Topological insulators are new states of quantum matter in which surface states residing in the bulk insulating gap of such systems are protected by time-reversal symmetry. The study of such states was originally inspired by the robustness to scattering of conducting edge states in quantum Hall systems. Recently, such analogies have resulted in the discovery of topologically protected states in two-dimensional and three-dimensional band insulators with large spin–orbit coupling. So far, the only known three-dimensional topological insulator is BixSb1−x, which is an alloy with complex surface states. Here, we present the results of first-principles electronic structure calculations of the layered, stoichiometric crystals Sb2Te3, Sb2Se3, Bi2Te3 and Bi2Se3. Our calculations predict that Sb2Te3, Bi2Te3 and Bi2Se3 are topological insulators, whereas Sb2Se3 is not. These topological insulators have robust and simple surface states consisting of a single Dirac cone at the Γ point. In addition, we predict that Bi2Se3 has a topologically non-trivial energy gap of 0.3 eV, which is larger than the energy scale of room temperature. We further present a simple and unified continuum model that captures the salient topological features of this class of materials.
Recently, the subject of time-reversal-invariant topological insulators has attracted great attention in condensed-matter physics1,2,3,4,5,6,7,8,9,10,11,12. Topological insulators in two or three dimensions have insulating energy gaps in the bulk, and gapless edge or surface states on the sample boundary that are protected by time-reversal symmetry. The surface states of a three-dimensional (3D) topological insulator consist of an odd number of massless Dirac cones, with a single Dirac cone being the simplest case. The existence of an odd number of massless Dirac cones on the surface is ensured by the Z2 topological invariant7,8,9 of the bulk. Furthermore, owing to the Kramers theorem, no time-reversal-invariant perturbation can open up an insulating gap at the Dirac point on the surface. However, a topological insulator can become fully insulating both in the bulk and on the surface if a time-reversal-breaking perturbation is introduced on the surface. In this case, the electromagnetic response of three-dimensional (3D) topological insulators is described by the topological θ term of the form , where E and B are the conventional electromagnetic fields and α is the fine-structure constant10. θ=0 describes a conventional insulator, whereas θ=π describes topological insulators. Such a physically measurable and topologically non-trivial response originates from the odd number of Dirac fermions on the surface of a topological insulator.
Soon after the theoretical prediction5, the 2D topological insulator exhibiting the quantum spin Hall effect was experimentally observed in HgTe quantum wells6. The electronic states of the 2D HgTe quantum wells are well described by a 2+1-dimensional Dirac equation where the mass term is continuously tunable by the thickness of the quantum well. Beyond a critical thickness, the Dirac mass term of the 2D quantum well changes sign from being positive to negative, and a pair of gapless helical edge states appears inside the bulk energy gap. This microscopic mechanism for obtaining topological insulators by inverting the bulk Dirac gap spectrum can also be generalized to other 2D and 3D systems. The guiding principle is to search for insulators where the conduction and the valence bands have the opposite parity, and a ‘band inversion’ occurs when the strength of some parameter, say the spin–orbit coupling (SOC), is tuned. For systems with inversion symmetry, a method based on the parity eigenvalues of band states at time-reversal-invariant points can be applied13. On the basis of this analysis, the BixSb1−x alloy has been predicted to be a topological insulator for a small range of x, and recently, surface states with an odd number of crossings at the Fermi energy have been observed in angle-resolved photoemission spectroscopy (ARPES) experiments12.
As BixSb1−x is an alloy with random substitutional disorder, its electronic structures and dispersion relations are only defined within the mean field, or the coherent potential approximation. Its surface states are also extremely complex, with as many as five or possibly more dispersion branches, which are not easily describable by simple theoretical models. Alloys also tend to have impurity bands inside the nominal bulk energy gap, which could overlap with the surface states. Given the importance of topological insulators as new states of quantum matter, it is important to search for material systems that are stoichiometric crystals with well-defined electronic structures, preferably with simple surface states, and describable by simple theoretical models. Here, we focus on layered, stoichiometric crystals Sb2Te3, Sb2Se3, Bi2Te3 and Bi2Se3. Our theoretical calculations predict that Sb2Te3, Bi2Te3 and Bi2Se3 are topological insulators, whereas Sb2Se3 is not. Most importantly, our theory predicts that Bi2Se3 has a topologically non-trivial energy gap of 0.3 eV, larger than the energy scale of room temperature. The topological surface states for these crystals are extremely simple, described by a single gapless Dirac cone at the k=0 Γ point in the surface Brilloiun zone. We also propose a simple and unified continuum model that captures the salient topological features of this class of materials. In this precise sense, this class of 3D topological insulators shares the great simplicity of the 2D topological insulators realized in the HgTe quantum wells.
