Inertial MEMS: Principles and Practice

This page intentionally left blank

Inertial MEMS Principles and Practice A practical and systematic overview of the design, fabrication, and testing of MEMS-based inertial sensors, this comprehensive and rigorous guide shows you how to analyze and transform application requirements into practical designs, and helps you to avoid potential pitfalls and to cut design time. With this book you’ll soon be up to speed on the relevant basics, including MEMS technologies, packaging, kinematics and mechanics, and transducers. You’ll also get a thorough evaluation of different approaches and architectures for design and an overview of key aspects of testing and calibration. Unique insights into the practical difficulties of making sensors for real-world applications make this up-to-date description of the state of the art in inertial MEMS an ideal resource for professional engineers in industry, managers, and application engineers, as well as for students looking for a complete introduction to the area. Volker Kempe has more than 40 years of experience in research and development in both academia and industry. He led the microelectronics engineering department at Austria Mikro Systems for over 10 years. In 2003 he co-founded, and became Vice President of, SensorDynamics AG, and his current interests focus on the functionality, technology, and application of inertial MEMS.

Inertial MEMS Principles and Practice Volker Kempe Sensor Dynamics AG, Austria

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ a o Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521766586 ° C

Cambridge university Press 2011

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Kempe, Volker. Inertial MEMS : principles and practice / Volker Kempe. p. cm. Includes bibliographical references and index. ISBN 978-0-521-76658-6 (hardback) 1. Microelectromechanical systems. 2. Inertial navigation systems. TK7875.K46 2011 629.04′ 5 – dc22 2010037668 ISBN

3. BioMEMS.

978-0-521-76658-6 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

I. Title.

Contents

Preface Acknowledgments Notation

page xiii xv xvi

1

Introduction 1.1 A short foray through the pre-MEMS history 1.2 Applications and market 1.3 The ingredients of inertial MEMS References

1 1 6 9 11

2

Transducers 2.1 Anisotropic material properties, tensors, and rotations 2.1.1 Stress, strain, and piezoresistivity Hooke’s law Normal stresses Shear stresses Stress and strain tensors The stress–strain relation for anisotropic materials The piezoresistance of silicon 2.1.2 Rotation of coordinate systems Coordinate frames The rotation tensor Transformation of tensors of second order 2.2 Piezoresistive transducers 2.2.1 Piezoresistors 2.2.2 Piezoresistors on silicon Thin piezoresistors Temperature compensation in piezoresistors 2.2.3 Piezoresistors on polysilicon 2.3 Piezoelectric transducers 2.3.1 The piezoelectric effect 2.3.2 Piezoelectric equations Piezoelectric sensors in MEMS

13 13 14 14 14 15 17 21 24 25 26 27 30 33 33 34 35 36 38 39 39 42 43

vi

Contents

2.4

Capacitive transducers 2.4.1 Electrostatic forces 2.4.2 Parallel-plate capacitors Capacitance sensing The pull-in effect 2.4.3 Tilting-plate capacitors Nonlinear distortions Instabilities of a singular tilting plate Instabilities of the tilting capacitance pair 2.4.4 Comb capacitors Unidirectional linear combs Bidirectional actuation Radial combs Frame-based capacitors 2.4.5 Levitation Comb levitation Levitation in drive combs Reduction of levitation forces References 3

Non-inertial forces 3.1 Springs 3.1.1 Beams 3.1.2 The stiffness matrix 3.1.3 The bending equation for beams Internal forces and moments Differential relations of a bent beams 3.1.4 Cantilever beams Cantilevers under different loads Skew beam bending and asymmetric suspensions Residual stress in bending beams 3.1.5 Torsion springs Cylindrical torsion bars Torsion bars with arbitrary cross-section Rectangular bars Cylindrical bars 3.1.6 Stress concentration 3.1.7 Suspensions Parallel and serial spring connections Beam chains Plate suspension 3.2 Damping forces 3.2.1 Fluid-flow models

47 49 51 51 53 55 56 58 58 62 62 65 66 67 69 69 71 73 75 79 79 80 81 82 83 83 87 87 90 93 96 96 97 100 101 101 103 103 104 105 108 108

Contents

Continuous viscous flow Viscosity of gases Continuous-flow equations Slide damping Couette flow for slowly moving plates Stokes flow for rapidly oscillating plates Squeeze damping Reynolds’ equation Low-frequency squeeze damping High-frequency squeeze damping The impact of perforation Drag forces Free molecular flow Structural damping

111 112 114 117 118 120 124 125 127 131 137 143 145 147 148

MEMS technologies 4.1 Microfabrication of inertial MEMS 4.1.1 Basic microelectronic fabrication steps Deposition Patterning Doping 4.1.2 Etching Isotropic wet etching Anisotropic wet etching Electrochemical etch stop 4.1.3 Dry etching Reactive-ion etching Deep reactive-ion etching 4.2 Wafer bonding 4.2.1 Zero-level packaging and wafer bonding 4.2.2 Wafer-bonding processes Fusion bonding Anodic bonding Glass-frit bonding Metallic-alloy seal bonding Polymer bonding Thermocompression bonding 4.3 Integrated processes 4.3.1 Bulk micromachining 4.3.2 Surface micromachining A thick polysilicon process Cavity sealing using SMM

152 153 154 155 159 161 162 162 164 167 168 168 170 172 172 174 175 176 177 179 180 180 180 182 184 185 188

3.2.2

3.2.3

3.2.4 3.2.5 3.2.6 References 4

vii

viii

Contents

4.3.3

SOI-MEMS processes MEMS prototyping processes 4.3.4 CMOS-MEMS Pre-CMOS MEMS Intra-CMOS MEMS Post-CMOS MEMS References

189 192 192 193 195 196 199

5

First-level packaging 5.1 FLP packages 5.2 FLP technologies 5.2.1 Dicing and die separation 5.2.2 Die attachment Packaging materials Die-attachment-induced stress 5.2.3 Electrical interconnection 5.2.4 Encapsulation Overmolded plastic packages Pre-molded plastic packages References

205 206 209 210 211 212 213 217 220 221 223 225

6

Electrical interfaces 6.1 Sensing electronics – building blocks 6.1.1 The MOS transistor Drain current The small-signal model 6.1.2 Operational and transconductance amplifiers A simple transconductance amplifier Models of operational and transconductance amplifiers The real Op Amp Instrumentation amplifiers 6.2 Sensor interfaces 6.2.1 Resistive interfaces 6.2.2 Piezoelectric interfaces 6.2.3 Capacitive interfaces Principles of capacitive sensing Current sensing Voltage sensing Charge sensing Comparison and improvements Switched-capacitor sensing 6.3 Data converters 6.3.1 Sampling and hold

227 228 229 229 231 234 234 236 240 246 247 247 249 253 254 255 258 259 260 260 266 266

Contents

6.3.2 6.3.3

7

Single-sample conversion in the amplitude domain Time-domain conversion Pulse-width and pulse-density modulation Σ∆ Converters

ix

References

268 269 269 271 280

Accelerometers 7.1 General measurement objectives 7.2 The spring–mass system 7.2.1 The transfer functions The trade-off between sensitivity and bandwidth 7.2.2 Accelerometer imperfections A simplified accelerometer model with imperfections Cross-coupling 7.2.3 Accelerometer feedback control The linearized feedback model The signal-to-noise ratio Closed-loop dynamics 7.2.4 Feedback control with nonlinear actuators Bidirectional capacitive actuators Single-sided actuators Linearization and embedded Σ∆ converters 7.3 Resonant accelerometers 7.3.1 Resonant beams Resonance vibration – exact solution Resonance frequencies by the energy method 7.3.2 Resonant accelerometer systems 7.4 Beam accelerometers 7.4.1 Beam dynamics The principle of virtual work Eigenmode expansion Damping and electrostatic forces Static deflection 7.4.2 Model implementation The impact of nonlinear damping Feedback control 7.5 Various other accelerometer principles 7.5.1 Tunneling accelerometers 7.5.2 Convective and bubble accelerometers 7.6 From 1D to 6D accelerometers 7.6.1 1D accelerometers Piezoresistive accelerometers Capacitive accelerometers

283 283 284 286 291 292 296 297 298 300 305 307 309 310 312 313 319 320 320 323 324 326 328 329 330 332 333 333 335 336 337 337 339 341 342 343 346

x

Contents

Piezoelectric accelerometers 2D and 3D accelerometers Parallel implementation 2D and 3D accelerometers with multi-DOF sensing elements 7.6.3 6D accelerometers References

348 350 350 353 355 358

Gyroscopes 8.1 Some basic principles 8.2 Kinematics of gyroscopes 8.2.1 Platform rotation and angular velocity 8.2.2 Body rotation in a non-inertial system 8.2.3 The angular-momentum theorem 8.2.4 The momentum equation 8.2.5 The small-angle approximation 8.3 The performance of gyroscopes 8.4 Rate-integrating gyroscopes 8.4.1 Two-DOF gyroscopes 8.4.2 The principle of angular gyroscopes 8.4.3 An imperfection model 8.4.4 Imperfection in angular gyroscopes 8.4.5 Gyroscope control 8.5 Rate gyroscopes 8.5.1 System architecture The drive resonator Sensing 8.5.2 Resonance sensing 8.5.3 Non-resonant sensing 8.5.4 Noise 8.5.5 The zero-rate output Mechanical bias sources Q-bias The impact of transducer imperfections R-bias Other bias sources 8.5.6 Bias stability 8.5.7 Acceleration suppression and tuning forks Anti-phase-driven identical gyroscopes Tuning-fork gyroscopes 8.5.8 Drive-motion control and spring nonlinearities The phase-locked loop The amplitude loop Spring nonlinearities and the resonator transfer function

364 364 367 369 371 373 375 376 378 380 380 381 385 387 388 389 391 393 393 395 398 400 402 403 405 408 409 410 411 413 414 415 418 418 419 421

7.6.2

8

Contents

9

xi

8.6

Gyroscope architectures 8.6.1 Mode-decoupling architectures 8.6.2 z-Gyroscopes Mode decoupling by frame-based architectures Doubly decoupled z-gyroscopes 8.6.3 In-plane-sensitive gyroscopes In-plane-sensitive linear gyroscopes Linear–rotatory gyroscopes 8.6.4 Torsional gyroscopes 1D torsional gyroscopes Decoupled torsional gyroscopes 8.7 Non-planar MEMS gyroscopes 8.7.1 Beam gyroscopes 8.7.2 Quartz tuning forks 8.7.3 Ring gyroscopes 8.7.4 Bulk acoustic-wave gyroscopes 8.8 2D and 3D gyroscopes and ways towards a 6D IMU 8.8.1 Single-mass multiple-DOF inertial sensors Gyroscope-free, single-mass IMUs Single-mass, gyroscope-based IMUs 5D inertial sensors 8.8.2 2D gyroscopes 8.8.3 3D gyroscopes A fully decoupled 3D gyroscope and extension towards an IMU References

424 424 424 424 427 429 429 430 431 432 433 438 438 440 441 443 444 445 445 445 447 448 449 450 452

Test and calibration References

460 464

Concluding remarks References

466 467

Index

468

Th e co l o r p la t es a re t o be fo und bet w een pa ges 240 a nd 241.

Preface

Inertial microelectromechanical sensors – commonly abbreviated to inertial MEMS – have a history of more than two decades of intense research, development, and commercialization. Sometimes unperceived, they left the shadow of military and space-related utilities and entered daily life hidden in products surrounding us. Cars with airbag-release sensors and electronic stability control have become a matter of course. Activity monitoring of pacemaker patients and stabilization of platforms such as transport robots and cameras are now improving our quality of life. The creation of easy-to-use human–machine interfaces has helped many people to conquer complicated equipment around us, not only computer games. The penetration of inertial MEMS – often merged with other sensor systems – into new application areas is a trend that is still gaining momentum. The intention of this book is to reflect the interdisciplinary complexity of inertial MEMS. It will try to give a systematic survey of the design, fabrication, and performance evaluation of MEMS-based inertial sensors, with emphasis on the practical problems arising from the impact of technological imperfections and of often harsh environmental conditions. A product going to the market has to be guaranteed to have a certain level of reliability against failure throughout its lifetime. The basic concepts and the theoretical background of inertial measurements will be presented. However, the book has evolved not from academic activity but rather from conceptual and development work within industry. It is intended to address the symbiosis of practice and theory. Consequently, the analysis and transformation of application requirements into design concepts plays a significant role. Considerable space is devoted to the analysis and modeling of parasitic effects, of shock and vibration robustness, of the stability of the main performance parameters and so on, since this is necessary for practical work. The book contains nine chapters. Six of them – including the introduction – cover various aspects of MEMS, with a special focus on inertial MEMS. The first chapter describes the most important transducers and their properties. The second one is dedicated to non-inertial forces such as spring forces and damping forces that play a crucial role for designing inertial MEMS. The next two chapters cover the main MEMS technologies, including packaging, while the electronic interfaces are presented in a further chapter. These six chapters may be

xiv

Preface

interesting not only for people working with inertial MEMS but also for everybody who is looking for a general introduction into mechanical MEMS. The following two main chapters are devoted to the two representatives of inertial MEMS – accelerometers and gyroscopes. Here the focus is on the basic principles, the methods and models to describe the dynamic behavior, and a comprehensive presentation of different approaches and architectures, including their pros and cons. A short overview on test and calibration is added as a separate chapter. The book is written on an engineering level. Where possible, effects and processes are described analytically by mathematical models in order to impart a feeling for the order of magnitude of different effects. The book should be useful not only for specialists developing, manufacturing, and using inertial sensors but also for people working in the application field, for product managers, and for sales people looking for background knowledge in their area. The book can serve as a starting point for further academic investigations, for instance in the area of shock-impact analysis of an entire packaged gyro, including the effect of signal processing. In the experience of the author, many engineers, physicists, and mathematicians are thankful for an exact but comprehensible presentation of the complex and difficult world of MEMS-based inertial sensors, where the effects and models behind the practical problems are reflected without improper simplifications or phenomenological descriptions. The book is a modest attempt to meet some of these challenges. Having worked with many specialists in the production, testing, and design of inertial sensors, the author is convinced that the book can meet actual needs, and hopes to elicit the broad interest of practitioners and scientists in this area. For interested people, including students, the book may also serve as an introduction to the world of mechanical MEMS.

