Abstract
Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the semiconductor SnTe by identifying its nonzero topological index. We predict that as a manifestation of this nontrivial topology, SnTe has metallic surface states with an even number of Dirac cones on highsymmetry crystal surfaces such as {001}, {110} and {111}. These surface states form a new type of highmobility chiral electron gas, which is robust against disorder and topologically protected by reflection symmetry of the crystal with respect to {110} mirror plane. Breaking this mirror symmetry via elastic strain engineering or applying an inplane magnetic field can open up a continuously tunable band gap on the surface, which may lead to wideranging applications in thermoelectrics, infrared detection and tunable electronics. Closely related semiconductors PbTe and PbSe also become topological crystalline insulators after band inversion by pressure, strain and alloying.
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Introduction
The discovery of topological insulators^{1,2,3} has attracted much interest in topological states of matter beyond the existing Z_{2} material class. In searching for new phases, crystal symmetries have a multifaceted role by either constraining the band topology^{4,5} or engendering new ones^{7,8}. Topological crystalline insulators^{8} are such new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Thanks to the complexity and richness of crystal structures, the study of topological crystalline insulators has just begun and a large number of topological crystalline insulators awaits discovery—both theoretically and experimentally.
In this work, we theoretically demonstrate that SnTe is a topological crystalline insulator with mirror symmetry. As a consequence, it is predicted to have robust surface states with an even number of Dirac cones on crystal surfaces such as {001}, {110} or {111}, which are symmetric about {110} mirror planes. The notation {hkl} refers to the (hkl) plane and all those that are equivalent to them by virtue of the crystal symmetry (a similar convention is used with directions: n_{1}n_{2}n_{3} refers to, collectively, the [n_{1}n_{2}n_{3}] direction and its equivalent ones). The existence of these surface states is dictated by a nonzero integer topological invariant—the mirror Chern number^{9}.
Results
Crystal structure and mirror symmetry
SnTe has a simple rocksalt structure (Fig. 1a); its fundamental band gaps are located at four equivalent L points in the facecenteredcubic Brillouin zone. It has long been established that the ordering of the conduction and valence bands at L points in SnTe is inverted relative to PbTe, so that the band gap of the alloy Pb_{1−x}Sn_{x} Te closes and reopens as x increases^{10}. As this band inversion occurs at an even number of points, neither SnTe nor PbTe in the rocksalt structure is a topological insulator with Z_{2} topological order.
However, the above band inversion has significant consequences when the mirror symmetry of the facecenteredcubic lattice is taken into consideration. Consider the plane ΓL_{1}L_{2} in momentum space, defined by the three points Γ, L_{1} and L_{2} (Fig. 1b). Crystal momenta on this plane are invariant under reflection about the {110} mirror planes in real space. This allows us to label the Bloch wavefunctions on this plane by their eigenvalues ±i under the mirror operation M (which satisfies M^{2}=−1 for spin 1/2 electrons). Each class of M= ±i mirror eigenstates has an associated Chern number n_{±i}, and the mirror Chern number n_{M}^{9} is defined by n_{M}=(n_{+i}−n_{−i})/2. Provided that mirror symmetry is present, n_{M} is an integer topological invariant. A nonzero mirror Chern number defines a topological crystalline insulator with mirror symmetry.
Mirror Chern numbers of SnTe and PbTe
We first demonstrate that SnTe and PbTe have mirror Chern numbers that differ by two, and hence, one of them is a topological crystalline insulator. This is established by considering the band inversion at four L points between SnTe and PbTe. The k·p theory of the band structure near a given L point is given by Mitchell and Wallis^{11}:
Here k_{1},k_{2},k_{3} form an orthogonal system with k_{3} along ΓL and k_{1} along the [110] direction perpendicular to the mirror plane; σ_{z}= ±1 corresponds to the porbital on the cation (Sn or Pb) and anion (Te), respectively; s_{3}= ±1 labels the total angular momentum j=±1/2 along ΓL. A positive m means that the conduction and valence bands at L are respectively derived from the cation and anion, and vice versa for negative m. The form of H in equation (1) is uniquely determined by the D_{3d} point group symmetries, which leave an L point invariant^{12}. In particular, reflection about the (110) mirror plane is represented by M= −is_{1}. On the mirrorinvariant plane ΓL_{1}L_{2} (k_{1}=0), H reduces to . Due to mirror symmetry, H_{0} decomposes into the s_{1}=1 (M=−i) and s_{1}=−1 (M=i) subspaces:
each of which describes a twodimensional massive Dirac fermion.
