A Graph Random Walk Method for Calculating Time-of-Flight Charge Mobility in Organic Semiconductors from Multiscale Simulations

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 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 on charge dynamics of single and multiple carriers in diffusive and drift-diffusive (due to an external electric field) regimes using a surrogate lattice model of a realistic material whose properties have been simulated within a multiscale model framework combining quantum-mechanical and molecular-mechanics methods. The surrogate model allows varying types and strengths of energetic disorder from the reference baseline. A comparison with results from the master equation confirms the theoretical equivalence of both approaches also in numerical implementations. We further show that KMC results indicate 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 using sparse matrix methods.