Anomalous Frequency-Dependent Conductivity in Disordered One-Dimensional Systems

Abstract

Conduction by classical charge carriers is investigated for a one-dimensional system interrupted by barriers with a random distribution of activation energies. Under certain conditions the transport properties are dominated by disorder, and the model predicts an anomalous low-frequency conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{-}i\ensuremath{\omega})}^{\ensuremath{\nu}}$, with $\ensuremath{\nu}=\frac{(1\ensuremath{-}\frac{T}{{T}_{m}})}{(1+\frac{T}{{T}_{m}})}$, for $T<{T}_{m}$, and a mobility transition at $T={T}_{m}$. The model is shown to describe recent experiments on the one-dimensional superionic conductor hollandite (${\mathrm{K}}_{1.54}$${\mathrm{Mg}}_{0.77}$${\mathrm{Ti}}_{7.23}$${\mathrm{O}}_{16}$).