Exploring strong and weak topological states on isostructural substitutions in TlBiSe2

Ankita Phutela; Preeti Bhumla; Manjari Jain; Saswata Bhattacharya

doi:10.1038/s41598-022-26445-y

. 2022 Dec 20;12:21970. doi: 10.1038/s41598-022-26445-y

Exploring strong and weak topological states on isostructural substitutions in TlBiSe $_{2}$

Ankita Phutela ^1,^✉, Preeti Bhumla ¹, Manjari Jain ¹, Saswata Bhattacharya ^1,^✉

PMCID: PMC9767939 PMID: 36539475

Abstract

Topological Insulators (TIs) are unique materials where insulating bulk hosts linearly dispersing surface states protected by the Time-Reversal Symmetry. These states lead to dissipationless current flow, which makes this class of materials highly promising for spintronic applications. Here, we predict TIs by employing state-of-the-art first-principles based methodologies, viz., density functional theory and many-body perturbation theory (G $_{0}$ W $_{0}$ ) combined with spin-orbit coupling effects. For this, we take a well-known 3D TI, TlBiSe $_{2}$ and perform complete substitution with suitable materials at different sites to check if the obtained isostructural materials exhibit topological properties. Subsequently, we scan these materials based on SOC-induced parity inversion at Time-Reversal Invariant Momenta. Later, to confirm the topological nature of selected materials, we plot their surface states along with calculation of Z $_{2}$ invariants. Our results show that GaBiSe $_{2}$ is a strong Topological Insulator, besides, we report six weak Topological Insulators, viz., PbBiSe $_{2}$ , SnBiSe $_{2}$ , SbBiSe $_{2}$ , Bi $_{2}$ Se $_{2}$ , TlSnSe $_{2}$ and PbSbSe $_{2}$ . We have further verified that all the reported TIs are dynamically stable, showing all real phonon modes of vibration.

Subject terms: Topological insulators, Electronic structure

Introduction

Since the discovery of Topological Insulators (TIs) about a decade ago, there has been an enormous increase in interest towards topological condensed matter systems^1–13. TIs show great potential applications in quantum computing and spintronics due to the insensitivity of the transport property towards non-magnetic perturbations^13,14. TIs also pave the way for realizing advanced quantum phenomena such as Weyl semimetals¹⁵, Majorana-fermions¹⁶ and Higgs mechanism¹⁷. These alluring materials are insulating in bulk but support the flow of electrons on their surface. As a result, their surface consists of linear states that are protected by the Time-Reversal Symmetry (TRS)¹⁸. A necessary condition for the appearance of these states is the inversion of bands, which takes place at the Time-Reversal Invariant Momenta (TRIM) in the bulk Brillouin Zone (BZ)^19,20. The natural ordering of the energy levels forming the edges of the gap is inverted owing to the strong Spin-Orbit Coupling (SOC) associated with heavy elements. The topology of TRS invariant insulators is characterized by the Z $_{2}$ index, $ν_{0}$ , which can either be 0 or 1, depicting a topologically trivial or non-trivial phase, respectively. However, it has been recently reported that even when $ν_{0}$ = 0, the system can show non-trivial characteristics²¹. Based on whether the material hosts odd or even number of Dirac cones in the electronic structure of its surface, TIs are further classified as Strong Topological Insulator (STI) having Z $_{2}$ invariant, $ν_{0}$ = 1 or Weak Topological Insulator (WTI) with Z $_{2}$ invariant, $ν_{0}$ = 0^22,23. Nevertheless, the complete characterization of 3D TIs requires a set of, in total, four Z $_{2}$ numbers: ( $ν_{0}$ ; $ν_{1} ν_{2} ν_{3}$ ). The indices $ν_{1}$ , $ν_{2}$ and $ν_{3}$ are called weak indices and are believed to be nonrobust quantities^24,25. Therefore, the calculation of surface states and Z $_{2}$ invariant gives complete information about the topological nature of material.

First-principles calculations led to the prediction of a large number of 2D and 3D TIs^22,26–34. Among the various well established families of 3D TIs, Bi $_{2}$ Se $_{3}$ and Bi $_{2}$ Te $_{3}$ have been most widely studied for investigating the topological states and their properties^35–37. Their crystal structures consist of quintuple layers held together by weak van der Waals (vdW) forces, providing natural cleavage planes without breaking strong bonds. The band structure calculations in Bi $_{2}$ Se $_{3}$ have shown that the Dirac point of surface state lies close to the Valence Band Maximum (VBM)³⁸. This leads to the opening of electron scattering channel from surface states to bulk continuum states, and the topological transport regime begins to collapse. Therefore, there is a strong need for materials with ideal and relatively isolated Dirac cones. A variety of candidates with non-trivial electronic states including, HgTe³⁹, InAs⁴⁰, ternary tetramytes Ge $_{m}$ Bi $_{2 n}$ Te $_{(m + 3 n)}$ , half-Huesler compounds^41,42, LiAuSe honeycomb lattice⁴³, $β$ -Ag $_{2}$ Te⁴⁴ to non-centrosymmetric BiTeX (X=Cl, Br, I)⁴⁵ have been suggested. Theoretical studies have shown that Tl-based ternary chalcogenides, viz., TlSbTe $_{2}$ , TlBiSe $_{2}$ and TlBiTe $_{2}$ are 3D TIs with a single Dirac cone surface state at the $Γ$ point, which is well isolated from the bulk continuum⁴⁶. Tl-based materials have a 3D character because each Tl (Bi) layer is sandwiched between two Se layers with strong coupling between neighboring atomic layers instead of weak vdW forces as in Bi $_{2}$ Se $_{3}$ . The electronic structure of many Tl-based TIs, viz., TlAB $_{2}$ (A = Sb, Bi and B = Se, Te, S) have been investigated by Density Functional Theory (DFT) calculations^20,47. The role of surface termination has also been explored in TlBiSe $_{2}$ and TlBiTe $_{2}$ ⁴⁸. Following this, the isostructural substitution of the above base material is been endeavoured to retain its topological properties. For example, In-based compounds like InBiTe $_{2}$ and InSbTe $_{2}$ , crystallizing in the TlBiSe $_{2}$ like crystal structure have been studied. Intriguingly, these materials lack the Dirac cone feature, which depicts their topologically trivial nature⁴⁹. Therefore, despite TlAB $_{2}$ (A = Sb, Bi and B = Se, Te, S) showing topologically non-trivial band structure, any other isostructural substitution to retain its topological nature is hitherto unknown.

