Introduction

Topological insulators (TIs) are a group of materials with interesting electronic properties. They can exhibit insulating state in the bulk, but support spin-momentum locked gapless states at the boundaries. For three-dimensional TIs,1,2 the metallic topological surface states induced by spin–orbit coupling (SOC) are protected by time-reversal symmetry, known as Kramer’s pair. Time-reversal symmetry forbids the elastic backscattering of Kramer’s pair surface states, and the dispassion-less transport of the topological surface states is robust against nonmagnetic weak disorder. Topological crystalline insulators3 (TCIs) are new topological phases of matter in which the topological surface states are protected by crystal symmetries,4 such as the mirror symmetry of the crystal with respect to {110} plane in the rocksalt (RS) structure of semiconductor SnTe, instead of time-reversal symmetry in TIs. The SnTe-like TCI state is characterised by a new topological invariant called the mirror Chern number, instead of the Z2 invariant in general TIs.

The RS structure IV–VI narrow band gap semiconductors such as GeTe, SnTe and PbTe and their alloys have been extensively explored in developing infrared optoelectronic devices5–8 and as high-performance thermoelectric materials.9 Besides their wide range of applications, the IV–VI materials also exhibit exotic electronic structures. For example, SnTe was recently found to be the prototype of TCIs. The topological surface states of SnTe and related IV–VI group alloys Pb1−xSnxTe and Pb1−xSnxSe have been theoretically predicted by performing the first-principles calculations4 and experimentally observed by carrying out angle-resolved photoemission spectroscopy,10–13 scanning tunnelling microscope,14,15 as well as transport measurements.16,17 Surprisingly, although SnTe is a well-known TCI, both of its lighter isovalent GeTe and heavier isovalent PbTe counterparts are normal insulators. This non-monotonic behaviour of IV–Te compounds is quite unexpected and the origin has not been fully explained.

In this paper, performing first-principles calculations, we unravel the origin of the anomalies by studying the electronic structures of GeTe, SnTe and PbTe. We find that in the cubic (RS, β-phase) the (low lying) cation s lone-pair state, which are valence electrons that are not shared with another atom, couples strongly with (high lying) anion valence p state that leads to band inversion in SnTe, but the coupling is not large enough for GeTe due to its low 4s orbital energy and for PbTe due to its large Pb atom size and relativistic effects, so they remain to be normal semiconductors. Surprisingly, we find that the strong admixture of cation p valence state and anion p conduction state in rhombohedral phase, which does not have inversion symmetry, leads to loss of band inversion and makes the low-temperature phase α-SnTe a normal semiconductor. This raises an interesting question about the validity of previous theoretical4 and experimental10–17 studies about the topological behaviour of SnTe, suggesting that more studies are needed to clarify this issue.

Results and Discussion

Figure 1 shows the calculated band structures of GeTe, SnTe and PbTe in ideal RS structure. They share common features over a large energy range of 20 eV, despite some systematic changes from compound to compound arising from the chemical shift of cations, changes in lattice constant, as well as spin–orbit splitting. Specifically, in the displayed energy window, all the conduction and valence band levels at high symmetry k-points occur in the same order, except near the fundamental band gap, which is at the L point. II–VI semiconductors (e.g., CdTe) are usually more stable in the zinc-blende structure and IV–VI semiconductors (e.g., PbTe) are usually more stable in RS structure, although both have similar face-centred cubic BZ. Unlike conventional II–VI semiconductors where the cation s band becomes the lowest conduction band and is unoccupied, the large electronegativity of the group-IV elements results in their cation s band (dashed lines in Figure 1) being occupied and below the anion p band, forming the so-called lone-pair s band. Because of this, II–VI semiconductors usually have conduction band minimum (CBM) state occurring at Γ point with mostly cation s orbital character, whereas IV–VI semiconductors usually have CBM state at L point with mostly cation p orbital character. The s–p coupling between cation s band and anion valence p band is forbidden at the high symmetry Γ point but is allowed at the L point. In IV–VI compounds, the occupied lone-pair s band is low lying. The s–p level repulsion pushes the valence p band up remarkably at L point in energy, making it the valence band maximum (VBM) over the whole BZ. Thus, IV–VI semiconductors often have a direct band gap at the L point despite the fact that all II–VI compounds have direct band gap at the Γ point.18 In the following, we will discuss in more detail the band structures and chemical trends of GeTe, SnTe and PbTe.

Figure 1
figure 1

Calculated band structures of cubic (a) GeTe, (b) SnTe and (c) PbTe. The energy zero sets at the top of the valence band.

