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| {{distinguish|Fermi level}}
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| The '''Fermi energy''' is a concept in [[quantum mechanics]] usually referring to the energy difference between the highest and lowest occupied single-particle states, in a quantum system of non-interacting [[fermion]]s at [[absolute zero]] [[temperature]].
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| In a [[Fermi gas]] the lowest occupied state is taken to have zero kinetic energy, whereas in a [[metal]] the lowest occupied state is typically taken to mean the bottom of the [[conduction band]].
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| Confusingly, the term "Fermi energy" is often used to refer to a different but closely related concept, the [[Fermi level|Fermi ''level'']] (also called [[electrochemical potential]]).<ref>The use of the term "Fermi energy" as synonymous with [[Fermi level]] (a.k.a. [[electrochemical potential]]) is widespread in semiconductor physics. For example: [http://books.google.com/books?id=n0rf9_2ckeYC&pg=PA49 ''Electronics (fundamentals And Applications)''] by D. Chattopadhyay, [http://books.google.com/books?id=lmg13dHPKg8C&pg=PA113 ''Semiconductor Physics and Applications''] by Balkanski and Wallis.</ref>
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| There are a few key differences between the Fermi level and Fermi energy, at least as they are used in this article:
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| * The Fermi energy is only defined at absolute zero, while the Fermi level is defined for any temperature.
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| * The Fermi energy is an energy ''difference'' (usually corresponding to a [[kinetic energy]]), whereas the Fermi level is a total energy level including kinetic energy and potential energy.
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| * The Fermi energy can only be defined for [[Fermi-Dirac statistics|non-interacting fermions]] (where the potential energy or band edge is a static, well defined quantity), whereas the Fermi level (the electrochemical potential of an electron) remains well defined even in complex interacting systems, at thermodynamic equilibrium.
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| Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state,
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| then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature.
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| ==Introduction==
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| ===Context===
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| In [[quantum mechanics]], a group of particles known as [[fermion]]s (for example, [[electron]]s, [[proton]]s and [[neutron]]s) obey the [[Pauli exclusion principle]]. This states that two fermions cannot occupy the same [[quantum state]]. Since an idealized non-interacting Fermi gas can be analyzed in terms of single-particle [[stationary state]]s, we can thus say that two fermions cannot occupy the same stationary state. These stationary states will typically be distinct in energy. To find the ground state of the whole system, we start with an empty system, and add particles one at a time, consecutively filling up the unoccupied stationary states with the lowest energy. When all the particles have been put in, the '''Fermi energy''' is the kinetic energy of the highest occupied state.
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| What this means is that even if we have extracted all possible energy from a Fermi gas by cooling it to near [[absolute zero]] temperature, the fermions are still moving around at a high speed. The fastest ones are moving at a velocity corresponding to a kinetic energy equal to the Fermi energy. This is the '''Fermi velocity'''. Only when the temperature exceeds the '''Fermi temperature''' do the electrons begin to move significantly faster than at absolute zero.
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| The Fermi energy is one of the important concepts in the [[solid state physics]] of metals and superconductors. It is also a very important quantity in the physics of [[Superfluid|quantum liquid]]s like low temperature [[helium]] (both normal and superfluid <sup>3</sup>He), and it is quite important to [[nuclear physics]] and to understand the stability of [[White dwarf|white dwarf stars]] against [[gravitational collapse]].
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| ===Advanced context===
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| The Fermi energy (''E<sub>F</sub>'') of a system of non-interacting [[fermion]]s is the increase in the [[ground state]] [[energy]] when exactly one particle is added to the system, minus the potential energy of that particle.
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| It can also be interpreted as the maximum kinetic energy of an individual fermion in this ground state. The [[internal chemical potential]] at zero temperature is equal to the Fermi energy.
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| ==Illustration of the concept for a one dimensional square well==
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| The one dimensional [[Particle in a box|infinite square well]] of length ''L'' 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
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| :<math>E_n = E_0 + \frac{\hbar^2 \pi^2}{2 m L^2} n^2. \,</math>
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| where <math>E_0</math> is the potential energy level inside the box.
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| 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 not more than two particles can have the same energy, i.e., two particles can have the energy of <math>E_1</math>, two other particles can have energy <math>E_2</math> and so forth. The reason that two particles can have the same energy is that a 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. In the configuration for which the total energy is lowest (the ground state), all the energy levels up to n=N/2 are occupied and all the higher levels are empty.
