CoChat
Sign Up
Cited by 52Thermal properties of materialsThermoelastic and Magnetoelastic PhenomenaThermography and Photoacoustic Techniques

Radial quasiballistic transport in time-domain thermoreflectance studied using Monte Carlo simulations

Open access
Ding, D., Chen, X., Minnich, A. J.·2014·Applied Physics Letters
DOI 10.1063/1.4870811
Open this paper in CoChat — free accountOpen PDF

Abstract

Recently, a pump beam size dependence of thermal conductivity was observed in Si at cryogenic temperatures using time-domain thermal reflectance (TDTR). These observations were attributed to quasiballistic phonon transport, but the interpretation of the measurements has been semi-empirical. Here, we present a numerical study of the heat conduction that occurs in the full 3D geometry of a TDTR experiment, including an interface, using the Boltzmann transport equation. We identify the radial suppression function that describes the suppression in heat flux, compared to Fourier's law, that occurs due to quasiballistic transport and demonstrate good agreement with experimental data. We also discuss unresolved discrepancies that are important topics for future study.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper uses deviational Monte Carlo to solve the Boltzmann transport equation in the full 3D geometry of a TDTR experiment and identifies a radial suppression function for quasiballistic phonon transport in Si. The radial suppression function agrees well with room-temperature TDTR data but overpredicts thermal conductivity at cryogenic temperatures below 150 K.

Key findings

  • Radial heat flux suppression depends on pump beam diameter, while cross-plane suppression is independent of pump diameter; the effective thermal conductivity decreases with decreasing pump size, consistent with prior TDTR observations.
  • For a pump diameter of 0.8 μm at 300 K, the Monte Carlo decay fits an effective thermal conductivity of 65 W/mK, compared with bulk Si \(k_{\text{bulk}} = 148.2\) W/mK.
  • The extracted radial suppression function is robust to artificial noise and constraint removal to within 5%, and it predicts room-temperature TDTR data well in the absence of cross-plane suppression.
  • At cryogenic temperatures down to 60 K, the calculation predicts the same trend but overpredicts effective thermal conductivity for all pump diameters below \(T = 150\) K.

Methods

  • The authors solve the transient BTE using a deviational Monte Carlo method in a 3D TDTR geometry: a 10 nm Al transducer on semi-infinite Si, with a Gaussian initial temperature profile and diffuse interface transmission/reflection.
  • Phonon transport uses Si [100] dispersion discretized into 1000 bins, acoustic phonons only, relaxation times from Ref. 22; the Al transducer is assigned a constant 1 ps relaxation time, giving \(k \approx 3\) W/mK, and electrons are neglected.
  • Effective thermal conductivities are obtained by fitting Monte Carlo temperature decays with an isotropic Fourier-law model; the radial suppression function is then extracted using convex optimization and compared with new room-temperature TDTR data and previously reported cryogenic data.

Limitations

  • The paper states that only a single pulse is simulated rather than the modulated pulse train of an actual TDTR experiment, though radial effects are expected to be the same.
  • The paper states that the isotropic fit is used despite anisotropic heat flux because concentric pump-probe TDTR often cannot extract anisotropic conductivity and radial sensitivity decreases at large pump sizes.
  • The paper states that cryogenic discrepancies below 150 K may be partly due to isotope/defect differences and an unaccounted cross-plane effect; the text also makes evident additional modeling choices such as a model Si dispersion, acoustic phonons only, no electron heat conduction in the metal, and a constant Al relaxation time.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1063/1.4870811

Cites

34 of 34 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

20 of 24 cited sources checked · 14 supported · 0 flagged · accuracy not yet measured

  • Interpretationown data

    The pump size is a key variable that sets the thermal length scale for radial transport.

    “The diameter dependence of the radial heat flux demonstrates that the pump size is a key variable that sets the thermal length scale for radial transport, confirming previous explanations for observations of a pump-beam size dependent thermal conductivity.14”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] 2011, Phys. Rev. Lett. source
  • Findingcitation

    The thermal decay predicted by the Boltzmann transport equation is slower than Fourier's law predicts.

