Diffusion of a mono-energetic implanted species with a Gaussian profile

The implanted profile in an isotropic substrate of a mono-energetic ion species is usually very near a Gaussian profile. An exact solution to the time-dependent Fick diffusion equation of an initially Gaussian profile is presented. This solution is a general one also covering the diffusion within the two limiting cases usually considered in solutions to the Fick equation, viz. a perfect sink at the surface and a perfectly reflecting surface plane at the surface. An analysis of the solutions for these two cases shows that at small diffusion times the main effect of annealing is a nearly symmetric broadening of the implanted profile. At the origin and for longer diffusion times the profile deviates significantly from Gaussian. A review is also given of past attempts to extract diffusion coefficients by fitting experimental data to approximate equations based on simplified initial profiles.