Introduction

Since the discovery of Topological Insulators (TIs) about a decade ago, there has been an enormous increase in interest towards topological condensed matter systems1,2,3,4,5,6,7,8,9,10,11,12,13. TIs show great potential applications in quantum computing and spintronics due to the insensitivity of the transport property towards non-magnetic perturbations13,14. 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, \(\nu _0\), 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 \(\nu _0\) = 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, \(\nu _0\) = 1 or Weak Topological Insulator (WTI) with Z\(_2\) invariant, \(\nu _0\) = 022,23. Nevertheless, the complete characterization of 3D TIs requires a set of, in total, four Z\(_2\) numbers: (\(\nu _0\);\(\nu _{1}\nu _{2}\nu _{3}\)). The indices \(\nu _1\), \(\nu _2\) and \(\nu _3\) are called weak indices and are believed to be nonrobust quantities24,25. 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 TIs22,26,27,28,29,30,31,32,33,34. 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 properties35,36,37. 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\(_{2n}\)Te\(_{(m+3n)}\), half-Huesler compounds41,42, LiAuSe honeycomb lattice43, \(\beta\)-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 \(\Gamma\) 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) calculations20,47. 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 (\(\epsilon _{xc}\)) 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

Figure 1
figure 1

(a) Primitive crystal structure of TlBiSe\(_2\), (b) 3D BZ for primitive unit cell with four time-reversal invariant points \(\Gamma\), 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.

Full size image

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, \(\Gamma\), 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 \(\epsilon _{xc}\) functional. Therefore, we have used hybrid \(\epsilon _{xc}\) functional (HSE06) with default \(\alpha\) = 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) \(\epsilon _{xc}\) 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 \(\epsilon _{xc}\) functional (see Section I of Supplementary Information). Therefore, we have used PBE \(\epsilon _{xc}\) 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 \(\Gamma\) 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.

Table 1 Band gap of TlBiSe\(_2\) using different \(\epsilon _{xc}\) functionals.
Full size table
Figure 2
figure 2

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.

Full size image

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 \(\Gamma\) (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 \(\Gamma\) and F points, unlike the former. In this BZ, there are 8 TRIMs, i.e., \(\Gamma\), 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.

Figure 3
figure 3

Phonon band structures of (a) GaBiSe\(_2\) and (b) PbBiSe\(_2\).

Full size image
Table 2 Band gap of different materials using G\(_0\)W\(_0\)@PBE+SOC.
Full size table

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.

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 (\(\nu _0\);\(\nu _{1}\nu _{2}\nu _{3}\)) 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 \(\nu _0\) = 1. This type of system is robust against weak time-reversal invariant perturbations. However, when \(\nu _0\) = 0, and at least one of the indices out of \(\nu _1\), \(\nu _2\) or \(\nu _3\) is nonzero, then the material is WTI. It can be viewed as a stacking of 2D TIs, and is less robust against perturbations.

Figure 4
figure 4

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.

Full size image

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 \(\Gamma\) point in the projected 2D BZ. Alongside, we have calculated the topological invariant, \(\nu _0\), 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.

Table 3 Calculated Z\(_2\) invariants for different materials.
Full size table

Afterwards, we have performed surface band structure calculation for PbBiSe\(_2\). We have obtained even (two) number of surface states lying at \(\Gamma\) and M along (111) direction (see Fig. 4e and f). The presence of even number of surface states yields \(\nu _0\) = 0, and the weak indices come out to be \(\nu _1\) = 0, \(\nu _2\) = 0 and \(\nu _3\) = 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.

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 \(\Gamma\) 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 DFT54,55 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 \(\Gamma\)-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 \(\epsilon _{xc}\) 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 package61,62. 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.