Unified description for hopping transport in organic semiconductors including both energetic disorder and polaronic contributions

Abstract

We developed an analytical model to describe hopping transport in organic semiconductors including both energetic disorder and polaronic contributions due to geometric relaxation. The model is based on a Marcus jump rate in terms of the small-polaron concept with a Gaussian energetic disorder, and it is premised upon a generalized effective medium approach yet avoids shortcomings involved in the effective transport energy or percolation concepts. It is superior to our previous treatment [Phys. Rev. B 76, 045210 (2007)] since it is applicable at arbitrary polaron activation energy ${E}_{a}$ with respect to the energy disorder parameter $\ensuremath{\sigma}$. It can be adapted to describe both charge-carrier mobility and triplet exciton diffusion. The model is compared with results from Monte Carlo simulations. We show (i) that the activation energy of the thermally activated hopping transport can be decoupled into disorder and polaron contributions whose relative weight depend nonlinearly on the $\ensuremath{\sigma}/{E}_{a}$ ratio, and (ii) that the choice of the density of occupied and empty states considered in configurational averaging has a profound effect on the results of calculations of the Marcus hopping transport. The $\ensuremath{\sigma}/{E}_{a}$ ratio governs also the carrier-concentration dependence of the charge-carrier mobility in the large-carrier-concentration transport regime as realized in organic field-effect transistors. The carrier-concentration dependence becomes considerably weaker when the polaron energy increases relative to the disorder energy, indicating the absence of universality. This model bridges a gap between disorder and polaron hopping concepts.