Rock-salt chalcogenides represent a significant group of functional materials, in which many novel properties are discovered, including superconductivity,^[1] thermoelectricity,^[2,3] ferroelectricity,^[4] optoelectronics^[5,6] and spintronics.^[7,8] Recently, this class of materials has attracted considerable attention again due to the prediction and realization of the nontrivial topological properties in SnTe,^[9,10] which is termed TCI.^[11,12] In contrast to the widely studied Z₂ topological insulator (TI),^[13–20] Dirac cones lying on the topological crystalline insulator (TCI) surfaces are protected by the mirror symmetry, rather than time reversal (TR) symmetry. Therefore, TCI phase is associated with a new topological invariant called mirror Chern number, which classifies and distinguishes TCI from TI and an ordinary insulator. Importantly, a quantized π-Berry phase induced by the band inversion is needed and essential in TCI to characterize its topological properties such as Dirac cones.

Though the discovery of TCI enriches people’s understanding of the topological classification, few TCIs are realized in experiment. In order to satisfy the requirement of the potential device applications, it is urgent to find more TCIs, especially those with large band gap and high working temperature. As we all know, most lead chalcogenides adopt the same structure as SnTe, where the band inversion at L point is driven by the spin–orbit coupling (SOC). Considering that lead is much heavier than tin, it is natural to postulate that PbX may be a large band gap TCI due to the much stronger SOC. However, until now, no TCI phase is realized in PbX. Therefore, it is important to figure out the main factors of the band gap evolution in rock-salt chalcogenides with the purpose of achieving large band gap TCIs. Based on previous studies, there is a famous empirical relation^[21] between band gap E_G and lattice constant a₀, which states that has a linear relationship with for a series of relevant rock-salt semiconductors. However, this relation is invalid in lead chalcogenides, where PbTe exhibits a well-known anomaly that E_G(PbS)> E_G(PbTe)>E_G(PbSe),^[22,23] even though the lattice constant a₀(PbTe) is much larger than a₀(PbSe). This band gap anomaly has been a long-standing question and remains under debate.

In this paper, by combining ab initio calculations and TB method, we have studied the band evolution and band gap anomaly in PbX. Our studies show that the band gap anomaly in PbTe is closely related to the delocalized 5s electrons of Te. Comparing to S and Se, the 5s electrons of Te show much low binding energy (high on-site energy) and very extended distribution caused by the huge screening effect of the numerous interior electrons. As a result, Te 5s orbital pushes up the Pb p orbital through the huge s–p hybridization, leading to a large band gap for PbTe. Moreover, our calculations show that the SOC energy gap E_G(PbPo) roughly agrees with the linear relation formed by PbS and PbSe, and it is a negative number, which means that band inversion happens at L point in PbPo. Our detailed non-local Heyd–Scuseria–Ernzerhof (HSE) hybrid functional calculations show that PbPo is an indirect band gap semiconductor with a quantized π-Berry phase, which clearly indicates that PbPo is a TCI. The calculated mirror Chern number and surface states double confirm this conclusion.

This paper is arranged as follows. In Section 2, we will introduce the details of the ab initio calculations and TB model. In Section 3, based on the TB model, we will study the origin of the band gap anomaly. In Section 4, we will focus on the electronic structure and topological properties of PbPo. Finally, Section 5 contains a summary of this work.

Our ab initio calculations are carried out by the projector augmented wave (PAW) method^[24,25] implemented in Vienna ab initio simulation package (VASP).^[26,27] The crystal structure of PbX is shown in Fig. 1(a). Experimental lattice constants, with a₀ = 5.942,^[28] 6.124,^[29] 6.460,^[30] and 6.59 Å^[31] for PbS, PbSe, PbTe, and PbPo are adopted in our calculations. The exchange and correlation potential is treated within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof type.^[32] Considering the possible underestimation of the band gap by GGA, HSE hybrid functional^[33] is further supplemented to improve the accuracy of the band gap. The cutoff energy of the plane wave expansion is 500 eV, and 11 × 11 × 11 k-point grids are used in the self-consistent calculations. SOC is consistently considered in the calculations. Modified Becke–Johnson (mBJ)^[34] calculations are performed using the all-electron full-potential linearized augmented plane-wave (FP-LAPW) method implemented in the WIEN2k package.^[35]

Fig. 1.

