Concentration-dependent diffusion of boron and phosphorus in silicon

A calculation is made of the tail of the impurity concentration profile resulting from concentration-dependent diffusion from a constant surface concentration into a semi-infinite medium. The calculation predicts that if the concentration dependence at low impurity concentrations is negligible, the low concentration portion of the doping profile should still take the familiar form,
C = C'_{s} erfc (x/2 D_{i^{\frac{1}{2}}}t^{\frac{1}{2}})
. D
i
is the commonly known diffusion coefficient at low impurity concentrations, while
C'_{s}
is the "apparent" surface concentration.
C'_{s}
depends on the actual surface concentration and also depends on how the diffusion coefficient varies with impurity concentration at high concentrations. It is a constant for a given diffusion system but could be orders of magnitude higher than the actual surface concentration. Empirical data have been obtained for boron and phosphorus diffusions in silicon and found to be in good agreement with this prediction.