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| Defining the reference for the Fermi energy to be <math>E_0</math>, the Fermi energy is therefore given by
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| :<math>E_F=E_{N/2}-E_0=\frac{\hbar^2 \pi^2}{2 m L^2} (N/2)^2,</math>
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| for an even number of electrons (''N''), or an odd number of electrons (''N''-1).
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| ==Three-dimensional case==
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| The three-dimensional [[isotropic]] case is known as the '''Fermi sphere'''.
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| 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.
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| The states are now labeled by three quantum numbers n<sub>x</sub>, n<sub>y</sub>, and n<sub>z</sub>. The single particle energies are
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| ::<math>E_{n_x,n_y,n_z} = E_0 + \frac{\hbar^2 \pi^2}{2m L^2} \left( n_x^2 + n_y^2 + n_z^2\right) \,</math>
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| ::n<sub>x</sub>, n<sub>y</sub>, n<sub>z</sub> are positive integers.
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| There are multiple states with the same energy, for example <math>E_{211}=E_{121}=E_{112}</math>. Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case where N is large.
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| If we introduce a vector <math>\vec{n}=\{n_x,n_y,n_z\}</math> then each quantum state corresponds to a point in 'n-space' with energy
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| :<math>E_{\vec{n}} = E_0 + \frac{\hbar^2 \pi^2}{2m L^2} |\vec{n}|^2 \,</math>
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| With <math> |\vec{n}|^2 </math> denoting the square of the usual euclidian length <math> (\sqrt{n_x^2+n_y^2+n_z^2})^2 </math>
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| The number of states with energy less than E<sub>F</sub> + E<sub>0</sub> is equal to the number of states that lie within a sphere of radius <math>|\vec{n}_F|</math>{{why}} in the region of n-space where n<sub>x</sub>, n<sub>y</sub>, n<sub>z</sub> are positive. In the ground state this number equals the number of fermions in the system.
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| :<math>N =2\times\frac{1}{8}\times\frac{4}{3} \pi n_F^3 \,</math>
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| [[Image:Fermi energy momentum.svg|thumb|The free fermions that occupy the lowest energy states form a [[sphere]] in [[momentum]] space. The surface of this sphere is the [[Fermi surface]].]]
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| 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.
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| We find
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| :<math>n_F=\left(\frac{3 N}{\pi}\right)^{1/3} </math>
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| so the Fermi energy is given by
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| :<math>E_F = \frac{\hbar^2 \pi^2}{2m L^2} n_F^2 = \frac{\hbar^2 \pi^2}{2m L^2} \left( \frac{3 N}{\pi} \right)^{2/3}</math>
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| Which results in a relationship between the Fermi energy and the [[Particle number density|number of particles per volume]] (when we replace L<sup>2</sup> with V<sup>2/3</sup>):
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>E_F = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3} \,</math>
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| |}
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| The total energy of a Fermi sphere of <math>N</math> fermions is given by
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| :<math>E_t = N E_0 + {\int_0}^{N} E_F dN^\prime = ({3\over 5} E_F + E_0)N</math>
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| Therefore, the average energy of an electron is given by:
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| :<math> E_{av} = E_0 + \frac{3}{5} E_F </math>
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| ==Related quantities==
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| A related quantity is '''Fermi temperature''' <math>T_F</math>, defined as <math>E_F/k_B</math>, where <math>k_B</math> is the [[Boltzmann constant]] and <math>E_F</math> the '''Fermi energy'''. Other quantities defined in this context are '''Fermi momentum''', <math>p_F</math>, and '''Fermi velocity''', <math>v_F</math>, the [[momentum]] and [[group velocity]], respectively, of a [[fermion]] at the [[Fermi surface]]. (These quantities are ''not'' well-defined in cases where the Fermi surface is non-spherical). In the case of the quadratic dispersion relations given above, they are given by:<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html Fermi level and Fermi function], from [[HyperPhysics]]</ref>
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| :<math> p_F = \sqrt{2 m_e E_F} </math>
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| :<math> v_F = \frac{p_F}{m_e}</math>
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| where <math> m_e </math> is the mass of the electron.
