# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Particles and Transitions
#### 3.3.2 Excitons  
An exciton is an intramolecular excited state between a Coulomb-bound electron-hole pair [64]. Compared to the weakly-bound Wannier-Mott excitons present in inorganic semiconductors, strongly-bound Frenkel excitons are formed in organic materials due to the low dielectric constant [3; 81]. The excited state is a singlet (triplet) state, if the spins of the electron and the hole are anti-parallel (parallel). We will now describe all physical rates modeled for both singlets and triplets, as well as the conversion rates between these excitons. For a detailed discussion on the photophysical processes, please refer to [64; 82]. A schematic overview of the rates and the corresponding energetic levels are depicted in Figure 5. The singlet (blue) and triplet excitons (red), residing at the energetic levels \(S_{1}\) and \(T_{1}\), respectively, as well as the separated electron-hole pair forming a charge-transfer (CT) state and the ground state \(S_{0}\) are visualized. Singlet excitons can be generated upon light absorption. Absorption of a photon excites one electron, resulting in a singlet exciton. The optical generation profile of singlets is modeled using a transfer matrix model (TMM) [83; 84; 85]. The TMM is used to calculate the generation profile in the active layer under an AM1.5 illumination spectrum. For every layer in penetration direction \(z\), a constant generation rate \(a_{\text{gen}}^{s}\) is obtained. As the TMM does not provide the generation profile perpendicular to the penetration direction, the \(x\) and \(y\) position of the generated singlet is chosen randomly [10; 13]. The generation of both singlets and triplets can occur by electrical excitation. The rate \(a_{\text{exg}}^{s/t}\) for electrical generation of a singlet/triplet is enabled if two charges of different polarity reside at neighboring sites and if the respective spins, a constant property of the charge carriers, show a singlet/triplet configuration. Depending on the origin of charges, we assign their spin states following a distinguished procedure. If a charge is injected, we allocate the spin following the expected singlet  
Figure 5: (Left) Bulk-heterojunction including dopant sites with a volume concentration of 0.8 vol %. (Right) Jablonski diagram illustrating the exciton transitions. All processes considering triplet excitons are visualized in red color, while singlet transitions are highlighted in blue. The diagram shows the lowest excited singlet (\(S_{1}\)) and lowest excited triplet state (\(T_{1}\)), the singlet ground state (\(S_{0}\)) and charge transfer state (\(CT\)). Our excitonic model consists of the optical singlet exciton generation (\(a_{\text{gen}}^{s}\)) upon the absorption of an incident photon \(\gamma\), singlet/triplet exciton separation (\(a_{\text{exg}}^{s/t}\)), electrical generation (\(a_{\text{exg}}^{s/t}\)) and decay (\(a_{\text{exd}}^{s/t}\)), triplet-triplet annihilation (\(a_{\text{TTA}}\)) and intersystem crossing (\(a_{\text{ISC}}\)). Two triplet excitons can annihilate with a rate \(a_{\text{TTA}}\) forming an intermediate state \(TT^{*}\), which relaxes to either the lowest singlet \(S_{1}\) or lowest triplet state \(T_{1}\). to triplet ratio of 1:3. If an exciton dissociates, the spin states of the charges reflect the spins of the singlet or triplet excitation. We use the convention that holes always have a spin with eigenvalue \(-\hbar/2\) (spin-down), while the spin of the electrons is assigned such that the above criteria are fulfilled. Following [PERSON] et al. [23], the electrical generation rate can be calculated as a hopping process of the electron on a hole-occupied neighboring site using the Miller-Abrahams equation (Equation (11)) [74]. In comparison to [PERSON] et al., our model only considers electron hops for the electrical exciton generation. The energy difference for the singlet/triplet generation is given by:  
\[\Delta E^{s/t}=E_{\text{B}}^{s/t}+\Delta E_{\text{LUMO}}-E_{\text{CT,B}}\,, \tag{15}\]  
where \(E_{\text{B}}^{s/t}\) denotes the exciton binding energy.