Transport Quantities in One-Dimensional Disordered Systems

Abstract

In discussing transport quantities in one-dimensional disordered systems, care must be taken to assure that averaged quantities are statistically meaningful. ANDERSON, THOULESS, ABRAHAMS, and FISHER [1] have recently pointed to a, the inverse localization length, as a proper quantity to average. The mathematical properties of a have been discussed in the past by FURSTENBURG [2] and O’CONNOR [3], for example. The connection of a to resistance and conductance was made by LANDAUER [4]. He considered a collection of random barriers of total length L, and described by transmission coefficient T and reflection coefficient R = 1-T. By using the Einstein relation between conductivity and the diffusion constant, he showed that the conductance, G, is given by

$$ G = \frac{{{e^2}}} {{2\pi h}}{\kern 1pt} \frac{T} {R} $$

((1))

where e is the electronic charge and h is Planck’s constant. LANDAUER [4] and ANDERSON et al. [1] agree that averaging the resistances of a collection of random systems gives a quantity proportional to exp(2αL)-1, where a is a parameter independent of length. While this accurately describes average resistance its usefulness is questionable, for if we instead average G, we will find that 1/<G> ≠ <1/G>.