Acknowledgments

I would like to express my gratitude to all my colleagues at SensorDynamics AG (Austria) and the Institut f¨ ur Silizium Technologie (ISIT) of the Fraunhofer Society (Germany) for creating an atmosphere that has helped to solve the manifold problems of MEMS industrialization. I would like to thank the ‘Inertial Micro Sensor Systems’ team, with whom I have had the great privilege of working even during the childhood of the newly founded company SensorDynamics AG. This time was most fruitful, flooding us all with new problems and insights into how to solve them. My thanks go to my colleagues from SensorDynamics for providing me with such necessary illustrative material for the book as SEM photographs and measurement plots. Gottfried Frais, Manfred Heller, Christian Rossadini, J¨ org Sch¨ onbacher, Ute Stotter, and Johann Wagner prepared a lot of material from which I could select the most appropriate items. Gerd Radl and his team accompanied me during all my mistakes with new hardware and software. Drago Strle from the University of Ljubljana deserves my special thanks for the close cooperation throughout all Σ∆-related issues. Peter Merz from the ISIT gave me invaluable feedback on all technology-related questions. Professor Karl Wohlhart from the University of Graz helped me to gain a deeper understanding of the kinematics of gyroscopes. Hanno Hammer took on the burden of proofreading the first chapters and supplying me with valuable feedback. Last but not least, warmest thanks go to my family for supporting me despite all the personal loads that each of us has had to carry. Julia and Oded helped me to resolve quickly the countless difficulties in adapting appropriate documentcreating tools to my needs. Vera, Ian, and Marius gave valuable advice on English phrasing. My dear wife took on additional duties despite her own excessive workload in working with very ill patients. Thank you.

Notation

1. A convention employed in this book is the slightly lax usage of “s” as differential operator, s = d/dt, as argument of the Laplace transformation, and as argument of the Fourier transformation, s = jω. The case-dependent unambiguous or multivalent meaning is usually clear from the context. Accordingly, a filter function is described by f = f (s), which means that in a transfer function this expression has to be interpreted as a Laplace or Fourier transformation and within a differential equation as a rational fraction of two polynomial differential operators. Correspondingly, a variable like x has to be treated as a representant in the time domain if s = d/dt is supposed, or as a Laplace/Fourier-transformed function if s is meant as the argument of such a transformation. 2. Unless stated otherwise, coordinate systems pertain to the platform carrying the inertial sensor. In this case the x- and y-axes lie in the plane of the platform, while z is the out-of-plane axis. Out-of-plane and z-axis orientation are used synonymously.

1

Introduction

An inertial sensor is an observer who is caught within a completely shielded case and who is trying to determine the position changes of the case with respect to an outer inertial reference system. Inertial sensors exploit inertial forces acting on an object to determine its dynamic behavior. The basic dynamic parameters are acceleration along some axis and the angular rate. External forces acting on a body cause an acceleration and/or a change of its orientation (angular position). The rate of change of the angular position is the angular velocity (angular rate). A speedometer is not an inertial sensor because it is able to measure a constant velocity of a body that is not exposed to inertial forces. An inertial sensor is unable to do so; however, if the initial conditions of the body are known, their evolution can be calculated by integrating the dynamic equation on the basis of the measured acceleration and rate signals. In the overwhelming majority of practical applications, such as vibrational measurements, active suspension systems, crash-detection systems, alert systems, medical activity monitoring, safety systems in cars, and computer-game interfaces, the short-term dynamic changes of the object are of interest. But there are also many applications where inertial sensors are used for determination of the positions and orientations of a body, as in robotics, general machine control, and navigation. Owing to the necessity of integrating the corresponding dynamic equations, the accuracy requirements in these applications are usually higher because the measurement errors and instabilities of the sensors are accumulated over the integration time. Often inertial sensors are used in conjunction with other measurement systems, as in the case of robotics, where they are used together with position and force/torque sensors, or in the case of the integration of Inertial Navigation Systems (INS) with Global Positioning Systems (GPS) in cars. The accuracy of INS measurements can be improved significantly by correcting them with the GPS data using Kalman filtering procedures. The INS can then aid navigation even when the GPS is degraded or interrupted because of jamming or interference.

1.1

A short foray through the pre-MEMS history The history of inertial sensors is relatively short. Despite the fundamental role played by inertial sensors in controlling the movement of a body, very little is

2

Introduction

known about early applications. This is even more remarkable given that most of the ingredients for building acceleration and angular-rate sensors, such as fine mechanics and precise spring technologies, were available from the late Middle Ages on and were used in the construction of, for instance, beautiful precision clocks.

Accelerometers One of the most likely reasons for the late appearance of acceleration sensors (or “accelerometers” for short) was the lack of indicator technologies, or, in modern phrasing, the lack of interfaces. This is certainly the reason why some former applications of acceleration switches that needed only very simple mechanical interfaces can be found. An acceleration switch initiated an action at a certain level of acceleration, as in the activation of a detonator in some bombs during the First World War. The first commercial accelerometer for broader application is credited to B. McCollum and O. S. Peters and was developed around 1920 [McCullom and Peters 1924]. It was based on a tension–compression resistance of a Wheatstone half-bridge where the resistances were formed by carbon rings. The next technological step was the use of strain-gauge transducers starting from around 1938, followed by the introduction of piezoelectric and piezoresistive transducers at the end of the 1940s. These transducers could capture the forces caused by the displacement of an elastically mounted mass within the sensor structure. Miniaturization and the high robustness of this type of sensor paved the way for broad applications in industry, terrestrial transport, aerospace, military uses, seismology, science, and so on. The piezoelectric and piezoresistive transducer principles were also among the first to be employed at the beginning of the entry into the world of inertial microelectromechanical systems (MEMS) – the world of the combination of micrometer- and nanometer-scale mechanical elements, sensors, actuators, and electronic circuits on one carrier or even on one chip. This entry was prepared in the late 1970s, for instance with the demonstration of a batchfabricated silicon accelerometer with piezoresistive transducers. The silicon bulk micromachined proof mass was bonded between two glass wafers [Roylance and Angell 1979]. The commercialization of similar devices began around 10 years later and was very soon based on a variety of available transducer principles such as the sensing of capacitance changes between fixed and movable plates, the frequency measurement of resonant devices, the stabilization of a tunneling current by a closed-loop system, the sensing of thermal changes between a heater and a movable heat sink, and the sensing of changes of the thermal distribution within an air bubble. This broad invasion of new and old ideas in the world of microelectronic technologies has opened the way to inexpensive mass applications of inertial sensors in industry, cars, medicine, consumer goods, and so on. Everybody knows the pioneering role of MEMS-based 50g accelerometers used in airbag ignition devices, which became the first high-volume product in the area of inertial MEMS. It was especially encouraging that within these successful highvolume products an example of the full monolithic integration of sensor and

1.1 The pre-MEMS history

3

signal processing on one chip could be found. Analog Devices, supported by strong governmental funding, developed a special BiCMOS-MEMS process combining a known microelectronic process with a polysilicon deposition, etching, and release technology. Various inertial sensors were developed on the basis of this process, of which the first was the 50g accelerometer [Analog Devices 1993].

Gyroscopes The stabilizing effect of rapidly rotating disks has been known for a long time and was used in ancient times for yo-yo-like toys and for ceremonies. Real angularrate sensors emerged quite late but have had a remarkably long history compared with accelerometers. This obviously is due to the much lower early requirements on the speed of the interfaces and, of course, to the moderate values of the signals to be measured. For instance, the Earth’s rotation is characterized by a rate signal of 360◦ /24 hours or 0.1◦ /s. The first technical realization of an angular-rate sensor took place around 1817 with the mechanical gyroscope designed by Johann Gottfried Friedrich von Bohnenberger in T¨ ubingen in Germany. It was not a true sensor but a demonstrator of rotational effects. The system was based on the spinning top and – not surprisingly – was used to demonstrate the mechanism of the Earth’s axis’ very slow precession accelerated in duration from a full cycle of 25 800 years to a small amount of seconds or minutes. Similar demonstrators were built around 1830 by the American Walter Johnson (Johnson’s Rotascope). The principle underlying the emergence of Coriolis forces within a rotating non-inertial coordinate system was demonstrated in 1851 by the French scientist Leon Foucault by building a 67-m-long pendulum with a mass of 28 kg within the Paris Pantheon (Fig. 1.1). This was the first real rotation sensor which measured the rotation of the Earth. Incidentally, a similar experiment was first performed in 1661 by the Italian physicist Vincenco Viviani and, after Foucault, implemented in countries all over the world. Under the influence of the Earth’s rotation the oscillation plane of such a pendulum changes by 360◦ sin ϕ in 24 hours. The angle ϕ is the geographic latitude of the experimental location. At the equator, the Foucault Pendulum does not show any reaction; at the poles, the rotation would be the full 360◦ /24 hours. At all other places the tip of the pendulum will draw nice rosettes on the floor. The Coriolis force was introduced by Gaspard-Gustave Coriolis, a French scientist, who described it in 1835. The Coriolis force appears in the equation of motion of an object in a rotating frame of reference and depends on the linear or angular velocity of the moving object. It will be considered in more detail in Chapter 8. Using this principle, in 1852 Foucault built a spinning-top gyroscope (“Meridiankreisel”), which can be considered the basis of modern spinning-top gyros. The term “gyroscope” was introduced at this time (“gyros” – rotation, “skopein” – vision). However, the rise of rotation sensors was preceded by the use

4

Introduction

Figure 1.1 Foucault’s Pendulum, in the Pantheon, Paris.

of the stabilizing ability of spinning wheels in torpedoes and cannon ammunition. Only in 1904 was the technical principle of the fast-spinning-top gyroscope patented by the German art historian Hermann Ansch¨ utz-K¨ ampfe [Schell 2005], who developed the idea of using the gyroscope within a compass (in 1908). The spinning wheel in a gyroscope is mounted on gimbals so that the wheel’s axis is free to orient itself. The key element for a compass was the introduction of a mechanism that results in an applied torque whenever the compass’ axis is not pointing North. The precession returns the compass’ axis towards the true North if it is disturbed towards another orientation. Other inventors such as the American Elmer Ambrose Sperry (1910) followed Ansch¨ utz-K¨ ampfe, and the acceptance of the gyro-compasses by the navy led to a quick penetration first on large ships and later on smaller ones and on airplanes. Inertial navigation and platform stabilization in aircraft and naval vessels remained the domain of spinning-top gyroscopes for a very long time. Driven by the requirements for cost reduction and miniaturization, around 1960 new types of gyros like the vibrating-string gyro [Quick 1964], the tuning-fork gyro [Hunt and Hobbs 1964] and the vibrating-shell resonator emerged and opened the way to a drastic size and weight reduction, which finally ended with the transfer of these principles into the world of MEMS. The vibrating-string principle is based on the action of the Coriolis force on a simple oscillator such as a mass on a string or a vibrating beam; the tuning-fork principle is based on balanced oscillators; and the vibrating shell uses the two vibration modes of a ring or a cylinder as in the classic wineglass effect. These principles will be presented in Chapter 8.

1.1 The pre-MEMS history

5

The new miniaturized vibrating gyros captured marked shares of the market step by step. However, for high-precision applications a rival for the classical gyros emerged around the end of the 1970s: the optical gyroscope. These gyros based on optical ring resonators and later on fiber optics have dominated, for instance, the aircraft navigator market since 1980. The transition from miniaturized gyros to the MEMS gyros was smooth. In miniaturized gyros it has become more and more difficult to realize bearings for endlessly rotating objects. The same problem applies for the early MEMS technologies and – with respect to specific friction, wear, and reliability – is still present today despite all the progress in this area. Only in 2008 did the first rotating MEMS gyro, developed by the Japanese company Tokimec, emerge on the market. Previous MEMS gyros used not the spinning-top principle, but rather the vibration or oscillation of masses within small linear or angular intervals as prepared by the miniaturized mechanical constructions. These so-called Coriolis vibratory gyroscopes (CVGs) – irrespective of whether or not they are based on MEMS – use at least two vibration modes of the structure, in which the Coriolis force excited by the interaction of the external rotation with the socalled primary mode causes an energy transfer to the secondary vibration mode. The vibrating elements are joined to hinges or anchors via springs. Such springs can be outstandingly formed in silicon because silicon possesses not only excellent mechanical and thermal properties in comparison with classical metals but also outstanding machinability. However, the full implementation of a complete gyroscope structure by using only microelectronic or emerging MEMS technologies was not the first step towards MEMS-based gyroscopes. Instead the designers first tried to use MEMS technologies to create key components for miniaturized gyros, such as the quartz tuning forks with piezoelectric actuators and piezoresitive transducers developed by Systron Donner [Soderkvist 1990] and the silicon-based rings including the spring suspension for the vibrating-shell systems developed by British Aerospace System and Equipment [Hopkin 1997]. Such components were then mounted on appropriate carriers. Concurrently, typical MEMS-technology-based devices formed and bonded on wafer level were proposed around 1986 by the Charles Draper Laboratory [Greiff et al. 1991] and demonstrated first in 1991 with a bulk-micromachined tuning-fork gyroscope and a little bit later, in 1993, with a silicon-on-glass tuning-fork gyroscope [Weinberg et al. 1994]. In 1998, researchers at the University of Michigan demonstrated a polysilicon-ring gyroscope produced with a trench-refill technology [Ayazi and Najafi 1998]. Predecessors of different batch-fabricated gyros that used more exotic technology exist, but these designs could not prevail against products based on technologies that were becoming mainstream MEMS technologies. One of the most interesting of the non-mainstream products was the vibrating-ring gyroscope of the University of Michigan produced using metal electroforming of nickel into a thick polyimide mold on a silicon substrate, which was demonstrated in 1994 [Putty and Najafi 1994].

