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Semiconductor: Cz Crystal Growth

  1. Sep 13, 2010 #1
    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 1018 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
     
  2. jcsd
  3. Sep 14, 2010 #2
    Never mind: Csmax is a value for the solid crystal, so I just used densities and molar masses of solid silicon and arsenic to get C0 in units of cm-3.

    Also, to correct a mistake in last post: I made an algebra error while isolating Csmax.

    The second-to-last equation should read
    [tex]\left(\frac{C_{s}_{max}}{kC_{0}}\right)^{\frac{1}{k-1}}=(1-X_{max})[/tex]

    and similarly for the last equation.
     
  4. Sep 16, 2010 #3
    In order to calculate C_0, do you think we should use solid densities or liquid densities? C_0 is the initial melt concentration.

    You are right about the fact that pressure and temperature of the process are not given. But I think using liquid densities will yield more accurate result.

    Thanks for taking the initiative to discuss the problem.

    Regards

    Hasib
     
  5. Sep 17, 2010 #4
    Yes, but from my understanding, melts are generally made by putting solid components into a crucible and then heating them until they melt (and perhaps a bit beyond).
    But I hear what you are saying, that C0 is supposed to describe a liquid.
    Perhaps for most accurate results, we should convert Csmax, which is for a solid, into a molar ratio.
     
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