Electron Localization in Disordered Systems

The general features of the electronic structure of disordered materials are investigated using Edwards's model of an electron in the presence of dense weak random scatterers. This model is equivalent to that of Zittartz and Langer which considers an electron in a Gaussian random potential. A self-consistent-field (SCF) formulation, which generalizes and clarifies that of earlier works, is used to demonstrate the presence of a mobility edge, a transition between localized and extended states at energy $E_c$. Particular emphasis is placed upon the symmetry and analytic structure of the SCF and upon how the SCF theory introduces the symmetry breaking which leads the localized states. In particular, the localization probability, the probability density that an electron with energy $E$ returns to its initial position after infinite time, is found to vary as ${|E-E_c|}^{\frac{13}{6}}$ for $E\le E_c$ and to vanish for $E>\mathrm{}{E}_c$. Below the mobility edge the size of the localized states is found to vary as ${|E-E_c|}^{-\frac{2}{3}}$. Although questions concerning electron localization can rigorously be answered from a consideration of quantities which are the averages of a product of two Green's functions, such as the localization probability, the SCF theory obtained from the average Green's function alone gives rise to the same analytic structure as the SCF theory which is based upon the localization probability. This indicates that it may generally be possible to extract information concerning electron localization from the simpler average Green's function. The SCF theory is also generalized to consider quantities which are related to electron mobility. Although the mathematical difficulties encountered in this case resemble those of the general three-body problem, a proof is given that the lowlying states do, in fact, give a vanishing mobility. In addition, the SCF is used to derive the model which Mott employs to show that the mobility due to electrons in localized states vanishes as ${\omega }^2\ln \omega$ as the frequency $\omega$ tends to zero.