Abstract
We predict planar Sb/Bi honeycomb to harbor a twodimensional (2D) topological crystalline insulator (TCI) phase based on firstprinciples computations. Although buckled Sb and Bi honeycombs support 2D topological insulator (TI) phases, their structure becomes planar under tensile strain. The planar Sb/Bi honeycomb structure restores the mirror symmetry, and is shown to exhibit nonzero mirror Chern numbers, indicating that the system can host topologically protected edge states. Our computations show that the electronic spectrum of a planar Sb/Bi nanoribbon with armchair or zigzag edges contains two Dirac cones within the band gap and an even number of edge bands crossing the Fermi level. Lattice constant of the planar Sb honeycomb is found to nearly match that of hexagonalBN. The Sb nanoribbon on hexagonalBN exhibits gapped edge states, which we show to be tunable by an outoftheplane electric field, providing controllable gating of edge state important for device applications.
Introduction
Topological crystalline insulators (TCIs) are a recently discovered novel phase of quantum matter in which topological protection of states arises from a combination of timereversal and crystalline symmetries^{1,2}. TCIs are to be contrasted sharply from the more common topological insulators (TIs) where timereversal symmetry (TRS) alone is the source of topological protection of states^{3,4,5,6}. In particular, the 2D TIs, also referred to as quantum spin Hall (QSH) insulators, can support dissipationless edge currents since the only available scattering channel (backscattering) is forbidden by symmetry constraints^{3,4,5,6}. In the panorama of topological materials, the tunable surface/edge states of the TCIs could provide unique platforms for designing next generation electronics and energy and information technologies.
SnTe class of materials was theoretically predicted to harbor the threedimensional (3D) TCI phase^{2}, and this class of material still remains the only such phase that has been realized experimentally. Properties of SnTe^{7,8,9}, Pb_{1−x}Sn_{x}Te^{10,11}, and Pb_{1−x}Sn_{x}Se^{12,13,14} have been investigated theoretically and experimentally. In a recent report, the freestanding monolayer PbSe is predicted to be 2D TCI^{15}, although this has not so far been realized experimentally. On the other hand, many thin films of groups IV and V, IIIV compounds^{16,17,18,19,20,21,22,23,24,25}, and their films passivated by hydrogens or halogens^{26,27,28,29,30,31,32,33,34,35} are predicted to harbor the 2DTI order in the honeycomb structure. This provides strong motivation for exploring the possibility that TCI phases could be hidden in 2D honeycombs of thin films of other materials. Strained Sb and Bi buckled honeycombs are especially germane in this connection because they assume a planar structure under strain^{22}, and thus restore the mirror symmetry of the pristine honeycomb, opening up a new playground for the discovery of 2DTCI phases.
In this study, we identify the pristine Sb/Bi honeycomb as a 2DTCI in its planar form via firstprinciples calculations. With increasing inplane strain, the buckled Sb/Bi honeycomb first undergoes a transition from a Z_{2} nontrivial to a trivial phase through a band inversion at the M point, before it assumes the planar structure. Further increase in strain leads to the appearance of the TCI phase in the planar Sb/Bi honeycomb, which is protected by the reflection symmetry to the planar structure. Since the corresponding Z_{2} invariant is zero, a nonzero mirror Chern number, thus supports its 2DTCI character. We also investigated nanoribbons of Sb/Bi planar honeycomb with both armchair as well as zigzag edges, and found the presence of two Dirac cones with nodes lying within the band gap with an even number of edge bands crossing the Fermi level as signature of TCIs. Keeping in mind that a film must eventually be placed or grown on a substrate, we propose that hexagonalBN (hBN) is a good candidate substrate because the lattice constants of planar Sb honeycomb and hBN are nearly commensurate. In this connection, we consider the electronic structure of a planar Sb nanoribbon on hBN, and found that it supports two gapped Dirac cones, reflecting the substrateinduced symmetry breaking. This substrateinduced gap, however, is shown to be tunable with an outoftheplane electric field, providing a useful materials platform for applications.
Results
The structural phase transition of freestanding Sb/Bi honeycomb from buckled to planar structure under a tensile strain, as well as the crystal structure and the associated 2D Brillouin zone (BZ), are shown in Fig. 1(a). The equilibrium lattice constants of the buckled and planar Sb honeycombs are 4.12 Å and 5.04 Å^{25}, respectively; while the corresponding lattice constants for Bi honeycombs are 4.33 Å (buckled) and 5.27 Å (planar)^{22}. Concerning topological properties, note that previous studies based on an analysis of parities of states at high symmetry points in the BZ, have shown that planar Sb and Bi honeycombs are both Z_{2} topologically trivial^{22,24}. Since we employ the general method of ref. 36 for computing the Z_{2} invariant throughout this study, we have verified that the band topologies of Sb/Bi planar honeycombs we obtain are indeed Z_{2} trivial, as is also the case for the buckled Sb honeycomb.
