Variable-range hopping: Role of Coulomb interactions

Abstract

The effect of Coulomb interactions on hopping conduction in the variable-range hopping regime is analyzed within a linear-response formalism. Here the conductivity and the dielectric function are related to the density-density response function for which a generalized master equation (GME) can be derived using the Mori-Zwanzig projector formalism. The GME can be thought of as a random resistor network with frequency-dependent internode conductances, whose values can be determined from a function related to the current-current correlator at the two nodes. We evaluate the internode conductances using a diagrammatic perturbation formalism. For a single electron hop with all the other charges frozen, we obtain hop rates correct to all orders in Coulomb interaction. This gives us a finite temperature generalization of existing results for the interacting system. We then incorporate relaxation effects that accompany electron hops, using a dynamical model of the Coulomb gap. We argue that the parameter that governs the local relaxation is related to the conductivity itself. These internode conductances are then used to calculate the dc conductivity of the network by effective- medium approximation. We show that a crossover from Efros-Shklovskii's ${T}^{1/2}$ behavior to Mott's ${T}^{1/4}$ behavior occurs due to the relaxation effects, as the temperature is increased. At low temperatures the relaxation is slow so that electrons hop in a frozen charge background and thereby sense the Coulomb gap. This gives the ${T}^{1/2}$ behavior. At higher temperatures the relaxation gets faster and the Coulomb gap is alleviated leading to Mott's behavior.