Band structure and parity analysis
Bi2Se3, Bi2Te3, Sb2Te3 and Sb2Se3 share the same rhombohedral crystal structure with the space group D3d5 () with five atoms in one unit cell. We take Bi2Se3 as an example and show its crystal structure in Fig. 1a, which has layered structures with a triangle lattice within one layer. It has a trigonal axis (three-fold rotation symmetry), defined as the z axis, a binary axis (two-fold rotation symmetry), defined as the x axis, and a bisectrix axis (in the reflection plane), defined as the y axis. The material consists of five-atom layers arranged along the z-direction, known as quintuple layers. Each quintuple layer consists of five atoms with two equivalent Se atoms (denoted as Se1 and Se1′ in Fig. 1c), two equivalent Bi atoms (denoted as Bi1 and Bi1′ in Fig. 1c) and a third Se atom (denoted as Se2 in Fig. 1c). The coupling is strong between two atomic layers within one quintuple layer but much weaker, predominantly of the van der Waals type, between two quintuple layers. The primitive lattice vectors t1,2,3 and rhombohedral unit cells are shown in Fig. 1a. The Se2 site has the role of an inversion centre and under an inversion operation, Bi1 is changed to Bi1′ and Se1 is changed to Se1′. The existence of inversion symmetry enables us to construct eigenstates with definite parity for this system.
Ab initio calculations for Sb2Te3, Sb2Se3, Bi2Te3 and Bi2Se3 are carried out in the framework of the Perdew–Burke–Ernzerhof-type14 generalized gradient approximation of the density functional theory. The BSTATE package15 with the plane-wave pseudo-potential method is used with a k-point grid taken as 10×10×10 and the kinetic energy cutoff fixed to 340 eV. For Sb2Te3, Bi2Te3 and Bi2Se3, the lattice constants are chosen from experiments, whereas for Sb2Se3, the lattice parameters are optimized in the self-consistent calculation for rhombohedral crystal structure (a=4.076 Å, c=29.830 Å), owing to the lack of experimental data.
Our results are consistent with the previous calculations16,17. In particular, we note that Bi2Se3 has an energy gap of about 0.3 eV, which agrees well with the experimental data (about 0.2–0.3 eV; refs 18, 19). In the following, we take the band structure of Bi2Se3 as an example. Figure 2a and b show the band structure of Bi2Se3 without and with SOC, respectively. By comparing the two figure parts, one can see clearly that the only qualitative change induced by turning on SOC is an anti-crossing feature around the Γ point, which thus indicates an inversion between the conduction band and valence band due to SOC effects, suggesting that Bi2Se3 is a topological insulator. To firmly establish the topological nature of this material, we follow the method proposed by Fu and Kane13. Thus, we calculate the product of the parities of the Bloch wavefunction for the occupied bands at all time-reversal-invariant momenta Γ,F,L,Z in the Brillouin zone. As expected, we find that at the Γ point, the parity of one occupied band is changed on turning on SOC, whereas the parity remains unchanged for all occupied bands at the other momenta F,L,Z. As the system without SOC is guaranteed to be a trivial insulator, we conclude that Bi2Se3 is a strong topological insulator. The same calculation is carried out for the other three materials, from which we find that Sb2Te3 and Bi2Te3 are also strong topological insulators, and Sb2Se3 is a trivial insulator. The parity eigenvalues of the highest 14 bands below the Fermi level and the first conduction band at the Γ point are listed in Fig. 2d. From this table we can see that the product of parities of occupied bands at the Γ point changes from the trivial material Sb2Se3 to the three non-trivial materials, owing to an exchange of the highest occupied state and the lowest unoccupied state. This agrees with our earlier analysis that an inversion between the conduction band and valence band occurs at the Γ point.
