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Cited by 172Thermal properties of materialsThermography and Photoacoustic TechniquesThermoelastic and Magnetoelastic Phenomena

Universal phonon mean free path spectra in crystalline semiconductors at high temperature

Open access
Freedman, Justin P., Leach, Jacob H., Preble, Edward A., Sitar, Zlatko, Davis, Robert F., Malen, Jonathan A.·2013·Scientific Reports
DOI 10.1038/srep02963PMC PMC3797423
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Abstract

Thermal conductivity in non-metallic crystalline materials results from cumulative contributions of phonons that have a broad range of mean free paths. Here we use high frequency surface temperature modulation that generates non-diffusive phonon transport to probe the phonon mean free path spectra of GaAs, GaN, AlN and 4H-SiC at temperatures near 80 K, 150 K, 300 K and 400 K. We find that phonons with MFPs greater than 230 ± 120 nm, 1000 ± 200 nm, 2500 ± 800 nm and 4200 ± 850 nm contribute 50% of the bulk thermal conductivity of GaAs, GaN, AlN and 4H-SiC near room temperature. By non-dimensionalizing the data based on Umklapp scattering rates of phonons, we identified a universal phonon mean free path spectrum in small unit cell crystalline semiconductors at high temperature.

AI snapshot

Written from the paper’s full text.

TL;DR

Broadband frequency-domain thermoreflectance was used to measure phonon mean free path accumulation functions in GaAs, GaN, AlN, and 4H-SiC near 80, 150, 300, and 400 K. After non-dimensionalizing by Umklapp scattering rates, the accumulation functions collapse to a universal phonon mean free path spectrum for small-unit-cell crystalline semiconductors at high temperature.

Key findings

  • Phonons with MFPs greater than 230 ± 120 nm, 1000 ± 200 nm, 2500 ± 800 nm, and 4200 ± 850 nm contribute 50% of the bulk thermal conductivity of GaAs, GaN, AlN, and 4H-SiC near room temperature, respectively.
  • The measured accumulation functions for GaAs, Si, GaN, AlN, and 4H-SiC collapse to a single universal thermal conductivity accumulation function when plotted against a non-dimensional phonon MFP based on Umklapp scattering rates.
  • Approximately 90% of thermal conductivity in these crystalline semiconductors results from phonons with MFPs 1–200 times the MFP of Brillouin zone edge acoustic phonons, \(l_{\min}\).
  • The collapse of GaAs and GaN data indicates that isotope scattering did not strongly affect the \(\tau^{-1} \propto \omega^2\) scattering of long-MFP phonons probed by BB-FDTR.

Methods

  • BB-FDTR optical pump-probe measurements used 488 nm pump and 532 nm probe lasers, heterodyning with a second electro-optic modulator, and heating frequencies swept from 200 kHz to 200 MHz with a 2.65 ± 0.13 μm \(1/e^2\) spot radius.
  • Samples were undoped semiconductor-grade (100) GaAs, (0001) GaN, (0001) AlN, and (0001) 4H-SiC bulk wafers with Cr/Au transducer layers; measurements were taken near 80 K, 150 K, 300 K, and 400 K.
  • Phase-lag data were fit to a layered heat diffusion model; thermal conductivity was extracted in overlapping 13-point frequency windows and plotted against thermal penetration depth to generate accumulation functions, then normalized by bulk thermal conductivity and compared with truncated Debye and Born-von Karman Slack model forms.

Limitations

  • The paper states that assuming ballistic phonons do not contribute to measured thermal conductivity equates \(l^*\) to the thermal penetration depth; this convention is “not rigorously proven” and neglects spot-size effects that have influenced TDTR measurements in Si.
  • Bulk thermal conductivity values are an additional uncertainty not reflected in the error bars, and an isotropic thermal conductivity model was assumed for all materials; for 4H-SiC, an anisotropic fit changed results by less than 11% of \(k_{\text{bulk}}\) at 407 K and 304 K and less than 4% at 151 K and 81 K.
  • The paper states that because the data are based on small-unit-cell, high-thermal-conductivity materials, additional experiments are needed to determine whether alloys and large-unit-cell crystals adhere to a different form of \(k_{\text{universal}}\).