In going from PbTe to SnTe (increasing x in Pb_{1−x}Sn_{x}Te), the cation/anion character of the conduction/valence bands becomes switched at L, which in the k·p theory corresponds to m→−m. The sign reversal of m at one L point changes the Chern number of the s_{1}= ±1 subspace by ±1, and hence the mirror Chern number changes by one. Furthermore, L_{1} and L_{2} are related by a twofold rotation around the axis. Under this rotation, both the spin polarization s_{1} and the orientation of the ΓL_{1}L_{2} plane are flipped. As a result, the Berry curvatures at L_{1} and L_{2} are related by
so that
that is, both L_{1} and L_{2} contribute equally to the change in mirror Chern number. The net result is that the band inversion changes the mirror Chern number for the ΓL_{1}L_{2} plane by two. From this, we deduce that SnTe and PbTe are topologically distinct phases, as long as the reflection symmetry with respect to any of the six equivalent {110} mirror planes is present.
SnTe is topologically nontrivial
To determine which of the two is topologically nontrivial, we analyse their band structures obtained from firstprinciples density functional theory calculations^{13,14,15,16,17,18}, using the generalized gradient approximation^{19}. The results we obtain are consistent with several previous studies^{20,21,22,23}. In particular, it is generally agreed that the band ordering at L points is correct at the totalenergy optimized volume.
On the basis of two firstprinciples findings, we now demonstrate that, of the two materials, SnTe is the topologically nontrivial one. First, the conduction and valence bands of PbTe throughout the Brillouin zone are primarily derived from the porbitals of Pb and Te atoms, respectively (Fig. 2b). This suggests that PbTe is smoothly connected to the atomic limit, in which Pb orbitals are empty and Te orbitals filled due to their onsite energy difference. In contrast, the orbitals in the band structure of SnTe are switched near L points: the conduction band edge is derived from Te and the valence band edge from Sn (Fig. 2a). Therefore, SnTe has an intrinsically inverted band structure (m<0). Our conclusion is further supported by the dependence of the band gap on the lattice constant obtained in firstprinciples calculations (Fig. 2c) and measured via pressure coefficients^{24}. As the lattice constant increases, the band gap of PbTe increases monotonically, whereas that of SnTe decreases to zero and then reopens. This gap closing signals a topological phase transition, in which SnTe at ambient pressure is a topological crystalline insulator with the aforementioned mirror Chern number n_{M} = −2. The sign of n_{M} has been discussed in detail in ref. 9.
Surface states of SnTe
The nonzero mirror Chern number in SnTe dictates the existence of surface states on any crystal surface symmetric about the {110} mirror planes. Three common surface terminations satisfying this condition are: {001}, which is symmetric about two equivalent {110} mirror planes; {111}, which is symmetric about three equivalent mirror planes; and {110}, which is symmetric about the mirror plane. According to the bulkboundary correspondence, we now infer the topology of these surface bands. For the {001} surface, the plane ΓL_{1}L_{2} in the bulk Brillouin zone projects onto the symmetry line in the surface Brillouin zone, with both L_{1} and L_{2} projecting onto . The mirror Chern number n_{M} = −2 dictates that there must exist two pairs of counterpropagating, spinpolarized surface states with opposite mirror eigenvalues along the line . By rotational symmetry, such surface states also appear along the line . But they are absent along any other mirrorinvariant line. The crossing of two mirror branches creates an anisotropic twodimensional Dirac point with different velocities along the parallel and perpendicular direction. Therefore, the {001} surface states have four Dirac points located on the four equivalent lines. Similar considerations apply to the other two surfaces. For the {111} surface, the plane ΓL_{1}L_{2} projects onto the line ; so there are two Dirac points along each of the three equivalent lines . For the {110} surface, the plane ΓL_{1}L_{2} projects onto the line , on which there are two Dirac points. In all cases, surface states of SnTe have an even number of Dirac points, which can be easily distinguished from Z_{2} topological insulators having an odd number. Our theoretical prediction of these surface states is the main result of this work.
Using firstprinciples calculations, we now explicitly demonstrate the above surface states in a slab geometry along the [001] axis. Results for other surfaces will be published elsewhere. As predicted by the above topological band theory, two surface bands with opposite mirror eigenvalues are found to cross each other and form a Dirac point on the line (Fig. 3a). The Dirac velocity is found to be 1.7×10^{5} m/s in the direction. Interestingly, these surface states exhibit a Lifshitz transition—a change of Fermi surface topology as a function of Fermi energy. Figure 3b shows a set of Fermi surfaces at different energies. As the Fermi energy decreases from the Dirac point towards the valence band, the Fermi surface initially consists of two disconnected hole pockets outside ; the two pockets then touch each other and reconnect to form a large hole and a small electron pocket, both centred at .
