ti}>E_{ph.}$, where $E_{\rm corr.}$ and $E_{ph.}$ correspond to the typical energy scale of electronic correlations and electron-phonon interaction, respectively. Though the superconductivities in BaTi2$Pn_{2}$O ($Pn$ = Sb, Bi) have been thought to be conventional BCS type Subedi (2013); Kitagawa et al. (2013); Nozaki et al. (2013); von Rohr et al. (2013), the present result implies the possibility of some exotic superconducting mechanism with heavier $Pn$. We also note that it was pointed out Kamusella et al. (2014) that the superconductivity in Ba1-xNaxTi2Sb2O is at the verge of unconventional superconductivity by the Uemura classification scheme Uemura et al. (1991).

IV Conclusion

We performed ab-initio phonon calculations for Na2Ti${}_{2}Pn_{2}$O ($Pn$ = As, Sb) to investigate if the experimentally observed superlattice peaks can be explained by negative frequency modes. For $Pn$ = As, we obtained a $C$2/$m$ (monoclinic No.12) superlattice that is completely consistent with the experimentally observed structure. The consistency indicates that simple electron-phonon interaction can explain the phase transition for $Pn$ = As. On the other hand, we obtained a $Cmce$ (orthorhombic No.64) superlattice for $Pn$ = Sb that is inconsistent with the experimentally observed structure $Cmcm$ (orthorhombic No.63). The discrepancy would be intrinsic so that it should be accounted for by those mechanism beyond electron-phonon interaction, such as strong electronic correlations. Such a discrepancy is also found in superconducting BaTi${}_{2}Pn_{2}$O when the pnictogen gets heavier. These facts imply a systematic dependence on pnictogen to tune the mechanism of the phase transition and superconductivity from conventional to exotic.

Acknowledgements.

The computation in this work has been mainly performed using the facilities of the Center for Information Science in JAIST. K.H. is grateful for financial support from a KAKENHI grant (15K21023), a Grant-in-Aid for Scientific Research on Innovative Areas (16H06439), PRESTO and the Materials research by Information Integration Initiative (MI2I) project of the Support Program for Starting Up Innovation Hub from Japan Science and Technology Agency (JST). R.M. is grateful for financial support from MEXT-KAKENHI grants 26287063 and that from the Asahi glass Foundation.

Appendix A : Electronic structures

We described electronic structures for undistorted Na2Ti2$Pn_{2}$O ($Pn$ = As, Sb) structure in order to carefully examine the artifacts due to the choice of pseudo potentials (PP). The obtained electronic structures shown in Fig. 6 are consistent with the previous calculations Suetin and Ivanovskii (2013); Yan and Lu (2013).

Appendix B : Brillouin zones

The primitive Brillouin zone for undistorted Na2Ti${}_{2}Pn_{2}$O ($I$4/$mmm$) is shown in Fig. 7 (a). The special $k$ and $q$ points are ${\Gamma}$ = $(2\pi/a)\cdot$(0, 0, 0), $N$ = $(2\pi/a)\cdot$(1/2, 0, $a/2c$), $P$ = $(2\pi/a)\cdot$(1/2, 1/2, $a/2c$), $X$ = $(2\pi/a)\cdot$(1/2, 1/2, 0), $\Sigma^{\prime}$ = $(2\pi/a)\cdot$(1/2, 0, 0), $Z$ = $(2\pi/a)\cdot$(0, 0, $a/c$) in cartesian axis, where ${a}$ and ${c}$ are conventional lattice constants for the undistorted structures.

The primitive Brillouin zone for Na2Ti2Sb2O superlattice ($Cmce$) is also shown in Fig. 7 (b). The special $k$ and $q$ points are ${\Gamma}$ = $(2\pi/a^{\prime})\cdot$(0, 0, 0), $X$ = $(2\pi/a^{\prime})\cdot$(1, 0, 0), $S$ = $(2\pi/a^{\prime})\cdot$(1/2, $a^{\prime}/2b^{\prime}$, 0), $Z$ = $(2\pi/a^{\prime})\cdot$(0, 0, $a^{\prime}/2c^{\prime}$), $A$ = $(2\pi/a^{\prime})\cdot$(1, 0, $a^{\prime}/2c^{\prime}$), $R$ = $(2\pi/a^{\prime})\cdot$(1/2, $a^{\prime}/2b^{\prime}$, $a^{\prime}/2c^{\prime}$) in cartesian axis, where ${a^{\prime}}$, ${b^{\prime}}$ and ${c^{\prime}}$ are conventional lattice constants for the superlattice structure.

Appendix C : Geometry optimizations

The results of geometry optimizations are shown in Table 1-3. The conventional lattice vectors of the superlattice structure for $Pn$ = As ($C$2/$m$) are redefined as ${\vec{a}}^{\prime}=2\left({\vec{a}}-{\vec{b}}\right)$, ${\vec{b}}^{\prime}=2\left({\vec{a}}+{\vec{b}}\right)$ and ${\vec{c}}^{\prime}=1/2\left(-{\vec{a}}+{\vec{b}}+{\vec{c}}\right)$, those for $Pn$ = Sb ($Cmce$) are redefined as ${\vec{a}}^{\prime}={\vec{c}}$, ${\vec{b}}^{\prime}=\left({\vec{a}}-{\vec{b}}\right)$ and ${\vec{c}}^{\prime}=\left({\vec{a}}+{\vec{b}}\right)$, and those for $Pn$ = Sb ($Cmcm$) are redefined as ${\vec{a}}^{\prime}=2\left({\vec{a}}+{\vec{b}}\right)$, ${\vec{b}}^{\prime}=2\left({\vec{b}}-{\vec{a}}\right)$ and ${\vec{c}}^{\prime}={\vec{c}}$, where ${\vec{a}}$, ${\vec{b}}$ and ${\vec{c}}$ are conventional lattice vectors of the undistorted structures.

For $Pn$ = As ($C$2/$m$), we need to take special care when comparing our calculated geometry with an experimental one due to Davis et al.[28] We found that the atomic positions listed in their Table I are incompatible with their superlattice vectors, ${\vec{a}}^{\prime}=2\left({\vec{a}}+{\vec{b}}\right)$, ${\vec{b}}^{\prime}=2\left({\vec{b}}-{\vec{a}}\right)$, and ${\vec{c}}^{\prime}=1/2\left({\vec{a}}+{\vec{b}}+{\vec{c}}\right)$ , described in their main text [28]: (1) If the description in the main text is assumed to be correct, Na and Sb positions in the superlattice are respectively located unlikely far from their undistorted positions. (2) O is not located at (0.0,0.0) in their Fig. 6 (a), being inconsistent with their Table I. Accordingly, we speculate that they again define the superlattice vectors compatible with the atomic positions in their Table I, which differs from that in their main text. We also found $y$ = 0.0 for As3 in their Table I must be a typo and correctly $y$ = 0.25 because As3 should occupy 8$j$ Wyckoff site and its undistorted position is $y$ = 0.5. Assuming our speculation is correct, our optimized geometry parameters agree well with their experimental values.

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