DEVELOPMENT OF AN ULTRASONIC PIEZOELECTRIC MEMS-BASED RADIATOR FOR NONLINEAR ACOUSTIC APPLICATIONS By BENJAMIN ANDREW GRIFFIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 °c 2009 Benjamin Andrew Gri±n 2 To my wife, Elizabeth, with much love and appreciation. Isaiah 40:31 ...but those who hope in the Lord will renew their strength. They will soar on wings like eagles; they will run and not grow weary, they will walk and not be faint. 3 ACKNOWLEDGMENTS Financial support for this work has been provided by graduate fellowships from the National Science Foundation and the University of Florida. I thank my advisors, Mark Sheplak and Louis N. Cattafesta III, for their many helpful technical discussions, as well as their career and personal advice. I am also grateful to my committee members, Havana V. Sanka, David Arnold, and Nab Ho Kim, for their expertise and assistance in the success of this project. I am especially grateful to my many colleagues in the Interdisciplinary Microsystems Group. I would like to thank my predecessors, Venkataraman Chandrasekaran and Guiqin Wang, for establishing a ¯rm foundation upon which this work was built. Former colleagues David Martin and Stephen Horowitz are greatly appreciated for their mentorship and train- ing during our concurrent association with the Interdisciplinary Microsystems Group. I have much gratitude for contemporaries Brian Homeijer and Vijay Chandrasekharan as we have \come of age" as graduate students together. Their engaging technical discussions, friend- ship, and comradery have been a sustaining force in my graduate career. I am also indebted to Matthew Williams, who I have worked closely with on this project. Without his addition to the Interdisciplinary Microsystems Group, this undertaking would not have been as suc- cessful. In addition, I am grateful to Chase Co®man, Dylan Alexander, and John Gri±n for their assistance with experimental setups, package fabrication, and data acquisition. I would also like to acknowledge all of the Interdisciplinary Microsystems group whose contributions are too numerous to list. I am particularly thankful to Avago Technologies Limited for the access to their fab- rication facilities. Special thanks goes to David Martin and Osvaldo Buccafusca at Avago for their special attention and personal time devoted to this project. I also acknowledge Dynatex International for their skill in wafer separation. I am especially grateful for the excellent machining work performed by Ken Reed at TMR engineering. The Mechanical and Aerospace Engineering departmental sta® is thanked for their kind assistance. 4 I thank my parents, Mike and Barbara Gri±n, for instilling in me a good work ethic and perseverance; without which, this project would not have been as successful. I would also like to express appreciation to my brother John and sister-in-law Karen who never hesitated to provide any assistance and support they could give. Above all, I am grateful to my wife, Elizabeth, for her patience, encouragement, support, and love. 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES ..................................... 9 LIST OF FIGURES .................................... 11 ABSTRACT ........................................ 17 CHAPTER 1 Introduction ...................................... 19 1.1 Parametric Arrays ............................... 19 1.2 Transducer Issues ................................ 22 1.2.1 Current Limitations ........................... 23 1.2.2 Potential Transducer Solution ..................... 24 1.3 Research Objectives ............................... 24 1.4 Dissertation Overview ............................. 24 2 Nonlinear Acoustics .................................. 25 2.1 Finite Perturbation Acoustic Theory ..................... 25 2.2 Parametric Array ................................ 30 2.2.1 Model Equations of Nonlinear Acoustic Theory ............ 31 2.2.2 Sound Beam Solutions ......................... 34 2.2.3 Existing Implementations ........................ 38 2.2.4 MEMS Parametric Arrays ....................... 47 2.3 Conclusion .................................... 50 3 Air-Coupled MEMS Ultrasonic Transducers .................... 51 3.1 Principles of Transmitter Operation ...................... 51 3.2 Acoustic Sources ................................ 55 3.2.1 Planar Radiation ............................ 55 3.2.2 Array of Sources ............................. 61 3.2.3 Acoustic Attenuation .......................... 63 3.2.4 Summary ................................. 66 3.3 MEMS Actuators ................................ 66 3.3.1 Electrostatic Transduction ....................... 67 3.3.2 Piezoelectric Transduction ....................... 