Fermi level in semiconductors

Question

I have a question about fermi level (chemical potential). It is stated that the fermi level is the thermodynamic work required to add one electron to the body of a material. How is it possible that, regardles of temperature, fermi level is in the band gap? Since those energies aren't allowed, how can they be the work required to add one electron to the body of a semiconductor. I now there are other definitons, but how is this problem explained?

Answer 1

The problem is in the original statement. It is wrong.

Going to Ashcroft and Mermin's excellent Solid State Physics book, pg. 43 in the second edition:

We shall see shortly that for metals the chemical potential remains equal to the Fermi energy to a high degree of precision, all the way up to room temperature. As a result, people frequently fail to make any distinction between the two when dealing with metals. This, however, can be dangerously misleading. In precise calculations it is essential to keep track of the extent to which $\mu$, the chemical potential, differs from its zero temperature value, $\varepsilon_{F}$.

So, no, the Fermi energy is not strictly equal to the chemical potential, even in a metal.

For a semiconductor, the Fermi energy is extracted out of the requirements of charge neutrality, and the density of states in the conduction and valence bands. Yes, it is in the gap (generally speaking). And, this does cause some problems if you interpret it as the chemical potential as you have noted. Going back to Ashcroft and Mermin (p. 573n in the second edition):

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 on-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 occupied 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...

Often we introduce physics concepts at one level, as an easy 'blanket' statement. When you encounter problems with the interpretation, you usually need to look under the rug and untangle what was swept under there.

Answer 2

A semiconductor is a solid that is defined as having its Fermi level (chemical potential) in a band gap of the electronic structure of the solid, which is not too large compared to the thermal energy, so that, according to the Fermi distribution, at room temperature you have a significant number of electrons and/or holes in the conduction and valence band available for conduction. In cooling a semiconductor down to absolute zero, you obtain an insulator because the thermal energy kT becomes much smaller than the band gap. At zero T, the valence band is completely filled with electrons (no holes present) and the conduction band is completely empty. In a metal, the Fermi level is positioned in a band and you have electrons (and possibly holes) available for conduction down to zero T. The chemical potential (Fermi level) is a thermodynamic quantity of the solid which has no relation to available energy states at that energy.