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

1 Abstract 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 and their energies are dependent on the crystal length L but not on the crystal boundary 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 the crystal boundary location τ but not on the crystal length L. This state is either a constant energy confined band edge state or a surface state in the band gap. A slight change of the boundary location τ could change the property and the energy of this state dramatically. 2 Bloch theorem plays a central role in our current understanding on the electronic structures in modern solid state physics. However, any real crystal always has a finite size and does not have a hypothetical infinite size or the periodic boundaries which Bloch theorem is based on[1]. The difference between the electronic structure of a real crystal of finite size and the electronic structure obtained based on the translational invariance becomes more significant as the crystal size decreases. A clear understanding of the properties of electronic states in real crystals of finite size has both theoretical and practical significant importance. A straightforward way is to obtain exact solutions for crystals of finite size. However, to obtain exact solutions of the Schrödinger equation with a general periodic potential for a finite crystal with boundaries has been being considered as a rather difficult problem: the lack of translational invariance in the crystal of finite size is a major obstacle. It is the use of the translation invariance that greatly simplifies the mathematics in solving the Schrödinger equation with a periodic potential. Without a such simplification the corresponding problem for finite crystals with boundaries could become rather difficult mathematically. Thus most previous theoretical investigations on this subject were based on approximate and/or numerical approaches and were usually on a specific material and/or on a specific model[2]. Historically, many of our current fundamental understandings on the electronic structures of crystals were obtained through the analysis of one dimensional crystals[3, 4, 5]. Among the …