Motivated with this idea, in this article, we have explored the possibility of having materials belonging to same class of ternary chalcogenides via a thorough isostructural substitution approach. First, we have performed substitution at suitable sites of TlBiSe $_{2}$ and scanned for those materials whose band structure shows band inversion at odd/even number of TRIMs. After that, to determine the accurate band gap of TlBiSe $_{2}$ , we have employed various exchange-correlation ( $ϵ_{xc}$ ) functionals, viz., PBE+SOC, HSE06+SOC, G $_{0}$ W $_{0}$ @PBE+SOC and G $_{0}$ W $_{0}$ @HSE06+SOC. The band gap obtained from G $_{0}$ W $_{0}$ @PBE+SOC functional is in close agreement with the experimental value. Therefore, we have further calculated the band gap of all materials using G $_{0}$ W $_{0}$ @PBE+SOC. Subsequently, we have examined the potential materials for their dynamical stability. The stable materials are then characterized as STI/WTI depending on whether they show odd/even number of surface states, respectively. To confirm their topological nature, we have also calculated the Z $_{2}$ topological invariants.

Results and discussion

Bulk electronic structure and band inversion

TlBiSe $_{2}$ belongs to the Tl-family of compounds having a rhombohedral crystal structure with space group R $\bar{3}$ m⁴⁶. There are four atoms in the primitive unit cell which are placed in layers normal to the three-fold axis with the sequence -Tl-Se-Bi-Se-, i.e., along [111] axis of rhombohedral unit cell (see Fig. 1a). The 3D BZ for rhombohedral unit cell having high symmetry points F, $Γ$ , L and Z, along with its projected (111) surface BZ is shown in Fig. 1b. The structure has inversion symmetry where both Bi and Tl act as inversion centers. We have first estimated the band gap of TlBiSe $_{2}$ using PBE+SOC functional. The calculated band gap is 215 meV (direct), whereas the experimental band gap is 350 meV⁵⁰. The band gap is thus underestimated due to the DFT limitation arising from the approximations used in the $ϵ_{xc}$ functional. Therefore, we have used hybrid $ϵ_{xc}$ functional (HSE06) with default $α$ = 0.25, i.e., incorporating 25 $%$ of Hartree-Fock exact exchange to capture the electron’s self-interaction error along with SOC. It gives a direct band gap of 85 meV, which is also not in accordance with previously reported theoretical calculations⁴⁸. Therefore, we have performed G $_{0}$ W $_{0}$ calculations on top of the orbitals obtained from the PBE+SOC (G $_{0}$ W $_{0}$ @PBE+SOC) and HSE06+SOC (G $_{0}$ W $_{0}$ @HSE06+SOC) $ϵ_{xc}$ functional. The respective band gaps are 280 meV and 249 meV (see Table 1). G $_{0}$ W $_{0}$ @PBE+SOC gives the most accurate band gap, however, the band profile is not much affected by the choice of $ϵ_{xc}$ functional (see Section I of Supplementary Information). Therefore, we have used PBE $ϵ_{xc}$ functional to plot the band structures in view of its low computational cost. The band structure of TlBiSe $_{2}$ with the projected wavefunctions to atomic orbitals is shown in Fig. 2a. The Conduction Band (CB) is dominated by Bi-p and Tl-p orbitals, while Se-p orbitals dominate the Valence Band (VB). The inclusion of SOC has led to an increase in the band gap around the $Γ$ point. The valence and conduction band edges switch their orbital character around this point, indicating the band inversion. The p orbitals of Se and Bi are involved in this band inversion, as can be clearly seen from Fig. 2b.

(a) Primitive crystal structure of TlBiSe $_{2}$ , (b) 3D BZ for primitive unit cell with four time-reversal invariant points $Γ$ , Z, F and L along with the projected surface BZ, and (c) complete substitution with various elements at Tl, Bi, Se and Tl, Bi sites simultaneously.

Table 1.

Band gap of TlBiSe $_{2}$ using different $ϵ_{xc}$ functionals.

PBE+SOC (meV)	HSE06+SOC (meV)	G $_{0}$ W $_{0}$ @PBE+SOC (meV)	G $_{0}$ W $_{0}$ @HSE06+SOC (meV)
215	85	280	249

Open in a new tab

The band structures for TlBiSe $_{2}$ , GaBiSe $_{2}$ and PbBiSe $_{2}$ without SOC are shown in (a), (c), (e) and with SOC are shown in (b), (d), (f), respectively. Insets show band inversion at respective high symmetry points.

The similar band inversion has also been observed in TlBiTe $_{2}$ and TlSbSe $_{2}$ ⁴⁸. Following the trend, we have carried out complete substitution in TlBiSe $_{2}$ at Tl, Bi or Se sites and Tl, Bi sites simultaneously to obtain different materials belonging to the same class (see Fig. 1c). The band structures of these materials are plotted to see the effect of SOC on the orbital contribution projected on the bands lying near Fermi level. Firstly, Ga is substituted at the Tl site (Ga $_{Tl}$ ) to form GaBiSe $_{2}$ . It also crystallizes in R $\bar{3}$ m phase with lattice parameters given as a = b = c = 7.18 Å (in rhombohedral setting) and yields the band structure shown in Fig. 2c. SOC driven inversion of energy levels along with opening of band gap takes place at high symmetry point $Γ$ (Fig. 2d). The projected wavefunctions to atomic orbitals show that p orbitals of all atoms contribute near the Fermi level. The inversion involves Bi-p and Se-p orbitals (for details see Section II of Supplementary Information), giving an initial indication that the material can harbor non-trivial topological phase. The indirect band gap as calculated by G $_{0}$ W $_{0}$ @PBE+SOC is 183 meV.

After GaBiSe $_{2}$ , we have substituted Pb $_{Tl}$ to get PbBiSe $_{2}$ having lattice parameters as a = b = c = 8.27 Å, and crystallizing in the R $\bar{3}$ m phase. G $_{0}$ W $_{0}$ @PBE+SOC yields an indirect band gap of 3 meV. The VB and CB are mainly composed of p orbitals of Pb, Bi and Se, as shown in Fig. 2e and f. The parity inversion occurs at $Γ$ and F points, unlike the former. In this BZ, there are 8 TRIMs, i.e., $Γ$ , Z (non-degenerate) and F, L (triply-degenerate). Therefore, the inversion is occurring at even number of TRIMs, which means that the system should be in trivial state. However, it has been found that if even number of band inversions occur in the first quadrant, but if one or more BZ sides possess odd number of band inversions, then a WTI can be obtained³¹.

A similar type of parity inversion at even number of TRIMs is obtained for SnBiSe $_{2}$ , SbBiSe $_{2}$ , Bi $_{2}$ Se $_{2}$ , TlSnSe $_{2}$ and PbSbSe $_{2}$ (refer Section III and Section IV of Supplementary Information for band structures and optimized lattice parameters, respectively). The band gaps of all these materials calculated using G $_{0}$ W $_{0}$ @PBE+SOC are given in Table 2. After screening the systems for SOC-induced inversion in the band structures, we have analyzed them for dynamical stability. Figure 3 shows the phonon band structures for GaBiSe $_{2}$ and PbBiSe $_{2}$ . The absence of negative frequencies confirms the dynamical stability of these materials. For other systems showing band inversion, the dynamical stability is given in Section V of Supplementary Information.

