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?
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$\begingroup$ I don't get what "are not called to be present at the Fermi energy level" means. Fermi energy is the highest occupied energy at absolute zero temperature. Above that there is always a finite probability that energy is above Fermi energy. Check the Fermi Dirac distribution has a tail above 0K. $\endgroup$– Siddhartha DamJun 25, 2019 at 8:33
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3 Answers
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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.
answered Jun 25, 2019 at 13:29
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Fermi energy is only defined at $T=0$. For a finite temperature we refer instead to the chemical potential. Thus, thermally excited electrons are technically "above" the Fermi level and not "on" the Fermi level. In other words, thermally excited electrons don't define what the Fermi level is, only the highest occupied states at $T=0$.
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$\begingroup$ I'm not sure, but I think the OP wants to know how a Fermi level can be defined in an energy gap when there are no electrons there. $\endgroup$– garypJun 25, 2019 at 11:04
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$\begingroup$ Well technically it is not defined in a gap but rather at the band edge for an insulator. Placing a Fermi level in a band gap is just a matter of convention and won't change the physics anyway (since the density of states is 0). $\endgroup$– fgoudraJun 25, 2019 at 13:18
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Don't forget that to calculate the number of electrons (or electron density) in any energy range you need to multiply the density of states by the occupancy number (F-D function). In the band gap there are no energy levels (as the name shows) so the density of states is zero. So what the F-D distribution tells you is that if you had a level right in the middle of the gap (for intrinsic semiconductor), the occupancy number would have been 1/2. But then it won't be a gap, would it? In the conduction band you have non-zero density of states so you can have electrons. Not so many as in a metal, as the values of F-D function are quite small in that region.
