Despite recent progress in the first-principles calculations and measurements of phonon mean-free-paths (ℓ), contribution of low-energy phonons to heat conduction in silicon is still inconclusive, as exemplified by the discrepancies as large as 30% between different first-principles calculations. Here, we investigate the contribution of low-energy phonons with ℓ > 0.8 μm by accurately measuring the cross-plane thermal conductivity (Λcross) of crystalline silicon films by time-domain thermoreflectance (TDTR), over a wide range of film thicknesses 1 ≤ h f ≤ 10 μm and temperatures 100 ≤ T ≤ 300 K. We employ a dual-frequency TDTR approach to improve the accuracy of our Λcross measurements. We find from our Λcross measurements that phonons with ℓ > 0.8 μm contribute 53 W m−1 K−1 (37%) to heat conduction in natural Si at 300 K, while phonons with ℓ > 3 μm contribute 523 W m−1 K−1 (61%) at 100 K, >20% lower than first-principles predictions of 68 W m−1 K−1 (47%) and 717 W m−1 K−1 (76%), respectively. Using a relaxation time approximation model, we demonstrate that macroscopic damping (e.g., Akhieser's damping) eliminates the contribution of phonons with mean-free-paths >20 μm at 300 K, which contributes 15 W m−1 K−1 (10%) to calculated heat conduction in Si. 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 differences between our measurements and calculations. Our work provides an important benchmark for future measurements and calculations of the distribution of phonon mean-free-paths in crystalline silicon.
We sincerely thank R. B. Wilson, Professor D. A. Broido, and Professor D. G. Cahill for many useful discussions. This material is based upon work supported by the Singapore Ministry of Education Academic Research Fund Tier 1 Start-up Grant and Singapore Ministry of Education Academic Research Fund Tier 2, under Award No MOE2013-T2-2-147. L.L. acknowledges support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division.
I. INTRODUCTION II. EXPERIMENTAL DETAILS III. RESULTS AND DISCUSSION IV. SUMMARY