Temperature related fermi-levels

Zymandia

Considering the effect on fermi-levels of a semi-conductor bandgap or fermion forbidden zone (FFZ) in an idealized density of states (DOS).
Let's assume there is a bandgap just below the fermi-level say 4 meV, and that the temp. is such that FD distribution predicts +/- 10meV of thermal energy range. But -10meV is forbidden, -4meV is the lowest allowed by the FFZ.
It seems clear that the FD distribution can no longer be symmetric about the 0 K fermi-level, suggesting that the average must rise. So the fermi-level must rise with temperature.
A similar process with a FFZ just above the fermi-level inhibitting higher thermal energies should result in a fermi-level that decreases with temperature.
What is this effect called and have any experimental examples of either been found?

kanato

Just being pedantic: the Fermi energy is independent of temperature, because it's defined as the chemical potential at zero temperature. But your intuition is correct; the chemical potential is a function of temperature. You don't even need a gap near the Fermi level for this to happen, structure in the density of states will cause the chemical potential to shift as temperature changes.

For real solids this is rarely a concern. Typically in order to see a measurable shift in the chemical potential for Fermions you need to have a temperature well above the boiling point of your material.

Zymandia

I'm aware that there is a slight temperature dependence of chemical energy in normal lattices, however it is the change in what would normally be a fermi-dirac distribution caused by a FFZ close to the 0 K fermi level that interests me. For instance it should give an odd specific heat curve.