Fermi energy¶

Fermi energy is a concept in quantum mechanics that usually refers 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. It is applied in determining the electrical and thermal characteristics of the solids. It is often used interchangeably with Fermi level, although both the terms are equal at absolute zero temperature, they are different at other temperatures.

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.

The reason for the existence of this energy level is due to Pauli exclusion principle, which states two fermions cannot occupy that same quantum state. So, if a system has more than one fermion, each fermion has a different set of magnetic quantum numbers associated with it. The Fermi Temperature(\(T_F\)) can be defined as the energy of the Fermi level divided by the Boltzmann’s constant. It is also the temperature at which the energy of the electron is equal to the Fermi energy. It is the measure of the electrons in the lower states of energy in metal.

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