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Mechanics of Microelectromechanical Systems This page intentionally left blank Nicolae Lobontiu Ephrahim Garcia Mechanics of Microelectromechanical Systems KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 0-387-23037-8 1-4020-8013-1 ©2005 Springer Science + Business Media, Inc. Print ©2005 Kluwer Academic Publishers Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.kluweronline.com http://www.springeronline.com To our families This page intentionally left blank TABLE OF CONTENTS Preface ix 1 1 1 1 6 14 STIFFNESS BASICS 1 INTRODUCTION 2 STIFFNESS DEFINITION 3 DEFORMATIONS, STRAINS AND STRESSES 4 MEMBERS, LOADS AND BOUNDARY CONDITIONS 5 LOAD-DISPLACEMENT CALCULATION METHODS: CASTIGLIANO’S THEOREMS 6 COMPOSITE MEMBERS 7 PLATES AND SHELLS Problems 21 43 58 60 2 MICROCANTILEVERS, MICROHINGES, MICROBRIDGES 1 INTRODUCTION 2 MICROCANTILEVERS 3 MICROHINGES 4 COMPOUND MICROCANTILEVERS 5 MICROBRIDGES Problems 65 65 66 97 103 114 126 3 MICROSUSPENSIONS 1 INTRODUCTION 2 MICROSUSPENSIONS FOR LINEAR MOTION 3 MICROSUSPENSIONS FOR ROTARY MOTION Problems 131 131 131 170 179 MICROTRANSDUCTION: ACTUATION AND SENSING 1 INTRODUCTION 2 THERMAL TRANSDUCTION 3 ELECTROSTATIC TRANSDUCTION 4 ELECTROMAGNETIC/MAGNETIC TRANSDUCTION 5 PIEZOELECTRIC (PZT) TRANSDUCTION 6 PIEZOMAGNETIC TRANSDUCTION 7 SHAPE MEMORY ALLOY (SMA) TRANSDUCTION 8 BIMORPH TRANSDUCTION 9 MULTIMORPH TRANSDUCTION 10 OTHER FORMS OF TRANSDUCTION Problems 183 183 184 195 212 223 230 232 238 249 256 257 4 viii STATIC RESPONSE OF MEMS 1 INTRODUCTION 2 SINGLE-SPRING MEMS 3 TWO-SPRING MEMS 4 MULTI-SPRING MEMS 5 DISPLACEMENT-AMPLIFICATION MICRODEVICES 6 LARGE DEFORMATIONS 7 BUCKLING 8 COMPOUND STRESSES AND YIELDING Problems 263 263 263 271 285 286 307 315 330 335 MICROFABRICATION, MATERIALS, PRECISION AND SCALING 1 INTRODUCTION 2 MICROFABRICATION 3 MATERIALS 4 PRECISION ISSUES IN MEMS 5 SCALING Problems 343 343 343 363 365 381 390 Index 395 5 6 PREFACE This book offers a comprehensive coverage to the mechanics of microelectromechanical systems (MEMS), which are analyzed from a mechanical engineer’s viewpoint as devices that transform an input form of energy, such as thermal, electrostatic, electromagnetic or optical, into output mechanical motion (in the case of actuation) or that can operate with the reversed functionality (as in sensors) and convert an external stimulus, such as mechanical motion, into (generally) electric energy. The impetus of this proposal stems from the perception that such an approach might contribute to a more solid understanding of the principles governing the mechanics of MEMS, and would hopefully enhance the efficiency of modeling and designing reliable and desirably-optimized microsystems. The work represents an attempt at both extending and deepening the mechanical-based approach to MEMS in the static domain by providing simple, yet reliable tools that are applicable to micromechanism design through current fabrication technologies. Lumped-parameter stiffness and compliance properties of flexible components are derived both analytically (as closed-form solutions) and as simplified (engineering) formulas. Also studied are the principal means of actuation/sensing and their integration into the overall microsystem. Various examples of MEMS are studied in order to better illustrate the presentation of the different modeling principles and algorithms. Through its objective, approach and scope, this book offers a novel and systematic insight into the MEMS domain and complements existing work in the literature addressing part of the material developed herein. Essentially, this book provides a database of stiffness/compliance models for various spring-type flexible connectors that transmit the mechanical motion in MEMS, as well as of the various forces/moments that are involved in microtransduction. In order to predict their final state, the microsystems are characterized by formulating, solving and analyzing the static equilibrium equations, which incorporate spring, actuation and sensing effects. Chapter 1 gives a succinct, yet comprehensive review of the main tools enabling stiffness/compliance characterization of MEMS as it lays the foundation of further developments in this book. Included are basic topics from mechanics of materials and statics such as load-deformation, stressstrain or structural members. Presented are the Castigliano’s theorems as basic tools in stiffness/compliance calculation. Straight and curved line elements are studied by explicitly formulating their compliance/stiffness characteristics. Composite micromembers, such as sandwiched, serial, parallel, and hybrid (serial-parallel) are also treated in detail, as well as thin plates and shells. All the theoretical apparatus presented in this chapter is illustrated with examples of MEMS designs. Chapter 2 is dedicated to characterizing the main flexible components that are encountered in MEMS and which enable mechanical mobility through