[Physics] Fermi level in metals and semicoductors

Question

As we read about Fermi energy, it is the energy of the highest occupied electrons. But if we look at the energy level diagram of semiconductor the Fermi level is situated somewhere between the valence band and the conduction band. The probability of the occupancy of the Fermi energy level is zero. If there enter some electrons in the conduction band due to thermal excitation conduction happens. My question is that how does the energy of the electrons whose energy is above the Fermi level, against the definition of the Fermi energy, are not called to be present at the Fermi energy level?

Answer 1

To quote from a note in Ashcroft and Mermin's Solid State Physics book (p. 573 in the 1976 version):

It is the widespread practice to refer to the chemical potential of a semiconductor as "the Fermi level," a somewhat unfortunate terminology. Since the chemical potential almost always lies in the energy gap, there is no one-electron level whose energy is actually at "the Fermi level" (in contrast to the case of a metal). Thus the usual definition of the Fermi level (that energy below which the one-electron levels are occupies and above which they are unoccupied in the ground state of a metal) does not specify a unique energy in the case of a semiconductor: Any energy in the gap separates occupied from unoccupied levels at $T=0$. The term "Fermi level" should be regarded as nothing more than a synonym for "chemical potential" in the context of semiconductors.

Now, you might also want to look deeper into The chemical potential of an ideal intrinsic semiconductor (Mark R. A. Shegelski, American Journal of Physics 72, 676 (2004)) for a deeper look at the behavior of the chemical potential as a function of energy. In that paper it is shown that as $T \rightarrow 0$ the chemical potential goes to the bottom of the conduction band.