A mesh free floating random walk method for solving diffusion imaging problems

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Highlights

  • A stochastic random walk based method for diffusion imaging is developed.

  • A probabilistic representation for mixed boundary value problems in semi-infinite domains is constructed.

  • A logarithmic cost and convergence for the method suggested are shown.

  • Applications in dislocation imaging in semiconductors are suggested.

Abstract

We suggest a new mesh free random walk method for solving boundary value problems in semi-infinite domains with mixed boundary conditions. The method is based on a probabilistic interpretation of the diffusion processes. Our simulations show that the suggested algorithm is extremely efficient for solving diffusion imaging problems, in particular, for calculating the defect contrast in cathodoluminescence (CL) and electron beam-induced current (EBIC) techniques. The method avoids to simulate the long diffusion trajectories. Instead, it exploits exact probability distributions of the first passage and survival probabilities.

Introduction

Mesh-free methods for solving high-dimensional problems are of great interest, especially for domains with complicated shape of boundaries, for unbounded domains, for problems with high solution gradients near the boundaries, etc., where the implementation of a mesh is very complicated and operates with extremely high dimensional matrices. Probabilistic representations for partial differential equations and related stochastic algorithms are well developed mainly for parabolic and elliptic partial differential equations (e.g., see Dynkin, 1965, Sabelfeld, 1991, Friedman, 1976). Due to deep intrinsic relation between diffusion random processes and the parabolic equations the solutions to the relevant boundary value problems are represented in the form of expectations of functionals over trajectories of the random diffusion processes. But of particular interest are special cases where the solution to a PDE can be represented not through the whole continuous trajectory of the diffusion process but through much simpler random events only, like a survival probability in a domain, the first passage time, known as the exit time from a domain, splitting probabilities describing the probabilities to hit different parts of the boundary, the exit positions, and many others. This kind of probabilistic representations for solutions to elliptic and parabolic equations can be used to construct such mesh-free methods, and the most frequently applied stochastic methods are the Random Walk on Spheres (RWS) (Sabelfeld, 1991, Ermakov et al., 1989), and the Random Walk on Boundary (RWB) algorithms (Sabelfeld and Simonov, 1994). Both these methods are mesh-free, the solution at any desired point is obtained by averaging over an ensemble of Markov chains living in a continuous phase space. These methods however are not a perfect choice if one needs the whole solution field as in the case of diffusion imaging, e.g., in the cathodoluminescence (CL) and electron beam induced current (EBIC) semiconductor analysis techniques performed in scanning electron and transmission microscopes (Donolato, 1998, Farvacque and Sieber, 1990, Parish and Russell, 2006). Moreover, these methods fail in many practically interesting cases when on different parts of the boundary, mixed Dirichlet–Neumann–Robin boundary conditions are prescribed.

In this paper we suggest a new floating mesh free random walk on semi-cylinders algorithm. The method is extremely efficient for evaluation of the CL and EBIC signals, and is applied to analyze the minority carrier diffusion and collection in a semiconductor containing a regular array of straight dislocations perpendicular to the junction plane. These contrast signals are calculated as fluxes to the relevant parts of the boundary (EBIC signal), or the concentration of carriers (CL signal) absorbed inside the semiconductor.

Section snippets

Governing equations

Let us first consider the case of one circular dislocation. Thus we deal with a steady state exterior boundary value problem of diffusion from a source point in a rotationally symmetric domain, namely, a half space V bounded by a semi-cylinder of radius R and the plane z=0. The plane boundary of the cylinder coincides with the plane z=0.

The Green function which presents the solution for a source point is defined by the equation Δrφ(r,z)+d2φdz2λ2φ(r,z)=1Dδ(rr0) where Δr is the radial part of

Green’s function representation

To construct a Markov chain which calculates the CL and EBIC signals in the very general case, we derive a Poisson integral relation for the interior Dirichlet problem in a semi-infinite cylinder Δφ(r,ψ,z)λ2φ(r,ψ,z)=0φ=f1(ψ,z)forr=R;φ=f2(r,ψ)forz=0. Using a separation of radial and vertical coordinates we find the Green function representation φ(r,ψ,z)=R02π0f1(η,ζ)[ξG(r,ψ,z;ξ,η,ζ)]|ξ=Rdζdη+02π0Rf2(ξ,η)[ζG(r,ψ,z;ξ,η,ζ)]|ζ=0ξdξdη where the Green function reads G(r,ψ,z;ξ,η,ζ)=12πR2n=0

Floating random walk on semi-cylinders

Now we define a Markov chain which we call a random walk on semi-cylinders. This random walk is constructed for the case of Dirichlet boundary value problems in a half-cylindrical domain V which is bounded by a set of half-cylindrical dislocations of arbitrary cross-sections, both for interior and exterior problems, for the stationary diffusion equation Δφλ2φ=0. The boundary Γ of V is composed by the side surfaces of the dislocations denoted here by Γ1,,Γm. We define also a Γε as an ε

Robin boundary conditions on the end cap

The described algorithm is easily extended to the case when on the plane boundary z=0, the Robin boundary conditions are imposed φ|r=R=f1(ψ,z),r=R,φz+κ2φ=f2(r,ψ)z=0. The Green function representation derived analogously to (13) reads φ(r,ψ,z)=R02π0f1(η,ζ)[ξG(r,ψ,z;ξ,η,ζ)]|ξ=Rdζdη+02π0Rf2(ξ,η)G(r,ψ,z;ξ,η,0)ξdξdη where the Green function is the same, with the only difference that Fm(z,ζ)=2exp(βmz)[βmcosh(βmζ)+κ2sinh(βmζ)]βm(βm+κ2),z>ζ,Fm(z,ζ)=2exp(βmζ)[βmcosh(βmz)+κ2sinh(βmz)]βm(βm+κ2

Acknowledgment

The author was supported by the Russian Science Foundation under Grant 14-11-00083.

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