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In physics, the '''thermal conductance quantum''' <math>g_0</math> describes the rate at which [[heat]] is transported through a single [[Ballistic conduction|ballistic]] [[phonon]] channel with temperature <math>T</math>. 

It is given by

:<math>g_{0} = \frac{\pi^2 {k_{\rm B}}^2 T}{3h} \approx (9.464\times10^{-13} {\rm W/K}^{2})\;T</math>.

The [[Thermal conductivity|thermal conductance]] of any electrically insulating structure that exhibits ballistic phonon transport is a positive integer multiple of <math>g_0.</math> The thermal conductance quantum was first measured in 2000.<ref name="Schwab">{{cite journal|title=Measurement of the quantum of thermal conductance|journal=Nature|year=2000|first=K.|last=Schwab|author2=E. A. Henriksen |author3=J. M. Worlock |author4=M. L. Roukes |volume=404|pages=974–7|doi=10.1038/35010065|pmid=10801121|issue=6781|bibcode=2000Natur.404..974S|s2cid=4415638}}</ref> These measurements employed suspended [[silicon nitride]] ({{chem|Si|3|N|4}}) nanostructures that exhibited a constant thermal conductance of 16 <math>g_0</math> at temperatures below approximately 0.6 [[kelvin]].

== Relation to the quantum of electrical conductance ==
For ballistic electrical conductors, the [[electron]] contribution to the thermal conductance is also quantized as a result of the electrical [[conductance quantum]] and the [[Wiedemann–Franz law]], which has been quantitatively measured at both cryogenic (~20 mK) <ref name=Jezouin>{{cite journal|title=Quantum Limit of Heat Flow Across a Single Electronic Channel|journal=Science|year=2013|first=S. |display-authors=etal |last=Jezouin |volume=342|issue=6158|pages=601–604|doi=10.1126/science.1241912|pmid=24091707|arxiv=1502.07856|bibcode=2013Sci...342..601J|s2cid=8364740}}</ref> and room temperature (~300K).<ref name=Cui>{{cite journal|title=Quantized thermal transport in single-atom junctions
|journal=Science|year=2017|first=L. |display-authors=etal |last=Cui|volume=355|issue=6330|pages=1192–1195|doi=10.1126/science.aam6622|pmid=28209640|bibcode=2017Sci...355.1192C|s2cid=24179265|url=https://kops.uni-konstanz.de/bitstream/123456789/38281/1/Cui_0-399840.pdf|doi-access=free}}</ref><ref>{{cite journal |last1=Mosso |first1=N. |display-authors=etal |title=Heat transport through atomic contacts |journal=Nature Nanotechnology |date=2017 |volume=12 |issue=5 |pages=430–433 |doi=10.1038/nnano.2016.302 |pmid=28166205 |arxiv=1612.04699 |bibcode=2017NatNa..12..430M |s2cid=5418638 }}</ref>

The thermal conductance quantum, also called quantized thermal conductance, may be understood from the Wiedemann-Franz law, which shows that

:<math>
{\kappa \over \sigma} = LT,
</math>

where <math>L</math> is a universal constant called the [[Wiedemann–Franz law|Lorenz factor]],

:<math>
L = {\pi^2 k_{\rm B}^2 \over 3e^2}.
</math>

In the regime with quantized electric conductance, one may have

:<math>
\sigma = {n e^2 \over h},
</math>

where <math>n</math> is an integer, also known as TKNN number. Then

:<math>
\kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n,
</math>

where <math>g_0</math> is the thermal conductance quantum defined above.
==See also==
* [[Thermal properties of nanostructures]]

==References==
{{reflist}}

[[Category:Mesoscopic physics]]
[[Category:Nanotechnology]]
[[Category:Quantum mechanics]]
[[Category:Condensed matter physics]]
[[Category:Physical quantities]]
[[Category:Heat conduction]]