First-principles prediction of a new family of layered topological insulators

TL;DR

This paper uses first-principles calculations to predict a new family of layered topological insulators and metals among V$_2$IV$_2$VI$_6$ compounds, expanding the potential materials for spintronic applications.

Contribution

It predicts nine strong topological insulators and two topological metals in a family of layered compounds, many of which can be experimentally synthesized.

Findings

Nine strong topological insulators predicted with energy gaps 0.04-0.2 eV.

Small strains can convert trivial insulators into topological insulators.

Several compounds are already synthesized, facilitating experimental validation.

Abstract

Dozens of layered V$_2$IV$_2$VI$_6$ (V=P, As, Sb, Bi, IV=Si, Ge, Sn, Pb, VI=S, Se, Te) materials are investigated, several of which have been successfully synthesized in experiment. Among them, we predict nine strong topological insulators (TIs), two strong topological metals (TMs) and nearly twenty trivial insulators at their equilibrium structures. The TIs are in the (1;111) topological class, with energy gaps ranging from 0.04 to 0.2 eV. The strong TMs and the trivial insulators belong to the (1;111) and (0;000) topological classes, respectively. Small compressive strains easily turn some of the trivial insulators into strong TIs. This study enriches not only the family of topological materials but also the family of van der Waals layered materials, providing promising candidates for the future spintronic devices.

Tables

Table 1. Table 1: Theoretical parameters of nontrivial V 2 IV 2 VI 6 crystals: lattice constants a and c , dynamically stable (Y) or unstable (N), energy gap E g , and topological invariants ( ν 0 ; ν 1 ​ ν 2 ​ ν 3 subscript 𝜈 0 subscript 𝜈 1 subscript 𝜈 2 subscript 𝜈 3 \nu_{0};\nu_{1}\nu_{2}\nu_{3} ). Minus gaps of As 2 Sn 2 Te 6 and P 2 Sn 2 Te 6 mean topological metals. The superscripts d and i on the gap data indicate the direct and indirect gap nature, respectively.

	a (Å)	c (Å)	Stability	Eg (eV)	$Z_2$ class
As2Sn2Se6	6.757	20.011	Y	0.04i	(1;111)
Sb2Ge2Se6	6.816	19.949	Y	0.06i	(1;111)
Sb2Ge2Te6	7.302	21.010	Y	0.12i	(1;111)
Sb2Sn2Se6	6.974	20.153	Y	0.12i	(1;111)
Sb2Sn2Te6	7.433	21.467	Y	0.15i	(1;111)
Bi2Ge2Se6	6.909	20.197	Y	0.04d	(1;111)
Bi2Ge2Te6	7.392	21.159	Y	0.19i	(1;111)
Bi2Sn2Se6	7.064	20.334	Y	0.13i	(1;111)
Bi2Sn2Te6	7.521	21.637	Y	0.20i	(1;111)
P2Sn2Te6	7.147	20.880	Y	$-0.07$	(1;111)
As2Sn2Te6	7.223	21.092	Y	$-0.01$	(1;111)

Table 2. Table 2: Theoretical parameters of trivial V 2 IV 2 VI 6 crystals: lattice constants a and c , dynamically stable (Y) or unstable (N), energy gap E g , and topological Z 2 invariants. The superscripts d and i on the gap data indicate the direct and indirect gap nature, respectively.

