Fractal character of eigenstates in weakly disordered three-dimensional systems

Abstract

The electronic wave functions of simple cubic systems with weak Gaussian disorder are determined by a direct diagonalization method in the whole energy range. We demonstrate that the states have a fractal character. The fractal dimensionality follows from an analysis of the inverse participation ratio and its dependence on the sample size. In the intermediate region between uniformly extended and exponentially localized states the results can be unambiguously explained by the assumption that the wave functions fall off as a power law. The mobility edge is reached when the wave function turns from power-law localization to a scattered wave, or, equivalently, when the fractal dimension equals two. The thus defined critical disorder at the center of the band is in good agreement with other investigations, and it drops as a monotonous function of the energy towards the band edge.