To get a better understanding of the inversion and the parity exchange, we start from the atomic energy levels and consider the effect of crystal-field splitting and SOC on the energy eigenvalues at the Γ point. This is summarized schematically in three stages (I), (II) and (III) in Fig. 3a. As the states near the Fermi surface are mainly coming from p orbitals, we will neglect the effect of s orbitals and start from the atomic p orbitals of Bi (6s26p3) and Se (4s24p4). In stage (I), we consider the chemical bonding between Bi and Se atoms within a quintuple layer, which is the largest energy scale in the current problem. First we can recombine the orbitals in a single unit cell according to their parity, which results in three states (two odd, one even) from each Se p orbital and two states (one odd, one even) from each Bi p orbital. The formation of chemical bonding hybridizes the states on Bi and Se atoms, thus pushing down all of the Se states and lifting up all of the Bi states. In Fig. 3a, these five hybridized states are labelled as |P1x,y,z±〉, |P2x,y,z±〉 and |P0x,y,z−〉, where the superscripts +,− stand for the parity of the corresponding states. In stage (II), we consider the effect of the crystal-field splitting between different p orbitals. According to the point-group symmetry, the pz orbital is split from the px and py orbitals whereas the last two remain degenerate. After this splitting, the energy levels closest to the Fermi energy turn out to be the pz levels |P1z+〉 and |P2z−〉. In the last stage (III), we take into account the effect of SOC. The atomic SOC Hamiltonian is given by Hso=λl·S, with l,S being the orbital and spin angular momentum, and λ is the SOC parameter. The SOC Hamiltonian mixes spin and orbital angular momenta while preserving the total angular momentum, which thus leads to a level repulsion between |P1z+,↑〉 and |P1x+iy+,↓〉, and similar combinations. Consequently, the |P1z+,↑(↓)〉 state is pushed down by the SOC effect and the |P2z−,↑(↓)〉 state is pushed up. If the SOC is large enough (λ>λc), the order of these two levels is reversed. To see this inversion process explicitly, we also calculate the energy levels |P1z+〉 and |P2z−〉 for a model Hamiltonian of Bi2Se3 with artificially rescaled atomic SOC parameters λ(Bi)=xλ0(Bi), λ(Se)=xλ0(Se), as shown in Fig. 3b. Here, λ0(Bi)=1.25 eV and λ0(Se)=0.22 eV are the realistic values of Bi and Se atomic SOC parameters, respectively20. From Fig. 3b, one can see clearly that a level crossing occurs between |P1z+〉 and |P2z−〉 when the SOC is about 60% of the realistic value. As these two levels have opposite parity, the inversion between them drives the system into a topological insulator phase. Therefore, the mechanism for the 3D topological insulator in this system is exactly analogous to the mechanism in the 2D topological insulator HgTe. In summary, through the analysis above we find that Bi2Se3 is topologically non-trivial due to the inversion between two pz orbitals with opposite parity at the Γ point. Similar analyses can be carried out on the other three materials, from which we see that Sb2Te3 and Bi2Te3 are qualitatively the same as Bi2Se3, whereas the SOC of Sb2Te3 is not strong enough to induce such an inversion.
Topological surface states
The existence of topological surface states is one of the most important properties of the topological insulators. To see the topological features of the four systems explicitly, we calculate the surface states of these four systems on the basis of an ab initio calculation. First we construct the maximally localized Wannier function (MLWF) from the ab initio calculation using the method developed by Marzari and co-workers21,22. We divide the semi-infinite system into a surface slab with finite thickness and the remaining part as the bulk. The MLWF hopping parameters for the bulk part can be constructed from the bulk ab initio calculation, and the ones for the surface slab can be constructed from the ab initio calculation of the slab, in which the surface correction to the lattice constants and band structure have been considered self-consistently and the chemical potential is determined by the charge neutrality condition. With these bulk and surface MLWF hopping parameters, we use an iterative method23,24 to obtain the surface Green’s function of the semi-infinite system. The imaginary part of the surface Green’s function is the local density of states (LDOS), from which we can obtain the dispersion of the surface states. The surface LDOS on the  surface for all four systems is shown in Fig. 4. For Sb2Te3, Bi2Se3 and Bi2Te3, one can clearly see the topological surface states that form a single Dirac cone at the Γ point. In comparison, Sb2Se3 has no surface state and is a topologically trivial insulator. Thus, the surface-state calculation agrees well with the bulk parity analysis, and confirms conclusively the topologically non-trivial nature of the three materials. For Bi2Se3, the Fermi velocity of the topological surface states is vF≃5.0×105 m s−1, which is similar to that of the other two materials.
