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Cited by 4Thermal properties of materialsThermoelastic and Magnetoelastic PhenomenaThermography and Photoacoustic Techniques

Boundary conditions dictate frequency dependence of thermal conductivity in silicon

Liu, Yizhe, Li, Qinshu, Liu, Fang, Wang, Xinqiang, Sun, Bo·2025·Applied Physics Reviews
DOI 10.1063/5.0254248
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Abstract

Non-Fourier thermal transports have drawn significant attention for decades. Among them, the frequency-dependent thermal conductivity has been extensively explored by pump-probe techniques, such as time-domain thermoreflectance, which is employed to probe the spectra of phonon mean free paths. However, previous studies on silicon have not exhibited apparent frequency dependence despite its broad phonon distribution. Here, we report the frequency-dependent thermal transport in Al/Si with an atomically sharp interface, where the matched Debye temperatures preserve the temperature difference between low- and high-energy phonons in Si and contribute as additional non-equilibrium thermal resistance. The dependence vanishes in Al/SiO2/Si at room temperature, since the SiO2 interlayer facilitates phonon scattering and destroys thermal non-equilibrium. At 80 K, frequency dependence reemerges in Al/SiO2/Si due to reduced interfacial phonon scattering, which is not sufficient to destroy the temperature difference between low- and high-energy phonons. The frequency dependence is weakened in the Al/Si sample at 500 K, originating from the enhanced phonon scattering rate in Si. Our findings highlight the significance of boundary conditions in frequency-dependent thermal conductivity.

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Written from the paper’s full text.

TL;DR

TDTR measurements on Al/Si with an atomically sharp interface reveal frequency-dependent thermal conductivity in Si at room temperature, whereas Al/SiO₂/Si with a diffuse interface shows no dependence at room temperature but reemerges at 80 K. The results show that boundary conditions control whether phonon nonequilibrium produces frequency dependence and enable extraction of Si phonon mean-free-path spectra.

Key findings

  • At room temperature, Al/Si apparent Λ decreases from 135 W m⁻¹ K⁻¹ at 0.47 MHz to 106 W m⁻¹ K⁻¹ at 10.1 MHz, while G increases from 302 to 385 MW m⁻² K⁻¹. Al/SiO₂/Si shows no apparent frequency dependence: Λ ranges from 129 to 136 W m⁻¹ K⁻¹ and G from 80.1 to 82.2 MW m⁻² K⁻¹.
  • At 80 K, Al/Si still shows frequency dependence with a 26.2% variation in Λ, and Al/SiO₂/Si reemerges with an 18.1% change in Λ. Maximum measured Λ is 745 W m⁻¹ K⁻¹ for Al/Si and 890 W m⁻¹ K⁻¹ for Al/SiO₂/Si, both below the bulk Si reference.
  • Two-channel fits for Al/Si at room temperature give Λ₁ = 66 W m⁻¹ K⁻¹, Λ₂ = 75 W m⁻¹ K⁻¹, G₁ = 58 MW m⁻² K⁻¹, G₂ = 400 MW m⁻² K⁻¹, and g = 1.2 × 10¹⁴ W m⁻³ K⁻¹; total Λ = 141 W m⁻¹ K⁻¹ and total G = 458 MW m⁻² K⁻¹, with channel contributions of 46.8% and 53.2%. At 80 K, Λ₁ = 630 W m⁻¹ K⁻¹, Λ₂ = 320 W m⁻¹ K⁻¹, G₁ = 22 MW m⁻² K⁻¹, G₂ = 162 MW m⁻² K⁻¹, g = 5 × 10¹² W m⁻³ K⁻¹; total Λ = 950 W m⁻¹ K⁻¹ and total G = 184 MW m⁻² K⁻¹, with Channel 1 contributing 66.5%.
  • Phonon MFP spectra extracted from Al/Si agree with first-principles calculations at room temperature and 80 K, while Al/SiO₂/Si cumulative Λ deviates substantially from first-principles results because the diffuse interface reduces the ballistic phonon population.

