Diffusion Currents in Large Electric Fields for Discrete Lattices

A derivation for the steady-state current J produced by a large homogeneous electric field E0 in the presence of a concentration gradient is presented which includes explicitly the effects due to lattice discreteness. The resulting equation is J=4aν exp(−W/kBT) sinh(ZeE0a/kBT)[C(L)−C(0) exp(ZeE0L/kBT)]/[1−exp(ZeE0L/kBT)],where C(0) and C(L) are the boundary concentrations of the diffusing species at the interfaces of the planar film at positions x=0 and x=L; e, the electronic-charge magnitude; Ze, the charge per particle of the diffusing species; 2a, the distance between adjacent potential minima; v, the frequency at which the ion attempts energy barriers which have height W in zero field; kB, the Boltzmann constant; and T, the absolute temperature. A derivation valid in the limit of a continuum model is also presented, and the results are compared numerically. The equations for the discrete and continuum models reduce to the results predicted by the ordinary linear diffusion equation for electric fields below approximately 105 V/cm. The relevance of the equations to the phenomena of anodic and thermal oxidation and to thin-film current-voltage devices is briefly described.