# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 4 Applications
### Charge Carrier Transport through Polymer Chains  
The charge carrier mobility is a crucial figure of merit for the applicability of organic semiconductors. While for some devices such as OLEDs, a low mobility is critical to optimize the performance, OSCs and OTFTs demand a high mobility. In semiconducting polymers, charge carrier mobility strongly depends on the morphology [54; 55], but its interplay is not well understood [52; 96]. Investigating the impact of different transport mechanisms, such as intrachain hopping, on carrier mobility can help the development of organic materials and improve device performance. A single material cubic lattice of \(50\,\mathrm{nm}\times 50\,\mathrm{nm}\times 50\,\mathrm{nm}\) with periodic boundary conditions in all three spatial dimensions is used for the following investigation of the charge carrier mobility under the influence of intrachain hopping. For the energetic disorder, a Gaussian density of states with \(\sigma=0.1\,\mathrm{eV}\) is used. The Miller-Abrahams rate given in Equation (11) is used for the hopping process with an attempt-to-escape frequency \(a_{0}=3\times 10^{12}\,\mathrm{s}^{-1}\) for inter-chain transitions and an inverse localization length of \(\gamma=2\,\mathrm{nm}^{-1}\). Charges are allowed to hop to the six nearest neighbors on the cubic lattice. We run the simulation with four electrons placed in the simulation box resulting in a charge density of \(n=3.2\times 10^{16}\,\mathrm{cm}^{-3}\). This charge density is constant throughout the simulation because of the periodic boundary conditions. To account for the faster intrachain hopping, an attempt-to-escape frequency of \(a_{\mathrm{chain}}=m\times a_{0}\), \(m\in\mathbb{R}^{+}\), is used to calculate the rates. Further simulation parameters are summarized in Table 1. For each electric field \(F\), three simulation runs for a simulation time of \(6\,\mathrm{\mu s}\) are performed. The charge carrier mobility in the field direction is calculated by:  
\[\mu_{z}=\frac{\langle\vec{v}\cdot\vec{F}\rangle}{F^{2}}=\frac{d_{z}}{t_{ \mathrm{sim}}F}\,, \tag{20}\]with the electric field \(\vec{F}=F\vec{e_{z}}\) in the \(z\)-direction \(\vec{e_{z}}\) and the average carrier velocity \(\vec{v}\) given by the distance \(d_{z}\) traveled during the simulation time \(t_{\text{sim}}\) in the field direction. The results represent the mean value and the standard deviation of the mobility. For this study, different morphologies with chain length \(l_{c}=20\,\text{nm}\) are generated. We study both Gaussian distributed chains with \(\eta=0\) and worm-like chains with \(\eta=0.7,0.9\) oriented in the \(z\)-direction. The multiplication factor \(m\) is varied between one and \(100\). A factor of \(m=1\) means that hopping along polymer chains is not preferred and the chain morphology does not have any impact. Thus, the blue curve in Figure 7a is a reference for comparing the impact of intrachain hopping. Because of the relatively high energetic disorder of \(\sigma=0.1\,\text{eV}\), the carriers are slowed down due to the high energetic barriers between sites. At low fields, this slowing effect is dominant, and transport is mainly determined by diffusion. For higher fields, the impact of the energetic barriers becomes weaker, and transport is dominated by the drift field. For high fields, the carrier velocity saturates, and the mobility decreases with \(1/F\). Figure 7a shows the carrier mobility for completely undirected Gaussian chains. For higher intrachain hopping rates, the mobility increases slightly, but when increasing intrachain hopping rates by a factor of \(100\), mobility only increases by a factor of \(2.2\). When including very fast hopping along undirected polymer chains, only a very small amount contributes to the carrier mobility along the field direction. The limiting factor for carrier mobility is the slower inter-chain hopping. Polymer chain orientation the in field direction, on the other hand, shows a larger influence on carrier mobility.