From Shockley Surface States to Topological Insulators

The 1939 paper of Shockley [1] showed that for a 1D crystal a transition in the bulk band structure, where bands "cross" and exchange eigenvectors at the high symmetry points k=0 and k=pi/2, leads to emergence of a surface state in the gap. In a model with s and p states, the transition marks the change from atomic-like to bonding sp character, leaving a dangling bond, i.e., a half-filled surface state. In higher dimensions this becomes a surface band where the states at each k|| parallel to the surface are determined by a 1D problem with parameters that vary with k||. As Shockley pointed out, the mid-gap states should occur on metals but for insulators surface conditions can eliminate states in the gap, e.g, by passivation. Presumably for that reason this work is largely ignored in the literature of insulators. The purpose of this talk is to point out the prescience aspects of Shockley's work and a way to understand the bands of topological insulators by replacing the p state by a spin-orbit coupled p+ or p- state. The result is the model used to describe HgCdTe quantum wells and graphene can be described by a transformation to a two-site model.

[1] W. Shockley, "On the surface states associated with a periodic potential", Phys. Rev. 56:317–323, 1939.