Dispersive diffusion transport and noise, time-dependent diffusion coefficient, generalized Einstein-Nernst relation, and dispersive diffusion-controlled unimolecular and bimolecular reactions

Abstract

Starting from the same fundamental bases as a recently advanced unified theory of low-frequency fluctuation-dissipation properties of condensed matter, we have approached the various problems associated with diffusion phenomena. In this approach the classical Markovian diffusion transport is a limiting case ($n=0$) of an in-general non-Markovian diffusion transport ($n\ensuremath{\ne}0$). The theory involves a single parameter $n$, the infrared divergent exponent, which ranges from 0 to 1. For any value of $n$ in this range, the form of the diffusion transport equation is completely specified through a universal time correlation function $\ensuremath{\psi}(t)$ whose form is dependent on $n$ only. Since the present treatment is one branch of the unified theory, the quantity $n$, although it cannot be easily calculated for a given material and process at this time, can be determined by or inferred from measurements in another low-frequency response provided the same diffusion process governs both phenomena. From the availability of large amounts of experimental data in dielectric, mechanical, and viscoelastic relaxations, in NMR spin-lattice relaxations, in dispersive transient transport, etc., and the already-proven successes of the unified theory when specialized to these subjects, $n$ should not be considered as a free parameter that cannot be cross checked or cannot be determined. The new results on diffusion derived in this work for each $n$ value include (1) the diffusion noise spectra which tend to $\frac{1}{f}$ spectra when $n$ approaches unity; (2) a modified Einstein relation between mean-square displacement and time of diffusion; (3) time-dependent diffusion coefficient and mobility with their ratio now obeying a generalized Nernst-Einstein formula, and (4) the time evolutions of diffusion-controlled unimolecular and bimolecular reactions or recombinations. We have compared these predictions with recent experiments on amorphous Si and achieved good agreement between them. In contrast, the results of our analysis of diffusion in another well-known model due to Scher and Montroll, based on a stochastic-event time distribution function $\ensuremath{\psi}(t)\ensuremath{\equiv}{t}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\alpha}}$ at large times, fail to account for the time dependence of bimolecular recombination of electrons and holes that is controlled by dispersive diffusion.