    “As in prior works, 22,24 the thermal decay predicted by the BTE is slower than Fourier's law predicts.”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [22] 2011, Phys. Rev. B source
    • cites [24] 2011, Phys. Rev. B source

      The Simulation results section compares the BTE simulation with a Fourier-law solution, and the Fig. 14 caption attributes their difference to ballistic effects. Neither passage states which prediction decays more slowly, so our copy does not establish the claimed direction of the difference. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: The metal transducer's phonons are the sole heat carrier; electron heat conduction is neglected.
    • The authors state this as an assumption: Radial effects are the same for the impulse response and the multi-pulse response.
  • Findingcitation

    The effective thermal conductivity decreases with decreasing pump beam size.

    “The results show the experimentally observed trend of decreasing effective thermal conductivity with decreasing pump beam size.14”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] 2011, Phys. Rev. Lett. source
  • Interpretationcitation

    The reduction in thermal conductivity is due to the suppressed cross-plane heat flux in Fig. 1(b).

    “The reduction in thermal conductivity is due to the suppressed cross-plane heat flux in Fig. 1(b), which apparently does not occur in the actual experiment but is consistent with the earlier simulations.22,24”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [22] 2011, Phys. Rev. B source
    • cites [24] 2011, Phys. Rev. B source

      The “Influence of nanopore alignment” section (page unavailable) reports that pore geometry reduces simulated thermal conductivity by almost a factor of two. But it measures conductivity in the y direction, and “Heat flux and thermal conductivity in a thin slab” (page unavailable) likewise describes y-directed heat flux. Neither passage attributes the reduction to suppressed cross-plane heat flux. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
  • Premisecitation

    Solving the transient Boltzmann transport equation in the full 3D geometry of a TDTR experiment is extremely challenging using previously reported numerical methods due to its large spatial extent and 3D geometry.

    “Solving the transient BTE in this domain is extremely challenging using previously reported numerical methods 22,27,29,30 due to its large spatial extent and the 3D geometry.”
    Quote in the paperSubstantial claim1 of 4 sources checked
    • cites [22] 2011, Phys. Rev. B source

      In “A. Calculation of the equilibrium distribution” (page unavailable), the paper says that a large domain with a fine spatial grid makes the BTE solution computationally demanding. Its Introduction describes a one-dimensional simulation; our copy does not state that the full 3D geometry of a TDTR experiment makes previously reported numerical methods extremely challenging. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • cites [27] 1993, J. Heat Transfer source
    • cites [29] 2008, J. Heat Transfer source
    • cites [30] 2003, J. Heat Transfer source
  • Classified as probably not a substantial claim. 13
  • May combine two claims. 5

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Further computational efficiency in deviational Monte Carlo can be obtained by linearizing the scattering distribution, eliminating the need for spatial and temporal discretization.

    “Further computational efficiency can be obtained by linearizing the scattering distribution, eliminating the need for spatial and temporal discretization.26”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [26] 2012, Appl. Phys. Lett. source

  • Classified as probably not a substantial claim.

    Premisecitation

    The interface transmissivity can be related to the interface conductance G specified in Fourier's law according to the model of Ref. 22.

    “At the interface, phonons have a probability to be transmitted or reflected diffusely; this transmissivity can be related to the interface conductance G specified in Fouriers law according to the model of Ref. 22.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] 2011, Phys. Rev. B source

  • Classified as probably not a substantial claim.

    Findingcitation

    Assigning phonons a constant relaxation time of 1 ps in the metal transducer yields a low thermal conductivity of around 3 W/mK.

    “Following Ref. 22, we assign phonon to have a constant relaxation time of 1 ps, yielding a low thermal conductivity of around 3 W/mK.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] 2011, Phys. Rev. B source
    • The authors state this as an assumption: The metal transducer's phonons are the sole heat carrier; electron heat conduction is neglected.

  • Classified as probably not a substantial claim.

    Premisecitation

    The mean free path distribution can be reconstructed from measurements of quasiballistic heat conduction using a method introduced by Minnich, provided that the quasiballistic transport in the experiment can be accurately simulated.

