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.

• Thermal transport in nanostructures