Stochastic Simulation Algorithms for Solving Transient Anisotropic Diffusion-recombination Equations and Application to Cathodoluminescence Imaging

A meshless Random Walk on arbitrary parallelepipeds simulation algorithm is developed and implemented for solving transient anisotropic diffusion-reaction equations. In contrast to the conventional Feynman-Kac based algorithm the suggested method does not use small time step simulations of the relevant diffusion processes. Instead, exact simulation of large random jumps over a set of appropriately constructed parallelepipeds in the domain is carried out. This decreases the cost of simulations considerably especially for domains with complicated boundary shape. Application to the problem of time-resolved cathodoluminescence intensity calculations for semiconductor materials with a set of threading dislocations is given. Important issues are the construction of an efficient sampling method from the first passage time density and the position distribution on the surface of an arbitrary parallelepiped. We combine a rejection algorithm and a probability density tabulation approach to construct optimal sampling methods from different densities including the random time a particle spends in a parallelepiped before it is absorbed inside it. We present in the last section results of computer simulation for the evaluation of the exciton flux to dislocations and a plane substrate, the cathodoluminescence intensity for threading dislocations imaging, and the concentration of the survived excitons. In addition, to validate the developed algorithms we have compared the computer simulations with the exact results, and obtained a perfect agreement.