Topological phase transitions can be analyzed by considering the evolution of band structure as a function of the lattice constant. In the case of the Sb honeycomb, two successive band inversions are seen with increasing tensile strain in Fig. 1(b–f). [Here, the black circles are proportional to the contribution of s orbitals]. The first inversion, which is seen to occur at the Γpoint, is associated with the slightly larger lattice constant a = 4.325 Å, while the second inversion takes place at the Mpoint with a = 4.905 Å, before the formation of the planar structure. The Bi honeycombs also undergo two similar band inversions (not shown in Fig. 1 for brevity)^{22}. We find that the buckled Sb honeycomb at the equilibrium lattice constant is topologically trivial (Z_{2} = 0), and that after the first band inversion, the system becomes nontrivial. Interestingly, the system returns to the trivial phase once again after the second band inversion at the Mpoint. In sharp contrast, the Bi honeycomb is Z_{2} nontrivial in the equilibrium state, but the inversion at the Mpoint under tensile strain gives rise to a trivial topological phase.
We emphasize that once the planar structure is formed, the film not only possesses the full sixfold rotational symmetry but that its mirror symmetry is also restored in that the honeycomb behaves as a mirror plane with reflection symmetry. Accordingly, we examine the associated mirror Chern numbers of the planar honeycombs. For this purpose, at each k point, Bloch states can be classified into two groups by their mirror eigenvalues: and , where +(−) denotes the mirror eigenvalue +i(−i). The Chern number for each mirror sector can then be computed via^{37}
where E_{f} is the Fermi energy, and the integration is over the whole 2D BZ. In this way, we obtained mirror chern number of 2 for both the Sb and Bi planar honeycombs, which indicates that these films are TCIs with an even number of edge bands crossing the Fermi level.
Insight into the nature of the protected edge states is obtained by constructing nanoribbons of planar Sb and Bi honeycombs armchair and zigzag edges, see Fig. 2(c,f). Here, instead of using the equilibrium lattice constant, we used a lattice constant of 5.229 Å, which is comparable with the lattice constant of 2 × 2 hBN. [These results will also be helpful in comparing the edge states of Sb/Bi planar honeycomb on a 2 × 2 hBN substrate in the discussion below]. The widths of ribbons with zigzag and armchair edges were set at 108.7 Å and 73.2 Å (14 × a), respectively. These values are large enough so that interactions between the two edges of the ribbon can be neglected. The resulting band structures are shown in Fig. 2 in which contributions of the left and right hand side edges are identified along with the bulk band structures.
The armchair as well as the zigzag edges in Fig. 2 are seen to exhibit the presence of two edge states related Dirac cones whose nodes lie within the bulk band gap. Recall that the Z_{2} topological phase can be identified by counting the number of edge bands crossing the Fermi level in half the BZ. An odd number of crossings between two timereversal invariant momentum points in the BZ indicates a nontrivial nature of the band structure, whereas an even number of such crossings corresponds to the Z_{2} trivial order. Here, however, we see in Fig. 2, that there is an even number of edge bands cutting across the Fermi level for both Sb and Bi ribbons. This is consistent with the computed value of , further verifying that our planar Sb and Bi honeycombs support the 2DTCI phase.
A film must eventually be placed or grown on a substrate. In this connection, we note that the 2 × 2 hBN substrate has a lattice constant of 5.229 Å, which is quite close to the lattice constants of 5.04 Å and 5.27 Å for planar Sb and Bi honeycombs, respectively. Also, we find the total energy of the 2DTCI films of Sb and Bi on the hBN substrate to be insensitive to the buckling of honeycombs. [The energy difference between the planar and buckled Sb/Bi honeycomb on 2 × 2 hBN is only 1 meV per supercell]. These considerations lead us to suggest that 2 × 2 hBN would be a good substrate for supporting 2DTCI films of Sb and Bi. Accordingly, we placed planar Sb and Bi ribbons of Fig. 2 on 2 × 2 hBN substrate as shown in Fig. 3(a,c). Substrateinduced band gaps can be seen at Γ as well as in the Dirac cones lying between the Γ and M points (see Fig. 3(b,d)). Since the Dirac cone at Γ is protected by timereversal symmetry, the gap opening at Γ reflects size effects resulting from the finite width of the ribbon. In contrast, the Dirac cone lying between Γ and M points is protected by the mirror symmetry, and for this reason, the gap opening is now also due to the breaking of the mirror symmetry in the presence of the substrate. Note that as the width of the ribbon is increased in the computations, we would expect the gap at Γ to become smaller and eventually vanish, while the gap in the Dirac cone between Γ and M will remain.