Methods

  • Time-domain thermoreflectance (TDTR) used a pulsed femtosecond laser at 785 nm and 80 MHz repetition rate; the pump beam was modulated from 0.47 MHz to 10.1 MHz, the probe was chopped at 200 Hz, and both beams were focused to a 1/e² radius of 30 µm. Measurements were performed at room temperature and 80 K.
  • Two samples were compared: Al/Si with an atomically sharp interface grown by molecular beam epitaxy of Al(111) on Si(111), with 138 nm Al thickness; and Al/SiO₂/Si with a native oxide interlayer and magnetron-sputtered Al, with 70 nm Al thickness.
  • Apparent Λ and G were first extracted with a one-channel stratified thermal model based on Fourier’s law. Frequency dependence was then interpreted with a two-channel model separating low-energy (<3 THz) and high-energy (>3 THz) phonons, with coupling factor g and heat capacities C₁ and C₂ derived from Si phonon density of states.

Limitations

  • Stated: the two-channel coupling factor g has relatively low sensitivity across all frequencies at both temperatures, indicating larger uncertainty; reported uncertainties include ±28.8% for g at room temperature and ±47.5% at 80 K. One-channel uncertainties are less than ±4.5% for Λ and G at room temperature, while at 80 K G reaches ±5.2% and Λ is less than ±8.3%.
  • Stated: at 80 K the total Λ is lower than previously reported Si values, probably due to suppressed in-plane Λ from a not-large-enough laser spot size. The Al/Si 80 K MFP spectrum deviates for short-MFP phonons because Al/Si phonon DOS mismatch causes high-frequency Si phonons above 10 THz to be reflected or decomposed at the interface. Al/SiO₂/Si cumulative Λ deviates from first-principles calculations because the diffuse interface reduces ballistic phonon population.
  • Evident: the study tests only two boundary conditions and one transducer metal (Al), over a limited modulation range of 0.47–10.1 MHz; the suggestions that Ta or AuPd may be preferable for suppressing frequency dependence are explicitly postulated rather than experimentally tested here.

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

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Registry

Crossref · 10.1063/5.0254248

Cites

41 of 41 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

13 of 22 cited sources checked · 1 supported · 0 flagged · accuracy not yet measured

  • Interpretationcitation
    § A. Frequency dependence at room temperature · p. 4

    The absence of frequency dependence in Al/SiO2/Si at room temperature suggests that a diffuse thermal interface inhibits non-equilibrium heat flow in high-frequency TDTR measurements, preventing frequency dependence.

    “The absence of frequency dependence in the Al/SiO 2 /Si sample agrees with prior TDTR studies on Al/Si structures, suggesting that a diffuse thermal interface inhibits non-equilibrium heat flow in highfrequency TDTR measurements, thereby preventing the emergence of frequency dependence [20].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [20] 2022, Nat. Commun. source

      The Results (page unavailable) say that diffuse Al/Si interfaces have smaller phonon non-equilibrium and diminished inelastic phonon transport, supporting the proposed mechanism in general. The paper also describes preparing an Al/SiO2/Si sample, but the passages found do not report an absence of TDTR modulation-frequency dependence at room temperature or establish that this mechanism prevents it; the reported dependence is on temperature. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
  • Findingown data
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    The total thermal conductance G1 + G2 = 458 MW m−2 K−1 at room temperature matched the DMM prediction, even without considering serial interfacial thermal resistance from electron-phonon interactions within the Al layer.

    “The total thermal conductance G 1 + G 2 = 458 MW m -2 K -1 matched the value predicted by the DMM [20], even without considering the serial interfacial thermal resistance from electron-phonon interactions within the Al layer [34].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [20] 2022, Nat. Commun. source
    • cites [34] 1979, Phys. Rev. B source
  • Study design and conductcitation
    § B. Frequency dependence at 80 K · p. 4

    The 1/e2 beam radius of 30 μm in the measurement results in negligible sensitivity of TDTR signals to in-plane thermal conductivity Λ// at frequencies higher than 4 MHz.