6

Introduction

There were hundreds of different demonstrators from the University of Berkeley, from Samsung and Murata, from the Hahn-Schickard Gesellschaft– IMIT (Germany) and many others, but only a few were commercially successful. An excellent overview of the emerging MEMS-based inertial sensors can be found in Yazdi et al. [1998], which can be complemented by reading Shkel [2001].

1.2

Applications and market The applications of the classical accelerometers are vibrometry, shock detection, tilt measurement, dynamometry, seismology and other areas related to test and evaluation of devices exposed to inertial forces. Some of these applications coincide with the main application areas of classical gyroscopes – inertial navigation and platform stabilization. MEMS technologies dramatically changed this relatively peaceful picture. Nearly every month a new application is created and checked for commercial attractiveness and realizability.

Some general trends The trends of the inertial-MEMS market’s development are not significantly different from those of the entire MEMS market if one excludes the two leading and very old and stable products – ink-jet heads for printers and write/read heads for magnetic and optical memory disks. These two market segments alone occupied around an estimated 25% of the about 10 Bn $ MEMS market in 2010. MEMS addressed very fragmented markets that have had predominantly low-volume and only a few large to truly high-volume applications. This market is transforming more and more into a high-volume market with steadily expanding size, and, crucially, with a growing number of different applications. Compound annual growth rates (CAGRs) of 5% to 20% are typical for these different applications. At present the number of companies manufacturing MEMS products is around 270 in addition to 150 fab-less companies. Around 90 R&D industrial facilities, which are able to develop prototypes and to perform small-volume production, round off the picture of today’s MEMS community. It should be borne in mind that in the 1990s three to five years were needed to develop new MEMS designs and five to eight years from the prototypes to volume production. In the case of safety-critical applications the time lapse was even longer. Now the overall time from design start to volume production has decreased by a factor of two to three and is reducing further. Consequently, the interest in research and development is enormous and is still growing. It will continue to grow as long as MEMS products penetrate all areas of human activity. Today the distance which has thus far been covered on the way to all-encompassing applications of MEMS and inertial sensors is almost negligible in comparison with the distance still to go.

1.2 Applications and market

7

The inertial-MEMS market Within the MEMS market modern inertial sensors – accelerometers and gyroshave gained a considerable share, exceeding 20% of the expected 12 Bn $ MEMS market in 2011 (∼6 Bn $ in 2009). In 2005 nearly 80% of all applications were related to automotive safety functions such as automatic break systems (ABSs), airbag sensors, rollover sensors, electronic stabilization systems (ESP), and other anti-skid systems as well as to navigation. Starting with 50g accelerometers in airbag safety systems, the next step – the introduction of electronic stabilization control (ESC) by Bosch and Systron Donner in 1994 – was significantly accelerated by the disastrous elk test of the newly invented Mercedes-Benz A-Class in 1997. ESC had to rescue the reputation of the brand name of one of the leading makers of high-quality cars. In ESC, yaw-rate gyroscopes and low-g sensors are usually the decisive information sources for controlling the finely allotted brake forces on the different wheels to avoid accidents. Rollover protection, highly sophisticated front and side airbags combined with safety-belt control, and suspension control, especially for trucks, and many other applications have not only expanded the market but also forced the development of new low-g accelerometers, of gyroscopes sensitive in different axes and with different accuracy levels, and – importantly – the co-integration of two or more sensors in one package or even on one chip. Remarkably, the market shares of gyroscopes and accelerometers nearly equalized around 2005. New accelerometer applications mainly in the consumer market have shifted the relation to a stable 40% share of gyroscopes within the inertialMEMS market. After 2012–13 the picture may change because the killer applications in the consumer market, which with an expected 1 Bn $ contribution will then be at least comparable in size to the other segments, have not yet been defined and are hard to predict. Today, large companies are fighting for their share in the market of automotive inertial MEMS sensors. Among these, the world’s largest MEMS-sensor manufacturer is Bosch in Germany, which has put considerable effort into the production of MEMS gyroscopes for automotive applications. However, British Aerospace Equipment Silicon Sensor Systems (BAE SSS), BEI Systron Donner, DelphiDelco, Murata, Matsushita, and Samsung have also long been very successful and delivered many millions of gyros to component manufacturers in the car industry. Within the accelerometer market the Norwegian company Sensonor, with its 50g accelerometers, and the Swedish VTI Technolgies, which was long the leader in the low-g accelerometer market, have made surprisingly dominant contributions to the emerging killer applications within the automotive industry. However, the number of high-volume players in the inertial sensor market is quite limited, encompassing Bosch, Analog Devices, Freescale, and almost a dozen others. Besides the automotive industry, the remainder of the market is dictated by consumer applications, which in recent years have shown dramatic growth, with CAGR 25%–30%. Accepted and growing applications are related to the use of

8

Introduction

inertial sensors in hand-held cameras for picture stabilization, in personal computers for hard-disk protection against mechanical shocks, in pedometers for motion sensing, and in more exotic products such as the two-wheel Human Transporter of Segway. Probably one of the most interesting applications is the motion sensing integrated into mobile phones, game controllers, toys and other human–machine interfaces. The Nintendo Wii’s motion-sensing remote control has attracted the broad interest of the public to inertial MEMS. The consumer and information technology (IT) sector has increased from about 10% in 2005 to about a 45% share in 2009/10. Within the consumer market the companies Analog Devices, Kionix, ST Microelectronics, and MEMSIC are the dominating players within the accelerometer business, while Panasonic and Murata have long led the gyro market. However, every year new companies are entering the market for inertial sensors, and the established manufacturers of inertial MEMS as well as newcomers are focusing their attention more and more on the consumer market. New systems such as very small and cheap one-axis sensors as well as more highly integrated multidimensional accelerometers and gyroscopes are now on the market. ST Microelectronics introduced in 2009 a three-axis high-performance gyroscope, whereas VTI announced a three-dimensional (3D) accelerometer combined with a 1D gyroscope [MEMSentry 2009]. Sensordynamics introduced in 2008 a combined one-axis accelerometer and one-axis gyroscope [Micronews 2009] and announced a three-axis accelerometer plus one-axis gyro combination. The race into the world of multi-axis inertial sensors, including completely integrated (six-axis) inertial measurement units (IMUs), is fully under way. An IMU measures the accelerations and rotation rates on all three axes and, in principle, represents the (functionally) ultimate inertial sensor. Applications are numerous, ranging from medical 3D gesture and motion recognition via human– machine interfaces (HMIs) for game controllers and mobile phones to personal navigation systems. The high-volume application of inertial MEMS in the automotive and consumer markets was for a long time in some contrast with aeronautic and defense applications. Here the high added value guaranteed a large benefit for the customer, and, consequently, the low quantities have been to some extent compensated for by good prices. Remarkably, within the last few years many inertialMEMS products developed for the aeronautic and defense sector have reached commercialization. This sector has doubled within the last five years, approaching now around 50% of the size of the automotive segment. As with aeronautics and defense markets, the industrial and medical segments of inertial MEMS’ applications are also at the stage of entry to high-volume markets. Applications such as activity monitoring in pacemakers have considerable dissemination throughout the world. It can be expected that the inertial control of robots and machine parts with more than one degree of freedom (DOF) will achieve a quite broad distribution.

1.3 The ingredients of inertial MEMS

1.3

9

The ingredients of inertial MEMS Inertial sensors convert the inertial forces caused by the input acceleration or rate signal into some physical changes such as deflection of masses or deviations of stresses, which then are captured by a corresponding transducer and transformed into an electrical signal. The electrical signal is subjected to some estimation procedures such as linear or nonlinear filtering in order to derive an estimate of the input signal. The final output represents the calibrated value of the measured acceleration or rate. Of course, not only electrical output signals are feasible; however, only in such exceptional cases as for instance in highly explosive environments are other forms of the output, such as optical signals, used. Within this book only sensors with electrical output signals are treated. Accelerations and angular velocities are vectorial signals possessing absolute values and orientations. If only one component of the vector should be measured the sensor is denoted 1D or one-axis. If two or all three components of the acceleration or the rate signal should be captured, the sensor is called a 2D or 3D accelerometer, or a rate sensor. Today a MEMS-based accelerometer or gyroscope is understood as a complete product that is packaged, calibrated, and tested, and has to be delivered to the customer, who wants to integrate this component with minimal effort into a higher-level measurement or control system. The level of accuracy required depends on the application. The environmental conditions for the integration of inertial MEMS at the customer site may be also quite different and usually are divided into classes with respect to the applicable temperature range and the exposure to humidity and aggressive chemicals as well as to vibration and shocks. The length of the lifetime, the reliability, and the safety against failures during operation may vary significantly, and to a large extent determine the product’s price. Consequently, orders of magnitude may separate the complexity and the price of an inertial MEMS for different applications even if the underlying sensor principles are identical. In order to make the following explanations and terminology systematic, it is meaningful to sketch a general structure of a sensor system with emphasis on inertial sensors. The system consists of not only the sensor itself but also additional components such as transducers and electronics. In Fig. 1.2 a very crude representation of the whole system is shown. The intrinsic sensor transforms the input signal – acceleration or rate – into a physical objective, which can be gathered by the transducer and transformed into an electrical signal. The sensor and transducer are subject to interactions with the package. In inertial MEMS the main interactions are stress and heat transfer. Environmental factors may be transferred via the package to the sensor and transducer, changing their behavior. The electronic part consists of an input stage that amplifies the transformed signal into a conveniently manageable form. The electronics also may generate

10

Introduction

Electronics Input signal

Sensor

Transducer

Output signal Input stage

Actuator

Figure 1.2 The general architecture of a sensor system.

excitation and control signals that are necessary for bias setting and for operational conditioning. Actuating stimuli may be used, for instance, for feedback control, as well as for test signals. In practice the borders between these generic blocks are quite fuzzy. The transducer and the electronic input stage often form an indivisible object where the input stage provides the necessary biasing and excitation for the transducer and, vice versa, parts of the input stage may act as components of the transducer. The transducer is often directly integrated into the intrinsic sensor. In general, the application of a certain transducer principle usually entails not only the choice of a certain transducer element but also the adaption of the sensor and the electronics. Nevertheless, the generic structure shown allows us to systematize the understanding of the main interactions between the components.

r Feedback control is beneficial with respect to linearity and optimization of transfer characteristics. However, it requires actuators and, thus, additional effort. Therefore, not all sensors have feedback components and, where possible, sensors operate in an open-loop mode. r The system performance is to a large extent determined by noise and disturbances. Typical noise sources exist within inertial MEMS. So, the intrinsic inertial sensor exhibits mechanical noise caused mainly by friction with the usually gaseous environment. For not-too-ambitious performance targets the dominant noise stems from the second noise source – the transducer and electronic stages, and here mainly from the electronic input stage. In accord with the system structure illustrated, the various basic sensor principles, the transducer mechanisms, the corresponding governing models, and, last but not least, the typical error sources and performance parameters of the overall system will be covered by this book. The seemingly central role of the sensor principle within inertial MEMS quickly dissolves when one looks at the many stages of creating the final product.

r A working environment for the sensor is usually established by the so-called zero-level packaging.

r The sensor signals must be acquired and processed.

References

11

r In many cases, as for instance in all vibratory gyroscopes, the sensor has to be driven in a primary movement that must be controlled to high accuracy.

r For high-performance applications, on-line-monitoring functions are integrated.

r The output signals for the external communication – including, if applicable, the results of in-build tests – and monitoring must be formed.

r The sensor and the signal-processing unit have to be packaged and bonded to connections with the external pins or solder balls of the (first-level) package.

r The whole product must be calibrated, which requires special equipment in order to create the necessary accelerations or rate signals with high accuracy. Selected representative functions are tested, usually within the entire temperature range of application. r Last but not least, more or less comprehensive qualification procedures must be passed before production may be initiated. Similarly to most other MEMS applications, the key for success within the area of inertial MEMS is the comprehensive development and integration of all the mentioned components necessary for these multidisciplinary products: MEMS technologies, first- and second-level packaging, application analysis, system design considering the specific effects of technology tolerances and forces in a micro- to nano-scale environment, test methods for high-volume production, and, finally, reliability and lifetime guarantees on an unprecedented scale, especially for automotive and medical applications. Huge investments were necessary to build up high-volume fabs and to run a high-yield production. Partial solutions have reached some maturity. For instance, selected surface and bulk micromachining technologies of leading manufacturers and of R&D facilities have become something like standard processes and are free to be used by the scientific community for prototype development. However, most of the subareas are still in a phase of rapid growth and improvement. Correspondingly, product cycles are changing rather quickly despite the stringent stability requirements of the automotive industry. The third generation of acceleration sensors and gyroscopes is now on the market, the next generation ante portas.

References Analog Devices (1993). ADXL50 – monolithic accelerometer with signal conditioning. Data sheet, Analog Devices, Norwood, MA. Ayazi, F. and Najafi, K. (1998). Design and fabrication of a high performance polysilicon vibrating ring gyroscope, in Proceedings of the IEEE Micro Electro Mechanical Systems Workshop (MEMS ’98), Heidelberg, pp. 621–626. Greiff, P., Boxenhorn, B., King, T., and Niles, L. (1991). Silicon monolithic micromechanical gyroscope, in Proceedings of the IEEE 1991 International

12

Introduction

Conference on Solid State Sensors and Actuators, San Francisco, pp. 966– 968. Hopkin, I. (1997). Performance and design of a silicon micromachined gyro, in Proceedings of the Symposium on Gyro Technology, Stuttgart, pp. 1.0–1.10. Hunt, G. W. and Hobbs, A. E. W. (1964). Development of an accurate tuningfork gyroscope, in Symposium on Gyros, Proceedings of the Institute of Mechanical Engineers (London), 1964–65, 179(3 E). McCullom, B. and Peters, O. (1924). A new electric telemeter. Technology Papers of the National Bureau of Standards, 17(247). MEMSentry (2009). ST’s new 3-axis analog gyroscope. MEMSentry, December 2009, 47:5. Micronews (2009). Inside the first combination inertial sensor. Micronews, May 2009, 80:7. Putty, M. and Najafi, K. (1994). A micromachined vibrating ring gyroscope, in Technical Digest Solid-State Sensor Actuator Workshop, Hilton Head Island, SC, pp. 213–220. Quick, W. H. (1964). Theory of the vibrating string as an angular motion sensor. Transactions of the ASME Journal of Applied Mechanics, 523–534. Roylance, L. M. and Angell, J. A. (1979). A batch-fabricated silicon accelerometer. IEEE Trans. Electron Devices, 26:1911–1917. Schell, B. (2005). 100 Years of Anschuetz gyro compasses – 100 years of innovations in nautical technology, in Symposium Gyro Technology 2005, Stuttgart, pp. 1–20. Shkel, A. M. (2001). Micromachined gyroscopes: challenges, design solutions, and opportunities. Proceedings of the SPIE, 4334:74–85. Soderkvist, J. (1990). Design of solid-state gyroscopic sensor made of quartz. Sensors and Actuators A, 21/23:293–296. Weinberg, M., Bernstein, J., Cho, S. et al. (1994). A micromachined comb-drive tuning fork gyroscope for commercial applications. Proceedings of the Sensor Expo, Cleveland, OH, pp. 187–193. Yazdi, N., Ayazy, F., and Najafi, K. (1998). Micromachined inertial sensors. Proceedings of the IEEE, 86(8):1640–1659.

