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Research Papers: Micro/Nanoscale Heat Transfer

# Size Effect on the Thermal Conductivity of Thin Metallic Films Investigated by Scanning Joule Expansion Microscopy

[+] Author and Article Information
Siva P. Gurrum

Semiconductor Packaging Technology Research, Texas Instruments Incorporated, Dallas, TX 75243

William P. King

Department of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign, Urbana, IL 61801

Yogendra K. Joshi

G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

Koneru Ramakrishna

Package Material Technology Development, Analog & Mixed Signal Technologies, Technology Solutions Organization,  Freescale Semiconductor, Inc., Austin, TX 78735

J. Heat Transfer 130(8), 082403 (May 30, 2008) (8 pages) doi:10.1115/1.2928014 History: Received March 07, 2007; Revised February 05, 2008; Published May 30, 2008

## Abstract

A technique to extract in-plane thermal conductivity of thin metallic films whose thickness is comparable to electron mean free path is described. Microscale constrictions were fabricated into gold films of thicknesses $43nm$ and $131nm$. A sinusoidal voltage excitation across the constriction results in a local temperature rise. An existing technique known as scanning joule expansion microscopy, measures the corresponding periodic thermomechanical expansion with a $10nm$ resolution and determines the local temperature gradient near the constriction. A three-dimensional finite-element simulation of the frequency-domain heat transfer fits the in-plane thermal conductivity to the measured data, finding thermal conductivities of $82±7.7W∕mK$ for the $43nm$ film and $162±16.7W∕mK$ for the $131nm$ film, at a heating frequencies of $100kHz$ and $90kHz$, respectively. These values are significantly smaller than the bulk thermal conductivity value of $318W∕mK$ for gold, showing the electron size effect due to the metal-dielectric interface and grain boundary scattering. The obtained values are close to the thermal conductivity values, which are derived from electrical conductivity measurements after using the Wiedemann–Franz law. Because the technique does not require suspended metal bridges, it captures true metal-dielectric interface scattering characteristics. The technique can be extended to other films that can carry current and result in Joule heating, such as doped single crystal or polycrystalline semiconductors.

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## Figures

Figure 1

A schematic of the constriction in a metal thin film

Figure 2

Effect of skin depth on heat generation profiles is shown for different frequencies. The inset zooms a portion of the plot to resolve lower frequencies. For the frequencies considered in this work, the skin effect can be neglected. It is assumed that L→∞.

Figure 3

(a) Comparison of expansion amplitude and temperature amplitude over the metal line. The two-dimensional structure modeled under the plane strain assumption is shown in the inset. (b) Deformation amplitude is shown by the displaced structure. Temperature amplitude is plotted using color shading. The deformation is artificially scaled by a large factor for clarity.

Figure 4

A comparison of numerical and experimental expansion amplitudes as the frequency is varied. The block arrows in the left column figures show the viewing direction. Viewing in this direction shows that the temperature amplitude surface gradually becomes more concave (for example, look at the yellow band of contours), which is a clear signature of in-plane thermal penetration depth.

Figure 5

Error in numerical fit at different frequencies as the thermal conductivity of metal film is varied for Constriction A (a) and Constriction B (b)

Figure 6

Comparison of experimental and numerical temperature amplitude profiles along the centerline near the constriction after minimizing the error in numerical fit

Figure 7

A summary of the extracted thermal conductivities for two constrictions and their comparison with bulk value and WFL predictions

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