Discrete stochastic simulation of the electrons and holes recombination in the $\mathrm{2D}$ and $\mathrm{3D}$ inhomogeneous semiconductor

Stochastic models of electron-hole recombination in $\mathrm{2D}$ and $\mathrm{3D}$ inhomogeneous semiconductors based on a discrete cellular automata approach are presented in the paper. These models are derived from a Monte Carlo algorithm based on spatially inhomogeneous nonlinear Smoluchowski equations with the random initial distribution density used to simulate the annihilation of spatially separate electrons and holes in a disordered semiconductor characterized by the heterogeneous properties of the material. Recombination kinetics in different regimes such as a pure diffusion, diffusion in vicinity of tunneling and diffusion in the presence of recombination centers are investigated by a cellular automata simulation. Statistical characteristics of the recombination process (particle concentrations and the radiative intensity) obtained by the cellular automaton models are compared with the theoretically known asymptotics derived for a pure diffusion case. The results obtained for a two-dimensional domain correspond to the theoretical asymptotics, whereas in three-dimensional case, they differ from the exact asymptotics. It is found out by simulations that a spatial electron and hole separation (segregation) occurs under certain conditions on the diffusion and tunneling rates. The electron-hole spatial segregation in $\mathrm{2D}$ and $\mathrm{3D}$ semiconductors is analyzed by using the probability density of the electron-hole separation. In addition, the execution time of the codes implementing the cellular automaton model of the recombination in $\mathrm{2D}$ and $\mathrm{3D}$ semiconductors is studied in dependence on the number of simulated electron-hole pairs and the size of the semiconductor domain. It is shown that the execution time for semiconductors of dimension $d$ is proportional to a polynomial of order $d$.