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

Integrity signals

Registry

Crossref · 10.1038/srep02963

Cites

43 of 43 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

8 of 36 cited sources checked · 6 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    It is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity (the phonon MFP spectrum).

    “Recent studies have shown that it is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity–the phonon MFP spectrum12345.”
    Quote in the paperSubstantial claim2 of 3 sources checked
    • cites [1] Regner, K. T. et al. Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance. Nat. Commun. 4, 1640 (2013). source
    • cites [2] Koh, Y. K. & Cahill, D. G. Frequency dependence of the thermal conductivity of semiconductor alloys. Phys. Rev. B 76, 075207 (2007). source
    • cites [3] Minnich, A. J. et al. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Phys. Rev. Lett. 107, 095901 (2011). source
  • Findingown data

    A constant thermal conductivity over all heating frequencies under predicts thermal conductivity in single crystal Si and in the GaAs, GaN, AlN, and 4H-SiC crystals studied here.

    “On the other hand, a constant thermal conductivity over all heating frequencies was found to under predict thermal conductivity in single crystal Si1, and likewise in the GaAs, GaN, AlN, and 4H-SiC crystals studied here.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [1] Regner, K. T. et al. Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance. Nat. Commun. 4, 1640 (2013). source
  • Findingcitation

    Density functional theory driven simulations of kaccum in GaAs at T = 300 K agree with the measured data.

    “Density functional theory driven simulations of kaccum in GaAs at T = 300 K agree with the measured data in Fig. 2(a)19.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [19] Luo, T., Garg, J., Shiomi, J., Esfarjani, K. & Chen, G. Gallium arsenide thermal conductivity and optical phonon relaxation times from first-principle calculations. EPL 101, 16001 (2013). source

      The Results and discussion text reports a first-principles calculation of GaAs thermal-conductivity accumulation at 300 K (Fig. 4(a)). It separately compares total thermal conductivity with measurements and calls the agreement good, despite a roughly 10% discrepancy at 300 K. Our copy does not report a comparison between the calculated accumulation curve and measured accumulation data. (LLM, conf=0.88)

      Experimental — accuracy not yet measured
  • Premisecitation

    Heat sources with dimensions smaller than phonon MFPs perceive a locally suppressed thermal conductivity.

    “Heat sources with dimensions smaller than phonon MFPs also perceive a locally suppressed thermal conductivity323.”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [3] Minnich, A. J. et al. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Phys. Rev. Lett. 107, 095901 (2011). source
    • cites [23] Chen, G. Nonlocal and nonequilibrium heat conduction in the vicinity of nanoparticles. J. Heat Transfer 118, 539–545 (1996). source
  • Findingown data

    Phonons with MFPs greater than 4200 ± 850 nm contribute 50% of the bulk thermal conductivity of 4H-SiC near room temperature.

    “We find that phonons with MFPs greater than 230 ± 120 nm, 1000 ± 200 nm, 2500 ± 800 nm, and 4200 ± 850 nm contribute 50% of the bulk thermal conductivity of GaAs, GaN, AlN, and 4H-SiC near room temperature.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Ballistic phonons did not contribute to the measured value of thermal conductivity.
  • Classified as probably not a substantial claim. 9
  • May combine two claims. 9

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

    Premisecitation

    First principles calculations have been used to evaluate kaccum of GaAs, though the predictions have not been experimentally confirmed.

    “First principles calculations have been used to evaluate kaccum of GaAs, though the predictions have not been experimentally confirmed19.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [19] Luo, T., Garg, J., Shiomi, J., Esfarjani, K. & Chen, G. Gallium arsenide thermal conductivity and optical phonon relaxation times from first-principle calculations. EPL 101, 16001 (2013). source

      The paper’s Abstract and discussion of Fig. 4(a) report first-principles calculations of GaAs thermal-conductivity accumulation, supporting the first part of the claim. The paper compares total thermal conductivity with measurements, but the passages do not establish that the accumulation predictions had not been experimentally confirmed; those are distinct quantities. Page numbers are unavailable in our copy. (LLM, conf=0.83)

      Experimental — accuracy not yet measured

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

    Premisecitation

    Transient grating experiments have varied the grating period to study related ballistic phonon transport effects in Si membranes.