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Figure 2 shows the band structures of RS SnTe calculated with and without SOC. We first examine the effects of the lone-pair s band on the band structure of IV–VI semiconductors by neglecting the SOC, as shown in Figure 2a. At Γ point, the anion p orbital valence band belongs to the triply degenerate Γ15 irreducible representation (single group notation19 and chosen the cation sites as origin) but the low-lying cation lone-pair s band belongs to the Γ1 irreducible representation, thus their coupling is strictly forbidden by symmetry since only states with the same symmetry can couple to each other.20 But the valence band Γ15 state can couple with the high-lying cation p derived conduction band Γ15 state, pushing the valence band state down and the conduction band state up in energy. At X point, the lone-pair s band has X1 symmetry, whereas the three-fold degenerate valence p orbital bands split into two bands with non-degenerate X ′ 4 X ′ 5 symmetry, respectively, and thus their coupling to the lone-pair band is also forbidden. Like at Γ point, the p–p coupling pushes the valence band ( X ′ 5 X ′ 5 state) up in energy. Therefore, at both Γ and X points the low-lying lone-pair s band could not directly influence the valence p band, but the p–p coupling leads to a level repulsion that lowers the valence p band and raises the conduction p band. However, at the L point, the valence anion p orbital bands split into a doubly degenerate L3 state and a non-degenerate L1 state, and the lone-pair s band also has the L1 symmetry, leading to inter-band coupling between p orbital L1 state and lone-pair s orbital L1 state and resulting in a strong level repulsion. The emergent s–p coupling at L point significantly pushes the L1 branch of the valence p states upward to a much higher energy than ones at Γ point, making it the VBM. On the other hand, the p–p coupling, which is allowed at Γ and X points, is forbidden by symmetry at L point as shown in Figure 2a, and thus the conduction band energy at the L point is low. The conduction bands at L point are further pushed down by high-lying conduction bands (with anion d character) of the same symmetry, making it the CBM. The SOC lowers the crystal symmetry and further lifts the degeneracy of p orbital bands, as shown in Figure 2b, and may introduce additional inter-band coupling. However, the SOC introduced inter-band coupling is usually weak compared with the crystal field induced coupling unless the coupled states are nearly degenerate. Therefore, the general band structure is unaffacted by including SOC. This explains why the IV–VI RS compounds have a direct band gap occurs at the L point.

Figure 2
figure 2

Calculated electronic band structures of SnTe (a) without SOC and (b) with SOC. The wave function squares as well as atomic orbital components (in percentage) of the CBM and VBM states are given in c and d for calculation without SOC and with SOC, respectively. Inset shows the schematic plot of s–p coupling in SnTe.

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Note that for conventional II–VI semiconductors in zinc-blende structure, the cation s state is unoccupied and forms the conduction band. The lowest conduction s band has different symmetry at Γ point but same symmetry at X and L points as the valence p derived states. The allowed s–p coupling between conduction band and valence band at X and L points repels substantially the conduction band upwards and valence band downwards. This s-p coupling thus makes most of the zinc-blende II–VI semiconductors direct bandgap, having the CBM at the Γ point and the VBM at the Γ point.

We now turn to the abnormal trend of band ordering of the fundamental band gap among group-IV tellurides. Figure 1 shows that for GeTe and PbTe, the ordering of the bands at L point is normal in the sense that they have a cation p derived L 6 − L 6 + VBM state. However, for SnTe this band order is reversed, known as the Dimmock reversal,21 despite the fact that Sn element is in the middle of Ge and Pb in the group 14 column of the periodic table. Specifically, the CBM is a L 6 + L 6 − state, as shown in Figures 1 and 2b. It is usually expected that physical properties will change monotonically as the atomic number of the isovalent elements changes. For example, the band gaps of ZnTe, CdTe and HgTe are reduced from 2.4 to 1.5 eV and to −0.3 eV.18 Table 1 shows the calculated band gaps of GeTe, SnTe and PbTe, which are in good agreement with experimental results. Such band reversal can also be illustrated by examining the charge density distribution and the atomic orbital characters22 of the VBM and CBM states. Figure 3 shows clearly a switch of wavefunctions between CBM and VBM states from GeTe to SnTe and a switch back from SnTe to PbTe. This switch back indicates that the band gap order of SnTe is opposite to both of its lighter (GeTe) and heavier (PbTe) isovalent neighbours. By projecting the CBM and VBM wavefunctions onto the atomic orbitals, we find in Figure 3 that the CBM of GeTe and PbTe and the VBM of SnTe are cation p predominated with minor contributions from anion s and anion d orbitals, whereas the VBM of GeTe and PbTe and the CBM of SnTe are mainly from anion p with remarkable contributions from cation s. This is consistent with the wave function distribution of CBM and VBM among GeTe, SnTe and PbTe. Therefore, the CBM symmetry of both GeTe and PbTe is L 6 − L 6 + . On the other hand, the order is reversed for the SnTe, i.e., the CBM is L 6 + L 6 − . Because of the band inversion, SnTe was recently found to be a TCI, but GeTe and PbTe are normal semiconductors.