    “However, the 1/e 2 beam radius of 30 µ m in our measurement results in a negligible sensitivity of signals to Λ // at f higher than 4 MHz (see Supplemental Material [25]).”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [25] 2024, Phys. Rev. Lett. source

      The TDTR measurement passage (section labeled Abstract; page unavailable) mentions a sensitivity analysis but identifies metal-transducer thickness and heat capacity as the main sources of uncertainty. Searches for the 30 μm beam radius, frequencies above 4 MHz, and sensitivity to in-plane thermal conductivity found no passage addressing the claimed relationship. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
  • Findingcitation
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    The obtained coupling factor g at room temperature lies in the order of 10^14 W m−3 K−1, as expected from scattering rates of low-frequency phonons.

    “The obtained coupling factor g lies in the order of 10 14 Wm -3 K -1 , which is expected by the scattering rates of low-frequency phonons [35, 36].”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [35] 2002, J. Appl. Phys. source
    • cites [36] 2014, Sci. Rep. source

      The Methods section on phonon–phonon scattering and the Discussion address phonon scattering rates, including those of low-frequency phonons. Neither reports a coupling factor g or relates a value of 10^14 W m−3 K−1 to those rates (searched for the factor, its value, and low-frequency phonon scattering). (LLM, conf=0.77)

      Experimental — accuracy not yet measured
  • Findingcitation
    § C. Cumulative thermal conductivity with phonon mean free path · p. 5

    At 80 K, the Al/Si measurements align closely with first-principles results interpolated from Minnich et al. at 100 K and 60 K, with a slight deviation for phonons with short mean free paths.

    “At 80 K, our measurements align closely with first-principles results, calculated through interpolation of calculations by Minnich et al. at 100 K and 60 K [14], with a slight deviation for phonons with short MFPs.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] 2011, Phys. Rev. Lett. source

      The main text and Fig. 3 caption (p. 3) identify Minnich et al.’s first-principles calculations of silicon thermal conductivity versus phonon mean free path, so the citation supports the source of the calculations. Our copy does not identify calculations specifically at 100 K and 60 K. The 80 K interpolation, agreement with the citing authors’ measurements, and short-MFP deviation are presented as the citing authors’ own analysis, not findings attributed to Minnich et al. (LLM, conf=0.78)

      Experimental — accuracy not yet measured
  • May combine two claims. 30
  • Classified as probably not a substantial claim. 5

  • May combine two claims.

    Study design and conductcitation
    § A. Frequency dependence at room temperature · p. 3

    The maximum Λ and G values obtained for the Al/Si sample agree with literature values within the error range, validating the accuracy of the measurements.

    “The obtained maximum values of Λ and G in the Al/Si sample are in excellent agreement with literature values within the error range [20, 26], validating the accuracy of our measurements.”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [20] 2022, Nat. Commun. source

      The Results (Fig. 1 caption and accompanying text; page unavailable) compare Al/Si thermal conductance G with earlier measurements and report a similar trend at lower temperatures. They do not report a maximum Λ value or establish that both maxima agree with literature values within error; the paper describes its room-temperature G as record high and a different trend at higher temperatures. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • cites [26] 2025, Mod. Phys. Lett. B source

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

    Findingcitation
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    The 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].”
    Quote in the paperNot a substantial claim1 of 3 sources checked
    • cites [18] 2014, Nat. Commun. source

      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 measured
    • cites [26] 2025, Mod. Phys. Lett. B source
    • cites [32] 2015, J. Appl. Phys. source

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    Both 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.”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [17] 2013, Phys. Rev. B source
    • cites [18] 2014, Nat. Commun. source

      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 measured

  • May combine two claims.

