In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which prevents fermions from all crowding into the same state. Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Hot metal work from a blacksmith In chemistry, a metal (Greek: Metallon) is an element that readily forms positive ions (cations) and has metallic bonds. ... Together with the metals and nonmetals, the metalloids (in Greek metallon = metal and eidos = sort - also called semimetals) form one of the three categories of chemical elements as classified by ionization and bonding properties. ... A semiconductor is a solid whose electrical conductivity can be controlled over a wide range, either permanently or dynamically. ... Rose des Sables (Sand Rose), formed of gypsum crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In solid state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is forbidden or allowed to have. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state simultaneously. ...

Theory

Formally speaking, the Fermi surface is a surface of constant energy in -space where is the wavevector of the electron. At absolute zero temperature the Fermi surface separates the unfilled electronic orbitals from the filled ones. The energy of the highest occupied orbitals is known as the Fermi energy E_F or 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 determined by the valence electron concentration where is reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. When a materials' Fermi level falls in a bandgap, there is no Fermi surface. A wave vector is a vector representation of a wave. ... The electron is a fundamental subatomic particle that carries an electric charge. ... Absolute zero is the point on the thermodynamic (absolute) temperature scale where all kinetic motion in the particles comprising matter ceases and they are at complete rest in the “classic” (non-quantum mechanical) sense. ... It has been suggested that Fermi_temperature be merged into this article or section. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... // Definition An Insulator is a material or object which resists the flow of electric charge. ... In solid state physics and related applied fields, the band gap is the energy difference between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. ...

A view of the graphite Fermi surface at the corner H points of the Brillouin zone showing the trigonal symmetry of the electron and hole 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 direction. Often in a metal the Fermi surface radius k_F 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 is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown modulo where a is the lattice constant. Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets, Jahn-Teller distortions and spin density waves. Image File history File links GraphiteFS.png‎ File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Fermi surface ... Image File history File links GraphiteFS.png‎ File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Fermi surface ... Graphite (named by Abraham Gottlob Werner in 1789 from the Greek γραφειν: to draw/write, for its use in pencils) is one of the allotropes of carbon. ... In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Lattice constant, or a, defines the distance between atoms in crystal lattice. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ... The Jahn-Teller effect, sometimes also known as Jahn-Teller distortion, describes the geometrical distortion of the electron cloud in a non-linear molecule under certain situations. ... Spin-density wave (SDW) and charge-density wave (CDW) are names for two similar low-energy ordered states of solids. ...

The state occupancy of fermions 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. In particle physics, fermions are particles with half-integer spin. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ...

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 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 1 / H 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 where ω_c = eH / m ^* c is called the cyclotron frequency, e is the electronic charge, m ^* is the electron effective mass and c is the speed of light. In a famous result, Lars Onsager proved that the period of oscillation ΔH is related to the cross-section of the Fermi surface (typically given in ) perpendicular to the magnetic field direction by the equation . Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface. The de Haas-van Alfven effect {dHvA} was discovered in 1930 by Wander Johannes de Haas and PM van Alfven. ... In electrical engineering, the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. ... Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of electric current. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Electron cyclotron resonance is a phenomenon observed both in plasma physics and condensed matter physics. ... In solid state physics, a particles effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. In metric units, c is exactly 299,792,458 metres per second (1,079,252,848. ... Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian physical chemist, winner of the 1968 Nobel Prize in Chemistry. ...

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 at usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic Field Laboratory in France, the Tsukuba Magnet Laboratory in Japan or the National High-Field Magnet Lab in the United States. For sound waves in an enclosure, the mean free path is the average distance the wave travels between reflections off of the enclosures walls. ...

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.

Angle resolved photoemission. 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. Image File history File linksMetadata Fermi_surface_of_BSCCO_exp. ... Image File history File linksMetadata Fermi_surface_of_BSCCO_exp. ... Fig. ... Angle resolved photoemission spectroscopy (ARPES), also known as ARUPS (angle resolved ultraviolet photoemission spectroscopy), is a direct experimental technique to observe the distribution of the electrons (more precisely, the density of single particle electronic excitations) in the reciprocal space of solids. ... In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ... Bismuth strontium calcium copper oxide, or BSCCO, chemical formula Bi2Sr2CanCun+1O2n+6, is a family of high-temperature superconductors. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ... Angle resolved photoemission spectroscopy (ARPES), also known as ARUPS (angle resolved ultraviolet photoemission spectroscopy), is a direct experimental technique to observe the distribution of the electrons (more precisely, the density of single particle electronic excitations) in the reciprocal space of solids. ... Angle resolved photoemission spectroscopy (ARPES), also known as ARUPS (angle resolved ultraviolet photoemission spectroscopy), is a direct experimental technique to observe the distribution of the electrons (more precisely, the density of single particle electronic excitations) in the reciprocal space of solids. ... Fig. ... Angle resolved photoemission spectroscopy (ARPES), also known as ARUPS (angle resolved ultraviolet photoemission spectroscopy), is a direct experimental technique to observe the distribution of the electrons (more precisely, the density of single particle electronic excitations) in the reciprocal space of solids. ...

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.
• VRML Fermi Surface Database