# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 4 Applications
### Impact of Dopant Volume Concentration on Exciton Dynamics  
Using the presented kMC framework, we study the impact of the dopant concentration on the exciton dynamics in an organic bulk material. We focus on the diffusion length and lifetime of excitons and analyze the number of singlets converted into triplets. With this simulation, which is rather theoretical, we aim to mimic the exciton dynamics and interplay of excitons within a single organic material without any interfaces, defects and contacts. As for the charge transport simulation, a single material cubic lattice of \(50\,\mathrm{nm}\times 50\,\mathrm{nm}\times 50\,\mathrm{nm}\) with a lattice constant of \(1\,\mathrm{nm}\) and periodic boundaries in all spatial dimensions is generated. Rates for Dexter energy transfer and random walk are computed for the six nearest-neighbors, while FRET is considered for all neighbors within the Forster radius. Simulations are executed for different dopant volume concentrations varying from \(0\,\mathrm{vol}\%\) up to \(10\,\mathrm{vol}\%\). As no contacts are included, charges cannot be injected or collected. Due to the lack of interfaces, exciton splitting cannot occur. Singlet excitons are generated with a constant optical generation rate \(a_{\mathrm{gen}}^{s}=$7\,\mathrm{s}^{-1}\mathrm{nm}^{-3}$\). Triplet excitons can be induced upon intersystem crossing close to the phosphorescent dopants. In Table 2, the simulation parameters used are summarized. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameter** & **Value** \\ \hline Gaussian energetic disorder & \(\sigma\) & \(0.1\,\mathrm{eV}\) \\ Attempt-to-escape frequency & \(a_{0}\) & \(3\times 10^{12}\,\mathrm{s}^{-1}\) \\ Inverse localization length & \(\gamma\) & \(2\,\mathrm{nm}^{-1}\) \\ Charge density & \(n\) & \(3.2\times 10^{16}\,\mathrm{cm}^{-3}\) \\ Temperature & \(T\) & \(298\,\mathrm{K}\) \\ Simulation time & \(t_{\mathrm{sim}}\) & \(6\,\mathrm{\mu s}\) \\ Applied field & \(F\) & \(5\times 10^{5}\)–\(5\times 10^{6}\,\mathrm{V}\,\mathrm{cm}^{-1}\) \\ Dielectric constant & \(e_{r}\) & \(3.5\) \\ Intersite distance & \(a\) & \(1\,\mathrm{nm}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for mobility considering polymer chains. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Description** & **Parameter** & **Value** & **Ref.** \\ \hline \multicolumn{4}{l}{**Common Parameters**} \\ \hline Phosphorescent dopant concentration & \(c\) & \(0\,\mathrm{vol}\%-10\,\mathrm{vol}\%\) & - \\ Energetic disorder excitons & \(\sigma^{s/t}\) & \(0\,\mathrm{eV}\) & - \\ Inverse localization length & \(\gamma\) & \(2\,\mathrm{nm}^{-1}\) & [11; 12] \\ Simulation time & \(t_{\mathrm{sim}}\) & \(1\,\mathrm{ms}\) & - \\ \hline \multicolumn{4}{l}{**Exciton parameters**} \\ \hline Optical generation rate & \(a_{\mathrm{gen}}^{s}\) & \(7\,\mathrm{s}^{-1}\,\mathrm{nm}^{-3}\) & - \\ Hopping rate (RW) & \(a_{\mathrm{ex,RW}}^{s}\) & \(2.0\times 10^{10}\,\mathrm{s}^{-1}\) & [13] \\ Singlet decay rate & \(a_{\mathrm{ex,d}}^{s}\) & \(2.0\times 10^{9}\,\mathrm{s}^{-1}\) & [12; 17; 97] \\ Singlet binding energy & \(E_{\mathrm{B}}^{t}\) & \(0.5\,\mathrm{eV}\) & - \\ ISC radius phosphorescent dopant & \(r_{\mathrm{INC}}\) & \(2\,\mathrm{nm}\) & - \\ ISC rate & \(a_{\mathrm{ISC}}^{0}\) & \(1.0\times 10^{12}\,\mathrm{s}^{-1}\) & - \\ Forster radius dopant & \(r_{\mathrm{F,dopant}}\) & \(1.5\,\mathrm{nm}\) & [98] \\ Dexter rate & \(a_{\mathrm{D}}\) & \(2.9\times 10^{8}\,\mathrm{s}^{-1}\) & [22] \\ Decay rate dopant & \(a_{\mathrm{ex,dopant}}^{t}\) & \(1.0\times 10^{6}\,\mathrm{s}^{-1}\) & - \\ Decay rate host material & \(a_{\mathrm{ex,host}}^{t}\) & \(1.0\times 10^{6}\,\mathrm{s}^{-1}\) & [24; 63] \\ Triplet binding energy & \(E_{\mathrm{B}}^{t}\) & \(1.0\,\mathrm{eV}\) & - \\ TTA rate Dexter type & \(a_{\mathrm{D,TTA}}\) & \(2.9\times 10^{8}\,\mathrm{s}^{-1}\) & [22] \\ TTA Forster radius & \(r_{\mathrm{F,TTA}}\) & \(1.5\,\mathrm{nm}\) & [98] \\ \hline \hline \end{tabular}
\end{table}