Probability distributions in a two-parameter scaling theory of localization

Probability distributions for the resistance of two- and three-dimensional disordered conductors are studied using a Migdal-Kadanoff–type scaling transformation together with the author’s previously derived distributions in one dimension. The present treatment differs from earlier work in two respects: On one hand, it includes the effect of an average potential barrier V experienced by an electron originating from the perfect leads which connect the conductor to a constant-voltage source; on the other hand, the input distribution for one-dimensional systems is based on an exact solution for the effect of the random potential on the complex reflection amplitude of an electron at a certain energy. The scaling equation for probability distributions and for their successive moments are parametrized in terms of the mean resistance, ρ¯, and of a fixed parameter γ related to V. Hence they correspond to a special form of two-parameter scaling. A mobility edge, ρ¯≡${\rho }_c$, exists only for d>2 and, for d=3, detailed results for ${\rho }_c$, for the conductivity exponent ν, and for the fixed resistance distribution at ${\rho }_c$ as a function of γ are presented. The asymptotic distribution of resistance away from the mobility edge for d=3, and in both small- and large-resistance regimes for d=2 are also studied. In the metallic regime for d>2 our treatment yields two distinct distributions, one of which is characterized by Ohm’s law for the mean resistance and the other one by Ohm’s law for the mean conductance. In the latter case the fluctuations of conductivity are independent of sample size for large samples. The calculated distributions are generally broad and in the localized regime, for d=3 and d=2, the rms values of resistance dominate the mean values in the infinite-sample limit.