Hopping transport on site-disorderedd-dimensional lattices

We consider here the hopping of an electron among a band of localized electronic states on a d-dimensional lattice. The hopping rates are assumed to be stochastic variables determined by some probability distribution. We restrict our attention to nearest-neighbor transport in the limit in which the fluctuations in the hopping rates are large. In this limit we construct an exact expansion for the frequency-dependent diffusion coefficient D(ɛ) that is applicable to a wide range of transport phenomena (d=1 conductors, trapping phenomena, molecularly based electronic devices, etc.) in any spatial dimension. For the case of hopping transport with d=1, our method confirms earlier results that strong fluctuations in the hopping rates give rise to a non-Markovian ${\varepsilon }^{1/2}$ correction to normal diffusion. In two dimensions, we establish explicitly the existence of a non-Markovian logarithmic correction ɛ lnɛ to D(ɛ). The formalism is then extended to d dimensions and the frequency corrections are discussed. For d=3, two frequency corrections must be retained. One is linear in ɛ and the other proportional to ${\varepsilon }^{3/2}$. It is shown that only the ${\varepsilon }^{3/2}$ correction contributes to the long-time tail $t^{\mathrm{-}3/2}$ in the time-dependent diffusion coefficient D(t). From these results we show that the long-time tail in the velocity autocorrelation function which is a consequence of the strong fluctuations in the hopping rates is of the form $t^{\mathrm{-}(1+d/2)}$. Comparison is made with earlier results.