Localization length in Dorokhov’s microscopic model of multi-channelwires

Abstract

We derive exact quantum expressions for the localization lengthLcfor weak disorder in two- and three-chain tight-binding systems coupled byrandom nearest-neighbour interchain hopping terms and including randomenergies of the atomic sites. These quasi-1D systems are the two- andthree-channel versions of Dorokhov’s model of localization in a wire ofNperiodically arranged atomic chains. We find that, for weak disorder, Lc−1 = Nξ−1 for the systemsconsidered with N = (1, 2, 3),where ξis Thouless’ quantum expression for the inverse localization length in asingle 1D Anderson chain. The inverse localization length is defined fromthe exponential decay of the two-probe Landauer conductance, which isdetermined from an earlier transfer matrix solution of the Schrödinger equationon a Bloch basis. Our exact expressions above differ qualitatively fromDorokhov’s localization length, identified as the length-scaling parameter inhis scaling description of the distribution of the participation ratio. ForN = 3we also discuss the case where the coupled chains are arranged on a strip ratherthan periodically on a tube. From the transfer matrix treatment we also obtainmatrices of reflection coefficients, which allow us to find mean free paths and todiscuss their relation to localization lengths in the two- and three-channel systems.