- Fermi surface
In

condensed matter physics , the**Fermi surface**is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties ofmetal s,semimetal s, and dopedsemiconductor s. The shape of the Fermi surface is derived from the periodicity and symmetry of thecrystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of thePauli exclusion principle , which allows a maximum of one electron per quantum state.**Theory**Formally speaking, the Fermi surface is a surface of constant energy in $vec\{k\}$-space where $vec\{k\}$ is the

wavevector of theelectron . Atabsolute zero temperature the Fermi surface separates the unfilled electronic orbitals from the filled ones. The energy of the highest occupied orbitals is known as theFermi energy $E\_F$ which, in the zero temperature case, resides on the Fermi level. The linear response of a metal to an electric, magnetic or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy. Free-electron Fermi surfaces are spheres of radius $k\_F\; =\; sqrt\{2\; m\; E\_F\}/hbar$ determined by the valence electron concentration where $hbar$ is thereduced Planck's constant . A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of thebandgap . When a material's Fermi level falls in a bandgap, there is no Fermi surface.Fermi surface at the corner H points of the

Brillouin zone showing the trigonal symmetry of the electron andhole pockets.]Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure illustrates the

anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energy along the $vec\{k\}\_z$ direction. Often in a metal the Fermi surface radius $k\_F$ is larger than the size of the firstBrillouin zone which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface can be displayed in an extended-zone scheme where $vec\{k\}$ is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown modulo $frac\{2\; pi\}\; \{a\}$ where a is thelattice constant . Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to formground state s where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states aresuperconductor s,ferromagnet s, Jahn-Teller distortions andspin density wave s.The state occupancy of

fermion s like electrons is governed byFermi-Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.**Experimental determination****de Haas-van Alphen effect.**Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields $H$, for example thede Haas-van Alphen effect (dHvA) and theShubnikov-De Haas effect (SdH). The former is an oscillation inmagnetic susceptibility and the latter inresistivity . The oscillations are periodic versus $1/H$ and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted byLev Landau . The new states are called Landau levels and are separated by an energy $hbar\; omega\_c$ where $omega\_c\; =\; eH/m^*c$ is called the cyclotron frequency, $e$ is the electronic charge, $m^*$ is the electroneffective mass and $c$ is thespeed of light . In a famous result,Lars Onsager proved that the period of oscillation $Delta\; H$ is related to the cross-section of the Fermi surface (typically given in $AA^\{-2\}$) perpendicular to the magnetic field direction $A\_\{perp\}$ by the equation $A\_\{perp\}\; =\; frac\{2\; pi\; e\; Delta\; H\}\{hbar\; c\}$. Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface.Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a

mean free path . Therefore dHvA and SdH experiments are usually performed at high-field facilities like the [*http://www.hfml.ru.nl/ High Field Magnet Laboratory*] in Netherlands, [*http://ghmfl.grenoble.cnrs.fr/ Grenoble High Magnetic Field Laboratory*] in France, the [*http://akahoshi.nims.go.jp/TML/english/ Tsukuba Magnet Laboratory*] in Japan or the [*http://www.magnet.fsu.edu/ National High Magnetic Field Laboratory*] in the United States.**Angle resolved photoemission.**The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (seereciprocal lattice ), and, consequently, the Fermi surface, is theangle resolved photoemission spectroscopy (ARPES ). An example of theFermi surface of superconducting cuprates measured byARPES is shown in figure.**Two photon positron annihilation.**Withpositron annihilation the two photons carry the momentum of the electron away; as the momentum of a thermalized positron is negligible, in this way also information about the momentum distribution can be obtained. Because the positron can bepolarized , also the momentum distribution for the two spin states in magnetized materials can be obtained. Another advantage with De Haas-Van Alphen-effect is that the technique can be applied to non-dilute alloys. In this way the first determination of a "smeared Fermi surface" in a 30% alloy has been obtained in 1978.**ee also***

Fermi energy

*Brillouin zone

*Fermi surface of superconducting cuprates **References***N. Ashcroft and N.D. Mermin, "Solid-State Physics," ISBN 0-03-083993-9.

*W.A. Harrison, "Electronic Structure and the Properties of Solids," ISBN 0-486-66021-4.

* [*http://www.phys.ufl.edu/fermisurface/ VRML Fermi Surface Database*]**External links**

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