Table 2.

Band gap of different materials using G $_{0}$ W $_{0}$ @PBE+SOC.

Material	Indirect band gap (meV)
GaBiSe $_{2}$	183
PbBiSe $_{2}$	3
SnBiSe $_{2}$	20
SbBiSe $_{2}$	118
Bi $_{2}$ Se $_{2}$	327
TlSnSe $_{2}$	44
PbSbSe $_{2}$	41

Open in a new tab

Phonon band structures of (a) GaBiSe $_{2}$ and (b) PbBiSe $_{2}$ .

Topological surface states and Z $_{2}$ invariant

The presence of band inversion on inclusion of SOC is a telltale signature of possibility of non-trivial phase. However, an inverted band structure cannot be considered as a sole criterion to assure the existence of a non-trivial phase. Therefore, further analysis is required to establish its topological nature. Previous studies have reported that non-trivial band topology generates metallic surface states, which are the hallmark of TIs⁵¹. In view of this, we have computed the spectrum of surface states by considering a semi infinite slab of 3D material. These lattice surfaces possess Dirac cones, lying at the same k-point where the band inversion has occurred in the corresponding bulk band structure. Furthermore, to elucidate the topological nature of the materials, Z $_{2}$ topological invariants are calculated. TRS yields four distinct Z $_{2}$ invariants ( $ν_{0}$ ; $ν_{1} ν_{2} ν_{3}$ ) in 3D case. Each of these four invariants takes up value either 0 or 1, indicating a total of 16 phases with three general classes: a normal insulator, an STI and a WTI²⁵. An ordinary or trivial insulator is obtained when all four invariants are zero, i.e., (0;000), while an STI is obtained when $ν_{0}$ = 1. This type of system is robust against weak time-reversal invariant perturbations. However, when $ν_{0}$ = 0, and at least one of the indices out of $ν_{1}$ , $ν_{2}$ or $ν_{3}$ is nonzero, then the material is WTI. It can be viewed as a stacking of 2D TIs, and is less robust against perturbations.

It has already been established that TlBiSe $_{2}$ is a strong 3D TI⁵². On that account, we first obtain its surface band structure. For this, a tight-binding Hamiltonian with MLWFs considering the projection of p orbitals of Bi/Se and sp orbitals of Tl is constructed. Since the left and right surface for TlBiSe $_{2}$ terminates with different atoms (Tl and Se, respectively), therefore, we have plotted surface state spectra of (111) surface for both the surface terminations. Figure 4a and b show a single surface state protected by TRS at the $Γ$ point in the projected 2D BZ. Alongside, we have calculated the topological invariant, $ν_{0}$ , which comes out to be 1, confirming that TlBiSe $_{2}$ is an STI. Following this, we have explored GaBiSe $_{2}$ for topological properties. In this case, the p orbitals of Bi/Se and sp orbitals of Ga are considered in constructing the tight-binding Hamiltonian. A single Dirac cone protected by TRS has been observed for (111) surface (see Fig. 4c and d), and the corresponding topological invariant is (1;000), which is a signature of non-trivial topology. This validates that GaBiSe $_{2}$ is an STI.

Surface states for the left surface of TlBiSe $_{2}$ , GaBiSe $_{2}$ and PbBiSe $_{2}$ are shown in (a), (c), (e) and for the right surface are shown in (b), (d), (f), respectively. Here, the sharp red curves represent surface states, whereas, the shaded regions show the spectral weight of projected bulk bands.

Afterwards, we have performed surface band structure calculation for PbBiSe $_{2}$ . We have obtained even (two) number of surface states lying at $Γ$ and M along (111) direction (see Fig. 4e and f). The presence of even number of surface states yields $ν_{0}$ = 0, and the weak indices come out to be $ν_{1}$ = 0, $ν_{2}$ = 0 and $ν_{3}$ = 1. As even number of surface states lead to scattering and are not topologically protected, hence this material is categorized as a WTI. The similar calculations are performed for SnBiSe $_{2}$ , SbBiSe $_{2}$ , Bi $_{2}$ Se $_{2}$ , TlSnSe $_{2}$ and PbSbSe $_{2}$ . All of them show even number of surface states along (111) direction, as shown in Section VI of Supplementary Information, and corresponding Z $_{2}$ invariants are given in Table 3. These materials belong to the class of Z $_{2}$ WTI. Nowadays, WTIs are also getting attention as it has been found that their surface states are robust against imperfections, owing to the delocalization of surface electrons⁵³.

Table 3.

Calculated Z $_{2}$ invariants for different materials.

Material	Z $_{2}$ :( $ν_{0}$ ; $ν_{1} ν_{2} ν_{3}$ )	Number of surface states	Type
GaBiSe $_{2}$	(1;000)	1	STI
PbBiSe $_{2}$	(0;001)	2	WTI
SnBiSe $_{2}$	(0;001)	2	WTI
SbBiSe $_{2}$	(0;001)	2	WTI
Bi $_{2}$ Se $_{2}$	(0;001)	2	WTI
TlSnSe $_{2}$	(0;001)	2	WTI
PbSbSe $_{2}$	(0;111)	2	WTI

Open in a new tab

Conclusions

In summary, we have performed isostructural substitution of materials based on SOC-induced parity inversion in the band structure. Using first-principles based methodologies, viz., PBE, HSE06, and many-body perturbation theory (G $_{0}$ W $_{0}$ ), we have systematically studied the electronic structure of topological materials belonging to R $\bar{3}$ m space group. The band gap calculated using G $_{0}$ W $_{0}$ @PBE+SOC is in close agreement with the experimental value. We have confirmed that GaBiSe $_{2}$ is an STI as its surface accommodates a single crossing at the $Γ$ point. PbBiSe $_{2}$ , SnBiSe $_{2}$ , SbBiSe $_{2}$ , Bi $_{2}$ Se $_{2}$ , TlSnSe $_{2}$ and PbSbSe $_{2}$ are WTI catering even number of surface states within the bulk band gap. The absence of negative frequencies in the phonon band structures indicates dynamical stability. The calculated Z $_{2}$ invariants are in accordance with the surface state plots confirming their topological nature. We believe, exploring these materials both theoretically and experimentally will offer a great platform for studying intriguing quantum effects.

