For metals, the electron chemical potential and the Fermi level are similar, at reasonable temperatures (they are identical at 0K and it depends how smeared the Fermi-Dirac distribution is at a given temperature).

When we discuss the Fermi level of insulators/semiconductors, we actually mean the chemical potential; as you point out, it usually falls in the gap and there wouldn't be any electrons at that energy (=the definition of the Fermi energy in metals).

As long as T>0K, there will be some electrons in the conduction band and some holes in the valence band. The chemical potential reflects this balance, and it depends on the occupation of the bands, and their DOS (which can be approximated by the effective masses for large enough DOS). 

>> a) what determines the Fermi level for an insulators/semiconductor,

>> b) does the Fermi level have phyiscal meaning ?

As we deal with pure (intrinsic) semiconuctor, the Fermi level for it has a physical meaning already noted above - it is the chemical potential F of a system (which appears in the Fermi-Dirac distribution function f(E,T)=(exp[(E-F)/kT]+1)-1. Actually, this chemical potential slightly depends on T, but it is often ignored for simplicity.

And the position of Fermi level in pure semiconductor is quite definite - in the middle of band gap (if the effective masses of electrons and holes are close to each other) - it follows from electrical neutrality requirements - the quantity of thermally excited electrons in the conductivity band should be equal to the quantity of holes in the valence band at any temperature.

However, note that in the investigations devoted to the first-principles modeling of semiconductors band structure at T=0 K (which do not deal with thermal effects) the point specified above is often ignored, and Fermi level has a quite conventional meaning - simply as the maximal energy of one-electron state, which is occupied yet at T=0 K. Of course, for pure semiconductor this "conventional" Fermi level is at the top of filled valence band.

>>3) I can compare the calculated Fermi level of one system using different density-functionals (all other settings are the same)

Oh... This kind of comparison requires a VERY cautious approach. Because the different kinds of exchange-correlation functionals may provide different... so to say, "zero level of energy" (actually it often depends on their specific implementation in various packages).

So, first you should compute the electronic structure of the concrete compound using the different functionals, then to compare the total energies obtained in each scheme. If they are the same (or the difference is negligible with respect to measure of the effects you want to model), it makes a sense to think about it

Dear Lior and V. V. Bannikov,

Thanks for your responses and kind help.

Another related is the chemical potential and the density-functional employed.

As the Fermi level is related to chemical potential, one expects that the the standard DFT (LDA, GGAs) provides in general too small chemical potentials (the gap-problem). To have a more reliable chemical potential value, one needs to do ' beyond DFT' calculations.

All of what is written above is correct as far as I can see.

However, best is perhaps to explain a bit about why you want to know the Fermi level.  Namely, what kind of system are you modeling?  A point defect in the semiconductor?  A surface?  Or the actual bulk material?

Here is why I ask with an example.  Let's say you are studying a point defect in a semiconductor with ab initio methods that has localized states whose energies are in the band gap.  You are wondering about the charge state of the defect (neutral, +1 positive (missing an electron), -1 negative, -2 negative, etc.)  How to model this?

Well, if you just take the bulk semiconductor and make a big periodic supercell and remove one atom (a point defect for example) and run the calculation, you'll be describing the neutral defect since you removed a neutral atom.  This is basically assuming the Fermi level in the bulk semiconductor is intrinsic (i.e. neutral perfect semiconductor with Fermi level somewhere in the gap).  But that may be incorrect from the actual experimental viewpoint.

The Fermi level in practice depends on the doping state of the material both intentional doping as well as native defects that one can't control.  So the Fermi level is set to some value in the bulk semiconductor due to other considerations not in the calculation.  So let's say the semiconductor is n-type (e.g. typical GaN experimental sample).  Then the Fermi level is "high" close to the conduction band edge and very likely the defect will take electrons from the Fermi level and become negatively charged.  The Fermi level is fixed by thermodynamic bulk issues and won't move due to a single defect.

If you decide to p-type dope the semiconductor and study the same defect, then maybe the defect is now positively charged.

Thus, the energy of the defects depends strongly on the Fermi level and charge state.

For such modeling, we can safely stay at T=0 but the Fermi level moves around in the gap depending on the material properties.  You could move to T>0 and have a chemical potential, but that is secondary for this type of calculation.

All these examples want to illustrate is the "Fermi level" depends on the system you are studying.

If you are not doing defects but studying intrinsic bulk, then you should set T>0 and compute the chemical potential which has a unique value in the band gap (for a specific chosen exchange correlation approximation).  T=0 in the intrinsic bulk has the Fermi level anywhere in the bulk band gap.  Again, why would we care?  And what are we trying to understand?

Dear Dr. Sohrab Ismail-Beigi,

Thank you very much for your detailed description about defects and related states and charged systems. I agree with you that charge reservors will play an important role in such cases.