# Fermi velocity in a non-degenerate semiconductor

1. Feb 7, 2009

### BeauGeste

Hi,

Ok, so let's say we have a non-degenerate n-type semiconductor such that the Fermi-level/chemical potential is somewhere in the bandgap (probably needs to be low temperature). Typically in a metal you would say that the Fermi velocity is $$\hbar k_F/m_e$$. But since the Fermi-energy is below the conduction band, that doesn't seem to make sense.

My thought would be that since we are at a non-zero temperature, there are some conduction electrons from the donors simply due to thermodynamics. So my thought would be to find the conduction electron density due to thermalized donors and use that as $$n_c$$ and then use the standard expressions for Fermi wavevector and Fermi velocity.

What do you think?

Thanks.

2. Feb 8, 2009

### snapback

Hi BeauGeste,

The concept of the fermi-velocity is clearly defined at T=0 when you have a degenerate gas, and all electron levels are filled up to EF (Fermi-Dirac distribution). But in the case you are describing, the electrons in the conduction band are distributed according to Boltzmann, and this means there is really no sharp edge in the electron distribution, So I do not see how to define a meaningful Fermi velocity. What are you trying to calculate ?

Regards

a remark: in a non-degenerate semiconductor, the Fermi-Level is always (almost by definition) in the bandgap, especially at higher temperatures. At sufficiently high temperatures, all semiconductors become intrinsic and in this region, the Fermi level is somewhere in the middle of the bandgap

3. Feb 9, 2009

### BeauGeste

I would like to determine the (average, I guess) velocity of electrons in the conduction band in the system I described. So I guess I would have to average $$k_F$$ over the Boltzmann distribution since there will be a distribution of velocities.

4. Feb 9, 2009

### snapback

i would suppose that averaging the velocities (velocity should be something like gradkEn(k)/$$\hbar$$ for an electron in a level specified by a band index n and wave vector k) over Boltzmann would give more meaningful results.

the concept of Fermi-velocity (or Fermi-impulse) is only properly defined for a degenerate system of fermions, and for me, a non-degenerate semiconductor is pretty far away from being exactly that

cheers