An exact analytical theory was 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 was 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 were due to the dimensionality of space or due to was considered, where analytical calculations and computer simulations were carried out for two- and three-dimensional systems.

Effect of Electric Field on Diffusion in Disordered Materials. I. One-Dimensional Hopping Transport. A.V.Nenashev, F.Jansson, S.D.Baranovskii, R.Österbacka, A.V.Dvurechenskii, F.Gebhard: Physical Review B, 2010, 81[11], 115203