# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Energetic Landscape  
The energetic landscape of an organic semiconductor plays a significant role in the charge transport and exciton dissociation [10; 27; 28]. In contrast to inorganic semiconductors, the molecular orbital energies inside an organic material are strongly disordered due to the non-crystalline arrangement of the molecules and polymers [28]. Energetic disorder values of \(\sigma=0.1\) eV are present in many materials [6]. It is widely accepted that the shape of the density of states (DOS) can be modeled using a Gaussian distribution [15; 27; 36; 58; 69; 70]. Some experimental observations require a correlation in the site energies, as described in the correlated disorder model (CDM) [15; 36; 71; 72], while many studies are based on the uncorrelated, Gaussian disorder proposed by the Gaussian disorder model (GDM) [6; 58; 72].  
We implement a correlated disorder model based on a simple averaging method of site energies. Figure 4 visualizes the CDM for an organic system consisting of PCBM (black dots) and P3 HT (red dots). First, for every site, we choose an energy randomly from a Gaussian distribution:  
\[g(\epsilon)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(\epsilon-E^{0})^{2}}{ 2\sigma^{2}}\right)\,, \tag{7}\]  
with the molecular orbital energy \(E^{0}\) and the global energetic disorder \(\sigma\). Herewith, the energy distribution in the GDM is obtained. For the CDM, the following procedure is performed.  
1. Starting from the randomly Gaussian-distributed energetic landscape, we compute the weighted average of the energies \(E_{k}\) of all sites \(k\) with \(r_{ik}=|\mathbf{r}_{i}-\mathbf{r}_{k}|\leq r_{\mathrm{corr}}\) for every site \(i\) using: \[E_{i}^{*}=\frac{1}{\sum_{k}\frac{1}{|r_{ik}|}}\sum_{k}E_{k}\frac{1}{|r_{ik}|}\,.\] (8) Note that the average is only performed over sites \(k\) representing the same material as site \(i\). The average energies \(E_{i}^{*}\) are stored in an intermediate vector.
2. Replace the site energies by the average energies \(E_{i}\to E_{i}^{*}\).  
Scaling of the site energies: The averaging procedure reduces the energetic disorder. To keep the initially chosen \(\sigma\), we compute the energetic disorder \(\sigma^{*}\) after the averaging process. Then, the site energies are scaled by: \[E_{i}\to E_{i}\sqrt{\frac{\sigma}{\sigma^{*}}}-\left(\sqrt{\frac{\sigma}{\sigma^ {*}}}-1\right)E_{i}^{0}\,.\] (9)
4. The level of correlation can be varied by either changing \(r_{\mathrm{corr}}\) or repeating Steps 1-3.  
A measure of the level of correlation is defined by the correlation function \(c(r)\). The correlation of sites \(i\) and \(j\) can be obtained with:  
\[c_{i,j}=\frac{\left(E_{i}-E_{i}^{0}\right)\left(E_{j}-E_{j}^{0}\right)}{\sigma ^{2}}\,. \tag{10}\]  
If we take the average of the correlation of the ensemble \(\langle\cdot\rangle\) of pairs \((i,j)\) with the same distance \(r_{ij}\), we obtain a measure of the correlation of the whole organic material. Using this correlation mechanism, we have shown that a high correlation in the site energies gives rise to filamentary structures of the local currents [39]. For distances below \(2\,\mathrm{nm}\), the correlation function obtained by atomistic simulations [73] has been reproduced with our mechanism, while for long-range hopping, other procedures, e.g., a dipole-based correlation [36], give better results. While a dipole-based correlation is obtained by summation over all dipoles within the simulation box [36; 71], the presented method is computationally favorable as only sites within a certain radius are considered.