# Band gaps and Fermi level

Does the (intrinsic) Fermi level of an insulator HAVE to lie very near the middle of the band gap? I know it might deviate slightly if electrons and holes have different effective masses, e.g. in Si. But can it be radically different? For example, are there insulators with 5 eV band gaps that have the Fermi level 4 eV above the valence band (so 1 eV from the conduction band)? Or is it a very general result that the Fermi level ends up very close to the middle?

It's fairly common. The definition of the Fermi level is $E_F = \lim_{T \rightarrow 0} \mu(T)$ where $\mu$ is the chemical potential. I think you might get a more unusual value for a charge-transfer insulator, where the valence states are wide, like O 2p bands, and the conduction states are narrow, like a 3d or 4f type of band.

But usually it doesn't matter, since for most purposes the Fermi level could be taken to be any arbitrary value within the gap and you wouldn't know the difference for the vast majority of low temperature calculations.

It matters significantly for the calculations I'm trying to do. I want to get the equilibrium band alignment of a metal-insulator-metal structure (i.e. a tunnel junction). Depending on how the insulator is aligned with the metals, the electrons at the metal Fermi level see significantly different barrier heights, which impacts conductivity exponentially.

I thought that the Fermi level for wide band gap insulators (>5 eV) would be a well-defined quantity as it is for the common semiconductors. But from reading around, increasingly it seems that the quantity is almost meaningless and people read it off ab initio DOS rather arbitrarily. I guess that at the interface, there must be gap states that determine the alignment even if the bulk insulator Fermi level is meaningless. Is that what I should be concerned with for determining barrier height?

Does the (intrinsic) Fermi level of an insulator HAVE to lie very near the middle of the band gap? I know it might deviate slightly if electrons and holes have different effective masses, e.g. in Si. But can it be radically different? For example, are there insulators with 5 eV band gaps that have the Fermi level 4 eV above the valence band (so 1 eV from the conduction band)? Or is it a very general result that the Fermi level ends up very close to the middle?

For an insulator, you can be quite sure that it's in the middle, in semiconductors it depends if there are added donors/acceptors.

n_i² = np =

n_i=N_c.exp]-(E_c-E_i)/kT]
With Nc the effective density (not sure if they call it like that in English), E_c energy of conductingband, E_i intrinstik Fermi level and np the amount of conducting electrons/holes

I'm curious if Dreak's point answers your question. For wide-bandgap materials, as with any, the position of the Fermi level is highly dependent on how much activated dopant there is. Or is this too basic an explanation to address your question?

Of course, the value of a nominally intrinsic (undoped) material is usually not perfectly centered between Ec and Ev because of unintentional doping. Perhaps that's why you read different values? For instance, AlN is quite easily grown as an insulating material normally, but if you accidentally get Se impurity, you quickly have a p-type semiconductor on your hands.