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Fig. 1. (a) The crystal structure of PbX with space group (b) First Brillouin zone (BZ) of bulk and the projected surface BZ on (001) and (111) planes. There are four L points (L1, L2, L3, L4) in the bulk first BZ. For the (001) face, L1 and L2 are projected to the , while L3 and L4 are projected to another . For the (111) face, L3 is projected to , while L1, L2 and L4 are projected to The light yellow region is invariant under the mirror symmetry .

In order to study the band evolution and the band gap anomaly accurately, a general eight-band Slater–Koster^[36] TB model with bases is constructed, in which, besides the p–p hoppings, the s–p hybridizations are taken into account too

When SOC is taken into account, the Hamiltonian size will be doubled. The final Hamiltonian with SOC can be written in the form

Table 1.

Table 1.

Table 1. Fitted TB parameters (in eV). . PbS PbSe PbTe PbPo A1 −1.1333 −2.0072 −1.6603 −2.2198 s,s(000) C−C A2 0.0991 0.0891 0.0959 0.0857 s,s(110) C−C A3 −0.0012 −0.0215 −0.0406 −0.0623 s,x(110) C−C A6 7.7801 6.9910 7.2422 6.7530 x,x(000) C−C A7 −0.0002 −0.0259 −0.0574 −0.0830 x,x(110) C−C A8 −0.0169 −0.0283 −0.0419 −0.0535 x,x(011) C−C A9 0.3623 0.3110 0.2693 0.2157 x,y(110) C−C C1 −6.9050 −7.8493 −5.0836 −7.5326 s,s(000) A−A C2 −0.0793 −0.0848 −0.0907 −0.0894 s,s(110) A−A C3 0.1584 0.1689 0.1788 0.1875 s,x(110) A−A C6 3.3582 3.1139 4.2744 4.2034 x,x(000) A−A C7 0.1611 0.1686 0.1740 0.1810 x,x(110) A−A C8 −0.0170 −0.0188 −0.0203 −0.0217 x,x(011) A−A C9 0.1480 0.1356 0.1201 0.1106 x,y(110) A−A B1 −0.1537 −0.1257 −0.1082 −0.0779 s,s(200) C−C B2 0.0855 0.0974 0.1113 0.1247 s,x(200) C−C B4 0.5301 0.4891 0.4579 0.4131 x,x(200) C−C B5 0.1262 0.1050 0.0870 0.0605 y,y(200) C−C E1 −0.2421 −0.2215 −0.1947 −0.1701 s,s(200) A−A E2 −0.0854 −0.1308 −0.1733 −0.2168 s,x(200) A−A E4 0.1210 0.1409 0.1516 0.1713 x,x(200) A−A E5 0.0298 0.0330 0.0360 0.0393 y,y(200) A−A D1 −0.6586 −0.6076 −0.5565 −0.5015 s,s(100) C−A D2 1.3365 1.2714 1.2389 1.1757 s,x(100) C−A D3 −1.6336 −1.4507 −1.4907 −1.2878 x,s(100) C−A D4 1.8308 1.8474 1.8633 1.8795 x,x(100) C−A D5 −0.2662 −0.2950 −0.3287 −0.3675 y,y(100) C−A F1 0.3581 0.3100 0.2688 0.2239 s,s(111) C−A F2 −0.0504 −0.0491 −0.0481 −0.0470 x,s(111) C−A F3 −0.0071 −0.0069 −0.0067 −0.0065 s,x(111) C−A F4 0.1154 0.1013 0.0868 0.0736 x,x(111) C−A F5 0.0315 0.0425 0.0564 0.0680 x,y(111) C−A

Table 1. Fitted TB parameters (in eV). .

