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Condensed matter physics From Wikipedia, the free encyclopedia Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the condensed phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the bonding and electromagnetic force between atoms. More exotic condensed phases include the superfluid and the Bose-Einstein condensate found in certain atomic systems at very low temperatures, the superconducting phase exhibited by conduction electrons in certain materials, and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics. Much progress has also been made in theoretical condensed matter physics. By one estimate, one third of all American physicists identify themselves as condensed matter physicists. Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term condensed matter physics was apparently coined by Philip Anderson and Volker Heine when they renamed their research group at Cavendish Laboratory - previously solid-state theory - in 1967. In 1978, the Division of Solid State Physics at the American Physical Society was renamed as the Division of Condensed Matter Physics. Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.One of the reasons for calling the field condensed matter physics is that many of the concepts and techniques developed for studying solids actually apply to fluid systems. For instance, the conduction electrons in an electrical conductor form a type of quantum fluid with essentially the same properties as fluids made up of atoms. In fact, the phenomenon of superconductivity, in which the electrons condense into a new fluid phase in which they can flow without dissipation, is very closely analogous to the superfluid phase found in helium 3 at low temperatures.Fermi energyThe 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 absolute zero temperature. This article requires a basic knowledge of quantum mechanics.Introduction In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle. This principle states that two identical fermions can not be in the same quantum state. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. To determine the lowest energy a system of fermions can have, we first group the states in sets with equal energy and order these sets by increasing energy. Starting with an empty system, we then add particles one at a time, consecutively filling up the unoccupied quantum states with lowest-energy. When all the particles have been put in, the Fermi energy is the energy of the highest occupied state. What this means is that even if we have extracted all possible energy from a metal by cooling it down to near absolute zero temperature (0 kelvins), the electrons in the metal are still moving around, the fastest ones would be moving at a velocity that corresponds to a kinetic energy equal to the Fermi energy. This is the Fermi velocity. The Fermi energy is one of the important concepts of condensed matter physics. It is used, for example, to describe metals, insulators, and semiconductors. It is a very important quantity in the physics of superconductors, in the physics of quantum liquids like low temperature helium (both normal 3He and superfluid 4He), and it is quite important to nuclear physics and to understand the stability of white dwarf stars against gravitational collapse.The Fermi energy (EF) of a system of non-interacting fermions is the increase in the ground state energy when exactly one particle is added to the system. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The chemical potential at zero temperature is equal to the Fermi energy.Illustration of the concept for a one dimensional square wellThe one dimensional infinite square well is a model for a one dimensional box. It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labeled by a single quantum number n and the energies are given by. Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1/2. Then only two particles can have the same energy i.e. two particles can have the energy of , or two particles can have energy E2 = 4E1 and so forth. The reason that two particles can have the same energy is that a spin-1/2 particle can have a spin of 1/2 (spin up) or a spin of -1/2 (spin down), leading to two states for each energy level. When we look at the total energy of this system, the configuration for which the total energy is lowest (the ground state), is the configuration where all the energy levels up to n=N/2 are occupied and all the higher levels are empty. The Fermi energy is therefore. The three-dimensional caseThe three-dimensional isotropic case is known as the fermi sphere.Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers nx, ny, and nz. The single particle energies arenx, ny, nz are positive integers. There are multiple states with the same energy, for example E100 = E010 = E001. Now lets put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case for N is large.If we introduce a vector then each quantum state corresponds to a point in n-space with EnergyThe number of states with energy less than Ef is equal to the number of states that lie within a sphere of radius in the region of n-space where nx, ny, nz are positive. In the ground state this number equals the number of fermions in the system.The free fermions that occupy the lowest energy states form a sphere in momentum space. The surface of this sphere is the Fermi surface.the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We findso the Fermi energy is given by Which results in a relationship between the fermi energy and the number of particles per volume (when we replace L2 with V2/3): The total energy of a fermi sphere of N0 fermions is given by Typical fermi energies White dwarfsStars known as White dwarfs have mass comparable to our Sun, but have a radius about 100 times smaller. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a White dwarf are on the order of 1036 electrons/m3. This means their fermi energy is:Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:where A is the number of nucleons. The number density of nucleons in a nucleus is therefore:Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus, and vice versa.So the fermi energy of a nucleus is about:The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the fermi energy usually given is 38 MeV. Fermi levelThe Fermi level is the highest occupied energy level at absolute zero, that is, all energy levels up to the Fermi level are occupied by electrons. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a Fermi sea of electron energy states. 1 In this state (at 0 K), the average energy of an electron is given by: where Ef is the Fermi energy.The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by: where me is the mass of the electron.This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy.The Fermi velocity is the velocity of fermions at the Fermi surface. It is defined by: where me is the mass of the electron.Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by: where k is the Boltzmann constant.Quantum mechanicsAccording to quantum mechanics, fermions - particles with a half-integer spin, usually 1/2, such as electrons - follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO); however, within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV. Pinning of Fermi levelWhen the energy density of surface states is very high (1013/cm2), the position of the Fermi level is determined by the neutral level of the Surface states 2 and becomes independent of Work Function 3 variations.Free electron gasIn the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labeled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the quasi-momentum or crystal momentum (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.The Fermi energy of the free electron gas is related to the chemical potential by the equationwhere EF is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature EF/k. The characteristic temperature is on the order of 105 K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.References Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.Table of fermi energies, velocities, and temperatures for various elements. a discussion of fermi gases and fermi temperatures.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. 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 allows a maximum of one electron per quantum state.TheoryFormally 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 EF 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 determined by the valence electron concentration where is the reduced Plancks 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 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 kF 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.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.Experimental determinationde 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 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 H 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.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 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 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.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. 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.ReferencesN. Ashcroft and N.D. Mermin, Solid-State