Low-energy effective model
As the topological nature is determined by the physics near the Γ point, it is possible to write down a simple effective Hamiltonian to characterize the low-energy long-wavelength properties of the system. Starting from the four low-lying states |P1z+,↑(↓)〉 and |P2z−,↑(↓)〉 at the Γ point, such a Hamiltonian can be constructed by the theory of invariants25 for the finite wave vector k. On the basis of the symmetries of the system, the generic form of the 4×4 effective Hamiltonian can be written down up to the order of O(k2), and the tunable parameters in the Hamiltonian can be obtained by fitting the band structure of our ab initio calculation. The important symmetries of the system are time-reversal symmetry T, inversion symmetry I and three-fold rotation symmetry C3 along the z axis. In the basis of (|P1z+,↑〉,|P2z−,↑〉,|P1z+,↓〉,|P2z−,↓〉), the representation of the symmetry operations is given by , and , where is the complex conjugation operator, σx,y,z and τx,y,z denote the Pauli matrices in the spin and orbital space, respectively. By requiring these three symmetries and keeping only the terms up to quadratic order in k, we obtain the following generic form of the effective Hamiltonian: with k±=kx±iky, and . By fitting the energy spectrum of the effective Hamiltonian with that of the ab initio calculation, the parameters in the effective model can be determined. For Bi2Se3, our fitting leads to M=0.28 eV, A1=2.2 eV Å, A2=4.1 eV Å, B1=10 eV Å2, B2=56.6 eV Å2, C=−0.0068 eV, D1=1.3 eV Å2, D2=19.6 eV Å2. Except for the identity term ε0(k), the Hamiltonian (1) is nothing but the 3D Dirac model with uniaxial anisotropy along the z-direction and k-dependent mass terms. From the fact M,B1,B2>0, we can see that the order of the bands |T1z+,↑(↓)〉 and |T2z−,↑(↓)〉 is inverted around k=0 compared with large k, which correctly characterizes the topologically non-trivial nature of the system. Such an effective Dirac model can be used for further theoretical study of the Bi2Se3 system, as long as the low-energy properties are considered. For example, as one of the most important low-energy properties of the topological insulators, the topological surface states can be obtained from diagonalizing the effective Hamiltonian equation (1) with an open boundary condition, with the same method used in the study of the 2D quantum spin Hall insulator26. For a surface perpendicular to the z-direction (that is, the  direction), kx,ky are still good quantum numbers but kz is not. By substituting −i∂z for kz in equation (1), one can write down the 1D Schrödinger equations for the wavefunctions ψkx,ky(z). For kx=ky=0, there are two renormalizable surface-state solutions on the half infinite space z>0, denoted by |ψ0↑〉,|ψ0↓〉. By projecting the bulk Hamiltonian (1) onto the subspace of these two surface states, to the leading order of kx,ky we obtain the following surface Hamiltonian in the basis of |ψ0↑〉,|ψ0↓〉. Here, the surface-state wavefunction |ψ0↑(↓)〉 is a superposition of the |P1z+,↑(↓)〉 and |P2z+,↑(↓)〉 states, respectively. For A2=4.1 eV Å obtained from the fitting, the Fermi velocity of the surface states is given by vF=A2/ℏ≃6.2×105 m s−1, which agrees reasonably well with the ab initio results shown in Fig. 4c. In summary, the effective model of the surface states equation (2) characterizes the key features of the topological surface states, and can be used in the future to study the surface-state properties of the Bi2Se3 family of topological insulators.
The topological surface states can be directly verified by various experimental techniques, such as ARPES and scanning tunnelling microscopy. In recent years, evidence of surface states has been observed for Bi2Se3 and Bi2Te3 in ARPES (ref. 27) and scanning tunnelling microscopy28 experiments. In particular, the surface states of Bi2Te3 observed in ref. 27 had a similar dispersion to what we obtained in Fig. 4d, which were also shown to be quite stable and robust, regardless of photon exposure and temperature. Near the completion of this work, we became aware of the ARPES experiment29 on Bi2Se3, which measures a Dirac cone near the Γ point of the surface Brilloiun zone. These experimental results support the main conclusion of our theoretical work. Moreover, the 3D topological insulators are predicted to exhibit the universal topological magneto-electric effect10 when the surface is coated with a thin magnetic film. Compared with the Bi1−xSbx alloy, the surface states of the Bi2Se3 family of topological insulators contain only a single Fermi pocket, making it easier to open up a gap on the surface by magnetization and to observe the topological Faraday/Kerr rotation10 and image magnetic monopole effect30. If observed, such effects would be an unambiguous experimental signature of the non-trivial topology of the electronic properties.
We would like to thank B. F. Zhu for the helpful discussion. This work is supported by the NSF of China, the National Basic Research Program of China (No. 2007CB925000), the International Science and Technology Cooperation Program of China (No. 2008DFB00170) and by the US Department of Energy, Office of Basic Energy Sciences under contract DE-AC02-76SF00515.
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Topological quantum states of matter in iron-based superconductors: from concept to material realization
National Science Review (2019)