Discussion
To understand both the connection between the bulk and surface bands, and the effect of potential perturbations at a microscopic level, we introduce a simplified tightbinding model for SnTe, detailed in Methods. Using the tightbinding model, we now study the electronic properties of the {001} surface states under various perturbations. Similar analysis applies to other surfaces. The doubly degenerate surface states ψ_{α}(K_{j}) at the four Dirac points K_{j} have opposite mirror eigenvalues iα, and hence opposite expectation values of spin polarization perpendicular to the K_{j} direction. For convenience, we choose a natural basis in which the relative phases between the wavefunctions at different Dirac points are fixed by the fourfold rotation relating them
In this basis, the k · p Hamiltonians at four Dirac points take an identical form:
Here k_{1} and k_{2} form a local righthanded coordinate system centred at each K_{j}, with k_{2} parallel to K_{j}.
It is important to note that the four branches of surface Dirac fermions have the same chirality, defined by the relative sign of and v_{} in equation (6). As such, the surface states of the topological crystalline insulator SnTe form a “chiral” Dirac metal protected by crystal symmetries, thereby defining a new symmetry/topology universality class. In particular, provided that mirror symmetry is present (k_{1} → −k_{1}, s_{2} → − s_{2}), the Dirac points here line, but cannot annihilate with each other.
We now consider ways to engineer a band gap on the surface. Perturbations which break mirror symmetries can generate Dirac mass term m_{j}s_{3} and thus open up energy gaps E_{j}=2m_{j} at the Dirac points. The nature of the gapped phase depends on the relative signs of m_{j}, which is determined by the symmetry of the perturbation. For example, a perpendicular magnetic field B couples to electron's spin and yields Dirac masses of the same sign due to rotational invariance: m_{1}=m_{2}=m_{3}=m_{4}B. This drives the surface states into an integer quantum Hall state.
The perturbations which break the four fold rotation symmetry of the {001} surface are particularly interesting. One example is a structure distortion with atoms being displaced by a vector u, which can be introduced by strain and possibly external electric field. Also, SnTe is known to spontaneously distort along the [111] direction into a rhombohedral structure at low temperature. This perturbation is timereversal invariant but breaks rotational symmetry, leading to a ferroelectric phase. Although this distortion has negligible effect on the band structure in the bulk, it can dramatically affect the Dirac surface states. The effect of the distortion can be captured by adding a modulated hopping term to the tight binding model (see Methods). By symmetry analysis (equation (5)), one can show that the distortion gives the following anisotropic Dirac mass terms:
where is the surface normal. As a result, the metallic surface acquires a band gap, which is linearly proportional to the magnitude of structural distortion and depends on its direction. For the rhombohedral distortion u[111], the two Dirac points along the direction are gapped, but the other two along the [110] direction remain gapless due to the unbroken (110) mirror plane. The ability to continuously tune the surface band gap via applying strain suggests potential electronic and optoelectronic device applications based on topological crystalline insulators such as SnTe.
Likewise, an inplane magnetic field in the u direction generates Dirac mass terms given by m_{j}u·K_{j}. In particular, when the magnetic field is perpendicular to one of the mirror planes, the system is still symmetric about this mirror plane and hence continues to exhibit gapless surface states despite the broken time reversal symmetry. Therefore, topological crystalline insulators defined by mirror Chern number also exist in magnetic systems such as Mn and Crdoped SnTe^{25}], contrary to existing Z_{2} topological insulators.
As equation (7) shows, when the displacement vector u points along the mirror invariant line , the surface states remain gapless at two Dirac points. Therefore, there are four distinct types of fully gapped surface ferroelectric phases depending on the direction of u (Fig. 4). These ferroelectric phases correspond to breaking crystal symmetry in “different directions” and their electronic structures cannot be adiabatically connected. Now consider two such gapped ferroelectric regions (A and B), which are spatially adjacent to each other. The domain wall between them turns out to be particularly interesting if u_{A} lies in the I–III quadrants and u_{B} the IIIV quadrants. Owing to their different displacement vectors, a pair of Dirac masses at opposite momenta change sign in going from A to B. As a result, there exists a timereversed pair of counterpropagating onedimensional gapless states bound to the domain wall. Similar to the edge states of quantum spin Hall insulators, the domain wall states here are protected from elastic backscattering^{26}, and form perfectly conducting channels.
The existence of such domain wall states explains the robustness of these surface states against timereversalinvariant disorder. By definition, disorder breaks mirror symmetry in a random way locally, not macroscopically. For the sake of argument, let us assume that the disorder potential is slowly varying. The surface is then an equalweight mixture of the aforementioned type I–III and type II–IV ferroelectric domains, with domain walls percolating throughout the surface. As the domain wall is a perfectly conducting channel, the entire surface must be conducting. A similar situation occurs in disordered weak topological insulators^{27,28,29}.
