# A Study on Parametric Amplification in a Piezoelectric MEMS Device

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. MEMS Device

#### 2.2. Measurement Scheme

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Manzaneque, T.; Ruiz-Díez, V.; Hernando-García, J.; Wistrela, E.; Kucera, M.; Schmid, U.; Sánchez-Rojas, J.L. Piezoelectric MEMS resonator-based oscillator for density and viscosity sensing. Sens. Actuators A Phys.
**2014**, 220, 305–315. [Google Scholar] [CrossRef] - Calleja, M.; Kosaka, P.M.; San Paulo, Á.; Tamayo, J. Challenges for nanomechanical sensors in biological detection. Nanoscale
**2012**, 4, 4925. [Google Scholar] [CrossRef] [PubMed][Green Version] - Patimisco, P.; Sampaolo, A.; Dong, L.; Tittel, F.K.; Spagnolo, V. Recent advances in quartz enhanced photoacoustic sensing. Appl. Phys. Rev.
**2018**, 5, 011106. [Google Scholar] [CrossRef] - Patimisco, P.; Sampaolo, A.; Mackowiak, V.; Rossmadl, H.; Cable, A.; Tittel, F.K.; Spagnolo, V. Loss Mechanisms Determining the Quality Factors in Quartz Tuning Forks Vibrating at the Fundamental and First Overtone Mode. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2018**, 65, 1951–1957. [Google Scholar] [CrossRef] [PubMed] - Gonzalez, M.; Seren, H.R.; Ham, G.; Buzi, E.; Bernero, G.; Deffenbaugh, M. Viscosity and Density Measurements Using Mechanical Oscillators in Oil and Gas Applications. IEEE Trans. Instrum. Meas.
**2018**, 67, 804–810. [Google Scholar] [CrossRef] - Karabacak, D.M.; Yakhot, V.; Ekinci, K.L. High-Frequency Nanofluidics: An Experimental Study Using Nanomechanical Resonators. Phys. Rev. Lett.
**2007**, 98, 254505. [Google Scholar] [CrossRef] [PubMed] - Franosch, T.; Grimm, M.; Belushkin, M.; Mor, F.M.; Foffi, G.; Forró, L.; Jeney, S. Resonances arising from hydrodynamic memory in Brownian motion. Nature
**2011**, 478, 85–88. [Google Scholar] [CrossRef] [PubMed] - Paul, M.R.; Cross, M.C. Stochastic Dynamics of Nanoscale Mechanical Oscillators Immersed in a Viscous Fluid. Phys. Rev. Lett.
**2004**, 92, 235501. [Google Scholar] [CrossRef] - González, M.; Zheng, P.; Garcell, E.; Lee, Y.; Chan, H.B. Comb-drive micro-electro-mechanical systems oscillators for low temperature experiments. Rev. Sci. Instrum.
**2013**, 84, 025003. [Google Scholar] [CrossRef] - González, M.; Jiang, W.G.; Zheng, P.; Barquist, C.S.; Chan, H.B.; Lee, Y. Temperature dependence of viscosity in normal fluid He3 below 800 mK determined by a microelectromechanical oscillator. Phys. Rev. B
**2016**, 94, 014505. [Google Scholar] [CrossRef] - Zheng, P.; Jiang, W.; Barquist, C.; Lee, Y.; Chan, H. Critical Velocity in the Presence of Surface Bound States in Superfluid He 3—B. Phys. Rev. Lett.
**2017**, 118, 065301. [Google Scholar] [CrossRef] - Kokavecz, J.; Horváth, Z.L.; Mechler, A. Dynamical properties of the Q-controlled atomic force microscope. Appl. Phys. Lett.
**2004**, 85, 3232–3234. [Google Scholar] [CrossRef] - Moran, K.; Burgner, C.; Shaw, S.; Turner, K. A review of parametric resonance in microelectromechanical systems. Nonlinear Theory Appl. IEICE
**2013**, 4, 198–224. [Google Scholar] [CrossRef][Green Version] - Shaw, S.W. Nonlinearity and parametric pumping in sensors: Opportunities and limitations. In Proceedings of the 2017 IEEE SENSORS, Glasgow, UK, 29 October–1 November 2017; pp. 1–3. [Google Scholar] [CrossRef]
- José, J.; Saletan, E. Classical Dynamics: A Contemporary Approach; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Rugar, D.; Grütter, P. Mechanical parametric amplification and thermomechanical noise squeezing. Phys. Rev. Lett.
**1991**, 67, 699–702. [Google Scholar] [CrossRef] - Karabalin, R.B.; Masmanidis, S.C.; Roukes, M.L. Efficient parametric amplification in high and very high frequency piezoelectric nanoelectromechanical systems. Appl. Phys. Lett.
**2010**, 97, 183101. [Google Scholar] [CrossRef][Green Version] - Moreno-Moreno, M.; Raman, A.; Gomez-Herrero, J.; Reifenberger, R. Parametric resonance based scanning probe microscopy. Appl. Phys. Lett.
**2006**, 88, 193108. [Google Scholar] [CrossRef][Green Version] - Prakash, G.; Raman, A.; Rhoads, J.; Reifenberger, R.G. Parametric noise squeezing and parametric resonance of microcantilevers in air and liquid environments. Rev. Sci. Instrum.
**2012**, 83, 065109. [Google Scholar] [CrossRef] - Papariello, L.; Zilberberg, O.; Eichler, A.; Chitra, R. Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators. Phys. Rev. E
**2016**, 94, 022201. [Google Scholar] [CrossRef] - Zhang, W.; Baskaran, R.; Turner, K.L. Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor. Sens. Actuators A Phys.
**2002**, 102, 139–150. [Google Scholar] [CrossRef] - Carr, D.W.; Evoy, S.; Sekaric, L.; Craighead, H.G.; Parpia, J.M. Parametric amplification in a torsional microresonator. Appl. Phys. Lett.
**2000**, 77, 1545–1547. [Google Scholar] [CrossRef] - Collin, E.; Moutonet, T.; Heron, J.S.; Bourgeois, O.; Bunkov, Y.M.; Godfrin, H. Nonlinear parametric amplification in a triport nanoelectromechanical device. Phys. Rev. B
**2011**, 84, 054108. [Google Scholar] [CrossRef][Green Version] - Thomas, O.; Mathieu, F.; Mansfield, W.; Huang, C.; Trolier-Mckinstry, S.; Nicu, L. Piezoelectric parametric amplifiers with integrated actuation and sensing capabilities. In Proceedings of the 2013 IEEE 26th International Conference on Micro Electro Mechanical Systems (MEMS), Taipei, Taiwan, 20–24 January 2013; pp. 588–591. [Google Scholar]
- Thomas, O.; Mathieu, F.; Mansfield, W.; Huang, C.; Trolier-McKinstry, S.; Nicu, L. Efficient parametric amplification in micro-resonators with integrated piezoelectric actuation and sensing capabilities. Appl. Phys. Lett.
**2013**, 102, 163504. [Google Scholar] [CrossRef][Green Version] - Jia, Y.; Du, S.; Seshia, A.A. Twenty-Eight Orders of Parametric Resonance in a Microelectromechanical Device for Multi-band Vibration Energy Harvesting. Sci. Rep.
**2016**, 6, 30167. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wu, S.; Sheng, J.; Zhang, X.; Wu, Y.; Wu, H. Parametric excitation of a SiN membrane via piezoelectricity. AIP Adv.
**2018**, 8, 015209. [Google Scholar] [CrossRef] - Cowen, A.; Hames, G.; Glukh, K.; Hardy, B. PiezoMUMPs Design Handbook; MEMSCAP Inc.: Durham, NC, USA, 2014. [Google Scholar]
- Lifshitz, R.; Cross, M.C. Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators. In Reviews of Nonlinear Dynamics and Complexity; Schuster, H.G., Ed.; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2008; pp. 1–52. [Google Scholar]