    Interpretationcitation
    § B. Frequency dependence at 80 K · p. 5

    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.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [29] 2014, Appl. Phys. Lett. source

      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

  • May combine two claims.

    Findingown data
    § A. Frequency dependence at room temperature · p. 4

    In the Al/SiO2/Si sample at room temperature, the measured Λ ranges from 129 to 136 W m−1 K−1 and G from 80.1 to 82.2 MW m−2 K−1, with no apparent frequency dependence within experimental uncertainty.

    “In this sample, the measured Λ ranges from 129 W m -1 K -1 to 136 W m -1 K -1 at varying frequencies, while G varies from 80.1 MW m -2 K -1 to 82.2 MW m -2 K -1 , demonstrating no apparent frequency dependence within experimental uncertainty.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § A. Frequency dependence at room temperature · p. 3

    The one-channel model fits the Al/Si TDTR data well at 0.47 MHz but overestimates the data at higher frequencies beyond the error range.

    “Applying one set of parameters (Λ=135 W m -1 K -1 and G =302 MW m -2 K -1 ), the one-channel model fits well for data at 0.47 MHz but shows an obvious overestimation for data at higher frequencies, exceeding the error range.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    The total thermal conductivity at 80 K is 950 W m−1 K−1, still lower than previously reported values in Si.

    “The total Λ of 950 W m -1 K -1 is still lower than previously reported values in Si, which probably arises from the suppressed Λ // due to a not large enough laser spot size.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    At 80 K, the two-channel model yielded Λ1 = 630 W m−1 K−1, Λ2 = 320 W m−1 K−1, G1 = 22 MW m−2 K−1, G2 = 162 MW m−2 K−1, and g = 5 × 10^12 W m−3 K−1.

    “Additionally, at 80 K, we obtained Λ 1 = 630 W m -1 K -1 , Λ 2 = 320 W m -1 K -1 , G 1 = 22 MW m -2 K -1 , G 2 = 162 MW m -2 K -1 , and g = 5 × 10 12 W m -3 K -1 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    In the Al/Si sample at room temperature, the two-channel model yielded Λ1 = 66 W m−1 K−1, Λ2 = 75 W m−1 K−1, G1 = 58 MW m−2 K−1, G2 = 400 MW m−2 K−1, and g = 1.2 × 10^14 W m−3 K−1.

    “In the Al/Si sample, we obtained Λ 1 = 66 W m -1 K -1 , Λ 2 = 75 W m -1 K -1 , G 1 = 58 MW m -2 K -1 , G 2 = 400 MW m -2 K -1 , and g = 1.2 × 10 14 W m -3 K -1 at RT.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § B. Frequency dependence at 80 K · p. 4

    In the Al/Si sample at 80 K, frequency dependence is observed with a variation of 26.2% in Λ, larger than at room temperature.

    “In the Al/Si sample with Boundary 1, we still observed the frequency dependence, with a relatively larger variation of 26.2 % in Λ, compared to the RT values.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    At 80 K, the partial thermal conductivity contributed by Channel 1 increases to 66.5%, consistent with the expectation that low-energy phonons play a more significant part at lower temperatures.

    “The partial Λ contributed by Channel 1 increases to 66.5 %, which is consistent with the expectation that low-energy phonons play a more significant part in Λ at lower temperatures.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § B. Frequency dependence at 80 K · p. 4

    At 80 K, the measured maximum Λ in the Al/Si sample (745 W m−1 K−1) is lower than in the Al/SiO2/Si sample (890 W m−1 K−1), and both are smaller than the bulk Si reference value measured by the steady-state method.

    “The measured maximum Λ in the Al/Si sample (745 Wm -1 K -1 ) is lower than that in the Al/SiO 2 /Si sample (890 W m -1 K -1 ) while both were smaller than the reference value measured using the steady-state method.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § B. Frequency dependence at 80 K · p. 4

    In the Al/SiO2/Si sample at 80 K, frequency dependence is observed with a change of 18.1% in Λ, in contrast to room temperature.