The above analysis of disorder also accounts for surface roughness. Surfaces without any symmetry are generically gapped. Similar to structural distortions, different crystal faces correspond to breaking crystal symmetry in different directions. By the same reasoning, the aforementioned onedimensional domain wall states exist on edges connecting them. As a result, a surface is guaranteed to be metallic, if it preserves mirror symmetry on average in the sense of exhibiting sharp Bragg diffraction peaks as the clean surface. A comprehensive theory of disordered topological crystalline insulators is in progress.
On the experimental side, the surface states of SnTe we predicted can be readily detected in angleresolved photoemission spectroscopy and tunnelling spectroscopy experiments. The underlying mirror Chern number can be deduced from the spin polarization of surface states^{30}. Moreover, SnTebased thin films and superlattices have remarkably high mobility, exceeding 2500 cm^{2} /Vs at room temperature^{31,32}, which provide a promising platform for device applications.
Finally, we relate our work to a wider class of materials, including PbTe and PbSe. Although both are topologically trivial at ambient pressure, our firstprinciples calculation (Fig. 2c) shows that decreasing the lattice constant by 2% inverts the band gap and drives PbTe into a topological crystalline insulator. This band inversion is realized under moderate pressure (around 3 GPa in PbTe and 2 GPa in PbSe^{24,22}). Alternatively, one can achieve the topological regime by growing these materials on substrates with smaller lattice constants. As a precedent, highquality PbTe quantum wells have been fabricated and exhibit ballistic transport^{33,34}. It is also known that the alloys Pb_{1−x}Sn_{x}Te and Pb_{1−x}Sn_{x}Se undergo band inversion as Sn composition increases^{35,36}, so that they become topological crystalline insulators on the inverted side.
We briefly comment on how our work relates to early pioneering fieldtheoretic studies, which predicted the existence of two dimensional massless Dirac fermions at the interface of PbTe and SnTe^{37,38}, or domain wall of PbTe^{39}. Our work has made it clear that only interfaces symmetric about the {110} mirror plane have protected gapless states, which are solely derived from the topological crystalline insulator SnTe and exist even when PbTe is removed. In light of ther topological nature, which we identified, SnTe material class in IV–VI semiconductors is likely to lead a new generation of topological materials.
Methods
Tight binding model
The tightbinding model for SnTe is constructed from the Wannier functions of the conduction and valence bands, which are primarily three porbitals of Sn and Te atoms. The Hamiltonian H_{tb} is given by
Here r labels the site, j=1,2 labels the Sn or Te atom, α=↑,↓ labels electron's spin. The components of vectors c^{†} and c correspond to the three porbitals. In the Hamiltonian (equation (8)), m is the onsite potential difference between Sn and Te; t_{12}=t_{21} is the nearestneighbour hopping amplitude between Sn and Te; t_{11} and t_{22} are the next nearestneighbour hopping amplitudes within a sublattice; is the unit vector connecting site r to r′. The second line thus represents σbond hopping (headtotail) of the porbitals. The λ_{1,2} term is L·s atomic spinorbit coupling, where is the orbital angular momentum in porbital basis. The bulk and surface bands of the above tightbinding Hamiltonian nicely reproduce the essential features of the firstprinciples calculation, and additional terms such as πbond hopping of the porbitals can be added to improve the fit.
The effect of a structural distortion in which atoms are displaced by u can be captured by adding the modulated hopping
to the tight binding model.
Additional information
How to cite this article: Hsieh, T.H. et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 3:982 doi: 10.1038/ncomms1969 (2012).
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Acknowledgements
This work is supported by the NSF Graduate Research Fellowship number 0645960 (T.H.) and startup funds from MIT (L.F.). The work at Northeastern University is supported by the Division of Materials Science and Engineering, Basic Energy Sciences, US Department of Energy, through Grant number DEFG0207ER46352 and benefitted from the allocation of supercomputer time at the NERSC and Northeastern University's Advanced Scientific Computation Center. J.L. and W.D. acknowlege the financial support from the Ministry of Science and Technology of China (Nos. 2011CB921901 and 2011CB606405), the National Natural Science Foundation of China and NSF Grant number DMR1005541.
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T.H. and L.F. made theoretical analysis of SnTe and related IVVI semiconductors. H.L., J.L., W.D. and A.B. performed abinitio calculations. L.F. conceived the idea for topological crystalline insulators in the SnTe material class, supervised the whole project, and wrote part of the manuscript with contributions from all authors.
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Hsieh, T., Lin, H., Liu, J. et al. Topological crystalline insulators in the SnTe material class. Nat Commun 3, 982 (2012). https://doi.org/10.1038/ncomms1969
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DOI: https://doi.org/10.1038/ncomms1969
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