Abstract
The quantum spin Hall (QSH) phase is an exotic phenomena in condensedmatter physics. Here we show that a minimal basis of three orbitals (s, p_{x}, p_{y}) is required to produce a QSH phase via nearestneighbour hopping in a twodimensional trigonal lattice. Tightbinding model analyses and calculations show that the QSH phase arises from a spin–orbit coupling (SOC)induced s–p band inversion or p–p bandgap opening at Brillouin zone centre (Γ point), whose topological phase diagram is mapped out in the parameter space of orbital energy and SOC. Remarkably, based on firstprinciples calculations, this exact model of QSH phase is shown to be realizable in an experimental system of Au/GaAs(111) surface with an SOC gap of ∼73 meV, facilitating the possible roomtemperature measurement. Our results will extend the search for substrate supported QSH materials to new lattice and orbital types.
Introduction
Currently, there are two prevailing theoretical models for the quantum spin Hall (QSH) phase in a twodimensional (2D) system: the Kane–Mele model^{1} and the Bernevig–Hughes–Zhang (BHZ) model^{2,3}. In Kane–Mele model, the QSH phase is realized by any finite SOCinduced bandgap opening at Dirac point, as a generalization of Haldane’s model^{4} to spinful system with time reversal symmetry in a hexagonal lattice. In BHZ model, the QSH phase is realized by SOCinduced band inversion at time reversal invariant momenta between two bands of different parities, originally derived from a square lattice of HgTe quantumwell system. However, 10 years after these original theoretical proposals, only two material systems with a tiny bandgap (HgTe/CdTe (ref. 3) and InAs/GaSb (ref. 5)) have been confirmed experimentally for the BHZ quantumwell model, while no experiment has yet to confirm the Kane–Mele model in a real material of hexagonal lattice, despite many material systems have since been theoretically predicted^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23}. Therefore, there remains an intensive search for QSH materials, especially with a large gap, and it is highly desirable to expand such search beyond the original hexagonal and square lattice to increase the feasibility for experimental realization.
Here we prescribe a discrete lattice model for QSH phase in a trigonal lattice. We consider a minimal basis of three orbitals (s, p_{x}, p_{y}) per lattice site of trigonal symmetry with nearestneighbour hopping, solve an effective tightbinding Hamiltonian and develop a generic phase diagram for the nontrivial band topology in the parameter space of orbital energy and SOC. Depending on the order of s versus p orbital energies, a QSH phase may arise from either strong or weak SOC. Most remarkably, based on firstprinciples calculations, this exact QSH model is shown to be possibly realizable in an experimental system of Au/GaAs(111)^{24} with a large nontrivial SOC gap of ∼73 meV.
Results
The minimal basis QSH lattice model
In general, a minimal basis of three orbitals (s, p_{x}, p_{y}) in a trigonal lattice is shown in Fig. 1a, which can also be equivalently transformed into three sp^{2} hybridized orbitals, as shown in Fig. 1b. The corresponding tightbinding Hamiltonian in the basis of (s, p_{x}+ip_{y}, p_{x}−ip_{y}) is shown in Supplementary Note 1. Around the Γ point, expending to the firstorder of k the Hamiltonian for spinup band reduces to
where ɛ_{s} and ɛ_{p} are onsite energies for s and p orbitals, respectively. t_{ssσ}, t_{spσ}, t_{ppσ} and t_{ppπ} are nearestneighbour hopping parameters and λ is SOC strength. k_{x} and k_{y} are momentum along x and y direction. Solving equation (1), the three eigenvalues are E_{s}=ɛ_{s}+6t_{ssσ} and E_{p}^{±λ}=E_{p}±λ=ɛ_{p}+3(t_{ppσ}+t_{ppπ})±λ. Clearly, the s orbital is independent of SOC and the two p orbitals are degenerate without SOC. Depending on the order of E_{s} versus E_{p}, there are two different types of bands.