2

Transducers

The basic building blocks of inertial MEMS are sensing elements to acquire the reaction of the measurement system, actuators to excite the mechanical system, and other components of the mechanical system such as proof masses, beams, springs, and suspensions. The properties and dimensions of all these components are decisive for their application within a given sensor system. Silicon, with its outstanding mechanical properties (e.g. Gad-el-Hak [2002] and Franssila [2004]), plays a key role for building these blocks. It is as strong as steel. It is ideally elastic, not exhibiting plastic deformations up to the yield point as do most metals. The elastic modulus E may be as large as 190 GPa, depending on crystal orientation, and the yield strength is about 7 GPa. With appropriate doping (boron, phosphorus) concentration the resistivity can be changed by eight orders of magnitude between 10−4 and 104 Ω cm; thus structures such as conductive plates or comb fingers can be manufactured. The density of silicon is 2300 kg/m3 , the thermal conductivity is 1.57 W/(cm K), and the thermal expansion coefficient CT = 2.33 × 10−6 /K [Kovacs 1998]. Polycrystalline silicon, or “polysilicon” for short, which is made up of small single-crystal domains of silicon (grains), has similar properties and is the most popular building material for surface-micromachined devices (SMMs). Monocrystalline silicon is anisotropic and exhibits pronounced orientationdependent properties such as piezoresistivity and piezo-Hall effects. The mechanical and electrical properties of polysilicon are slightly inferior to those of monocrystalline silicon but are very similar to those of an isotropic material; thus, the material is much easier to handle than monocrystalline silicon. While the main focus of this chapter is on transducers, some material properties that are relevant to both transducers and general mechanical components of inertial MEMS will be summed up first in order to better understand the relevant transducer mechanisms.

2.1

Anisotropic material properties, tensors, and rotations Anisotropic material properties play a prominent role in inertial MEMS. A suitable mathematical description of these properties is provided by the concepts of stress and strain. Considering the fundamental role of these terms in the

14

Transducers

(a)

(b)

Figure 2.1 Axial and shear forces acting on a body. (a) Beam elongation under axial force. (b) Cube deformation under shear load.

engineering literature, a short introduction to the matter is given. Readers familiar with these concepts can skip this section.

2.1.1

Stress, strain, and piezoresistivity Hooke’s law The concept of stress and strain was developed in order to describe the interrelation between external forces, acting on a solid body, and the internal forces between the volume elements of the body, which cause the deformations. Another objective was to reflect the effects a strained condition can induce on material properties such as resistivity. Stress is a force distribution and strain is the distribution of deformations.

Normal stresses The stress–strain relation is a generalization of Hooke’s law, which, when applied to a metal bar orientated along the x-axis and with length lx0 and cross-section Ax , can be expressed as (see Fig. 2.1(a)): Fx = EAx

∆lx lx0

or

σx = Eεx ,

(2.1)

where σx is the axial stress, i.e. the applied axial force per unit area that is normal to the surface, σx =

Fx , Ax

(2.2)

and εx is the (normal) strain, i.e. the dimensionless relative elongation εx =

lx1 − lx0 ∆lx = . lx0 lx0

(2.3)

E is the elastic modulus or Young modulus of the given material. It expresses the material’s resistance to elastic deformations. A large elastic modulus characterizes a stiff material and a small E characterizes a highly elastic object. If

2.1 Anisotropic material properties

15

the applied forces cause an elongation of the object, the resulting stress is called tensile; in the opposite case the stress is called compressive. For ordinary materials, the elongation of the bar is accompanied by a reduction of its lateral dimensions (contraction of the bar), i.e. by a transverse strain εy y = εz z =

∆ly ∆lz = . ly lz

(2.4)

The ratio of the lateral to the axial strain εt /εa is Poisson’s ratio ν=−

εz z εy y =− . εx εx

(2.5)

Within the elastic range of the medium, ν is constant, and for most materials its value is between 0.2 and 0.35. An upper limit of 0.5 exists and pertains to ideally incompressible media. For an isotropic elastic object, which is subject to applied stresses along all three orthogonal axes, the strain in any of the three directions is the sum of the elongation in that direction minus the contraction caused by the stresses in the two remaining orthogonal axes, for instance 1 [σx − ν(σy + σz )]; E 1 εy = [σy − ν(σx + σz )]; E 1 εz = [σz − ν(σx + σy )]; E

εx =

E [(1 − ν)εx + ν(εy + εz )], (1 + ν)(1 − 2ν) E σy = [(1 − ν)εy + ν(εx + εz )], (1 + ν)(1 − 2ν) E σz = [(1 − ν)εz + ν(εx + εy )]. (1 + ν)(1 − 2ν) (2.6)

σx =

On the right-hand side the equation is resolved for the case in which the strains are known. For “non-standard” materials such as foams and synthetic compounds Poisson’s ratio can take negative values; however, such materials are not considered in this book.

Shear stresses Besides the normal stresses, which are perpendicular to the body’s surfaces, shear stresses exist, particularly in twisted bodies. A shear force acts parallel to a given surface as shown in Fig. 2.1(b). For instance, Fz x denotes a force in the x-direction acting parallel to a surface whose normal vector is oriented along the z-direction. The shear stress is the shear force per unit area. Shear forces create shear strain, i.e. deformations along the direction of the shear force. In Fig. 2.1(b) the shear strain εz x is the ratio between the displacement ∆lx and the height of the cube lz . Shear stress and shear strain are related to each other by the shear modulus G: Fz x = GAz

∆lx lz

or

σz x = Gεz x .

(2.7)

16

Transducers

(a)

(b)

Figure 2.2 Torsion of a hollow cylinder. (a) A twisted cylinder. (b) Cross-sections of

untwisted and twisted hollow cylinders.

The basic experiment to determine the relation between shear stress and shear strain is shown in Fig. 2.2(a). A homogeneous hollow cylinder with thin walls is exposed to a torsional moment at both sides, and the resulting shear angle γ is measured. This angle determines the torsion or twist angle ϕ of the front face of the cylinder according to the obvious relation L , (2.8) R where L is the length of the cylinder, t is the thickness of the wall, and R is the radius of the cylinder. Assuming that the stress across the cross-section of the cylinder, σt , which is orientated in the tangential direction, is constant, the moment created is Rσt A, where A is the area of the cross-section. This moment is balanced by the applied Mt , ϕ=γ

Mt = 2πR2 tσt .

(2.9)

By measuring Mt and the shear angle γ, the ratio σt /γ can be determined. Within the limits of elastic deformation without warping it has been found that σt = Gγ.

(2.10)

The shear angle γ is the deformation angle of the elementary rectangles of the cylinder as shown in Fig. 2.2(b). For the marked rectangle (in the middle of the bottom part of the cylinder) the stress lies in the plane x = constant and is directed along the y-axis: σt = σxy .1 The shear angle γ is exactly double the shear strain γ = 2εxy . Using this definition and applying the results to an arbitrary 1

The natural coordinate system is here a cylindrical system in which σ t = σ x , ϕ .

2.1 Anisotropic material properties

17

volume element of an isotropic body, the shear stress–strain relation can be written in the more common notation εxy =

σxy ; 2G

εxz =

σxz ; 2G

εy z =

σy z , 2G

(2.11)

which together with Eq. (2.6) represents the generalized form of Hook’s law. It is valid for small deformations, where the superposition principle remains in force, i.e. the resulting deformation caused by different forces can be substituted by the sum of the deformations caused by these forces. A deeper insight into the fundamentals of the theory of elasticity reveals that for isotropic materials the original tension test of Hooke without any additional torsion test is sufficient to derive not only the deformations caused by normal stresses, but also the relation between shear forces and corresponding deformations. This leads to a well-defined interrelation between Young and shear moduli presented below (Eq. (2.36)). However, the torsion test is illustrative with respect to torsional components of inertial sensors and allows us to introduce the basic ideas for the analysis of torsional springs.

Stress and strain tensors Usually, objects subject to external forces are not built as simply as the ones considered so far, i.e. bars or torsional cylinders. They may have a complicated geometry and may be exposed to a combination of normal and shear stresses. To analyze the reaction of the body, the external forces have to be related to the internal forces and deformations at different inner points. Consequently, the concepts of stress and strain must be brought down to infinitesimal length scales, where balance of force and momentum is applied on each face of such an entity. On the basis of such a representation, partial differential equations for the relation among stress, strain, and global and local forces within the body can be derived; as usual, they must be complemented by appropriate boundary conditions. These equations can be numerically simulated or – in some exceptional cases – solved analytically.

The stress tensor To derive an infinitesimal description of the stress or strain distribution, infinitesimal volume elements of the medium, pertaining to a Cartesian coordinate frame, are considered; for such coordinates, the volume elements are cubes, as can be seen in Fig. 2.3 [Popov 1968, 1999], The faces of the cube have areas dA. The medium is supposed to be in a state of mechanical equilibrium; this implies that forces and moments either originating from interaction with neighboring volume elements or being imparted by external sources must be balanced at each face of each volume element. At the boundary of the body, external load forces and reaction forces of the supports have to be added.

18

Transducers

Figure 2.3 The stress tensor.

For convenience the x-, y-, and z-axes are often defined by the vectors e¯1 , e¯2 , and e¯3 , and the corresponding coordinates are x ⇒ x1 ,

y ⇒ x2 ,

z ⇒ x3 .

(2.12)

A force or a stress component acting on the face with normal vector e¯i and oriented along direction e¯j is termed ∆Fij . The stress is the limit σij = ∆Fij /dAi |dA i →0 . The cube under consideration is in mechanical equilibrium with adjacent cubes so that the total force exerted on a face of the cube is compensated for by the corresponding force of the adjacent cube. The stress at a face σ¯i = ∆F¯i /dA is decomposed into the orthogonal components σii , σij , and σik . The normal stress σii is caused by the force perpendicular to the face and tends to compress or elongate the cube. Shear stresses act parallel to the face and tend to shift the corner points of the face, thus distorting the cube into a rhomboid. Mechanical equilibrium implies that the total force and the total torque acting on the cube must be zero. This has two implications. 1. The normal stresses at opposite faces have the same values but opposite directions σii′ = −σii . 2. Zero total torque about any axis requires that the stress components must satisfy a symmetry condition, σij = σj i .

(2.13)

Stress components at opposite faces have opposite orientations by definition. A complete description of the state of stress at point P (x, y, z) is then given by nine quantities σij (x, y, z), and is subject to the symmetry condition (2.13), which reduces the number of independent quantities to six. This collection of quantities σij (x, y, z) at P (x, y, z) defines the coordinates of the so-called stress tensor at P (x, y, z). The stress tensor represented within the given coordinate system can be represented by the coordinate matrix σ with nine components

2.1 Anisotropic material properties

19

Table 2.1. Index transfer by Voigt’s notation

Matrix indices Vector index

11 1

22 2

33 3

23, 32 4

∂v dy ∂y

13, 31 5

12, 21 6

∂u dy ∂y ∂u dx ∂x

β x, y

α

∂u ∂v ; ∂x ∂y

1 − 2 2

(

x, y

∂v dx ∂x

) = 12 (α + β )

Figure 2.4 Deformation of an elementary cube.

σij ; i, j = 1, 2, 3: σ = {σij }.

(2.14)

In this book, tensor quantities are usually represented in a given coordinate system. As a consequence, matrix algebra can be used to express tensor relations. Since the symmetry property (2.13) reduces the number of independent stress components from nine to six, it is convenient to express the stress status not by a 3 × 3 stress matrix but by a stress vector σ ¯ using Voigt’s notation, which is based on the equivalence of indices (shown in Table 2.1): σ ¯ T = (σ1 = σ11 , σ2 = σ22 , σ3 = σ33 , σ4 = σ23 , σ5 = σ13 , σ6 = σ12 ).

(2.15)

The superscript T denotes the transpose of a vector or a matrix.