80 3.3.3 Thermoelastic Actuation ........................ 95 3.4 Conclusion .................................... 100 6 4 Ultrasonic Radiator Design ............................. 102 4.1 Avago's FBAR Process ............................. 102 4.2 Fabrication ................................... 104 4.3 Package ..................................... 105 4.4 Conclusion .................................... 108 5 Modeling ....................................... 109 5.1 Equivalent Circuit ............................... 109 5.1.1 Acoustical Domain ........................... 112 5.1.2 Electrical Domain ............................ 118 5.1.3 Transduction ............................... 118 5.1.4 Equivalent Circuit ............................ 119 5.1.5 Approximate Performance ....................... 121 5.1.6 Example Device ............................. 122 5.2 Nonlinear Acoustic Modeling .......................... 130 5.3 Conclusion .................................... 132 6 Design Optimization ................................. 133 6.1 Methodology .................................. 133 6.2 Radiator Optimization ............................. 134 6.2.1 Limitations-Constraints ......................... 135 6.2.2 Problem Formulation .......................... 137 6.3 Results ...................................... 138 6.4 Alternate Designs ................................ 142 6.5 Conclusions ................................... 143 7 Experimental Setup and Results ........................... 145 7.1 Fabrication Results ............................... 145 7.2 Electrical Characterization ........................... 148 7.2.1 Setup ................................... 148 7.2.2 Results .................................. 149 7.3 Device Topography ............................... 151 7.4 Electromechanical Characterization ...................... 153 7.4.1 Setup ................................... 155 7.4.2 Frequency Response Function ..................... 157 7.4.3 Diaphragm Resonance ......................... 159 7.4.4 Linearity ................................. 160 7.4.5 Variable Back Cavity .......................... 163 7.4.6 Vacuum Experiments .......................... 167 7.5 Electroacoustic Characterization ........................ 171 7.5.1 Setup ................................... 171 7.5.2 Results .................................. 174 7.6 Performance as a Parametric Array ...................... 176 7 7.7 Conclusion .................................... 178 8 Conclusion and Future Work ............................. 180 8.1 Conclusions ................................... 180 8.2 Recommendations for Future Work ...................... 182 8.3 Recommendations for Future Design ..................... 183 APPENDIX A NONLINEAR ACOUSTIC MODELING ...................... 186 A.1 Westervelt Parametric Array Solution ..................... 186 A.2 Berktay Solution ................................ 189 B PLATE MODEL ................................... 191 B.1 Basic Assumptions ............................... 192 B.2 Static Equilibrium ............................... 192 B.3 Constitutive Equations ............................. 194 B.4 Governing Di®erential Equations ....................... 196 B.5 General Solution ................................ 197 B.6 Boundary and Matching Conditions ...................... 198 B.7 Incremental Plate Deflection .......................... 200 C UNCERTAINTY ANALYSIS ............................ 201 C.1 Electrical Characterization Uncertainties ................... 201 C.1.1 Electrical Impedance .......................... 201 C.1.2 Element Extraction ........................... 202 C.2 Electromechanical Characterization Uncertainties .............. 206 C.2.1 Velocity ................................. 206 C.2.2 Volume Velocity ............................. 207 C.2.3 Resonant Frequency ........................... 208 C.2.4 Damping Coe±cient Estimation .................... 208 C.2.5 Variable Back Cavity .......................... 210 REFERENCES ....................................... 213 BIOGRAPHICAL SKETCH ................................ 227 8 LIST OF TABLES Table page 2-1 CMUT transducer array speci¯cations and performance presented by Wygant et al. [15]. ........................................ 48 2-2 Transducers used in parametric array experiments. ................ 49 3-1 Air-coupled cMUT characteristics. ......................... 81 3-2 Piezoelectric ¯lm properties. ............................. 87 3-3 Air-coupled pMUT characteristics. ......................... 96 3-4 Thermoelastic MEMS characteristics. ........................ 100 4-1 Typical epoxy dispense and cure parameters used for die attachment. PCB boards