    “Importantly, despite the presence of a SiO 2 interlayer in the Al/SiO 2 /Si sample, frequency dependence was observed at 80 K, with a change of 18.1 % in Λ, in sharp contrast to the room temperature behavior.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    The total thermal conductance at 80 K is 184 MW m−2 K−1, confirming DMM predictions within measurement uncertainty.

    “Besides, the total G =184 MW m -2 K -1 confirms DMM predictions within the measurement uncertainty.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation
    § D. Two-channel model for non-equilibrium thermal transport · p. 6

    At room temperature, low-frequency phonons in Channel 1 contribute 46.8% of the total thermal conductivity and high-frequency phonons contribute 53.2%, consistent with theoretical predictions.

    “Low-frequency phonons in Channel 1 contribute 46.8 % of the total Λ while high-frequency phonons contribute 53.2 %, consistent with theoretical predictions [33].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [33] 2012, Appl. Phys. Lett. source

  • May combine two claims.

    Findingown data
    § 1.3. Sensitivity analysis · p. 11

    Sensitivity to in-plane thermal conductivity Λ// is negligible for both samples at room temperature but notable for modulation frequencies lower than 4 MHz at 80 K.

    “The sensitivity to inplane thermal conductivity Λ // is negligible for both samples at RT while exhibiting a notable value for modulation frequencies lower than 4 MHz at 80 K.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § A. Frequency dependence at room temperature · p. 3

    The atomically sharp interface in the Al/Si sample preserves non-equilibrium effects and reveals the frequency dependence in the TDTR measurements.

    “The atomically sharp interface in our Al/Si sample preserves the non-equilibrium effects and reveals the frequency dependence in our TDTR measurements.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § A. Frequency dependence at room temperature · p. 4

    The diffuse interface introduces additional phonon scattering sources such as atomic intermixing, roughness, oxide interlayer, and contamination, facilitating energy exchange between different phonon branches and suppressing phonon non-equilibrium.

    “As illustrated in Fig. 3, the diffuse interface introduces additional sources of phonon scattering, such as atomic intermixing, roughness, oxide interlayer, and contamination, facilitating sufficient energy exchange between different phonon branches and consequently significantly suppressing phonon non-equilibrium.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § I. INTRODUCTION · p. 1

    The atomically sharp Al/Si interface, with matched phonon density of states, minimizes interfacial phonon scattering and preserves thermal nonequilibrium between low- and high-energy phonons in Si.

    “The atomically sharp Al/Si interface, with matched phonon density of states (DOS), minimizes the interfacial phonon scattering, preserving the phonon non-equilibrium in Si.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. Frequency dependence at 80 K · p. 5

    At 80 K, the interface even with a SiO2 layer is insufficient to scatter heat-carrying phonons, so the boundary condition transforms from Boundary 2 to Boundary 1 for the low-temperature case.

    “Therefore, the interface, even with a SiO 2 layer, is insufficient to scatter heat-carrying phonons, and the boundary condition transforms from Boundary 2 to Boundary 1 for the low-temperature case.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given
    § C. Cumulative thermal conductivity with phonon mean free path · p. 5

    Ta has a higher density of states for low-energy phonons than Al, leading to a smaller temperature difference between low- and high-energy phonons in Si.

    “Secondly, Ta has a higher density of states for low-energy phonons, leading to a smaller temperature difference between low- and high-energy phonons in Si, compared to the Al transducer.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § A. Frequency dependence at room temperature · p. 4

    The value of G is not a decisive factor in mediating frequency dependent thermal transport; a sharp interface with appropriate phonon spectra of constituent materials constitutes the proper boundary conditions.

