Random walk on spheres method for solving anisotropic drift-diffusion problems

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Anisotropic Diffusion ◽  
Stochastic Simulation Algorithm ◽  
Local Variation ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Mesh Free ◽  
Electron Beam Induced Current ◽  
Defects In Semiconductors ◽  
Diffusion Problems

AbstractWe suggest a random walk on spheres based stochastic simulation algorithm for solving drift-diffusion-reaction problems with anisotropic diffusion. The diffusion coefficients and the velocity vector vary in space, and the size of the walking spheres is adapted to the local variation of these functions. The method is mesh free and extremely efficient for calculation of fluxes to boundaries and the concentration of the absorbed particles inside the domain. Applications to cathodoluminescence (CL) and electron beam induced current (EBIC) methods for the analysis of dislocations and other defects in semiconductors are discussed.

Random walk on spheres method for solving anisotropic drift-diffusion problems

2018 ◽  
Vol 24 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Anisotropic Diffusion ◽  
Stochastic Simulation Algorithm ◽  
Local Variation ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Mesh Free ◽  
Electron Beam Induced Current ◽  
Defects In Semiconductors ◽  
Diffusion Problems

Abstract We suggest a random walk on spheres based stochastic simulation algorithm for solving drift-diffusion-reaction problems with anisotropic diffusion. The diffusion coefficients and the velocity vector vary in space, and the size of the walking spheres is adapted to the local variation of these functions. The method is mesh free and extremely efficient for calculation of fluxes to boundaries and the concentration of the absorbed particles inside the domain. Applications to cathodoluminescence (CL) and electron beam induced current (EBIC) methods for the analysis of dislocations and other defects in semiconductors are discussed.

Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems

2017 ◽  
Vol 23 (3) ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Walk On Spheres Algorithm

AbstractWe suggest in this paper a Random Walk on Spheres (RWS) method for solving transient drift-diffusion-reaction problems which is an extension of our algorithm we developed recently [

A random walk on small spheres method for solving transient anisotropic diffusion problems

2019 ◽  
Vol 25 (3) ◽  
pp. 271-282
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Anisotropic Diffusion ◽  
First Passage Time ◽  
Three Dimensional ◽  
Passage Time ◽  
Exit Point ◽  
Stochastic Algorithm ◽  
Probability Densities ◽  
Spatially Varying ◽  
Diffusion Problems

Abstract A meshless stochastic algorithm for solving anisotropic transient diffusion problems based on an extension of the classical Random Walk on Spheres method is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have derived approximations of the probability densities for the first passage time and the exit point on a small sphere. The method can be conveniently applied to solve diffusion problems with spatially varying diffusion coefficients and is simply implemented for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We present some simulation results in the case of cathodoluminescence and electron beam induced current in the vicinity of a dislocation in a semiconductor material.

Random walk on rectangles and parallelepipeds algorithm for solving transient anisotropic drift-diffusion-reaction problems

2019 ◽  
Vol 25 (2) ◽  
pp. 131-146 ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Three Dimensional ◽  
Passage Time ◽  
Diffusion Reaction ◽  
Exit Point ◽  
Drift Diffusion ◽  
Continuity Equations ◽  
Reaction Equations ◽  
Reaction Problem

Abstract In this paper a random walk on arbitrary rectangles (2D) and parallelepipeds (3D) algorithm is developed for solving transient anisotropic drift-diffusion-reaction equations. The method is meshless, both in space and time. The approach is based on a rigorous representation of the first passage time and exit point distributions for arbitrary rectangles and parallelepipeds. The probabilistic representation is then transformed to a form convenient for stochastic simulation. The method can be used to calculate fluxes to any desired part of the boundary, from arbitrary sources. A global version of the method we call here as a stochastic expansion from cell to cell (SECC) algorithm for calculating the whole solution field is suggested. Application of this method to solve a system of transport equations for electrons and holes in a semicoductor is discussed. This system consists of the continuity equations for particle densities and a Poisson equation for electrostatic potential. To validate the method we have derived a series of exact solutions of the drift-diffusion-reaction problem in a three-dimensional layer presented in the last section in details.

