Adiabatic polaron theory of electron hopping in crystals: A reaction-rate approach

Abstract

The paper presents an application of reaction-rate theory to nonradiative electron transitions in polar crystals based on a "two-site" model. For this purpose localized electronic states are described by using the adiabatic approximation. Both "large" polarons in ionic crystals and "small" polarons in molecular crystals are considered from a unified point of view. By assuming a single frequency of the lattice vibrations (Einstein model) one can calculate the transition probability for both the limits of adiabatic and nonadiabatic electron transfer as well as for the whole intermediate range. The relevant expressions, earlier derived in treating electron transfer in solution, are reviewed and discussed in order to give a justification of their application to polaron hopping in crystals. In a similar way, a review is made of the reaction-rate approach to electron transfer to obtain the appropriate rate equations for the low- and high-temperature limits and the intermediate temperature range as well. New expressions for the polaron-hopping rate are also derived. The conditions of its validity are discussed. This paper provides an essentially new approach to nonradiative electron transfer in polar crystals. It permits one to overcome the limitations of the usual methods, such as the multiphonon approach, entirely based on time-dependent perturbation theory and/or the Franck-Condon approximation which restrict their applicability only to nonadiabatic polaron hopping. The reaction-rate treatment exactly reproduces the results of the more rigorous multiphonon theory of nonadiabatic transitions. Moreover, it yields correct rate expressions for adiabatic transitions, in particular, in the high-temperature range, in which the classical occurrence-probability approach greatly overestimates the activation energy. Some problems concerning the Einstein one-frequency oscillator model assumed and the irreversibility of the electron transfer are discussed with some details from standpoint of both reaction-rate theory and multiphonon theory. The advantages of the reaction-rate approach are emphasized.