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TECHNICAL PAPERS

Measurement of Biaxial Stress States in Silicon Using Micro-Raman Spectroscopy

[+] Author and Article Information
Peter A. Gustafson

 University of Michigan, Ann Arbor, MI 48109petegus@umich.edu

Stephen J. Harris, Ann E. O’Neill

Physical and Environmental Science Department,  Ford Research and Advanced Engineering Center, Dearborn, MI 48121

Anthony M. Waas

 University of Michigan, Ann Arbor, MI 48109dcw@umich.edu

J. Appl. Mech 73(5), 745-751 (Jan 18, 2006) (7 pages) doi:10.1115/1.2187527 History: Received December 16, 2004; Revised January 18, 2006

Micro-Raman spectroscopy is used to determine the multiaxial stress state in silicon wafers using a strategy proposed by Narayanan, (J. Appl. Phys.82, 2595–2602 (1997)) Previously, this strategy was validated when silicon was subjected to uniaxial stress in the laboratory frame (Harris, J. Appl. Phys.96, 7195–7201 (2004)). In the present work, silicon wafers have been analyzed that were subjected to biaxial stress states in the laboratory frame. The predicted curves for the initially degenerate F2g peaks were found to fall within the variability of the measured curves. Stress ratios were found to be predictable. Stress magnitudes were also found to be predictable, but are subject to uncertainty greater than 25%. To perform these tests, an apparatus has been developed which can provide controlled ratios of biaxial stress in a simple and compact test geometry. This fixture was used under a microscope, enabling in situ measurement of biaxial stress states.

FIGURES IN THIS ARTICLE
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Copyright © 2006 by American Society of Mechanical Engineers
Topics: Stress , Silicon
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References

Figures

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Figure 1

Predictions of observed Raman shift (Δω¯) per stress σxx for several ratios of σyy∕σxx, based on Anastassakis PDPs

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Figure 2

Stresses σrr and σθθ in an isotropic, clamped, circular plate (diameter 2a, thickness h, Poisson’s ratio ν) subjected to uniform pressure load (q). There is an area in the center of the plate subjected to a uniform biaxial stress field.

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Figure 3

Theoretical relationship between stress ratio and ellipse ratio, plotted for a range of Poisson’s ratios

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Figure 4

Isometric view of the ABAQUS model. The silicon and rubber gaskets are modeled with solid elements.

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Figure 5

Elliptical plate loading device, consisting of a steel base plate with air feed and an elliptical clamping plate. Rubber gaskets are used to seat the clamped silicon wafer.

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Figure 6

Measured Raman shift Δω¯ plotted as a function of applied stress σxx with an in-line 0 deg and 90 deg polarizer. These data, taken with the 2:1 elliptical plate, are an example of the output for each experiment.

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Figure 7

Raman sensitivity Δω0¯∕σxx plotted versus stress ratio σyy∕σxx for measured data (0 deg polarizer) and for predicted result using Anastassakis PDPs. The predictions fall within the variability in the measured data.

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Figure 8

Raman sensitivity Δω90¯∕σxx plotted versus stress ratio σyy∕σxx for measured data (90 deg polarizer) and for predicted result using Anastassakis PDPs. The predictions fall within the variability in the measured data.

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Figure 9

Measured Raman shift ratio Δω0¯∕Δω90¯ plotted as a function of stress ratio σyy∕σxx. The variability present in the shift ratio data is smaller than the variability of the sensitivity data from which it was calculated.

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