Resistance of Sample w/ Impurities: Proving Constant?

[Corneo]

If we have a N-type Silicon $N_d = 10^{16}$. If the sample has impurities which give rise to 10^10 recombination levels. Then carrier lifetime is now 100ns. Is the resistance different from when there are no impurities?

My guess is no since charge carrier concentration will stay constant, thus resistance is still the same whether there are impurities or not. How would I go about proving this though?

Thanks.

 

[WalterContrata]

Please Clarify

I take it that the densities are in cm^-3?

When we talk about semiconductors, I think that "impurity" often refers to donors and acceptors. Is it 10^16 donor impurities / cm^-3?

And you have of second impurity that results in a 10^10 deep levels/cm, and this level facilitates recombination such that the minority carrier lifetime is 100 ns. Is that the situation?

And you are considering whether the resistivity would be different if the second impurity where not present?

I think you are probably correct. Resistivity depends on the product of carrier concentration and mobility. If you can show that these are unchanged, then you are done. Mobility depends on effective mass and scattering rate. Should any of these be different if the impurities were not present, and if so, how?

 

[Corneo]

I take it that the densities are in cm^-3?
When we talk about semiconductors, I think that "impurity" often refers to donors and acceptors. Is it 10^16 donor impurities / cm^-3?
And you have of second impurity that results in a 10^10 deep levels/cm, and this level facilitates recombination such that the minority carrier lifetime is 100 ns. Is that the situation?

That is correct.
There is a formula that states $$\mu_n = \frac {q \tau_n}{m_e^*}$$
However if $\tau_n$ is reduced to 100ns, then $\mu_n$ would also reduced. Since $\sigma = q (\mu_n n + \mu_p p)$, we can drop the mu_p p term since we have N type silicon. Thus, $\sigma = q (\mu_n n)$, however I think this reasoning is not right. I still think the resistance of the sample is left unchanged, I just don't know how to show that mu and n stays constant.

 

[WalterContrata]

Great. The Tau_n in the formula for mu_n should be the average time between scatterings for an electron. What is the dominant scattering mechanism? Does it involve the recombination levels?

Carrier lifetime usually refers to something different. Suppose that there is a hole in this n-type Si. The carrier lifetime is the average time until that hole combines with an electron, and that is the end of it.

So, if you can show that tau_n is dominated by something other than the recombination sites, then mu_n is independent of the recombination site desnity. All that is left is to show that n is independent, and p is negligible. Does your text show how to calculate these carrier densities, given the donor impurity concentration? How much does it depend on the deep recombination level?

 

[Corneo]

Thanks for your help thus far WalterContrata.
I believe as the problem stands, the only scattering mechanism is caused by temperature. However I recall that imperfections can interact with the electrons through distortions in the periodic potential variation of the crystal.
I am not sure if I can show that n is independent of these recombination sites. I know that n would depend on the doping concentration with the approximation
[tex]n \approx N_d[/itex][/tex]