Editing Fermi level (section)

==Technicalities==

===Nomenclature===

The term ''Fermi level'' is mainly used in discussing the solid state physics of electrons in [[semiconductor]]s, and a precise usage of this term is necessary to describe [[band diagram]]s in devices comprising different materials with different levels of doping.
In these contexts, however, one may also see Fermi level used imprecisely to refer to the ''band-referenced Fermi level'', ''μ''&nbsp;−&nbsp;''ϵ''<sub>C</sub>, called ''ζ'' above.
It is common to see scientists and engineers refer to "controlling", "[[Fermi level pinning|pinning]]", or "tuning" the Fermi level inside a conductor, when they are in fact describing changes in ''ϵ''<sub>C</sub> due to [[doping (semiconductor)|doping]] or the [[field effect (semiconductor)|field effect]].
In fact, [[thermodynamic equilibrium]] guarantees that the Fermi level in a conductor is ''always'' fixed to be exactly equal to the Fermi level of the electrodes; only the band structure (not the Fermi level) can be changed by doping or the field effect (see also [[band diagram]]).
A [[Electrochemical potential#Conflicting terminologies|similar ambiguity]] exists between the terms, ''[[chemical potential]]'' and ''[[electrochemical potential]]''.

It is also important to note that Fermi ''level'' is not necessarily the same thing as [[Fermi energy|Fermi ''energy'']].
In the wider context of quantum mechanics, the term [[Fermi energy]] usually refers to ''the maximum kinetic energy of a fermion in an idealized non-interacting, disorder free, zero temperature [[Fermi gas]]''.
This concept is very theoretical (there is no such thing as a non-interacting Fermi gas, and zero temperature is impossible to achieve). However, it finds some use in approximately describing [[white dwarf]]s, [[neutron star]]s, [[atomic nuclei]], and electrons in a [[metal]].
On the other hand, in the fields of semiconductor physics and engineering, ''Fermi energy'' often is used to refer to the Fermi level described in this article.<ref>For example: {{cite book|url=https://books.google.com/books?id=n0rf9_2ckeYC&pg=PA49 |title=Electronics (fundamentals And Applications)|author= D. Chattopadhyay|isbn=978-81-224-1780-7|year=2006|publisher=New Age International }} and {{cite book|url=https://books.google.com/books?id=lmg13dHPKg8C&pg=PA113| title=Semiconductor Physics and Applications|author=  Balkanski and Wallis|isbn=978-0-19-851740-5|date=2000-09-01| publisher=OUP Oxford}}</ref>

===Fermi level referencing and the location of zero Fermi level===

Much like the choice of origin in a coordinate system, the zero point of energy can be defined arbitrarily. Observable phenomena only depend on energy differences.
When comparing distinct bodies, however, it is important that they all be consistent in their choice of the location of zero energy, or else nonsensical results will be obtained.
It can therefore be helpful to explicitly name a common point to ensure that different components are in agreement.
On the other hand, if a reference point is inherently ambiguous (such as "the vacuum", see below) it will instead cause more problems.

A practical and well-justified choice of common point is a bulky, physical conductor, such as the [[electrical ground]] or earth.
Such a conductor can be considered to be in a good thermodynamic equilibrium and so its ''μ'' is well defined.
It provides a reservoir of charge, so that large numbers of electrons may be added or removed without incurring charging effects.
It also has the advantage of being accessible, so that the Fermi level of any other object can be measured simply with a voltmeter.

====Why it is not advisable to use "the energy in vacuum" as a reference zero====

[[File:Work function mismatch gold aluminum.svg|thumb|300 px|When the two metals depicted here are in thermodynamic equilibrium as shown (equal Fermi levels E<sub>F</sub>), the vacuum [[electrostatic potential]] ''ϕ'' is not flat due to a difference in [[work function]].]]

In principle, one might consider using the state of a stationary electron in the vacuum as a reference point for energies.
This approach is not advisable unless one is careful to define exactly where ''the vacuum'' is.<ref group=Note>Technically, it is possible to consider the vacuum to be an insulator and in fact its Fermi level is defined if its surroundings are in equilibrium. Typically however the Fermi level is two to five electron volts ''below'' the vacuum electrostatic potential energy, depending on the [[work function]] of the nearby vacuum wall material. Only at high temperatures will the equilibrium vacuum be populated with a significant number of electrons (this is the basis of [[thermionic emission]]).</ref> The problem is that not all points in the vacuum are equivalent.

