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Thermal conductance quantum

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

It is given by

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

κ σ = L T , {\displaystyle {\kappa \over \sigma }=LT,} {\displaystyle {\kappa \over \sigma }=LT,}

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

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

In the regime with quantized electric conductance, one may have

σ = n e 2 h , {\displaystyle \sigma ={ne^{2} \over h},} {\displaystyle \sigma ={ne^{2} \over h},}

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

κ = L T σ = π 2 k B 2 3 e 2 × n e 2 h T = π 2 k B 2 3 h n T = g 0 n , {\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,} {\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 g 0 {\displaystyle g_{0}} {\displaystyle g_{0}} is the thermal conductance quantum defined above.

See also

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. OSTI 1434837. 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.