Tuning Frequency Stability in Micromechanical Resonators with Parametric Pumping

In this work we experimentally investigate the influence of parametric amplification and parametric suppression on the frequency stability of micromechanical resonators. We isolate the influence of phase slope tuning from changes in the vibrational amplitude and find that parametric suppression improves the frequency stability in the thermal noise regime by over threefold, while parametric amplification degrades the frequency stability by nearly a factor of two.

Introduction

Frequency stability is a key figure of merit that affects the signal-to-noise ratio (SNR) of resonant sensors and timing references. In microelectromechanical (MEM) resonators, frequency stability is known to depend on nonlinearity [1], [2], environmental noise [3], and resonator parameters such as the quality factor [4], [5]. A variety of techniques such as parametric amplification and thermal-piezoresistive pumping have been studied for tuning the effective quality factor of MEM resonators [6], [7], and are known to affect the thermomechanical noise of a resonator [8], but the exact role they play in improving sensor and oscillator performance is still under investigation. These techniques have typically been used to construct oscillators [9] or im-prove the sensitivity of amplitude-modulated sensors by in-creasing the vibration amplitude [10], [11], but recently there is interest in using effective quality factor tuning mecha-nisms to improve the performance of frequency-shift sen-sors. When placed in a phase-locked loop, frequency-shift sensors transduce signals that induce a shift in the resonant frequency of the device to a change in the loop phase via the resonator's phase-frequency relationship. In this oper-ational mode, a larger phase slope results in more phase change for a given input signal. Effective quality factor tuning mechanisms have been used to improve the performance of resonant sensors using this principle [12], [13], but the ef-fect on a resonator's fundamental frequency stability under the influence of tuning mechanisms is under investi-gation [14].

$$\begin{equation*} \sigma_{a}(\tau)\approx\left(\omega_{0}\left\vert \frac{\partial\phi}{\partial\omega}\right\vert _{\omega_{0}}\right)^{-1}\sqrt{\frac{V_{n}^{2}}{V_{s}^{2}}}\sqrt{\frac{1}{2\pi\tau}},\tag{1}\end{equation*}$$ where is the phase slope at resonance, is the reso-nant frequency, is the white noise floor, and is the sig-nal amplitude. Effective quality factor tuning mechanisms modify the transfer function of the resonator, which changes the observed thermomechanical noise, but in most practical cases the noise floor of the system is dominated by another source, such as noise in the readout method or an amplifier. In the case where the tuning method does not modify the noise of floor of the combined system and the amplitude of the device remains constant, Eq. (1) demonstrates that in-creasing the phase slope at resonance improves frequency stability. Counterintuitively, this results in a scenario where parametric suppression improves the frequency stability of the resonator and parametric amplification, which reduces the resonator's linewidth, degrades frequency stability. Figure 1:

The setup for measuring the open loop response of the cantilevered beam resonator under test. The motion of the resonant beam, held at a bias voltage, , induces a current in the sensing electrode which is transduced into a measurable voltage by a transimpedance amplifier (tia). We apply a drive voltage, , at a frequency near the res-onant frequency of the cantilever, and a parametric pump voltage, , at twice the frequency of the drive voltage.