† Corresponding author. E-mail:
Project supported by the Ministry of Science and Technology of China (Grant No. 2016YFA0301000) and the National Natural Science Foundation of China (Grant No. 11334006).
The rise of topological insulators in recent years has broken new ground both in the conceptual cognition of condensed matter physics and the promising revolution of the electronic devices. It also stimulates the explorations of more topological states of matter. Topological crystalline insulator is a new topological phase, which combines the electronic topology and crystal symmetry together. In this article, we review the recent progress in the studies of SnTe-class topological crystalline insulator materials. Starting from the topological identifications in the aspects of the bulk topology, surface states calculations, and experimental observations, we present the electronic properties of topological crystalline insulators under various perturbations, including native defect, chemical doping, strain, and thickness-dependent confinement effects, and then discuss their unique quantum transport properties, such as valley-selective filtering and helicity-resolved functionalities for Dirac fermions. The rich properties and high tunability make SnTe-class materials promising candidates for novel quantum devices.
In the past decade, a new field dubbed topological insulator (TI)[1–4] has caused enormous attentions in the condensed matter physics. Different from the earliest topological states in quantum Hall effect[5,6] which requires a strong magnetic field, TI is time-reversal-symmetry protected, and can be characterized by the Z2 topological invariant.[3,7–9] In TIs, the bulk insulating states are accompanied by metallic helical Dirac-like boundary states which are related to its bulk topology and protected by the time-reversal symmetry. With the novel properties of the edge states, such as high mobility, absence of backscattering and spin-polarized conductivity channels, TIs provide a promising platform for realizing new electronics and spintronics applications. The discovery of TIs also results in the realization of quantum anomalous Hall effect.[10,11] Moreover, the proximity effect between TI and superconductor hosts non-Abelian Majorana fermion and opens a new venue for topological quantum computations.[12] On the materials front, all TI materials are narrow-gap semiconductors with inverted band gap at an odd number of time-reversal-invariant momenta (TRIMs).[8,9] Two-dimensional (2D) TIs (also known as quantum spin Hall states) range from HgTe/CdTe[4,13] and InAs/GaSb[14,15] quantum well structures to the layered honeycomb lattice materials such as silicene,[16,17] germanene,[16,17] stanene,[17–19] their halogenides,[18,20] and ultrathin Bi films.[21,22] The 3D TIs materials include Bi1−xSbx alloys,[9,23] Bi2Se3-class materials,[24–26] half-Heusler compounds,[27,28] TlBiSe2 family chalcogenides,[29–31] strained HgTe,[9,32] α-Sn,[9,33] bismuth-based III–V semiconductors,[34] and so forth.[35–37] Additional ways could convert the normal insulators into TIs, such as external strain[38] and chemical doping.[39]
The discovery of TIs stimulates the identification and searching for more topological quantum states. In TIs, the two branches of helical edge/surface states are related to each other by time-reversal symmetry, and they are degenerate at the TRIMs due to the Kramers’ theorem thus forming the Dirac points. In principle, degeneracies can also come from other types of symmetries, such as particle–hole symmetry, crystalline symmetry, etc. Therefore, finding new topological phases protected by other symmetries is a new hot topic in this rapidly developing field.[40,41] In 2011, Fu[42] proposed that crystalline symmetry can protect new types of topological states, which are called topological crystalline insulators (TCIs). The first theoretically predicted and experimentally realized TCI materials are IV–VI semiconductors, with SnTe as a representative.[43–46] The symmetry responsible for their topological character is the mirror symmetry. Due to the simple crystal structure, SnTe-class TCI materials have caused extensive concerns. Later, some other TCI materials are theoretically proposed in transition metal oxides with a pyrochlore structure[47] and anti-perovskite materials.[48] Both of these two classes of materials are also topologically protected by mirror symmetry. Additionally, crystalline symmetry has been extended from the point group to non-symmorphic space group and a new class of TCIs named topological non-symmorphic crystalline insulator is proposed.[49] Considering the richness and complexity of crystal operations, recently some researchers[41,50,51] have made great efforts to classify the TCIs systematically. However, these proposals ask for the support of realistic materials, and to date the only TCI materials realized in experiments are the IV–VI semiconductors. In this article, we mainly review the recent research on the SnTe-class TCI materials, which may facilitate future researches to search for more TCI materials and particularly to explore their potential applications.
