Boundary Conditions and Surface States Spectra in Topological Insulators

TL;DR

This paper investigates how boundary conditions influence surface state spectra in 2D and 3D topological insulators, revealing that boundary parameters can significantly alter the existence and dispersion of surface states.

Contribution

It introduces a class of time-reversal invariant boundary conditions with phenomenological parameters affecting surface state spectra in topological insulators.

Findings

Boundary parameters can alter the energy spectrum of surface states.

Surface states may cross the bulk gap at points other than the zone center.

Existence of surface states depends strongly on boundary conditions.

Abstract

We study spectra of surface states in 2D topological insulators (TIs) based on HgTe/(Hg,Cd)Te quantum wells and 3D Bi$_2$Se$_3$-type compounds by constructing a class of feasible time-reversal invariant boundary conditions (BCs) for an effective ${\bf k}{\bf p}$-Hamiltonian and a tight-binding model of the topological insulators. The BCs contain some phenomenological parameters which implicitly depend on both bulk Hamiltonian parameters and crystal potential behavior near the crystal surface. 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 not only an energy spectrum but even the very existence of these states inside the bulk gap near the Brillouin zone center. Nevertheless, we reveal in frames of the tight-binding model 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.