Why n*p always equal to ni square? (semiconductor)

[paulzhen]

Why n*p always equal to ni square?? (semiconductor)

Hi,

For you guys who studied semiconductor physics must be familiar with the equation:

np=n_i²

I can understand why this is true for the intrinsic case (the broken bonds would always provide electron and hole in pairs )

But why is this still true for dopped semiconductors? Take Si for example, n=p=n_i=10¹⁰ in intrinsic case (that's we all know). However, if dope 10¹⁵ N_d into the material, then n~10¹⁵ and p~10⁵. The highlighted part is my confusion! Why p become smaller? Where do the holes go?

Thanks for helping!

 

[rollingstein]

Dopant brings in extra electrons that quenches the holes?

 

[paulzhen]

Dopant brings in extra electrons that quenches the holes?

Don't think so. I have also thought of that but if this is true there must be some relevant theories to explain the "dopping and quenching relationship", or, do the scientists consider this is a neglectable question?

 

[Ratch]

paulzhen,

Don't think so. I have also thought of that but if this is true there must be some relevant theories to explain the "dopping and quenching relationship", or, do the scientists consider this is a neglectable question?

You are correct in being suspicious of that explanation. See the attachment for a mathematical explanation.

Ratch

 

[paulzhen]

paulzhen,
You are correct in being suspicious of that explanation. See the attachment for a mathematical explanation.

Ratch

Thanks a lot Ratch. Actually I saw a similar maths explanation before, you both use the fact of neutrality to form the equation.
But could you tell me where do the holes go? Are they being eliminate by extra electrons? This is also the only physical explanation i can imagine. But since Nd>>p, the explanation of eliminating seems ... just not right, cause the eliminating efficiency will be too small to be (felt) right.
Please enlighten me

 

[Ratch]

paulzhen,

But could you tell me where do the holes go? Are they being eliminate by extra electrons? This is also the only physical explanation i can imagine. But since Nd>>p, the explanation of eliminating seems ... just not right, cause the eliminating efficiency will be too small to be (felt) right.
Please enlighten me

In intrinsic semiconductor material, the electron and hole concentration is caused by thermal recombination and generation (R-G). At room temperature, the balance for silicon is Nd = Na =10^10. If more mobile electrons are added to the semiconductor by doping, then the R-G equilibrium changes, and does not need to generate as many electrons thermally to support the equation np=ni^2. Since thermally generated charge carriers come in pairs, then the hole generation is also reduced. So the holes don't go anywhere, it is just that fewer of them are generated by the R-G thermal process, which has slowed down due to the excess of electrons. That is what those equations mean, and it the basis for their derivation. Ratch

 

[paulzhen]

paulzhen,



In intrinsic semiconductor material, the electron and hole concentration is caused by thermal recombination and generation (R-G). At room temperature, the balance for silicon is Nd = Na =10^10. If more mobile electrons are added to the semiconductor by doping, then the R-G equilibrium changes, and does not need to generate as many electrons thermally to support the equation np=ni^2. Since thermally generated charge carriers come in pairs, then the hole generation is also reduced. So the holes don't go anywhere, it is just that fewer of them are generated by the R-G thermal process, which has slowed down due to the excess of electrons. That is what those equations mean, and it the basis for their derivation. Ratch

I think I got what you mean, thanks!