# Semiconductor: Cz Crystal Growth

1. Sep 13, 2010

### mbrmbrg

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

$$C_{s}=kC_{0}(1-X)^{(k-1)}$$

Where $$C_{s}$$ is the concentration in the solid, k is the segregation coefficient $${C_s}/{C_l}$$, $$C_0$$ 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 $$C_s$$ increases as X increases.
I am trying to find X when $$C_{s}_{max}=10^{18}cm^{-3}$$

$$C_{s}_{max}=kC_{0}(1-X_{max})^{(k-1)}$$

$$\frac{C_{s}_{max}}{kC_{0}}=(1-X_{max})^{(k-1)}$$

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

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

From here on out it's plug-n-play with my one show-stopper: I am given $$C_{0}$$ as a molar ratio (unitless), and $$C_{s}_{max}$$ as a volume ratio (#/cm^3). I need my final answer to be unitless. How do I convert $$C_{s}_{max}$$ 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. Sep 14, 2010

### mbrmbrg

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.

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

and similarly for the last equation.

3. Sep 16, 2010

### hasib_eee

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

4. Sep 17, 2010

### mbrmbrg

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.