Methods

The calculations are performed using DFT^54,55 with the Projected Augmented Wave (PAW)^56,57 method implemented in Vienna Ab initio Simulation Package (VASP)⁵⁸ code. All the structures are optimized with the Generalized Gradient Approximation (GGA) of Perdew-Burke-Ernzerhof (PBE)⁵⁹ until the Hellmann-Feynman forces are smaller than 0.001 eV/Å. The plane wave basis is used with 400 eV cutoff energy. The $Γ$ -centered 6 $\times$ 6 $\times$ 4 k-grid is used to sample the irreducible BZ of rhombohedral phase with the R $\bar{3}$ m space group. SOC is included in all calculations except in ionic optimization. The advanced hybrid $ϵ_{xc}$ functional (HSE06) including SOC and many-body perturbation methods, G $_{0}$ W $_{0}$ @PBE+SOC and G $_{0}$ W $_{0}$ @HSE06+SOC are used for the better estimation of the band gap⁶⁰. For this, we have used 4 $\times$ 4 $\times$ 4 k-grid, and the number of bands is set to six times the number of occupied bands. The phonon calculations are performed with 4 $\times$ 4 $\times$ 4 supercells using the PHONOPY package^61,62. In order to investigate the topological properties of the materials, we have performed DFT calculations using fully relativistic norm-conserving pseudopotentials as implemented in the QUANTUM ESPRESSO code⁶³. The results of these DFT calculations are then fed as input to WANNIER90⁶⁴ for constructing a tight-binding model based on Maximally Localized Wannier Functions (MLWFs) with p orbitals of Se, Bi and sp orbitals of Ga, Tl, Sn, Sb, Pb. The surface states and topological invariants are then calculated using the Green’s function method as implemented in the Wannier-TOOLS package⁶⁵.

Supplementary Information

Supplementary Information.^{(11.4MB, pdf)}

Acknowledgements

A.P. acknowledges IIT Delhi for the junior research fellowship. P.B. acknowledges UGC, India, for the senior research fellowship [Grant no. 1392/(CSIR-UGC NET JUNE 2018)]. M.J. acknowledges CSIR, India, for the senior research fellowship [Grant No. 09/086(1344)/2018-EMR-I]. S.B. acknowledges financial support from SERB under a core research grant (Grant no. CRG/2019/000647) to set up his High Performance Computing (HPC) facility “Veena” at IIT Delhi for computational resources.

Author contributions

S.B. and A.P. conceived the project. S.B. supervised overall. A.P. performed all the calculations. A.P., P.B. and M.J. got involved in various discussion to analyze the data. All authors took part in finalizing the manuscript.

Data availibility

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Footnotes

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Ankita Phutela, Email: ankita@physics.iitd.ac.in.

Saswata Bhattacharya, Email: saswata@physics.iitd.ac.in.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-022-26445-y.

