Editing Fermi level (section)

===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>