Celebrating the 100th anniversary of the Stoney equation for film stress: Developments from polycrystalline steel strips to single crystal silicon wafers

Nonlinear effects in single-crystal silicon microresonators are analyzed with the focus on mechanical nonlinearities. The bulk acoustic wave (BAW) resonators are shown to have orders-of-magnitude higher energy storage capability than flexural beam resonators. The bifurcation point for the silicon BAW resonators is measured and the maximum vibration amplitude is shown to approach the intrinsic material limit. The importance of nonlinearities in setting the limit for vibration energy storage is demonstrated in oscillator applications. The phase noise calculated for silicon microresonator-based oscillators is compared to the conventional macroscopic quartz-based oscillators, and it is shown that the higher energy density attainable with the silicon resonators can partially compensate for the small microresonator size. Scaling law for microresonator phase noise is developed.

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Nonlinear limits for single-crystal silicon microresonators

The use of ultrasonic measurements under modest pressure to estimate compression at high pressure

The third-order elastic constants of copper, silver, and gold have been measured at room temperature using high-purity single crystals. A sensitive ultrasonic interference method was employed to measure small changes of wave velocities in crystals deformed both uniaxially and hydrostatically at low stress levels. The results of the measurements were analyzed using finite elasticity theory to obtain the complete sets of six third-order constants. The values of ${C}_{111}$ are largest (of the order of ${10}^{13}$ dyn/${\mathrm{cm}}^{2}$), ${C}_{112}$ and ${C}_{166}$ are about half of ${C}_{111}$ in magnitude, and all are negative in sign. Almost all of the values of ${C}_{123}$, ${C}_{144}$, and ${C}_{456}$ are negative and small compared with the other constants. It is shown that the closed-shell repulsive interaction between nearest-neighbor atoms makes the dominant contribution to the higher order elastic constants in noble metals. In this sense, the anharmonic properties are simpler than the harmonic properties of these materials. The fourth-order elastic constants are also estimated on this basis.

Anharmonicity in Noble Metals; Higher Order Elastic Constants

Structural reliability of integrated circuit (IC) chips in electronic packages continues to be a major concern due to ever-increasing die size, circuit densities, power dissipation, operating temperatures, and the use of a wide range of low-cost packaging materials. A powerful method for experimental eval- uation of silicon die stress distributions is the use of test chips incorporating integral piezoresistive sensors. In this paper, a review is made of the state-of-the-art in the area of silicon piezore- sistive stress sensor test chips. Developments in sensor theory, calibration methods, and packaging applications are presented. In the absence of die failure, packaging-induced stresses result in changes in the parametric performance of circuitry on the die, and the theory discussed here can be used to predict such changes. Index Terms—Electronic packaging, piezoresistive, stress sensor, test chip.

Silicon piezoresistive stress sensors and their application in electronic packaging

Experimental techniques for determining third‐order moduli of single crystals by means of ultrasonic wave propagation are described. Results for silicon and germanium demonstrate a good degree of self‐consistency among the basic experimental data obtained with both hydrostatic and uniaxial pressure.In units of 1012 dyn/cm2 and for 25°C, the third‐order moduli are: ModulusValue for SiValue for GeC111−8.25±0.10−7.10±0.06C112−4.51±0.05−3.89±0.03C123−0.64±0.10−0.18±0.06C144+0.12±0.25−0.23±0.16C166−3.10±0.10−2.92±0.08C456−0.64±0.20−0.53±0.07 Thermodynamic definitions for the moduli (K. Brugger, Phys. Rev. 133, A1611 (1964)] have been used.

P. Andreatch

Papers

Measurement of Third‐Order Moduli of Silicon and Germanium