Fermi surface

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. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across A semimetal is a material with a small overlap in the energy of the conduction band and Valence bands However the bottom of the conduction band is A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that 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. 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 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. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925

1 Theory
2 Experimental determination
3 See also
4 References
5 External links

Theory

Formally speaking, the Fermi surface is a surface of constant energy in $\vec{k}$ -space where $\vec{k}$ is the wavevector of the electron. A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J At absolute zero temperature the Fermi surface separates the unfilled electronic orbitals from the filled ones. Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point The energy of the highest occupied orbitals is known as the Fermi energy $E F$ which, in the zero temperature case, resides on the Fermi level. The Fermi energy is a concept in Quantum mechanics usually referring to the energy of the highest occupied Quantum state in a system of Fermions at 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 the reduced Planck's constant. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. An insulator, also called a Dielectric, is a material that resists the flow of Electric current. In Solid state physics and related applied fields a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states When a material's Fermi level falls in a bandgap, there is no Fermi surface.

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. The Mineral graphite, as with Diamond and Fullerene, is one of the Allotropes of carbon. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the

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. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity 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. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the 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 the lattice constant. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers The Lattice Constant refers to the constant distance between Unit cells in a Crystal lattice. 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. In Quantum mechanics, a stationary state is an Eigenstate of a Hamiltonian, or in other words a state of definite energy Examples of such ground states are superconductors, ferromagnets, Jahn-Teller distortions and spin density waves. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it 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 Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that 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 the de Haas-van Alphen effect (dHvA) and the Shubnikov-De Haas effect (SdH). The de Haas–van Alphen effect, often abbreviated to dHvA, was discovered in 1930 by Wander Johannes de Haas and PM van Alphen An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense time varying Magnetic fields, the Shubnikov-de Haas effect (ShdH The former is an oscillation in magnetic susceptibility and the latter in resistivity. In Electromagnetism the magnetic susceptibility ( Latin: susceptibilis “receptiveness” is the degree of Magnetization of a material in response Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of Electric current. 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. Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 The new states are called Landau levels and are separated by an energy $\hbar \omega_c$ where $ω c = e H / m * c$ is called the cyclotron frequency, $e$ is the electronic charge, $m *$ is the electron effective mass and $c$ is the speed of light. Electron cyclotron resonance is a phenomenon observed both in Plasma physics and Condensed matter physics. In Solid state physics, a particle's effective mass is the Mass it seems to carry in the semiclassical model of transport in a Crystal. In a famous result, Lars Onsager proved that the period of oscillation $Δ H$ 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 $A_{\perp} = \frac{2 \pi e \Delta H}{\hbar c}$ . Lars Onsager ( November 27, 1903 &ndash October 5, 1976) was a Norwegian – American physical chemist and 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. In Physics the mean free path of a particle is the average distance covered by a particle ( Photon, Atom or Molecule) between subsequent impacts Therefore dHvA and SdH experiments are 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 Magnetic Field Laboratory in the United States.

Fermi surface of BSCCO measured by ARPES. The electronic structure of superconducting cuprates also called high temperature superconductors ( HTSC) is highly anisotropic Angle resolved photoemission spectroscopy ( ARPES), also known as ARUPS -angle resolved Ultraviolet Photoemission spectroscopy - is 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. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the Bismuth strontium calcium copper oxide, or BSCCO (pronounced "bisko" is a family of High-temperature superconductors having the generalized chemical

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). In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 Angle resolved photoemission spectroscopy ( ARPES), also known as ARUPS -angle resolved Ultraviolet Photoemission spectroscopy - is Angle resolved photoemission spectroscopy ( ARPES), also known as ARUPS -angle resolved Ultraviolet Photoemission spectroscopy - is An example of the Fermi surface of superconducting cuprates measured by ARPES is shown in figure. The electronic structure of superconducting cuprates also called high temperature superconductors ( HTSC) is highly anisotropic Angle resolved photoemission spectroscopy ( ARPES), also known as ARUPS -angle resolved Ultraviolet Photoemission spectroscopy - is

Two photon positron annihilation. 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 states in magnetized materials can be obtained. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin 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.

References

N. The Fermi energy is a concept in Quantum mechanics usually referring to the energy of the highest occupied Quantum state in a system of Fermions at In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the The electronic structure of superconducting cuprates also called high temperature superconductors ( HTSC) is highly anisotropic 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

External links

Dictionary

-noun

(physics) An abstract boundary, of constant energy, useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors.

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