Although this volume is basically about metallic conductors, it is relevant to a discussion of electron states in disordered materials to give here some general background, plus relevant references, to the theory of localization of noninteracting electrons in a random potential.

The pioneering paper in this area was that of Anderson (1958) on the absence of diffusion in certain random lattices. But there was subsequently a lot of effort on this problem of noninteracting electrons in a static disordered potential before there was agreement on, at least, the answers to some important questions in this general area.

One might, at first sight, think it fruitful to compare the theory of electrons in disordered systems with the Bloch wave theory of the behaviour of electrons in a regular lattice. As Thouless (1979) emphasizes in his survey article, the theory of electrons in disordered systems is much more closely analogous to and owes much more to the theory of critical phenomena (see Chapter 9). In a sense then, developments in electron states in disordered systems awaited the synthesis of the theory of critical behaviour.

Following Thouless (1979), it is useful to group approaches into a number of areas, the first category being perturbative methods.

Perturbative methods

Anderson's original paper (1958) was based on the application of perturbation theory to a system that was strongly localized by a lot of disorder. In some ways, it resembles the Ursell-Mayer approach to phase transitions by examining the convergence of a high temperature perturbation series.