Chemical Potential & Fermi Energy

[Chiz]

Hi there,
i've already read some topics in this forum about the fermi-energy/chemical potential. I've also read the article "The chemical potential of an ideal intrinsic semiconductor" from Mark R. A. Shegelski which made the whole thing a little bit more clear to me. but there are some questions left:
The chemical potential ist defined as :
$$\mu(N,T,V)=F(N,T,V)-F(N+1,T,V)$$
So it is the change in the free energy by adding one particle to the ensemble. in my interesting case: adding one electron to the semiconductor

By T=0K, the chemical potential must lie (if the valence band is totaly filled) on the bottom edge of the conduction band $$E_c$$, because by adding one electron, the energy change is $$E_c$$. that's what is written in the article of Shegelski and what's wrong in many textbooks. there's often said "at T=0K the chemical potenial lies in the middle of the band gap".
But at higher temperatures, when there are some electrons in the conduction band, the chemical potential lies round about the middle of the gap. but why? for me it doesn't make sense, that by adding one electron the energy changes about a value $$\approx E_{gap}/2$$ because i can't add an electron whit that energy.
I think I must mix up something here.

On the other side the fermi energy/chemical potential is often defined as the energy where the fermi distribution has the value 1/2. but for me this has nothing to do with the thermodynamic definition
$$\mu(N,T,V)=F(N,T,V)-F(N+1,T,V)$$
are both definitions related with each other?

maybe someone can make the whole thing a bit more clear to me!

All this confusion came up to me by thinking about the pn-contact. I've read several times "in contact, the chemical potential must be the same everywhere in the system".

 

[kanato]

That definition of the chemical potential makes sense when the spectrum is not gapped, so that the energy change is the same for adding or removing a particle (in the thermodynamic limit), so what's often used is $$\mu = dF/dN$$. A better definition would be the average of the energy changes for adding or removing a particle,
$$\mu(N) = \frac12 [(F(N+1) - F(N)) + (F(N) - F(N-1))]$$
(which looks like a slightly better finite difference approximation of a derivative). Then for a semiconductor the chemical potential would sit in the middle of the gap at T = 0, because the energy change to add a particle is E_c and the energy change for the last particle added was E_v, so the chemical potential is 1/2 (E_c + E_v).

 

[Chiz]

Ok, thanks. So by defining $$\mu(N)$$ with that approximation for the derivative, it makes sense to me, that the chemical potential lies inside the band gap at T=0 K.
But now the next question is, what happens at a temperature where there are some electrons inside the conduction band? the chemical potential changes a little bit, but is still somewhere inside the bandgap, right?
with your definition it would lie at higher energies than the $$E_c$$, when the elections are simply filled on top of each other inside the conduction band.
but the electrons also occupy the free states inside the valence band and because of that the chemical potential lies still somewhere inside the band gap?

 

[kanato]

Well it gets a bit more hazy here for me but I will do my best.

At finite temperature, some electrons are depleted from the valence band and some are added to the conduction band. Then if you add an electron, there is some probability that it will go into the valence band and some probability it will wind up in the conduction band. Which direction the chemical potential shifts depends on the density of states above and below the gap; if the density of states above the gap is higher than it is below the gap, then the chemical potential will go up as temperature is increased because more states above the gap are thermally accessible than below the gap.