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

TL;DR

This work addresses calculating time-of-flight mobilities in disordered organic semiconductors without full dynamical simulations. It introduces a Graph Random Walk (GRW) framework that discretizes the material as a directed graph and uses a configuration-space graph to handle multiple carriers under Pauli exclusion, enabling direct computation of hitting times via sparse linear systems. The GRW approach yields results in close agreement with Master Equation analyses and avoids sampling issues inherent to KMC, while delivering field-dependent mobility consistent with Poole–Frenkel behavior for a range of disorder and carrier counts. The method offers a scalable, accurate alternative for multiscale material studies, with strong implications for efficient design and analysis of organic electronic devices.

Abstract

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. Our approach allows direct evaluation of the expected hitting time of the collecting nodes in terms of a sparse, linear system, avoiding numerically cumbersome and potentially fluctuations-prone methods based on explicit time evolution from solutions of a high-dimensional Master Equation or from kinetic Monte Carlo (KMC). We validate the GRW approach by numerical studies of charge dynamics of single and multiple carriers in diffusive and drift-diffusive 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. 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.

Figures (11)

• Figure 1: Schematic representations of time-of-flight setups used in experiment and simulations of charge transport: (a) basic setup indicating the electrode setting and the externally applied electric field; (b) circles introduced to emphasize the disordered nature of the molecular material; (c) all-atom molecular detail of an organic semiconductor; (d) center-of-mass view of the molecular material. In all panels, the yellow line indicated an effective charge carrier trajectory.
• Figure 1: The ten states of a connected 5-site system with blue circles representing unoccupied sites and red circles for occupied sites. The numbers on the top left figure give the site indices. The state notations are the brackets containing 0 and 1 below each state plot. If site $5$ is the sink, then the sink states are $\mathbf{s}_4, \mathbf{s}_7, \mathbf{s}_9, \mathbf{s}_{10}$
• Figure 1: Calculated ToF $\tau$ (in []s) for $N_c=1$ depending on disorder strength in systems with uncorrelated (a) and spatially correlated (b) site energies, obtained from MEq (filled symbols) and GRW (open symbols), respectively. Each data point $\tau$ represents the sample average of the ten realizations of the Gaussian distributed energy landscapes.
• Figure 1: Convergence behavior of the ToF calculation using the GRW method for $N_c=1$.
• Figure 2: Distributions of KMC steps until absorption to sink ($N_\text{ToF}$) and the corresponding first hitting time $\tau$ obtained via KMC for a single realization of the lattice model with no site-energy disorder (a), Gaussian-distributed site-energies without (b) and with spatial correlations (c). The density histogram consists of $10^7$ KMC sample points grouped into 100 bins.