	a (Å)	c (Å)	Stability	Eg (eV)	Z2 invariants
P2Sn2Se6	6.543	23.491	Y	0.45d	0; (000)
As2Si2S6	6.124	20.613	Y	1.58i	0; (000)
As2Si2Se6	6.467	19.902	Y	0.49d	0; (000)
As2Si2Te6	6.986	20.931	Y	0.42d	0; (000)
As2Ge2S6	6.209	21.035	Y	1.15d	0; (000)
As2Ge2Se6	6.564	19.779	Y	0.08i	0; (000)
As2Ge2Te6	7.069	20.774	Y	0.08d	0; (000)
As2Sn2S6	6.370	22.051	Y	0.88d	0; (000)
Sb2Si2S6	6.428	19.454	Y	0.88d	0; (000)
Sb2Si2Se6	6.722	20.155	Y	0.31d	0; (000)
Sb2Si2Te6	7.223	21.236	Y	0.15d	0; (000)
Sb2Ge2S6	6.534	19.378	Y	0.40d	0; (000)
Sb2Sn2S6	6.721	19.298	Y	0.01d	0; (000)
Bi2Si2S6	6.537	19.608	Y	0.71d	0; (000)
Bi2Si2Se6	6.821	20.419	Y	0.30d	0; (000)
Bi2Si2Te6	7.316	21.438	Y	0.07d	0; (000)
Bi2Ge2S6	6.630	19.536	Y	0.43d	0; (000)
Bi2Sn2S6	6.807	19.499	Y	0.10d	0; (000)
Bi2Pb2Se6	-	-	N	-	-
Bi2Pb2Te6	-	-	N	-	-

Full text

I Introduction

About a decade ago, topological insulators (TIs) were theoretically predicted Fu et al. (2007); Moore and Balents (2007); Roy (2009); Fu and Kane (2007) and experimentally realized in real materials Xia et al. (2009); Zhang et al. (2009). Since then, TIs have attracted tremendous attention due to their fundamental interest as well as potential applications Hasan and Kane (2010); Qi and Zhang (2011). A great many three-dimensional (3D) TIs Agarwala and Shenoy (2017); Pi et al. (2017); Pielnhofer et al. (2017); Young et al. (2017); Feng and Yao (2012); Lin et al. (2010); Kuroda et al. (2010, 2012) have been proposed, but only a few of them have been experimentally synthesized and veryfied Lin et al. (2010); Kuroda et al. (2010, 2012), among which only the TIs of the Bi2Se3 family have been widely and intensively studied. This family of TIs have large energy gaps (typically, 0.3 eV for Bi2Se3), enough for room temperature utilizations; they are van der Waals (vdW)-layered, so defective surface states can be largely avoided and the lattice matching requirement can be significantly reduced in epitaxial growth on substrates. However, these TIs suffer from intrinsic doping. Bi2Se3 and Bi2Te3 are usually $n$-type Hsieh et al. (2009); Chen et al. (2009); Xia et al. (2009) while Sb2Te3 shows $p$-type Hsieh et al. (2009). Though the Fermi levels can be tuned by extrinsic doping or alloying Ren et al. (2011, 2012); Zhang et al. (2011), such experiments must be carefully controlled and the studies are still on the way. Searching or predicting new TIs, especially those with large energy gaps and layered crystal structures, are sitll of great interest.

Recently, ternary compounds Bi2Si2Te6 and Sb2Si2Te6 have been successfully synthesized in experiment Seidlmayer (2010). These compounds are vdW-layered, belonging to the V2IV2VI6 (V=P, As, Sb, Bi, IV=Si, Ge, Sn, Pb, VI=S, Se, Te) family. Here we show that Bi2Si2Te6 and Sb2Si2Te6 can be easily turned into strong TIs by small compressive strains. Moreover, tens of the V2IV2VI6 family of compounds are intensively investigated. Based on the calculated $Z_{2}$ invariants ($\nu_{0};\nu_{1}\nu_{2}\nu_{3}$), we predict nine strong TIs, two strong topological metals (TMs), and nearly twenty trivial insulators at equilibrium lattice constants. More strong TIs and TMs can be obtained by applying small compressive strains to the trivial insulators. The TIs have energy gaps ranging from several tens of meV to 0.2 eV, and support one single Dirac cone at the topological surface states.

