Fermi surface

Fermi surface [′fer·mē ‚sər·fəs]

(solid-state physics)

A constant-energy surface in the space containing the wave vectors of states of members of an assembly of independent fermions, such as electrons in a semiconductor or metal, whose energy is that of the Fermi level.

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Fermi surface

The surface in the electronic wavenumber space of a metal that separates occupied from empty states. Every possible state of an electron in a metal can be specified by the three components of its momentum, or wavenumber. The name derives from the fact that half-integral spin particles, such as electrons, obey Fermi-Dirac statistics and at zero temperature fill all levels up to a maximum energy called the Fermi energy, with the remaining states empty. See Fermi-Dirac statistics

The fact that such a surface exists for any metal, and the first direct experimental determination of a Fermi surface (for copper) in 1957, were central to the development of the theory of metals. A surprise arising from the earliest determined Fermi surfaces was that many of the shapes were close to what would be expected if the electrons interacted only weakly with the crystalline lattice. The long-standing free-electron theory of metals was based upon this assumption, but most physicists regarded it as a serious oversimplification. See Free-electron theory of metals

Fermi surface of copper, as determined in 1957; two shapes were found to be consistent with the original data, and the other, slightly more deformed version turned out to be correct

The momentum p of a free electron is related to the wavelength λ of the electronic wave by the equation below,

where ℏ is Planck's constant divided by 2&pgr;. The ratio 2&pgr;/λ, taken as a vector in the direction of the momentum, is called the wavenumber k. If the electron did not interact with the metallic lattice, the energy would not depend upon the direction of k, and all constant-energy surfaces, including the Fermi surface, would be spherical.

The Fermi surface of copper was found to be distorted (see illustration) but was still a recognizable deformation of a sphere. The polyhedron surrounding the Fermi surface in the illustration is called the Brillouin zone. It consists of Bragg-reflection planes, the planes made up of the wavenumbers for which an electron can be diffracted by the periodic crystalline lattice. The square faces, for example, correspond to components of the wavenumber along one coordinate axis equal to 2&pgr;/a, where a is the cube edge for the copper lattice. For copper the electrons interact with the lattice so strongly that when the electron has a wavenumber near to the diffraction condition, its motion and energy are affected and the Fermi surface is correspondingly distorted. The Fermi surfaces of sodium and potassium, which also have one conducting electron per atom, are very close to a sphere. These alkali metals are therefore more nearly free-electron-like. See Brillouin zone

In transition metals there are electrons arising from atomic d levels, in addition to the free electrons, and the corresponding Fermi surfaces are more complex than those of the nearly free-electron metals. However, the Fermi surfaces exist and have been determined experimentally for essentially all elemental metals.

The motion of the electrons in a magnetic field provides the key to experimentally determining the Fermi surface shapes. The simplest method conceptually derives from ultrasonic attenuation. Sound waves of known wavelength pass through the metal and a magnetic field is adjusted, yielding fluctuations in the attenuation as the orbit sizes match the sound wavelength. This measures the diameter of the orbit and Fermi surface. The most precise method uses the de Haas-van Alphen effect, based upon the quantization of the electronic orbits in a magnetic field. Fluctuations in the magnetic susceptibility give a direct measure of the cross-sectional areas of the Fermi surface. See De Haas-van Alphen effect, Skin effect (electricity);, Ultrasonics