At thermodynamic equilibrium, it is typical for electrical potential differences of order 1 V to exist in the vacuum ([[Volta potential]]s).
The source of this vacuum potential variation is the variation in [[work function]] between the different conducting materials exposed to vacuum.
Just outside a conductor, the electrostatic potential depends sensitively on the material, as well as which surface is selected (its crystal orientation, contamination, and other details).

The parameter that gives the best approximation to universality is the Earth-referenced Fermi level suggested above. This also has the advantage that it can be measured with a voltmeter.


===Discrete charging effects in small systems===
In cases where the "charging effects" due to a single electron are non-negligible, the above definitions should be clarified. For example, consider a [[capacitor]] made of two identical parallel-plates. If the capacitor is uncharged, the Fermi level is the same on both sides, so one might think that it should take no energy to move an electron from one plate to the other. But when the electron has been moved, the capacitor has become (slightly) charged, so this does take a slight amount of energy. In a normal capacitor, this is negligible, but in a [[nanotechnology|nano-scale]] capacitor it can be more important.

In this case one must be precise about the thermodynamic definition of the chemical potential as well as the state of the device: is it electrically isolated, or is it connected to an electrode?

* When the body is able to exchange electrons and energy with an electrode (reservoir), it is described by the [[grand canonical ensemble]]. The value of chemical potential {{math|''μ''}} can be said to be fixed by the electrode, and the number of electrons {{math|''N''}} on the body may fluctuate. In this case, the chemical potential of a body is the infinitesimal amount of work needed to increase the ''average'' number of electrons by an infinitesimal amount (even though the number of electrons at any time is an integer, the average number varies continuously.): <math display="block">\mu(\left\langle N \right\rangle,T) = \left(\frac{\partial F}{\partial \left\langle N \right\rangle}\right)_T,</math> where <math>F(N,T) = \Omega(N,T) + \mu N</math> is the [[Helmholtz free energy]] of the grand canonical ensemble.
* If the number of electrons in the body is fixed (but the body is still thermally connected to a heat bath), then it is in the [[canonical ensemble]]. We can define a "chemical potential" in this case literally as the work required to add one electron to a body that already has exactly {{math|''N''}} electrons,<ref name=Shegelski> {{Cite journal
 |doi         = 10.1119/1.1629090
 |volume      = 72
 |issue       = 5
 |pages       = 676–678
 |last        = Shegelski
 |first       = Mark R. A.
 |title       = The chemical potential of an ideal intrinsic semiconductor
 |journal     = American Journal of Physics
 |date        = May 2004
 |bibcode     = 2004AmJPh..72..676S
 |doi-access= free
 }}</ref> <math display="block">\mu'(N, T) = F(N + 1, T) - F(N, T),</math> where {{math|''F''(''N'', ''T'')}} is the free energy function of the canonical ensemble, alternatively, <math display="block">\mu''(N, T) = F(N, T) - F(N - 1, T) = \mu'(N - 1, T).</math>

These chemical potentials are not equivalent, {{math|''μ'' ≠ ''μ''&prime; ≠ ''μ''&Prime;}}, except in the [[thermodynamic limit]]. The distinction is important in small systems such as those showing [[Coulomb blockade]],<ref>{{Cite journal | last1 = Beenakker | first1 = C. W. J. | title = Theory of Coulomb-blockade oscillations in the conductance of a quantum dot | doi = 10.1103/PhysRevB.44.1646 | journal = Physical Review B | volume = 44 | issue = 4 | pages = 1646–1656 | year = 1991 | pmid =  9999698|bibcode = 1991PhRvB..44.1646B | hdl = 1887/3358 | url = https://openaccess.leidenuniv.nl/bitstream/handle/1887/3358/172_063.pdf?sequence=1 | hdl-access = free }}</ref> but technically affects large sized semiconductors at zero temperature, at least ideally.<ref name=Shegelski/>
The parameter, {{math|''μ''}}, (i.e., in the case where the number of electrons is allowed to fluctuate) remains exactly related to the voltmeter voltage, even in small systems.
To be precise, then, the Fermi level is defined not by a deterministic charging event by one electron charge, but rather a statistical charging event by an infinitesimal fraction of an electron.<ref>{{cite journal|doi=10.1007/s10955-005-8067-x |title=The Chemical Potential |date=2006 |last1=Kaplan |first1=T. A. |journal=Journal of Statistical Physics |volume=122 |issue=6 |pages=1237–1260 |bibcode=2006JSP...122.1237K }}</ref>