- Fermi energy
The

**Fermi energy**is a concept inquantum mechanics usually referring to the energy of the highest occupiedquantum state in a system offermion s atabsolute zero temperature . This article requires a basic knowledge of quantum mechanics.Note that the term "Fermi energy" is often, confusingly, used to describe a different but closely-related concept, the

chemical potential . [*The use of the term "Fermi energy" to describe the chemical potential is widespread in books and papers on semiconductor physics. As just one example, Pankove, "Optical Processes in Semiconductors", ISBN 0486602753 (1971), states: "The Fermi level is the energy at which the expectation of finding a state occupied by an electron is 1/2." (page 6)*] The Fermi energy and chemical potential are the same at absolute zero, but differ at other temperatures, as described below.**Introduction****Context**In

quantum mechanics , a group of particles known asfermion s (for example,electron s,proton s andneutron s are fermions) obey thePauli exclusion principle , which states that no two fermions can occupy the samequantum 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 ametal by cooling it down to near absolute zero temperature (0kelvin s), 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 ofcondensed matter physics . It is used, for example, to describe metals,insulator s, andsemiconductor s. It is a very important quantity in the physics of superconductors, in the physics ofquantum liquid s like low temperaturehelium (both normal and superfluid^{3}He), and it is quite important tonuclear physics and to understand the stability of white dwarf stars againstgravitational collapse .**Advanced context**The Fermi energy ("E

_{F}") of a system of non-interactingfermion s is the increase in theground 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. Thechemical potential at zero temperature is equal to the Fermi energy.**Illustration of the concept for a one dimensional square well**The 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:$E\_n\; =\; frac\{hbar^2\; pi^2\}\{2\; m\; L^2\}\; n^2\; ,$.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 $E\_1=frac\{hbar^2\; pi^2\}\{2\; m\; L^2\}$, or two particles can have energy $E\_2=4\; E\_1$ 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 :$E\_f=E\_\{N/2\}=frac\{hbar^2\; pi^2\}\{2\; m\; L^2\}\; (N/2)^2\; ,$.**The three-dimensional case**The 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 n_{x}, n_{y}, and n_{z}. The single particle energies are ::$E\_\{n\_x,n\_y,n\_z\}\; =\; frac\{hbar^2\; pi^2\}\{2m\; L^2\}\; left(\; n\_x^2\; +\; n\_y^2\; +\; n\_z^2\; ight)\; ,$ ::n_{x}, n_{y}, n_{z}are positive integers.There are multiple states with the same energy, for example $E\_\{100\}=E\_\{010\}=E\_\{001\}$. 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 for N is large.If we introduce a vector $vec\{n\}=\{n\_x,n\_y,n\_z\}$ then each quantum state corresponds to a point in 'n-space' with Energy :$E\_\{vec\{n\; =\; frac\{hbar^2\; pi^2\}\{2m\; L^2\}\; |vec\{n\}|^2\; ,$The number of states with energy less than E

_{f}is equal to the number of states that lie within a sphere of radius $|vec\{n\}\_f|$ in the region of n-space where n_{x}, n_{y}, n_{z}are positive. In the ground state this number equals the number of fermions in the system.:$N\; =2\; imesfrac\{1\}\{8\}\; imesfrac\{4\}\{3\}\; pi\; n\_f^3\; ,$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 find:$n\_f=left(frac\{3\; N\}\{pi\}\; ight)^\{1/3\}$so the Fermi energy is given by:$E\_f\; =frac\{hbar^2\; pi^2\}\{2m\; L^2\}\; n\_f^2$ ::$=\; frac\{hbar^2\; pi^2\}\{2m\; L^2\}\; left(\; frac\{3\; N\}\{pi\}\; ight)^\{2/3\}$

Which results in a relationship between the fermi energy and the number of particles per volume (when we replace L

^{2}with V^{2/3}):::The elimination of $L$ in favor of $V$::$L^2\; =\; V^\{frac\{2\}\{3$

::