References

1.Fan Z, Liang Q-F, Chen Y, Yao S-H, Zhou J. Transition between strong and weak topological insulator in ZrTe $_{5}$ and HfTe $_{5}$ . Sci. Rep. 2017;7:1–7. doi: 10.1038/srep45667. [DOI] [PMC free article] [PubMed] [Google Scholar]
2.Feng L-Y, et al. Prediction of topological Dirac semimetal in Ca-based Zintl layered compounds CaM2X2 (M= Zn or Cd; X= N, P, As, Sb, or Bi) Sci. Rep. 2022;12:1–8. doi: 10.1038/s41598-022-08370-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
3.Moore JE. The birth of topological insulators. Nature. 2010;464:194–198. doi: 10.1038/nature08916. [DOI] [PubMed] [Google Scholar]
4.Fu L, Kane CL, Mele EJ. Topological insulators in three dimensions. Phys. Rev. Lett. 2007;98:106803. doi: 10.1103/PhysRevLett.98.106803. [DOI] [PubMed] [Google Scholar]
5.Niu C, et al. Realization of tunable Dirac cone and insulating bulk states in topological insulators (Bi $_{(1 - x)}$ Sb $_{x}$ ) $_{2}$ Te $_{3}$ . Sci. Rep. 2012;2:1–7. doi: 10.1038/srep00976. [DOI] [PMC free article] [PubMed] [Google Scholar]
6.Kou L, et al. Robust 2d topological insulators in van der waals heterostructures. ACS Nano. 2014;8:10448–10454. doi: 10.1021/nn503789v. [DOI] [PubMed] [Google Scholar]
7.Ma Y, Dai Y, Kou L, Frauenheim T, Heine T. Robust two-dimensional topological insulators in methyl-functionalized bismuth, antimony, and lead bilayer films. Nano Lett. 2015;15:1083–1089. doi: 10.1021/nl504037u. [DOI] [PubMed] [Google Scholar]
8.Chen Y, et al. High-pressure phase transitions and structures of topological insulator BiTeI. J. Phys. Chem. C. 2013;117:25677–25683. doi: 10.1021/jp409824g. [DOI] [Google Scholar]
9.Qi XL, Hughes TL, Zhang SC. Topological field theory of time-reversal invariant insulators. Phys. Rev. B. 2008;78:195424. doi: 10.1103/PhysRevB.78.195424. [DOI] [Google Scholar]
10.Moore JE, Balents L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B. 2007;75:121306. doi: 10.1103/PhysRevB.75.121306. [DOI] [Google Scholar]
11.Hasan MZ, Kane CL. Colloquium: Topological insulators. Rev. Mod. Phys. 2010;82:3045. doi: 10.1103/RevModPhys.82.3045. [DOI] [Google Scholar]
12.Chiu CK, Teo JC, Schnyder AP, Ryu S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 2016;88:035005. doi: 10.1103/RevModPhys.88.035005. [DOI] [Google Scholar]
13.Qi XL, Zhang SC. Topological insulators and superconductors. Rev. Mod. Phys. 2011;83:1057. doi: 10.1103/RevModPhys.83.1057. [DOI] [Google Scholar]
14.Wan X, Turner AM, Vishwanath A, Savrasov SY. Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B. 2011;83:205101. doi: 10.1103/PhysRevB.83.205101. [DOI] [Google Scholar]
15.Singh B, et al. Topological electronic structure and weyl semimetal in the TlBiSe $_{2}$ class of semiconductors. Phys. Rev. B. 2012;86:115208. doi: 10.1103/PhysRevB.86.115208. [DOI] [Google Scholar]
16.Wilczek F. Majorana returns. Nat. Phys. 2009;5:614–618. doi: 10.1038/nphys1380. [DOI] [Google Scholar]
17.Sato T, et al. Unexpected mass acquisition of dirac fermions at the quantum phase transition of a topological insulator. Nat. Phys. 2011;7:840–844. doi: 10.1038/nphys2058. [DOI] [Google Scholar]
18.Kane CL, Mele EJ. Z $_{2}$ topological order and the quantum spin hall effect. Phys. Rev. Lett. 2005;95:146802. doi: 10.1103/PhysRevLett.95.146802. [DOI] [PubMed] [Google Scholar]
19.Eremeev S, Koroteev YM, Chulkov EV. Ternary thallium-based semimetal chalcogenides Tl-V-VI2 as a new class of three-dimensional topological insulators. JETP Lett. 2010;91:594–598. doi: 10.1134/S0021364010110111. [DOI] [Google Scholar]
20.Singh B, Lin H, Prasad R, Bansil A. Topological phase transition and quantum spin hall state in TlBiS $_{2}$ . J. Appl. Phys. 2014;116:033704. doi: 10.1063/1.4890226. [DOI] [Google Scholar]
21.Lee K, et al. Discovery of a weak topological insulating state and van hove singularity in triclinic RhBi $_{2}$ . Nat. Commun. 2021;12:1–8. doi: 10.1038/s41467-021-22136-w. [DOI] [PMC free article] [PubMed] [Google Scholar]
22.Rusinov IP, et al. Mirror-symmetry protected non-trim surface state in the weak topological insulator Bi $_{2}$ TeI. Sci. Rep. 2016;6:1–7. doi: 10.1038/srep20734. [DOI] [PMC free article] [PubMed] [Google Scholar]
23.Majhi K, et al. Emergence of a weak topological insulator from the Bi $_{x}$ Se $_{y}$ family. Appl. Phys. Lett. 2017;110:162102. doi: 10.1063/1.4981875. [DOI] [Google Scholar]
24.Imura KI, Takane Y, Tanaka A. Weak topological insulator with protected gapless helical states. Phys. Rev. B. 2011;84:035443. doi: 10.1103/PhysRevB.84.035443. [DOI] [Google Scholar]
25.Noguchi R, et al. A weak topological insulator state in quasi-one-dimensional bismuth iodide. Nature. 2019;566:518–522. doi: 10.1038/s41586-019-0927-7. [DOI] [PubMed] [Google Scholar]
26.Zhao M, Dong W, Wang A. Two-dimensional carbon topological insulators superior to graphene. Sci. Rep. 2013;3:1–6. doi: 10.1038/srep03532. [DOI] [PMC free article] [PubMed] [Google Scholar]
27.Li J, et al. Two-dimensional topological insulators with tunable band gaps: Single-layer HgTe and HgSe. Sci. Rep. 2015;5:1–9. doi: 10.1038/srep14115. [DOI] [PMC free article] [PubMed] [Google Scholar]
28.Lohani H, et al. Band structure of topological insulator BiSbTe1.25Se1.75. Sci. Rep. 2017;7:1–10. doi: 10.1038/s41598-017-04985-y. [DOI] [PMC free article] [PubMed] [Google Scholar]
29.Chuang FC, et al. Prediction of large-gap two-dimensional topological insulators consisting of bilayers of group III elements with Bi. Nano Lett. 2014;14:2505–2508. doi: 10.1021/nl500206u. [DOI] [PubMed] [Google Scholar]
30.Fu B, Ge Y, Su W, Guo W, Liu C-C. A new kind of 2D topological insulators BiCN with a giant gap and its substrate effects. Sci. Rep. 2016;6:1–7. doi: 10.1038/srep30003. [DOI] [PMC free article] [PubMed] [Google Scholar]
31.Das, T. A Pedagogic Review on Designing Model Topological Insulators. arXiv:1604.07546. (2016).
32.Zhou L, et al. New family of quantum spin hall insulators in two-dimensional transition-metal halide with large nontrivial band gaps. Nano Lett. 2015;15:7867–7872. doi: 10.1021/acs.nanolett.5b02617. [DOI] [PubMed] [Google Scholar]
33.Hsieh TH, et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 2012;3:1–7. doi: 10.1038/ncomms1969. [DOI] [PubMed] [Google Scholar]
34.Kou L, Ma Y, Sun Z, Heine T, Chen C. Two-dimensional topological insulators: Progress and prospects. J. Phys. Chem. Lett. 2017;8:1905–1919. doi: 10.1021/acs.jpclett.7b00222. [DOI] [PubMed] [Google Scholar]
35.Edmonds MT, et al. Stability and surface reconstruction of topological insulator Bi $_{2}$ Se $_{3}$ on exposure to atmosphere. J. Phys. Chem. C. 2014;118:20413–20419. doi: 10.1021/jp506089b. [DOI] [Google Scholar]
36.Yang J, et al. Robust topological states in Bi $_{2}$ Se $_{3}$ against surface oxidation. J. Phys. Chem. C. 2020;124:6253–6259. doi: 10.1021/acs.jpcc.0c00458. [DOI] [Google Scholar]
37.Zhang H, et al. Topological insulators in Bi $_{2}$ Se $_{3}$ , Bi $_{2}$ Te $_{3}$ and Sb $_{2}$ Te $_{3}$ with a single Dirac cone on the surface. Nat. Phys. 2009;5:438–442. doi: 10.1038/nphys1270. [DOI] [Google Scholar]
38.Reid TK, Alpay SP, Balatsky AV, Nayak SK. First-principles modeling of binary layered topological insulators: Structural optimization and exchange-correlation functionals. Phys. Rev. B. 2020;101:085140. doi: 10.1103/PhysRevB.101.085140. [DOI] [Google Scholar]
39.Brüne C, et al. Quantum hall effect from the topological surface states of strained bulk HgTe. Phys. Rev. Lett. 2011;106:126803. doi: 10.1103/PhysRevLett.106.126803. [DOI] [PubMed] [Google Scholar]
40.Lin H, et al. Adiabatic transformation as a search tool for new topological insulators: Distorted ternary Li $_{2}$ AgSb-class semiconductors and related compounds. Phys. Rev. B. 2013;87:121202. doi: 10.1103/PhysRevB.87.121202. [DOI] [Google Scholar]
41.Feng W, Xiao D, Zhang Y, Yao Y. Half-Heusler topological insulators: A first-principles study with the Tran-Blaha modified Becke-Johnson density functional. Phys. Rev. B. 2010;82:235121. doi: 10.1103/PhysRevB.82.235121. [DOI] [Google Scholar]
42.Xiao D, et al. Half-heusler compounds as a new class of three-dimensional topological insulators. Phys. Rev. Lett. 2010;105:096404. doi: 10.1103/PhysRevLett.105.096404. [DOI] [PubMed] [Google Scholar]
43.Zhang H-J, et al. Topological insulators in ternary compounds with a honeycomb lattice. Phys. Rev. Lett. 2011;106:156402. doi: 10.1103/PhysRevLett.106.156402. [DOI] [PubMed] [Google Scholar]
44.Lee S, et al. Single crystalline $β$ -Ag $_{2}$ Te nanowire as a new topological insulator. Nano Lett. 2012;12:4194–4199. doi: 10.1021/nl301763r. [DOI] [PubMed] [Google Scholar]
45.Landolt G, et al. Bulk and surface rashba splitting in single termination BiTeCl. New J. Phys. 2013;15:085022. doi: 10.1088/1367-2630/15/8/085022. [DOI] [Google Scholar]
46.Lin H, et al. Single-Dirac-cone topological surface states in the TlBiSe $_{2}$ class of topological semiconductors. Phys. Rev. Lett. 2010;105:036404. doi: 10.1103/PhysRevLett.105.036404. [DOI] [PubMed] [Google Scholar]
47.Zhang Q, Cheng Y, Schwingenschlögl U. Emergence of topological and topological crystalline phases in TlBiS $_{2}$ and TlSbS $_{2}$ . Sci. Rep. 2015;5:1–7. doi: 10.1038/srep08379. [DOI] [PMC free article] [PubMed] [Google Scholar]
48.Singh B, Lin H, Prasad R, Bansil A. Role of surface termination in realizing well-isolated topological surface states within the bulk band gap in TlBiSe $_{2}$ and TlBiTe $_{2}$ . Phys. Rev. B. 2016;93:085113. doi: 10.1103/PhysRevB.93.085113. [DOI] [Google Scholar]
49.Eremeev S, et al. Ab initio electronic structure of thallium-based topological insulators. Phys. Rev. B. 2011;83:205129. doi: 10.1103/PhysRevB.83.205129. [DOI] [Google Scholar]
50.Sato T, et al. Direct evidence for the Dirac-cone topological surface states in the ternary chalcogenide TlBiSe $_{2}$ . Phys. Rev. Lett. 2010;105:136802. doi: 10.1103/PhysRevLett.105.136802. [DOI] [PubMed] [Google Scholar]
51.Hsieh D, et al. A topological Dirac insulator in a quantum spin hall phase. Nature. 2008;452:970–974. doi: 10.1038/nature06843. [DOI] [PubMed] [Google Scholar]
52.Chen Y, et al. Experimental realization of a three-dimensional topological insulator phase in ternary chalcogenide TlBiSe $_{2}$ . Phys. Rev. Lett. 2010;105:146801. doi: 10.1103/PhysRevLett.105.146801. [DOI] [PubMed] [Google Scholar]
53.Ringel Z, Kraus YE, Stern A. Strong side of weak topological insulators. Phys. Rev. B. 2012;86:045102. doi: 10.1103/PhysRevB.86.045102. [DOI] [Google Scholar]
54.Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys. Rev. 1964;136:B864. doi: 10.1103/PhysRev.136.B864. [DOI] [Google Scholar]
55.Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965;140:A1133. doi: 10.1103/PhysRev.140.A1133. [DOI] [Google Scholar]
56.Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B. 1999;59:1758. doi: 10.1103/PhysRevB.59.1758. [DOI] [Google Scholar]
57.Blöchl PE. Projector augmented-wave method. Phys. Rev. B. 1994;50:17953. doi: 10.1103/PhysRevB.50.17953. [DOI] [PubMed] [Google Scholar]
58.Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 1996;54:11169. doi: 10.1103/PhysRevB.54.11169. [DOI] [PubMed] [Google Scholar]
59.Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996;77:3865. doi: 10.1103/PhysRevLett.77.3865. [DOI] [PubMed] [Google Scholar]
60.Hedin L. New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys. Rev. 1965;139:A796. doi: 10.1103/PhysRev.139.A796. [DOI] [Google Scholar]
61.Togo A, Oba F, Tanaka I. First-principles calculations of the ferroelastic transition between rutile-type and CaCl $_{2}$ -type SiO $_{2}$ at high pressures. Phys. Rev. B. 2008;78:134106. doi: 10.1103/PhysRevB.78.134106. [DOI] [Google Scholar]
62.Togo A, Tanaka I. First principles phonon calculations in materials science. Scr. Mater. 2015;108:1–5. doi: 10.1016/j.scriptamat.2015.07.021. [DOI] [Google Scholar]
63.Giannozzi P, et al. Quantum Espresso: A modular and open-source software project for quantum simulations of materials. J. Phys.: Condens. Matter. 2009;21:395502. doi: 10.1088/0953-8984/21/39/395502. [DOI] [PubMed] [Google Scholar]
64.Mostofi AA, et al. Wannier90: A tool for obtaining maximally-localised wannier functions. Comput. Phys. Commun. 2008;178:685–699. doi: 10.1016/j.cpc.2007.11.016. [DOI] [Google Scholar]
65.Wu Q, Zhang S, Song H-F, Troyer M, Soluyanov AA. Wanniertools: An open-source software package for novel topological materials. Comput. Phys. Commun. 2018;224:405–416. doi: 10.1016/j.cpc.2017.09.033. [DOI] [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supplementary Information.^{(11.4MB, pdf)}

