Nonlocal theory for heat transport at high frequencies
Abstract
We develop a nonlocal theory for heat conduction under high-frequency temperature fields and apply the theory to explain reductions of the apparent thermal conductivity observed in recent experiments. Our nonlocal theory is an analytical solution of the Boltzmann transport equation for phonons in a semi-infinite solid, similar to a prior nonlocal theory for heat conduction under a high-temperature gradient but subjected to periodic heating at the surface. The boundary condition of periodic heating, as opposed to prior calculations of heating by a single laser pulse, better mimics time-domain thermoreflectance (TDTR) and broadband frequency-domain thermoreflectance (BB-FDTR) measurements. We find that, except for pure crystals at high frequencies, the effective thermal conductivity derived using the nonlocal theory compares well with calculations of a modified Callaway model that includes an upper limit on the phonon mean-free path at twice the thermal penetration depth. For pure crystals, however, the effective thermal conductivity derived from the out-of-phase calculations are independent of frequency, in agreement with prior TDTR measurements, due to the countereffect of reduced heat flux and diminished relative phase between the heat flux and temperature oscillations at high frequencies. Our results suggest that empirical interpretation of ballistic phonons not contributing to heat conduction is not general and can only be applied to measurements on alloys and not pure crystals, even when a large laser spot size is used in the experiments and the interfacial thermal resistance is negligible.
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Koh and Cahill were the first to observe a modulation-frequency-dependent thermal conductivity in some semiconductor alloys measured using TDTR, and they attributed it to the non-diffusive phonon transport.
“Koh and Cahill 90 were the first to observe a modulation-frequency-dependent thermal conductivity in some semiconductor alloys measured using TDTR, and they attributed it to the non-diffusive phonon transport, i.e. , phonons with Λ ݀ (and Λ 2 ݀ in their later publication 209 ) do not contribute to the apparent thermal conductivity measured by TDTR.”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [209]
The simple interpretation that phonons with Λ > dp do not contribute to the apparent thermal conductivity works as a good approximation only for semiconductor alloys due to their extremely broad MFP distributions but not for pure crystals.
“This simple interpretation, however, works as a good approximation only for semiconductor alloys due to their extremely broad MFP distributions 209 but not for pure crystals, as frequency dependence was not observed in TDTR measurements of crystalline Si despite its very long MFPs of phonons 185, 187 .”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [209]
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- Authors
- Koh, Y. K., Cahill, D. G., & Sun, B.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- National University of Singapore (SG)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2014-11-11
- Volume
- 90
- Issue
- 20
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W2315339812
- Cited by
- 60
- References
- 32
- Citations by year
- 2026: 1 · 2025: 8 · 2024: 4 · 2023: 2 · 2022: 3 · 2021: 2 · 2020: 5 · 2019: 8 · 2018: 8 · 2017: 8 · 2016: 7 · 2015: 4
- FWCI
- 2.88
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review B (publishedVersion)
- Funders
- Ministry of Education - Singapore (MOE2013-T2-2-147)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
- Physical Sciences › Engineering › Mechanics of Materials › Thermography and Photoacoustic Techniques
- Keywords
- nonlocal heat conduction, phonon Boltzmann transport equation, effective thermal conductivity, time-domain thermoreflectance, Callaway model, phonon mean free path, ballistic phonons, pure crystals, alloys, thermal penetration depth
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