# Location of Fermi level in semiconductors

Hi

Strictly from the definition of the Fermi level as the highest energy occupied at zero temperature, it seems that in the presence of a band gap the Fermi level (Ef) could be placed fairly arbitrarily anywhere between the conduction (Ec) and valence (Ev) bands, since the density of states is zero in that region.

At least, that's what I always thought.

What if I apply a constant voltage V to the sample? When V reaches Ec-Ef I should start getting a current even at zero temperature, and so measure Ec-Ef.

Of course, the difference between the Ec-Ev could be measured in many ways, for example, absorption spectra.

And so, Ef can be determined!

Am I right? If so, I'd be glad to hear what is the physics involved in determining Ef... How is it defined if not by the Fermi-Dirac distribution?

P.S
I know the Ef usually appears in the middle of the band gap in the literature. As I said, I think it is arbitrary in undoped materials, and would be glad to be taught otherwise.

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A better definition of Ef IMO is zero temperature limit of the chemical potential: $$E_F = \lim_{T \rightarrow 0} \mu(T)$$

The chemical potential will be well defined in a semiconductor at finite temperature, and it will deviate from the center of the gap depending on what the shape of the density of states for the valence and conduction bands are. But in practice, if one is not considering finite temperature effects, then you are right the Fermi level can be arbitrarily located within the gap.

Just applying a voltage won't work like you want. Otherwise you'd only need 10 V to get conduction through just about any material, since band gaps are rarely larger than that.

Well, you are never really at zero temperature, and there are always other mechanisms to get a current going before reaching such a large voltage.

So, of course my supposed experiment will not really work in a laboratory.

But, theoretically, can one give meaning to the Fermi level in this case? Why is it always placed in the middle of the gap in the literature?

Most of the articles I read place the gap at the top of the valence band.

Your experiment won't work the way you describe not because we're always working at finite temperature. If you apply a voltage bias so that the valence overlaps the conduction band, this results in breakdown conduction. But it's tied to the size of the band gap, not the location of the Fermi level within the band gap. Also it requires that electrons can tunnel the distance the voltage is applied over, so it depends on the thickness of the material. So it's more appropriate to talk about a breakdown electric field.