Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments
Open accessAbstract
The applicability of Fourier's law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature profile. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (<1 μm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature profile and interfacial-phonon scattering by time-domain thermoreflectance experiments on Si, Si0.99Ge0.01, boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.
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Previous TDTR measurements on Si at room temperature did not reveal apparent frequency dependence, regardless of the transducer/Si boundary conditions.
“Nevertheless, previous TDTR measurements on Si at room temperature (RT) did not reveal apparent frequency dependence, regardless of the transducer/Si boundary conditions.”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [18]
The ‘Anisotropic failure of Fourier theory in Si’ section reports that Si’s apparent thermal conductivity had insignificant frequency dependence between 1 and 10 MHz. The same section and the Fig. 6 caption examine different transducer/Si interfaces at 300 K, but the retrieved text does not establish that frequency dependence was absent under every boundary condition; it reports that some interface conditions produced values outside the error bars of bulk conductivity. (LLM, conf=0.84)
Experimental — accuracy not yet measuredThe total thermal conductivity Λ1 + Λ2 = 141 W m−1 K−1 at room temperature aligns well with previously reported values.
“The total thermal conductivity Λ 1 + Λ 2 = 141 W m -1 K -1 aligns well with previously reported value [18, 26, 32].”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [18]
The Figure 6 caption (Methods; page unavailable) gives a bulk Si thermal conductivity of 142 W m−1 K−1, which is close to the citing sentence’s 141 W m−1 K−1. The cited paper does not report a Λ1 + Λ2 decomposition or the total of 141 W m−1 K−1; it supports the comparison, not that specific result. (LLM, conf=0.86)
Experimental — accuracy not yet measuredBoth the two-channel model and the ballistic/diffusive model reveal the critical influence of the interface on frequency dependent thermal transport.
“Importantly, both theories reveal the critical influence of the interface on frequency dependent thermal transport, thus underscoring the significance of the transducer/sample boundary condition.”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [18]
The Discussion (page unavailable) says the ballistic/diffusive model links interfacial-phonon scattering to frequency-dependent TDTR measurements in alloys. The Methods (page unavailable) also describe different interface boundary conditions for ballistic and diffusive phonons. Our copy does not establish the same conclusion for a separate two-channel model, so it supports only that part of the attributed claim. (LLM, conf=0.85)
Experimental — accuracy not yet measuredCare should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs).
“However, care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs). 90-92, 112”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [112]
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References 41 · Cited by 244
Research works printed but not linked: 11
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- Garg, J., Bonini, N. & Marzari, N. in Length-Scale Dependent Phonon Interactions Vol. 128 Topics in Applied Physics. (eds Shinde ´, Subhash L. & Srivastava, Gyaneshwar P.) Chapter 4, 115-136 (Springer, 2014).
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- Authors
- Wilson, R. B., & Cahill, D. G.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Wilson, R. B.
- Venue
- Nature Communications
- Published
- 2014-10-01
- Volume
- 5
- Issue
- 1
- Pages
- 5075-5075
- ISSN
- 2041-1723 (ISSN-L)
- OpenAlex
- W2064434146
- Cited by
- 244
- References
- 41
- Citations by year
- 2026: 8 · 2025: 27 · 2024: 18 · 2023: 12 · 2022: 19 · 2021: 16 · 2020: 17 · 2019: 16 · 2018: 35 · 2017: 24 · 2016: 22 · 2015: 27 · 2014: 3
- FWCI
- 11.50
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Nature Communications (publishedVersion)
- PubMed (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Funders
- U.S. Department of Defense (FA9550)
- University of Illinois at Urbana-Champaign
- National Defense Science and Engineering Graduate
- Division of Materials Research
- Air Force Office of Scientific Research (AF FA9550‐12‐1‐0073, FA9550-, AF FA9550, FA9550-12, FA9550‐12‐1, FA9550-12-1-0073)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
- Physical Sciences › Engineering › Civil and Structural Engineering › Thermal Radiation and Cooling Technologies
- Keywords
- anisotropic thermal transport, time-domain thermoreflectance, nanoscale heat transfer, phonon mean free path, interfacial phonon scattering, radiation boundary conditions, heat equation, silicon-germanium alloys, boron-doped silicon, MgO, three-dimensional heat transfer
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