Non-metallic crystalline materials conduct heat by the transport of quantized atomic lattice vibrations called phonons. Thermal conductivity depends on how far phonons travel between scattering events-their mean free paths. Due to the breadth of the phonon mean free path spectrum, nanostructuring materials can reduce thermal conductivity from bulk by scattering long mean free path phonons, whereas short mean free path phonons are unaffected. Here we use a breakdown in diffusive phonon transport generated by high-frequency surface temperature modulation to identify the mean free path-dependent contributions of phonons to thermal conductivity in crystalline and amorphous silicon. Our measurements probe a broad range of mean free paths in crystalline silicon spanning 0.3-8.0 mm at a temperature of 311 K and show that 40±5% of its thermal conductivity comes from phonons with mean free path 41 mm. In a 500 nm thick amorphous silicon film, despite atomic disorder, we identify propagating phonon-like modes that contribute 435±7% to thermal conductivity at a temperature of 306 K.

We derive an analytical solution to the Boltzmann transport equation (BTE) to relate nondiffusive thermal conductivity measurements by thermoreflectance techniques to the bulk thermal conductivity accumulation function, which quantifies cumulative contributions to thermal conductivity from different mean free path energy carriers (here, phonons). Our solution incorporates two experimentally defined length scales: thermal penetration depth and heating laser spot radius. We identify two thermal resistances based on the predicted spatial temperature and heat flux profiles. The first resistance is associated with the interaction between energy carriers and the surface of the solution domain. The second resistance accounts for transport of energy carriers through the solution domain and is affected by the experimentally defined length scales. Comparison of the BTE result with that from conventional heat diffusion theory enables a mapping of mean-free-path-specific contributions to the measured thermal conductivity based on the experimental length scales. In general, the measured thermal conductivity will be influenced by the smaller of the two length scales and the surface properties of the system. The result is used to compare nondiffusive thermal conductivity measurements of silicon with first-principles-based calculations of its thermal conductivity accumulation function.

This paper describes the instrumentation for broadband frequency domain thermoreflectance (BB-FDTR), a novel, continuous wave laser technique for measuring the thermal conductivity accumulation function. The thermal conductivity accumulation function describes cumulative contributions to the bulk thermal conductivity of a material from energy carriers with different mean free paths. It can be used to map reductions in thermal conductivity in nano-devices, which arise when the dimensions of the device are commensurate to the mean free path of energy carriers. BB-FDTR uses high frequency surface temperature modulation to generate non-diffusive phonon transport realized through a reduction in the perceived thermal conductivity. By controlling the modulation frequency it is possible to reconstruct the thermal conductivity accumulation function. A unique heterodyning technique is used to down-convert the signal, therein improving our signal to noise ratio and enabling results over a broader range of modulation frequencies (200 kHz-200 MHz) and hence mean free paths. © 2013 AIP Publishing LLC. [http://dx

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We develop a solution to the two-temperature diffusion equation in axisymmetric cylindrical coordinates to model heat transport in thermoreflectance experiments. Our solution builds upon prior solutions that account for two-channel diffusion in each layer of an N-layered geometry, but adds the ability to deposit heat at any location within each layer. We use this solution to account for non-surface heating in the transducer layer of thermoreflectance experiments that challenge the timescales of electron-phonon coupling. A sensitivity analysis is performed to identify important parameters in the solution and to establish a guideline for when to use the two-temperature model to interpret thermoreflectance data. We then fit broadband frequency domain thermoreflectance (BB-FDTR) measurements of SiO2 and platinum at a temperature of 300 K with our two-temperature solution to parameterize the gold/chromium transducer layer. We then refit BB-FDTR measurements of silicon and find that accounting for non-equilibrium between electrons and phonons in the gold layer does lessen the previously observed heating frequency dependence reported in Regner et al. [Nat. Commun. 4, 1640 (2013)] but does not completely eliminate it. We perform BB-FDTR experiments on silicon with an aluminum transducer and find limited heating frequency dependence, in agreement with time domain thermoreflectance results. We hypothesize that the discrepancy between thermoreflectance measurements with different transducers results in part from spectrally dependent phonon transmission at the transducer/silicon interface.

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