II Crystal Structure and Method

The V2IV2VI6 (V=P, As, Sb, Bi, IV=Si, Ge, Sn, Pb, VI=S, Se, Te) family of compounds share the same rhombohedral structure with a space group of $R\overline{3}$ (No. 148) with ten atoms in each primitive cell. The atomic structure and the Brillouin zone of the primitive cell are shown in Fig. 1 (a) and (b), respectively. Along the trigonal axis or the [111] direction, VI3-IV-V2-IV-VI3 ten-atom layers are periodically stacked at five different heights, forming quintuple layers (QLs) as Fig. 1 (c) shows. The conventional cell contains 30 atoms and three QLs. The inter-QL coupling is strong, while the intra-QL coupling is much weaker, forming vdW-type interactions, similar to the Bi2Se3 case Zhang et al. (2009).

This work is performed using the first-principles method within the framework of density functional theory of generalized gradient approximations (DFT-GGA) Hohenberg and Kohn (1964); Perdew et al. (1996) at *High *precision, as implemented in the VASP codes Kresse and Furthmüller (1996). $8\times 8\times 8$ and $8\times 8\times 2$ K-point meshes are used for primitive and conventional cells, respectively. The atoms and the cell size are fully relaxed until the force acting on each ion is less than 0.01 eV/Å. Surface states are calculated within slab supercells containing nine QLs of V2IV2VI6 and a vacuum thicker than 20 Å. Phonon spectra are obtained by the calculation of the interatomic force constants (IFCs) in VASP, with the help of the Phonopy package Togo and Tanaka (2015). The $Z_{2}$ topological invariants ($\nu_{0};\nu_{1}\nu_{2}\nu_{3}$) Fu et al. (2007); Moore and Balents (2007); Roy (2009) are calculated using the parity method proposed by Fu and Kane Fu and Kane (2007).

III Results

We first consider Bi2Si2Te6 and Sb2Si2Te6, which have been successfully synthesized in experiment Seidlmayer (2010); Arabbagheri (2014). The calculated lattice constants of Bi2Si2Te6 (Sb2Si2Te6) are $a=7.317$ Å, $c=21.438$ Å ($a=7.223$ Å, $c=21.236$ Å), in good agreement with the experimental data $a=7.269$ Å, $c=21.293$ Å ($a=7.169$ Å, $c=21.186$ Å) Arabbagheri (2014). The band structure of Bi2Si2Te6 is shown in Fig. 2 (a) as an example. Its valence band maximum (VBM) and conduction band minimum (CBM) are located at T point in the Brillouin zone, forming a direct energy gap of 0.07 eV. The VBM and the CBM are mainly contributed by the Te-p and the Bi-p states, respectively. The band structure of Sb2Si2Te6 is similar, but having a larger direct gap of 0.15 eV. According to the calculated $Z_{2}$ topological invariants, both materials are in the (0;000) topological class, so they are trivial insulators.

Strains can remarkably change the band structure of Bi2Si2Te6 and Sb2Si2Te6 as well as their topological nature. Fig. 2 (b) shows the gap-dependence of Bi2Si2Te6 on the hydrostatic strain $\varepsilon$, where negative and positive $\varepsilon$ values represent compressive and tensile strains, respectively. A small compressive strain of 2% is enough to turn the material into a strong TI of the (1;111) topological class. The band structure of Bi2Si2Te6 at $\varepsilon=-2\%$ is shown in Fig. 2 (c). The CBM and the VBM are mainly composed of the Bi-p and Te-p states, respectively. A band inversion obviously occurs at T point, consistent with the topological phase transition.

Sb2Si2Te6 behaves differently under strains. Compressive strains less than 2% cannot change the topological nature of Sb2Si2Te6. At $\varepsilon=-3\%$, the VBM remains at T point, while the CBM shifts from T to F point and becomes energetically lower than the VBM, making the material metallic. However, local energy gaps exist everywhere around the Fermi level in the Brillouin zone. The calculated $Z_{2}$ invariants indicate the material now belongs to the (1;111) topological class, i.e., it becomes a strong TM.