The sizes and nature of gaps in TIs can be tuned by an outoftheplane electric field^{38}. This can also be expected for Dirac cones in TCIs. For example, in the case of a armchair ribbon, we have noted above that the hBN substrate can open a gap in the Dirac cone by breaking the mirror symmetry of the film, providing a pathway for an external on/off control of edge current via gating. Keeping in mind that the hBN substrate is a good insulator, we show schematically in Fig. 4 a proposed design of a field effect transistor based on multilayers of hBN and Sb/Bi planar honeycomb in which multiple TCI edge states can be used for transport.
Conclusions
Using firstprinciples computations, we have explored the viability of realizing a 2DTCI phase in films of Sb and Bi. Keeping in mind that the protection of topological states in a TCI is provided by a combination of timereversal and crystalline symmetries and not just the timereversal symmetry as is the case in a TI. We focus on planar Sb/Bi honeycombs in view of their mirror symmetry. Such a planar honeycomb stabilizes under tensile strain, even though the pristine Sb/Bi honeycombs assume a buckled structure, which does not possess the mirror symmetry. In order to analyze topological properties of planar Sb/Bi honeycombs, we evaluate the associated mirror Chern numbers, and find their value to be 2, indicating the presence of a TCI phase. Further insight into the nature of this TCI phase is obtained by computing the edge state spectrum of ribbons constructed from planar Sb/Bi honeycombs with armchair as well as zigzag edges, where we find two separate Dirac cones lying within the bulk band gap. Our analysis suggest that hBN with a lattice constant nearly commensurate with that of Sb/Bi planar honeycombs would be a suitable substrate for maintaining the 2DTCI phase. Our study predicts that planar Sb/Bi honeycombs harbor a 2DTCI phase with gaps controllable with an external outoftheplane electric field, and throws open the possibility of using 2DTCIs as novel applications platforms.
Methods
Firstprinciples calculations were carried out within the generalized gradient approximation (GGA) to the densityfunctional theory (DFT)^{39,40} using the projectoraugmentedwave (PAW) method^{41} as implemented in the Vienna AbInitio Simulation Package (VASP)^{42,43}. The kinetic energy cutoff was set at 400 eV and atomic positions were relaxed until the residual forces were less than 10^{−3} eV/Å. The convergence criteria for selfconsistent iterations was set at 10^{−6} eV for electronic structure calculations with or without spinorbit coupling (SOC). In order to simulate the buckled or planar honeycombs, a vacuum of at least 20 Å was included in the outofplane (z) direction, and a Γcentered 30 × 30 × 1 MonkhorstPack grid^{44} was used to sample the 2D Brillouin zone. In the case of the nanoribbons, a vacuum of at least 20 Å along both the y and z directions was used, and 12 × 1 × 1 and 30 × 1 × 1 grids were used for the armchair and zigzag ribbons, respectively.
Additional Information
How to cite this article: Hsu, C.H. et al. Twodimensional Topological Crystalline Insulator Phase in Sb/Bi Planar Honeycomb with Tunable Dirac Gap. Sci. Rep. 6, 18993; doi: 10.1038/srep18993 (2016).
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Acknowledgements
F.C.C acknowledges support from the National Center for Theoretical Sciences and the Ministry of Science and Technology of Taiwan under Grant Nos. MOST1042112M110002MY3, MOST1032112M110008MY3 and MOST1012218E110003MY3. He is also grateful to the National Center for Highperformance Computing for computer time and facilities. The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences grant number DEFG0207ER46352 (core research), and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC), the NERSC supercomputing center through DOE grant number DEAC0205CH11231, and support (applications to layered materials) from the DOE EFRC: Center for the Computational Design of Functional Layered Materials (CCDM) under grant number DESC0012575. H.L. acknowledge the Singapore National Research Foundation for support under NRF Award No. NRFNRFF201303.
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Affiliations
Department of Physics, National Sun YatSen University, Kaohsiung 804, Taiwan
 ChiaHsiu Hsu
 , ZhiQuan Huang
 , Christian P. Crisostomo
 , LiangZi Yao
 & FengChuan Chuang
Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, Singapore 117546
 YuTzu Liu
 , Baokai Wang
 , ChuangHan Hsu
 , ChiCheng Lee
 & Hsin Lin
Department of Physics, National University of Singapore, Singapore 117542
 YuTzu Liu
 , Baokai Wang
 , ChuangHan Hsu
 , ChiCheng Lee
 & Hsin Lin
Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
 Baokai Wang
 & Arun Bansil
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Contributions
F.C.C. and H.L. conceived and initiated the study. C.H.H., Z.Q.H., L.Z.Y. and Y.T.L. performed first principles calculations. B.W. and C.H.H. perform the mirror Chern number calculation. C.H.H., Z.Q.H., C.P.C., F.C.C., H.L. and A.B. performed the detailed analysis and contributed the discussions. C.H.H., Z.Q.H, C.P.C., F.C.C., B. W., C.C.L., H.L. and A.B. wrote the manuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to FengChuan Chuang.
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