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Cited by 14Thermal properties of materialsAdvanced MEMS and NEMS TechnologiesThermoelastic and Magnetoelastic Phenomena

Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges

English, Timothy S., Phinney, Leslie M., Hopkins, Patrick E., Serrano, Justin R.·2013·Journal of Heat Transfer
DOI 10.1115/1.4024357
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Abstract

Accurate thermal conductivity values are essential for the successful modeling, design, and thermal management of microelectromechanical systems (MEMS) and devices. However, the experimental technique best suited to measure the thermal conductivity of these systems, as well as the thermal conductivity itself, varies with the device materials, fabrication processes, geometry, and operating conditions. In this study, the thermal conductivities of boron doped single-crystal silicon microbridges fabricated using silicon-on-insulator (SOI) wafers are measured over the temperature range from 80 to 350 K. The microbridges are 4.6 mm long, 125 μm tall, and either 50 or 85 μm wide. Measurements on the 85 μm wide microbridges are made using both steady-state electrical resistance thermometry (SSERT) and optical time-domain thermoreflectance (TDTR). A thermal conductivity of 77 Wm−1 K−1 is measured for both microbridge widths at room temperature, where the results of both experimental techniques agree. However, increasing discrepancies between the thermal conductivities measured by each technique are found with decreasing temperatures below 300 K. The reduction in thermal conductivity measured by TDTR is primarily attributed to a ballistic thermal resistance contributed by phonons with mean free paths larger than the TDTR pump beam diameter. Boltzmann transport equation (BTE) modeling under the relaxation time approximation (RTA) is used to investigate the discrepancies and emphasizes the role of different interaction volumes in explaining the underprediction of TDTR measurements.

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Written from the paper’s full text.

TL;DR

The paper compares steady-state electrical resistance thermometry (SSERT) and optical time-domain thermoreflectance (TDTR) for measuring thermal conductivity of boron-doped single-crystal silicon microbridges from 80 to 350 K. Both techniques agree near room temperature at 77 W m⁻¹ K⁻¹, but TDTR underreports below 300 K, primarily due to ballistic thermal resistance from phonons with mean free paths larger than the TDTR pump beam diameter.

Key findings

  • At room temperature, a thermal conductivity of 77 W m⁻¹ K⁻¹ is measured for both 50 and 85 µm wide microbridges, with SSERT and TDTR agreeing.
  • Discrepancies between SSERT and TDTR increase with decreasing temperature below 300 K; TDTR measures lower thermal conductivity.
  • The reduction is primarily attributed to ballistic thermal resistance from phonons with mean free paths larger than the TDTR pump beam diameter (24 µm), not to near-surface dopant effects alone.
  • BTE modeling under the relaxation time approximation, excluding phonons with mean free paths >24 µm, trends closely with the TDTR data; near-surface dopant concentrations up to 5.4 × 10¹⁹ cm⁻³ cannot explain the large reduction.

Methods

  • Boron-doped single-crystal silicon microbridges fabricated from SOI wafers: 4.6 mm long, 125 µm tall, and either 50 or 85 µm wide; dopant concentration 3.8 × 10¹⁹ cm⁻³, with 5.4 × 10¹⁹ cm⁻³ within the first 50 nm from the surface tapering over the next 100 nm. SSERT tested dies 1–4; TDTR tested die 5 (85 µm wide).
  • SSERT used four-point probe measurements in a cryostat with a 1D heat diffusion model and corrected temperatures via method 1; TDTR used an 88 nm Al transducer, pump/probe beam radii of ~12 µm, 11 MHz modulation, and an axially symmetric layered thermal model over 80–350 K.
  • Thermal conductivity was modeled using the Boltzmann transport equation under the relaxation time approximation with an exact dispersion relation along [100], an isotropic Brillouin zone, and Matthiessen’s rule combining phonon-phonon, boundary, impurity, strain, and free hole-phonon scattering; the impurity scattering coefficient was fit from SSERT high-temperature data.