The strain tensor The strain tensor can be introduced in the same way as the stress tensor. Again, a non-deformed cube is considered at point P (x, y, z) (see Fig. 2.4). The deformation of the cube is described by the displacements each of its corner points

20

Transducers

is undergoing. Thus, a¡ point with coordinates (x, y,¢z) in the undeformed state will take the location u(x, y, z), v(x, y, z), w(x, y, z) after deformation. A point (x + ∆x, y + ∆y, z + ∆z) nearby will move to        ∂u/∂x ∂u/∂y ∂u/∂z ∆x u u(x + ∆x, y + ∆y, z + ∆z)  v(x + ∆x, y + ∆y, z + ∆z)  =  v  + ∂v/∂x ∂v/∂y ∂v/∂z   ∆y  ∂w/∂x ∂w/∂y ∂w/∂z ∆z w w(x + ∆x, y + ∆y, z + ∆z)     u ∆u =  v  +  ∆v  . (2.16) w ∆w

Deformations are supposed to be small so that the principle of superposition holds. From Eq. (2.16) it follows that the elongation of the face at z = 0 in the xdirection is (∂u/∂x)dx, and accordingly that in the y-direction is (∂v/∂y)dy. The diagonal strain components are now defined as the relative elongations in the respective directions: εxx = ε11 =

∂u ; ∂x

εy y = ε22 =

∂v ; ∂y

εz z = ε33 =

∂w . ∂z

(2.17)

The shear strain characterizes the deformation of the quadratic faces into parallelograms. It is defined as the average of the two angles β ∼ = ∂u/∂y and α∼ = ∂v/∂x of the resulting parallelogram (see Fig. 2.4), i.e. as the average of the angular distortions of the z–x-plane and the z–y-plane: µ ¶ 1 1 ∂u ∂v εxy = ε12 = (α + β) = + . (2.18) 2 2 ∂y ∂x Accordingly, for the other faces µ ¶ 1 ∂u ∂w + , 2 ∂z ∂x µ ¶ 1 ∂v ∂w = + . 2 ∂z ∂y

εxz = ε13 =

(2.19)

εy z = ε23

(2.20)

Again, the symmetry relation εij = εj i

for

i, j = 1, 2, 3

(2.21)

holds because for non-rotating solid bodies (i.e. for bodies not exposed to additional centrifugal, centripetal or Coriolis forces) ∂u ∂v ∂u ∂w ∂w ∂v − = − = − = 0. ∂y ∂x ∂z ∂x ∂y ∂z

(2.22)

In other words, without rotation, εxy and εy x are the average deformation angles for the same face and thus identical. Consequently, in Eq. (2.16) the mixed partial derivatives can be substituted by the average shear stresses, e.g. µ ¶ ∂u 1 ∂u ∂v = + = εxy . ∂y 2 ∂y ∂x

2.1 Anisotropic material properties

Thus, the deformation equation (2.16) can be rewritten         ∆x ∆x ε1 ε6 ε5 ∆u εxx εxy εxz  ∆v  =  εxy εy y εy z   ∆y  =  ε6 ε2 ε4   ∆y  . ∆z ε5 ε4 ε3 ∆z εxz εy z εz z ∆w

21

(2.23)

The 3 × 3 strain matrix ε = {εij } is the representation of the strain tensor and – owing to the symmetry relation – can be reduced to the six-dimensional strain vector using again Voigt’s index correspondence (Table 2.1) ε¯T = (ε1 , ε2 , ε3 , ε4 , ε5 , ε6 ).

(2.24)

In engineering disciplines like structural mechanics the shear strain components are defined as twice the values used in physics and correspond to the values used in the torsion test of the section “Shear stresses”: γxy = 2εxy = 2ε6 ;

γxz = 2εxz = 2ε5 ;

γy z = 2εy z = 2ε4 .

(2.25)

The stress–strain relation for anisotropic materials Assuming small deformations, the general relation between the strain and stress vectors (2.15) and (2.24) can be described by the linear equation σ ¯ = E · ε¯ + α ¯ ∆T,

(2.26)

where E = {Eij } is the elasticity matrix with 6 × 6 coefficients. The first contribution to the stress derives from the strain, but in the case of temperature differences ∆T a temperature-induced stress term α ¯ ∆T must be added. The vector α ¯ represents the thermal expansion coefficients α ¯ T = (αxx , αy y , αz z , αy z , αxz , αxy ).

(2.27)

The incremental change of the elastic energy δU of a unit cube is given by the sum of the deformation work 2 δU = Fxx δεxx + Fy y δεy y + Fz z δεz z + Fy z δεy z + Fxz δεxz + Fxy δεxy . (2.28) The forces are given by the stress components Fij = σij dA. Owing to the identity of the mixed partial derivatives 2 ∂ 2 U/∂εij ∂εk l = −∂Fij /∂εk l = −∂Fk l /∂εij

(2.29)

the matrix of elastic moduli is symmetric. Moreover, it has to be noted that the initial relation between the two 3 × 3 strain and stress tensors of second order should be represented by a tensor of fourth order with 9 × 9 = 81 coefficients. The symmetry relation of stress and strain has reduced this complexity to a tensor with 36 non-zero coefficients or to the equivalent compliance matrix with 6 × 6 coefficients. The last symmetry consideration has decreased this number to 21. In the light of desired analytical expressions such a large number of entities is still difficult to handle. Fortunately, for most of the relevant materials used in MEMS technologies, additional symmetry relations hold, allowing us to reduce the number of non-zero coefficients further.

22

Transducers

Figure 2.5 Silicon orientation on wafer.

The elasticity matrix of silicon As is well known, silicon has a diamond structure based on a cubic crystal lattice. The wafers used in microelectronics are usually cut along the (100) plane for CMOS and bulk micromachined devices.2 As shown in Fig. 2.5 the surface of the wafer is in the (001) plane, and the orientation flat is directed in the [110] direction. This means that a device on the wafer that is orientated perpendicularly to the flat has an angle of 45◦ with the cube orientation axis [100]. If the different rotational and mirror symmetries of a cubic lattice are considered, the number of independent coefficients of the elasticity matrix reduces from 21 to 3. The elasticity matrix for a material under stress, orientated along the [100] axis, is   E11 E12 E12 0 0 0  E E E  12 11 12 0 0 0    0 0 0  E E E (2.30) E =  12 12 11   0 0 0 E44 0 0     0 0 0 0 E44 0  0 0 0 0 0 E44

with only 12 non-zero moduli of elasticity. This structure reflects a general property of a parameter matrix of anisotropic materials with cubic symmetry and holds, for instance, also for the parameters describing the piezoresistivity effect. For stress orientations differing from the [100] axis, the coefficients of the rotated matrix have to be calculated. This will be done in Section 2.1.2. The inverse relation between stress and strain can be described by the compliance matrix, which is the inverse of the matrix of elastic moduli: ε¯ = S · (¯ σ−α ¯ ∆T ). 2

(2.31)

(100) Planes are perpendicular to [100] directions. The set of six equivalent directions is denoted by h100i, and the set of equivalent planes by {100}.

2.1 Anisotropic material properties

S = {sij } has the same structure as the E matrix,   s11 s12 s12 0 0 0 s s s   12 11 12 0 0 0    0 0 0  s s s S =  12 12 11  = E−1 ,  0 0 0 s44 0 0     0 0 0 0 s44 0  0 0 0 0 0 s44

23

(2.32)

with the following coefficients: s11 =

1 ; E

s12 = −

ν ; E

s44 =

1 . 2G

(2.33)

E and ν are the Hookean Young modulus and Poisson ratio. G is the shear modulus or modulus of rigidity, defined by the ratio σij G= , i 6= j. (2.34) 2εij For silicon, the following data, referred to the [100] crystal orientation, are usually quoted [Gad-el-Hak 2002, Brantley 1973]: E=

1 = 131 GPa; s11

ν=−

E11 = 166 GPa;

E12 = 64 GPa;

s12 = 0.28; s11

G=

1 = 80 GPa; 2s44

E44 = 80 GPa.

(2.35)

Despite its significantly lower yield and fracture strength, a mechanically isotropic polysilicon layer has elastic properties similar those of to bulk silicon. The Young modulus is usually specified as Ep oly = 160 MPa, and the Poisson ratio νp oly = 0.23; however, the spread within materials with different grain structures may be quite significant, and also the spread within different dies need not be negligible. To get a feeling for the orders of magnitude, a very simple example is considered. A silicon cube of dimensions 1 cm × 1 cm × 1 cm is loaded with 1000 kg on the top face. The cube is then compressed by 0.07% or 7 µm, and the lateral extension is 0.02% or 2µm.

Elasticity of isotropic materials For isotropic materials the structure of Eqs. (2.26) and (2.31) does not change. However, G is no longer an independent parameter, but given by the relation [Macke 1962, Chou and Pagano 1967] G=

E . 2(1 + ν)

(2.36)

This equation follows from the anisotropy definition αanisotropy =

2E44 , E11 − E12

(2.37)

24

Transducers

which for isotropic materials (αanisotropy = 1) fixes the interrelation among E11 , E12 , and E44 . It can also be derived from the identity of the principal stress and strain axes in isotropic bodies.3

Elastic strain energy If stress and strain within a material are known, it is often convenient to represent the status by the elastic strain energy Uelastic within a given volume V . Equilibrium states can be found by minimizing the energy, forces can be derived as gradients of the energy, and so on. The elastic energy Uelastic is the accumulated work for the generation of all displacements, i.e. the sum of all displacements multiplied by the forces needed for this displacement: Z Z Z 1X Uelastic = εij σij dx dy dz 2 i,j V Z 1 = dV (σx εx + σy εy + σz εz + 2σxy εxy + 2σxz εxz + 2σy z εy z ). (2.38) 2 V For isotropic materials the application of Hooke’s law yields Z n1 1 Uelastic = dV [σ 2 + σy2 + σz2 − 2ν(σx σy + σx σz + σy σz )] 2 V E x o 1 2 2 + (σxy + σxz + σy2 z ) . G

(2.39)

The piezoresistance of silicon Piezoresistors are excellent stress transducers and have found broad application within inertial MEMS. They change their resistance depending on the applied stress. ¯ and The material laws of Maxwell’s equation state that the electrical field E current density J¯ in an isotropic medium relate to each other as ¯ = ρJ¯ E

and

¯ J¯ = σ E

where

σ=

1 . ρ

(2.40)

Here ρ is the resistivity and σ the conductivity of the material. In silicon or other crystals resistivity and conductivity are anisotropic; ρ and σ become tensors of second order, e.g.      Jx ρxx ρxy ρxz Ex ¯ ¯      (2.41) E = ρJ ⇒ Jy  . Ey = ρxy ρy y ρy z Jz ρxz ρy z ρz z Ez

Here the symmetry relations ρij = ρj i , i 6= j, were used [Smith 1958, Nye 1985]. Using again Voigt’s notation ρ1 = ρxx , ρ2 = ρy y , ρ3 = ρz z , ρ4 = ρy z , ρ5 = ρxz , 3

The principal stress (strain) axes are defined by coordinate systems at the point P (x, y, z), for which the shear-stress (shear-strain) components vanish.

2.1 Anisotropic material properties

25

Table 2.2. Piezoresistivity coefficients for high-resistivity silicon in units of (100 GPa)−1

p-Silicon n-Silicon

ρ0 (Ω cm) 7.8 11.7

π11 6.6 −102.2

π12 −1.12 53.4

π44 138.1 −13.6

and ρ6 = ρxy , the resistivity can be expressed as the sum of the resistivity component in the absence of stress plus the changes induced by the stress4        ρ1 ρ0 π11 π12 π12 0 0 0 σ1 ρ  ρ  π π π     2  0  12 11 12 0 0 0   σ2          ρ3   ρ0   π12 π12 π22 0 0 0   σ3  (2.42)   =   + ρ0   .  ρ4   0   0 0 0 π44 0 0   σ4          ρ5   0   0 0 0 0 π44 0   σ5  ρ6 0 0 0 0 0 0 π44 σ6

As before, the symmetries of a cubic crystal with its axis aligned along the h100i axis were taken into consideration. The πij are the coefficients of the piezoresistivity tensor and are usually cited from Smith’s early work [Smith 1954], as shown in Table 2.2. For p-Si the approximation π11 = π12 = 0 is often used, and for n-Si π11 = −2π12 ; π44 = 0. At high temperatures doped piezoresistor structures on bulk silicon are corrupted by leakage currents to the substrate, which limits their applicability. Isolated polysilicon layers eliminate this drawback and are a useful alternative to crystalline silicon. The polysilicon layer always has a random grain distribution in the x–y-plane. There may be preferred z-orientations of the grains. In any case, the piezoresistivity coefficients for a planar piezoresistor with arbitrary orientation have to be calculated by taking the average of the piezo-coefficients over all grains and grain orientations. This task can be simplified considerably by considering only thin resistors with negligible height, as is the case in piezoresistive MEMS transducers (see Section 2.2.1).

2.1.2

Rotation of coordinate systems There are two occasions when the mathematical tools for describing the rotation of coordinate systems are needed.

r The orientation-dependency of stress, strain, and anisotropic material properties. Here, the two coordinate systems have fixed, time-independent orientations. 4

The components of the stress tensor σ i should not be confused with the notation for the conductivity of the isotropic medium σ.

26

Transducers

Figure 2.6 A vector in two different coordinate systems.

r The rotation as source of virtual inertial forces, e.g. for the description of the dynamics of gyroscopes. Here, one coordinate system rotates with respect to another with a certain, non-zero, speed. In this subsection the static transformation rules for vectors and tensors of second order in rotated coordinate systems are considered. In view of dynamic effects this subsection is formulated in a way that allows a natural extension towards rotating coordinate systems as considered in Chapter 8.

Coordinate frames In Fig. 2.6 the vector r¯ is represented in two different coordinate systems ¯x , E ¯y , E ¯z and e¯x , e¯y , e¯z , which have the same origin but are rotated with respect E to each other. More precisely, the two coordinate frames before – ΣE – and after – Σe – rotation are defined by their basis vectors ¯ T = [E ¯1 , E ¯2 , E ¯3 ] ΣE : E

and

Σe :

e¯T = [¯ e1 , e¯2 , e¯3 ],

(2.43)

¯T E ¯j = δij , and satisfy the right-handedness conwhich are orthonormal, e.g. E i T ¯2 × E ¯3 = +1. A¯ · B ¯ = A¯T B ¯ is the scalar or dot product of two ¯ E dition: e.g. E 1 T ¯ ¯ ¯ is their vector prodvectors A = [A1 , A2 , A3 ] and B = [B1 , B2 , B3 ]T . A¯ × B uct. The superscript T denotes a vector or matrix transposition, and δij is the Kronecker symbol. Underlined letters characterize a column arrangement of corresponding items, in Eq. (2.43), of the vectors e¯i . The product of a matrix D with an underlined vector, Du, is interpreted as a usual matrix multiplication irrespective of whether the elements of u are numbers or vectors. The orthonormality of the base yields ¯·E ¯T = E ¯T E ¯ = e¯ · e¯T = e¯T e¯ = I, E where I is the unit matrix.

