# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Particles and Transitions
#### 3.3.2 Excitons
The Boltzmann factor vanishes in the case of no energetic disorder in the binding energy. Both transfer mechanisms are shown schematically in Figure 6.  
To extract electrical energy from the absorbed photons in organic solar cells, singlets need to be dissociated. This process is only possible if singlets reach an acceptor-donor interface. The dissociation process is reported to happen on extremely fast timescales \(\tau_{\text{eos}}^{s}\) in BHJs [93]. A separation rate \(a_{\text{eos}}^{s}=1/\tau_{\text{eos}}^{s}\) is enabled if the singlet is located at an interface. Dissociation of a triplet exciton is interpreted as an electron hopping from its current position towards an unoccupied neighboring site [21; 22; 23]. The separation rate \(a_{\text{eos}}^{t}\) is calculated with the Miller-Abrahams rate using an energy difference of \(\Delta E=\Delta E_{\text{LUMO}}+E_{\text{CT,B}}-E_{\text{B}}^{t}\). The dissociation is probable if the state following a dissociation process is energetically lower than the triplet binding energy. Regarding high spin-orbit coupling, intersystem crossing from a singlet to a triplet state is possible. The reverse process is not considered here. In our model, a transition from singlets to triplets is only possible within a certain radius \(r_{\text{ISC}}\) around a dopant molecule. The radius of intersystem crossing is a material dependent property. If an exciton resides at a site within \(r_{\text{ISC}}\), the intersystem crossing rate \(a_{\text{ISC}}\) is enabled. The intersystem crossing rate is given by:  
\[a_{\text{ISC}}=a_{\text{ISC}}^{0}\exp\left(-\frac{2r_{i\to d}}{r_{\text{ISC}}} \right)\,, \tag{19}\]  
with an intersystem crossing frequency \(a_{\text{ISC}}^{0}\). The intersystem crossing decreases exponentially with the distance \(r_{i\to d}\) to a dopant site \(d\). If the intersystem crossing occurs, the singlet is converted into a triplet exciton, residing at the corresponding site. Figure 6: Schematic diagram illustrating the hopping processes of singlet (blue spheres) and triplet (red spheres) excitons. The black dots represent the organic material, while gray dots label dopant sites. FRET is shown for a triplet exciton hopping between dopant sites within the Forster radius \(r_{F}\). The Dexter type energy transfer is possible between all adjacent sites. The intersystem crossing is visualized for a singlet exciton in proximity to the dopant as a conversion of a singlet to a triplet exciton. With rising concentration of triplets, another undesired process occurs, the triplet-triplet annihilation (TTA). If two triplet excitons meet, they either create a singlet or a triplet exciton [20; 21; 94; 95]. The spin state of the resulting excitons depends on the spin angular momentum of the two involved triplet excitons. If a triplet exciton resides at a neighboring site or two involved triplet excitons reside at two dopant sites within the Forster radius, TTA is enabled. As explained before, TTA is interpreted as a triplet exciton hopping on a site occupied by a second triplet. If this transition is performed, at least one triplet exciton is removed. If the TTA process is Dexter-mediated, with a ratio of 1:3, the particle at the final site is either a singlet or a triplet, respectively. In the case of the Forster mediated TTA, one triplet is excited to a higher triplet level \(T_{\mathrm{n}}\), followed by a relaxation to the lowest triplet exciton level \(T_{1}\). We assume the relaxation to happen immediately. The second triplet decays to the singlet ground state.