    “Hence, the value of G is not a decisive factor in mediating frequency dependent thermal transport, while a sharp interface with appropriate phonon spectra of constituent materials consists of the proper boundary conditions.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Study design and conductnone given
    § 1.1. TDTR setup · p. 10

    The steady-state temperature rises on the sample surface were kept below 5 K for both room-temperature and cryogenic measurements by choosing laser powers of 250 mW pump and 150 mW probe.

    “We carefully chose the laser powers (250 mW for the pump beam and 150 mW for the probe beam) to make sure that the steady-state temperature rises on the sample surface are less than 5 K for both room-temperature (RT) and cryogenic measurements.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § C. Cumulative thermal conductivity with phonon mean free path · p. 5

    Al is a better transducer material than Ta for measuring the spectral phonon mean free paths of Si.

    “For example, we postulate that Al is a better transducer material, compared to Ta, to measure the spectral phonon MFPs of Si.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § C. Cumulative thermal conductivity with phonon mean free path · p. 5

    To accurately probe spectral phonon mean free paths, not only an atomically sharp interface but also a metal transducer with a matched or larger Debye temperature is required.

    “Specifically, not only an atomically sharp interface but also a metal transducer with a matched or larger Debye temperature is required.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. SUMMARY · p. 6

    Imperfections at the interface facilitate energy exchange between different phonon branches and prohibit non-equilibrium effects.

    “In this context, imperfections at the interface would facilitate energy exchange between different phonon branches and prohibit the non-equilibrium effects.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § IV. SUMMARY · p. 6

    The frequency dependence is previously buried due to the lack of high-quality sharp interfaces, as validated by measurements on the Al/SiO2/Si sample.

    “The dependence is previously buried due to the lack of high-quality sharp interfaces, as validated by measurements on the Al/SiO 2 /Si sample.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § C. Cumulative thermal conductivity with phonon mean free path · p. 5

    Ta has a more dissimilar Debye temperature to Si than Al, leaving a higher interfacial phonon scattering rate, especially for high-energy phonons.

    “Firstly, Ta shows a more dissimilar Debye temperature to Si, leaving a higher interfacial phonon scattering rate, especially for high-energy phonons.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. Frequency dependence at 80 K · p. 5

    The change in boundary condition at 80 K leads to a pronounced temperature difference between low- and high-energy phonons and observable non-equilibrium thermal resistance.

    “This change leads to a pronounced temperature difference between lowand high-energy phonons and consequently observable non-equilibrium thermal resistance.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductown data
    § 1.4. Uncertainty · p. 11

    The uncertainty of Λ and G is less than ±4.5% for both samples at room temperature; at 80 K, the uncertainty of G has a maximum of ±5.2% and that of Λ is less than ±8.3%.

    “The uncertainty of Λ and G is less than ± 4.5 % for both samples at RT. At 80 K, the uncertainty of G shows a maximum value of ± 5.2 %, while that of Λ is less than ± 8.3 %.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductown data
    § 2.3. Sensitivity analysis · p. 12

    At 80 K, Λ1, Λ2 and G2 show significant sensitivity for all frequencies, while G1 exhibits sensitivity only at frequencies higher than 1 MHz.

    “When it comes to 80 K, Λ 1 , Λ 2 and G 2 show significant sensitivity for all frequencies, while G 1 exhibitsonly at frequencies higher than 1 MHz.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § 1.3. Sensitivity analysis · p. 11

    The sensitivity analysis indicates that Λ// affects low-frequency TDTR measurements at cryogenic temperatures, and this effect would be suppressed by limited laser spot size, affecting measured apparent Λ.

    “The sensitivity analysis highlights the role of Λ // for low-frequency TDTR measurements at cryogenic temperatures, which would be suppressed due to the limited laser spot size, thus affecting the measured apparent Λ [ ? ].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [ ? ]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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

    Study design and conductown data
    § 2.3. Sensitivity analysis · p. 13

    The coupling factor g presents relatively low sensitivity across all frequencies at both temperatures, suggesting a larger degree of uncertainty in the measurements.