(2.44)

2.1 Anisotropic material properties

27

¯ T = A¯ ⊗ B ¯ is interpreted as the dyadic product of the vectors A¯ and Here, A¯B ¯ ¯ T = [B1 , B2 , B3 ] is defined by B. The dyadic product of A¯T = [A1 , A2 , A3 ] and B   A1 B1 A1 B2 A1 B3 ¯ =  A2 B1 A2 B2 A2 B3  . (2.45) A¯ ⊗ B A3 B1 A3 B2 A3 B3

The dot sign between the vectors in Eq. (2.44) means that the components of the dyadic product are connected by scalar multiplication. The vector r¯ is seen as the same entity in both coordinate systems and remains unchanged. Thus X X ¯i = re,i e¯i . (2.46) rE ,i E r¯ = i

i

For convenience the column vectors rE and re are introduced: rE = [rE ,1 , rE ,2 , rE ,3 ]T ;

re = [re,1 , re,2 , re,3 ]T .

(2.47)

They represent the vectors of coordinates of r¯ in ΣE and in Σe , respectively, and should not be confused with the actual vector r¯ itself, which can now be represented in a compact form: ¯ T r¯E = e¯T r¯e , r¯ = E

¯ r¯ and r¯e = e¯r¯. r¯E = E

(2.48)

The tensor concept should be briefly evoked. A tensor K of second order is an arrangement of vectors according to the following scheme: o nXX o nXX ¯ T KE E ¯= ¯j = e¯T Ke e¯ = ¯i E ¯j . (2.49) ¯i E K=E KijE E Ke E ij

i

j

i

j

It allows us to express vector relations independently of their representation in different coordinate systems. The corresponding coordinate matrices KE and Ke ¯ and e¯ are defined by the equality (2.49). On applying a tensor in two different E ¯ : r¯ = E ¯ T rE , one gets to a vector r¯, say within the coordinate system E T

T

T

¯ KE E ¯·E ¯ rE = E ¯ KE r E . K · r¯ = E

(2.50)

This means that the application of a tensor to a vector within a given coordinate system transforms the coordinate vector within this system by matrix multiplication with the coordinate matrix of the tensor represented within the same coordinate system.

The rotation tensor The tensor of rotation represented by its coefficient matrix in Σe , namely S = e, rotates the unit vectors of ΣE into the unit vectors of Σe . Strictly speaking, e¯T S¯ the rotation matrix S should be labeled by the superscript E, which will be omitted.

28

Transducers

The rotation is determined by ¯ e¯ = SE

or

e¯i =

X

¯j . Sˆij E

(2.51)

j

T

¯ the rotation By means of right-hand multiplication of the first equation with E matrix S can be explicitly written ¯T . S = e¯ E

(2.52)

Equations (2.48) and (2.52) deliver immediately rE = ST re

and correspondingly

re = SrE .

(2.53)

The coefficients of the matrix S = {Sˆij } follow from the second equation in ¯k , (2.51) after transposition and right-hand multiplication with E ¯ Sˆi,j = e¯T i Ej .

(2.54)

¯ ¯j . The scalar product e¯T ¯i onto E i Ej is the direction cosine, i.e. the projection of e T T −1 T T ¯ ¯ The inverse rotation is, by definition, S = E e¯ = (¯ e E ) = S . Thus, S−1 = ST .

(2.55)

Additionally, it is easy to show that det S = +1. Analogous equations are valid for all tensors of rotation. Since ST S = I, where I is the unit matrix, it follows that X X X 2 Sˆi,k = 1. (2.56) Sˆk2 ,i = ⇒ Sˆk ,i Sˆk ,j = δij k

k

k

This equation may be useful; it reduces the nine directional cosines to the three independent values.

Separation of partial rotations – Euler angles In order to derive the coefficients of the matrix S, the projections according to (2.54) have to be calculated. The most convenient way to do this goes back to Euler. Euler partitioned the whole rotation into three partial rotations, each partial rotation taking place about a well-defined axis by a certain angle. The fact that any rotation can be partitioned into three subsequent rotations shows that the matrix S has only three independent parameters. There are different ways to define the sequence of partial rotations to get a total rotation. Euler introduced the so-called standard Euler angles φ, θ, and ψ for the following (standard) sequence of rotations.

r First, counterclockwise rotation about the z-axis (E¯3 -axis) of the initial coordinate system by the angle φ, transforming the coordinates of a vector endpoint {1} {1} {1} (r1 , r2 , r3 ) into the new coordinates (r1 , r2 , r3 = r3 ). r Second, counterclockwise rotation about the new E¯ {1} -axis by the angle θ. This 2 {1} {1} {1} {2} {2} {1} {2} rotation transforms the coordinates (r1 , r2 , r3 ) into (r1 , r2 = r2 , r3 ).

2.1 Anisotropic material properties

29

Figure 2.7 z-Rotation of the x–y-coordinate system. x′ = x cos φ + y sin φ, y ′ = −x sin φ + y cos φ, z ′ = z, and φ = θ3 .

r Third, counterclockwise rotation about the new E¯ {2} -axis by the angle ψ. This {2}

{2}

3 {2}

rotation transforms the coordinates (r1 , r2 , r3 ) into the final coordinates {2} {3} {3} {3} (r1 = r1′ , r2 = r2′ , r3 = r3 = r3′ ). The described sequence converts the coordinate frames step by step according to {3} {2} {1} ¯ {2} ) is called ¯3 , E ¯ {1} , E ΣE → ΣE → ΣE → ΣE = Σe . The triad of vectors (E 3 2 the Euler basis.5

Bryan angles Throughout this book the Bryan or Cardan angles (e.g. Wittenburg [2008]) are used. They are represented by slightly modified, non-standard Euler angles ψ1 , ψ2 , ψ3 . Their corresponding Euler basis is ¯ {2} ), ¯1 , E ¯ {1} , E (E 3 2

(2.57)

i.e. the first counterclockwise rotation is about the x-axis by ψ1 = ψx , the second about the y-axis of the rotated frame by ψ2 = ψy , and the third about the z-axis of the secondly rotated coordinate system by ψ3 = ψz . In a rotation sequence – starting with rotation about the x-axis – only the third rotation differs from a standard Euler sequence. All coordinate transformations follow the same rules. For example, the third rotation about the z-axis by the angle ψ3 , shown in Fig. 2.7, transforms the coordinates of a given vector endpoint according to    {1}   x cθ 3 sθ 3 0 x  y {1}  =  −sθ 3 cθ 3 0   y  = S3 (ψ3 ). (2.58) {1} 0 0 1 z z Throughout this text the abbreviations sα for sin α and cα for cos α shall be used. Correspondingly, in cases of unambiguity sin ψi = si and cos ψi = ci . The

5

There are different standard Euler bases. So, the sequence of rotations may start from any of the axes. For the second rotation there are always two possibilities. The only requirement is the identity of the first and third rotational matrices.

30

Transducers

following transformations can be derived by a simple cyclic permutation of the axes of the coordinate system, i.e. for the second rotation by the permutation z ⇒ y; y ⇒ x; x ⇒ z. The following sequence of transformations results: r{1} = S1 (ψ1 )r, r{2} = S2 (ψ2 )r{1} ,

(2.59)

r{3} = S3 (ψ3 )r{2} = r′ , where the partial transformations are defined by     c3 s3 0 c2 0 −s2 S3 (ψ3 ) =  −s3 c3 0  ; S2 (ψ2 ) =  0 1 0  ; s2 0 c2 0 0 1

 1 0 0 S1 (ψ1 ) =  0 c1 s1  . 0 −s1 c1 (2.60) 

The overall coordinate transformation can now be written as re = r{3} = S3 (ψ3 )S2 (ψ2 )S1 (ψ1 )rE = SrE ,

(2.61)

with  c3 c2 c3 s2 s1 + s3 c1 −c3 s2 c1 + s3 s1 S(ψ1 , ψ2 , ψ3 ) = {Sˆij } =  −s3 c2 −s3 s2 s1 + c3 c1 s3 s2 c1 + c3 s1  s2 −c2 s1 c2 c1   l1 m1 n1 (2.62) =  l2 m2 n2  . l3 m3 n3 

In contrast to Euler angles, the Bryan angles give one the possibility to linearize the rotation matrix R about the zero angles, while the Euler transformation consists of members of order ψi ψj and higher with missing linear terms. Instead of doubly subscripted coefficients Sˆi,j , it is convenient to introduce the factors li , mi , and ni , for the sake of visual clarity (e.g. Gad-el-Hak [2002]). The reader should be reminded that, according to Eq. (2.56), the sum of the squares along a column or a row is equal to one.

Transformation of tensors of second order Up to now the transformation of vectors between rotated frames has been considered. However, the consideration of elasticity, piezoelectricity, moments of inertia, and so on requires the description of material properties governed by second-order tensors (or matrices) within rotated coordinate systems. Piezoresistivity may serve as a representative example for the transformation of material properties from one coordinate system into another. According to (2.40), ¯ = ρJ, ¯ E

(2.63)

2.1 Anisotropic material properties

31

¯ and current density J¯ within an the relation between the electrical field E anisotropic material, is valid in a given coordinate frame Σ. In the case of orientation-dependent properties such as piezoresistance this coordinate system is inseparably linked to the material orientation, here to the (100) plane and [100] axis of the cubic silicon crystal. If the material which possesses orientationdependent properties is rotated with respect to the applied field or current (or vice versa), the impact of the material properties changes. In order to describe ¯ and J¯ have to be expressed in the rotated coordinate system. these changes, E ¯ ′ = SE ¯ and J¯′ = SJ¯ in the rotated frame Σ′ , Eq. (2.63) can With the fields E be rewritten using property (2.55) as ¯ ′ = SρST J¯′ = ρ′ J¯′ . E

(2.64)

Thus, the coordinate matrix of the resistivity tensor ρ′ in the rotated coordinate system is ρ′ = SρST .

(2.65)

Since the new matrix {ρ′ij } is also symmetric and, thus, has only six known independent components, Voigt’s notation can be applied, transforming ρ′ into the vector representation ρ¯′T = [ρ′1 = ρ′xx , ρ′2 = ρ′y y , ρ′3 = ρ′z z , ρ′4 = ρ′y z , ρ′5 = ρ′xz , ρ′6 = ρ′xy ]. On performing this slightly cumbersome transformation for the matrix equation (2.65), one obtains ρ¯′ = Γ¯ ρ,

(2.66)

where Γ is defined by   2m1 n1 2l1 n1 2l1 m1 n21 m21 l12  l2  2m2 n2 2l2 n2 2l2 m2 n22 m22  2   2  2m3 n3 2l3 n3 2l3 m3 n23 m23  l3  Γ=  . (2.67)  l2 l3 m2 m3 n2 n3 m2 n3 + m3 n2 n2 l3 + n3 l2 m2 l3 + m3 l2     l1 l3 m1 m3 n1 n3 m3 n1 + m1 n3 n3 l1 + n1 l3 m3 l1 + m1 l3  l1 l2 m1 m2 n1 n2 m1 n2 + m2 n1 n1 l2 + n2 l1 m1 l2 + m2 l1 The matrix of piezo-coefficients in Eq. (2.42),  π11 π12 π12 0 0 π π π  12 11 12 0 0  0 0 π π π Π =  12 12 22  0 0 0 π44 0   0 0 0 0 π44 0 0 0 0 0

 0 0    0  , 0   0  π44

(2.68)

represents a tensor of fourth order relating the two tensors of second order, namely the stress tensor σ and the resistivity tensor ρ; more precisely it represents the tensor of resistivity changes ∆ρ under stress: ∆ρ =

ρ − ρ0 I = Πσ. ρ0

(2.69)

32

Transducers

The vector representation of the tensors of second order σ and ρ by σ ¯T = (σ1 , σ2 , σ3 , σ4 , σ5 , σ6 ) and ρ¯ allows us to describe the transformation of the tensor coefficients for a rotation of the coordinate system by a simple matrix relation: ρ¯′ = Γ¯ ρ;

¯ ′ ρ = Γ∆ρ; ¯ ∆

σ ¯ ′ = Γ¯ σ.

(2.70)

On substituting these expressions into Eq. (2.69) it follows that ∆′ ρ = ΓΠΓ−1 σ ′ .

(2.71)

Consequently, the piezoresistive tensor in the rotated coordinate system is given by the matrix Π′ = ΓΠΓ−1 .

(2.72)

Unfortunately, during the transfer of second-order tensors to vectors the nice unitarity property of the rotation tensor S−1 = ST has not been handed over to the corresponding matrix Γ. Γ−1 must be calculated straightforwardly:   2l2 l3 2l1 l3 2l1 l2 l32 l22 l12   m2 m2 m2 2m2 m3 2m1 m3 2m1 m2   3 2 1   2 2 2 2n2 n3 2n1 n3 2n1 m3 n3 n2   n1 −1 Γ = .  m1 n1 m2 n2 m3 n3 m2 n3 + m3 n2 m3 n1 + m1 n3 m1 n2 + m2 n1     l1 n1 l2 n2 l3 n3 n2 l3 + n31 l2 l1 n3 + l3 n1 n1 l2 + n2 l1  l1 m1 l2 m2 l3 m3 m2 l3 + m3 l2 l1 m3 + l3 m1 m1 l2 + m 2 l1 (2.73) In summary, the rotated matrix of piezo-coefficients Π′ no longer has only 12 non-zero components, but, rather, all 36 are now non-vanishing. The calculation of Π′ is simplified by the block structure of Γ as well as of Π. For the most important cases, where the coordinate system is rotated only about the axis perpendicular to the (100) plane of silicon by the angle ψ3 , the angles are ψ2 = 0 and ψ1 = 0, so that ′ {πij }=  π11 − 21 π0 sin2 (2ψ3 ) π11 + 21 π0 sin2 (2ψ3 ) π12  π + 1 π sin2 (2ψ ) π − 1 π sin2 (2ψ ) π 3 11 3 12  11 2 0 2 0  π12 π12 π11    0 0 0   0 0 0 1 − 41 π0 sin(4ψ3 ) π sin(4ψ ) 0 0 3 2

0 0 0 π44 0 0

 0 − 21 π0 sin(4ψ3 )  1 0  2 π0 sin(4ψ3 )  0 0  ,  0 0   π44 0 0 π44 + π0 sin2 (2ψ3 ) (2.74)

where π0 = π11 − π12 − π44 .

