Effect of electric field on diffusion in disordered materials. I. One-dimensional hopping transport

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, nonanalytic 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, nonanalytic 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 nonanalytic 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, where analytical calculations and computer simulations are carried out for two- and three-dimensional systems.