Quasiballistic heat removal from small sources studied from first principles

Abstract

Heat sources whose characteristic dimension $R$ is comparable to phonon mean free paths display thermal resistances that exceed conventional diffusive predictions. This has direct implications to (opto)electronics thermal management and phonon spectroscopy. Theoretical analyses have so far limited themselves to particular experimental configurations. Here, we build upon the multidimensional Boltzmann transport equation (BTE) to derive universal expressions for the apparent conductivity suppression $S(R)={\ensuremath{\kappa}}_{\text{eff}}(R)/{\ensuremath{\kappa}}_{\text{bulk}}$ experienced by radially symmetric 2D and 3D sources. In striking analogy to cross-plane heat conduction in thin films, a distinct quasiballistic regime emerges between ballistic (${\ensuremath{\kappa}}_{\text{eff}}\ensuremath{\sim}R$) and diffusive (${\ensuremath{\kappa}}_{\text{eff}}\ensuremath{\simeq}{\ensuremath{\kappa}}_{\text{bulk}}$) asymptotes that displays a logarithmic dependence ${\ensuremath{\kappa}}_{\text{eff}}\ensuremath{\sim}ln(R)$ in single crystals and fractional power dependence ${\ensuremath{\kappa}}_{\text{eff}}\ensuremath{\sim}{R}^{2\ensuremath{-}\ensuremath{\alpha}}$ in alloys (with $\ensuremath{\alpha}$ the L\'evy superdiffusion exponent). Analytical solutions and Monte Carlo simulations for spherical and circular heat sources in Si, GaAs, ${\mathrm{Si}}_{0.99}{\mathrm{Ge}}_{0.01}$, and ${\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}$, all carried out from first principles, confirm the predicted generic tendencies. Contrary to the thin film case, common approximations like kinetic theory estimates ${\ensuremath{\kappa}}_{\text{eff}}\ensuremath{\simeq}\ensuremath{\sum}{S}_{\ensuremath{\omega}}^{\text{grey}}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\kappa}}_{\ensuremath{\omega}}$ and modified Fourier temperature curves perform relatively poorly. Up to threefold deviations from the BTE solutions for sub-$100\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$ sources underline the need for rigorous treatment of multidimensional nondiffusive transport.