In the research field of topological materials, the theoretical calculations play an important role in the material predictions and provide a reliable guidance for the experimental observations.[37] So in the following (Sections 2 and 3), we will first describe the topological identification of the bulk and surface states properties of SnTe from a computational point of view, and then we introduce the experimental observations. Due to the crystalline symmetry protection and multiple Dirac surface states, TCIs have more tunable properties than TIs. In Sections 4 to 6, we will demonstrate the electronic properties of TCIs under various perturbations, carrier types control or superconductivity by chemical doping, strain-tuned surface states and topological phase transition, and thickness-dependent confinement effects on 2D topological states. Some novel quantum transport properties, such as valley-selective filtering and helicity-resolved functionalities for Dirac fermions, are also discussed in Sections 5 and 6. Finally, we give the conclusions and outlook in Section 7.
SnTe-class IV–VI semiconductors have a rocksalt structure (Fig.
However, the band inversions of SnTe can give new non-trivial topology if we take the crystalline symmetry into account. Face-centered-cubic structure has the mirror symmetry with respect to the {110} planes, one of which is shown by the green plane in Fig.
Hsieh et al.[43] introduced a low-energy effective model near the L point to discuss the different topology between PbTe and SnTe, and they found that the band inversion of SnTe at L changes the mirror Chern number by one. Because there are two L points (L1 and L2) in the Γ L1L2 plane and these two L points have the same contributions to the mirror Chern number,[43] finally the band inversion changes the mirror Chern number for the Γ L1L2 plane by two. As a result, PbTe is topologically trivial, while SnTe is a topological crystalline insulator with the mirror Chern number CM = −2. Actually, there are six equivalent {110} mirror planes for SnTe, and each possesses two L points, so the mirror Chern number for every plane is −2.
The bulk–boundary correspondence indicates the existence of topological surface states. Since band inversion occurs at four L points in bulk BZ of SnTe, there should be four Dirac cones in the surface BZ. However, the gapless surface states of TCIs require the underlying crystalline symmetry to be preserved on the boundary. So not all crystal surfaces of SnTe have gapless surface states, and only those surfaces that keep the mirror symmetry with respect to the (110) planes of rocksalt structure are gapless. Three common surface terminations are (001), (111), and (110),[43,54,55] and they meet the corresponding symmetry. Interestingly, there are two types of TCI surface states with qualitatively different electronic properties,[55] which depend on the surface orientation and are schematically shown in Fig.
![]() | Fig. 2. The projections of bulk BZ towards (001) and (111) surfaces of the rocksalt crystal structure, and the schematic locations of four Dirac cones in these two surface BZs.[56] |
The first-principles calculated band structure of (001) surface of SnTe is shown in Fig.
![]() | Fig. 3. The (001) surface states of SnTe. (a) Band structure along high-symmetry lines from first-principles calculation. Inset: band structure of k·p model of Eq. ( |
This type of TCI surface states can be understood as follows.[55,57] From Fig.
![](cpb_25_11_117313/cpb160880ieqn8.gif)
![](cpb_25_11_117313/cpb160880ieqn9.gif)
![]() | Fig. 4. The ARPES measurements of the (001) surface states of SnTe-class materials. (a) ARPES intensity for SnTe along the cuts A–E in the surface BZ and mapping in a 2D wave-vector plane at various EB.[44] (b) The temperature dependence of the ARPES data for the (001) surface of Pb0.77Sn0.23Se monocrystals. They clearly show the evolution of the gapped surface states (for T ≥ 100 K) into the Dirac-like states on lowering the temperature (T = 9 K).[45] (c) Up panel: ARPES iso-energetic contour mapping (EB = 0.02 eV) of Pb0.6Sn0.4Te covering the first surface BZ. Down panel: ARPES dispersion map along the mirror line direction. Inset: measured spin polarization profile is shown by the green and blue arrows on top of the ARPES iso-energetic contour at binding energy EB = 0.06 eV.[46] |
As the cleavage plane, the (001) surface of SnTe and its family materials has been successfully prepared and the corresponding type-II surface states have been confirmed by experiments.[44–46] Tanaka et al.[44] grew the high-quality single crystal of SnTe, and observed the signature of a metallic Dirac-cone surface band by the angle-resolved photoemission spectrum (ARPES), with its Dirac point slightly away from the edge of the surface BZ (Fig.
