Numerical modeling of a moving boundary diffusion/drift problem

We have modeled numerically the redistribution by diffusion and drift of impurities during epitaxial growth of semiconductors. We use a Crank-Nicholson scheme with dynamically adjusted time increment. The coupling between the redistribution of the charged impurities and the electric field is accounted for by solving the nonlinear Shockley-Poisson equation for an arbitrary doping profile by means of a quasi-linearization scheme. The combination of large gradients and a moving boundary necessitates a dynamically adjusted nonuniform mesh in the finite difference schemes. Except for some occasionally occurring spurious ripples, which are accounted for by stability arguments, the results are physically real and can explain some experimentally observed features[1].