To control junction depth in transistors, precise quantities of impurities are introduced at relatively shallow depths by ion implantation and diffused 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 of phosphorus in silicon at 1100°C is 6.5 x 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 depth of 1 μm?

Transcribed Image Text:

c(x, t): Q VT Dt exp -X 4Dt