On the Theory of Atomic Diffusion after Ion Implantation

An analytical solution to the problem of atomic diffusion after ion implantation (taking into account the drift of impurity atoms) in a semi-infinite medium characterized by a nonuniform distribution of nonequilibrium vacancies is obtained using asymptotic methods in the theory of differential equations. This solution can be used at annealing times that are longer than the setting time for the steady-state nonuniform distribution of vacancies originating from the implantation region, but correspond to a diffusion length that is shorter than the characteristic scale of variation of the impurity diffusion coefficient. This work analyzes the applicability of this solution in the general form. It has been compared to the results of numerical calculations for an exponential coordinate dependence of the vacancy density with the drift of atoms, which results from the nonuniform distribution of nonequilibrium vacancies, either taken into account or neglected. The concentration profiles of implanted aluminum in silicon at the initial stage of diffusion are calculated within the proposed approach.