Modeling and design of composite free–free beam piezoelectric resonators

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Abstract

A model is presented and experimentally validated for the design of asymmetrically layered piezoelectric free–free beam micromechanical resonators. The thin film resonators are composed of quad-symmetric torsional anchors connected to the nodal points of a free–free beam vibrating in its fundamental bending mode. The model provides analytical approximations for the resonance frequencies of the composite anchors and the free–free beam. Since dissipation in these devices is minimized when the mechanical impedances of the coupled structures are matched at resonance, i.e. when bending beam and torsional anchor resonance frequencies are equal, the model may be used to design resonators with optimized mechanical quality factors. The model is shown to successfully predict resonance frequencies in agreement with the results of finite element analysis and experimental results obtained for multi-layered lead zirconate titanate (PZT) piezoelectric microresonators with first natural frequencies in the range of 448 kHz to 1.1 MHz. The model is also validated as a tool for optimizing resonator quality factor.

Introduction

Micromechanical resonators offer the potential for significant benefits in signal processing systems, including reduced size and power consumption, the potential for direct integration with electronics, and the ability to design for a wide range of resonance frequencies on a single monolithic chip. Micromechanical resonators have been widely demonstrated based on electrostatic actuation and capacitive sensing in polysilicon [1], [2], [3], [4], [5] and silicon carbide [6], [7] thin films. More recently, piezoelectric transduction has been used to realize microresonators using active materials including ZnO [8], PZT [9], AlN [10], and Al1−xGaxAs [11], [12], [13], [14], [15]. Compared to electrostatically actuated resonators, piezoelectric microbeam resonators are attractive due to the potential for lower insertion loss, in particular at higher frequencies [8].

Clamped–clamped structures have received much attention, since they offer large stiffness to mass ratios, making high resonance frequencies possible [2]. However, clamped–clamped structures tend to experience significant energy loss to the substrate through their anchors, resulting in reduced mechanical quality factors (Qs). In contrast, free–free beam designs have been shown to lead to significantly higher Q values [2], [3], [4]. In its simplest form, the free–free resonator design features quad-symmetric torsional anchors connecting the nodal points of a free–free structure to the fixed substrate. To illustrate this geometry, a typical lead zirconate titanate (PZT) microresonator used in this study is shown in Fig. 1. By matching the resonance frequencies of the anchors with the beam, the mechanical impedance mismatch is minimized, thus reducing energy loss.

For electrostatic resonators that consist of a single material, such as polysilicon, achieving impedance matching between the bending primary beam structure and the torsional anchors is straightforward, since the required eigenfrequencies for both elements can be determined from readily available analytic expressions. In contrast, for a piezoelectric resonator design, the analysis is complicated by the composite nature of the structure, which is required to take advantage of the piezoelectric transduction. These resonators typically consist of four layers, namely the piezoelectric film, an elastic layer to offset the neutral axis of the beam, and an electrode layer on each side of the piezoelectric film, with each layer possessing different mechanical properties. Furthermore, the structure must possess an asymmetric cross-section in order for the beam element to operate in a bending mode.

In the following, an analytic model is presented for the design of composite piezoelectric free–free beam microresonators. Model validation is performed using finite element analysis and experimental data. For the considered PZT piezoelectric resonators, a comparison between model predictions and experimental data is provided to illustrate how this work can be used to carry out Q-factor optimization.

Section snippets

Resonator model

A cross-section of a generic 4-layer piezoelectric beam is shown in Fig. 2. This figure depicts a composite rectangular cross-section of total height H, total width W, and an ith layer with thickness ti. Each layer has a Young's modulus Ei, Poisson's ratio νi, and mass density ρi. The distance from the bottom layer of the cross-section to the top of the ith layer is denoted as hi. Although only four layers are shown in this figure, the development of this section applies to structures

