Prediction of Large-Gap Two-Dimensional Topological Insulators

Apr 15, 2014 - Ceng-Ceng Ren , Wei-Xiao Ji , Shu-Feng Zhang , Chang-Wen Zhang .... Hui Zhao , Wei-xiao Ji , Chang-wen Zhang , Ping Li , Shu-feng Zhang...
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Prediction of Large-Gap Two-Dimensional Topological Insulators Consisting of Bilayers of Group III Elements with Bi Feng-Chuan Chuang,*,† Liang-Zi Yao,† Zhi-Quan Huang,† Yu-Tzu Liu,† Chia-Hsiu Hsu,† Tanmoy Das,‡ Hsin Lin,*,‡ and Arun Bansil§ †

Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan Graphene Research Centre and Department of Physics, National University of Singapore, Singapore 117542 § Department of Physics, Northeastern University, Boston, Massachusetts 02115, United States ‡

ABSTRACT: We use first-principles electronic structure calculations to predict a new class of two-dimensional (2D) topological insulators (TIs) in binary compositions of group III elements (B, Al, Ga, In, and Tl) and bismuth (Bi) in a buckled honeycomb structure. We identify band inversions in pristine GaBi, InBi, and TlBi bilayers, with gaps as large as 560 meV, making these materials suitable for room-temperature applications. Furthermore, we demonstrate the possibility of strain engineering in that the topological phase transition in BBi and AlBi could be driven at ∼6.6% strain. The buckled structure allows the formation of two different topological edge states in the zigzag and armchair edges. More importantly, isolated Dirac-cone edge states are predicted for armchair edges with the Dirac point lying in the middle of the 2D bulk gap. A room-temperature bulk band gap and an isolated Dirac cone allow these states to reach the long-sought topological spin-transport regime. Our findings suggest that the buckled honeycomb structure is a versatile platform for hosting nontrivial topological states and spin-polarized Dirac fermions with the flexibility of chemical and mechanical tunability. KEYWORDS: 2D topological insulators, topological phase transition, quantum spin Hall effect, III−V semiconductor thin films, electronic structures, first-principles calculations

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silicene,8 2D TIs in tin-based compounds (called stanene and stanane),9 and also 3D TIs.10 Here we systematically study the crystal and electronic structure evolution of another family consisting of binary compositions of a group III element (B, Al, Ga, In, and Tl) with bismuth (Bi) in the buckled honeycomb structure. By identifying band inversions in the band structure, we predict nontrivial 2D TI phases with a sizable gap in several pristine materials including GaBi, InBi, and TlBi bilayers. The three new 2D TIs identified here all have large band gaps that exceed the thermal energy at room-temperature, making them suitable for room-temperature spintronics applications. Notably, the band gap of TlBi is 560 meV, which is larger than that in any experimentally realized or theoretically predicted topological insulator. Moreover, the bands are shown to be inverted in BBi and AlBi for 6.68% and 6.63% strain, respectively. A single Dirac-cone edge state is predicted for armchair edges where the Dirac point lies in the middle of the 2D bulk gap, allowing these states to reach the long-sought topological spin-transport regime.11−13 These materials and their variants including thinfilm versions will thus be attractive for applications in the semiconductor industry.14−16

wo-dimensional (2D) topological insulators (TIs), also known as quantum spin Hall (QSH) insulators, are novel materials in which even though the bulk system is insulating, the edge can support spin-polarized gapless states with a Diraccone-like linear energy dispersion.1 The edge states of 2D TIs are more robust against nonmagnetic impurities than in 3D TIs since the only available backscattering channel is forbidden, making 2D TIs better suited for coherent spin transport related applications. While several 3D TIs have been synthesized experimentally, the materials realization of 2D TIs is so far limited to the quantum well systems2−6 with a small band gap that does not survive at room temperature. Therefore, there is a great need to find new 2D TI materials with large band gaps. It has been long recognized that the 2D honeycomb lattice provides a plethora of Dirac cones which can be manipulated from a sublattice induced pseudospin basis in graphene7 to the edge states of a 2D topological insulator (TI) induced by spin− orbit coupling (SOC), to time-reversal symmetry breaking the quantum anomalous Hall state supporting protected conducting channels.1 However, this interesting pathway is not feasible in graphene due to the lack of sufficient strength of the SOC but becomes possible in the buckled honeycomb structure, which allows large material tunability. The buckled honeycomb lattice has thus regenerated intense interest because it can provide a common host for many interesting physical properties, including fully spin-polarized Dirac excitations in © 2014 American Chemical Society