    “The MFP distribution can be reconstructed from these measurements using a method introduced by Minnich provided that the quasiballistic transport in the experiment can be accurately simulated.19”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [19] 2012, Phys. Rev. Lett. source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Variance reduction in deviational Monte Carlo methods is achieved by simulating only the deviation from a known Bose-Einstein distribution.

    “Variance reduction in deviational MC methods is achieved by simulating only the deviation from a known Bose-Einstein distribution.24”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [24] 2011, Phys. Rev. B source

  • May combine two claims.

    Premisecitation

    The deviational Monte Carlo method is orders of magnitude faster than previous algorithms while requiring minimal memory.

    “Here, we use the recently introduced deviational Monte Carlo (MC) method 24 that is orders of magnitude faster than previous algorithms while requiring minimal memory.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [24] 2011, Phys. Rev. B source

      Section V (Computational Efficiency; page not available) reports speedups of approximately four to six orders of magnitude over standard non-variance-reduced Monte Carlo methods for small temperature signals. The claim omits that condition, and our copy does not establish that the method requires minimal memory. (LLM, conf=0.88)

      Experimental — accuracy not yet measured

  • Classified as probably not a substantial claim.

    Premisecitation

    Minnich recently introduced a framework in which the MFP distribution of the substrate can be reconstructed from effective thermal conductivities and a suppression function that describes the difference in heat flux between quasiballistic and Fourier predictions.

    “Minnich recently introduced a framework in which the MFP distribution of the substrate can be reconstructed from the effective thermal conductivities, as shown in Fig. 2(a)), and a suppression function that describes the difference in heat flux between the quasiballistic and Fourier predictions.19”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [19] 2012, Phys. Rev. Lett. source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Previously, this equation was used to find the MFP distribution.

    “Previously, this equation was used to find the MFP distribution.19”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [19] 2012, Phys. Rev. Lett. source

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation

    The suppression function depends on the experimental geometry and mathematically describes how the heat flux curves differ from the Fourier's law curve.

    “This function depends on the experimental geometry and mathematically describes how the heat flux curves in Fig. 1(c) differ from the Fourier's law curve.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [19] 2012, Phys. Rev. Lett. source

  • May combine two claims.

    Findingnone given

    S_z is independent of D_i and does not affect the radial suppression function.

    “S z is independent of D i and does not affect the radial suppression function.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductreasoning

    The low thermal conductivity of the metal transducer eliminates any possible artificial quasiballistic effects in the metal transducer, attributing all quasiballistic effects to the Si substrate.

    “This change eliminates any possible artificial quasiballistic effects in the metal transducer, attributing all quasiballistic effects to the Si substrate.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The metal transducer's phonons are the sole heat carrier; electron heat conduction is neglected.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductreasoning

    The approach would work with any phonon dispersion because the comparison is made to predictions of Fourier theory based on the same dispersion.

    “We have chosen this dispersion as it is a model dispersion for Si, but our approach would work with any dispersion because we compare to the predictions of Fourier theory based on the same dispersion.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Study design and conductnone given

    Our analysis remains valid because we are able to decouple the radial and cross-plane directions.

    “However, our analysis remains valid because we are able to decouple the radial and cross-plane directions.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Most TDTR measurements are taken using concentric pump and probe, for which extracting anisotropic thermal conductivity is not always possible.
    • Rests on an assumption: The sensitivity of the decay to radial thermal conductivity k_r decreases with increasing pump beam size, leading to large uncertainties in the fitted k_r.

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The radial suppression function describes the discrepancy in heat flux compared to the Fourier's law prediction.

    “further identify the radial suppression function that describes the discrepancy in heat flux compared to the Fouriers law prediction”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    Because f and k_i are known, the equation can also be solved for S to find the suppression function.

    “However, because here f and k i are known, this equation can also be solved for S to find the suppression function.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductown data

    We verified the robustness of the solution by adding artificial noise to k_i and by removing different constraints in the convex optimization.

