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?