Power Spectral Density of Thermal Noise at Low Frequencies in Thermal Conductance

We derive the power spectral density (PSD) of thermal flux fluctuation at low frequencies for a thermal conductance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${G}_{\text {th}}$ </tex-math></inline-formula> based on the Green-Kubo relation for thermal conductivity and the heat equation under nonequilibrium conditions. The result is given by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${4}\textit {kT}_{e}^{{2}}{G}_{\text {th}}$ </tex-math></inline-formula>, where k is Boltzmann’s constant, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}_{e}$ </tex-math></inline-formula>, the effective temperature, is shown to depend on the terminal temperatures of the thermal conductor and the temperature coefficient of thermal conductivity. In particular, if the thermal conductor is isothermal at temperature T, the flux fluctuation simplifies to the known <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${4}\textit {kT}^{{2}}{G}_{\text {th}}$ </tex-math></inline-formula>. Using a polysilicon resistor as an example, we demonstrate that thermal noise arising from thermal conductance can exceed the noise from electrical conductance under certain conditions, which is driven by the self-heating effect (SHE).