Studies are extended to dozens of similar compounds, including P2Sn2Se6, P2Sn2Te6, and 27 V2IV2VI6 materials with V=As, Sb, Bi, IV=Si, Ge, Sn, VI=S, Se, Te. Phonon dispersions of all these crystals are calculated, and no imaginary modes are found, indicating they are all dynamically stable. However, when several compounds containing Pb atoms including Bi2Pb2Se6 and Bi2Pb2Te6 are considered, they show obviously imaginary modes, thus they are dynamically unstable. The phonon dispersions of Bi2Ge2Te6 are shown in Fig. 1 (d) as an example, while those of more compounds can be found in Fig. LABEL:ext1-spho in the Supplementary Materials.

Band structure calculations indicate that these stable materials are either insulators or such metals with local energy gaps everywhere around Fermi level in the Brillouin zone. Then $Z_{2}$ topological invariants ($\nu_{0};\nu_{1}\nu_{2}\nu_{3}$) Fu et al. (2007); Moore and Balents (2007); Roy (2009) are calculated using a parity method proposed by Fu and Kane Fu and Kane (2007), according to which the materials are classified into three topological classes, i.e., TIs, TMs and trivial insulators. The topological phases and energy gaps of these materials are shown in Fig. 3, where the subscripts in the molecular formulas are neglected for clarity. The lattice constants, gap values and $Z_{2}$ topological invariants of TIs and TMs are listed in Table 1, while those of the trivial insulators are listed in Table  2.

Fig. 3 shows two metals, P2Sn2Te6 and As2Sn2Te6. Band structure of P2Sn2Te6 is shown in Fig. 2 (d) as an example, and that of As2Sn2Te6 is similar. The CBM of P2Sn2Te6 is located at F point, about 0.07 eV lower than the VBM, which is close to T point. The compound is a metal. However, local energy gaps exist around the Fermi level everywhere in the Brillouin zone, rendering the computation of topological invariants possible. According to the calculation results listed in Table 1, both P2Sn2Te6 and As2Sn2Te6 are in the nontrivial (1;111) class, thus they are strong TMs. Recently, multiple Dirac cones have been reported in Zr2Te2P Ji et al. (2016), which is a 3D strong TM that have the same symmetry group as P2Sn2Te6 and As2Sn2Te6.

Fig. 3 also shows nine TIs, including As2Sn2Se6 and eight V2IV2VI6 crystals with V=Sb, Bi, IV=Ge, Sn, VI=Se, Te. These TIs have narrow energy gaps ranging from 0.04 to 0.20 eV and include heavy elements with strong SOC, which are usually essential for TIs Hasan and Kane (2010); Qi and Zhang (2011). However, not all the materials containing heavy elements are TIs. Sb2Si2Te6, Bi2Si2Te6 and Bi2Sn2S6 are trivial insulators due to the involvement of the light Si or S atoms. As2Ge2Se6 is also topologically trivial because both As and Ge are not heavy enough. From Table 1, these TIs are all in the (1;111) $Z_{2}$ class, indicating they are 3D strong TIs. As comparisons, the Bi1-xSbx alloy and Bi2Se3 are 3D strong TIs in the (1,111) and (1;000) topological classes Teo et al. (2008); Feng et al. (2012), respectively.

The inplane lattice constants of the TIs range from about 6.7 to 7.5 Å, covering the lattice regions of Cr2Ge2Te6 and CrI3, which are typically around 7.0 Å Hou and Wu (2018); Zeisner et al. (2019). Cr2Ge2Te6 Burch et al. (2018) and CrI3 Song et al. (2018); Klein et al. (2018); Huang et al. (2017) are vdW-layered ferromagnetic insulators (FMIs) that have attracted very much attention recently. FMI/TI heterostructures such as Cr2Ge2Te6/Bi2Te3 Ji et al. (2013), Cr2Ge2Te6/BiSbTeSe2 Chong et al. (2018) and CrI3/Bi2Se3 Hou and Wu (2018) have been investigated to realize exotic quantum phenomena such as quantum anomalous Hall effect and so on. Compared with the Bi2Se3 family of TIs, whose inplane lattice constants vary from 4.1 to 4.3 Å Zhang et al. (2010), the current TIs match Cr2Ge2Te6 and CrI3 well. Hence, it is interesting to study the magnetic proximity effects between these FMIs and current TIs.

