Dispersive energy transport and relaxation in the hopping regime

A method for investigating relaxation phenomena for charge-carrier hopping between localized tail states is developed. It allows us to consider both charge and energy dispersive transport. The method is based on the idea of quasielasticity: the typical energy loss during a hop is much less than all other characteristic energies. We investigate two models with different density-of-state energy dependencies with our method. In general, we find that the motion of a packet in energy space is affected by two competing tendencies. First, there is a packet broadening, i.e., dispersive energy transport. Second, there is a narrowing of the packet if the density of states is depleting with decreasing energy. It is the interplay of these two tendencies that determines the overall evolution. If the density of states is constant, only broadening exists. In this case a packet in energy space evolves into a Gaussian one, moving with a constant drift velocity and mean-square deviation increasing linearly in time. If the density of states depletes exponentially with decreasing energy, the motion of the packet slows down tremendously with time. For large times the mean-square deviation of the packet becomes constant, so that the motion of the packet is "solitonlike."