  Despite recent progress in the first-principles calculations and measurements
of phonon mean-free-paths (MFPs), contribution of low-energy phonons to heat
conduction in silicon is still inconclusive, as exemplified by the
discrepancies between different first-principles calculations. Here we
investigate the contribution of low-energy phonons with MFP>0.8 um by
accurately measuring the cross-plane thermal conductivity of crystalline
silicon films by time-domain thermoreflectance (TDTR), over a wide range of
film thickness 1-10 um and temperature 100-300 K. We employ a dual-frequency
TDTR approach to improve the accuracy of our cross-plane thermal conductivity
measurements. We find from our cross-plane thermal conductivity measurements
that phonons with MFP>0.8 um contribute 53 W/m-K (37%) to heat conduction in Si
at 300 K while phonons with MFP>3 um contribute 523 W/m-K (61%) at 100 K, >20%
lower than the first-principles predictions by Lindsay et al. of 68 W/m-K (47%)
and 695 W/m-K (77%), respectively. Using a relaxation times approximation (RTA)
model, we demonstrate that macroscopic damping (e.g., Akhieser's damping)
eliminates the contribution of phonons with mean-free-paths >30 um at 300 K,
which contributes 15 W/m-K (10%) to heat conduction in Si according to Lindsay
et al. Thus we propose that omission of the macroscopic damping for low-energy
phonons in the first-principles calculations could be one of the possible
explanations for the observed discrepancy between our measurements and
calculations by Lindsay et al. Our work provides an important benchmark for
future measurements and calculations of the distribution of phonon
mean-free-paths in crystalline silicon.
