We derive a macroscopic drift-diffusion equation for charge carrier mobilities in disordered materials. Our procedure takes site blocking and internal field effects explicitly into consideration. The final expression contains several drift terms originating from spatial inhomogeneities and energetic disorder. Thus even a spatial variation of the hopping rates is allowed, if local detailed balance still holds. The main advantage of our approach is a time-dependent numerical solution of the drift-diffusion equation for all positions within a sample without resort to a lattice description or a finite element analysis. We discuss our numerical results for important cases of spatial disorder and energy density of states within a sample and follow the space-time evolution of local charge distributions. The occurrence of shock waves is discussed both theoretically by giving asymptotic solutions and in simulations.
We study the effect of energetic and spatial disorder, anisotropy and sample orientation on the field-dependent mobility of charge carriers using a dynamic Monte Carlo simulation. Our transfer rate is based on a polaronic model of phonon-assisted hopping in an effective diabatic potential (Marcus-theory). We find that our simulations, in contrast to the Gaussian Disorder Model or the Correlated Disorder Model, neither require unphysical model parameters nor correlated disorder to explain experimental data for the field and temperature dependence of mobilities. Our simulations show, that no energetic disorder is necessary to fit experiments. A clear transition from a 3-D diffusion and drift limited mobility to a quasi 1-D drift limited process with increasing external fields in the presence of spatial disorder can be observed. A well-controlled degree of disorder can under certain conditions increase carrier mobility. Simulation of mobilities on a regular lattice are found to strongly depend on the direction of the external field with respect to the lattice in a non-trivial and field-dependent manner. This usually neglected effect is highly sensitive to the choice of the hopping rate and the underlying lattice and can easily modify mobilities by 25% or more.