Theoretically, the (111) surface states of SnTe are more simple than those of (001) surface. As shown in Fig.
![]() | Fig. 5. Band structures of the (111) surface for Sn and Te terminations. There are four Dirac pockets in the surface BZ: one at ![]() |
The ideal (111) surface of SnTe-class materials is a polar surface with unpaired surface electrons. In principle, the huge electrostatic potential induced by the dipole accumulation will cause the instability of the surface.[59] Using first-principles calculations, Wang et al.[60] studied the (111) surface morphology and the associated electronic structures of SnTe under different growth conditions. They found that three different surfaces can stably exist, with the increasing of chemical potential of Sn: unreconstructed Te-terminated surface,
![]() | Fig. 6. (a) Band structure of (2 × 1)-reconstructed (111) surface, where the surface states are denoted by red lines. Inset: surface atomic configuration, where only one of original three {110} mirror symmetries (indicated by yellow line) remains after reconstruction. (b) The surface Brillouin zones (SBZs): SBZ1 (blue hexagon) for original unreconstructed (111) surface and SBZ2 (red rectangle) for (2 × 1)-reconstructed surface structure. The iso-energetic contour with energies at 5 meV (30 meV) above the Fermi energy is represented by red circle (green dotted circle and shaded area). (c) Band structure of ![]() ![]() |
In experiments, the observation of (111) surface states is truly more challenging owing to the difficulty in the preparation of non-cleavage surface. To date, epitaxial layer growth is an effective method for obtaining the (111) film. Figures
![]() | Fig. 7. The ARPES measurements of the (111) surface states of SnTe-class materials. (a) Near-EF ARPES intensity of SnTe measured along several cuts around the ![]() ![]() ![]() |
As mentioned above and observed by experiments, the as-grown SnTe crystals are usually heavily hole doped, which hinders the detection and investigations of the surface states and possible applications. Meanwhile, PbTe, the cousin material, can be either n or p type, depending on the growth conditions. Wang et al.[64] studied the role of native point defects in the doping behavior of these two materials and explained the microscopic origin of the p-type conductivity of SnTe.
As shown in Fig.
![]() | Fig. 8. Formation energy as a function of Fermi level for native point defects in (a) SnTe and (b) PbTe under Sn/Pb rich and Te-rich conditions. The slope of each segment indicates the charge on the defect. Zero of the Fermi level is set to be the VBM of the host, and the Fermi level range spans the bulk gap.[64] |
The large difference in the nature of defects between SnTe and PbTe originates from the different work functions or the distinction of the VBM of these two materials, as the formation energies of charged defects depend explicitly on the Fermi level referenced to the VBM. The VBM of SnTe is 0.5 eV higher than that of PbTe, inducing that SnTe has a lower formation energy of the negatively charged defects than PbTe. Meanwhile, alloying SnTe with Pb can continuously lower the VBM and CBM, so as to modify the conductivity type. So tuning the Pb/Sn ratio in Pb1−xSnxTe alloy could obtain either n-type or p-type conductivity, which is a suitable way to realize bulk insulating and observe the topological surface states.
When doping the magnetism into thin films of TIs, the topological electronic states with a Chern number ±1 can be formed and the quantum anomalous Hall effect (QAHE) can be realized,[10] which is experimentally observed in Cr doped Bi2Te3 class of TIs.[11] Here in TCIs, Fang et al.[65] proposed that when an out-of-plane ferromagnetic order is introduced into a TCI film, the QAHE with a tunable large Chern number can be produced. However, magnetic doping in SnTe-class TCIs is not easily achievable. In the early years, some experimental efforts have been made to study various types of magnetism in SnTe crystals doped with 3d transition metals (TMs), which are found to be degenerate magnetic semiconductor with small magnetic moments for these TM atoms.[66] These systems show complex magnetic properties: for example, doping with Mn, Cr, and Fe act as ferromagnetic behavior, while doping with Co and Ni are magnetically ineffective.[66] More peculiarly, a strong effect of carrier concentration on the magnetic properties of these diluted magnetic semiconductors has been demonstrated, and it can be understood on the basis of the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction mechanism.[67] Recently, some calculations investigated the electronic and magnetic properties of V, Cr, or Mn-doped SnTe.[68] However their doping concentration is very high, and it is likely to damage the TCI phase of SnTe. Another theoretical study comes to a conclusion that the doped TM atoms in SnTe have comparatively high formation energies, and predicts that the uniform TM doping with a higher concentration in SnTe will be difficult unless clustering.[69] To date, the effective magnetic doping in SnTe-class TCI materials is still a challenge, and the further investigations of magnetic effect on TCIs are required.
