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.

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:

[tex]\frac{C_{s}-C{x}}{C_{s}-C_{o}}=erf(x/2\sqrt{D*t})[/tex]

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,

[tex]\frac{C_{s}-C{x}}{C_{s}-C_{o}}=\frac{1018atoms/cm3-1016atoms/cm3}{1018atoms/cm3}=0.001964[/tex]

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

Then we solve for x:

[tex]x=0.00174055*2*\sqrt{D*t} =0.00174055*2*\sqrt{2*10^{-12}cm^{2}/s*8h*3600s/h}=8.35464*10^{-7}cm[/tex]