Ion implantation is used to create a phosphorous$-$rich $(n-$type$)$ surface layer in one side of a $p-$tipe

silicon wafer $1mm$ thick. For the purposes of this problem, we'll assume the resulting phosphorous$-$

rich layer has uniform phosphorous concentration ${10}^{21}\frac{1\text{a}\text{t}\text{o}\text{m}\text{s}}{c{m}^3}$ from the surface to a depth of $0\mathrm{.}1m.$ The

wafer begins with boron doping $(p-$type$)$ at a uniform concentration ${10}^{19}\frac{\text{a}\text{t}\text{o}\text{m}\text{s}}{c{m}^3}.$

At time $t=0,$ this wafer is heated to a temperature $1200C,$ at which the diffusivity is $2\mathrm{.}49{10}^{-12}\frac{c{m}^2}{(s)}$

$($which is orders of magnitude larger than it was before$).$ The phosphorous diffuses into the wafer,

making a thicker $n-$type layer with lower concentration. This step is called "drive$-$in$"$ diffusion.

$($a$)$ Sketch the concentration as a function of depth into the wafer, showing:

i$.$ the uniform boron concentration $C_B$

ii$.$ the inital phosphorous layer at concentration $C_{P0}$

iii. the time evolution of phosphorous concentration, at "long" times $($the initial distribution isn't

actually a uniform layer, so the erf$-$like short$-$time solution isn't very helpful$).$

$($b$)$ For how long can the $1mm$ thick wafer be considered "semi$-$infinite"?

$($c$)$ The silicon is $n-$type where the phosphorous concentration is greater than the boron concentration.

How thick is that $n-$type layer after $30$ mitutes $(1800$ seconds$)?$ After $90$ minutes $(5400$ seconds$)?$