Another important doping is In-doped SnTe, which hosts the superconducting behavior.[70,71] In particular, Sn1−xInxTe with an indium content of x = 0.045 preserving the topological surface states has been proved a topological superconducting state by the signals of surface Andreev bound states in the point-contact spectroscopy, and the critical temperature is 1.2 K.[71,72] Furthermore it has recently been reported that the superconducting TC of Sn1−xInxTe continues to increase with a higher level of doping, such as 4.5 K with In content of x = 0.45.[73–75] However, the carrier density does not vary monotonically with Indium content. In fact at high Indium content, Indium has a mixed oxidation state in the system, i.e., it is neither In1+ nor In3+.[76] So superconducting Sn1−xInxTe cannot be viewed as a simple hole doped semiconducting material.
Protected by the crystal symmetry and possessing multiple branches of Dirac surface states, TCIs have a much wider range of tunable electronic properties than TIs under various perturbations. Strain is an important controllable method.[77] Not only can strain drive topological phase transition in the bulk system, but it can also modulate the electronic properties of the Dirac surface states.
Strain is an effective way to tune the band gap of semiconductors. For the narrow-gap semiconductors whose VBM and CBM have the opposite parity or bonding types, the strain will influence their levels differently, and even induce the band inversion.[38] As has been proposed in Bi2Se3 family compounds, strain can induce the topological phase transition from a normal insulator (NI) to a TI and vice verse.[38,78,79] This manipulation can also be applied to TCIs. As mentioned by Hsieh et al.,[43] with the lattice constant decreasing, PbTe can be changed from a NI to a TCI. Such topological phase transition also occurs for other lead chalcogenides under external pressure or volume compression.[80,81] While for the volume expansion, SnS and SnSe can be transformed from an ambient pressure TCI to a topologically trivial insulator.[82]
As has been known to us, there are four L points in the bulk BZ; the isotropic strain cannot distinguish them and will change the band orders of four valleys simultaneously. So the isotropic strain, such as hydrostatic pressure for the rocksalt structure, can only induce the topological phase transition between a NI and a TCI. However, an uniaxial or biaxial strain, which could be applied in the epitaxial films, can offer a diverse manipulation of the electronic properties. A systematic study of SnTe with various in-plane biaxial strains perpendicular to the [001] direction has been made by Qian et al.[83] Since the unit cell fitting for this type of strain is along the [001] direction, there is a folding of bulk BZ. As depicted in Fig.
![]() | Fig. 9. (a), (b) The effect of [001]-oriented strain on the SnTe bulk. (a) BZs of SnTe for primitive cell (in black) and conventional cell along the [001] direction (in blue). (b) Band gap of bulk SnTe at the R point as a function of in-plane biaxial strain. (c)–(f) The effect of [111]-oriented strain on the SnTe bulk. (c) Primitive cell (in black) and conventional cell along the [111] direction (in red). (d) BZs of SnTe for primitive cell (in black) and conventional cell along the [111] direction (in green). (e), (f) Strain-dependent bulk band-gap evolution at the A (red squares) and L (blue circles) points under in-plane and out-of-plane strains, respectively. In (b), (e), and (f), the pink, light green, and white regions indicate the TCI, TI, and NI phases, respectively. Figures (c)–(f) are adapted with permission from Ref. [84], and copyrighted by the American Physical Society. |
A special strain which can distinguish the four L points is the [111]-oriented strain. Zhao et al.[84] studied the band evolution of SnTe with the in-plane biaxial strain and out-of-plane uniaxial strain in the [111] direction. The unit cell and BZ are shown in Figs.