Table 1 Calculated band gaps at L[E( L 6 − )−E( L 6 + )]Γ[E( Γ 6 − )−E( Γ 8 − )] points for GeTe, SnTe and PbTe at the experimental lattice constants a
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Figure 3
figure 3

Wave function square (or charge distribution) of CBM and VBM states that occur at the L point for cubic (a and d) GeTe, (b and e) SnTe and (c and f) PbTe. The site and angular momentum projected wave function characters within the projector augmented wave spheres of the CBM and VBM states in percentage are listed for each state.

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The anomalous band gap ordering among GeTe, SnTe and PbTe can be explained as follows. Although the Sn element sits between Ge and Pb elements in the group 14 column of the periodic table, the energy of Sn 5s orbital is 1.15 and 1.26 eV higher than the Ge 4s and Pb 6s orbitals, respectively, and thus more delocalised in real space and closer in energy to the high-lying Te 4p orbital (Table 1). The abnormally low Ge 4s orbital in energy is due to the incomplete screening of the nucleus by the 3d orbitals, whereas, the low Pb 6s orbital energy is due to the large relativistic effect of Pb.23 The high energy and delocalised character of Sn 5s orbital lead to stronger s–p coupling in SnTe than in other two compounds. In fact, the s–p coupling (Figure 2a inset) in SnTe is so strong that the L 6 + (Figures 1b and 2b) state is pushed up even above the L 6 − state and leads to the band inversion. Such s–p coupling also appears in the other two compounds, but it is not strong enough to leading to band inversion in GeTe and PbTe.

Besides the cation s atomic energy levels, there are several other factors that also have consequences for the band ordering. First, the cation atomic radius increases in the order r(Ge)<r(Sn)<r(Pb) (Table 1) so are the bond lengths in the IV–Te compounds. The shorter the bond length is, the larger the s–p repulsion could be. Among the three compounds, GeTe has the shortest bond length, so this factor will favour GeTe to have inverted band gap. Second, SOC is generally regarded as the main driving force to invert band order for three-dimensional TIs.2 The SOC in Pb is the largest among the three cation elements as evidenced by the fact that the atomic p orbital spin–orbit splitting is 0.29, 0.71 and 2.4 eV for Ge, Sn and Pb,24 respectively, and the spin–orbit splitting of the conduction band at the Γ point [ E ( Γ 8 − ) − E ( Γ 6 − ) ] is 0.55, 0.81 and 2.1 eV in GeTe, SnTe and PbTe, respectively, which are in good agreement with the atomic trend. The spin–orbit splitting pushes the CBM down and VBM up and therefore will favour PbTe to having band inversion. Furthermore, the Ge 4p is 0.18 and 0.55 eV lower in energy than Sn 5p and Pb 6p orbitals, respectively. The lower energy of the cation p orbital also favors the band inversion. The net effect of the ‘balancing acts’ is that all the three IV–Te compounds have similar band structures with SnTe being band inverted.

It is interesting to see that the band gaps at Γ point E g ( Γ 6 − − Γ 8 − ) also show non-monotonic behaviour of the three compounds, but in the exactly opposite way, that is, E g ( Γ 6 − − Γ 8 − ) decreases from SnTe to GeTe to PbTe (Table 1). From Figure 1, it can be seen that E g ( Γ 6 − − Γ 8 − ) is between the bonding and anti-bonding states of anion p and cation p orbitals. SnTe has the largest E g ( Γ 6 − − Γ 8 − ) because Sn 5p orbital has higher energy than that of Ge 4p orbital energy. For PbTe, although Pb has even higher 6p orbital energy than Sn 5p, the larger atomic size and strong SOC of Pb reduces E g ( Γ 6 − − Γ 8 − ) of PbTe.

For GeTe and SnTe, the stable structure below the transition temperature Tc is actually the rhombohedral distorted α phase. The Tc is 670 K and 98 K, respectively, for GeTe and SnTe25 in the low carrier concentration limit. For PbTe, the ferroelectric phase transition does not occur at low temperature.26 The lattice parameters of the primitive rhombohedral cell27–30 are ar=4.293 Å, αr=58.032° for GeTe and ar=4.475 Å, αr=59.879° for SnTe. The RS-to-rhombohedral ferroelectric phase transition is accompanied by a relative shift τ (0.028 for GeTe and 0.014 for SnTe) of the cation and anion sublattices along the cubic <111> direction that breaks the space inversion symmetry. The four degenerate L points in RS BZ split into three L points with symmetry Cs and one Z point with symmetry C3v in the rhombohedral BZ.

