Predicted Growth of Two-Dimensional Topological Insulator Thin Films of III-V Compounds on Si(111) Substrate

We have carried out systematic first-principles electronic structure computations of growth of ultrathin films of compounds of group III (B, Al, In, Ga, and Tl) with group V (N, P, As, Sb, and Bi) elements on Si(111) substrate, including effects of hydrogenation. Two bilayers (BLs) of AlBi, InBi, GaBi, TlAs, and TlSb are found to support a topological phase over a wide range of strains, in addition to BBi, TlN, and TlBi which can be driven into the nontrivial phase via strain. A large band gap of 134 meV is identified in hydrogenated 2 BL film of InBi. One and two BL films of GaBi and 2 BL films of InBi and TlAs on Si(111) surface possess nontrivial phases with a band gap as large as 121 meV in the case of 2 BL film of GaBi. Persistence of the nontrivial phase upon hydrogenations in the III-V thin films suggests that these films are suitable for growing on various substrates.

and TlSb, which are found to survive over a wide range of strains. BBi, TlN, and TlBi films support a nontrivial phase for certain values of strain. Nontrivial topological phases are also found in 1 and 2 BL films of GaBi as well as 2 BL films of InBi and TlAs on Si(111) substrate. For the 2 BL film of GaBi on Si(111), the band gap can be as large as 121 meV, well above the room-temperature thermal excitation energy. Si(111) would thus provide a viable substrate for growing the proposed topological III-V films.

Results
We assumed the investigated III-V films to be stacked along the [111] direction in zincblende (ZB) and wurtzite (WZ) structures. We define one BL as a III-V pair of layers in the buckled honeycomb structure. The 2 BL film can assume ZB and WZ as two possible structures. The top view of the atomic structure (1 × 1 unit cell is outlined) for a 2 BL film based on the ZB structure is shown in Fig. 1(a). The surface Brillouin zone is presented in Fig. 1(b). Side views of a 2 BL film in ZB and WZ are shown in Fig. 1(c,d), respectively. The band structures for the equilibrium crystal structures were calculated including the spin-orbit coupling. In order to ascertain the band topologies, we computed the Z 2 invariant following the method of ref. 28.
Previous studies [12][13][14] have shown that the only 1 BL films of III-V compounds, which exhibit nontrivial phases, involve Bi from group V and Tl from group III. With this in mind, even though we investigated all 25 combinations of group III (B, Al, In, Ga, and Tl) and group V (N, P, As, Sb, and Bi) elements, we will focus on 2 BL films of Bi and Tl, many of which are 2D TIs in the freestanding case. The optimized structures of these nine 2 BL III-V films before and after hydrogenation are summarized in Table 1. Also included in Table 1 are the preferred structure, the system band gap (defined as the energy difference between the lowest conduction band and the highest valence band), and the Z 2 invariant, along with the bulk lattice constant, a 0 , the optimized lattice constant, a, of the film, buckling distances in the upper (d 1 ) and lower (d 2 ) layer, and the distance (D) between the two layers in the 2 BL film. The hydrogenated 2 BL film of InBi is seen in Table 1 to possess the largest band gap of 134 meV at its equilibrium structure.
A reference to Table 1 shows that the unhydrogenated 2 BL films of AlBi, GaBi, and InBi are nontrivial semi-metals with Z 2 = 1 and a negative system gap. TlN, TlAs, and TlSb, in contrast, are nontrivial insulators with band gaps of 18 to 51 meV. Upon hydrogenation, only the TlN film changes from nontrivial to trivial phase, where this transition is partially driven by the 4% (3.739 Å to 3.9 Å) change in lattice constant. Since all of the 2 BL films in Table 1 are seen to prefer the ZB rather than the WZ structure around the equilibrium bulk lattice constant, for the remainder of this study, we will only consider the ZB structure.
In order to simulate the effects of the supporting substrate, nine hydrogenated films with ZB structure in Table 1 were further analyzed over a range of lateral strains around the equilibrium structures (0% strain); the results are summarized in Fig. 2. We see from Fig. 2(a) that the hydrogenated 2 BL films of TlAs, TlSb, AlBi, GaBi, and InBi maintain their nontrivial phase over a wide range of strains; in fact, certain strain values yield even larger band gaps, see Fig. 2(b). The TlBi film, even though it is trivial phase at its equilibrium lattice constant, exhibits nontrivial phase at strain values greater than 2% and less than − 8%. It is remarkable that the nontrivial phase is robust for selected hydrogenated III-V films in that it is maintained over a wide range of strains, suggesting that this nontrivial phase is more likely to be realized when the film is grown on a suitable substrate. Notably, this family of films harbors band gaps large enough for possible room temperature applications. InBi and GaBi films, for example, possess band gaps up to 0.4 eV under compressive strain of around − 5%.
Turning to the question of practical realization of our proposed films, Si(111), may be a good candidate substrate to grow TlAs, AlBi, GaBi, and InBi films since the hydrogenated equilibrium lattice constants of these 2 BL films in × 3 3 supercell (7.85, 7.95, 7.93, and 8.44 Å) are close to the lattice constant (7.73 Å) of 2 × 2 Si(111); also, as noted already in connection with Fig. 2