Among the nine TIs, only Bi2Ge2Se6 has a direct energy gap. Its band structure, and zoomed-in band structures are shown in Fig. 4 (a) and (b), respectively. Both the VBM and the CBM are located at T point in the reciprocal space, forming a direct gap of 0.04 eV. When spin-orbit coupling (SOC) is considered, the VBM and the CBM of Bi2Ge2Se6 are mainly contributed by the Bi-p and the Te-p states, respectively. If SOC is turned off, the VBM and the CBM mainly come from the Te-p and the Bi-p states, respectively. Hence, SOC results in the band inversion, which is a common mechanism of TIs Hasan and Kane (2010); Qi and Zhang (2011).

Other TIs are indirectly gapped. As2Sn2Se6 has the smallest energy gap of 0.04 eV, equal to that of Bi2Ge2Se6. Bi2Ge2Te6 and Bi2Sn2Te6 have the largest energy gaps, 0.19 and 0.20 eV, respectively, in the same order as those of the Bi2Se3 family of TIs Zhang et al. (2009), and large enough for room temperature utilizations. Consider, for example, the case of Bi2Ge2Te6. Its CBM is located at T point, but the VBM shifts away along the TF direction, as shown in Fig. 4 (c), forming an indirect energy gap. When not considering SOC, the CBM and the VBM of Bi2Ge2Te6 mainly come from the Te-p and the Bi-p states. When considering SOC, the energy bands near the Fermi level are inverted around T point, very similar to the case of Bi2Ge2Se6.

Strains can significantly change the electronic states of the current TIs. We discuss two examples, Bi2Ge2Se6 and Bi2Ge2Te6. Fig. 4 (d) shows the gap-dependence on the hydrostatic strain $\varepsilon$. In both cases, small compressive strains remarkably increase the energy gaps. For example, a 4% compressive strain increases the energy gap of Bi2Ge2Se6 from 0.04 to 0.22 eV. However, after that, the gap drastically decreases as the compressive strain increases. Similar behavior also occurs in Bi2Ge2Te6 after a compressive strain of 2%. This is because after the critical strains of 4% and 2%, the CBMs of the both materials shift from T to F point in the reciprocal space. If the compressive strain continues, energy levels of the new CBMs decrease more quickly with respect to their respective VBMs.

Under tensile strains, Bi2Ge2Se6 and Bi2Ge2Te6 behave differently. At $\varepsilon<1\%$, Bi2Ge2Se6 is gapped, corresponding to a nontrivial phase or a TI. At $\varepsilon=1\%$, the VBM and the CBM of Bi2Ge2Se6 touch at T point, so the gap closes, forming a semimetal. At $\varepsilon>1\%$, the gap reopens, producing an insulator. This gap closing and reopening strongly suggests the transition between nontrivial and trivial phases Zhang et al. (2010). We confirm this by computing the $Z_{2}$ invariants, which are (1;111) and (0;000) before and after the gap reopening, respectively. Thus, after the gap reopens, Bi2Ge2Se6 becomes a trivial insulator. At $\varepsilon=2\%$, the energy gap is about 0.03 eV, which increases to 0.07 eV at $\varepsilon=5\%$. In the case of Bi2Ge2Te6, the energy gap decreases with the increasing of the tensile strain, but that slows down as the strain becomes large. At $\varepsilon=8\%$, the energy gap is still 0.03 eV. Further analyses indicate that gap remains nonzero until the material becomes metallic at an extremely large tensile strain of 14%. In this process, there are no gap closing and reopening, thus no topological phase transitions. It illustrates the robustness of a TI against nonmagnetic perturbations like strains Hasan and Kane (2010); Qi and Zhang (2011).

