Semiconductor: Cz Crystal Growth

[mbrmbrg]

Homework Statement

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)

Homework 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.

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

 

[mbrmbrg]

Never mind: C_smax 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 C_smax.

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

and similarly for the last equation.

 

[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

 

[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 C_smax, which is for a solid, into a molar ratio.