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

[CR1] 1.Fan Z, Liang Q-F, Chen Y, Yao S-H, Zhou J. Transition between strong and weak topological insulator in ZrTe $_{5}$ and HfTe $_{5}$ . Sci. Rep. 2017;7:1–7. doi: 10.1038/srep45667. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR2] 2.Feng L-Y, et al. Prediction of topological Dirac semimetal in Ca-based Zintl layered compounds CaM2X2 (M= Zn or Cd; X= N, P, As, Sb, or Bi) Sci. Rep. 2022;12:1–8. doi: 10.1038/s41598-022-08370-2. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR3] 3.Moore JE. The birth of topological insulators. Nature. 2010;464:194–198. doi: 10.1038/nature08916. [DOI] [PubMed] [Google Scholar]

[CR4] 4.Fu L, Kane CL, Mele EJ. Topological insulators in three dimensions. Phys. Rev. Lett. 2007;98:106803. doi: 10.1103/PhysRevLett.98.106803. [DOI] [PubMed] [Google Scholar]

[CR5] 5.Niu C, et al. Realization of tunable Dirac cone and insulating bulk states in topological insulators (Bi $_{(1 - x)}$ Sb $_{x}$ ) $_{2}$ Te $_{3}$ . Sci. Rep. 2012;2:1–7. doi: 10.1038/srep00976. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR6] 6.Kou L, et al. Robust 2d topological insulators in van der waals heterostructures. ACS Nano. 2014;8:10448–10454. doi: 10.1021/nn503789v. [DOI] [PubMed] [Google Scholar]

[CR7] 7.Ma Y, Dai Y, Kou L, Frauenheim T, Heine T. Robust two-dimensional topological insulators in methyl-functionalized bismuth, antimony, and lead bilayer films. Nano Lett. 2015;15:1083–1089. doi: 10.1021/nl504037u. [DOI] [PubMed] [Google Scholar]

[CR8] 8.Chen Y, et al. High-pressure phase transitions and structures of topological insulator BiTeI. J. Phys. Chem. C. 2013;117:25677–25683. doi: 10.1021/jp409824g. [DOI] [Google Scholar]

[CR9] 9.Qi XL, Hughes TL, Zhang SC. Topological field theory of time-reversal invariant insulators. Phys. Rev. B. 2008;78:195424. doi: 10.1103/PhysRevB.78.195424. [DOI] [Google Scholar]

[CR10] 10.Moore JE, Balents L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B. 2007;75:121306. doi: 10.1103/PhysRevB.75.121306. [DOI] [Google Scholar]