For the first type, the sband is above the pband (E_{s}>E_{p}) without SOC, as shown in Fig. 1c. The red and blue colours indicate the components of s and p orbitals, respectively, and the parities for each subband at time reversal invariant momenta^{25} (Γ point and three M points) are labelled with +and − signs. It is found that the topological invariant^{25} is Z_{2}=0 for middle and bottom two subbands, indicating clearly a normal insulator phase. To realize the QSH phase, an SOCinduced s–p band inversion is needed (E_{s}<E_{p}^{λ}), that is, first closing of a trivial gap followed by reopening of a nontrivial gap by including the SOC. With the increasing SOC strength, one can see that the bandgap between top and middle subband first reduces and closes (Fig. 1d), and then reopens (Fig. 1e) at the Γ point. The s–p band inversion induces a parity exchange in this process, so that the topological invariant changes from Z_{2}=0 (Fig. 1c) to Z_{2}=1 (Fig. 1e) for the middle and bottom two subbands.
For the second type, the sband is below the pband (E_{s}<E_{p}) without SOC, as shown in Fig. 1f. Different from Fig. 1c, the topological invariant is already Z_{2}=1 for the middle and bottom two subbands, that is, the band order has already been inverted even without SOC. Thus, any finite SOC will open a nontrivial gap at Dirac point to turn the system into a QSH phase as the case for graphene^{1}, as shown in Fig. 1g. This second type of QSH phase in a trigonal lattice has also been recently discussed with multiple pbands in a k·p model^{26}. The parity does not change by including the SOC in this process. On the basis of the above band order analysis, a topological phase diagram can be constructed, as shown in Fig. 1h. Using Δ=E_{s}−E_{p} and λ as two independent parameters, normal insulator and QSH phase is divided by the bandgap closing line (E_{s}=E_{p}^{+λ}, dashed line in Fig. 1h). The parameters used for band structure in Fig. 1c–g are labelled with I–V. We have marked these data points in the phase diagram (Fig. 1h), with dashed arrows indicating the increasing SOC strength to distinguish these two different QSH phase realization processes. Thus, we introduce a discrete lattice model with minimal basis of three orbitals in a trigonal lattice to realize the QSH phase.
The effective QSH model
The band structure evolution in Fig. 1c–g (I–V) can be seen more intuitively from a threedimensional (3D) band plot around the Γ point, as shown in Fig. 2a,d for the first and second type of band order, respectively. It has been shown that different lattices models may be adiabatically connected to different classes of effective models of topological quantum field theory at the continuum limit, which generally requires only twoband inversion^{27}. Correspondingly, one can reduce the threeband Hamiltonian to an effective twoband Hamiltonian around the Γ point^{27} as H_{eff}=d_{0}(I)+d·σ, where I is the identity matrix and σ is the Pauli matrices, d_{0} is a parameter and d is a 3D vector field in momentum space (Supplementary Note 2). For the nontrivial band III (Fig. 2a) and V (Fig. 2d), has a vortex structure around the Γ point, as shown in Fig. 2b,e, respectively. At the Γ point, is along south pole in Fig. 2b and north pole in Fig. 2e. When k goes away from the Γ point, changes direction gradually from out of plane to inplane in both cases. In addition, the energy splitting between the two bands is characterized by the value of 2d, which is E_{s}–E_{p}^{+λ} and 2λ at the Γ point for band III and V, respectively. Generally, a vortex is a topological defect described by the Chern number (C). To confirm this, we also calculate the Berry curvature for band III and V, as shown in Fig. 2c,f, respectively. The Berry curvature is nonzero around the Γ point and gives C=1, confirming a nontrivial topological phase.
The firstprinciples calculations
After establishing the above QSH model, the next question is whether it can be realized in a real material? Instead of searching for a freestanding 2D material, which is generally metastable and whose intrinsic topological properties can be altered when it is placed on a substrate, we opt to focus on surfacebased 2D materials that is supported on a substrate, that is, it is atomically bonded to but electronically decoupled from the substrate^{15,16,17,18,19,28,29}, so that it is relatively more feasible for experimental realization, as well as easier to facilitate experimental measurement and device fabrication. Also, we note that usually metal atoms favour a closepacking geometry forming trigonal lattice in 2D rather than the open hexagonal lattice. In fact, many heavy metal atoms grown on a semiconductor surface have been found to exhibit a trigonal lattice symmetry, such as Bi and Tl on Si(111)^{30}, Au on Ge(111)^{31}.
