Effect of electric field on diffusion in disordered materials. II. Two- and three-dimensional hopping transport

In the previous paper [A. V. Nenashev, F. Jansson, S. D. Baranovskii, R. Oumlsterbacka, A. V. Dvurechenskii, and F. Gebhard, Phys Rev. B 81, 115203 (2010)] 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, nonanalytic 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.