Limitations

  • Stated: SSERT uncertainty is dominated by corrected temperatures, which are as large as 10% near room temperature; TDTR uncertainty arises from spot size, Al film thickness, and standard deviation of repeated measurements. The near-surface dopant concentration is treated as an upper bound.
  • Evident: TDTR measurements were made only on 85 µm wide microbridges (Table 1), so no TDTR comparison exists for the 50 µm width. The BTE model assumes an isotropic Brillouin zone and uses literature parameters for a similarly doped sample (1 × 10¹⁹ cm⁻³), with the impurity scattering coefficient fit to SSERT data.

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

Integrity signals

Registry

Crossref · 10.1115/1.4024357

Cites

57 of 57 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

7 of 30 cited sources checked · 5 supported · 0 flagged · accuracy not yet measured

  • Premisecitation
    § 8 Ballistic Thermal Resistance · p. 5

    The thermal conductivity reductions reported by Minnich et al. for comparable pump diameters in natural silicon agree in trend over the reported temperature range with the calculated reduction.

    “Furthermore, the thermal conductivity reductions reported by Minnich et al. [39] for comparable pump diameters in natural silicon agree in trend over the reported temperature range, as shown in Fig. 7.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [39] 2011, Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths, Phys. Rev. Lett. source
  • Interpretationreasoning
    § 8 Ballistic Thermal Resistance · p. 5

    The ballistic thermal resistance effects found in TDTR measurements in both this and previous works can contribute over a wide temperature range extending from cryogenic up to even room temperatures.

    “These findings are consistent with the conclusion that the ballistic thermal resistance effects found in TDTR measurements in both this and previous [39] works can contribute over a wide temperature range extending from cryogenic up to even room temperatures.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [39] 2011, Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths, Phys. Rev. Lett. source

      The paper’s TTR measurements section (page not specified in our copy) reports ballistic thermal resistance in silicon as temperature falls below about 200 K. It also says the measurements are independent of beam diameter and agree with literature values at room temperature. Thus it supports the cryogenic effect, but not the claim that the effects found in this work extend to room temperature. (LLM, conf=0.89)

      Experimental — accuracy not yet measured
  • Findingreasoning
    § 5 Thermal Conductivity Results · p. 3

    The thermal conductivity of both 50 and 85 μm wide microbridges is significantly reduced below that of bulk, high-purity, undoped, single-crystal silicon.

    “The thermal conductivity is reduced significantly below that of the bulk reference silicon in the case of both microbridge widths”
    Quote in the paperSubstantial claim
  • Findingown data
    § Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges · p. 1

    Increasing discrepancies between the thermal conductivities measured by SSERT and TDTR are found with decreasing temperatures below 300 K.

    “However, increasing discrepancies between the thermal conductivities measured by each technique are found with decreasing temperatures below 300 K.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § 5 Thermal Conductivity Results · p. 4

    The thermal conductivities of the 50 and 85 μm wide microbridges measured by SSERT are comparable from 80 to 350 K.

    “The thermal conductivity of the 50 and 85 l mwide microbridges measured by SSERT are comparable from 80 to 350 K.”
    Quote in the paperSubstantial claim
  • May combine two claims. 10
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Study design and conductown data
    § 5 Thermal Conductivity Results · p. 4

    Agreement among SSERT measurements, comparison to Raman thermometry, and continuum multiphysics modeling provides confidence in the SSERT measurements and repeatability across test structures.

    “The agreement amongst SSERT measurements, along with comparison to Raman thermometry [37,38] measurements and continuum multiphysics modeling not presented here, provides confidence in the SSERT measurements and repeatability across test structures.”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [37] 2006, Micro-Raman Thermometry of Thermal Flexure Actuators, J. Micromechan. Microeng. source
    • cites [38] 2007, Simultaneous Mapping of Temperature and Stress in Microdevices Using Micro-Raman Spectroscopy, Rev. Sci. Instrum. source

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

    Premisecitation
    § Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges · p. 1

    Electrical resistance thermometry has been applied to measure the thermal conductivity of low-dimensional solids in both direct and modulated current configurations.