Specifically, the system of Au/GaAs(111) has drawn our attention. Hilner et al.^{24} showed that at low coverage of Au on the Asterminated GaAs(111) surface, a trigonal superlattice structure was observed by scanning tunnelling microscopy. Furthermore, a theoretical model with one Au atom adsorbed on every third hexagonal closepacked threefold hollow site of the Ga lattice is shown to be the most energetically stable, with a simulated scanning tunnelling microscopy image in excellent agreement with the experiment (Fig. 1c in ref. 24). So, starting from the known structural model of Au/GaAs(111)^{24} (Fig. 3a), its topological properties are systematically studied. The band structure of Au/GaAs(111) without SOC is shown in Fig. 3b. The key feature of notice is the Dirac band around the Γ point above the Fermi level (a 3D band plotting is shown in the inset of Fig. 3b), which are well separated from the other bulk bands (shaded region). The s and p orbital components for band I, II and III (labelled in Fig. 3b) are shown by red and blue colours, respectively. At the Γ point, band I, II and III are mainly made of p, p and s orbitals. When k goes away from the Γ point, band I, II and III are mainly made of s, p and p orbitals. The overall band shape and orbital components are found to be consistent with the band IV (Fig. 1f), as obtained from the minimal basis model in a trigonal lattice discussed above. This is further confirmed by plotting the realspace charge density distribution of these three bands at the Γ point, as shown in Fig. 3c. For band I and II, the charge densities are mainly localized on top of three As atoms, showing an inplane p orbital shape. For band III, the charge densities are mainly localized on Au atom, showing an s orbital shape. So, effectively these three bands arise from three orbitals of (s, p_{x}, p_{y}) character (Fig. 1b) with nearestneighbour hopping in a trigonal lattice.
Quantitatively, to obtain a better fitting for the firstprinciples bands (band I, II and III) of Au/GaAs(111), a 3 × 3 Wannier Hamiltonian is constructed by using the maximally localized Wannier functions (MLWFs) in Wannier90 package^{32}. As shown in Fig. 3d, the Wannier band shows excellent agreement with the firstprinciples band. The three fitted MLWFs are shown in Fig. 3e, which are equivalent to each other and have a threefold rotational symmetry around centre Au atom. Each MLWF has mixed components from both Asp and Aus orbitals. Adding the components together, the overall orbital shape is shown in the last panel of Fig. 3e. An stype orbital is centred at Au atom, and three tilted ptype orbitals are centred at three surface As atoms along the As–Au bond direction, forming effectively an sp^{2} hybridization. This can be easily understood from the unhybridized surface orbitals. As schematically shown in Fig. 3f, each Au atom contributes one s orbital, each dangling bond of surface As atom contributes one sp^{3} orbital. The sp^{3} orbital can be further decoupled into the orbitals along and perpendicular to the Au–As bond directions. Because of the odd orbital symmetry, the perpendicular component has a negligible hopping integral with Aus orbital, while the other component along the bond has a large overlap with the Aus orbital to form a σ bond.
The analysis of nontrivial topological phase
Generally, when SOC is introduced to the Dirac band, a nontrivial topological phase is expected^{1}. Thus, we proceed by calculating the band structure of Au/GaAs(111) with SOC. As shown in Fig. 4a, an SOC gap is opened at the Γ point; the orbital components for band I, II and III remain the same as that without SOC (Fig. 3b). Because of the large SOC in Au and GaAs, the size of SOC gap is very significant of ∼73 meV (Fig. 4b), which facilitates possibly the roomtemperature measurement. In addition, the inversion symmetry is broken in Au/GaAs(111), so that the band degeneracy is lifted by Rashba splitting. Neglecting the Rashba effect, the SOC gap and bands can be well reproduced by including an onsite SOC term in our minimalbasis tightbinding band V, as shown in Fig. 1g. A further comparison between the fitted Wannier band and firstprinciples band is shown in Fig. 4b, showing excellent agreement.