Results and discussion

A range of resonator designs based on PZT as the active piezoelectric film were fabricated with resonance frequencies ranging from 448 kHz to 1.1 MHz. The device cross-section consists of four material layers, namely, 2 μm SiO2, 0.16 μm Ti/Pt, 1 μm PZT, and 0.1 μm Pt. The SiO2 acts to offset the neutral axis of the beam from the centerline of the PZT film, and the Ti/Pt and Pt layers are electrodes for sensing and actuation. The fabrication process has been reported in detail elsewhere [9]. Briefly,

Conclusion

A model has been presented for the analysis of asymmetric composite free–free beam piezoelectric resonators. PZT-based devices have been fabricated to validate the model, with resonance frequencies in the range of 448–1.1 MHz. Results of finite element analysis and experimental results are in good agreement with the composite model predictions for the resonance frequencies of both free–free beam elements and torsional anchors. The model can enable designers to accurately choose device dimensions

Acknowledgements

The support of DARPA/MTO and the NSA Laboratory for Physical Sciences is gratefully acknowledged. The authors would like to thank Toby Oliver (Laboratory for Physical Sciences/National Security Agency), and Rich Piekarz, John Conrad, Jeff Pulskamp, Luke Currano, and Ron Polcawich (Army Research Laboratory) for their assistance with PZT device fabrication.

Anthony Ferguson received the MS degree in Mechanical Engineering from the University of Maryland, College Park in 2004. His research interests are in microresonators for RF applications, and more broadly in the areas of microsystems fabrication and modeling.

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  • Cited by (0)

    Anthony Ferguson received the MS degree in Mechanical Engineering from the University of Maryland, College Park in 2004. His research interests are in microresonators for RF applications, and more broadly in the areas of microsystems fabrication and modeling.

    Lihua Li is a PhD student in the Center of Micro Engineering at the University of Maryland, College Park. She received her BS and MS degrees from Tsinghua University in 1997 and 2000, respectively. Her research is focused on the development of low-loss piezoelectric microresonators based on III-V materials for RF filters.

    Vengalattore Nagaraj has a Master's degree in Aerospace Engineering from Indian Institute of Science, Bangalore, India and a PhD in Aerospace Engineering from Loughborough University of Technology, England. He has worked on aircraft and helicopter design at Hindustan Aeronautics Limited, Bangalore, India. His current interests include helicopter design and helicopter dynamics.

    B. Balachandran is a professor of Mechanical Engineering at the University of Maryland, College Park, where he has been since 1993. His research interests include nonlinear dynamics, vibration and acoustics control, system identification, and signal analyses. The publications he has authored/co-authored include a Wiley book entitled “Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods” (1995) and a Brooks/Cole-Thomson book entitled “Vibrations” (2003). He is a fellow of ASME.

    Brett Piekarski received a BS in Mechanical Engineering from the University of North Dakota in 1987, was the distinguished graduate of the Army Material Command Production Engineering Program in 1988, received a Masters of Mechanical Engineering from Johns Hopkins University in 1992, and is currently pursuing a PhD in Mechanical Engineering at the University of Maryland, College Park. Mr. Piekarski joined the Army Research Laboratory (ARL) in 1988, and in 1997 joined the MEMS Team at ARL where he was responsible for MEMS design, fabrication, and testing including several RF MEMS projects. In 2002 Mr. Piekarski was named the ARL Specialty Electronic Materials and Sensors Cleanroom (SEMASC) Manager and currently leads a team responsible for daily operations, process development, and customer collaborations within the SEMASC facility.

    Don DeVoe received the PhD degree in Mechanical Engineering from U.C. Berkeley in 1997, with a specialization in microsystems technologies. Dr. DeVoe is currently an Associate Professor of Mechanical Engineering and Faculty Member of the Bioengineering Program at the University of Maryland, with research interests encompassing piezoelectric microsystems as well as microfluidic systems for genomic and proteomic analysis, single-molecule detection in nanofluidic systems, and novel microfabrication methods involving polymer replication techniques. Dr. DeVoe is a recipient of the Presidential Early Career Award for Scientists and Engineers from the National Science Foundation for advances in Microsystems technology, and holds six U.S. patents in this area.

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