These effective thermal conductivities vary as a function of experimental variables such as the laser heating modulation frequency, 1,2 the heat source diameter, 23 the grating spacing, 10,11 etc. 10–12,25 These experiments typically report effective thermal conductivities, extracted by applying the Fourier heat conduction equation to the experimental configuration, which are lower than those of the bulk materials. Recently, non-diffusive heat conduction has been observed experimentally with optical-based techniques such as time-domain thermoreflectance 4,5,19–23 (TDTR), frequency-domain thermoreflectance (FDTR), 2,24 and transient thermal grating (TTG). 16–18 It has been realized that the heat diffusion equation, which has been the workhorse of thermal transport science and engineering, becomes inadequate at distances comparable to the mean free path (MFP) of heat-carrying phonons. Recent years have witnessed intensifying research of phonon-mediated thermal transport on the micro- and nano-scale 1–13 stimulated by diverse technology drivers such as thermal management of microelectronic devices 14,15 and nanostructured thermoelectric materials. The results demonstrate the utility of the variational technique for analyzing non-diffusive phonon-mediated heat transport for nanostructures in multi-dimensional transport geometries, and will assist the probing of the mean free path distribution of materials via transient grating experiments. We also obtain a closed form expression for the effective thermal conductivity that demonstrates the full material property and heat transfer geometry dependence, and recovers the limits of the one-dimensional TTG expression for very thick films and the Fuchs-Sondheimer expression for very large grating spacings. We obtain an analytical expression for the thermal decay rate that shows excellent agreement with Monte Carlo simulations. We find a solution of the BTE in the thin film transient thermal grating (TTG) experimental geometry by using a recently developed variational approach with a trial solution supplied by the Fourier heat conduction equation. The phonon Boltzmann transport equation (BTE) is widely utilized to study non-diffusive thermal transport.
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