Energy Space Diffusion of Hopping Excitations in Non-Equilibrium

Abstract

The effective medium theory of hopping by Bleibaum, Bö ttger and Bryksin is generalized to finite temperatures. Starting point is the system of rate equations for the (localized) site occupation numbers of a disordered system. An effective generalized Fokker-Planck equation is derived thereof, which contains terms describing diffusion in real space, as well as terms describing drift and diffusion in energy space. The corresponding effective medium is determined self-consistently. The theory describes the relaxation in a system of localized excitations, where transitions are possible by hopping. It considers the behavior at high temperatures, corresponding to nearest neighbor hopping. IntroductionThe model under consideration consists of a number of localized states, which interact anharmonically with extended states, so that transitions between the localized states are possible. One example is the hopping motion of electrons in band tails. Another one is the hopping of localized vibrational states, discussed at the end of this paper. Of central interest is the behavior of excitations of the localized subsystem over time, especially their relaxation after an initial excitation. Such excitations can spatially diffuse away, as well as undergo a ''diffusion" in energy space, which superficially contradicts energy conservation, but can effectively occur, since the ''missing" energy is transferred into the extended subsystem, which is treated as a heat bath. This implies, that relaxation processes within this subsystem must be very fast compared with relaxation in the localized subsystem.The temporal behavior of the localized states is assumed to be adequately described by linearized rate equations for the occupation numbers of those states. Thus, the particles are considered as non-interacting, and the basic transport is incoherent and Markovian. But, the equations for the effective medium will be non-Markovian.For this problem an effective medium theory has been put forward by Bleibaum et al. [1][2][3]. We generalize this theory to finite temperatures. In addition to the drift term in energy space, which is already considered in Ref.[3], here we also have to take into account a diffusion term in energy space. Additionally, we allow the excitations to have a finite life-time.