In metals at room temperature Fermi energy corresponds to the highest filled level. In semiconductors at room temperature it is the level with probability of occupation half. Even though the probability of occupation is one half at FL this level is situated in the forbidden gap and hence no electrons there. Metals don't have a band gap, whereas semiconductors have a band gap ( and by a loose definition < 3 eV). The fermi level of intrinsic semiconductors will be practically at mid gap at room temperature.

Consider also that the Fermi Energy in SC is strongly dependent on the semiconductor's doping. This leads the Fermi Energy do shift above of under the mid gap dependening on the doping's concentration and type.

Abrevied def: the level in the distribution of electron energies in a solid at which a quantum state is equally likely to be occupied or empty.

Explaned def: In quantum mechanics, particles with a half-integer spin, usually spin 1/2 (for example electrons) follow the Pauli exclusion principle, which states that no two particles may occupy the same quantum state. Such particle are usually referred to as fermions, as opposed to bosons, whci can share the same quantum state. When a number of electrons are put into a system, in the ground state (i.e., at zero temperature) electrons will occupy higher energy levels when the lower ones are filled up. The Fermi energy (EF) is the energy of the highest occupied state at zero temperature. Fermi energy is a central concept for the whole solid state physics

We should distinguish between the Fermi energy and Fermi level. The Fermi energy is simply the energy difference between the HOMO and LUMO states, but only at zero temperature and for non-interacting fermions.

If you're talking about the Fermi level, I agree with the above definitions. On a related note, you can also think of the Fermi level as the electrochemical potential of the material/system. This may be a more intuitive answer depending on your background.

Dear Colleagues

Thank you very much for your contribution and valuable explanations.

Can we summarize what is said above as?

Fermi energy (let us use it in parallel with Fermi level, I found it in Wikipedia!) can be defined in semiconductor as an ideal level at which the probability of occupation with electrons is 50% which lies in the band gap at (T=0K) . While in metal it’s the top occupied energy level in the conduction band at (T=0K). Thus we have two definitions one for Semiconductors and the other for metals…

Why we don’t try to unify the definitions?

I mean let us go deep in the Fermi Dirac distribution function, the 50% occupation rule is obvious. What about the second definition of metal?

Metal's conduction band is practically a continues energy band, i. e. energy levels are very closely spaced such that practically they can be considered as continues ( I found it in Principle of solid state physics by Levy). Now what happen if we say that the first definition is also working in metals?

The top occupied level in metals is very close to the above unoccupied level (practically they are overlapped) now, the ideal energy level which lies between the top occupied level and the lower unoccupied level is practically the top occupied level !! Why we don't say that the first definition leads to the second ?

I hop that my idea is clear.

I appreciate any comments or notes about this topic

regards