    “Electrical resistance thermometry has been applied to measure the thermal conductivity of low dimensional solids (films, membranes, wires, and bridges) [13-15] in both direct [16-20] and modulated (1-omega, 2-omega, 3-omega) [21,22] current configurations.”
    Quote in the paperNot a substantial claim2 of 10 sources checked
    • cites [13] 1994, Solid Layer Thermal-Conductivity Measurement Techniques, Appl. Mech. Rev. source
    • cites [14] 2004, Experimental Techniques for Thin-Film Thermal Conductivity Characterization source
    • cites [15] 2011, Reduced Joule Heating in Nanowires, Appl. Phys. Lett. source

      The paper’s discussion on p. 1 models the temperature distribution in a nanowire under a steady current and notes implications for thermal measurements. The passages searched for resistance thermometry and omega-modulated methods do not say that either configuration was applied to measure thermal conductivity. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [16] 1988, Thermal Conductivity of Heavily Doped Low-Pressure Chemical Vapor Deposited Polycrystalline Silicon Films, J. Appl. Phys. source
    • cites [17] Reifenberg, J., Voss, R. E., Rao, P., Schmitt, W., Yang, Y., Shojaei-Zadeh, S., Liu, W., Sadeghipour, S. M., and Asheghi, M., 2003, “Thermal Conductivity Measurements of Thin Aluminum Layers Using Steady State Joule Heating and Electrical Resistance Thermometry in Suspended Bridges,” Proceedings of ASME IMECE03(42055), pp. 377–380. source
    • cites [18] Yang, Y., and Asheghi, M., 2004, “A Novel Technique for In-Plane Thermal Conductivity Measurements of Electrically Conductive Interconnects and Nanostructures,” Thermal and Thermomechanical Phenomena in Electronic Systems, 2004. ITHERM’04, The Ninth Intersociety Conference on, Vol. 2, pp. 564–569. source
    • cites [19] Shojaei-Zadeh, S., Zhang, S., Liu, W., Yang, Y., Sadeghipour, S., Asheghi, M., and Sverdrup, P., 2004, “Thermal Characterization of Thin Film Cu Interconnects for the Next Generation of Microelectronic Devices,” Thermal and Thermomechanical Phenomena in Electronic Systems, 2004. ITHERM'04, The Ninth Intersociety Conference on, Vol. 2, pp. 575–583. source
    • cites [20] 2010, Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr, ASME J. Heat Transfer source
    • cites [21] 1990, Thermal Conductivity Measurement From 30 to 750 K: the 3ω Method, Rev. Sci. Instrum. source
    • cites [22] 2005, 1ω, 2ω, and 3ω Methods for Measurments of Thermal Properties, Rev. Sci. Instrum. source

  • May combine two claims.

    Premisecitation
    § 8 Ballistic Thermal Resistance · p. 5

    Minnich et al.'s measurements were independent of modulation frequency but showed a pump diameter dependence.

    “This is consistent with the results of Minnich et al. whose measurements were independent of modulation frequency (i.e., varying thermal penetration depth), but instead showed a pump diameter dependence.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [39] 2011, Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths, Phys. Rev. Lett. source

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

    Findingcitation
    § 4 Time-Domain Thermoreflectance · p. 3

    The Al film thickness was 88 nm on average, confirmed with picosecond ultrasonics.

    “The thickness of the metal (Al) film was 88 nm on average, as confirmed with picosecond ultrasonics [31,32].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [31] 1994, Electron Diffusion in Metals Studied by Picosecond Ultrasonics, Phys. Rev. B source
    • cites [32] 1984, Coherent Phonon Generation and Detection by Picosecond Light Pulses, Phys. Rev. Lett. source

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

    Findingown data
    § Table 1 Test structures · p. 2

    The device layer of the SOI wafers is p-doped (boron) at a concentration of 3.8×10^19 cm−3, measured by SIMS.

    “The device layer is h 100 i silicon with a thickness of 125 6 1 l m, and is p-doped (boron) at a concentration of 3.8 -10 19 cm 3 measured by time-of-flight secondary ion mass spectrometry (SIMS).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges · p. 1

    A thermal conductivity of 77 W m−1 K−1 is measured for both 50 and 85 μm wide microbridges at room temperature, and the SSERT and TDTR techniques agree at that temperature.

