Effect of Electric Field on Diffusion in Disordered Materials I. One-dimensional Hopping Transport

TL;DR

This paper develops an exact analytical theory for charge carrier diffusion in one-dimensional disordered materials under electric fields, revealing non-analytic field dependence and violation of Einstein relation, with implications for higher-dimensional systems.

Contribution

It provides the first exact analytical expressions for diffusion coefficients in 1D disordered systems with electric fields, extending previous mobility theories.

Findings

Diffusion coefficient shows linear, non-analytic dependence on electric field at low fields.

Mobility exhibits parabolic, analytic dependence in the random-barrier model.

Einstein relation is violated at any finite electric field in these models.

Abstract

An exact analytical theory is developed for calculating the diffusion coefficient of charge carriers in strongly anisotropic disordered solids with one-dimensional hopping transport mode for any dependence of the hopping rates on space and energy. So far such a theory existed only for calculating the carrier mobility. The dependence of the diffusion coefficient on the electric field evidences a linear, non-analytic behavior at low fields for all considered models of disorder. The mobility, on the contrary, demonstrates a parabolic, analytic field dependence for a random-barrier model, being linear, non-analytic for a random energy model. For both models the Einstein relation between the diffusion coefficient and mobility is proven to be violated at any finite electric field. The question on whether these non-analytic field dependences of the transport coefficients and the concomitant violation of the Einstein's formula are due to the dimensionality of space or due to the considered models of disorder is resolved in the following paper [Nenashev et al., arXiv:0912.3169], where analytical calculations and computer simulations are carried out for two- and three-dimensional systems.