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In understanding the behavior of semiconductors, I'm coming across a description of the Fermi Energy here and at Wikipedia's page (Fermi Energy, Fermi Level). If I understand correctly, the Fermi Level refers to the energy state at which there's a 50% chance of finding an electron. This varies with temperature. The Fermi Energy is the highest occupied energy state of fermions at absolute zero.

I'm a little confused as to the relation of the two terms. Additionally, in a semiconductor, the Fermi Energy falls between the valence band and the conduction band. However, my understanding is that electrons cannot exist between the two bands -- so why isn't the Fermi Energy the top of the valence band?

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The reason for this apparent contradiction is that you have two "separate" quantum effects.

  1. Fermi-Dirac distribution describes the energies of single particles in a system comprising many identical particles that obey the Pauli exclusion principle. Distribution is calculated for potential-free space and is temperature dependant.

  2. You put electrons into the material, and in the material they feel potential of atomic cores. This potential restrict possible energetic states that are available for electrons, that is it makes bands, where electrons can behave almost freely (according to Fermi-Dirac distribution), but makes energetic states between the bands forbidden.

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  • $\begingroup$ I hope this isn't a completely silly question, but with regards to #1, what is a particle's "energy" if there is no potential? That is to say, how can an electron have energy in the absence of the electric potential energy from an atomic core? $\endgroup$ – Perrako May 15 '12 at 7:25
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    $\begingroup$ If your question would be about classical mechanics, this would be trivial, as particles can have also kinetic energy. However, for quantum mechanics this is no longer trivial. If the density of electrons/fermions is too large, particles cannot just have any energy, they are bound by quantum effects. Moreover since by Pauli exclusion principe two fermions cannot be in the exacty the same quantum state (in classical physics it would be possible to have two particles with say the same kinetic energy) you get Fermi-Dirac distribution of energies. $\endgroup$ – Pygmalion May 15 '12 at 7:55
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In despite of being closely related, the Fermi energy and Fermi level are two different concepts which apply to different situations. To former is applied at zero temperature whereas the latter makes sense at finite temperature.

To exemplify the difference let us consider a continuum of energies for a multi electronic system. At $0\, \mathrm K$ all the levels are completely filled from the bottom. The energy of the highest occupied level is then said to be the Fermi energy, although this is not its definition. At finite temperature some electrons are excited and the levels are no longer completely filled up to a given one which forbids us to use the concept of Fermi energy. The best one can do now is to define the Fermi level as the level which has an occupation probability (according to the Fermi-Dirac distribution) of $1/2$.

For semiconductors as well as insulators, the Fermi energy falls in the the band gap. This is actually general property of systems presenting discrete levels of energy. To understand it one has to use the precise definition of the Fermi energy which is the chemical potential at zero temperature. Consider a system with discrete energies, with $N$ occupied levels and at $0\, \mathrm K$. The Fermi-Dirac distribution is given by $$n=\frac{1}{\exp\left(\frac{E-E_F}{kT}\right)+1},$$ where $E_F$ is the the chemical potential at zero temperature, aka the Fermi energy. The occupation of the $N$th level is $1$ and from the above equation this gives $E_N<E_F$. On the other hand, the occupation of the $(N+1)$th level is $0$ which leads to $E_{N+1}>E_F$. Hence, $$E_N<E_F<E_{N+1}.$$ In particular, for insulators and semiconductors the Fermi level shall be in the gap between the valence band (whose last level is $E_N$) and the conduction band (whose first level is $E_{N+1}$). Note that only for continuum levels the Fermi energy is equivalent to the energy of the highest occupied level at zero temperature.

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