Here, maintaining the TCI phase, we discuss the effect of strain on the (001) surface states. The Dirac points on the (001) surface of SnTe are not pinned to TRIMs. A mechanical strain can shift the Dirac point positions in the momentum space just like an electromagnetic field acts on an electron.[85] Qian et al.[83] studied the strain-dependent electronic properties of SnTe (001) nanomembranes. Here we take the band structure of a 51-layer (001) membrane under the in-plane biaxial strain as an example. As shown in Fig.
The biaxial strain will equally shift the four Dirac points of the (001) surface states. However, a structural distortion with a relative displacement of atoms along the [110] direction on the (001) surface can break one of two mirror planes, and generate a mass to the Dirac cone. As a result, the two Dirac points along
The Dirac surface states on the (111) surface are noninteracting and located at well-separated TRIMs (i.e.,
Maintaining the TCI phase of the entire system, Zhao et al.[84] studied the strain-dependent Dirac valley evolution on the SnTe (111) surface under different strains. Figures
The transmission probabilities T(θ) through the strain-induced nanostructure with two different strain levels are shown in Figs.
![]() | Fig. 11. (a)–(c) DFT-calculated surface band structures of SnTe (111) surface under no strain, in-plane biaxial strain of −2% and out-of-plane uniaxial strain of −2%, respectively. The horizontal dotted lines denote the Fermi levels under different strains, and all bands have set the Fermi level in the unstrained system as the zero reference for energy. (d) Schematic of a strain-induced nanostructure in the SnTe (111) surface grown on a substrate. (e) Schematic of the energy spectra in the strain-induced nanostructure shown in (d). (f), (g) Transmission probabilities T(θ) of massless Dirac fermions at the ![]() |
When the thickness of SnTe film decreases, the top and bottom surface states will hybridize to open up an energy gap at the Dirac points. However, the hybridization is complex and multiple novel properties can emerge.
The Dirac point of SnTe (001) surface states is deviated from the X̄ point in the surface BZ. While at X̄, there is an inverted band gap, i.e., the valence band (conduction band) is derived from the cation (anion). When the thickness of SnTe (001) film is below the penetration length of surface states, the top and bottom surface states will hybridize and result in an energy splitting between the bonding and anti-bonding states. The conduction and valence bands of the 2D film at X will come from the bonding state of the anion at energy EA+(X) and anti-bonding state of the cation at energy EC−(X).[88] However, the band ordering of EA+(X) and EC−(X) depends on the competition between the hybridization of the two surfaces and the inverted gap of each surface. For thick films, the hybridization is weak so that EA+(X) > EC−(X), which inherits the inverted band gap of the 3D limit. While the strong hybridization in the thin film drives EC−(X) higher than EA+(X), leading to a trivial phase. Considering the SnTe (001) films with an odd number of atomic layers, which are symmetric under the reflection z → −z about the (001) plane in the middle, the band gap [EA+(X) − EC−(X)] at the X point as a function of film thickness is shown in Fig.
An important result of 2D TCI with mirror Chern number |NM| = 2 is that there are two pairs of counter-propagating edge states in the band gap. As shown in the left panel of Fig.
A similar mechanism appears in PbTe/SnTe superlattices along the [001] direction.[89] The folding of BZ due to the superlattice structure projects two L points in the original BZ to a single point L′, and their hybridizations form the bonding and anti-bonding states. The coupling strength between the states at equivalent L points depends on the relative thickness or the ratio m/2n in (PbTe)m(SnTe)2n−m superlattice. For a large range of this ratio, the strong coupling induces a band inversion between the bonding state of conduction band and anti-bonding state of valence band. Thanks to reduction in the number of L points from four to two, the band inversion at two L′ will lead to a weak TI phase in PbTe/SnTe superlattices.[89]
![]() | Fig. 12. (a) Band gap of SnTe at the X point as a function of film thickness. The yellow region indicates a 2D TCI phase. (b) Edge states with mirror eigenvalues of an 11-layer SnTe film (left) and electric-field-induced gap (right). (c) Proposed topological transistor device for using an electric field to tune charge and spin transport. Without the electric field, the spin-filtered edge states are gapless and it is in the “on” state (left). Applying a perpendicular electric field, the edge states are gapped and it is in the “off” state (right).[88] |
Another important film is the (111) thin film, which has been grown epitaxially in recent experiments.[61–63] Different from (001) surface states, four Dirac cones of (111) surface states are centered at TRIMs: one is at the Γ point and the other three are at the M points, and they are not all equivalent. When the thickness of (111) film decreases, the hybridization strength between the top and bottom surface states at Γ and M will be different. In fact, the penetration length of the surface states at M is much larger than that at Γ, hence the hybridization-induced gap at M is correspondingly larger than the one at Γ by orders of magnitude.[90] As a result of this inequivalence between Γ and M, the properties of SnTe (111) films are mainly determined by the low-energy physics at the Γ point. The magnitude and sign of the gap at Γ will depend on the film thickness, thus leading to a topological phase transition as a function of the film thickness.[90] Taking the films with an even number of layers as an example (Fig.