TIs distinguish from trivial insulators by their massless and linearly dispersed Dirac surface states, which are protected by time-reversal symmetry Hasan and Kane (2010); Qi and Zhang (2011). The predicted TIs support a single Dirac cone on the surface. Take Bi2Sn2Te6 as an example. Fig. 4 (e) shows the band structure of a Bi2Sn2Te6 slab of nine QLs, where the surface contribution is indicated by red symbols. A single Dirac cone is easily resolved, with the Dirac point at $\Gamma$ point, and it obviously comes from surface atoms. The projected wavefunction of the Dirac point in the real space shows that the surface states penetrate a depth of about two QLs, as demonstrated in Fig. 4 (f). Band structure of more TI slabs are shown in Fig. LABEL:ext1-sslab in Supplementary Materials.

Besides the above TMs and TIs, Fig. 3 shows 18 trivial insulators in the topological class (0;000), some parameters of which are listed in Table  2. They involve light atoms such as S, Si and P with weak SOC. Most of them are directly gapped. Both the CBM and the VBM are located at T point, as the example Bi2Si2Te6 shows in Fig. 2 (a). As2Si2S6 and As2Ge2Se6 are indirectly gapped. In the As2Si2S6 case, the CBM and the VBM are at L and T points, respectively. In the As2Ge2Se6 case, the CBM is also at T point, but the VBM slightly shifts away from T point along the TF direction. The lattice constants of these trivial insulators vary from about 6.1 to 7.3 Å, and their energy gaps range from 0.01 to 1.58 eV. The smallest and largest energy gaps occur in Sb2Sn2S6 and As2Si2S6, respectively.

Topological phase transitions can be easily controlled by applied strains in some trivial insulators. Take Sb2Sn2S6 as an example. At equilibrium lattice constants, Sb2Sn2S6 is trivial, with a small direct energy gap 0.01 eV. However, as Fig. 2 (b) shows it becomes nontrivial at $\varepsilon=-1\%$ according to the calculated $Z_{2}$ invariants.

IV Discussion

The studied materials are all vdW-layered, periodically stacked along the [111] direction by V2IV2VI6 QLs. Epitaxially grown thin films of such materials can mostly avoid defective surface defects, and slabs or films can also be exfoliated from their 3D bulk. Phonon dispersions (see Fig. LABEL:ext1-sslab in Supplementary Materials) show that slabs of these materials are dynamically stable down to one QL. These TIs have large energy gaps enough for room temperature utilizations. Their lattice constants vary from 6.7 to 7.5 Å, covering the lattice range of Cr2Ge2Te6 and CrI3 Hou and Wu (2018); Zeisner et al. (2019). Hence, they are promising candidates to study the topological-magnetic proximity effect with Cr2Ge2Te6 and CrI3. Under small compressive strains, more trivial insulators can be turned into TIs. Several compounds in the family have been successfully synthesized in experiment Seidlmayer (2010); Arabbagheri (2014), thus the synthesis of other materials of this family are quite promising and worth trying.

In conclusion, a number of 3D strong TIs and TMs are predicted based on first-principles calculations. They are vdW-layered, exhibit large energy gaps and wide range of lattice constants. This work greatly enriches the family of TIs, and provides ideal candidates for the study of topological-magnetic interactions with ferromagnetic insulators Cr2Ge2Te6 and CrI3. Synthesis of these materials is recommended to follow the successful approach synthesizing Bi2Si2Te6 and Sb2Si2Te6. We call for such efforts.

This work is supported by the NSF of China (Grant Nos. 21271007 and 11747016), the NSF of Anhui Province (Grant No. 1308085QA05) and the Doctoral Foundation of Anhui Jianzhu University (Grant No. 2017QD19).