    “A thermal conductivity of 77Wm 1 K 1 is measured for both microbridge widths at room temperature, where the results of both experimental techniques agree.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Table 1 Test structures · p. 2

    The dopant concentration is 5.4×10^19 cm−3 within the first 50 nm from the surface, tapering off to 3.8×10^19 cm−3 over the next 100 nm.

    “The dopant concentration measured by SIMS was also found to be 5.4 -10 19 cm 3 within the first 50 nm from the surface, tapering off to 3.8 -10 19 cm 3 over the next 100 nm.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 5 Thermal Conductivity Results · p. 4

    The thermal conductivity of the 85 μm wide microbridge measured by TDTR agrees with SSERT near room temperature but diverges at lower temperatures.

    “The thermal conductivity of the 85 l m wide microbridge measured by TDTR agrees with SSERT near room temperature, but diverges at lower temperatures.”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § Table 1 Test structures · p. 2

    The microbridge widths are slightly tapered (+2 μm top, −2 μm bottom), attributed to high aspect ratio deep reactive-ion etching.

    “Instead, the widths are slightly tapered ( þ 2 l m top, 2 l m bottom), which is attributed to the use of high aspect ratio deep reactive-ion etching.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 1-D Thermal Model · p. 3

    For the largest applied current, both the average and maximum temperature of the test structure increase with nominal cryostat temperature.

    “The departure from the nominal temperature (black line), demonstrates that both the average and maximum temperature increase with nominal cryostat temperature.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges · p. 1

    The thermal conductivity of thin single-crystal silicon layers has been reported for a range of thicknesses from 20 nm to 3 μm.

    “The thermal conductivity of thin single-crystal silicon layers has been reported for a range of thicknesses from 20 nm [8] to 3 l m [7,11,12].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [8] 2004, Phonon-Boundary Scattering in Ultrathin Single-Crystal Silicon Layers, J. Appl. Phys.

      Not checked in the free run.

    • cites [7] 2005, Thermal Conduction in Silicon Micro- and Nanostructures, Annu. Rev. Heat Transfer source
    • cites [11] 1997, Phonon-Boundary Scattering in Thin Silicon Layers, Appl. Phys. Lett. source
    • cites [12] 2002, Thermal Conduction in Doped Single-Crystal Silicon Films, J. Appl. Phys. source

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

    Study design and conductnone given
    § 5 Thermal Conductivity Results · p. 4

    The uncertainty associated with TDTR measurements is due to several sources, including uncertainties in the spot size and Al film thickness at the point of measurement on the sample surface.

    “The uncertainty associated with TDTR measurements is due to several sources, including uncertainties in the spot size and Al film thickness at the point of measurement on the sample surface.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5 Thermal Conductivity Results · p. 4

    Both the increased scattering of short-wavelength phonons and the escape of longer wavelength phonons would lead to lower measured thermal conductivities using TDTR.

    “Both of these phenomena would lead to lower measured thermal conductivities using TDTR”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductown data
    § 3 Steady-State Electrical Resistance Thermometry · p. 2

    A delay time of 10 s between successive current measurements was sufficient to provide adequate time to reach steady-state for a given current, confirmed by the repeatability of the R versus I curve.

    “A delay time of 10 s between successive current measurements was sufficient to provide adequate time to reach steady-state for a given current, which was confirmed by the repeatability of the R versus I curve.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 6 Thermal Conductivity Modeling · p. 4

    Similarly large values for the impurity scattering coefficient have been routinely used to model highly (>1e19 cm-3) boron doped single-crystal silicon.

    “Similarly large values for the impurity scattering coefficient have been routinely used to model highly ( > 1 -10 19 cm 3 ) boron doped single-crystal silicon [12,45].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [12] 2002, Thermal Conduction in Doped Single-Crystal Silicon Films, J. Appl. Phys. source
    • cites [45] 2005, Thermal Conduction in Ultrathin Pure and Doped Single-Crystal Silicon Layers at High Temperatures, J. Appl. Phys. source

  • May combine two claims.

    Interpretationreasoning
    § 9 Conclusion · p. 6

    The SSERT measurements best reflect the true intrinsic thermal conductivity of the structures, as these results are obtained in the limit of low frequency (dc measurements), or equivalently, an infinite spot size.

