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 p = (p(x), p(y)). The Hamiltonian of the model is a 2 x 2 matrix with the off-diagonal element t (k, 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, p). The surface states exist for those in-plane momenta p where the winding number of the function t (k, 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, 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 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 Bi2Se3 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.