# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Particles and Transitions
#### 3.3.2 Excitons
\(E_{\text{CT,B}}\) gives the charge transfer binding energy, corresponding to the Coulomb binding energy of an electron-hole pair residing at neighboring sites, and \(E_{\text{LUMO}}\) defines the LUMO energy level difference of the neighboring sites. Coulomb interactions with the surrounding charge carriers are omitted. If an electrical generation event is chosen, the electron and hole are removed, and a singlet/triplet exciton is generated at the site the former hole was located. All excited states have a certain lifetime before they decay to the ground state. The lifetime of triplets exceeds that of singlets due to the spin selection rules by far. The rate for the exciton decay is chosen as the inverse of average lifetime \(a_{\text{exd}}^{s/t}=1/\tau^{s/t}\). Due to the spin-orbit coupling of a phosphorescent dopant, triplets may decay due to the lifted spin-selection rules. Thus, the triplet lifetime is reduced within the radius \(r_{\text{ISC}}\) around dopant sites. Excitons are quasi-particles with a net charge of zero. Thus, they diffuse through the organic semiconductor by energy transfer between sites. Existing kMC models account for exciton diffusion using either Forster resonance energy transfer (FRET), Dexter type transfer or are based on a random walk [19; 86; 87; 88]. The diffusion of singlets and triplets can be modeled using FRET [89] or by a random walk based on the Einstein-Smoluchowksi model for the Brownian motion with a diffusion coefficient \(D\)[90; 91]. [PERSON] described a non-radiative energy transfer between an exciton donor and acceptor as a dipole-dipole interaction [89]. As the Coulomb interaction causes this energy transfer, FRET is a long-distance mechanism. For the random walk (RW), the hopping rate is computed as the inverse of the mean time interval for a hopping process \(\tau_{\text{hop}}\) between neighboring sites, which gives the rate:  
\[a_{\text{exh,RW}}^{s/t}=\frac{1}{\tau_{\text{hop}}^{s/t}}=\frac{6D^{s/t}}{l^{2 }}\,, \tag{16}\]  
where \(l\) is the next-neighbor distance. Diffusion rates by FRET are modeled with:  
\[a_{\text{exh,F}}^{s/t}=\Gamma\left(\frac{r_{F}}{r_{ij}}\right)^{6} \begin{cases} \exp\left(-\frac{\Delta E_{ij}}{k_{B}T}\right)&\text{for}\quad\Delta E_{ij} \geq 0,\\ 1&\text{for}\quad\Delta E_{ij}\leq 0,\end{cases}  \tag{17}\]  
where \(r_{F}\) is the Forster radius, \(\Gamma\) is the total decay rate, \(\Delta E_{ij}=E_{i}-E_{j}\) is the exciton energy difference between the initial and final configuration and \(r_{ij}\) denotes the distance between two sites. The computational effort of random walk is typically 2-3 orders of magnitude smaller compared to FRET as the number of sites considered for the FRET diffusion increases with \(r_{F}\), while RW diffusion only considers next-neighbors [19]. Especially for a low energetic disorder, RW and FRET give similar results [19], with rising energetic disorder FRET needing to be considered. In case of the triplet diffusion, FRET is a spin-forbidden process and can only occur between phosphorescent dopants within the Forster radius \(r_{F}\). A second mechanism for the diffusion comes into play. In contrast to the Forster rate, the Dexter type energy transfer rises due to the exchange interaction and may occur between next neighbors [92]. Following [22], the Dexter type hopping is modeled as a Miller-Abrahams hopping defined by:  
\[a_{\text{exh,D}}^{t}=a_{\text{D}}\exp\left(-2\gamma r_{ij}\right) \begin{cases} \exp\left(-\frac{\Delta E_{ij}}{k_{B}T}\right)&\text{for}\quad\Delta E_{ij} \geq 0,\\ 1&\text{for}\quad\Delta E_{ij}\leq 0,\end{cases}  \tag{18}\]  
with the exciton energy difference \(\Delta E_{ij}\) of initial state \(i\) and final state \(j\), the Dexter hop rate \(a_{\text{D}}\) in the zero distance limit and the inverse localization length \(\gamma\) of donor and acceptor sites [64]. The triplet binding energy \(E_{\text{B}}^{t}\) is usually assumed to be equal for every site, but can be modified by a disorder following a Gaussian distribution.