The band gaps calculated by different methods for PbX are shown in Fig. 2(a), in which the non-SOC band gap is defined as as shown in Fig. 3(a). In Fig. 2(a), the band gaps for PbTe show a distinct jump in all calculation methods, which means that the band gap anomaly in PbTe is intrinsic of the material, and irrelevant to SOC and the form of the exchange–correlation functional. As pointed out by Hummer et al.,^[39] HSE03 is performing extremely well for lead chalcogenides. Therefore, in the following, the band structures calculated by HSE03 without SOC are taken as reference to fit the parameters in Eq. (3), and all parameters are directly transferred to Eq. (4) by adding the isotropic SOC term to estimate the band gaps and dispersions for all compounds. The comparisons between the non-SOC band structures obtained by Eq. (3) and original HSE03 calculations are shown in Fig. 3, and all fitted parameters are listed in Table 1, in which their detailed definitions are given in the sixth and seventh columns. Taking D₃ as an example, “x,s(100)” and “C–A” mean D₃ is the hopping parameter from the p_x orbital (x) of cation (C) to the s orbital (s) of anion (A) along the vector a₀/2 (1,0,0). In Fig. 3, the non-SOC band structures obtained from HSE03 calculations and TB fittings by Eq. (3) are plotted with red lines and blue dots respectively. It is clear that our TB results agree with HSE03 calculations very well, which certifies that our model is accurate enough to study all details of the system.

Fig. 2.

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Fig. 2. (a) Band gaps at L point for PbX calculated by different methods. Red circles, green up triangles, purple squares, and blue down triangles represent the band gaps got from GGA no SOC, GGA SOC, HSE03 no SOC and mBJ no SOC calculations respectively. (b) SOC TB band gap EG for PbX as a function of (red circles). If we decrease C1 only, the band gap decreases about 0.2 eV as shown as the green up triangle. If we decrease C1 and increase D3 at the same time, the final band gap is shown as the purple square. Available experimental data at 4.2 K (blue pentagons) are plotted for comparison. (c) and (d) Outermost s-electron binding energy Es and atomic radius for S, Se, Te, and Po as a function of the atomic number Z.

Fig. 3.

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	Fig. 3. Non-SOC band structures of PbX. The red lines and the blue dots are obtained from HSE03 calculations and TB model fitting respectively. (a) PbS, (b) PbSe, (c) PbTe, (d) PbPo.

The estimated band gaps E_G (red circles) from Eq. (4) for all four compounds, as well as the experimental data at 4.2 K (blue pentagons), are summarized in Fig. 2(b). It is clear that our TB results almost overlap with the experimental data, which certifies that our model is accurate and powerful again. In order to study the origin of the band gap anomaly in PbTe, we have checked all the parameters in Table 1 carefully and found that the s orbital on-site energy of Te (C₁) is obviously higher than the other three. Taking A₁ (on-site energy of cation s orbital) as a referential energy, we get C₁ − A₁ equal to −5.7716, −5.8421, −3.4233, and −5.3128 for PbS, PbSe, PbTe and PbPo respectively. As a result, the band energy of Te 5s orbital at Γ point shown in Fig. 3(c) is about −13 eV, while it is nearly −15 eV for the other three compounds. Owing to this high on-site energy, Te 5s orbital will push up Pb 6p orbital through the s–p hybridization and result in a large band gap for PbTe. Now, we try to artificially decrease the on-site energy of Te 5s orbital and push down the band energy at Γ point until it reaches the “regular” level −15 eV as the other three. (For this purpose, C₁ needs to be decreased by 2 eV.) After this adjustment, though the band gap does decrease about 0.2 eV as shown as the green triangle in Fig. 2(b), the band gap anomaly is still evident. Therefore, there must be some other reasons responsible to the band gap anomaly in addition to the high on-site energy of Te 5s orbital. We then recheck all the parameters in Table 1, and find that D₃ also show some irregularity. Naturally, |D₃| should decrease monotonically from PbS to PbPo due to the increasing lattice constant. However, |D₃(PbTe)| is a little larger than |D₃(PbSe)|, even though a₀(PbTe) = 6.460 Å is obviously larger than a₀(PbSe) = 6.124 Å. Based on the above discussions, after decreasing C₁ by 2 eV and increasing D₃ by 0.08 eV to a “regular” number, the final band gap for PbTe is shown as the purple square in Fig. 2(b), which falls on the line formed by PbS–PbSe–PbPo almost. Therefore, we conclude that the band gap anomaly in PbTe is mainly related to the high on-site energy of Te 5s orbital and the irregularly large s–p hopping.

