Localized edge states in two-dimensional topological insulators: Ultrathin Bi films

We theoretically study the generic behavior of the penetration depth of the edge states in two-dimensional quantum spin Hall systems. We found that the momentum-space width of the edge-state dispersion scales with the inverse of the penetration depth. As an example of well-localized edge states, we take the Bi(111) ultrathin film. Its edge states are found to extend almost over the whole Brillouin zone. Correspondingly, the bismuth (111) 1-bilayer system is proposed to have well-localized edge states in contrast to the HgTe quantum well.

Figure 1

Penetration depth $\ell$ for the effective model with ribbon geometry. CB (VB) represents the bulk conduction (valence) band. The plot on the right corresponds to a more asymmetric situation, leading to a larger $\ell$ at the crossing point of the edge states.

Figure 2

(a) Unit cell and lattice vectors and (b) TRIMs of Bi(111) ultrathin film. The TRIMs consist of the $\Gamma$ point and the three $M$ points. (c) Bulk energy bands and the parity at the TRIMs for a Bi(111) 1-bilayer. (d) and (e): Energy bands of the Bi(111) zigzag- and armchair-edge ribbons, respectively, with a width of 20 unit cells, calculated from the TB model. (f) Energy bands of a eight-unit-cell-wide Bi(111) zigzag-edge ribbon from first-principles calculations. The size of the symbols corresponds to the weight of the states in the edge atoms.

Figure 3

Energy bands of Bi{012}: (a) zigzag- and (b) armchair-edge ribbons with a width of 20 unit cells. $U$ ($L$) means that the state is localized on the upper (lower) edge. The crystal structures near the upper edge are shown in the right panels with 1, 2, 3, and 4 representing the lattice sites. We note that these four sites do not lie on the same plane. The shaded regions are the bulk energy bands.

Figure 4

(a) Penetration depth of the edge states on the zigzag edge of the Bi(111). (b), (c) Examples of the edge states in the 2D QSH systems.