Analytic model of hopping mobility at large charge carrier concentrations in disordered organic semiconductors:: Polarons versus bare charge carriers

An analytical theory based on the effective medium approach (EMA) is formulated to describe the charge carrier mobility as a function of the charge carrier concentration in a disordered organic material with a Gaussian density-of-state distribution using jump rate expressions based on either the Miller-Abrahams or polaron model. In this study, we address the problem of how the carrier density dependence of charge mobility is affected by the type of jump rate and, consequently, by polaron effects. Our theoretical consideration employs the concept of the effective transport energy. Results of the EMA calculations in the framework of the Miller-Abrahams jump rate model show a considerable increase of the drift charge carrier mobility with increasing carrier concentration, in good agreement with previous theoretical studies, numerical simulation data, and experiment. At very large carrier densities, however, the theory predicts an abrupt decrease of the charge mobility. A key result of the present study is that a considerably weaker dependence of the mobility on the carrier concentration is found for the polaron jump rate model. Also, with this model, the polaron mobility dramatically decreases at very high carrier densities. An important implication of this study is that the common observation of a field-effect mobility that is orders of magnitude larger than time-of-flight (ToF) or space-charge-limited-current mobilities is incompatible with a polaron binding energy large compared to the width of the distribution of states. On the other hand, the existence of a significant polaron binding energy offers a plausible explanation why, in certain organic disordered materials, field-effect transistor and ToF mobilities are similar.