The calculated direct energy band gap at L point is 0.59 eV for α-GeTe (rhombohedral structure), much larger than 0.17 eV for β-GeTe (cubic RS structure) due to atom displacement in rhombohedral structure, which lowers the crystal symmetry by lifting the inversion symmetry and leads to reduced s–p coupling and admixture between the cation p valence state and anion p conduction state at L point, hence opens up a large gap between valence and conduction states. For α-SnTe, the reduced s–p coupling and strong admixture between cation p valence state and anion p conduction state leads to loss of band inversion and makes the material a normal semiconductor with a band gap of 0.20 eV (Figure 4). It is interesting to notice that if we assume there is no atom displacement in α-SnTe (i.e., τ=0), the character of VBM and CBM is same as in the β-SnTe and the band gap is still inverted.

Figure 4
figure 4

Calculated energy band structure of (a) GeTe and (b) SnTe in rhombohedral structure.

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The topological phase transition induced by the structure phase transition in SnTe is consistent with theoretical results in ref. 28. It is believed that the Tc depends strongly on the carrier concentration and the Tc can be much lower when there is high carrier concentration.31 Although some of the reported experiments10–17 were conducted at a temperature below the low carrier concentration Tc limit, topological phase transition induced by structure phase transition have not been observed to our knowledge. Further experimental confirmation is called for to test our prediction that there is a topological phase transition between RS β-SnTe and rhombohedral α-SnTe.

Conclusions

We have investigated the electronic structure of group-IV tellurides GeTe, SnTe and PbTe using first-principles calculation. The abnormal trend in their electronic band structures, i.e., a band inversion in SnTe but not in GeTe and PbTe, is explained by symmetry analysis and the atomic energy levels and sizes. We show that the low-lying cation s lone-pair band state couples strongly with the high-lying anion valence p band states at L point and this s–p coupling pushes the anion valence p band up to become the VBM. The Ge 4s orbital energy is lower than Sn 5s orbital energy due to the incomplete screening of the nucleus by the 3d orbitals. The Pb 6s orbital energy is lower than Sn 5s orbital energy due to the large relativistic effect of Pb atom. The highest cation s orbital energy of Sn among the three cation atoms, leading to the largest s–p coupling in SnTe with respect to GeTe and PbTe, makes SnTe band gap (at the L point) inverted, thus explains the peculiar chemical trends in the band structure among the three compounds. We also demonstrate that internal atom displacement in α-GeTe and α-SnTe induces strong admixture between CBM and VBM and reduces s–p coupling, thus increases the band gap. The α-SnTe is found to be a normal semiconductor because of reduced s–p coupling and admixture of p orbitals, which requires further experimental verification. Our study illustrates that besides reported SOC,2 strain,2 alloying32 and interface dipole field,33 inter-level coupling can also be an effective pathway to design TIs. Specifically, we could choose different atoms with different atomic orbital energies, apply pressure or strain to change the atom–atom bond length, regulate the charge state (Coulomb U) of an atom to control inter-level coupling.

Computational methods

The electronic structure and total energy are calculated using first-principles density functional theory as implemented in the Vienna Ab initio simulation package.34 The local density approximation35,36 is used to deal the correlation effects whereas the modified Becke and Johnson exchange potential (termed TB-mBJ)37 is adopted to improve the local density approximation description of band structures. The projector augmented wave method38,39 is used to treat the interaction between the valence electrons and the core electrons.39 The valence configurations of Ge, Sn, Pb and Te atoms considered in the calculations are 3d104s24p2, 4d105s25p2, 5d106s26p2 and 5s25p4, respectively. A plane-wave expansion up to 400 eV is applied and a Г-centred 8×8×8 Monkhorst-Pack40 k-mesh is used for the Brillouin zone (BZ) integration. Spin–orbit interaction, implemented in the Vienna Ab initio simulation package code following the approach of Kleinman and Bylander,41 is included in the self-consistent calculations because it involves heavy elements of Ge, Sn, Pb and Te and has an important role in the topological behaviour of the compounds. The experimental lattice constants (Table 1) are used in TB-mBJ band structure calculation as suggested in ref. 42. to avoid uncertainty in the calculated band structure due to a small error in calculated lattice constants.4 It yields band gaps of semiconductors with an accuracy comparable to GW methods but is computationally much less expensive. It is worth to note that the TB-mBJ approach remarkably improves over the original BJ approach for the description of band gaps37 and the local density approximation error for p–p like band gap such as those in IV–Te compounds is expected to be small.6