# Fermi Level

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## Fermi level

[′fer·mē ‚lev·əl]
(statistical mechanics)
The energy level at which the Fermi-Dirac distribution function of an assembly of fermions is equal to one-half. Also known as Fermi energy.

## Fermi Level

the energy level below which all energy states of the particles of a degenerate gas that obey Fermi-Dirac statistics are occupied at a temperature of absolute zero (seeSTATISTICAL MECHANICS). Particles of a degenerate gas that obey Fermi-Dirac statistics are called fermions. The existence of the Fermi level is a consequence of the Pauli exclusion principle, according to which not more than (2s + 1) particles, where s is the spin of a particle, may occupy a state with a specified momentum p. The Fermi level coincides with the values of the chemical potential of a fermion gas at T = 0°K.

The Fermi level ℰF can be expressed in terms of the number n of gas particles per unit volume: ℰF = [(2πℏ)2/2m] [3n/4π(2s + 1)], where m is the particle mass. The quantity is called the Fermi momentum. At T = 0°K, the particles occupy all states with momenta p < pF; all states with p > pF are empty. In other words, at T = 0°K, fermions occupy states in momentum space that lie within a sphere p2 = 2mF of radius pF; this sphere is called the Fermi sphere. Upon heating, some particles move from a state with p < pF to a state with p > pF. Vacant sites, called holes, occur within the Fermi sphere. The quantity is called the Fermi velocity and specifies the upper limit to the velocities of fermions at T = 0°K.

A degenerate gas consisting of conduction electrons in a solid at T = 0°K occupies surfaces of more complex shape in momentum space (see).

### REFERENCE

Landau, L. D., and E. M. Lifshits. Statisticheskaia fizika, 2nd ed. Moscow, 1964. (Teoreticheskaia fizika, vol. 5.)

M. I. KAGANOV

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