Thermal conductance quantum

In physics, the thermal conductance quantum ${\displaystyle g_{0}}$ describes the rate at which heat is transported through a single ballistic phonon channel with temperature ${\displaystyle T}$.

It is given by

${\displaystyle g_{0}={\frac {\pi ^{2}{k_{\rm {B}}}^{2}T}{3h}}\approx (9.464\times 10^{-13}{\rm {W/K}}^{2})\;T}$.

The thermal conductance of any electrically insulating structure that exhibits ballistic phonon transport is a positive integer multiple of ${\displaystyle g_{0}.}$ The thermal conductance quantum was first measured in 2000.^[1] These measurements employed suspended silicon nitride (Si
₃N
₄) nanostructures that exhibited a constant thermal conductance of 16 ${\displaystyle g_{0}}$ 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) ^[2] and room temperature (~300K).^[3][4]

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

${\displaystyle {\kappa \over \sigma }=LT,}$

where ${\displaystyle L}$ is a universal constant called the Lorenz factor,

${\displaystyle L={\pi ^{2}k_{\rm {B}}^{2} \over 3e^{2}}.}$

In the regime with quantized electric conductance, one may have

${\displaystyle \sigma ={ne^{2} \over h},}$

where ${\displaystyle n}$ is an integer, also known as TKNN number. Then

${\displaystyle \kappa =LT\sigma ={\pi ^{2}k_{\rm {B}}^{2} \over 3e^{2}}\times {ne^{2} \over h}T={\pi ^{2}k_{\rm {B}}^{2} \over 3h}nT=g_{0}n,}$

where ${\displaystyle g_{0}}$ is the thermal conductance quantum defined above.

See also

• Thermal transport in nanostructures

References

1. 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.
2. 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.
3. 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.
4. 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.