Well-localized edge states in two-dimensional topological insulator: bismuth film

We calculate the $Z_2$ topological numbers of bismuth (111) and \{012\} ultrathin films from 2D tight-binding Hamiltonians obtained by first-principle calculation. We find that Bi(111) 1-bilayer is the quantum spin Hall (QSH) phase, while Bi\{012\} 2-monolayer is not. We calculate the QSH edge states of the (111) 1-bilayer film with zigzag edges, and there are three Kramers pairs of edge states at the Fermi energy, resulting in the two terminal conductance $G=6e^2/h$. It will be reduced to $G=2e^2/h$ by increasing nonmagnetic disorder, but will not vanish because of the topological protection. Compared with the known two-dimensional quantum spin Hall systems such as HgTe quantum well, the decay length $\ell$ of edge states of bismuth (111) 1-bilayer system is much shorter and is of the order of a few lattice constant. This short $\ell$ is attributed to the edge-state dispersion traversing over the Brillouin zone. It is in strong contrast with HgTe quantum well, where $\ell$ might be as long as 50nm.