Coupled-flux nucleation modeling of oxygen precipitation in silicon

Experimental data for oxygen precipitation densities in Czochralski-grown silicon following multistep annealing treatments are compared with predictions from a coupled-flux model for time-dependent nucleation. This is a more correct model for diffusion-controlled nucleation processes than is the classical theory of nucleation since it directly couples the two stochastic fluxes of interfacial attachment and long-range diffusion. Quantitative agreement is obtained between the measured and calculated densities for nucleation temperatures greater than 650°C. Good agreement is obtained for lower temperatures if the oxygen diffusion rate is taken to be larger than is predicted from high-temperature diffusion data. The fit values for the diffusion coefficient from the nucleation data are in good agreement with recent results from dislocation-unlocking experiments. The oxygen loss calculated by coupled-flux nucleation and diffusion-limited growth agrees with the experimental observations. Classical theory nucleation calculations predict a much greater oxygen loss, signaling the failure of the theory to correctly treat nucleation when long-range diffusion is important, true in most solid-state precipitation processes.