(2.75)

2.2 Piezoresistive transducers

2.2

33

Piezoresistive transducers Transducers are the ultimate sensing and actuating elements and as such responsible for the transformation of movement of measuring components into an electrical signal or vice versa. There is a great variety of transducer principles for inertial MEMS such as piezoresistive, capacitive, thermal [Dauderstadt et al. 1995], optical [Hall et al. 2008], electromagnetic [Abbaspour-Sani et al. 1994], piezoelectric [Weinberg 1999, Zou et al. 2008, Yu and Lan 2001, DeVoe and Pisano 2001], tunneling-current sensing [Kubena et al. 1996, Yeh and Najafi 1997], floatinggate field-effect-transistor (FET) sensing [Kniffin et al. 1998], and strain FET sensing [Haronian 1999]. The most often used transducers in inertial MEMS are capacitors shaped as combs or plates. The piezoresistive transducer also plays a significant role, in particular in MEMS accelerometers. Piezoelectric devices are used more infrequently in MEMS but are again gaining popularity. All three principles sometimes are coupled with resonance excitations of the inertial structure, which results in high sensitivity and performance. Thus far the techniques for implementation of the remaining principles are not yet mature enough to find broad application. Within this chapter the focus will be on piezoresistive, piezoelectric, and capacitive transducers, including the actuating mechanisms. Some introductory remarks on other principles will be relegated to Chapter 7.

2.2.1

Piezoresistors A very popular, small-sized, and sensitive transformer of stress into electrical signals is the piezoresistor. Piezoresistivity is the (linear) change of the electrical resistivity under stress; it is not restricted to anisotropic materials such as those considered in Section 2.1.1. Piezoresistivity has a long history, starting with the discovery by Lord Kelvin that some metals increase their resistance under tension. However, there were no practical applications of the phenomenon before the first half of the twentieth century, starting with the development of strain gauges – metal foils that are mounted on elastic carriers, which are then glued onto the surface of the device undergoing deformation. Strain gauges were also the first sensors used for commercially available accelerometers [McCullom and Peters 1924]. Their sensitivity was characterized by the so-called gauge factor GF =

∆R/R ∆R = , ∆L/L εR

(2.76)

where ∆R/R is the relative change of resistance and ε = ∆L/L is the relative change of the length, which is, by definition, the applied strain. This classical definition is based on materials whose properties do not change when they are subjected to stress. This means that changes in resistivity are caused by the elastic deformation of the material.

34

Transducers

In general, considering a block of material (e.g. Fig. 2.1(a)), the resistance is given by l R=ρ , A

(2.77)

where A is the cross-sectional area. On using Eqs. (2.3), (2.4), and (2.5), the change of resistance dR can then be expressed as dR dρ dl dr dρ = + −2 = + εa (1 + 2ν). R ρ l r ρ

(2.78)

Thus the gauge factor is GF =

dρ + (1 + 2ν). εa ρ

(2.79)

The first term describes the change of the material’s specific resistivity under stress, while the second term characterizes the strain-dependent change in resistance owing to the geometric deformation of the material. Materials with predominantly strain-dependent resistivity change are termed strain gauges. Their gauge factors lie between 1.4 and 1.7 (ν = 0.2–0.35). In piezoresistive materials such as the semiconductors silicon and germanium, heterogeneous solids, thin-metal films, metal–insulator–metal (MIM) structures, and Schottky-barrier junctions as well as in superconductors, the term dρ/(εa ρ) sometimes exceeds the strain-gauge effect by orders of magnitude. For example, silicon may reach absolute values of gauge factor up to 200, while polysilicon can have values on the order of 30 [Beeby et al. 2004]. These are among the most widely used materials for piezoresistors in inertial MEMS.

2.2.2

Piezoresistors on silicon Piezoresistors are very popular as transducers in pressure sensors and bulk silicon accelerometers. They can be implemented by selective bulk doping (diffusion, ionimplantation), which creates junction-isolated resistors. In n-doped silicon, stress increases the effective mobility of majority charges (electrons) and consequently reduces the resistivity. The gauge factor becomes negative. For p-doping the opposite holds. Leakage currents through the junction diodes, which increase exponentially with increasing temperature, are an inherent drawback of these devices. Moreover, the sheet resistance of the junction-embedded resistor depends on the local bias across the isolation diode. The temperature coefficient is also very high (around 0.25%/◦ C), exceeding by far the stress-induced resistance changes. Optimized stress-dependent-arrangement techniques for the resistors in order to reduce the temperature impact, plus additional electronic compensation techniques, are required in order to overcome these deficiencies. Sometimes materials with crystalline silicon on an insulator substrate are used in order to reduce the isolation problems, especially at high temperature, namely over 100 ◦ C. However, a more cost-efficient alternative is the use of polysilicon

2.2 Piezoresistive transducers

35

Figure 2.8 The coordinate system and dimensions of a piezoresistor.

to form piezoresistors. It can be deposited onto the SiO2 /Si substrate and has a much lower temperature coefficient than that of doped silicon, typically around 0.05%/◦ C [Middelhoek and Audet 1989]. The design of piezoresistive transducers depends on the construction of the stress-creating mechanism. At a given location, this mechanism should translate the local stress into a measurement quantity (acceleration, pressure), by means of a well-defined functional relationship between the two. In any case, the piezoresistor should be as small as possible in order to help to eliminate the impact of stress gradients.

Thin piezoresistors A very simple model of a piezoresistor is shown in Fig. 2.8. The resistor is usually a thin, rectangular layer of silicon or polysilicon. The resistance change under stress can be derived by measuring the voltage and current changes. In practice, the applied voltage or the feed-in current is kept constant (low-ohmic voltage supply or high-ohmic current source) and the corresponding change of current, or voltage, respectively, is measured. The resistance layer has its own coordinate system (x′ , y ′ , z ′ ) that may be rotated with respect to the reference system of the material. Since H ≪ B and H ≪ L, the vertical field components Ez′ and Jz′ can be set to zero. The fielddensity relation corresponding to Eq. (2.41) simplifies to µ

Ex′ ′ Ey′ ′

¶

=

µ

ρ′1 ρ′6 ρ′6 ρ′2

¶µ

Jx′ ′ Jy′ ′

¶

.

(2.80)

where ρ′1 = ρ′x ′ x ′ , ρ′2 = ρ′y ′ y ′ , and ρ′6 = ρ′x ′ y ′ . If there are no contacts on the sidewalls of the resistor, no current can flow in the y ′ -direction. Thus, Jy′ ′ = 0 and Ex′ ′ = ρ′1 Jx′ ′ ,

(2.81)

Ey′ ′

(2.82)

=

ρ′6 Jy′ ′ .

36

Transducers

With the notation Vl = LEx′ ′ for the voltage difference along the resistor and i′ = i′x ′ = BHJx′ for the current through the resistor, the first equation in (2.81) is transformed into6 i′ = RVl ,

(2.83)

where the resistance R is given by R=

L ′ ρ . BH 1

Since R0 = R = [L/(BH)]ρ0 the resistance change becomes ∆R ρ′ − ρ0 = 1 = ∆′ ρ. R0 ρ0

(2.84)

The stress-induced resistance change should be calculated according to Eq. (2.71). However, since the resistor is thin, the significant stress components σi′ can develop in the surface plane only, so that σ3′ = σ4′ = σ5′ = 0. In this situation the resistance change is described by ∆R ′ ′ ′ = π11 σ1′ + π12 σ2′ + π16 σ6′ . R0

(2.85)

Very often, the in-plane shear stress σ6′ can also be neglected, and only the longitudinal and transverse stresses determine the variation of R.

Temperature compensation in piezoresistors For permanent operating loads, such as vibrations in accelerometers, the maximal stress should be more than two orders of magnitude less than the fracture strength, or the yield point, in order to avoid fatigue effects. That is, stresses larger than some 107 Pa must be avoided. Thus, since the maximal value of the πs is about 10−9 /Pa, the maximal resistance change will be on the order of some few percent. In contrast, the resistance change caused by a temperature variation of, say, 100 ◦ C is about 25% and will mask the much smaller stress-induced changes.

Temperature compensation in p-doped silicon An elegant temperature compensation may be applied on the basis of the large differences among the piezo-coefficients of p-doped silicon (see Table 2.2). In this case the approximation π11 ≈ π12 ≈ 0 holds. If two piezoresistors are arranged in the (100) plane of silicon, one, R1 , parallel to the [110] direction along the resistor axis, and the second one, R2 , perpendicular to it, then their resistance 6

For doped layers the doping profile is not constant. Thus, the specific conductivity changes depending on z ′ . Consequently, ρ0 = ρ0 (z ′ ), and the coefficients π i′ j depend on z ′ . However, the dependency of π i′ j (z ′ ) on z ′ is weak. Neglecting it, the integration of the current density over z ′ leads to slightly modified total currents for the stressed and unstressed situations, which cancels out for the relative resistance change ∆R/R.

2.2 Piezoresistive transducers

37

i

Vout

R − ∆R

V

R + ∆R

R + ∆R

R − ∆R

Figure 2.9 A full Wheatstone Bridge implemented within a silicon plate.

changes are approximately equal but have opposite signs. Indeed, the first resistor is rotated with respect to the [100] axis by ψ3 = 45◦ , the other one by −45◦ . The corresponding piezo-coefficients are, according to Eq. (2.74), 1 π44 ′ = π11 − π0 ≈ π11 ; 2 2

1 π44 ′ = π11 + π0 ≈ − π12 ; 2 2

′ = 0. π16

(2.86)

For R2 the places of normal stresses in Eq. (2.85) are interchanged, hence the resistance changes of the two resistors become ∆R1 π44 = (σ1 − σ2 ), R 2 ∆R2 π44 ∆R1 =− (σ1 − σ2 ) = − . R 2 R

(2.87) (2.88)

A well-known method for the measurement of small resistance changes with opposite signs is the usage of a full Wheatstone bridge as shown in Fig. 2.9. Four identical resistors are arranged in a rectangular layout: two along the [110] orientation and two perpendicular to it. As illustrated in Fig. 2.9, the stressinduced resistivity changes unbalance the bridge by ∆R, resulting in non-zero output signals that are temperature-independent: Vout =

∆R π44 V = (σ1 − σ2 )V R 2

(2.89)

in the case of a voltage source and Vout =

∆R π44 V = (σ1 − σ2 ) iR R 2

(2.90)

in the case of a current source. In practice, the bridge will have to be electronically complemented by small alignment resistors inside the bridge in order to correct the misalignment between the four not completely matching resistors, and by external resistors in order to compensate for the remaining temperature coefficient.

38

Transducers

Temperature compensation in n-doped silicon In n-doped silicon such efficient temperature compensation is not possible, since, according to Table 2.2, the piezo-coefficients π11 , π12 , and π44 are comparable in terms of their absolute values. In this case, two out of the four piezoresistors of the Wheatstone bridge are located on an unstressed piece of the substrate and a Wheatstone half-bridge with approximately half the sensitivity is used.

Hall-like piezoresistors An alternative way to reduce the impact of the large temperature coefficient of R is measurement of the lateral voltage along the resistor and analysis of the ratio of the lateral and longitudinal voltages. Two additional contacts in the middle of the sidewalls of the resistors (in Fig. 2.8, in the y ′ -direction) have to be implemented. The four-contact design forms a sensing element that resembles a Hall sensor and is therefore often called “Hall-like.” The ratio of the longitudinal ′ and lateral fields depends only on the piezo-coefficients πij , not on the strongly temperature-sensitive specific resistance ρ0 . It can be shown that such a “Halllike” element in (100) silicon has maximum sensitivity if it is oriented along the [100] direction (see Fig. 2.5).

2.2.3

Piezoresistors on polysilicon If polysilicon piezoresistors are used, the stress in the polysilicon layer is different from the stress in the underlaying Si substrate. The intermediate insulator layer may also distort the stress transfer. However, the thicknesses of the polysilicon and isolator layers are much smaller than that of the substrate. Thus, the strain in the polysilicon layer is practically the same as the substrate strain. It follows that the stress is transferred according to the stress–strain relations of the two materials: εp olysi = εsi

or

σp olysi = σsi

Ep olysi . Esi

(2.91)

The theoretical calculation of the piezo-coefficients of polysilicon is more complicated than for an undisturbed crystal lattice. Usually the polysilicon is modelled by randomly or partly orientated grains with “isolating” boundary regions in between, and with depletion layers at the grain boundaries. Owing to the quite different technologies used in their fabrication, the grain size, the dominant orientation, and the boundary regions may differ significantly for polysilicon layers of different manufacturers. It is of utmost importance that even for a completely random orientation of grains a significant piezoresistive effect remains [Bao 2000]. It has the same order of magnitude as the piezo-coefficients of doped silicon. Bao [2000] estimates the piezoresistive effect of polysilicon, on average, as 50% lower than that in silicon. The actual values of the πij for a given polysilicon technology should be derived from experimental data.

2.3 Piezoelectric transducers

39

Figure 2.10 The direct piezoelectric effect.

2.3

Piezoelectric transducers Since piezoelectric transducers require special materials or technologies they are not used so frequently as piezoresistors or capacitive transducers. However, they were present at the advent of early industrial MEMS gyroscopes based on a quartz tuning fork (e.g. from BEI/Systron-Donner, Madni et al. [2003]) and occupy a considerable share within inertial MEMS for consumer applications. The technological improvements in integrating piezoelectric devices in MEMS structures led also to a growing interest in the area of 3D accelerometers in particular (i.e. DeVoe and Pisano [2001], Yu et al. [2003], Bohua and Rui [2005], and Huang et al. [2003]), and recently also in the area of 3D angular-rate sensors (i.e. Kagawa et al. [2006] and Wakatsuki et al. [2004]).

