In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at absolute zero temperature.^[1]

Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of electronic energy bands. The existence of a Fermi surface follows directly from the Pauli exclusion principle, which allows only one electron per quantum state.^[2]^[3]^[4]

The study of Fermi surfaces is called fermiology.

Fermi surface and electron momentum density of copper measured using 2D ACAR techniques.

Theory

For an ideal Fermi gas, the occupation of quantum states is governed by the Fermi–Dirac distribution: $⟨ n_{i} ⟩ = \frac{1}{e^{(ϵ_{i} - μ) / k_{B} T} + 1}$

At zero temperature (Template:Math), this simplifies to: $⟨ n_{i} ⟩ = {\begin{cases} 1 & (ϵ_{i} < μ) \\ 0 & (ϵ_{i} > μ) \end{cases}$

All states below the Fermi energy are filled, while all above are empty. In momentum space, these occupied states form a sphere of radius Template:Math, whose boundary is the Fermi surface.

For a free electron gas: $k_{F} = \frac{\sqrt{2 m E_{F}}}{ℏ}$

The shape of the Fermi surface determines how electrons respond to electric, magnetic, and thermal fields. Therefore, many physical properties of metals—such as conductivity—are controlled by states near the Fermi surface.

Fermi surface of graphite showing anisotropic electron and hole pockets in the Brillouin zone.

In real materials, Fermi surfaces can be highly complex. For example, graphite exhibits both electron and hole pockets due to multiple bands crossing the Fermi level. In many metals, the Fermi surface extends beyond the first Brillouin zone and is folded back into it using the reduced-zone scheme.

Materials in which the Fermi level lies inside a band gap (such as semiconductors and insulators) do not have a Fermi surface.

Physical significance

The Fermi surface plays a central role in determining:

electrical conductivity
thermal conductivity
magnetic properties
stability of low-temperature phases

Systems with a high density of states at the Fermi level often become unstable and develop new ground states such as superconductivity, ferromagnetism, or spin density waves.

At finite temperatures, the sharp boundary of the Fermi surface becomes slightly blurred due to thermal excitations.

Experimental determination

Fermi surface of a cuprate superconductor measured using angle-resolved photoemission spectroscopy (ARPES).

Fermi surfaces can be measured experimentally using several techniques:

These methods rely on quantum oscillations or direct measurement of electron energies in momentum space.

A key result by Lars Onsager relates oscillation periods in magnetic fields to the cross-sectional area of the Fermi surface: $A_{⊥} = \frac{2 π e Δ H}{ℏ c}$

Physics:Quantum Fermi surfaces

Theory

Physical significance

Experimental determination

See also

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