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Resonator Sensitivity to Axial Stress

The resonant frequency of a fixed-fixed beam is a strong function of the axial stress in the beam. Beam resonators with anchors at both ends, including double-anchored double-ended tuning forks (DA DETF), are susceptible to axial stresses introduced by the differential thermal expansion of the die substrate, the resonator, and the electrical package used for testing. Devices bonded with a thermal cure epoxy ("hard attach") or photoresist ("soft attach") exhibit dramatically different TCf, because the soft-attach method transmits less stress into the silicon die from the thermal expansion of the package lead frame than the hard-attach method. A photograph of the cross-section of a packaged resonator and simulations of the resulting stress distribution are shown in Figure 1.

Silicon resonator in DIP package with wirebonds
Cross-section photo of packaged resonator
FEM simulation of stress in the packaged resonator
Figure 1 - (top) a resonator packaged and wirebonded in a Open-Pak DIP package for testing. The silicon die is approximately 1 mm square.
(middle) A photograph of the cross-section of a resonator die in the electrical package. The resonator itself is in the top 40 Ám of the silicon die and is not visible at this scale. The packaged device was cut with a diamond blade saw. Note the presence of a thick copper pad layer in the package.
(bottom) Simulated stress distribution in a "hard-attach" packaged resonator created by a 25 C temperature increase. Simulation by R. Melamud.

TCf results reported in the literature for silicon MEMS resonators of various types are typically 10-30 ppm/C [3,4,5]. Our silicon, double-anchored 1.3MHz resonators with a hard-attach packaging have TCf values of 120-200 ppm/C, while soft attach parts have values of 30-40 ppm/C. Single-anchored designs fabricated in the same technology have TCf values ~28 ppm/C. Typical measured TCf values and simulation results are shown in Figure 2.

Figure 2 - Measured and simulated TCf values for different packaging conditions. The black line shows the result of a temperature-control experiment with a soft-attach resonator.

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Page Created By:
Matt Hopcroft
Stanford University Department of Mechanical Engineering
Phone: (+1) 650-736-0044
Email: hopcroft _at_ stanford _ edu
Page Updated: 18 October, 2006