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