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In quantum mechanics, particles with a half-integer spin, usually spin 1/2 (for example electrons) follow the Pauli exclusion principle, which states that no two particles may occupy the same quantum state. Such particle are usually referred to as fermions, as opposed to bosons, whci can share the same quantum state. When a number of electrons are put into a system, in the ground state (i.e., at zero temperature) electrons will occupy higher energy levels when the lower ones are filled up. The Fermi energy (EF) is the energy of the highest occupied state at zero temperature. Fermi energy is a central concept for the whole solid state physics.
In periodic solids, quantum states of electrons are characterized by their quasimomentum; in periodic metals occupied one-electron states are separated from unoccupied ones by a surface in the momentum space which is called the Fermi surface, and whose volume is defined by the number of electrons in the system (this theorem is known as Luttinger theorem). Its topology (sometimes called Fermiology) is directly related to the transport properties of metals, such as electrical conductivity. Fermi surface of most known metals are well studied both theoretically and experimentally.
Fermi energy was named after Enrico Fermi who, with Paul Dirac, derived the Fermi-Dirac statistics. These statistics allow one to predict the behaviour of large numbers of electrons under certain circumstances, especially in solids. The equations of quantum mechanics would otherwise be too hard to solve in such situations.
The Fermi energy of a three-dimensional, non-interacting, non-relativistic Fermi gas (or free electron gas) is related to the chemical potential by the equation
- <math>\mu = \epsilon _F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{\epsilon _F}\right) ^2 + \frac{\pi^4}{80} \left(\frac{kT}{\epsilon _F}\right)^4 + ... \right] <math>
where εF 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 105K for a metal, hence at room temperature (300K), 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.
See also
solid state physics, semiconductors, electrical engineering, electronics, statistical mechanics, thermodynamics
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