(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A Czochralski growth process is begun by inserting 1000 moles of pure silicon and 0.01

mole of pure arsenic in a crucible. For this boule, the maximum permissible doping

concentration is 10^{18}cm^{-3}. What fraction (X) of the boule is usable? (k=0.3)

2. Relevant equations

[tex]C_{s}=kC_{0}(1-X)^{(k-1)}[/tex]

Where [tex]C_{s}[/tex] is the concentration in the solid, k is the segregation coefficient [tex]{C_s}/{C_l}[/tex], [tex]C_0[/tex] is the initial doping concentration in the melt, and X is the fraction of the boule that is solidified.

3. The attempt at a solution

In our case, k<1, so [tex]C_s[/tex] increases as X increases.

I am trying to find X when [tex]C_{s}_{max}=10^{18}cm^{-3}[/tex]

[tex]C_{s}_{max}=kC_{0}(1-X_{max})^{(k-1)}[/tex]

[tex]\frac{C_{s}_{max}}{kC_{0}}=(1-X_{max})^{(k-1)}[/tex]

[tex]\left(\frac{C_{s}_{max}}{kC_{0}}\right)^{(1-k)}=(1-X_{max})[/tex]

[tex]X_{max}=1-\left(\frac{C_{s}_{max}}{kC_{0}}\right)^{(1-k)}[/tex]

From here on out it's plug-n-play with my one show-stopper: I am given [tex]C_{0}[/tex] as a molar ratio (unitless), and [tex]C_{s}_{max}[/tex] as a volume ratio (#/cm^3). I need my final answer to be unitless. How do I convert [tex]C_{s}_{max}[/tex] to a unitless ratio? I'd play with density, but I don't know either the pressure or temperature at which this process is being carried out.

Thanks!

~Malka

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# Homework Help: Semiconductor: Cz Crystal Growth

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