# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Particles and Transitions  
#### 3.3.1 Charge Carriers  
In disordered organic materials, charge carriers are spatially localized due to the lack of long-range order. Charge transport is modeled as a hopping process from one localized state to a neighboring localized state. This state can be seen as local energy valleys where charges locate in a long-term, which e.g., represent a small molecule or a few monomers along a polymer chain. Figure 4: Correlation of site energies inside a blend. The black and red dots represent lattice sites of the small molecule PCBM and the polymer P3 HT, respectively. A weighted average is performed over all sites of the same material type within the radius of \(r_{\mathrm{corr},i}\). [PERSON] and [PERSON] describe hopping as a combination of thermal activation and tunneling [74]. The hopping rate between states \(i\) and \(j\) reads:  
\[a_{ij}=a_{0}\exp(-2\gamma r_{ij}) \begin{cases}\exp\left(-\frac{\Delta E_{ij}}{k _{B}T}\right)&\text{for}\quad\Delta E_{ij}\geq 0,\\ 1&\text{for}\quad\Delta E_{ij}\leq 0,\end{cases}  \tag{11}\]  
where \(a_{0}\) is the attempt-to-escape frequency, \(\gamma\) is the inverse localization length, \(k_{B}\) is the Boltzmann constant, \(T\) gives the temperature, \(r_{ij}\) the distance and \(\Delta E_{ij}=E_{i}-E_{j}\) the energy difference between the initial configuration \(i\) and the final configuration \(j\), before and after a hopping process, respectively. When hopping upwards in energy, the Boltzmann term accounts for thermal activation [74]. This rate model is usually used when the coupling between electrons and phonons is weaker [75; 76]. [PERSON] has introduced a model that accounts for stronger coupling between lattice and charges [77]. Because of the low permittivity of organic materials, charges lead to local polarizations of the lattice, which are described as quasi-particles called polarons. The formula for polaron transitions reads:  
\[a_{ij}=\frac{\left|I_{ij}\right|^{2}}{\hbar}\sqrt{\frac{\pi}{\lambda k_{B}T}} \exp\left(-\frac{(\Delta E_{ij}-\lambda)^{2}}{4\lambda k_{B}T}\right)\,, \tag{12}\]  
with the reduced [PERSON]'s constant \(\hbar\), transfer integral \(I_{ij}\) and the reorganization energy \(\lambda\) describing the reconfiguration of the molecule due to the deformation by the charge. The effect of a spatial disorder in an irregular Voronoi set is included via the site to site distance \(r_{ij}\) in Miller-Abrahams rates (Equation (11)) and via the distance-dependent transfer integral \(I_{ij}\) in Marcus rates (Equation (12)). The hopping rates are calculated for every particle and all free neighbors. If a particle already occupies a neighboring site, the hopping rate is zero. When a hopping transition occurs, the occupation information of origin and destination sites and the particle position have to be updated. To calculate hopping rates, it is necessary to compute the electric energy \(E_{i}\) for every site \(i\). The energy at site \(i\) is given by:  
\[E_{i}=E_{i}^{0}+E_{i}^{\sigma}+E_{i}^{F}+E_{i}^{C}\,, \tag{13}\]  
with \(E_{i}^{0}\) being the energy level of the molecular orbital and the energetic disorder \(E_{i}^{\sigma}\). The energy arising from the external applied field is referred to as \(E_{i}^{F}\) and \(E_{i}^{C}\) represents the Coulomb energy. The Coulomb interaction between all particles is calculated using the Ewald sum [12; 78]. The Ewald sum makes use of the principle of image charges. As the potential at the contact layers has to be constant, all the field vectors have to end perpendicular to contacts. This can be achieved by removing the contact and introducing an image charge of opposite sign and the same distance behind the contact. Since an OSC device has two contacts, every image charge behind one contact also induces an image charge behind the other contact, leading to a periodic scheme.