Aluminum atoms are to be diffused into a silicon wafer using both predeposition and drive$-$in heat treatments; the background concentration of $Al$ in this silicon material is known to be $3\mathrm{.}0{10}^{19}$ atoms $\frac{?}{m^3}.$ The drive$-$in diffusion treatment is to be carried out at $1050C$ for a period of $4\mathrm{.}0h,$ which gives a junction depth $x_j$ of $3\mathrm{.}0m.$ Compute the predeposition diffusion time at $950C$ if the surface concentration is maintained at a constant level of $2{10}^{25}$ atoms $\frac{?}{m^3}.$ For the diffusion of $Al$ in $Si,$ values of $Q_d$ and $D_0$ are $3\mathrm{.}41eV$ and $1\mathrm{.}38{10}^{-4}\frac{m^2}{s},$ respectively.

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