Received: January 17, 2014 Revised: March 31, 2014 Published: April 15, 2014 2505

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Methods. Calculations were carried out within the generalized gradient approximation (GGA) to the density functional theory (DFT)17,18 using the projector-augmentedwave (PAW) method19 as implemented in the Vienna Ab-Initio Simulation Package (VASP).20 The kinetic energy cutoff was set to 400 eV, and atomic positions were relaxed until the residual forces were less than 10−2 eV/Å. The convergence criteria for self-consistency in electronic structure calculations, with or without spin−orbit coupling (SOC), was set at 10−6 eV. To model the single bilayer, a vacuum of at least 20 Å was included along the z direction, while for the ribbons a vacuum of at least 20 Å along both the y and z directions was used. For the buckled honeycomb structure, a 24 × 24 × 1 Monkhorst− Pack grid21 was used to sample the 2D Brillouin zone, while for constructed ribbons 20 × 1 × 1 and 60 × 1 × 1 grids were used for armchair ribbons and zigzag ribbons, respectively. Results and Discussion. The unit cell of the buckled honeycomb bilayer structure with two atoms and its Brillouin zone are shown in Figure 1a and b, respectively. The III−Bi

orbit coupling (WSOC). By adjusting WSOC gradually from 0 to 1, the effect of SOC can be delineated. We illustrate this adiabatic process with the example of GaBi with Figure 2a−c,

Figure 2. Band structure of GaBi in low-buckled state with SOC weighting of (a) WSOC = 0, (b) WSOC = 0.3, and (c) WSOC = 1. The size of red circle is proportion to the contribution of the s-orbital in the wave function. (d) Band inversion strength, EI, at Γ for various III−Bi compounds.

which show results for several different values of WSOC. Here, sizes of red circles are proportional to the contribution of the sorbital in the wave function. When SOC is turned off at WSOC = 0, there exists a small band gap of 0.111 eV. At WSOC = 0.3, the conduction and valence bands touch each other, forming a critical point in topological phase diagram. For larger values of WSOC, a gap reopens with inverted bands at Γ. When the SOC is fully switched on, the band gap at Γ is 0.187 eV, while the indirect gap is 0.168 eV, both are much larger than thermal energies at room temperature. This closing and reopening of band gaps implies that the system undergoes a topological phase transition. The single band inversion at Γ in GaBi at its actual value of SOC indicates that GaBi is a 2D Z2 topological insulator. A further proof is given below by considering edge states of a GaBi ribbon. We now discuss the band inversion strength (BIS), EI, defined as the direct band gap at Γ, which provides a measure of how far the material is from a topological critical point.22 EI is positive when the band is inverted and negative otherwise. Figure 2d shows values of EI for various III−Bi systems. EI increases as the atomic number increases. The trivial insulators BBi and AlBi exhibit negative values due to the small strength of SOC in B and Al, while GaBi, InBi, and TlBi with stronger SOC have positive BIS values. Note that all three 2D topological insulators identified here have large BIS values and support a large band gap, which exceeds the thermal energy at room temperature. In particular, the band gap of 560 meV in TlBi is larger than in any experimentally realized or theoretically predicted topological insulator material. As in the case of Sb and Bi bilayers,22,23 it is possible to manipulate band topology by strain. The band inversion strength at Γ as a function of strain shown in Figure 3 highlights the role of strain in driving topological phase transitions. Here, both the crystal and electronic structures at each value of the lattice constant have been obtained by relaxing the atomic positions relaxed in the low-buckled state. The solid lines mark positive BIS values or systems in

