The statistics of the conductance of one-dimensional disordered chains

Abstract

The method of 'generalised transfer matrices' previously used to find the average of any power of the transmission coefficient, (t^x) is applied to finding ( mod t mod ^y) in the cases y=0, -2, -4 . . . and Re y>1. These cases include all the integer moments of the conductance, mod t mod ², and its reciprocal, the resistance. The conductance is the more physically important quantity and is found to have a complicated dependence on chain length. The formulae are valid for any distribution of disorder and all lengths of sample. Limiting cases and numerical results are presented. An anomaly in the energy dependence of the conductance of a binary alloy is discovered.