[CR11] 11.Hasan MZ, Kane CL. Colloquium: Topological insulators. Rev. Mod. Phys. 2010;82:3045. doi: 10.1103/RevModPhys.82.3045. [DOI] [Google Scholar]

[CR12] 12.Chiu CK, Teo JC, Schnyder AP, Ryu S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 2016;88:035005. doi: 10.1103/RevModPhys.88.035005. [DOI] [Google Scholar]

[CR13] 13.Qi XL, Zhang SC. Topological insulators and superconductors. Rev. Mod. Phys. 2011;83:1057. doi: 10.1103/RevModPhys.83.1057. [DOI] [Google Scholar]

[CR14] 14.Wan X, Turner AM, Vishwanath A, Savrasov SY. Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B. 2011;83:205101. doi: 10.1103/PhysRevB.83.205101. [DOI] [Google Scholar]

[CR15] 15.Singh B, et al. Topological electronic structure and weyl semimetal in the TlBiSe $_{2}$ class of semiconductors. Phys. Rev. B. 2012;86:115208. doi: 10.1103/PhysRevB.86.115208. [DOI] [Google Scholar]

[CR16] 16.Wilczek F. Majorana returns. Nat. Phys. 2009;5:614–618. doi: 10.1038/nphys1380. [DOI] [Google Scholar]

[CR17] 17.Sato T, et al. Unexpected mass acquisition of dirac fermions at the quantum phase transition of a topological insulator. Nat. Phys. 2011;7:840–844. doi: 10.1038/nphys2058. [DOI] [Google Scholar]

[CR18] 18.Kane CL, Mele EJ. Z $_{2}$ topological order and the quantum spin hall effect. Phys. Rev. Lett. 2005;95:146802. doi: 10.1103/PhysRevLett.95.146802. [DOI] [PubMed] [Google Scholar]

[CR19] 19.Eremeev S, Koroteev YM, Chulkov EV. Ternary thallium-based semimetal chalcogenides Tl-V-VI2 as a new class of three-dimensional topological insulators. JETP Lett. 2010;91:594–598. doi: 10.1134/S0021364010110111. [DOI] [Google Scholar]

[CR20] 20.Singh B, Lin H, Prasad R, Bansil A. Topological phase transition and quantum spin hall state in TlBiS $_{2}$ . J. Appl. Phys. 2014;116:033704. doi: 10.1063/1.4890226. [DOI] [Google Scholar]

[CR21] 21.Lee K, et al. Discovery of a weak topological insulating state and van hove singularity in triclinic RhBi $_{2}$ . Nat. Commun. 2021;12:1–8. doi: 10.1038/s41467-021-22136-w. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR22] 22.Rusinov IP, et al. Mirror-symmetry protected non-trim surface state in the weak topological insulator Bi $_{2}$ TeI. Sci. Rep. 2016;6:1–7. doi: 10.1038/srep20734. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR23] 23.Majhi K, et al. Emergence of a weak topological insulator from the Bi $_{x}$ Se $_{y}$ family. Appl. Phys. Lett. 2017;110:162102. doi: 10.1063/1.4981875. [DOI] [Google Scholar]

[CR24] 24.Imura KI, Takane Y, Tanaka A. Weak topological insulator with protected gapless helical states. Phys. Rev. B. 2011;84:035443. doi: 10.1103/PhysRevB.84.035443. [DOI] [Google Scholar]

[CR25] 25.Noguchi R, et al. A weak topological insulator state in quasi-one-dimensional bismuth iodide. Nature. 2019;566:518–522. doi: 10.1038/s41586-019-0927-7. [DOI] [PubMed] [Google Scholar]

[CR26] 26.Zhao M, Dong W, Wang A. Two-dimensional carbon topological insulators superior to graphene. Sci. Rep. 2013;3:1–6. doi: 10.1038/srep03532. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR27] 27.Li J, et al. Two-dimensional topological insulators with tunable band gaps: Single-layer HgTe and HgSe. Sci. Rep. 2015;5:1–9. doi: 10.1038/srep14115. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR28] 28.Lohani H, et al. Band structure of topological insulator BiSbTe1.25Se1.75. Sci. Rep. 2017;7:1–10. doi: 10.1038/s41598-017-04985-y. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR29] 29.Chuang FC, et al. Prediction of large-gap two-dimensional topological insulators consisting of bilayers of group III elements with Bi. Nano Lett. 2014;14:2505–2508. doi: 10.1021/nl500206u. [DOI] [PubMed] [Google Scholar]

[CR30] 30.Fu B, Ge Y, Su W, Guo W, Liu C-C. A new kind of 2D topological insulators BiCN with a giant gap and its substrate effects. Sci. Rep. 2016;6:1–7. doi: 10.1038/srep30003. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR31] 31.Das, T. A Pedagogic Review on Designing Model Topological Insulators. arXiv:1604.07546. (2016).

[CR32] 32.Zhou L, et al. New family of quantum spin hall insulators in two-dimensional transition-metal halide with large nontrivial band gaps. Nano Lett. 2015;15:7867–7872. doi: 10.1021/acs.nanolett.5b02617. [DOI] [PubMed] [Google Scholar]

[CR33] 33.Hsieh TH, et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 2012;3:1–7. doi: 10.1038/ncomms1969. [DOI] [PubMed] [Google Scholar]

[CR34] 34.Kou L, Ma Y, Sun Z, Heine T, Chen C. Two-dimensional topological insulators: Progress and prospects. J. Phys. Chem. Lett. 2017;8:1905–1919. doi: 10.1021/acs.jpclett.7b00222. [DOI] [PubMed] [Google Scholar]

[CR35] 35.Edmonds MT, et al. Stability and surface reconstruction of topological insulator Bi $_{2}$ Se $_{3}$ on exposure to atmosphere. J. Phys. Chem. C. 2014;118:20413–20419. doi: 10.1021/jp506089b. [DOI] [Google Scholar]

[CR36] 36.Yang J, et al. Robust topological states in Bi $_{2}$ Se $_{3}$ against surface oxidation. J. Phys. Chem. C. 2020;124:6253–6259. doi: 10.1021/acs.jpcc.0c00458. [DOI] [Google Scholar]

[CR37] 37.Zhang H, et al. Topological insulators in Bi $_{2}$ Se $_{3}$ , Bi $_{2}$ Te $_{3}$ and Sb $_{2}$ Te $_{3}$ with a single Dirac cone on the surface. Nat. Phys. 2009;5:438–442. doi: 10.1038/nphys1270. [DOI] [Google Scholar]

[CR38] 38.Reid TK, Alpay SP, Balatsky AV, Nayak SK. First-principles modeling of binary layered topological insulators: Structural optimization and exchange-correlation functionals. Phys. Rev. B. 2020;101:085140. doi: 10.1103/PhysRevB.101.085140. [DOI] [Google Scholar]

