# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 4 Applications
### Impact of Dopant Volume Concentration on Exciton Dynamics

Table 2: Simulation parameters for the exciton dynamics. In Figure 8, the dependence of the average lifetime \(\tau\) and effective diffusion length \(L_{\text{diff,eff}}\) on the dopant volume concentration \(c\) is illustrated for both singlet (a) and triplet excitons (b). The effective exciton diffusion length is determined using:  
\[L_{\text{diff,eff}}^{s/t}=\sqrt{\frac{\sum{(\text{d}L_{i}^{s/t})^{2}}}{N^{s/t}}}\,, \tag{21}\]  
where \(\text{d}L_{i}\) is the Euclidean distance between the generation and final site of exciton \(i\) and \(N^{s/t}\) denotes the number of either singlet or triplet excitons [82]. The presented results show the average effective diffusion lengths of eight configurations simulated for each dopant concentration, while the error bars indicate the standard deviation. For a concentration of \(c=0\) vol%, an average singlet exciton lifetime of \(\tau^{s}=501\pm 24\) ps is obtained, which corresponds to an effective diffusion length of \(L_{\text{diff,eff}}^{s}=7.6\pm 0.1\) nm. As can be seen in Figure 8, both \(\tau^{s}\) and \(L_{\text{diff,eff}}^{s}\) decrease with rising dopant concentrations. This is caused by the increased intersystem crossing of singlets into triplets within simulation times below the decay time. Effective triplet diffusion lengths of \(L_{\text{diff,eff}}^{t}=101\pm 3\) nm are obtained for a low dopant concentration, whereas for increasing dopant concentrations, the diffusion length decreases down to \(L_{\text{diff,eff}}^{t}\approx 65\) nm for \(c=10\) vol%. The average lifetime of triplets decreases from \(5.5\pm 0.1\) s for \(c=0.2\) vol% to \(2.3\pm 0.1\) s at \(c=10\) vol%. This strong decrease in both effective diffusion length and lifetime is caused by the higher decay rates of triplets within the radius \(r_{\text{ISC}}\) around a dopant. With rising volume concentration, the influence of dopants becomes increasingly visible. Figure 9 depicts the dependence of the percentage of singlets undergoing intersystem crossing and triplets annihilated by TTA on the dopant concentrations. Multiple counting of singlets back-converted from triplets upon a TTA event is omitted. For a dopant concentration of \(c=0.2\) vol%, the proportion of converted triplets to optical generated singlets is 30%. For higher concentrations, more than 80% of singlets are converted into triplets before decaying. Figure 9 displays the percentage of triplets annihilated upon TTA. Starting with a ratio of 8.1% at \(c=0.2\) vol%, the proportion of triplet-triplet annihilation decreases to a value of 2.3% for \(c=10\) vol%. Triplet-triplet annihilation becomes unlikely for higher concentrations. The decrease in TTA can be related to the decreasing lifetime of triplets and the reduced diffusion length. Thus, the probability of triplets to meet each other decreases tremendously. This is explicitly caused by the lowering of the spin-selection rules within the strong spin-orbit coupling domains of the dopants. Higher triplet densities would significantly increase the impact of TTA. In Figure 9a, the effective exciton lifetime defined by:  
\[\tau_{\text{eff}}=\text{ISC}\times\tau^{t}+(1-\text{ISC})\times\tau^{s} \tag{22}\]  
is visualized. For a low dopant concentration, the \(\tau_{\text{eff}}\) increases strongly as most of the excitons are not converted into triplets. With rising \(c\), the effective lifetime rises and reaches its maximum at \(c=1\) vol%. Beyond the maximum, the effective lifetime decreases and saturates at a value of \(\tau_{\text{eff}}\approx(2.09\pm 0.06)\)\(\mu\)s. A similar trend is observed by [PERSON] et al. for doped P3 HT:PCBM solar cells [24]. The results for exciton diffusion lengths and lifetime conform with the experimental values within ideal material setups, documented in [63; 64; 86; 99]. Similar to the trend observed in [24; 26], the lifetime of excitons decreases with dopant concentration. Firstly, for a small dopant concentration, the lifetime increases tremendously in comparison to singlet lifetimes caused by the conversion of singlets to the long-living triplets.