    “More recently, transient grating experiments have varied the grating period to study related ballistic phonon transport effects in Si membranes4.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [4] Johnson, J. A. et al. Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane. Phys. Rev. Lett. 110, 025901 (2013). source

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

    Premisecitation

    Further studies using TDTR have equated l* to the laser spot size on Si and the dimensions of nano-patterned heaters on sapphire.

    “Further studies using TDTR have equated l* to the laser spot size on Si3 and the dimensions of nano-patterned heaters on sapphire26.”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [3] Minnich, A. J. et al. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Phys. Rev. Lett. 107, 095901 (2011). source
    • cites [26] Siemens, M. E. et al. Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft x-ray beams. Nat. Mater. 9, 26–30 (2010). source

  • May combine two claims.

    Findingown data

    The observed thermal conductivity difference between anisotropic and isotropic data interpretations varied by less than 11% of kbulk at T = 407 K and 304 K and by less than 4% of kbulk at T = 151 K and 81 K.

    “The observed thermal conductivity difference between the anisotropic and isotropic data interpretations varied by less than 11% of kbulk at T = 407 K and 304 K and by less than 4% of kbulk at T = 151 K and 81 K (see Supplementary Information).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: An isotropic model of thermal conductivity (kin-plane = kcross-plane) was assumed in GaAs, GaN, AlN, and 4H-SiC across all heating frequencies.

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

    Findingown data

    The measured thermal interface conductance of AlN and GaN with chromium at T = 300 K is within 20% of prior values.

    “Interfaces of AlN and GaN with chromium have been measured at T = 300 K and our results are within 20% of the values29.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [29] Stevens, R. J., Smith, A. N. & Norris, P. M. Measurement of thermal boundary conductance of a series of metal-dielectric interfaces by the transient thermoreflectance technique. J. Heat Transfer 127, 315–322 (2005). source

  • May combine two claims.

    Interpretationown data

    At low modulation frequency, phonon transport was primarily diffusive; at high frequency, phonons with MFP > LP traveled ballistically.

    “At low f, phonon transport was primarily diffusive, while at high f, phonons with a MFP > LP traveled ballistically as illustrated in Fig. 1(d).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data

    The existence of a material independent kuniversal in GaAs, Si, GaN, AlN, and SiC suggests that the phonon mean free path spectrum is a universal feature of intrinsic crystalline semiconductors.

    “The existence of a material independent kuniversal in GaAs, Si, GaN, AlN, and SiC suggests that the phonon mean free path spectrum is a universal feature of intrinsic crystalline semiconductors.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Umklapp scattering dominates thermal resistance in Si, GaN, AlN, and 4H-SiC at T ≈ 300 K and 400 K and GaAs at T ≈ 150 K, 300 K, and 400 K.

  • May combine two claims.

    Interpretationown data

    As temperature decreases, phonon occupation and phonon-phonon scattering are reduced, and long MFP phonons (>1 μm) become the dominant heat carriers.

    “As the temperature decreases, phonon occupation and phonon-phonon scattering are reduced, and long MFP phonons (>1 μm) become the dominant heat carriers.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    Thin film architectures and nanostructures possess reduced thermal conductivities compared to bulk materials due to boundary scattering, which limits phonon MFPs.

    “Thin film architectures and nanostructures possess reduced thermal conductivities compared to bulk materials due to boundary scattering, which limits phonon MFPs202122.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [20] Cho, J. W., Bozorg-Grayeli, E., Altman, D. H., Asheghi, M. & Goodson, K. E. Low thermal resistances at GaN-SiC interfaces for HEMT technology. IEEE Electron Device Lett. 33, 378–380 (2012). source
    • cites [21] Koh, Y. K., Cao, Y., Cahill, D. G. & Jena, D. Heat-transport mechanisms in superlattices. Adv. Funct. Mater. 19, 610–615 (2009). source
    • cites [22] Su, Z. et al. Layer-by-layer thermal conductivities of the group III nitride films in blue/green light emitting diodes. Appl. Phys. Lett. 100, 201106 (2012). source

  • May combine two claims.

    Interpretationreasoning

    The collapse of kaccum data equivalently implies a universal phonon MFP spectrum in these materials, as the accumulation function is the integral of the MFP spectrum.

