Shockley model description of surface states in topological insulators

Sergey S. Pershoguba and Victor M. Yakovenko
Phys. Rev. B 86, 075304 – Published 2 August 2012

Abstract

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=(px,py). The Hamiltonian of the model is a 2×2 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 2π. 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.

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  • Received 13 March 2012

DOI:https://doi.org/10.1103/PhysRevB.86.075304

©2012 American Physical Society

Authors & Affiliations

Sergey S. Pershoguba and Victor M. Yakovenko

  • Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

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Issue

Vol. 86, Iss. 7 — 15 August 2012

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