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.

 

[DeeNos]

As a scientist, it is important to carefully consider the units and conversions in our calculations. In this case, we are dealing with concentrations, which can be expressed in different units such as molarity or number of particles per volume. In order to convert the given C_{s}_{max} from volume ratio to molar ratio, we would need to know the density of the silicon-arsenic mixture at the specific conditions of the Czochralski growth process. Without this information, it is not possible to accurately convert the given C_{s}_{max} to a unitless ratio.

Additionally, it is important to note that the segregation coefficient k is typically temperature-dependent and can vary depending on impurities and other factors. Therefore, the calculated fraction X may not be entirely accurate without considering these variables.

In conclusion, it is not possible to accurately determine the fraction X of the boule that is usable without knowing the density and temperature at which the process is being carried out. It is also important to consider the temperature-dependent segregation coefficient in order to obtain a more accurate result. Further research and experimentation may be necessary to obtain a more precise answer.