Statistical mechanics, structure of matter Definition

Fermi surface

Builds on Electronic band structure Brillouin zone Fermi–Dirac statistics

Definition

Consider non-interacting electrons in a crystal with band functions $E_m \colon B \to \R$ , $m \ge 1$ , on the Brillouin zone $B$ . Let $E_F$ be the Fermi energy, the chemical potential at zero temperature fixed by the electron density. The Fermi surface is the set $S_F = \set{k \in B : E_m(k) = E_F \text{ for some } m \ge 1}.$ At zero temperature, by Fermi–Dirac statistics, the states with $E_m(k) < E_F$ are occupied and those with $E_m(k) > E_F$ are empty. So $S_F$ is the boundary of the occupied region in each band.

Remarks

Example: free electrons in $\R^3$ with $E(k) = \hbar^2 \abs{k}^2 / 2m$ . The Fermi surface is the sphere $\abs{k} = k_F$ with $E_F = \hbar^2 k_F^2 / 2m$ . Counting two spin states per wave vector, the density is $n = 2 \cdot \tfrac{4}{3}\pi k_F^3 / (2\pi)^3 = k_F^3 / 3\pi^2$ , so $k_F = (3\pi^2 n)^{1/3}$ .

Example: for the one-dimensional tight-binding band $E(k) = -2t\cos k$ at half filling, $E_F = 0$ and the Fermi surface is the pair of points $k = \pm\pi/2$ .

Non-example: in a band insulator $E_F$ lies in a spectral gap, so no $E_m$ takes the value $E_F$ and the Fermi surface is empty.

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Tag	`E4VEYCFB25`
URI	`m2://E4VEYCFB25`
Kind	definition
Writing	W0 stub
MSC	82D35, 82D20
Defines	Fermi surface
Uses	Electronic band structure, Brillouin zone, Fermi–Dirac statistics
Changed	3 November 2026

Fermi surface

Definition

Remarks

Reading path

Further reading

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