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| In [[condensed matter physics]], the '''Fermi surface''' is an abstract boundary in [[reciprocal space]] useful for predicting the thermal, electrical, magnetic, and optical properties of [[metal]]s, [[semimetal]]s, and doped [[semiconductor]]s. The shape of the Fermi surface is derived from the periodicity and symmetry of the [[crystalline lattice]] and from the occupation of [[electronic band structure|electronic energy bands]]. The existence of a Fermi surface is a direct consequence of the [[Pauli exclusion principle]], which allows a maximum of one electron per quantum state.
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| ==Theory==
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| Consider a spinless ideal [[Fermi gas]] of <math>N</math> particles. According to [[Fermi–Dirac statistics]], the mean occupation number of a state with energy <math>\epsilon_i</math> is given by<ref name='Reif1965dist341'>{{harv|Reif|1965|p=341}}</ref>
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| <math>\langle n_i\rangle =\frac{1}{e^{(\epsilon_i-\mu)/k_BT}+1},</math>
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| where,
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| *<math>\left\langle n_i\right\rangle</math> is the mean occupation number
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| *<math>\epsilon_i</math> is the kinetic energy of the <math>i^{th}</math> state
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| *<math>\mu</math> is the ''[[internal chemical potential]]'' (at zero temperature, this is the maximum kinetic energy the particle can have, i.e. [[Fermi energy]] <math>\epsilon_F</math>)
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| Suppose we consider the limit <math>T\to 0</math>. Then we have,
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| <math>\left\langle n_i\right\rangle\approx\begin{cases}1 & (\epsilon_i<\mu) \\ 0 & (\epsilon_i>\mu)\end{cases}.</math>
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| By the [[Pauli exclusion principle]], no two particles can be in the same state. Therefore, in the state of lowest energy, the particles fill up all energy levels below <math>\epsilon_F</math>, which is equivalent to saying that ''<math>\epsilon_F</math> is the energy level below which there are exactly <math>N</math> states.
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| In momentum space, these particles fill up a sphere of radius <math>p_F</math>, the surface of which is called the '''Fermi surface'''<ref>K. Huang, ''Statistical Mechanics'' (2000), p244</ref>
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| 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
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| <math>k_F = \frac{\sqrt{2 m E_F}} {\hbar}</math>
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| determined by the valence electron concentration where <math>\hbar</math> is the [[reduced Planck's constant]]. A material whose Fermi level falls in a gap between bands is an [[Electrical insulation|insulator]] or semiconductor depending on the size of the [[bandgap]]. When a material's Fermi level falls in a bandgap, there is no Fermi surface.
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| [[Image:graphiteFS.png|thumb|A view of the [[graphite]] Fermi surface at the corner H points of the
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| [[Brillouin zone]] showing the trigonal symmetry of the electron and
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| hole pockets.]]
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| 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 <math>\vec{k}_z</math> direction. Often in a metal the Fermi surface radius <math>k_F</math> is larger than the size of the first [[Brillouin 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 <math>\vec{k}</math> is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown [[Modular arithmetic|modulo]] <math>\frac{2 \pi} {a}</math> (in the 1-dimensional case) where a is the [[lattice constant]]. In the three dimensional case the reduced zone scheme means that from any wavevector <math>\vec{k}</math> there is an appropriate number of reciprocal lattice vectors <math>\vec{K}</math> subtracted that the new <math>\vec{k}</math> now is closer to the origin in <math>\vec{k}</math>-space than to any <math>\vec{K}</math>. Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form [[ground state]]s where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are [[superconductor]]s, [[ferromagnet]]s, [[Jahn–Teller effect|Jahn–Teller distortions]] and [[spin density wave]]s.
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| The state occupancy of [[fermion]]s like electrons is governed by [[Fermi–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.
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| ==Experimental determination==
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| Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields <math>H</math>, for example the [[de Haas-van Alphen effect]] (dHvA) and the [[Shubnikov–de Haas effect]] (SdH). The former is an oscillation in [[magnetic susceptibility]] and the latter in [[resistivity]]. The oscillations are periodic versus <math>1/H</math> and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by [[Lev Landau]]. The new states are called Landau levels and are separated by an energy <math>\hbar \omega_c</math> where <math>\omega_c = eH/m^*c</math> is called the [[electron cyclotron resonance|cyclotron frequency]], <math>e</math> is the electronic charge, <math>m^*</math> is the electron [[effective mass (solid-state physics)|effective mass]] and <math>c</math> is the [[speed of light]]. In a famous result, [[Lars Onsager]] proved that the period of oscillation <math>\Delta H</math> is related to the cross-section of the Fermi surface (typically given in <math>\AA^{-2}</math>) perpendicular to the magnetic field direction <math>A_{\perp}</math> by the equation <math>A_{\perp} = \frac{2 \pi e \Delta H}{\hbar c}</math>. Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface.
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| 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.
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| [[Image:Fermi surface of BSCCO exp.jpg|thumb| [[Fermi surface of BSCCO]] measured by [[ARPES]]. The experimental data shown as an intensity plot in yellow-red-black scale. Green dashed rectagle represents the [[Brillouin zone]] of the CuO2 plane of [[BSCCO]].]]
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| The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see [[reciprocal lattice]]), and, consequently, the Fermi surface, is the [[angle resolved photoemission spectroscopy]] ([[ARPES]]). An example of the [[Fermi surface of superconducting cuprates]] measured by [[ARPES]] is shown in figure.
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| With [[positron 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 be [[polarized]], also the momentum distribution for the two [[Spin (physics)|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 was obtained in 1978.
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| ==See also==
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| *[[Fermi energy]]
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| *[[Brillouin zone]]
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| *[[Fermi surface of superconducting cuprates]]
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| *[[Kelvin probe force microscope]]
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| ==References==
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| {{reflist}}
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| *N. Ashcroft and N.D. Mermin, ''Solid-State Physics,'' ISBN 0-03-083993-9.
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| *W.A. Harrison, ''Electronic Structure and the Properties of Solids,'' ISBN 0-486-66021-4.
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| *[http://www.phys.ufl.edu/fermisurface/ VRML Fermi Surface Database]
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| *J. M. Ziman, ''Electrons in Metals: A short Guide to the Fermi Surface'' (Taylor & Francis, London, 1963), ASIN B0007JLSWS.
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| ==External links==
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| [[Category:Condensed matter physics]]
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| [[Category:Electric and magnetic fields in matter]]
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| [[Category:Enrico Fermi|Surface]]
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