**Typical fermi energies****White dwarfs**Stars known as

White dwarfs have mass comparable to ourSun , 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 degenerateelectron gas . The number density of electrons in a White dwarf are on the order of 10^{36}electrons/m^{3}. This means their fermi energy is:::$E\_f\; =\; frac\{hbar^2\}\{2m\_e\}\; left(\; frac\{3\; pi^2\; (10^\{36\})\}\{1\; mathrm\{m\}^3\}\; ight)^\{2/3\}\; approx\; 3\; imes\; 10^5\; mathrm\{eV\}\; ,$**Nucleus**Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:::$R\; =\; left(1.25\; imes\; 10^\{-15\}\; mathrm\{m\}\; ight)\; imes\; A^\{1/3\}$:where "A" is the number of

nucleons .The number density of nucleons in a nucleus is therefore:::$n\; =\; frac\{A\}\{egin\{matrix\}\; frac\{4\}\{3\}\; end\{matrix\}\; pi\; R^3\; \}\; approx\; 1.2\; imes\; 10^\{44\}\; mathrm\{m\}^\{-3\}$

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 theproton s in the nucleus, and vice versa.So the fermi energy of a nucleus is about:::$E\_f\; =\; frac\{hbar^2\}\{2m\_p\}\; left(\; frac\{3\; pi^2\; (6\; imes\; 10^\{43\})\}\{1\; mathrm\{m\}^3\}\; ight)^\{2/3\}\; approx\; 30\; imes\; 10^6\; mathrm\{eV\}\; =\; 30\; mathrm\{MeV\}$

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 level**The

**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. [*http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html*] In this state (at 0 K), the average energy of an electron is given by::$E\_\{av\}\; =\; frac\{3\}\{5\}\; E\_f$where $E\_f$ is the Fermi energy.

The

**Fermi momentum**is themomentum offermion s at theFermi surface . The "Fermi momentum" is given by::$p\_F\; =\; sqrt\{2\; m\_e\; E\_f\}$where $m\_e$ is the mass of the electron.This concept is usually applied in the case of

dispersion relation s between theenergy andmomentum 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::$V\_f\; =\; sqrt\{frac\{2\; E\_f\}\{m\_e$where $m\_e$ 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::$T\_f\; =\; frac\{E\_f\}\{k\}$where "k" is theBoltzmann constant .**Quantum mechanics**According to quantum mechanics, fermions -- particles with a

half-integer spin, usually 1/2, such aselectron s -- follow thePauli exclusion principle , which states that multiple particles may not occupy the samequantum state . Consequently, fermions obeyFermi-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 level**When the energy density of surface states is very high (>10

^{12}/cm^{2}), the position of the Fermi level is determined by the neutral level of theSurface states and becomes independent ofWork Function variations.**Free electron gas**In the

free electron gas , the quantum mechanical version of anideal gas of fermions, the quantum states can be labeled according to theirmomentum . Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of ametal , using something called the "quasi-momentum" or "crystal momentum" (seeBloch wave ). In either case, the Fermi energy states reside on a surface inmomentum space known as the. For the free electron gas, the Fermi surface is the surface of aFermi surface sphere ; for periodic systems, it generally has a contorted shape (seeBrillouin zone s). 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 aselectrical 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 equation:$mu\; =\; E\_F\; left\; [\; 1-\; frac\{pi\; ^2\}\{12\}\; left(frac\{kT\}\{E\_F\}\; ight)\; ^2\; -\; frac\{pi^4\}\{80\}\; left(frac\{kT\}\{E\_F\}\; ight)^4\; +\; cdots\; ight]$

where "E"

_{F}is the Fermi energy, "k" is theBoltzmann constant and "T" istemperature . Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic**Fermi temperature**"E_{F}"/"k". The characteristic temperature is on the order of 10^{5}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 inFermi-Dirac statistics .**ee also***

fermi gas

*Fermi statistics

*semiconductor s

*electrical engineering

*electronics

*thermodynamics

*Quasi Fermi level **References***cite book | author=Kroemer, Herbert; Kittel, Charles | title=Thermal Physics (2nd ed.) | publisher=W. H. Freeman Company | year=1980 | id=ISBN 0-7167-1088-9

* [*http://hyperphysics.phy-astr.gsu.edu/hbase/tables/fermi.html Table of fermi energies, velocities, and temperatures for various elements*] .

* [*http://physicsweb.org/articles/world/15/4/7 a discussion of fermi gases and fermi temperatures*] .**External links**

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