Fermi level
/ FAIR-mee LEV-ul /
Imagine pouring water into an irregular tank: it settles to one flat surface, and that water line tells you at a glance how full the tank is. Electrons in a solid fill up their available energy states in a similar way, from the bottom up. The Fermi level is the water line of that electron sea — the energy that marks, roughly, the boundary between filled states below and empty states above.
More precisely, the Fermi level is the energy at which the probability of finding an electron is exactly one-half. At absolute zero it sits sharply at the top of the filled states; warm the material up and the boundary blurs slightly, with a few electrons thermally bumped just above the level and a few empty seats just below. Whether the Fermi level lands inside a band or inside a gap is the single fact that decides whether a material is a metal or an insulator.
The Fermi level matters because it is the reference energy from which almost everything is measured, and lining up the Fermi levels of two touching materials is what drives currents across junctions and contacts. A frequent confusion is between the Fermi level and the Fermi energy: the Fermi energy is strictly the level at absolute zero, while the Fermi level is the more general, temperature-aware version that physicists also call the chemical potential.
Bring two different metals into contact and electrons flow from the one whose Fermi level sits higher into the one where it sits lower, until the two levels line up flush. That tiny equalising flow leaves a small built-in voltage at the join — the same effect that lets a thermocouple measure temperature from a junction of two metals.
Two metals in contact shuffle electrons until their Fermi levels line up.
In a semiconductor the Fermi level usually sits inside the band gap, where there are no electron states at all; this does not mean an electron lives there — it is simply the balance point of the occupation probabilities of the bands on either side.