When processing silicon single crystals for microelectronics applications, precise quantities of impurities are often introduced at relatively shallow depths by ion implantation and dif fused into the silicon substrate in a subsequent thermal treatment. This can be approximated as a finite source diffusion problem. Applying the appropriate boundary conditions, the solution to Fick's second law under these conditions is


Where Q is the initial surface concentration with units of atoms/cm2.
Assume that we implant 1014 atoms/cm2 of phosphorus at the surface of a silicon wafer with a background boron concentration of 1016 atoms/cm3 and this wafer is subsequently annealed at 1100 ˚C. The diffusion coefficient (D) of phosphorus in silicon at 1100 ˚C is 6.5 × 10-13 cm2/s
(a) Plot a graph of the concentration c (atoms/cm3) versus x (cm) for anneal times of 5 minutes, 10 minutes, and 15 minutes.
(b) What is the anneal time required for the phosphorus concentration to equal the boron concentration at a dept h of 1 μm?