Semiconductor devices are prepared by doping silicon to create the source, gate, and drain regions used in a typical transistor. This doping is done primarily by diffusion processes and occurs in two steps, predeposition and drive-in. In the predeposition stage we expose the pure silicon substrate to the dopant, assuming that the final doped layer will be much thinner than the substrate. We also assume a constant value of the diffusivity, \(D_{d s}\), and that we expose the substrate to a constant surface concentration of dopant, \(c_{d o}\).

a. Determine the dopant concentration profile in the silicon substrate as a function of time.

b. How much dopant was deposited per unit area of substrate?

(At \(900^{\circ} \mathrm{C}\), the diffusivity of boron and phosphorous in silicon is about \(1.5 \times 10^{-19} \mathrm{~m}^{2} / \mathrm{s}\) and the solubility of boron and phosphorous in silicon is about \(3.7 \times 10^{26}\) atoms \(/ \mathrm{m}^{3}\) and \(6 \times 10^{26}\) atoms \(/ \mathrm{m}^{3}\) respectively).