# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 3 kMC Framework for Organic Materials and Devices
### Modeling of the Organic Material  
#### 3.1.1 Voronoi Tessellation  
With the rising amount of computational power and powerful clusters, investigations on the detailed structure of organic materials and their influence on the physical properties using molecular dynamics (MD) have gained rising interest in the community [40; 41; 42]. This raises the demand of a generalized lattice representation for kMC studies [6; 43; 44; 45].  
For a representative model of the active organic layer, the use of the Voronoi tessellation generalizes the cubic lattice model [39; 46]. Using the Voronoi cell library of [PERSON] [47], next-neighbors of each lattice site, as well as the distance and all face-areas are computed. In Figure 2, the procedure is shown for the cubic lattice (left) and the non-cubic, Voronoi set (right) in a simple 2D representation. Starting from an arbitrary grid point, the perpendicular bisectors to all neighboring sites in proximity are computed. The algorithm obtains the volume closest to the considered starting point, named the Voronoi cell, as known from the Wigner-Seitz cell in periodic crystals. All neighboring sites sharing one face with the considered Voronoi cell are taken as next-neighbors. The knowledge of these is fundamental for the computation of all physical rates.  
Comparing the cubic with the off-lattice approach, some obvious differences and their importance are to be highlighted. Due to the spatial disorder of the sites, the number of next-neighbors increases. Moreover, the major drawback of an equidistant spacing between lattice points in the cubic model is overcome with the use of the Voronoi set [48]. The local distribution of lattice sites can be imported from MD simulations using the center of masses of the molecules, providing a sophisticated multi-scale simulation method comparable to [42; 43]. In the case of available quantum chemical simulations, the localized quantum state could directly be used as the input of the sites. Herewith, the information depth of the structural alignment of, e.g., polymers or small molecules down to the nanoscale can be represented on a mesoscopic level. Besides, amorphous domains in both polymers and small molecules can be explicitly investigated, which makes the study of the physical properties more realistic. In the case of the exciton diffusion (see Section 3.3.2), non-nearest neighbors are required. The Voronoi tessellation supports a fast iteration method to obtain all neighbors within a certain radius around the centroid.  
Existing off-lattice models take into account all adjacent molecules within a certain radius [43; 49]. Depending on the density of sites, this may cause a strong increase in simulation time. Using the Voronoi tessellation, only adjacent sites are considered, which keeps the computation time feasibly low. In the case of a strong disorder, a common radius might lead to the case that certain sites are  
Figure 2: Visualization of the next neighbor generation in a cubic lattice (left) and a Voronoi set (right). For most physical processes, only the nearest-neighbors, highlighted in red color, are considered. The blue lattice sites do not define the Voronoi cell; thus, they are not labeled as next neighbors.  
fully disconnected from the surrounding material. This is avoided with the Voronoi tessellation. The tessellation furthermore allows the computation of local current and charge densities as a well-defined area of the faces, and the volume of the Voronoi cells is obtained [39]. Besides all the advantages, it still needs to be considered that the cell boundaries are purely based on the distance between centroids and do not contain any physical information, which, depending on the modeled system, has to be evaluated carefully for the computation of local quantities.