Abstract
Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors. Various methods have been utilized to tune the effective quality factor of MEM/NEM resonators, including external proportional feedback control, optical pumping, mechanical pumping, thermal-piezoresistive pumping, and parametric pumping. This work reviews these mechanisms and compares the effective Q tuning using a position-proportional and a velocity-proportional force expression. We further clarify the relationship between the mechanical Q, the effective Q, and the thermomechanical noise of a resonator. We finally show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques.
Highlights
Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors
We show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques
We conclude with experiments on a micromechanical resonator that compare degenerate parametric pumping, a phase-dependent Qeff tuning mechanism, with thermal-piezoresistive pumping, a phase-independent Qeff tuning mechanism, in terms of the resonator’s transfer function, phase slope, and thermomechanical displacement noise
Summary
Q of a resonator is defined as the ratio of the stored energy over the dissipated energy per vibration cycle[18]. Even though the thermal noise force decreases with increasing Q, the mean-squared noise displacement, x2n , and the mean-squared velocity, x_2n , of the resonator are not influenced by the quality factor This is because the integrated area under the thermomechanical displacement and velocity power-spectral-densities (PSD) is constant at a given temperature. Damping in a resonator can alternately be represented by a complex spring constant, kanelastic 1⁄4 kð[1] þ i/ðxÞÞ, where /ðxÞ is the phase lag of the displacement behind the forcing.[41] This model yields a thermal noise displacement spectrum that is steeper in one power of x than that predicted by the velocity-proportional damping model.[40]. The preceding discussion, which considers the relationship between the feedback parameters in Eq (19) and the thermomechanical noise ASD and mean-squared noise, is applicable for all phase-independent Qeff tuning mechanisms in their linear regime. Like phase-independent Qeff tuning mechanisms, phase-dependent Qeff tuning mechanisms will initiate self-oscillations of the mode when the pump exceeds a threshold, but unlike phaseindependent Qeff tuning mechanisms, applying sufficient Qeff suppression will induce self-oscillations, because of the Qeff enhancement in the other quadrature
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