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We know from the band theory that in a semiconductor (of course for all metals, insulator) all energy states are not allowed. In a semiconductor there is a valence band and a conduction band, and in between energy states of them i.e energy levels laying in between valence band and conduction band are forbidden. Now when we calculate Fermi energy of the semiconductor we find that it lies in between valence band and conduction band. For intrinsic semiconductor at T=0k, Fermi energy lies exactly half way between valence band and conduction band. But we know energy levels laying in between valence band and conduction band is forbidden, and we also know that Fermi energy is the highest energy level of a material that an electron corresponds to, at T=0 k. So what is the meaning that Fermi energy lies in a forbidden region in semiconductor which corresponds to valid energy level of the material at T=0k ?

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  • $\begingroup$ This question is about the Fermi level not the Fermi energy. "The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. " source: wikipedia. $\endgroup$ – my2cts Apr 21 '20 at 12:25
  • $\begingroup$ @my2cts it is the Fermi energy because he explicitly stated at $T=0K$. $\endgroup$ – boyfarrell Apr 21 '20 at 14:39
  • $\begingroup$ @boyfarrell No it is the Fermi level. $\endgroup$ – my2cts Apr 21 '20 at 15:38
  • $\begingroup$ @my2cts I thought that the Fermi-energy is a special case of the Fermi-level when $T=0K$. I know there is a lot of overloading with these terms so happy to know if that’s wrong interpretation in your experience? $\endgroup$ – boyfarrell Apr 22 '20 at 11:57
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It is more precise to say that there are no energy levels between the valence and the conduction bands, rather than saying that they are forbidden.

The Fermi energy is not the energy of the highest level, but rather en energy characterizing the fact that the states below this energy are filled, whereas the states above this energy are empty, as described by Fermi-Dirac distribution: $$f(E) = \frac{1}{e^{\frac{E-E_F}{k_B T}} +1}.$$ Thus, the position of Fermi energy in the gap energy region reflects the fact that (at zero temperature) all lower energy states (i.e., the states in the valence band) are filled, whereas all the higher energy states (i.e., the states in the conduction band) are empty.

Remark
One thing to keep in mind is that in the context of semiconductors one often uses term Fermi energy to mean Fermi level, i.e. the chemical potential. In a free electron gas the two are the same: they designate the position of the Fermi surface at zero temperature in the continuum spectrum. In a semiconductor the notion of Fermi energy is not very useful - the states are filled up to the top of the valence band. The Fermi level (i.e. the chemical potential), entering the Fermi distribution, is meaningful. Note however that adding the last electrons to the valence band costs zero energy, whereas adding the first electrons to the conduction band costs the gap energy, $E_g$. This is why the Fermi level is placed in the middle of the gap (for an intrinsic semiconductor).

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  • $\begingroup$ From the formula it is clear that at the Fermi level can be seen as the level for which the occupancy is 1/2 at finite T. $\endgroup$ – my2cts Apr 21 '20 at 12:27
  • $\begingroup$ @my2cts In my opinion it is simply an energy coefficient in the formula. There is no reason why there should exist a real energy state corresponding to this energy (or any other specific value of energy). $\endgroup$ – Vadim Apr 21 '20 at 12:35
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    $\begingroup$ And, of course, the Fermi function is non-zero in regions of the band structure of metals were there are no states as well. We just don’t get so worked up about it. Sometimes I think we should teach semiconductor band structure first and see if people have issues with materials having states were the Fermi function is 1/2. $\endgroup$ – Jon Custer Apr 21 '20 at 13:11
  • $\begingroup$ @Vadim I think at:my2cts might mean the probability of there being a Fermion. The actually occupancy is given by the density of states multipled by the probability. $\endgroup$ – boyfarrell Apr 21 '20 at 14:42
  • $\begingroup$ @Vadim I am just following the definitions. The energy coefficient is the Fermi energy at T=0. $\endgroup$ – my2cts Apr 21 '20 at 16:29

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