Boundary Conditions for Effective Hamiltonian and Surface States in 2D and 3D Topological Insulators

V. V. Enaldiev, I. V. Zagorodnev, V. A. Volkov

Published 2014-07-03, updated 2014-09-03Version 2

We study a surface effect on the energy of surface states in 2D and 3D topological insulators based on HgTe/(Hg,Cd)Te quantum wells and Bi$_2$Se$_3$-type compounds. We construct a class of feasible time-reversal invariant boundary conditions, which contain unknown phenomenological parameters, for effective ${\bf k}{\bf p}$-Hamiltonian of the topological insulators. Space symmetry reduces the number of the boundary parameters to four real parameters in the 2D case and three in the 3D case. We found that the boundary parameters may strongly affect an energy spectrum and even the very existence of these states inside of the bulk gap near the Brillouin zone center. The gap inversion might not be a necessary or sufficient condition for the appearance of the massless Dirac surface states at small momenta. Nevertheless, in frames of a tight-binding model we reveal that when surface states do not exist in the bulk gap in the Brillouin zone center they cross the gap in other points of the Brillouin zone in agreement with the bulk-boundary correspondence.