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{{short description|Quantized unit of electrical conductance}}
The '''conductance quantum''', denoted by the symbol {{math|''G''<sub>0</sub>}}, is the quantized unit of [[electrical conductance]]. It is defined by the [[elementary charge]] ''e'' and [[Planck constant]] ''h'' as:

:<math>G_0 = \frac{2 e^2}{h}</math> = {{physconst|G0|after=.{{refn|group=Note|S is the [[Siemens (unit)|siemens]]}}}}

It appears when measuring the conductance of a [[quantum point contact]], and, more generally, is a key component of the [[Landauer formula]], which relates the electrical conductance of a quantum conductor to its quantum properties.  It is twice the reciprocal of the [[von Klitzing constant]] (2/''R''<sub>K</sub>).

Note that the conductance quantum does not mean that the conductance of any system must be an integer multiple of ''G''<sub>0</sub>. Instead, it describes the conductance of two quantum channels (one channel for spin up and one channel for spin down) if the probability for transmitting an electron that enters the channel is unity, i.e. if transport through the channel is [[ballistic transport|ballistic]]. If the transmission probability is less than unity, then the conductance of the channel is less than ''G''<sub>0</sub>. The total conductance of a system is equal to the sum of the conductances of all the parallel quantum channels that make up the system.<ref>S. Datta, ''Electronic Transport in Mesoscopic Systems'', Cambridge University Press, 1995, {{ISBN|0-521-59943-1}}</ref>

== Derivation ==
In a 1D wire, connecting two reservoirs of potential <math>u_1</math> and <math>u_2</math> [[adiabatic]]ally:

The density of states is 
<math display="block">\frac{\mathrm{d}n}{\mathrm{d} \epsilon} = \frac{2}{hv} ,</math>
where the factor 2 comes from electron spin degeneracy, <math>h</math> is the [[Planck constant]], and <math>v</math> is the electron velocity.

The voltage is:
<math display="block">V = -\frac{(\mu_1 - \mu_2)}{e} ,</math>
where <math>e</math> is the electron charge.

The 1D current going across is the current density:
<math display="block">j = -ev(\mu_1-\mu_2) \frac{\mathrm{d}n}{\mathrm{d} \epsilon} .</math>

This results in a quantized conductance:
<math display="block">G_0 = \frac{I}{V}  = \frac{j}{V} = \frac{2e^2}{h} .</math>

== Occurrence ==
Quantized conductance occurs in wires that are ballistic conductors,  when the elastic mean free path is much larger than the length of the wire: <math>l_{\rm el} \gg L </math>{{clarify|date=December 2018}}. B. J. van Wees et al. first observed the effect in a point contact in 1988.<ref>{{Cite journal|title = Quantized Conductance of Point Contacts in a Two-Dimensional Electron Gas|last = B.J. van Wees|date = 1988|journal = Physical Review Letters|volume = 60|issue = 9|pages=848–850|doi = 10.1103/PhysRevLett.60.848|pmid = 10038668|display-authors=etal|bibcode = 1988PhRvL..60..848V |hdl = 1887/3316|hdl-access = free}}</ref>  Carbon nanotubes have quantized conductance independent of diameter.<ref>{{Cite journal|title = Carbon Nanotube Quantum Resistors|author1=S. Frank |author2=P. Poncharal |author3=Z. L. Wang |author4=W. A. de Heer |date = 1998|journal = Science| volume= 280| issue =  1744–1746|doi = 10.1126/science.280.5370.1744 |pmid = 9624050 |pages=1744–6|bibcode = 1998Sci...280.1744F |citeseerx = 10.1.1.485.1769}}</ref> The [[quantum hall effect]] can be used to precisely measure the conductance quantum value. It also occurs in electrochemistry reactions<ref>{{cite journal|last=Bueno |first=P. R. |title=Electron transfer and conductance quantum |journal=Physical Chemistry Chemical Physics |year=2020 |volume=22 |number=45 |pages=26109–26112|doi=10.1039/D0CP04522E |pmid=33185207 |bibcode=2020PCCP...2226109B |s2cid=226853811 }}</ref> and in association with the quantum capacitance defines the rate with which electrons are transferred between quantum chemical states as described by the quantum rate theory.

==See also==
* [[Mesoscopic physics]]
* [[Quantum point contact]]
* [[Quantum wire]]
* [[Thermal conductance quantum]]

==Notes==
{{reflist|group=Note}}

==Reference==
{{reflist}}

{{DEFAULTSORT:Conductance Quantum}}
[[Category:Mesoscopic physics]]
[[Category:Quantum electronics]]
[[Category:Nanoelectronics]]
[[Category:Condensed matter physics]]
[[Category:Physical quantities]]