    “In this study, we believe the SSERT measurements best reflect the true intrinsic thermal conductivity of the structures, as these results are obtained in the limit of low frequency (dc measurements), or equivalently, an infinite spot size.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 8 Ballistic Thermal Resistance · p. 5

    Discrepancies in the thermal conductivity between the electrical and optical techniques in this study are due primarily to the ballistic thermal resistance measured via TDTR, with the near-surface dopant concentration playing a comparatively less important role with decreasing temperature.

    “This suggests that discrepancies in the thermal conductivity between the electrical and optical techniques in this study are due primarily to the ballistic thermal resistance measured via TDTR, with the nearsurface dopant concentration playing a comparatively less important role with decreasing temperature.”
    Quote in the paperSubstantial claim
17 of 17

  • The authors state this as an assumption.

    The SSERT thermal model assumes 1D conduction along the length of the microbridge and no losses to convection or radiation.

    “This model assumes 1D conduction occurs along the length of the microbridge and that there are no losses to convection or radiation.”

  • The authors state this as an assumption.

    The silicon microbridges appear as semi-infinite media to TDTR because the thermal penetration depth is more than an order of magnitude less than the microbridge film thickness.

    “the silicon microbridges appear as semi-infinite media. This assumption is based on the fact that the thermal penetration depth during TDTR heating is on the order of d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C = p kf ð Þ p , where C is the heat capacity of Si, k is the thermal conductivity, and d is more than an order of magnitude less than the microbridge film thickness.”

  • The authors state this as an assumption.

    Each constituent scattering mechanism is independent and combined using Matthiessen's rule.

    “Each constituent scattering mechanism is assumed to be independent and is combined using the Matthiessen's rule [44].”

  • The paper relies on this without support.

    At lower temperatures, the mean free path of phonons, the primary energy carrier contributing to thermal transport, increases.

    “We consider the fact that at lower temperatures, the mean-free-path (MFP) of phonons, the primary energy carrier contributing to thermal transport, increases.”

  • The authors state this as an assumption.

    The free hole concentration is equal to the doping concentration, and the scattering contribution of bound holes is negligible.

    “Thus, we assume that the free hole concentration is equal to the doping concentration, and neglect the scattering contribution of any bound holes.”
5 of 5

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. 2012, Thermal Microactuators, 415
  2. 1970, Thermophysical Properties of Matter—Specific Heat: Nonmetallic Solids
  3. 1970, Thermophysical Properties of Matter—Specific Heat: Metallic Elements and Alloys
  4. 1976, Solid State Physics

Cited by

Sorted by citations

Related

Authors
English, T. S., Phinney, L. M., Hopkins, P. E., & Serrano, J. R.
Affiliations
  • Sandia National Laboratories (US)
  • University of Virginia (US)
Corresponding author
  • English, Timothy S.
  • Phinney, Leslie M.
  • Hopkins, Patrick E.
  • Serrano, Justin R.
ORCID
Hopkins, Patrick E.
Venue
Journal of Heat Transfer
Publisher
ASME International
Published
2013-07-26
Volume
135
Issue
9
ISSN
  • 0022-1481 (ISSN-L)
  • 1528-8943
DOI
10.1115/1.4024357
OpenAlex
W2145602797
Semantic Scholar
03a4c4a7b64cc5c9e353464c3ead0cbbc2667bd1
Cited by
14
References
51
Citations by year
2026: 1 · 2024: 1 · 2020: 1 · 2019: 2 · 2018: 1 · 2017: 2 · 2016: 2 · 2015: 3 · 2014: 1
FWCI
0.89
Type
journal article
Language
en
Open access
closed
Locations
ASME Journal of Heat and Mass Transfer (publishedVersion)
Funders
  • U.S. Department of Energy
  • National Nuclear Security Administration
  • Basic Energy Sciences
  • Sandia National Laboratories
Topics
  • Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Advanced MEMS and NEMS Technologies
  • Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
Keywords
thermal conductivity, mean free path, phonons, time-domain thermoreflectance, silicon-on-insulator, low-temperature thermal transport, Boltzmann equation, relaxation time approximation, microelectromechanical systems, thermal management

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