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| The Fermi momentum can also be described as <math>p_F = \hbar k_F </math>, where <math>k_F</math> is the radius of the Fermi sphere and is called the '''Fermi wave vector'''. <ref>{{cite book | last = Ashcroft | first = Neil W. | last2 = Mermin | first2 = N. David | title = Solid State Physics | publisher = [[Henry Holt and Company|Holt, Rinehart and Winston]] | date = 1976 | isbn = 0-03-083993-9 }}</ref>
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| ==Arbitrary-dimensional case==
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| Using a volume integral on <math>d</math> dimensions, we can find the state density:
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| :<math>g(E)=2\int\frac{d^d\vec{k}}{(2\pi)^d/V}\delta\left(E-E_0-\frac{\hbar^2\vec{k}^2}{2m}\right)=V\frac{d\,m^{d/2}(E-E_0)^{d/2-1}}{(2\pi)^{d/2}\ \Gamma(d/2+1)\hbar^d}</math>
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| By then looking for the number of particles, we can extract the Fermi energy:
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| <math>n=\int_{E_0}^{E_0+E_F}g(E)dE</math>
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| To get:
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| :<math>E_F=\frac{2\pi\hbar^2}{m}\left(\tfrac{1}{2}\Gamma\left(\tfrac{d}{2}+1\right)n\right)^{2/d}</math>
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| ==Typical Fermi energies==
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| ===Metals===
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| The number density <math>N/V</math> of conduction electrons in metals ranges between approximately 10<sup>28</sup> and 10<sup>29</sup> electrons/m<sup>3</sup>, which is also the typical density of atoms in ordinary solid matter.
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| This number density produces a Fermi energy of the order:
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| ::<math>E_F = \frac{\hbar^2}{2m_e} \left( 3 \pi^2 \ 10^{28 \ \div \ 29} \ \mathrm{m}^{-3} \right)^{2/3} \approx 2 \ \div \ 10 \ \mathrm{eV} </math>
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| ===White dwarfs===
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| Stars known as [[white dwarfs]] have mass comparable to our [[Sun]], but have about a hundredth of its radius. The high densities means that the electrons are no longer bound to single nuclei and instead form a [[Degenerate matter|degenerate]] [[electron gas]]. The number density of electrons in a white dwarf is of the order of 10<sup>36</sup> electrons/m<sup>3</sup>.
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| This means their Fermi energy is:
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| ::<math>E_F = \frac{\hbar^2}{2m_e} \left( \frac{3 \pi^2 (10^{36})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 3 \times 10^5 \ \mathrm{eV} = 0.3 \ \mathrm{MeV}</math>
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| ===Nucleus===
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| Another typical example is that of the particles in a nucleus of an atom. The [[Nuclear size|radius of the nucleus]] is roughly:
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| ::<math>R = \left(1.25 \times 10^{-15} \mathrm{m} \right) \times A^{1/3}</math>
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| :where ''A'' is the number of [[nucleons]].
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| The number density of nucleons in a nucleus is therefore:
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| ::<math>n = \frac{A}{\begin{matrix} \frac{4}{3} \end{matrix} \pi R^3 } \approx 1.2 \times 10^{44} \ \mathrm{m}^{-3} </math>
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| 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 [[neutron]]s does not affect the Fermi energy of the [[proton]]s in the nucleus, and vice versa.
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| So the Fermi energy of a nucleus is about:
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| ::<math>E_F = \frac{\hbar^2}{2m_p} \left( \frac{3 \pi^2 (6 \times 10^{43})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 3 \times 10^7 \ \mathrm{eV} = 30 \ \mathrm{MeV} </math>
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| The [[Nuclear size|radius of the nucleus]] admits deviations around the value mentioned above, so a typical value for the Fermi energy is usually given as 38 [[MeV]].
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| ==See also==
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| * [[Fermi-Dirac statistics]]: the distribution of electrons over stationary states for a non-interacting fermions at ''non-zero'' temperature.
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| ==References==
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| {{reflist}}
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| *{{cite book | author=Kroemer, Herbert; Kittel, Charles | title=Thermal Physics (2nd ed.) | publisher=W. H. Freeman Company | year=1980 | isbn=0-7167-1088-9}}
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/tables/fermi.html Table of Fermi energies, velocities, and temperatures for various elements].
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| [[Category:Condensed matter physics]]
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| [[Category:Fermi–Dirac statistics]]
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