Anomalous Frequency-Dependent Conductivity in Disordered One-Dimensional Systems

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 $\sigma \left(\omega \right)\sim {(-i\omega )}^{\nu }$, with $\nu =\frac{(1-\frac{T}{T_m})}{(1+\frac{T}{T_m})}$, for $T<\mathrm{}{T}_m$, and a mobility transition at $T=T_m$. The model is shown to describe recent experiments on the one-dimensional superionic conductor hollandite (${\mathrm{K}}_{1.54}$ ${\mathrm{Mg}}_{0.77}$ ${\mathrm{Ti}}_{7.23}$ ${\mathrm{O}}_{16}$).