Figure 1. (a) Side and top views of the 2D buckled honeycomb structure. (b) 2D Brillouin zones with specific symmetry points labeled. (c) Total energy of bilayers of various materials (see legend) as a function of lattice constant a. The double-well energy state is found in all materials except BBi. The left-hand side well is referred to as the high-buckled (HB) state and the right-hand side well as the lowbuckled (LB) state.

bilayers were constructed, and the atomic positions were optimized at each given lattice constant. The total energy per unit cell is plotted in Figure 1c as a function of lattice constant, a. A double-well energy curve was found in all cases except the BBi bilayer. The quantum well state at a smaller lattice constant is called the high-buckled (HB) state, and that at a larger lattice constant is called the low-buckled (LB) state. In the HB state, the vertical layer distance is longer and in the range of 2.01− 2.83 Å, while in the LB state, the distance is shorter and in the range of 0.51−0.85 Å. The HB state is unstable in BBi, whereas, in all other materials, the two minima corresponding to HB and LB states in Figure 1c are separated by a significant energy barrier. The equilibrium lattice constants of LB honeycomb are 3.891, 4.523, 4.521, 4.805, and 4.928 Å for BBi, AlBi, GaBi, InBi, and TlBi, respectively. For the remainder of this study, we will focus on the LB state, which provides interesting topological electronic structures. To demonstrate the characteristics of the inverted electronic bands, band structures are first compared for the LB state with and without SOC. We find that the band inversion occurs in GaBi, InBi, and TlBi bilayers only. To confirm the topological nature, a scaling factor was added artificially in front of the SOC term in the computations, referred to as the weight of spin− 2506

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while that from the bottom edge is marked with blue circles. The orange filled region denotes projected 2D bulk bands on the 1D momentum space. The sizes of red crosses and blue circles are proportional to contributions from the edges. In Figure 4a, the zigzag ribbon has two different kinds of terminations, so that each of these two edges has its own edge bands with different band dispersions. On the other hand, since the armchair ribbon has the same type of edge structure on both sides, the edge bands of the two sides have the same band dispersion as shown in Figure 4b. All edge bands are seen to connect to the conduction and valence bands and span the 2D bulk energy gap, yielding a 1D ribbon with gapless edge states. The Z2 topological phase can be examined by counting the number of edge bands crossing the 2D bulk band gap. The zero energy is chosen to lie in the middle of the 2D bulk band gap. In Figure 4a, between zero and π/a momentum points, the zero energy level crosses three times for the edge states from the top side of the edge (red) and only one time for those from the bottom side (blue). These odd numbers of crossings between two time-reversal invariant points prove the nontrivial nature of these films. Remarkably, the edge states for the armchair edge exhibit a single Dirac cone with its node placed in the middle of the 2D band gap. Such placement of the Dirac cone is highly desirable for various applications11−13 and suggests that III−Bi films could provide a useful quantum spin Hall materials platform. We emphasize that the existence of gapless edge states in a 2D topological insulator is guaranteed for any type of edge as long as time-reversal symmetry is preserved. However, the energy position and dispersion of the Dirac cone will depend on details of the atomic structure of the edge. Even when the edge is neither zigzag nor armchair, we can thus expect the topologically protected gapless states to survive, although their transport properties will be modified by edge imperfections. Finally, we comment on possible routes for synthesizing III− Bi films in the buckled honeycomb structure. Two basic approaches can be taken. One is to self-assemble the film by using a substrate which bonds strongly with the film. Along this line, one could attempt to grow III−Bi films by introducing Bi on surfaces of AlN, InN, InP, InSb, GaN, or GaAs, since the top two layers of the[111] surfaces of these III−V semiconductors are in the buckled honeycomb structure.14,24 However, the disadvantage is that strong bonding with the substrate can significantly alter the band topology of the film.22,25−27 An alternative approach is to grow the film on a passivated or terminated substrate, which interacts with the film only weakly through van der Waals interactions. The candidate substrates here could be transition metal dichalcogenides (e.g., MoS2), hBN, metallic substrates (e.g., silicene on Ag), and hydrogen- or metal-chalcogenide-terminated substrates (e.g., SiC, Si, and Ge).25 Conclusions. In summary, using first-principles computations, we have systematically studied crystal and electronic structures of binary films of group III elements (B, Al, Ga, In, and Tl) with bismuth (Bi) in the buckled honeycomb structure. The nontrivial topological characteristics of inverted bands are found in pristine GaBi, InBi, and TlBi bilayers. A topological phase transition is found in BBi and AlBi films under 6.68% and 6.63% strain, respectively. Details of topologically protected edge states are identified in pristine GaBi, InBi, and TlBi bilayers. Notably, these two-dimensional topological insulators support a large enough band gap for room-temperature

