Two types of electronic states in one dimensional crystals of finite length

TL;DR

This paper provides exact results on the electronic states in finite one-dimensional crystals, revealing two distinct types of states: band-dependent states and boundary-sensitive surface or edge states.

Contribution

It introduces a comprehensive analysis of how finite crystal boundaries influence electronic states, including the identification of boundary-dependent surface states and their properties.

Findings

N-1 states per energy band depend on crystal length L

One state per band gap depends on boundary tau

Boundary changes can dramatically alter surface state energies

Abstract

Exact and general results on the electronic states in one dimensional crystals bounded at tau and tau+L - where L=Na, N is a positive integer and a is the potential period - are presented. Corresponding to each energy band of Bloch wave, there are N-1 states in the finite crystal and their energies are dependent on the crystal length L but not on the crystal boundary tau and map the energy band exactly; There is always one and only one electronic state correponding to each band gap of the Bloch wave, whose energy is dependent on the crystal boundary tau but not on the crystal length L. This state is either a constant energy confined state at a band edge or a surface state in the gap. A slight change of the boundary tau could change the property and energy of this state dramaticaly.