Conductance distribution in strongly disordered mesoscopic systems in three dimensions

TL;DR

This paper analytically investigates the conductance distribution in 3D strongly localized disordered systems, revealing significant deviations from log-normal behavior and providing a new phenomenological model for understanding these differences.

Contribution

The study introduces a generalized DMPK equation with a phenomenological matrix K to analytically derive conductance distributions in 3D disordered systems, extending previous models.

Findings

K can be modeled by a single parameter in strong disorder limit

Analytic distribution P(g) differs from log-normal in 3D insulators

Method may apply to the critical regime of the Anderson transition

Abstract

Recent numerical simulations have shown that the distribution of conductance P(g) in 3D strongly localized regiem differs significally from the expected log normal distribution. To understand the origin of this difference analytically, we used a generalized DMPK equation for the joint probablity distribution of the transmission eigenvalues which includes a phenomenological (disorder and dimensionality dependent) matrix K containing certain correlations of the transfer matrices. We first of all examine the assumptions made in the derivation if the generalized DMPK and find that to a good approximation they remain valid in 3D. We then evaluate the matrix K numerically for various strength of the disorder and various system sizes. In the strong disorder limit we find that K can be described by a simple model which, for a cubic system, depends on a single parameter. We use this phenomenological model to analytically evaluate the full distribution P(g) for Anderson insulators in 3D. The analytic results allow us to develop an intuitive understanding of the entire distribution, which differs qualitatively from the log-normal distribution of a Q1D wire. We also show that out method could be applicable in the critical regime of the Anderson transition.