Stable Nontrivial $Z_2$ Topology in Ultrathin Bi (111) Films: A First-Principles Study

Recently, there have been intense efforts in searching for new topological insulator materials. Based on first-principles calculations, we find that all the ultrathin Bi (111) films are characterized by a nontrivial $Z_2$ number independent of the film thickness, without the odd-even oscillation of topological triviality as commonly perceived. The stable nontrivial $Z_2$ topology is retained by the concurrent band gap inversions at multiple time-reversal-invariant $k$ points with the increasing film thickness and associated with the intermediate interbilayer coupling of the Bi film. Our calculations further indicate that the presence of metallic surface states in thick Bi (111) films can be effectively removed by surface adsorption.

Figure 1

(a) The hexagonal unit cell of single-crystal bismuth. (b) The top view of the Bi lattice. (c) The first Brillouin zone of the hexagonal lattice.

Figure 2

The band structure of (a) 2 BLs and (b) 3 BLs under different inter-BL spacing. (c) The energy level at the $\Gamma$ point of (left) 2 BLs and (right) 3 BLs as a function of the inter-BL spacing. The red (circle) lines indicate even-parity levels. The blue (square) lines indicate odd-parity levels.

Figure 3

(a) The direct ($\Gamma$ point) and indirect band gap as a function of the film thickness. (b) The band structure and the parity information of 5 BLs. (c) A schematic depiction of the electronic property of a Bi (111) film before and after the H termination. (d) The band structure and the parity information of H-terminated 5 BLs.