Two-scale theory of edge state

Abstract

The edge state is considered in the spectrum region where its branch splits from the bottom of a continuous conduction band. It is shown that in this region the electron wave function demonstrates two different scale behaviours: slow and fast, that enabled us to construct some simplified procedure for the analysis of the edge state. The slow wave function part obeys a simple Schrödinger equation the parameters of which are insensitive to the peculiarities of the electron dynamics, while the fast part that describes the details of electron behaviour in the primitive cell reveals itself only at the edge. The equation for this fast part was transformed into the boundary condition for the slow part equation. The proposed method is illustrated considering the simplest continuous model for a topological insulator and a tight binding model for graphene.