**Figure 1.**

**Top**: Optical image of typical devices with different supporting beam lengths. The device used has 1050 $\mathsf{\mu}$m length beams supporting an 800 $\mathsf{\mu}$m square plate.

**Bottom**: Coarse frequency sweep showing the two main modes. Displayed modeshapes were obtained from COMSOL simulations. Points “a” and “b” denote the location of the laser beam spot for the laser doppler vibrometer (LDV). The color code on the modeshapes indicates red for areas of maximum displacement and blue for no displacement.

**Figure 2.**Schematic diagram for measurement. A multi-frequency lock-in amplifier is used to generate the output signal to the micro-electro-mechanical systems (MEMS) and to detect the displacement and velocity signals from the LDV.

**Figure 3.**

**Top**: Effective quality-factor ${Q}_{eff}$ as a function of parametric excitation. The solid red line is a fit to Equation (3). The inset shows examples or resonance peaks at different voltages. The amplitude of the peaks was normalized by the fitted corresponding amplitude of each peak.

**Bottom**: Resonance peaks as the phase of the parametric signal was varied while the parametric excitation voltage was kept at 102 V. The inset shows the amplitude gain as a function of phase. The red line was fitted from Equation (2).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gonzalez, M.; Lee, Y. A Study on Parametric Amplification in a Piezoelectric MEMS Device. *Micromachines* **2019**, *10*, 19.
https://doi.org/10.3390/mi10010019

**AMA Style**

Gonzalez M, Lee Y. A Study on Parametric Amplification in a Piezoelectric MEMS Device. *Micromachines*. 2019; 10(1):19.
https://doi.org/10.3390/mi10010019

**Chicago/Turabian Style**

Gonzalez, Miguel, and Yoonseok Lee. 2019. "A Study on Parametric Amplification in a Piezoelectric MEMS Device" *Micromachines* 10, no. 1: 19.
https://doi.org/10.3390/mi10010019