    “Notably, g presents relatively low sensitivity, spanning all frequencies at both temperatures, suggesting a larger degree of uncertainty in our measurements.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductown data
    § 2.3. Sensitivity analysis · p. 12

    At room temperature, Λ1 exhibits pronounced sensitivity across the entire frequency range, whereas Λ2 demonstrates sufficient sensitivity only at low frequencies; G1 displays remarkable sensitivity at low frequencies while G2 has highest sensitivity at 10.1 MHz.

    “At RT, Λ 1 exhibits pronounced sensitivity across the entire frequency range, whereas Λ 2 demonstrates sufficient sensitivity only at low frequencies. Conversely, G 1 displays remarkable sensitivity at low frequencies while G 2 gives a highest sensitivity at 10.1 MHz.”
    Quote in the paperNot a substantial claim
34 of 34

  • The authors state this as an assumption.

    All energy carriers share the same temperature, with no distinct temperature gradients between different phonon branches within the material.

    “The model utilized a numerical solution of the heat diffusion equation, grounded in the local-equilibrium assumption, where all energy carriers share the same temperature. This assumption implies, for instance, that there are no distinct temperature gradients between different phonon branches within the material.”

  • The authors state this as an assumption.

    Phonons with mean free paths longer than the characteristic length undergo ballistic transport and do not contribute to the measured thermal conductivity.

    “where we assume phonons with MFPs longer than the characteristic length undergo ballistic transport and do not contribute to the measured Λ [10, 30].”
    “where we consider ballistic phonons with characteristic lengths larger than 2 d do not contribute thermal conductivity, as illustrated in Fig. 5.”

  • The authors state this as an assumption.

    In the one-channel model, all energy carriers are in thermal equilibrium and transport diffusively for the whole process.

    “This model holds the assumption that all energy carriers are in thermal equilibrium and transport diffusively for the whole process.”
3 of 3

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.

  • Different fixed conductivity values reported for the same fits

    Minor

    The text says the Al/Si fitting curves use Λ = 135 W m−1 K−1, while the Fig. 2 caption says Λ is fixed at 134 W m−1 K−1 for all six curves, including the Al/Si curves.

  • Fig. 1 attributes semiconductor data to a graphite reference

    Minor

    The Fig. 1 caption cites reference [11] for SiGe and In-based semiconductor data, but the paper's reference [11] is a study of graphite.

2 of 2

Cites

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Cited by

Sorted by citations

Authors
Liu, Y., Li, Q., Liu, F., Wang, X., & Sun, B.
Affiliations
  • Tsinghua Shenzhen International Graduate School (CN)
  • Peking University (CN)
ORCID
Venue
Applied Physics Reviews
Publisher
AIP Publishing
Published
2025-09-01
Volume
12
Issue
3
ISSN
1931-9401 (ISSN-L)
DOI
10.1063/5.0254248
OpenAlex
W7082268859
Semantic Scholar
2e28b97796dab85fc9a1f55134cfab10626c0f4b
Cited by
4
References
38
Citations by year
2026: 1 · 2025: 3
FWCI
0.83
Type
journal article
Language
en
Open access
closed
Locations
Applied Physics Reviews (publishedVersion)
Funders
  • National Natural Science Foundation of China (52161145502, 12004211)
  • Shenzhen Science and Technology Innovation Program (RCYX20200714114643187, WDZC20200821100123001)
  • Tsinghua Shenzhen International Graduate School (QD2021008N, JC2021008)
  • Tip-top Scientific and Technical Innovative Youth Talents of Guangdong Special Support Program (2023TQ07A273)
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
non-Fourier heat conduction, silicon, Al/Si interface, phonon mean free path, phonon scattering, interfacial thermal resistance, time-domain thermoreflectance, boundary conditions, low-temperature thermal transport

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 2026-10-22.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-22.

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 ›

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