Effect of Electric Field on Diffusion in Disordered Materials II. Two- and Three-dimensional Hopping Transport

TL;DR

This paper develops an analytical theory for how electric fields influence diffusion in 2D and 3D disordered materials, revealing a smooth parabolic dependence at low fields supported by simulations.

Contribution

It extends previous 1D theories to 2D and 3D systems, predicting a parabolic field dependence and showing the Einstein relation is violated even at low fields.

Findings

Diffusion coefficient exhibits a parabolic field dependence in 2D and 3D.

Monte Carlo simulations support the analytical predictions.

The Einstein relation between mobility and diffusivity is violated at low fields.

Abstract

In the previous paper [Nenashev et al., arXiv:0912.3161] an analytical theory confirmed by numerical simulations has been developed for the field-dependent hopping diffusion coefficient D(F) in one-dimensional systems with Gaussian disorder. The main result of that paper is the linear, non-analytic field dependence of the diffusion coefficient at low electric fields. In the current paper, an analytical theory is developed for the field-dependent diffusion coefficient in three- and two-dimensional Gaussian disordered systems in the hopping transport regime. The theory predicts a smooth parabolic field dependence for the diffusion coefficient at low fields. The result is supported by Monte Carlo computer simulations. In spite of the smooth field dependences for the mobility and for the longitudinal diffusivity, the traditional Einstein form of the relation between these transport coefficients is shown to be violated even at very low electric fields.