Table 1. Calculated equilibrium structure [zincblende (ZB)], system band gap defined as the energy difference between the lowest conduction level and the highest valence level, and the topological invariant Z 2 for nine 2 BL films of III-V compounds with and without hydrogenation.
Other structural parameters given are: bulk lattice constant (a 0 ); lattice constant a of the 2 BL film; interlayer distance D in the 2 BL film; and the buckling heights d 1 and d 2 of the two layers. Distances D, d 1 and d 2 are also marked in Fig. 1.  Figure 3 presents the crystal and band structures for these two cases in which the Bi layer lies on (a) top or (d) below the Ga/In layer. The corresponding band structures for Bi on top of Ga (b) and In (c) and for Bi below Ga (e) and In (f) are also shown. Table 2 summarizes the main results by giving the total energy, the system energy gap, and the Z 2 invariant for 1 BL and 2 BL Ga/In-Bi, AlBi, and TlAs films on Si(111). For 2 BL Ga/In-Bi films, both cases support nontrivial topological insulator phase and a band gap [121 meV for GaBi-Si(111)] large enough for room-temperature applications. Notably, 1 BL films of GaBi or BiGa on Si(111) also exhibit a nontrivial topological phase. The transition to trivial from nontrivial phase of 1 BL film of InBi or BiIn on Si(111) is consistent with our previous work 13 in which the hydrogenated 1 BL film of InBi was found to be trivial under -4% strain. Finally, we comment on the results for AlBi and TlAs films. As shown in Fig. 2(a), 2 BL film of AlBi becomes trivial below − 1% strain, while TlAs film maintains the nontrivial phase for less than 1% strain. [AlBi and TlAs have lattice mismatches of − 2.77% and − 1.53%, respectively.] When placed on Si(111), only the TlAs film exhibits a nontrivial phase as expected in view of its lattice mismatch with Si(111), see also Fig. 2. We note that all 1 BL films of AlBi and TlAs films are found to be trivial.

Conclusions
We have presented a first-principles study of the crystal and electronic structures of freestanding and hydrogenated multilayers of III-V compounds. Two BL films of hydrogenated AlBi, GaBi, InBi, TlAs, and TlSb are found to harbor the nontrivial insulator phase. BBi, TlN, and TlBi films can be driven into the nontrivial topological phase via strain, although these films are in the trivial phase at their equilibrium structures. Hydrogenated 2 BL films of GaBi and InBi exhibit band gaps of 99 meV and 134 meV, respectively. We have also explored the electronic structures and topological properties of 1 BL and 2 BL films of GaBi as well as 2 BL films of InBi and TlAs on a Si(111) substrate, and found a nontrivial phase with a large band gap of 121 meV in 2 BL films of GaBi. Our study suggests that III-V thin films can support nontrivial topological phases, which are robust against hydrogenation, strain and substrate effects, and would thus provide a viable materials platform for future room temperature applications.

Methods
First-principles calculations were performed within the density functional theory (DFT) utilizing the generalized gradient approximation (GGA) [30][31][32][33][34] . Projector-augmented-wave (PAW) 35 wave functions with an energy cut-off of 400 eV were used in the Vienna Ab-Initio Simulation Package (VASP) 36,37 . Atomic positions were optimized for each lattice constant value considered until the residual forces were no greater than 10 −3 eV/Å. The criteria for convergence for self-consistent electronic structure was set at 10 −6 eV. A vacuum layer of at least 20 Å along the z direction was used to simulate multilayer films. A Γ -centered Monkhorst-Pack 29 grid of 18 × 18 × 1 was used for 2D integrations in the Brillouin zone of the 1 × 1 buckled honeycomb lattice.