The similar results apply to the odd-layer thin film,[90] and even the SnSe (111) film.[91] The mechanism for realizing the QSH phase in TCI thin films through the intersurface coupling is similar to the thin films of Bi2Se3-class TIs.[92–94] However, due to the much larger penetration length of the SnTe/SnSe surface states, the hybridization gap is much larger than that in Bi2Se3 class of TIs.
![]() | Fig. 13. (a) The fundamental gap Eg and the gap at the Γ point Eg(Γ) as functions of the film thickness with even number layers. The blue region indicates a quantum spin Hall (QSH) state, and the orange region represents a normal insulator (NI). The red (blue) dots in the inset denote the contribution of Te (Sn) atoms. (b) The edge states of a 12-layer thin film (left) and 14-layer thin film (right).[90] |
The Dirac fermion of the surface states in the topological materials has the property of spin-momentum locking, which can also be defined as the helicity degree of freedom. As depicted in Fig.
![]() | Fig. 14. A simplified model of electrically tuned band evolution in a thin film of topological material, where the helicity degree of freedom in the system can be controlled by a perpendicular electric field.[95] |
Taking an odd-layer SnTe (111) film as examples, Zhao et al.[95] studied the helicity-resolved band structures and transports. They found that a moderate electric field can induce a giant helicity splitting to the degenerate bands in the vicinity of the
![]() | Fig. 15. (a) Schematic design of a dual-gated heterostructure tuned by the electric field. The incident helicity-degenerate Dirac electrons emitted from the source, and the transmitted helical electron waves are detected by the STM probe. (b) Pronounced helicity filtering when the Fermi level crosses the upper Dirac point. The colorful arrows illustrate the spin orientations at different transmission angles, where the color scale gives the out-of-plane spin component.[95] |
In this article, we have reviewed the electronic properties of SnTe-class TCI materials, from the intrinsic topology, orientation-dependent surface states, to tunable properties under various perturbations. Protected by the mirror symmetry and characterized by the mirror Chern number, SnTe-class TCI materials have multiple branches of Dirac surface states, whose types depend on the surface orientations and even the surface reconstructions. Chemical doping can realize the control of carrier types or concentrations, the topological superconductivity, and the large-Chern-number QAHE. Strain is an effective way to tune the electronic properties of IV–VI materials, including driving topological phase transition for the bulk system, shifting the Dirac point position of the surface states in the momentum space or energy space. The latter could be used for the valleytronic applications. For the thin film of SnTe-class materials, the thickness of film can influence the hybridization strength, and create diverse topological phases, for instance, 2D TCI, weak TI, QSH states, and NI. Among them, the (001) thin film with 2D TCI phase could be utilized to design a topological transistor device with a high on/off speed, and more generally, the helicity degree of freedom of Dirac fermions in the thin film can be controlled by an external electric field to realize the helicity-resolved functionalities.
These novel properties and potential applications are not limited to the IV–VI materials. In fact, more TCIs are proposed theoretically,[41,47–51] and a great deal of studies are needed to search for new TCI materials and explore their applications. Moreover, the crystal symmetry protection has been extended from the insulators to semimetals, such as the Dirac semimetal[97–102] and nodal-line semimetal,[102–106] which possess more exotic effects.[105–107] The interplay between the electronic topology and crystal symmetry has broadened the field of topological phase, and paves a path for developing the quantum electronic and spintronic devices.
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