Two Types of Electronic States in One-Dimensional Crystals of Finite Length

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