[CR39] 39.Brüne C, et al. Quantum hall effect from the topological surface states of strained bulk HgTe. Phys. Rev. Lett. 2011;106:126803. doi: 10.1103/PhysRevLett.106.126803. [DOI] [PubMed] [Google Scholar]

[CR40] 40.Lin H, et al. Adiabatic transformation as a search tool for new topological insulators: Distorted ternary Li $_{2}$ AgSb-class semiconductors and related compounds. Phys. Rev. B. 2013;87:121202. doi: 10.1103/PhysRevB.87.121202. [DOI] [Google Scholar]

[CR41] 41.Feng W, Xiao D, Zhang Y, Yao Y. Half-Heusler topological insulators: A first-principles study with the Tran-Blaha modified Becke-Johnson density functional. Phys. Rev. B. 2010;82:235121. doi: 10.1103/PhysRevB.82.235121. [DOI] [Google Scholar]

[CR42] 42.Xiao D, et al. Half-heusler compounds as a new class of three-dimensional topological insulators. Phys. Rev. Lett. 2010;105:096404. doi: 10.1103/PhysRevLett.105.096404. [DOI] [PubMed] [Google Scholar]

[CR43] 43.Zhang H-J, et al. Topological insulators in ternary compounds with a honeycomb lattice. Phys. Rev. Lett. 2011;106:156402. doi: 10.1103/PhysRevLett.106.156402. [DOI] [PubMed] [Google Scholar]

[CR44] 44.Lee S, et al. Single crystalline $β$ -Ag $_{2}$ Te nanowire as a new topological insulator. Nano Lett. 2012;12:4194–4199. doi: 10.1021/nl301763r. [DOI] [PubMed] [Google Scholar]

[CR45] 45.Landolt G, et al. Bulk and surface rashba splitting in single termination BiTeCl. New J. Phys. 2013;15:085022. doi: 10.1088/1367-2630/15/8/085022. [DOI] [Google Scholar]

[CR46] 46.Lin H, et al. Single-Dirac-cone topological surface states in the TlBiSe $_{2}$ class of topological semiconductors. Phys. Rev. Lett. 2010;105:036404. doi: 10.1103/PhysRevLett.105.036404. [DOI] [PubMed] [Google Scholar]

[CR47] 47.Zhang Q, Cheng Y, Schwingenschlögl U. Emergence of topological and topological crystalline phases in TlBiS $_{2}$ and TlSbS $_{2}$ . Sci. Rep. 2015;5:1–7. doi: 10.1038/srep08379. [DOI] [PMC free article] [PubMed] [Google Scholar]

[CR48] 48.Singh B, Lin H, Prasad R, Bansil A. Role of surface termination in realizing well-isolated topological surface states within the bulk band gap in TlBiSe $_{2}$ and TlBiTe $_{2}$ . Phys. Rev. B. 2016;93:085113. doi: 10.1103/PhysRevB.93.085113. [DOI] [Google Scholar]

[CR49] 49.Eremeev S, et al. Ab initio electronic structure of thallium-based topological insulators. Phys. Rev. B. 2011;83:205129. doi: 10.1103/PhysRevB.83.205129. [DOI] [Google Scholar]

[CR50] 50.Sato T, et al. Direct evidence for the Dirac-cone topological surface states in the ternary chalcogenide TlBiSe $_{2}$ . Phys. Rev. Lett. 2010;105:136802. doi: 10.1103/PhysRevLett.105.136802. [DOI] [PubMed] [Google Scholar]

[CR51] 51.Hsieh D, et al. A topological Dirac insulator in a quantum spin hall phase. Nature. 2008;452:970–974. doi: 10.1038/nature06843. [DOI] [PubMed] [Google Scholar]

[CR52] 52.Chen Y, et al. Experimental realization of a three-dimensional topological insulator phase in ternary chalcogenide TlBiSe $_{2}$ . Phys. Rev. Lett. 2010;105:146801. doi: 10.1103/PhysRevLett.105.146801. [DOI] [PubMed] [Google Scholar]

[CR53] 53.Ringel Z, Kraus YE, Stern A. Strong side of weak topological insulators. Phys. Rev. B. 2012;86:045102. doi: 10.1103/PhysRevB.86.045102. [DOI] [Google Scholar]

[CR54] 54.Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys. Rev. 1964;136:B864. doi: 10.1103/PhysRev.136.B864. [DOI] [Google Scholar]

[CR55] 55.Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965;140:A1133. doi: 10.1103/PhysRev.140.A1133. [DOI] [Google Scholar]

[CR56] 56.Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B. 1999;59:1758. doi: 10.1103/PhysRevB.59.1758. [DOI] [Google Scholar]

[CR57] 57.Blöchl PE. Projector augmented-wave method. Phys. Rev. B. 1994;50:17953. doi: 10.1103/PhysRevB.50.17953. [DOI] [PubMed] [Google Scholar]

[CR58] 58.Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 1996;54:11169. doi: 10.1103/PhysRevB.54.11169. [DOI] [PubMed] [Google Scholar]

[CR59] 59.Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996;77:3865. doi: 10.1103/PhysRevLett.77.3865. [DOI] [PubMed] [Google Scholar]

[CR60] 60.Hedin L. New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys. Rev. 1965;139:A796. doi: 10.1103/PhysRev.139.A796. [DOI] [Google Scholar]

[CR61] 61.Togo A, Oba F, Tanaka I. First-principles calculations of the ferroelastic transition between rutile-type and CaCl $_{2}$ -type SiO $_{2}$ at high pressures. Phys. Rev. B. 2008;78:134106. doi: 10.1103/PhysRevB.78.134106. [DOI] [Google Scholar]

[CR62] 62.Togo A, Tanaka I. First principles phonon calculations in materials science. Scr. Mater. 2015;108:1–5. doi: 10.1016/j.scriptamat.2015.07.021. [DOI] [Google Scholar]

[CR63] 63.Giannozzi P, et al. Quantum Espresso: A modular and open-source software project for quantum simulations of materials. J. Phys.: Condens. Matter. 2009;21:395502. doi: 10.1088/0953-8984/21/39/395502. [DOI] [PubMed] [Google Scholar]

[CR64] 64.Mostofi AA, et al. Wannier90: A tool for obtaining maximally-localised wannier functions. Comput. Phys. Commun. 2008;178:685–699. doi: 10.1016/j.cpc.2007.11.016. [DOI] [Google Scholar]

[CR65] 65.Wu Q, Zhang S, Song H-F, Troyer M, Soluyanov AA. Wanniertools: An open-source software package for novel topological materials. Comput. Phys. Commun. 2018;224:405–416. doi: 10.1016/j.cpc.2017.09.033. [DOI] [Google Scholar]

PERMALINK

Exploring strong and weak topological states on isostructural substitutions in TlBiSe $_{2}$

Ankita Phutela

Preeti Bhumla

Manjari Jain

Saswata Bhattacharya

Abstract

Introduction

Results and discussion