Localization length in Dorokhov's microscopic model of multichannel wires

TL;DR

This paper derives exact quantum expressions for the localization length in weakly disordered two- and three-channel tight-binding systems, extending Dorokhov's model and analyzing their conductance and reflection properties.

Contribution

It provides the first exact quantum derivations of localization lengths for multi-channel systems based on Dorokhov's microscopic model, including new insights into their scaling behavior.

Findings

Localization length scales linearly with the number of channels.

Exact expressions differ from Dorokhov's original scaling assumptions.

Reflection matrices enable analysis of mean free paths and their relation to localization.

Abstract

We derive exact quantum expressions for the localization length $L_c$ for weak disorder in two- and three chain tight-binding systems coupled by random nearest-neighbour interchain hopping terms and including random energies of the atomic sites. These quasi-1D systems are the two- and three channel versions of Dorokhov's model of localization in a wire of $N$ periodically arranged atomic chains. We find that $L^{-1}_c=N.\xi^{-1}$ for the considered systems with $N=(1,2,3)$, where $\xi$ is Thouless' quantum expression for the inverse localization length in a single 1D Anderson chain, for weak disorder. The inverse localization length is defined from the exponential decay of the two-probe Landauer conductance, which is determined from an earlier transfer matrix solution of the Schr\"{o}dinger equation in a Bloch basis. Our exact expressions above differ qualitatively from Dorokhov's localization length identified as the length scaling parameter in his scaling description of the distribution of the participation ratio. For N=3 we also discuss the case where the coupled chains are arranged on a strip rather than periodically on a tube. From the transfer matrix treatment we also obtain reflection coefficients matrices which allow us to find mean free paths and to discuss their relation to localization lengths in the two- and three channel systems.