Boltzmann Distribution: Feynman's treatment of p-n junction

[strauser]

In Vol III, 14-4 and 14-5 of the Feynman Lectures (http://www.feynmanlectures.caltech.edu/III_14.html), Feynman gives a discussion of the p-n junction, in which he derives the diode characteristic equation via a nice, simple and convincing application of the Boltzmann distribution to the relative numbers of charge carriers on either side of a junction with a potential difference V.

I am however completely confused: do electrons/holes in a semi-conductor not obey Fermi-Dirac statistics? If so, why does he not mention or rely on this, and if so, how is his derivation valid?

I'll point out that I know not a great deal about solid state physics, or stat. mech., so the explanation may be utterly trivial.

 

[phyzguy]

In a semiconductor, the conduction and valence bands are far from the Fermi level in units of kT. In this limit, Fermi-Dirac statistics are very closely equal to Maxwell-Boltzmann statistics:

Fermi-Dirac: $N \propto \frac{1}{\exp(\frac{E-E_F}{kT})+1}$

But in this case $\exp(\frac{E-E_F}{kT}) >> 1$, so we can ignore the 1 in the denominator. It then reduces to Maxwell-Boltzmann statistics to a sufficient approximation.

 

[strauser]

In a semiconductor, the conduction and valence bands are far from the Fermi level in units of kT. In this limit, Fermi-Dirac statistics are very closely equal to Maxwell-Boltzmann statistics:

Fermi-Dirac: $N \propto \frac{1}{\exp(\frac{E-E_F}{kT})-1}$

OK, thanks, things are slightly clearer.

However, I'm not sure either what your $N$ or the expression on the RHS are. Is $N$ related to the density of states (i.e. what I'd call $N(E)$), or is it the concentration of charge carriers? And the expression on the RHS looks suspiciously like the Fermi function, but that has a +1.

Could you clarify this a bit more, please?

But in this case $\exp(\frac{E-E_F}{kT}) >> 1$, so we can ignore the 1 in the denominator. It then reduces to Maxwell-Boltzmann statistics to a sufficient approximation.

Right. So is it correct to say that, to a very good approximation, electrons in the conduction band of a semiconductor obey Maxwell-Boltzmann statistics, which is Feynman's tacit starting point?

 

[phyzguy]

Right, the +1 vs -1 is my error. It is -1 for Bose-Einstein and +1 for Fermi-Dirac. I corrected my original post. In my post, N is the density of charge carriers.

So is it correct to say that, to a very good approximation, electrons in the conduction band of a semiconductor obey Maxwell-Boltzmann statistics, which is Feynman's tacit starting point?

Yes.

 

[strauser]

Right, the +1 vs -1 is my error. It is -1 for Bose-Einstein and +1 for Fermi-Dirac. I corrected my original post. In my post, N is the density of charge carriers.

OK, it's clear that this is trivial. Thanks. In fact, the only relevant text I have available (Streetman, Solid State Electronic Devices) does in fact treat this topic but without using the name Boltzmann anywhere - I skimmed the section in question without really noticing the result, which looks to be fairly important, I'd guess.