    “This collapse equivalently implies a universal phonon MFP spectrum in these materials, as the accumulation function is the integral of the MFP spectrum.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Umklapp scattering dominates thermal resistance in Si, GaN, AlN, and 4H-SiC at T ≈ 300 K and 400 K and GaAs at T ≈ 150 K, 300 K, and 400 K.

  • Classified as probably not a substantial claim.

    Premisecitation

    In-plane and cross-plane thermal conductivities of AlN varied by less than 5% at room temperature and above.

    “Slack et al.16 reason that in-plane and cross-plane thermal conductivities of AlN varied by less than 5% at room temperature and above.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16] Slack, G. A., Tanzilli, R. A., Pohl, R. O. & Vandersande, J. W. The intrinsic thermal conductivity of AlN. J. Phys. Chem. Solids 48, 641–647 (1987). source

  • May combine two claims.

    Premisecitation

    Some debate exists on the importance of isotope scattering in gallium-based semiconductors, though authors agree that it is diminished at high temperatures.

    “Some debate exists on the importance of isotope scattering in gallium-based semiconductors, though authors agree that it is diminished at high temperatures31383940.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [31] Lindsay, L., Broido, D. A. & Reinecke, T. L. Thermal conductivity and large isotope effect in GaN from first principles. Phys. Rev. Lett. 109, 095901 (2012). source
    • cites [38] Morelli, D. T., Heremans, J. P. & Slack, G. A. Estimation of the isotope effect on the lattice thermal conductivity of group IV and group III-V semiconductors. Phys. Rev. B 66, 195304 (2002). source
    • cites [39] Slack, G. A., Schowalter, L. J., Morelli, D. & Freitas, J. A. Some effects of oxygen impurities on AlN and GaN. J. Cryst. Growth 246, 287–298 (2002). source

  • Probably not a substantial claim: may not pass S1 (asserted).

    Premisecitation

    Koh and Cahill hypothesized that the thermal penetration depth limited the diffusive phonons interrogated by TDTR to those having a MFP less than LP.

    “Based on observations of suppressed thermal conductivity in semiconductor alloys, Koh and Cahill2 hypothesized that the thermal penetration depth, , limited the diffusive phonons interrogated by TDTR to those having a MFP less than LP, where f was the modulation frequency of the pump laser.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [2] Koh, Y. K. & Cahill, D. G. Frequency dependence of the thermal conductivity of semiconductor alloys. Phys. Rev. B 76, 075207 (2007). source

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    Experimental measurements of the intrinsic kaccum are required to understand boundary scattering and local suppression effects prevalent in GaAs, GaN, AlN, and SiC technologies.

    “Therefore, experimental measurements of the intrinsic kaccum are required to understand these effects, which are prevalent in GaAs, GaN, AlN, and SiC technologies.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Umklapp scattering is the dominant resistive scattering process above the temperature of peak thermal conductivity, which typically occurs at less than 10% of the Debye temperature in bulk crystalline materials with low impurity concentration.

    “Umklapp scattering is the dominant resistive scattering process above the temperature of peak thermal conductivity, which typically occurs at less than 10% of the Debye temperature in bulk crystalline materials with low impurity concentration36.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [36] Tien, C. L., Majumdar, A. & Gerner, F. M. Microscale Energy Transport, Taylor and Francis, Washington, DC, Chap. 1 p. 72 (1998).

      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 conductnone given

    Bulk values of thermal conductivity are an additional source of uncertainty not reflected in the error bars.

    “Bulk values of thermal conductivity are an additional source of uncertainty that are not reflected in the error bars on Fig. 2 (see Supplementary Information for justification of the chosen bulk values).”
    Quote in the paperNot a substantial claim
16 of 16

  • The authors state this as an assumption.

    Ballistic phonons did not contribute to the measured value of thermal conductivity.

    “Consistent with prior studies, we assumed that ballistic phonons did not contribute to the measured value of thermal conductivity123426.”

  • The authors state this as an assumption.

    An isotropic model of thermal conductivity (kin-plane = kcross-plane) was assumed in GaAs, GaN, AlN, and 4H-SiC across all heating frequencies.

    “An isotropic model of thermal conductivity, where kin-plane = kcross-plane, was assumed in GaAs, GaN, AlN, and 4H-SiC across all heating frequencies.”

  • The authors state this as an assumption.

    GaN exhibits quasi-isotropic behavior similar to AlN.

