Thermal conductance quantum
In physics, the thermal conductance quantum <math>g_0</math> describes the rate at which heat is transported through a single 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 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">Schwab, K.; E. A. Henriksen; J. M. Worlock; M. L. Roukes (2000). "Measurement of the quantum of thermal conductance". Nature. 404 (6781): 974–7. Bibcode:2000Natur.404..974S. doi:10.1038/35010065. PMID 10801121. S2CID 4415638.</ref> These measurements employed suspended silicon nitride (Si
3N
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>Jezouin, S.; et al. (2013). "Quantum Limit of Heat Flow Across a Single Electronic Channel". Science. 342 (6158): 601–604. arXiv:1502.07856. Bibcode:2013Sci...342..601J. doi:10.1126/science.1241912. PMID 24091707. S2CID 8364740.</ref> and room temperature (~300K).<ref name=Cui>Cui, L.; et al. (2017). "Quantized thermal transport in single-atom junctions" (PDF). Science. 355 (6330): 1192–1195. Bibcode:2017Sci...355.1192C. doi:10.1126/science.aam6622. PMID 28209640. S2CID 24179265.</ref><ref>Mosso, N.; et al. (2017). "Heat transport through atomic contacts". Nature Nanotechnology. 12 (5): 430–433. arXiv:1612.04699. Bibcode:2017NatNa..12..430M. doi:10.1038/nnano.2016.302. PMID 28166205. 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 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.