Figure 3. Band inversion strength for various bilayer materials as a function of strain. Dashed (solid) lines denote the trivial (nontrivial) phase.

topologically nontrivial phases, while dashed lines denote negative BIS values or systems that are in trivial phases. BBi and AlBi are seen to be in a trivial phase at zero strain but transform into a nontrivial phase at positive 6.68% and 6.63% strain, respectively. GaBi, InBi, and TlBi, on the other hand, are in the nontrivial phase at zero strain. GaBi and InBi transform into the trivial phase under negative strains of −3.54% and −4.16%, while TlBi remains in the nontrivial phase until strain reaches −10.15%. In order to investigate the edge states, we considered ribbons of all our III−Bi bilayers at their equilibrium lattice constants in the LB state. Topologically protected edge states were only found in GaBi, InBi, and TlBi ribbons, confirming that these systems are 2D-topological insulators in their pristine phase. In Figure 4, we show the crystal and band structures of GaBi ribbons for zigzag and armchair edges as illustrative examples. The width of ribbon with a zigzag edge is 77.01 Å and that with an armchair edge is 88.17 Å, which is large enough to ignore the interaction between the two sides of the ribbon. The contribution from the top edge is marked with red crosses,

Figure 4. Crystal and band structures of GaBi ribbons for a zigzag (a) and an armchair edge (b). Contribution from the top (bottom) edge is marked with red crosses (blue circles). The sizes of red crosses and blue circles are proportional to contributions from the edges. The region with orange filling denotes bulk bands. 2507

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(22) Huang, Z.-Q.; Chuang, F.-C.; Hsu, C.-H.; Liu, Y.-T.; Chang, H.R.; Lin, H.; Bansil, A. Phys. Rev. B 2013, 88, 165301. (23) Chuang, F.-C.; Hsu, C.-H.; Chen, C.-Y.; Huang, Z.-Q.; Ozolins, V.; Lin, H.; Bansil, A. Appl. Phys. Lett. 2013, 102, 022424. (24) Keen, B.; Makin, R.; Stampe, P. A.; Kennedy, R. J.; Sallis, S.; Piper, L. J.; McCombe, B.; Durbin, S. M. J. Electron. Mater. 2014, 43, 914. (25) Hsu, C.-H.; Ozolins, V.; Chuang, F.-C. Surf. Sci. 2013, 616, 149. (26) Hsu, C.-H.; Lin, W.-H.; Ozolins, V.; Chuang, F.-C. Appl. Phys. Lett. 2012, 100, 063115. (27) Riedl, C.; Coletti, C.; Iwasaki, T.; Zakharov, A. A.; Starke, U. Phys. Rev. Lett. 2009, 103, 246804.

applications. Isolated Dirac-cone edge states are predicted for armchair edges where the Dirac point lies in the middle of the 2D bulk gap, allowing these states to reach the long-sought topological spin-transport regime.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (F.-C.C.). *E-mail: [email protected] (H.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.-C.C. acknowledges support from the National Center for Theoretical Sciences and the Taiwan National Science Council under Grants Nos. NSC-101-2112-M110-002-MY3 and NSC101-2218-E-110-003-MY3. He is also grateful to the National Center for High-Performance Computing for computer time and facilities. The work at Northeastern University is supported by the US Department of Energy, Office of Science, Basic Energy Sciences contract DE-FG02-07ER46352, and benefited from theory support at the Advanced Light Source and the allocation of supercomputer time at NERSC through DOE grant number DE-AC02-05CH11231. T.D. and H.L. acknowledge the Singapore National Research Foundation for support under NRF Award No. NRF-NRFF2013-03.

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