Scaling studies of the resistance of the one-dimensional Anderson model with general disorder

Abstract
The resistance ρ of a one-dimensional Anderson model with both diagonal and off-diagonal disorder is studied by analytic and numerical techniques. A recursive method is developed and used to derive an exact scaling law for the average resistance at E=0 for arbitrary disorder, and for E0 in the limit of weak disorder. The average resistance grows exponentially with L, the length of the sample, in all cases. The typical resistance ρ̃=exp[ln(1+ρ)]1 is also found to grow exponentially with L in all cases, except for purely off-diagonal disorder at E=0, where ln(1+ρ)L. An explanation is given for the existence of this special case and it is shown that all our results are consistent with a lognormal probability distribution of the resistance for ρ>>1. Quantitative estimates are made of the reliability of numerically performed averages which show that a numerical average will converge only very slowly to the analytic result. This provides a qualitative explanation of the slower than linear growth of lnρ with L found in several numerical calculations; its consequences for experiment are also explored.