Invariant imbedding and the band theory of solids

Abstract

The mathematical theory of invariant imbedding has been described in numerous papers by Bellman and Kalaba and their co-workers [l]. The invariant imbedding technique has been applied to wave mechanics [2], and we will show here that one-dimensional band theory can also be analyzed by means of the technique. The band theory formulation is an extension of the work by Davies. We will be concerned with two problems in this paper. We will first show that the Kronig-Penney model of a rectangular barrier can be analyzed by invariant imbedding and that the same restrictions are reached on the energy levels of the electron. We will then show that the invariant imbedding technique gives a computational form which can be solved for any one-dimensional potential variation. The equations to be solved to determine the energy levels are numerical equations and can be solved by means of a high speed digital computer. There are many extensions of the work which we shall not describe in detail here. For example, the potential function can be modified by random changes in height and periodicity. A discussion of the use of invariant imbedding in problems with random properties will be found in the book by Adams and Denman [3]. The band structure of ordered and disordered alloys can be studied by using the method described in this paper.