Because SOC nearly has no contribution to the band gap anomaly as identified by our ab initio calculations, we would like to analytically discuss the band gap evolution and anomaly based on the non-SOC TB model in the following. At this case, the band gap ΔE can be easily written as

Next, we would like to discuss the origin of the irregularity of C₁ and D₃ in depth. Because both C₁ and D₃ are closely related to the s orbital of anion, it is natural to propose that the irregularity may be originated from the chalcogen atoms themselves. We have calculated the binding energies of the outermost s electrons E_s for all chalcogen atoms and plotted them with the atomic number Z in Fig. 2(c), in which the points for sulfur, selenium and polonium show an approximately straight line while the point of tellurium is much higher than the straight line. This result clearly demonstrates that the high on-site energy C₁ of PbTe is inherited from the low binding energy of tellurium 5s electrons. To study the irregularity of D₃ in PbTe, we show a plot of the atomic radius of the chalcogen atoms^[42] as a function of atomic number Z in Fig. 2(d). We can see that the atomic radius of tellurium does not lie on the straight line as the others do and its position is higher than the line, which means tellurium 5s orbital is more extended. As a result, the hybridization between 5s orbital of Te and p_x orbital of Pb, i.e., D₃, is abnormally larger than the “regular” value. Both the low binding energy and the extended distribution of the tellurium 5s electrons are related to the unusually strong screening effect of the V-period elements, as well as the penetration effect and relativistic effect. In general, the huge screening effect from the interior electrons will reduce the binding effect of nucleus and lead to a low binding energy and more extended distribution of Te 5s electrons. However, the detailed discussion of the atomic energy level and distribution are very complicated and out of the scope of this work. Finally, we claim that such 5s electrons’ irregularity is universal for the right side elements in the V row of the periodic Table of Elements,^[43] e.g., Sn, and our discussion is universal for other covalent systems with the same structure. For example, similar analysis would give the conclusion that the 5s orbital irregularity of Sn will obviously reduce the band gap of the tin chalcogenides, which may be the main reason why the band gap of tin chalcogenide is usually smaller than the same row lead chalcogenide.^[45,46]

One important result of our calculations is that E_G is negative for PbPo as shown in Fig. 2(b), which means that band inversion happens at L point of PbPo. Detailed HSE03 + SOC calculations have been performed to confirm this conclusion. The calculated band structure is shown in Fig. 4(a), in which one p orbital of Pb is obviously dropped down below the Fermi level at L point as represented by the blue circles. Furthermore, our calculations show that PbPo is an indirect band gap (6.5 meV) semiconductor (see inset of Fig. 4(a)), rather than a direct band gap semiconductor^[46] or semimetal.^[47] All these band characters of PbPo are very similar to SnTe, which implies that PbPo may be a TCI too.

Fig. 4.

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Fig. 4. (a) Band structures of PbPo calculated by HSE03+SOC. The size of the blue circles represents the weight of the projected p orbitals of Pb. Parity information for the top of the valence band and the bottom of the conduct band at L point has been given. The inset shows the top of the valence band and the bottom of the conduction band of PbPo. (b) and (c) The calculated surface states of PbPo on (111) and (001) planes.

There are four L points {L₁(0, π, 0), L₂(0, 0, π), L₃(π, π, π), L₄(π, 0, 0)} in the first BZ of PbPo (see Fig. 1(b)). Because there are even band inversion points, the topological property of PbPo is different with 3D strong TI.^[48,49] Actually, the topological property is protected by the mirror symmetry, and its topological invariant is the mirror Chern number C_M instead of Z₂.

The eigenvalue (m) of the mirror operator is a good quantum number for PbPo, so we can classify the Bloch wavefunctions on plane by m, and define the berry connection A^m (k) and berry curvature Ω^m(k) on the plane as follows:

In summary, we have studied the band evolution in PbX, and found that, though the s orbital of the anion is far away from the Fermi level, it has crucial influence on the band gap through the huge s–p hybridization. The high on-site energy of Te s orbital and its irregular extended distribution combining together result in an irregular big band gap in PbTe, i.e., the famous band gap anomaly in lead chalcogenides. Furthermore, our calculations show that PbPo is an indirect band gap (6.5 meV) semiconductor with negative E_G, which means that band inversion happens at L point in PbPo. The calculated mirror Chern number and surface states confirm that PbPo is a TCI.