Shockley model description of surface states in topological insulators

Surface states in topological insulators can be understood based on the well-known Shockley model, a one-dimensional tight-binding model with two atoms per elementary cell, connected via alternating tunneling amplitudes. We generalize the one-dimensional model to the three-dimensional case representing a sequence of layers connected via tunneling amplitudes $t$, which depend on the in-plane momentum $\mathit{p}=(p_x,p_y)$. The Hamiltonian of the model is a $2\times 2$ matrix with the off-diagonal element $t(k,\mathit{p})$ depending also on the out-of-plane momentum $k$. We show that the existence of the surface states depends on the complex function $t(k,\mathit{p})$. The surface states exist for those in-plane momenta $\mathit{p}$ where the winding number of the function $t(k,\mathit{p})$ is nonzero when $k$ is changed from 0 to $2\pi$. The sign of the winding number determines the sublattice on which the surface states are localized. The equation $t(k,\mathit{p})=0$ defines a vortex line in the three-dimensional momentum space. Projection of the vortex line onto the space of the two-dimensional momentum $\mathit{p}$ encircles the domain where the surface states exist. We illustrate how this approach works for a well-known model of a topological insulator on the diamond lattice. We find that different configurations of the vortex lines are responsible for the “weak” and “strong” topological insulator phases. A topological transition occurs when the vortex lines reconnect from spiral to circular form. We apply the Shockley model to Bi${}_2$Se${}_3$ and discuss applicability of a continuous approximation for the description of the surface states. We conclude that the tight-binding model gives a better description of the surface states.