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Research Papers

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

[+] Author and Article Information
Timothy S. English

Engineering Sciences Center,
Sandia National Laboratories,
Albuquerque, NM 87185;
Department of Mechanical Engineering,
Stanford University,
Stanford, CA 94305
e-mail: tsengli@sandia.gov;
englisht@stanford.edu

Leslie M. Phinney

Engineering Sciences Center,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: lmphinn@sandia.gov

Patrick E. Hopkins

Department of Mechanical and
Aerospace Engineering,
University of Virginia,
Charlottesville, VA 22904
e-mail: phopkins@virginia.edu

Justin R. Serrano

Engineering Sciences Center,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: jrserra@sandia.gov

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 26, 2012; final manuscript received January 25, 2013; published online July 26, 2013. Assoc. Editor: Zhuomin Zhang.

J. Heat Transfer 135(9), 091103 (Jul 26, 2013) (7 pages) Paper No: HT-12-1408; doi: 10.1115/1.4024357 History: Received July 26, 2012; Revised January 25, 2013

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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Figures

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Fig. 1

Image of a packaged die (microbridges span left to right) with no backside. A subset of microbridges are wired in a four-point probe configuration with two wirebonds connected to the bondpad at each end of the structure.

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Fig. 2

Resistance measured as a function of current for an 85 μm wide microbridge at a nominal cryostat temperature of 253 K, and best fit model resistance predictions. The average temperature rise of the beam is shown for the maximum and minimum measurement currents. (Inset) Resistance of a test structure from 77 to 350 K.

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Fig. 3

Average and maximum test structure temperature for the largest applied current at several nominal (cryostat) temperatures. The departure from the solid black line (no temperature rise) increases with nominal temperature due to the decrease in the test structure thermal conductivity.

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Fig. 4

Schematic of the TDTR instrument at Sandia National Laboratories

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Fig. 5

Thermal conductivity plots for the 50 μm and 85 μm microbridges

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Fig. 6

Thermal conductivity calculated via the Boltzmann transport equation considering phonon–phonon, boundary, impurity, strain, and free hole-phonon scattering times with a dopant concentration n = 3.8 × 1019 cm−3. Contributions from longitudinal and transverse acoustic phonon modes are shown.

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Fig. 7

Thermal conductivity calculated using the BTE and excluding phonons with MFPs larger than 24 μm, normalized to the SSERT measurements. Reduction in thermal conductivity reported in Ref. [39] for varying TDTR pump beam diameters on natural silicon are shown with select error bars from the original data.

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