    “GaN was assumed to exhibit similar quasi-isotropic behavior to that of AlN due to the fact that GaN and AlN have identical lattice structures and similar acoustic wave velocity deviation between their in-plane and cross-plane velocities3132.”

  • The authors state this as an assumption.

    The relaxation time for Umklapp phonon scattering takes a specific assumed form involving material-dependent constants P and CU and phonon frequency ω.

    “To derive an expression for kaccum at high temperatures, the following form for the relaxation time, τ, of phonon scattering due to Umklapp processes was assumed35, where P and CU are material dependent constants that describe Umklapp scattering rates, and ω is the phonon frequency.”

  • The authors state this as an assumption.

    Classical occupation is assumed for phonons.

    “The assumption of classical occupation is neccesary for deriving Eqn. (4).”

  • The authors state this as an assumption.

    The thermal interface conductance between the sample and the chromium-gold layer was frequency independent.

    “The thermal interface conductance between the sample and the chromium-gold layer was frequency independent.”

  • The authors state this as an assumption.

    The volumetric heat capacity was frequency independent.

    “The volumetric heat capacity was frequency independent.”

  • The authors state this as an assumption.

    Umklapp scattering dominates thermal resistance in Si, GaN, AlN, and 4H-SiC at T ≈ 300 K and 400 K and GaAs at T ≈ 150 K, 300 K, and 400 K.

    “The assumption of Umklapp dominated scattering is adequately met by Si, GaN, AlN, and 4H-SiC at T ≈ 300 K and 400 K and GaAs at T ≈ 150 K, 300 K, and 400 K given that the temperatures of their peak thermal conductivity occur at ~ 25 K37, ~ 55 K14, ~ 60 K16, ~ 50 K18, ~ 20 K12, respectively.”

  • The paper relies on this without support.

    Experimental determination of the phonon MFP spectrum has remained a challenge.

    “While the heat capacity and phonon group velocity can be predicted and measured, experimental determination of the phonon MFP spectrum has remained a challenge.”
9 of 9

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?

  • BB-FDTR was introduced to study phonon mean free path spectra in semiconductors, similar to frequency-dependent TDTR based on the physical picture of non-diffusive phonon transport.

    “For example, it was introduced to study phonon MFP spectra in semiconductors, 92, 147, 148 similar to the frequency-dependent TDTR that is based on the physical picture of non-diffusive phonon transport, see more details in Section IV(D).”

    From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [148]

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Related

Authors
Freedman, J. P., Leach, J. H., Preble, E. A., Sitar, Z., Davis, R. F., & Malen, J. A.
Affiliations
  • Carnegie Mellon University (US)
  • Kyma Technologies (United States) (US)
  • North Carolina State University (US)
ORCID
Venue
Scientific Reports
Publisher
Springer Science and Business Media LLC
Published
2013-10-16
Volume
3
Issue
1
Pages
2963-2963
ISSN
2045-2322 (ISSN-L)
DOI
10.1038/srep02963
PMC
PMC3797423
OpenAlex
W2078641765
Semantic Scholar
acb91c3230ddda706b4f400e292658ee5f8874f2
Cited by
171
References
42
Citations by year
2026: 15 · 2025: 13 · 2024: 13 · 2023: 13 · 2022: 16 · 2021: 9 · 2020: 19 · 2019: 14 · 2018: 11 · 2017: 6 · 2016: 15 · 2015: 14 · 2014: 12 · 2013: 1
FWCI
6.21
Type
journal article
Language
en
Open access
Licence
Locations
  • Scientific Reports (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • Europe PMC (PubMed Central) (submittedVersion, cc-by)
  • PubMed Central (submittedVersion, cc-by)
  • KiltHub Repository (submittedVersion, cc-by)
  • Figshare (submittedVersion, cc-by)
  • KiltHub Repository (cc-by)
Funders
  • National Science Foundation (1133394)
  • Carnegie Mellon University
  • Division of Chemical, Bioengineering, Environmental, and Transport Systems
Topics
  • Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
  • Physical Sciences › Engineering › Mechanics of Materials › Thermography and Photoacoustic Techniques
  • Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
Keywords
phonon mean free path, thermal conductivity, GaAs, GaN, aluminum nitride, 4H-SiC, Umklapp scattering, phonon scattering, crystalline semiconductors, high temperature

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