Electrons in disordered systems and the theory of localization

This paper gives a review of the theory of noninteracting electrons in a static disordered lattice. The introductory section gives a brief survey of the main aspects of the problem and of its relevance to the physics of amorphous and disordered crystalline solids. The second section is concerned with the methods which can be used to find the density of states, both in the main part of the band, where the coherent potential approximation can be used, and in the tail of the band, where other methods must be used. The third section gives a survey of the theory of localization. There is a detailed discussion of the qualitative differences between localized and extended states which enable a sharp distinction to be made between them. There is a brief survey of the theory of one-dimensional systems and of the percolation problem, and then the Anderson model and its self-consistent modification are discussed. There is also a discussion of numerical work on the Anderson model and the use of path-integral methods. In the final section a tentative theory is proposed to combine various features of the problem which have been revealed by some of the different approaches.