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) = κ eff (R)/κ 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 (κ eff ∼ R) and diffusive (κ eff κ bulk ) asymptotes that displays a logarithmic dependence κ eff ∼ ln(R) in single crystals and fractional power dependence κ eff ∼ R 2−α in alloys (with α the Lévy superdiffusion exponent). Analytical solutions and Monte Carlo simulations for spherical and circular heat sources in Si, GaAs, Si0.99Ge0.01 and Si0.82Ge0.18, all carried out from first principles, confirm the predicted generic tendencies. Contrary to the thin film case, common approximations like kinetic theory estimates κ eff S grey ω κω and modified Fourier temperature curves perform relatively poorly. Up to threefold deviations from the BTE solutions for sub-100 nm sources underline the need for rigorous treatment of multidimensional nondiffusive transport.

Engineering Nanoscale Thermal Transport: Size- and Spacing-Dependent Cooling of Nanostructures

“…This difference might explain the difference between the EUV nanometrology and the TDTR measurements. Other theoretical approaches are in development that are also very promising for explaining our observed return toward the diffusive prediction, including hydrodynamic and superdiffusive views of the BTE that show lower temperatures near nanoscale heat sources than the temperature predicted by effective Fourier models [44][45][46]. In addition, potential coherent effects can modify the thermal transport in nanostructured systems, even at room temperature, via local resonances [6,7].…”

Experimental Evidence of Superdiffusive Thermal Transport in Si0.4Ge0.6 Thin Films

“…As ω 1 should be much lower than ω D , the thermal conductivity is proportional to 1/ω 1 , and hence κ ∝ t 1/4 . We note that quite a few theoretical predictions have been published in the past decade on the power law size dependence of thermal conductivity of SiGe nanostructures, ,,,− and different exponents have been suggested. Our systematic study provides the first experimental evidence supporting the 1/4 power law dependence in alloy-scattering dominated systems.…”

“…Superdiffusive thermal transport represents a classical anomaly that attracts continued interest. − Unlike the diffusive process in bulk materials that renders a materials-intrinsic thermal conductivity ( k ), superdiffusive transport is characterized by a size-dependent thermal conductivity of κ ∝ L β , where L is the system size and β is a constant between 0 and 1. , Superdiffusive thermal transport was first predicted for one-dimensional (1D) lattices composed of single atomic chains, and the concept remained purely theoretical until a recent experimental observation of the κ ∝ L 1/3 length dependence in the range from 6 to >42.5 μm for ultrathin van der Waals crystal NbSe 3 nanowires . Superdiffusive thermal transport has also been predicted in three-dimensional (3D) alloys such as silicon–germanium (SiGe), − even though it is expected to only persist over a limited system size range, unlike in 1D materials where theoretically the length dependence could extend to infinity. While experimental validation of the theoretically predicted superdiffusive behavior in SiGe should be much easier than that for 1D lattices, surprisingly, no systematic experimental data directly demonstrating κ ∝ L β in SiGe have been reported so far.…”

Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

“…Advances in measurement 8 include coherent xray thermal probing of strip-line arrays [9][10][11] , transient thermal gratings [12][13][14][15] , and time-domain thermal reflectance [16][17][18] . Theory has become increasingly powerful [19][20][21] and evolves togther with experiment [22][23][24] .…”