# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 2 Kinetic Monte Carlo Method
If the coarse-graining is chosen accurately such that the correlation time of the environment in the continuous-time dynamics is much shorter than the typical rate constants, the discretized system evolution \(i_{1}\to i_{2}\to\cdots\to i_{N}\) can be regarded as Markovian. A single trajectory of state transitions  
Figure 1: (a) Coarse-graining of the system dynamics and illustration of long-term system states; (b) transition from initial state \(i\) to final states \(j\) as employed in the kinetic Monte Carlo (kMC) algorithm. does not yield the same system behavior as described by the master equation. Averaging over several trajectories provides an equivalent system behavior one would obtain by solving the master equation. The dynamical evolution of the system is governed by the transitions rates. The rates are inputs to the kMC model and can either come from experimental measurements or are based on underlying theoretical models. Given rates, the kMC algorithm is straightforward and can be reduced to the following scheme. For a detailed description of the derivation, we refer to [30; 34]. 1. Initialize the system, and define all parameters that are required for the transition rates of all processes in the kMC. Define termination conditions such as the maximum simulation time \(t_{\text{max}}\). 2. Identify the activated transitions for the current system state \(i\) and calculate the transition rates \(\{a_{ij}\}\) of all physical processes that can occur. The rates \(a_{ij}\) mirror the probability of the occurrence of the transition to state \(j\). The sum \(a=\sum_{j}a_{ij}\) of all rates is calculated. 3. Two uniformly-distributed random numbers \(r_{1}\in(0,a)\) and \(r_{2}\in(0,1)\) are chosen. The first number \(r_{1}\) chooses the transition \(\mu\) from the set of events \(\left\{a_{ij}\right\}\), which fulfills the condition: \[\sum_{j=1}^{\mu-1}a_{ij}<r_{1}\leq\sum_{j=1}^{\mu}a_{ij}\,.\] (2) The second number \(r_{2}\) gives the time passed before the transition is performed, given by: \[\tau=-\frac{\ln(r_{2})}{a}\,.\] (3) The derivation of Equations (2) and (3) is based on the direct method [30], which divides the probability density function (PDF) for the transition \(\mu\) and the corresponding time step \(\tau\) into two separate PDFs, each of which only depends on either the time \(\tau\) or the transition \(\mu\). 4. The simulation time \(t\) is advanced by \(\tau\), and the system configuration is updated following the transition \(\mu\). 5. If the simulation time exceeds \(t_{\text{max}}\), the simulation stops, and all particle trajectories and the counters tracking the events are evaluated. If \(t<t_{\text{max}}\), the kMC loop (Step 2-4) is repeated. For simple systems such as unipolar single layer devices, the master equation has been applied successfully [35; 36; 37]. However, critical properties such as the electrostatic Coulomb interaction between charge carriers can only be captured by an effective potential in the framework of the master equation. With the particular application to OSCs in mind, particles like excitons, electrons and holes reside in localized states of either monomers of a polymer or small organic molecules. Charge motion can occur by processes on shorter timescales, but as long as they do not perform a hopping process to another localized state, the particles are spatially assigned to that state. The overall dynamical behavior is determined by hopping events between localized states. This concept has been applied to model charge transport in disordered organic semiconductors, as well as organic devices such as OSCs or OLEDs. A significant contribution of describing charge transport has been conducted by [PERSON] [6]. Several models for OSCs have been set up investigating active layer morphology and exciton separation [11; 17], recombination [10; 14; 38], Coulomb interaction [12] and charge densities [13].