To clearly identify the nontrivial topology in Au/GaAs(111), we have calculated its spin Berry curvature, spin Hall conductance and edge states by using the fitted Wannier Hamiltonian. The spin Hall conductance () is defined based on the spin Chern number (C_{s}), which can be calculated from the spin Berry curvature by using standard Kubo formula^{10}. Here e is the elementary charge and π is the mathematical constant. Figure 4c shows the spin Berry curvature around the Γ point, which is comparable to that shown in Fig. 2f. Figure 4d shows the calculated spin Hall conductance as a function of Fermi level, which has a quantized value within the energy window of SOC gap, demonstrating the critical feature of QSH phase. The nonzero spin Chern number can also be manifested by the presence of gapless edge states within the SOC gap. Onedimensional (1D) ribbon band structure of Au/GaAs(111) with a width of 100 unit cells are calculated^{33}, as shown in Fig. 4e. A pair of gapless edge states with a Dirac cone at the Γ point, which are energydegenerate for opposite (left and right) edges, are seen within the SOC gap. The real space distribution of the edge states, at the energy marked by blue dot in Fig. 4e, is shown in Fig. 4f. Clearly, the degenerated edge states are spatially localized at the opposite edges (left and right) of the ribbon. The number of the edge states indicates the absolute value of the spin Chern number, which is consistent with the calculated C_{s}=1 in Fig. 4d.
Surface ndoping to tune Fermi level
We note that the Fermi level of Au/GaAs(111) is not in the nontrivial SOC gap, so ndoping is needed. Generally, ndoping can be more easily realized than pdoping in surface systems. For example, alkalimetal atoms have been widely used for surface ndoping in the experiments, including graphene^{34}, superconducting FeSe (ref. 35) and metal film^{30}. Thus, we have adopted the same strategy using K to ndope the Au/GaAs(111) surface. Our calculations show that K prefers to adsorb at the hexagonal closepacked hollow site and form a uniform distribution due to the Coulomb repulsion between charged K atoms^{15}. At 1/3 and 2/3 monolayer (ML) of K coverage, the Fermi level of Au/GaAs(111) can be tuned close to and below and above the SOC gap, respectively, and 1D ribbon calculations further demonstrate their nontrivial topological edge states (Supplementary Fig. 1). The Fermi level can be continuously moved upward, so at 5/12 ML (see atomic structure in Fig. 5a modelled by a larger 2 × 2 supercell), the Fermi level is moved into the Dirac band without SOC (Fig. 5b) or inside the gap with SOC (Fig. 5c). It is important to notice that K surface adsorption provides an ideal ndoping by moving the Fermi level without affecting the bands of host system in all the cases studied. Furthermore, 1D ribbon calculations demonstrate two nondegenerated gapless Dirac edge states within SOC gap, as shown in Fig. 5d. From the edgeprojected bands, the left (Fig. 5e) and right (Fig. 5f) Dirac edge states can be distinguished. Thus, Au/GaAs(111) can be tuned into an intrinsic 2D TI with surface alkalimetal doping. Moreover, we have done molecular dynamics simulations in a canonical assemble at room temperature to further demonstrate the stability of K atoms on GaAs surface (Supplementary Fig. 2).
Discussion
Currently, there are two experimental methods that can be used to detect topological edge states in 2D TIs. One is the transport measurement to measure the quantized conductance induced by topological edge state, and the other is scanning tunnelling spectroscopy measurement to image the realspace topological edge state within the energy window of SOC gap. For the second method, even if the Fermi level is below the nontrivial SOC gap (a nonintrinsic TI), its topological edge state can still be observed experimentally^{36,37,38}, because the empty state of TI far away from the Fermi level can be easily accessed by scanning tunnelling spectroscopy^{39}. Thus, we believe our work may stimulate new experimental studies of Au/GaAs(111) beyond previous efforts, especially for Kdoped systems. Without our theoretical prediction, however, this system could be easily overlooked because it has a trigonal (Au) lattice, which is not known to support topological state before.
In summary, we introduce a generic minimalbasis threeorbital QSH model in trigonal lattice, which can be reduced to an effective twoband model at the continuum limit. Furthermore, we suggest possible experimental realization of our model in the Au/GaAs(111) system that has already been experimentally grown previously. Our findings not only enrich the basic knowledge of QSH phase but also extend the search for QSH materials to new lattice and orbital types. We believe our proposed new classes of QSH materials can be generalized to other metal elements and semiconductor surfaces for experimental exploration.
