Band Structures of One-Dimensional Crystals with Square-Well Potentials

Abstract

The energy band structure for a one-dimensional periodic square-well potential is obtained in terms of the well depth for the whole range of possible ratios of well width to hill width. This model bears a closer resemblance to a real crystal since, as potential depth is varied for a fixed ratio of well width to hill width, the curves bounding distinct bands cross while in the case of a delta-function potential no such crossings occur. The location of these crossings is derived. The number of times that a given pair of boundary curves can cross is considered. For the set of boundary curves that belong to a given ratio of well-to-hill widths, this number is unbounded.