The second diffusion step in semiconductor processing is drive-in, where we no longer expose the wafer to dopant but allow the dopant to redistribute itself within the substrate. To model this region, we assume that the drive-in depth is much smaller than the substrate thickness and that we do not lose any dopant in the process.

a. What is the differential equation describing the process?

b. What are the initial condition and the boundary conditions?

c. Verify that the following solution holds, where \(M\) represents the moles of material originally present in the substrate before drive-in commenced.

\[c_{d}=\frac{M}{A_{c} \sqrt{\pi D_{d s} t}} \exp \left(-\frac{x^{2}}{4 D_{d s} t}\right) \quad M=\int_{0}^{\infty} c_{d}(t=0) A_{c} d x\]