Methods
The firstprinciples calculations
The firstprinciples calculations are carried out in the framework of generalized gradient approximation with Perdew–Burke–Ernzerhof functionals using the Vienna Ab initio simulation package^{40}. The GaAs(111) surface is simulated by sixbilayer slabs with lattice constant a=7.06 Å obtained from the optimized bulk GaAs. The bottom surface is saturated by pseudohydrogen atoms fractionally charged with 1.25e. All the calculations are performed with a planewave cutoff of 400 eV on the 9 × 9 × 1 Monkhorst–Pack kpoint mesh. The vacuum layer of 15 Å thick is used to ensure decoupling between neighbouring slabs. During structural relaxation, bottom twobilayer GaAs and hydrogen atoms are fixed. The Au and top fourbilayer GaAs atoms are relaxed until the forces are smaller than 0.01 eV Å^{−1}.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Wang, Z. F. et al. Quantum spin Hall phase in 2D trigonal lattice. Nat. Commun. 7:12746 doi: 10.1038/ncomms12746 (2016).
References
 1
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
 2
Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).
 3
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
 4
Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensedmatter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).
 5
Knez, I., Du, R.R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2011).
 6
Murakami, S. Quantum spin Hall effect and enhanced magnetic response by spinorbit coupling. Phys. Rev. Lett. 97, 236805 (2006).
 7
Liu, C.C., Feng, W. & Yao, Y. Quantum spin Hall effect in silicene and twodimensional germanium. Phys. Rev. Lett. 107, 076802 (2011).
 8
Weeks, C., Hu, J., Alicea, J., Franz, M. & Wu, R. Engineering a robust quantum spin Hall state in graphene via adatom deposition. Phys. Rev. X 1, 021001 (2011).
 9
Wang, Z. F., Liu, Z. & Liu, F. Organic topological insulators in organometallic lattices. Nat. Commun. 4, 1471 (2013).
 10
Wang, Z. F., Su, N. & Liu, F. Prediction of a twodimensional organic topological insulator. Nano Lett. 13, 2842–2845 (2013).
 11
Jin, K.H. & Jhi, S.H. Proximityinduced giant spinorbit interaction in epitaxial graphene on a topological insulator. Phys. Rev. B 87, 075442 (2013).
 12
Xu, Y. et al. Largegap quantum spin Hall insulators in tin films. Phys. Rev. Lett. 111, 136804 (2013).
 13
Chuang, F.C. et al. Prediction of largegap twodimensional topological insulators consisting of bilayers of group III elements with Bi. Nano Lett. 14, 2505–2508 (2014).
 14
Si, C. et al. Functionalized germanene as a prototype of largegap twodimensional topological insulators. Phys. Rev. B 89, 115429 (2014).
 15
Zhou, M. et al. Epitaxial growth of largegap quantum spin Hall insulator on semiconductor surface. Proc. Natl Acad. Sci. USA 111, 14378–14381 (2014).
 16
Zhou, M. et al. Formation of quantum spin Hall state on Si surface and energy gap scaling with strength of spin orbit coupling. Sci. Rep. 4, 7102 (2014).
 17
Hsu, C.H. et al. The nontrivial electronic structure of Bi/Sb honeycombs on SiC(0001). New J. Phys. 17, 025005 (2015).
 18
Crisostomo, C. P. et al. Robust large gap twodimensional topological insulators in hydrogenated III–V buckled honeycombs. Nano Lett. 15, 6568–6574 (2015).
 19
Xu, Y., Tang, P. & Zhang, S.C. Largegap quantum spin Hall states in decorated stanene grown on a substrate. Phys. Rev. B 92, 081112(R) (2015).
 20
Ren, Y., Qiao, Z. & Niu, Q. Topological phases in twodimensional materials: a review. Rep. Prog. Phys. 79, 066501 (2016).
 21
Ma, Y., Dai, Y., Kou, L., Frauenheim, T. & Heine, T. Robust twodimensional topological insulators in methylfunctionalized bismuth, antimony, and lead bilayer films. Nano Lett. 15, 1083–1089 (2015).
 22
Zhao, M., Zhang, X. & Li, L. Straindriven band inversion and topological aspects in antimonene. Sci. Rep. 5, 16108 (2015).
