Consider one three-dimensional, time-dependent problem to illustrate the power of a Green's function solution. We are interested in doping a semi-infinite boule of initially pure, single crystal silicon. By exposing the surface at \(z=0\) to the dopant, we manage to introduce an amount, \(m_{o}\), of material into the silicon.

a. Demonstrate that the governing equation for this process is:

\[\frac{\partial c_{a}}{\partial t}=D_{a b}\left(\frac{\partial^{2} c_{a}}{\partial x^{2}}+\frac{\partial^{2} c_{a}}{\partial y^{2}}+\frac{\partial^{2} c_{a}}{\partial z^{2}}\right)\]

b. What are the boundary conditions, the initial condition, and any other necessary constraints for this problem?

c. Show that the following Green's function satisfies the differential equation and the boundary conditions.

\[c_{a}=\frac{2 m_{o}}{\left(4 \pi D_{a b} t\right)^{3 / 2}} \exp \left[-\frac{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+\left(z-z^{\prime}\right)^{2}}{4 D_{a b} t}\right]\]