# What's the difference between Fermi Energy and Fermi Level?

I'm a bit confused about the difference between these two concepts. According to Wikipedia the Fermi energy and Fermi level are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level?

• – Qmechanic Jun 28 '12 at 17:58

If you consider a typical metal the highest energy band (i.e. the conduction band) is partially filled. The conduction band is effectively continuous, so thermal energy can excite electrons within this band leaving holes lower in the band.

At absolute zero there is no thermal energy, so electrons fill the band starting from the bottom and there is a sharp cutoff at the highest occupied energy level. This energy defines the Fermi energy.

At finite temperatures there is no sharply defined most energetic electron because thermal energy is continuously exciting electrons within the band. The best you can do is define the energy level with a 50% probability of occupation, and this is the Fermi level.

It depends on who you ask.

If you ask someone with solid-state physics background, they will probably answer along the lines of Colin McFaul or John Rennie: The fermi level is the same as chemical potential (or maybe one should say "electrochemical potential"), i.e. the energy at which a state has 50% chance of being occupied, while the fermi energy is the fermi level at absolute zero.

If you ask someone with semiconductor engineering background, they will probably give the same definition of "fermi level", but they will say that "fermi energy" means exactly the same thing as fermi level. (The obvious question is, "Then what term would a semiconductor engineer use to describe the fermi level at absolute zero? The answer is, they call it "the fermi level at absolute zero"!)

The Fermi energy is as you describe: it is the highest occupied level at absolute zero. The Fermi level is the chemical potential. It is the energy level with 50% chance of being occupied at finite temperature T. The Fermi energy does not depend on temperature; the Fermi level does depend on temperature.

• The fermi level is NOT necessarily higher than fermi energy. It is higher if the density of states is an increasing function of energy, or lower if the density of states is a decreasing function of energy. (I might have gotten that backwards.) – Steve Byrnes Jun 30 '12 at 2:26
• That is really interesting to learn! I just assumed that that the Fermi level would always be greater than or equal to the Fermi energy (even mis-wrote what I meant). I was unable to find a clear statement of that possibility either way in any of my books, so I will take your word for it. It seems very odd; do you have any reference to a system where the Fermi level is lower than the Fermi energy? – Colin McFaul Jul 2 '12 at 22:58
• Take a small-bandgap macroscopic pure semiconductor crystal with just two n-type dopant atoms and one p-type dopant atom. At room temperature the fermi level is almost exactly in the center of the bandgap--the dopant atoms are irrelevant. At absolute zero the fermi level (i.e. fermi energy) is at the n-type impurity level near the conduction band minimum, i.e. higher. You can switch the letters "n" and "p" to get a situation where fermi energy is lower. – Steve Byrnes Jul 3 '12 at 22:13
• Is it also true that doping concentration would alter both Fermi level and Fermi energy of a doped semiconductor? – dr3patel Jul 30 '14 at 4:21
• The Fermi level, or chemical potential, is actually lower than the Fermi energy for any system in which the density of states increases with energy--such as electrons in a metal at non-zero temperature--so Steve Byrnes did in fact have it backwards in the first comment. Reference: Schroeder, Thermal Physics, Fig. 7.14. – Richard H Downey Mar 31 '17 at 0:50

Fermi level as a state with 50% chance of being occupied by an electron for the given temperature of the solid and at absolute zero temperature occupancy is 100%.

Fermi energy is the corresponding energy of Fermi level.

Fermi energy is difference of energies of highest and lowest occupied single state particle. But, Fermi level is the sum of total energy of a particle (i.e) the sum of kinetic and potential energies.