Answer:
the depth below the silicon surface at which the aluminum concentration would be [tex]10^{16}[/tex] atoms/[tex]cm^{3}[/tex] is [tex]5.723*10^{-6}[/tex]
Explanation:
The concentrations of a diffusing element in a silicon wafer at any particular position below the surface and the time it takes to reach that concentration has a relationship given by Fick's second law
[tex]\frac{C_{s}-C_{s}}{C_{x}-C_{0}}=erf(\frac{x}{2\sqrt{Dt} } ) ...(1)[/tex]
Where
The concentration of the carbon on the gear surface is [tex]C_{s}[/tex]
The initial uniform (normal) concentration of carbon in the gear is [tex]C_{0}[/tex]
The concentration of carbon at a distance x below the gear surface after time t is [tex]C_{x}[/tex]
Distance below the gear surface is x
Diffusivity of carbon at a given temperature is D
Time is t
erf is a mathematical function called error function
The impurity diffusion is a method in which an impurity is diffused into a silicon wafer at a very high temperature.The given values are
[tex]C_{s}[/tex] = [tex]10^{18}[/tex] atoms/[tex]cm^{3}[/tex]
[tex]C_{0}[/tex] = 0 atoms/[tex]cm^{3}[/tex]
[tex]C_{x}[/tex] = [tex]10^{16}[/tex] atoms/[tex]cm^{3}[/tex]
t = 206 minutes = 12360 s [1 minutes= 60 seconds ]
For an aluminum diffusing in silicon at 11100°C Diffusivity is [tex]=2*10^{-12} cm^{2}/s[/tex]
Substituting these values we have
[tex]\frac{10^{18}-10^{16}}{10^{18}-0}[\frac{x}{2\sqrt{2*10^{-12}cm^{2}/s[\frac{10^{-4}m^{2}}{1cm^{2}} ]*12360s} } ][/tex]
[tex]0.99 =erf[\frac{x}{3.145*10^{-6}} ][/tex]
Let assume [tex]R = [\frac{x}{3.145*10^{-6}} ][/tex]
Hence
[tex]erf(R) = 0.99[/tex]
[tex]R = [\frac{x}{3.145*10^{-6}} ] ...(2)[/tex]
To determine for what number the error function is 0.99
The table on the first uploaded image is a table that error function for some number
Where z is the same thing as R
calculating the R (i.e z) value whose error function is 0.99 , using the interpolation method we have
[tex]\frac{R-1.8}{1.9-1.8}=\frac{0.9900-0.9891}{0.9928-0.9891}[/tex]
[tex]\frac{R-1.8}{0.1}=0.2432[/tex]
[tex]R = 1.82[/tex]
Substituting the value of R in the equation 2 we have
[tex]1.82 = [\frac{x}{3.145*10^{-6}} ][/tex]
[tex]x = 1.82*3.145*10^{-6}[/tex]
[tex]x = 5.723*10^-6m[/tex]
So therefore the value of x is [tex]5.723*10^{-6}[/tex]
Hence the depth below the silicon surface at which the aluminum concentration would be [tex]10^{16}[/tex] atoms/[tex]cm^{3}[/tex] is [tex]5.723*10^{-6}[/tex]