Random walk on spheres algorithm for solving steady-state and transient diffusion-recombination problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Steady State ◽  
Coupled Systems ◽  
Diffusion Reaction ◽  
Stochastic Algorithm ◽  
Monte Carlo Algorithms ◽  
Coupled Diffusion ◽  
Mesh Free ◽  
Integral Characteristics ◽  
Transport Problems

Abstract We further develop in this study the Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion-recombination equations first suggested in our recent article [K. Sabelfeld, First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations, Appl. Math. Lett. 88 2019, 141–148]. The random walk on spheres process mimics the isotropic diffusion of two types of particles which may recombine to each other. Our motivation comes from the transport problems of free and bound exciton recombination. The algorithm is based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. Therefore, the method is mesh free both in space and time. In this paper we implement the RWS algorithm for solving the diffusion-recombination problems both in a steady-state and transient settings. Simulations are compared against the exact solutions. We show also how the RWS algorithm can be applied to calculate exciton flux to the boundary which provides the electron beam-induced current, the concentration of the survived excitons, and the cathodoluminescence intensity which are all integral characteristics of the solution to diffusion-recombination problem.

Probability distribution of the life time of a drift-diffusion-reaction process inside a sphere with applications to transient cathodoluminescence imaging

2018 ◽  
Vol 24 (2) ◽  
pp. 79-92 ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Anastasiya Kireeva
Keyword(s):  
Broad Class ◽  
Life Time ◽  
Stochastic Simulation Algorithm ◽  
Reaction Process ◽  
Diffusion Reaction ◽  
Threading Dislocations ◽  
Time Behavior ◽  
Drift Diffusion ◽  
Robin Boundary ◽  
Cathodoluminescence Imaging

Abstract Exact representations for the probability density of the life time and survival probability for a sphere and a disc are derived for a general drift-diffusion-reaction process. Based on these new formulas, we suggest an extremely efficient stochastic simulation algorithm for solving transient cathodoluminescence (CL) problems without any mesh in space and time. The method can be applied to a broad class of drift-diffusion-reaction problems where the time behavior of the absorbed material is of interest. The important advantage of the method suggested is the ability to incorporate local inclusions like dislocations, point defects and other singular folds and complicated structures. General Robin boundary conditions on the boundary are treated in a probabilistic way. The method is tested against exact solutions for a series of examples with bounded and unbounded domains. An application to the dislocation imaging problem, which includes thousand threading dislocations, is given.

Random walk on semi-cylinders for diffusion problems with mixed Dirichlet–Robin boundary conditions

2016 ◽  
Vol 22 (2) ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Boundary Conditions ◽  
Diffusion Imaging ◽  
Boundary Structure ◽  
Robin Boundary Conditions ◽  
Stochastic Simulation Algorithms ◽  
Different Types ◽  
Electron Beam Induced Current ◽  
Robin Boundary ◽  
Diffusion Problems

AbstractWe suggest random walk on semi-infinite cylinders methods for solving interior and exterior diffusion problems with different types of boundary conditions which include mixed Dirichlet, Neumann, and Robin boundary conditions on different parts of the boundary. Based on probabilistic interpretation of the diffusion process, stochastic simulation algorithms take into account specific features of each boundary condition to optimally adjust the Markov chain distribution on the relevant boundary parts. In contrast to the conventional direct trajectory tracking method, the new method avoids to simulate the diffusion trajectories. Instead, it exploits exact probabilities of different events like the first passage, splitting, and survival probabilities inside the semi-infinite cylinders, depending on the domain and its boundary structure. Applications to diffusion imaging methods like the cathodoluminescence (CL) and electron beam induced current (EBIC) semiconductor analysis techniques performed in scanning electron and transmission microscopes, are discussed.

Random walk on spheres method for solving drift-diffusion problems

2016 ◽  
Vol 22 (4) ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Laplace Equation ◽  
Constant Velocity ◽  
Mean Value ◽  
Drift Diffusion ◽  
Spherical Mean ◽  
Classical Integral ◽  
Von Mises ◽  
Velocity Drift ◽  
Diffusion Problems

AbstractThe well-known random walk on spheres method (RWS) for the Laplace equation is here extended to drift-diffusion problems. First we derive a generalized spherical mean value relation which is an extension of the classical integral mean value relation for the Laplace equation. Next we give a probabilistic interpretation of the kernel. The distribution on the sphere generated by this kernel is then related to the von Mises–Fisher distribution on the sphere which can be efficiently simulated. The rigorous expressions are given for the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion problems with arbitrary varying drift velocity vector. Applications to cathodoluminescence and EBIC imaging of defects and dislocations in semiconductors are discussed.

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