 23
Weng, H., Yu, R., Hu, X., Dai, X. & Fang, Z. Quantum anomalous Hall effect and related topological electronic states. Adv. Phys. 64, 227–282 (2015).
 24
Hilner, E. et al. Au wetting and nanoparticle stability on GaAs(111)B. Appl. Phys. Lett. 89, 251912 (2006).
 25
Liu, Z., Liu, C.X., Wu, Y.S. & Duan, W. Stable nontrivial Z2 topology in ultrathin Bi (111) films: a firstprinciples study. Phys. Rev. Lett. 107, 136805 (2011).
 26
Liang, Q. F., Yu, R., Zhou, J. & Hu, X. Topological states of nonDirac electrons on a triangular lattice. Phys. Rev. B 93, 035135 (2016).
 27
Qi, X.L., Hughes, T. L. & Zhang, S.C. Topological field theory of timereversal invariant insulators. Phys. Rev. B 78, 195424 (2008).
 28
Zhou, M., Liu, Z., Ming, W., Wang, Z. F. & Liu, F. sd^{2} graphene: Kagome band in hexagonal lattice. Phys. Rev. Lett. 113, 236802 (2014).
 29
Wang, Z. F. & Liu, F. Selfassembled Si(111) surface states: 2D Dirac material for THz plasmonics. Phys. Rev. Lett. 115, 026803 (2015).
 30
Gruznev, D. V. et al. A strategy to create spinsplit metallic bands on silicon using a dense alloy layer. Sci. Rep. 4, 4742 (2014).
 31
Nakatsuji, K., Motomura, Y., Niikura, R. & Komori, F. Selective doping in a surface band and atomic structures of the Ge(111) Au surface. J. Phys. Condens. Matter 25, 045007 (2013).
 32
Mostofi, A. A. et al. Wannier90: A tool for obtaining maximallylocalised Wannier functions. J. Comput. Phys. Commun. 178, 685–699 (2008).
 33
Wang, Z. F., Chen, L. & Liu, F. Tuning topological edge states of Bi (111) bilayer film by edge adsorption. Nano Lett. 14, 2879–2883 (2014).
 34
Fedorov, A. V. et al. Observation of a universal donordependent vibrational mode in graphene. Nat. Commun. 5, 3257 (2014).
 35
Miyata, Y., Nakayama, K., Sugawara, K., Sato, T. & Takahashi, T. Hightemperature superconductivity in potassiumcoated multilayer FeSe thin films. Nat. Mater. 14, 775–779 (2015).
 36
Yang, F. et al. Spatial and energy distribution of topological edge states in single Bi(111) bilayer. Phys. Rev. Lett. 109, 016801 (2012).
 37
Drozdov, I. K. et al. Onedimensional topological edge states of bismuth bilayers. Nat. Phys. 10, 664–669 (2014).
 38
Wang, Z. F. et al. Topological edge states in a hightemperature superconductor FeSe/SrTiO3(001) film. Nat. Mater doi:10.1038/nmat4686 (2016).
 39
Yee, M. M. et al. Spinpolarized quantum well states on Bi2xFexSe3 . Phys. Rev. B 91, 161306(R) (2015).
 40
Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561(R) (1993).
Acknowledgements
Z.F.W. acknowledges financial support from Chinese Youth 1000 Talents Program and Fundamental Research Funds for the Central Universities. K.H.J. and F.L. acknowledge financial support from DOEBES (No. DEFG0204ER46148). We also thank Supercomputing Center at USTC, NERSC and CHPC at University of Utah for providing the computing resources.
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F.L. and Z.F.W. conceived the project; K.H.J. and Z.F.W. performed tightbinding model calculations; Z.F.W. performed firstprinciples calculations; Z.F.W., K.H.J. and F.L. prepared the manuscript.
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Correspondence to Z. F. Wang or Feng Liu.
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Supplementary Figure 12 and Supplementary Note 12 (PDF 579 kb)
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Wang, Z., Jin, K. & Liu, F. Quantum spin Hall phase in 2D trigonal lattice. Nat Commun 7, 12746 (2016). https://doi.org/10.1038/ncomms12746
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