Electronic Structure of Disordered One-Dimensional Chains

Abstract

The positions of the electronic levels in a variety of finite one‐dimensional models of disordered alloys and of liquids have been evaluated numerically over a wide range of energies. The disordered alloys consist of two types of potential wells thrown together in a random sequence, but in equal proportions. In the disordered case the tightly bound states give rise to narrow bands in which the distribution of states cannot be described by a smooth function. The distribution of states can be interpreted in terms of an approximation in which each electron is confined to its own group of adjacent identical atoms. At high energy levels the disordered alloys obey perturbation theory, at least in a qualitative fashion, and the ordered alloy differs from the disordered one because of the effect of Bragg reflections from the superlattice. At high energies the distribution of states can be described by a reasonably smooth density function. Between the high energy range and the tightly bound levels, in some of the cases a transition region occurs, in which the levels are distributed irregularly, but without well‐defined gaps and without being susceptible to the simple explanation of localized states. The total number of states (not counting the two spin possibilities as separate states) up to a well‐pronounced energy gap is always an integral multiple of one‐half the total number of atoms in the chain. The liquids consist of a series of identical wells separated by a variable well‐to‐well space. In a liquid the forbidden energy ranges are either smaller than in the strictly periodic crystal, or else nonexistent. The extent to which an energy gap in the strictly periodic crystal will be affected by the introduction of dispersion in the spacing depends on the original gap width as well as on the sensitivity of the gap position, in the strictly periodic case, to uniform dilations. In both the liquid cases and the disordered alloy models a distinction must be made between energy ranges in which the density of states is very low and ranges in which it is strictly zero.