  We present a graph random walk (GRW) method for the study of charge transport
properties of complex molecular materials in the time-of-flight regime. The
molecules forming the material are represented by the vertices of a directed
weighted graph, and the charge carriers are random walkers. The edge weights
are rates for elementary jumping processes for a charge carrier to move along
the edge and are determined from a combination of the energies of the involved
vertices and an interaction strength. Exclusions are built into the random walk
to account for the Pauli exclusion principle. In time-of-flight experiments,
charge carriers are injected into the material and the time until they reach a
collecting electrode is recorded. In this setting, our GRW approach allows
direct evaluation of the expected hitting time of the collecting nodes in the
graph in terms of a typically sparse, linear system, thereby avoiding
numerically cumbersome and potentially fluctuations-prone methods based on
explicit time evolution from solutions of a high-dimensional system of coupled
ordinary differential equations (the Master Equation) or from kinetic Monte
Carlo (KMC). We validate the GRW approach by conducting numerical studies of
charge dynamics of single and multiple carriers in diffusive and
drift-diffusive regimes using a surrogate lattice model. The surrogate model
allows varying types and strengths of energetic disorder from the reference
baseline. Comparison with results from the Master Equation confirms the
theoretical equivalence of both approaches also in numerical implementations.
We further show that KMC results show substantial deviations due to inadequate
sampling. All in all, we find that the GRW method provides a powerful
alternative to the more commonly used methods without sampling issues and with
the benefit of making use of sparse matrix methods.
