Frequency dependence of the conductivity in presence of an electric field in one dimension: Weak-disorder limit

Abstract

The frequency dependence of the electrical conductivity is examined for a classical hopping model of a random one-dimensional system in the presence of a superposed static electric field. The effect of the field is taken as a constant bias for the left-right jump rates. A general expression is derived for the mean velocity and frequency-dependent conductivity. Explicit evaluation of these equations is given for correlated and uncorrelated hopping rates: (1) in general for high frequencies and (2) to lowest order in the disorder for all frequencies. In the latter case, an initial decrease in $\sigma \left(\omega \right)$ for very small $\omega$ is found, ${\sigma }_R=a_0-a_1{\omega }^2$, ${\sigma }_I=a_2{\omega }^3$. For larger frequencies, the conductivity crosses over to the form $\sigma \left(\omega \right)=b_0+b_1{\left(i\omega \right)}^{\frac{1}{2}}$, previously calculated by Alexander and Orbach. The $a_i$,$b_i$ are constants which depend on the strength of the bias and the randomness. The crossover frequency increases with the bias. In addition, the variance in the autocorrelation function is calculated in the long-time limit, for weak disorder in the symmetric case. It is shown that fluctuations do not significantly affect the determination of this quantity under these conditions.