Variable Range Hopping Conduction in Complex Systems and a Percolation Model with Tunneling

TL;DR

This paper investigates the variable range hopping conduction in disordered quantum insulators, proposing a percolation model with tunneling that explains the experimentally observed variation of the conduction exponent with dopant concentration.

Contribution

It introduces a semi-classical RRTN percolation model that captures the doping-dependent variation of the VRH conduction exponent in complex disordered systems.

Findings

The RRTN model reproduces the doping dependence of the VRH exponent.

The model explains deviations from traditional VRH predictions.

It provides a unified framework for understanding conduction in disordered composites.

Abstract

For the low-temperature electrical conductance of a disordered {\it quantum insulator} in $d$-dimensions, Mott \cite{mott} had proposed his Variable Range Hopping (VRH) formula, $G(T) = G_0 {\rm exp}[-(T_0/T)^{\gamma}]$, where $G_0$ is a material constant and $T_0$ is a characteristic temperature scale. For disordered but non-interacting carrier charges, Mott had found that $\gamma = 1/(d+1)$ in $d$-dimensions. Later on, Efros and Shkolvskii \cite{esh} found that for a pure ({\it i.e.}, disorder-free) {\it quantum insulator} with interacting charges, $\gamma =1/2$, {\it independent of d}. Recent experiments indicate that $\gamma$ is either (i) larger than any of the above predictions; and, (ii) more intriguingly, it seems to be a function of $p$, the dopant concentration. We investigate this issue with a {\it semi-classical} or {\it semi-quantum} RRTN ({\it Random Resistor cum Tunneling-bond Network}) model, developed by us in the 1990's. These macroscopic {\it granular/ percolative composites} are built up from randomly placed meso- or nanoscopic coarse-grained clusters, with two phenomenological functions for the temperature-dependence of the metallic and the semi-conducting bonds. We find that our RRTN model (in 2D, for simplicity) also captures this continuous change of $\gamma$ with $p$, satisfactorily.