Asymptotic estimates of diffusion times for rapid thermal annealing

Temperature profiles for rapid thermal annealing of ion implanted material are analyzed using asymptotic methods. Although only rapid thermal annealing is discussed, these methods are also applicable to many other annealing processes. Formulas for effective diffusion distance and effective annealing time are given which correct for the deviations of actual annealing profiles from ideal annealing profiles. The three major deviations considered are (1) ramp terms, (2) rising plateaus, and (3) plateau overshoots. The analysis shows that even relatively small deviations can have sizable effects.