What happens to the Fermi level of a metal/semiconductor under bias?

Question

When a metal/semiconductor junction is placed under bias, the Fermi level of the semiconductor shifts horizontally up or down increasing or decreasing the potential barrier. I was wondering if this is only because the Fermi level of the semi-conductor changes, or does the Fermi level of the metal change as well.

I could imagine that the change in Fermi level could depend on the density of states (DOS) so a much smaller shift in Fermi level is expected in the metal as the DOS is much larger than in the semiconductor and hence only the shift in the semiconductor is usually depicted.

Thanks for helping out.

New question:

I have another question now I think I understand this. Could someone explain why the Fermi level shifts in the Schottky junction and tilts in the Ohmic junction? I think it has to do something with the depletion region created in the Schottky junction, creating an electric field that can oppose the applied potential. And the lack of the depletion region in the Ohmic contact which does not create an opposing electric field and hence can pass all current through. So because the semiconductor has a higher resistance the voltage drop is "devided" equally across the whole semiconductor width. But the Fermi level/chemical potential is thus also increasing as $E_F=E_{F0}+qVx$ with $E_{F0}$ the Fermi level/chemical potential at the metal/semiconductor interface, $q$ the charge, $V$ the applied voltage and $x$ the distance in the semiconductor or so?

Answer 1

Strictly speaking, what we are dealing with here is not Fermi level, but the electrochemical potential. Fermi level is unambiguously related to the charge concentration and the band structure, which change very little when a bias is applied to the junction. Furthermore, in equilibrium the same Fermi level would characterize metal and the semiconductor - but what we are dealing here is a non-equilibrium situation (in which a current may flow), and the difference in chemical potentials is a proxy for the potential difference applied to the junction. In essence, we assume that both semiconductor and the metal are locally in equilibrium, but the Fermi levels difference corresponds to the energy that an electron gains/loses when it is transferred from one material to another. Electrochemical potential is just a generalization of this logic to continuous systems, where we assume local equilibrium at each point, but overall change of the potential through the system.

Eventually, what matters here is that there is a difference between the two Fermi levels: which of them goes up and which goes down is not really that essential. However, with the metal having more charge careers efficiently screening any perturbation, it is usually the semiconductor, whose charge concentration, band structure, etc. are first to get modified as a result of the applied electric field. So the chemical potential of metal is usually taken as unchanged.

Answer 2

It isn't really possible to apply a bias to a metal, because in a metal the electric field will be screened by the free carriers.