If aluminum is diffused into a thick slice of silicon with no previous aluminum in it at a temperature of 1100˚c for 8 hours, what is the depth below the surface at which the concentration is 1016 atoms/cm3 if the surface concentration is maintained at 1018 atoms/cm3? use d = 2 x 10-12 cm2/s for aluminum diffusing in silicon at 1100˚c.

8.354 nanometers

Explanation:

To treat a diffusive process in function of time and distance we need to solve  2nd Ficks Law. This a partial differential equation, with certain condition the solution looks like this:

Where Cs is the concentration in the surface of the solid

Cx is the concentration at certain deep X

Co is the initial concentration of solute in the solid

and erf is the error function

Then we solve right side,

And we need to look up the inverse error function of 0.001964 resulting in: 0.00174055

Then we solve for x:

they are all solids.