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

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.