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

Abstract

The resistance $\ensuremath{\rho}$ 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 $E\ensuremath{\ne}0$ in the limit of weak disorder. The average resistance grows exponentially with $L$, the length of the sample, in all cases. The typical resistance $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\rho}}=\mathrm{exp}[〈\mathrm{ln}(1+\ensuremath{\rho})〉]\ensuremath{-}1$ is also found to grow exponentially with $L$ in all cases, except for purely off-diagonal disorder at $E=0$, where $〈\mathrm{ln}(1+\ensuremath{\rho})〉\ensuremath{\propto}\sqrt{L}$. 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 $\ensuremath{\rho}>>1$. 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 $\mathrm{ln}〈\ensuremath{\rho}〉$ with $L$ found in several numerical calculations; its consequences for experiment are also explored.