What Is the Fermi Level?


Fermi-Dirac Statistics, Fermi Level and Energy

f(E)=\frac{1}{1+exp\left (\frac{E-E_F}{kT}\right )}

To describe the states of electrons in a solid, Fermi-Dirac statistics must be used. Looking at this equation, we can see that as temperature increases, the chance of finding an electron at higher energies becomes more likely. By graphing this function, we see that the probability of an electron filling a state increases with decreasing energy. This makes sense, since electrons stabilize themselves as much as possible by first occupying the lowest energy states. If we set the energy equal to the EF, we find that the probability of finding an electron is one half, or 50%. The Fermi level is the energy level at which there is a 50% chance that an electron has filled the level. This is not to be confused with Fermi energy, where Fermi energy is the energy at which the highest level is filled at absolute zero. Below is a table with their differences:

Fermi EnergyFermi Level
Constant, based on material's electron concentrationVariable, changes temperature
Defined at absolute 0KDefined using a function of temperature
Probability not taken into consideration (technically, it's 100%)50% chance of finding an electron filling the corresponding energy level
Maximum energy for electrons to fill at 0K: definedIn intrinsic semiconductors and insulators, lies halfway in between the bandgap, so it doesn't exist

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