    “We verified the robustness of the solution by adding artificial noise to k i and by removing different constraints in the convex optimization.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    The blue errorbars indicate 10% uncertainty of the measurements.

    “The blue errorbars indicate 10% uncertainty of our measurements.”
    Quote in the paperNot a substantial claim
17 of 17

  • The authors state this as an assumption.

    The metal transducer's phonons are the sole heat carrier; electron heat conduction is neglected.

    “We take the metal transducer to have the same dispersion as the experimental dispersion of Al in the [100] direction and neglect heat conduction by electrons, instead considering phonons as the sole heat carrier.”

  • The authors state this as an assumption.

    Radial effects are the same for the impulse response and the multi-pulse response.

    “Because radial effects are expected to be the same for the impulse and multi-pulse response, for simplicity, we only consider a single pulse in our study.”

  • The paper relies on this without support.

    TDTR routinely measures the correct thermal conductivity for Si at room temperature with similar pump sizes.

    “This observation is puzzling because TDTR routinely measures the correct thermal conductivity for Si at room temperature with similar pump sizes.”

  • The paper relies on this without support.

    Most TDTR measurements are taken using concentric pump and probe, for which extracting anisotropic thermal conductivity is not always possible.

    “First, most TDTR measurements are taken using concentric pump and probe, for which extracting anisotropic thermal conductivity is not always possible.”

  • The paper relies on this without support.

    The sensitivity of the decay to radial thermal conductivity k_r decreases with increasing pump beam size, leading to large uncertainties in the fitted k_r.

    “Second, the sensitivity of the decay to radial thermal conductivity k r decreases with increasing pump beam size, leading to large uncertainties in the fitted k r .”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The re-emerged frequency dependence in Al/SiO2/Si at 80 K is attributed to weakened interactions between different phonon branches at lower temperatures, causing inefficient energy transfer and stronger phonon non-equilibrium.

    “Firstly, the strength of interactions between different phonon branches weakens at lower temperatures, resulting in inefficient energy transfer [29]. A stronger phonon non-equilibrium facilitates the emergence of frequency dependence.”

    From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [29]

    The Abstract (page unavailable) describes a pump-beam-size dependence in cryogenic Si and attributes it to quasiballistic transport. The methods and heat-flux discussion (section @e, page unavailable) do not address weakened interactions between phonon branches, inefficient energy transfer at lower temperatures, or the proposed non-equilibrium explanation; searches for those terms found no supporting passage. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Cross-plane heat flux is attributed to the radial panel

    Minor

    The text says the cross-plane heat flux is in Fig. 1(c), but the figure caption identifies Fig. 1(b) as cross-plane and Fig. 1(c) as radial.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. 2005, Thermoelectrics Handbook: Macro to Nano

Cited by

Sorted by citations

Authors
Ding, D., Chen, X., & Minnich, A. J.
Affiliations
California Institute of Technology (US)
ORCID
Minnich, A. J.
Venue
Applied Physics Letters
Publisher
AIP Publishing
Published
2014-04-07
Volume
104
Issue
14
ISSN
  • 0003-6951 (ISSN-L)
  • 1077-3118
DOI
10.1063/1.4870811
OpenAlex
W2070771590
Semantic Scholar
974143b67c20805e908a6fbd8534fd17ccd6df44
Cited by
51
References
33
Citations by year
2026: 2 · 2025: 4 · 2024: 2 · 2023: 3 · 2022: 2 · 2021: 2 · 2020: 2 · 2019: 2 · 2018: 12 · 2017: 4 · 2016: 7 · 2015: 5 · 2014: 4
FWCI
2.98
Type
journal article
Language
en
Open access
Locations
  • Applied Physics Letters (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CaltechAUTHORS (California Institute of Technology) (submittedVersion)
  • arXiv (Cornell University)
Funders
NSF (CBET 1254213)
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
time-domain thermoreflectance, thermal conductivity, silicon, Boltzmann equation, heat conduction, cryogenic temperatures, non-Fourier heat conduction, Monte Carlo simulation

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-20.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-20.

Cited sources are checked by gpt-6-sol.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review