Editing Fermi level (section)

==Voltage measurement==

[[File:Old Volt Meter pic3.JPG|thumb|A [[voltmeter]] measures differences in Fermi level divided by [[electron charge]].]]

Sometimes it is said that electric currents are driven by differences in [[electrostatic potential]] ([[Galvani potential]]), but this is not exactly true.<ref>{{cite journal|doi=10.1016/S0167-2738(96)00542-5|title=What does a voltmeter measure?|journal=Solid State Ionics|volume=95|issue=3–4|pages=327–328|year=1997|last1=Riess|first1=I}}</ref>
As a counterexample, multi-material devices such as [[p–n junction]]s contain internal electrostatic potential differences at equilibrium, yet without any accompanying net current; if a voltmeter is attached to the junction, one simply measures zero volts.<ref>{{cite book |title=Fundamentals of Solid-State Electronics |url=https://archive.org/details/fundamentalssoli00sahc_987 |url-access=limited |last1=Sah |first1=Chih-Tang |year=1991 |publisher=World Scientific |isbn=978-9810206376 |page=[https://archive.org/details/fundamentalssoli00sahc_987/page/n405 404]}}</ref>
Clearly, the electrostatic potential is not the only factor influencing the flow of charge in a material—[[Pauli repulsion]], carrier concentration gradients, electromagnetic induction, and thermal effects also play an important role.

In fact, the quantity called ''voltage'' as measured in an electronic circuit has a simple relationship to the [[chemical potential]] for electrons (Fermi level).
When the leads of a [[voltmeter]] are attached to two points in a circuit, the displayed voltage is a measure of the ''total'' work transferred when a unit charge is allowed to move from one point to the other.
If a simple wire is connected between two points of differing voltage (forming a [[short circuit]]), current will flow from positive to negative voltage, converting the available work into heat.

The Fermi level of a body expresses the work required to add an electron to it, or equally the work obtained by removing an electron.
Therefore, ''V''<sub>A</sub>&nbsp;−&nbsp;''V''<sub>B</sub>, the observed difference in voltage between two points, ''A'' and ''B'', in an electronic circuit is exactly related to the corresponding chemical potential difference, ''μ''<sub>A</sub>&nbsp;−&nbsp;''μ''<sub>B</sub>, in Fermi level by the formula<ref>{{cite book
 | isbn         = 9780521631457
 | title        = Quantum Transport: Atom to Transistor
 | last1        = Datta
 | first1       = Supriyo
 | author-link1  = Supriyo Datta
 | year         =  2005
 | publisher    = Cambridge University Press
 | page=7
}}</ref>
<math display="block"> V_\mathrm{A} - V_\mathrm{B} = \frac{\mu_\mathrm{A} - \mu_\mathrm{B}}{-e} </math>
where −''e'' is the [[electron charge]].

From the above discussion it can be seen that electrons will move from a body of high ''μ'' (low voltage) to low ''μ'' (high voltage) if a simple path is provided.
This flow of electrons will cause the lower ''μ'' to increase (due to charging or other repulsion effects) and likewise cause the higher ''μ'' to decrease.
Eventually, ''μ'' will settle down to the same value in both bodies.
This leads to an important fact regarding the equilibrium (off) state of an electronic circuit:
{{block indent|em=1.5|text=''An electronic circuit in [[thermodynamic equilibrium]] will have a constant Fermi level throughout its connected parts.''<ref>{{cite journal|doi=10.1021/acsenergylett.0c02443|title=Potentially Confusing: Potentials in Electrochemistry|journal=ACS Energy Letters|volume=6|issue=1|pages=261–266|year=2021|last1=Boettcher|first1=S. W.|last2=Oener|first2=S. Z.|last3=Lonergan|first3=M. C.|last4=Surendranath|first4=S.|last5=Ado|first5=S.|last6=Brozek|first6=C.|last7=Kempler|first7=P. A.}}</ref>}}

This also means that the voltage (measured with a voltmeter) between any two points will be zero, at equilibrium.
Note that [[thermodynamic equilibrium]] here requires that the circuit be internally connected and not contain any batteries or other power sources, nor any variations in temperature.