What are the requirements for describing charge carriers (e.g. electrons) in a semiconducting material by
– Fermi distribution
– Boltzmann distribution
When do we apply the one or the other?
If the explanation to this question is in Ashcroft/Mermin, reference to the relevant chapter would be appreciated.
I don't agree with both commentators so far.
Edit
Consider Eq. 21 in Sze, PHysics of Semiconductor Devices:
$$n = N_C exp\left(-\frac{E_C-E_F}{kT}\right)$$
or Eq. 2.16 in Pierret, Semiconductor device fundamentals.
Best Answer
Electrons are fermions (as are holes), therefore you should always use the fermi distribution, never boltzmann.
It is OK to use the boltzmann distribution as an approximation to the fermi distribution. (Makes the math simpler sometimes.) When is it a good approximation? When any given conduction-band state has probability << 1 of having an electron in it. (That's for n-type.)
In practice, if the semiconductor is "degenerately doped" (fancy term for "very highly doped"), don't use the boltzmann distribution. If you're using the semiconductor as a laser, don't use the boltzmann distribution. In most other circumstances, the boltzmann distribution is a pretty close approximation to the true fermi distribution.