Density of localized states in disordered solids

Calculations of a localized state density (DOS) are carried out in the framework of a well-known variant of the Anderson model, i.e., for the three-dimensional single-band Hamiltonian with diagonal disorder. Results of the calculations give three regions of the energies below the virtual crystal band edge EGᵛᶜ characterized by different behavior of the DOS. (I) The band-edge (BE) region is placed in the vicinity of EGᵛᶜ. Here the DOS on a linear scale is approximately a linear function of the localization energy. (II) At lower energies the Urbach law governs the DOS behavior. (III) At lower energies the DOS exhibits Lifshitz singularity dependence. Here the DOS has a very small value and rare deep centers (DC) can appear in the spectrum. The problem of the DC-band inhomogeneous broadening is also considered. Estimations of the number of localized states and of the mobility-edge position are presented. The data on single-electron-DOS energy dependence of {α}-Si:H are used to compare qualitatively the calculated DOS with the experiment results. Good agreement is reached both in DOS behavior and in the mobility-edge location. An aspect in the approach to the problem is an additional restriction of the trial-function class minimizing the one-instanton action. The additional restriction was obtained from the analysis of the localized state problem for the concrete realizations of the disordered system under consideration. A strong-scattering problem in the limit of small concentration of scatterers is studied, as well as the case of the three-component system consisting of a binary solution of weak scatterers with a third component comprising rare deep centers. In all of the cases considered the general expressions for the DOS including prefactors are found, as well as their approximate forms. It is shown that there are limits which allow for single-instanton solution of the problem to coincide with the exact one.