Fig. 4: Phonon transport mechanism.

a Temperature distribution at 100 K obtained from simplified phonon BTE calculations. We divide the temperature drop into two parts: the temperature drop inside the heating zone \(\varDelta {T}_{{{{{{\rm{heat}}}}}}}\) and the temperature drop from the heating-substrate interface to the bottom \(\varDelta {T}_{{{{{{\rm{sub}}}}}}}\), i.e., temperature drop in the substrate. b The temperature drop inside the heating zone \(\varDelta {T}_{{{{{{\rm{heat}}}}}}}\) and temperature drop in the substrate \(\varDelta {T}_{{{{{{\rm{sub}}}}}}}\) for different phonon-defect relaxation time. c Calculated directional phonon energy flux in the substrate. The x-axis represents the projection of the energy flux on the x-y plane. The y-axis represents the projection of the energy flux on the z direction. d Phonon transport mechanism in the ballistic regime. When the scattering is rare (i.e., in the ballistic regime) inside the uniform volumetric heating zone, the phonon mode propagating oblique to the z direction (mode 2) travels a much longer distance in the heating zone than the mode propagating along the z direction (mode 1), and thus receives a much larger amount of energy. Therefore, the phonons propagating in different directions exhibit large directional nonequilibrium (\({e}_{2}\) is much larger than \({e}_{1}\)). the local heat flux is only contributed by the energy flux projected along z directions (blue arrow). e Phonon transport mechanism in the diffusive regime. If sufficient scattering is present in the heating zone (i.e., in the diffusive regime), scattering can randomly redirect phonons and restore the equilibrium directional distribution.