2.3.1

The piezoelectric effect Piezoelectric devices produce charges under applied stress (the direct piezoelectric effect) and deform under applied electrical fields (the converse or inverse piezoelectric effect). The direct piezoelectric effect is illustrated in Fig. 2.10. The stressed piezoelectric material generates charges Q at the surfaces, which are collected by the two electrodes, consequently creating a voltage V = Q/C. C is the capacitance between the electrodes. Conversely, if a voltage is applied to the electrodes, the piezoelectric material undergoes a deformation according to the stress/strain produced. Thus, a piezoelectric sensor measures the charges or voltages generated while the actuator transforms the applied voltage into a deformation of the piezoelectric material. The direct piezoelectric effect was discovered in 1880 by the brothers Jacques and Pierre Curie, and the converse effect was predicted in 1881 by Gabriel Lippmann on the basis of thermomechanical considerations, and immediately verified by the Curies.

Natural piezoelectrics The separation of charges under stress and the deformation of materials under electrical fields have the same root cause: the non-centro-symmetric charge

40

Transducers

Figure 2.11 SiO2 in a quartz crystal.

distribution within a crystal lattice. In Fig. 2.11 the mechanism is demonstrated for a natural piezoelectric – the quartz crystal. The silicon oxide in a quartz lattice is arranged in such a way that the four valency electrons of the silicon atom each bind one valency electron of one of the four surrounding oxygen atoms, while the four second valency electrons of those oxygen atoms (not shown) are bound by the neighboring silicon atoms. The geometric structure of this interconnection determines how the quartz will crystallize – in a stable α-quartz configuration or in other possible forms, e.g. at very high temperatures in the β-quartz configuration. The unit cell of the quartz crystal is electrically neutral. If a force is applied, the cell is deformed, and the negative anions are shifted in an asymmetric way with respect to the central Si cation, because the crystal does not exhibit centro-symmetry (corners 1, 3, 5 and 7 are empty). In this case the charge redistribution leads to a polarization: an electrical dipole is created. This example shows that a necessary precondition of piezoelectricity is the noncentro-symmetry of the crystal. Silicon itself is centro-symmetric and thus not piezoelectric. Overall, of the 32 crystal classes, 21 are centro-symmetric and 20 are piezolectric. Ten classes possess a non-vanishing electrical dipole moment associated with their unit cell even in the absence of external stress (pyroelectric polar materials). Despite the numerous applications of quartz as frequency normals, accelerometer sensors etc., piezoelectric materials based on single crystals are not particularly attractive for MEMS applications. Their technological integration into MEMS devices is very difficult. If used, they act as standalone materials like in tuning-fork gyroscopes.

Ferroelectric materials Fortunately, there is another class of piezoelectrics that are MEMS-friendly and may exhibit much higher efficiency. The ferroelectric materials were discovered and developed during the 1940s and quickly found their way into such applications as microphones and loudspeakers in music, sound and ultrasound sensors

2.3 Piezoelectric transducers

41

Figure 2.12 Domain orientation before and after poling.

and generators in material science, medicine, and military uses, high-precision actuators in optics and microscopy, including scanning probe microscopy, highvoltage generators and piezoelectric transformers in consumer goods. The most often used ferroelectric materials are PZT (lead zirconate titanate – Pb[Zrx Ti1−x ]O3 ; 0 < x < 1) and barium titanate (BaTiO3 ). A wide variety of other metal oxides is known. Since they are polar materials, the unit cells of these materials exhibit electrical dipole moments below the Curie temperature TC , which – like in ferromagnetic materials – interact locally and create the so-called Weiss domains: areas with dipoles aligned more or less randomly, as indicated in the left-hand picture of Fig. 2.12 (see, for instance, Draganovic [1998]). This effect is called spontaneous polarization, which vanishes for temperatures higher than the Curie point. For instance, above TC PZT materials change their tetragonal structure, with a displacement of the central cation of the unit cell into the geometry of a cubic cell with a central cation and thus no net dipole moment. Typical Curie temperatures of PZT are around 200 ◦ C. All ferroelectric materials must be poled in order to obtain the piezoelectric behavior. A high voltage is applied at temperatures below the Curie point, which orientates the Weiss domains and in this way polarizes the material. After release, the domains are polarized as indicated in the right-hand picture of Fig. 2.12. The polarization is a typical hysteresis process – see Fig. 2.13. The corresponding strain changes are shown in Fig. 2.13 by dashed lines. It follows that poling always creates the same compression or elongation, independently of the polarity of the poling voltage. After poling, the remanent polarization determines the piezoelectric parameters of the material. The operating point is one of the two remanent polarization points on the hysteresis curve. Accordingly, if the material is compressed, the voltage generated has the same polarity as the poling voltage. If, on the other hand, the applied voltage has the same polarity as the poling voltage, the probe elongates. Depending on the movability of the domain walls, the remanent polarization may deteriorate over the lifetime of the device. The domain-wall motion can be manipulated by doping (e.g. Bassiri-Gharb et al. [2007]). Hard materials with

42

Transducers

Polarization and strain vs. electrical field 1

Remanant polarization

P-Polarization; S-Strain

0.8

0.6

Strain

0.4

0.2

0

-E

E

0

-0.2

0

-0.4

-0.6

-0.8

-1

-1.5

-1

-0.5

0

0.5

1

1.5

E - Electrical field

Figure 2.13 Polarization and strain hysteresis of a typical ferroelectric material.

low wall movability and hysteresis curve close to a rectangular shape have good longevity but lower efficiency. Ferroelectric materials are usually manufactured as ceramics by sintering fine powders. Within the MEMS environment the piezo-ceramic materials can be deposited onto silicon carriers by sputtering, sol–gel deposition (manufacturing of non-metallic, inorganic or hybrid polymer material starting from a colloidal dispersion – the sol), screen-printing processes, and other means. Some polymers also feature high piezoelectric constants. The best-known representative is polyvinylidene fluoride (PVDF). The piezoelectric effect of polymers is caused by the mutual interaction of the long-chain intertwined molecules which attract or repel each other under the impact of an electrical field.

2.3.2

Piezoelectric equations Once polarized, the piezoelectric material properties can be described in linear approximation by an extension of Maxwell’s and Hook’s equations: ¯ = ǫσ¯ E ¯+dσ D ¯ E¯

¯ ¯ + dT E ε¯ = S σ

direct piezoeffect,

(2.92)

inverse piezoeffect.

¯ is the vector of the electrical displacement, the matrix ǫ contains the coeffiD ¯ is the electrical field vector, ε¯ and σ ¯ are the cients of electrical permittivity,7 E strain and stress vectors, respectively, in the usual convention of Voigt’s nota¯ indicate that the corresponding matrices are tion. The superscripts σ ¯ and E ¯ respectively. d is determined for a zero or constant stress σ ¯ or electric field E, 7

For better distinction from the strain components, permittivity is denoted here by the symbol ǫ instead of the more common notation ε.

2.3 Piezoelectric transducers

43

the matrix of direct piezoelectric coefficients or piezoelectric charge constants and describes the polarization caused by unit stress, or the strain generated by unit applied electrical field. As a consequence of using Voigt’s notation for the stress and strain tensors, d is not quadratic but a 6 × 3 matrix. For correct interpretation of the piezoelectricity equations the convention regarding the axes is very important. The dominant axis is given by the direction of poling and denoted by e¯z = e¯3 . The data sheets provided by manufacturers usually contain the most important piezo-coefficients and permittivities. For ferroelectrics based on materials of the ditetragonal-pyramidal class (4mm group) (with four vertical mirror planes and one four-fold polar rotation axis) or of the dihexagonal-pyramidal class (6mm) the equations in (2.92) take the forms   σ1        σ2   0 0 0 0 d15 0   E1 ǫ11 0 0 D1  σ3   D2  =  0 ǫ22 0   E2  +  0 0 0 d24 0 0     (2.93) σ  d31 d32 d33 0 0 0  4  E3 0 0 ǫ33 D3  σ5  σ6 and    E¯ s11 ε1  ε   sE¯  2   21    E¯  ε3   s31  =  ε4   0     ε5   0 ε6 0

¯

sE 12 ¯ sE 22 ¯ sE 32 0 0 0

¯

sE 13 ¯ sE 23 ¯ sE 33 0 0 0

0 0 0 ¯ sE 44 0 0

   0 0 0 σ1     0 0   σ2   0    0 0   σ3   0   +    σ4   0 0 0    ¯   σ5   d15 sE 0 55 ¯ ¯ E E σ6 0 0 2(s11 − s12 )

0 0 0 d24 0 0

 d31 d32  E   1 d33     E2 . 0   E3 0  0 (2.94)

PZT and barium titanate belong to the described crystal classes. In view of the important role these materials play for MEMS applications, in particular for stress/strain sensing, their properties should be considered in more detail.

Piezoelectric sensors in MEMS If the piezoelectric material is used as a sensor, the electrical displacement field created by the direct piezoelectric effect has to be measured [Li Yang 2009, Wang et al. 2004, Iula et al. 1999, Pereyma 2008].

Different load conditions Assuming that the material is shaped as as H shown in Fig. 2.14, the free charges on ¯ ¯ S D dS = −D3 ∆A, according to Gauss’ surface of a small box enclosing the area

a bar with electrodes in the z-plane an area ∆A of the plates are Q∆A = law. Here S is the chosen Gaussian ∆A, and the vector dS¯ is orientated

44

Transducers

Figure 2.14 Different load conditions for a piezoelectric bar.

normal to the surface. Thus, D3 is the (negative) charge density at the capacitor plate. If the electrical displacement is constant, the charge of the capacitor is Q = −LBD3 .

(2.95)

Some of the different load conditions are illustrated in Fig. 2.14. If, for instance, a force F33 = LBσ3 in the z-direction is present at the bar in the absence of an electrical field (a longitudinal load in the poling direction), and if all other stress components are zero, the total charges are, in light of Eq. (2.93) (d33 mode) D3 = d33 σ3

⇒ Q = −d33 F33 .

(2.96)

If a transverse load is present, e.g. F11 = BHσ11 , and all other stresses are zero, the charges to be measured are (d31 mode) D3 = d31 σ1

⇒Q=−

L d31 F11 . H

(2.97)

In the case of shear stresses along the sidewalls, the piezoelectric bar will not react (d34 = d35 = d36 = 0). In order to measure shear stresses, the capacitor plates must be placed perpendicularly to the poling direction on the sidewalls under load. If, for instance, the shear stress is applied along the right and left crosssections (F13 = BHσ5 ), and the electrodes are placed at these cross-sections, the free charges are equal to D1 = d15 σ5

⇒ Q = −d15 F13 .

(2.98)

The actual values of dij for a given material may have quite a large spread and depend on the doping materials, the manufacturing technology etc. The coefficients d33 and d31 for PZT and barium titanate can be roughly estimated, with absolute values on the order of some hundred 10−12 C/N = 10−12 m/V up to 1000 × 10−12 C/N. The coefficients of the d matrix are strongly temperature-dependent and have the tendency to decrease with increasing temperature.

2.3 Piezoelectric transducers

45

Instead of dij coefficients, other sets of parameters are often used, such as, for instance, the e = eij matrix, which is defined as ¯ ¯ ∂Di ¯¯ ∂σi ¯¯ eij = =− (2.99) ∂εj ¯E¯ = constant ∂Ej ¯ε= ¯ constant

In view of Eq. (2.32), the eij terms can easily be calculated: X dik Ek l . eij =

(2.100)

k

For applications in MEMS devices the d31 mode is the most important one, because the piezoelectric material can be deposited most easily as a planar layer, which normally is poled in the z-direction. Poling in the plane of the wafer is difficult, because vertical electrodes are needed.

Piezoelectric sensor layers The use of the direct piezoelectric effect for sensing purposes is usually based on uniform cantilever-beam structures or membranes, which fully or locally are covered by a piezoelectric layer. The carrier layer, which forms the beam or membrane, transfers the stress from the movable mass to the deposited piezoelectric material. Usually the strain transfer is assumed to occur without losses within a possibly used adhesion layer. If this is not the case a correction taking into account the so-called shear-lag effect has to be considered [Sirohi and Chopra 2000]. Both layers – carrier and piezoelectric – are embedded between electrodes, or the carrier layer forms one of the electrodes. Such an arrangement in the form of a beam corresponds to Fig. 2.14 with laminated conducting, piezoelectric, and carrier layers instead of the homogeneous piezoelectric material. One or two ends of the console beam may be fixed to the substrate. The described structure may occupy only part of the beam at the place where the stress should be measured. The beam may be stressed in the x- and y-directions (σ2 = σ4 = σ6 = 0) and experiences strain ε1 = εxx , ε3 = εz z , and ε5 = εxz . The object of interest is the electrical displacement DT = [D1 , D3 ] as a function of the strain produced. On expressing the stress by the strain in Eq. (2.93) and using Eq. (2.94), the displacement current becomes   µ ′ ¶µ ¶ ¶ µ ¶ ε1 µ ǫ11 0 E1 0 0 e15   D1 (2.101) = ε3 + E3 0 ǫ′33 e13 e33 0 D3 ε5 with ∆ = s11 s33 − s13 s31 , e15 =

d15 ; s55

e13 =

d31 s33 − d33 s31 ; ∆

e33 =

−d31 s13 + d33 s11 ∆

(2.102)

and ǫ′11 = ǫ11 −

d215 ; s55

ǫ′33 = ǫ33 −

d231 s33 + d233 s11 − d31 d33 (s31 + s13 ) . (2.103) ∆

46

Transducers

(a)

(b)

Figure 2.15 Equivalent models of a piezoelectric sensor. (a) Charge model. (b) Voltage

model.

Piezosensor

Beam

Figure 2.16 A piezoelectric sensor for beam strain measurement.

This incorporates the impact of the electrical field, in particular between the plate electrodes. If a constant strain is applied, the free charges generated will inevitably be lost over time via unavoidable leakage currents. Hence piezoelectric sensors have a bandpass characteristic. However, present-day IC and packaging technologies allow one to reduce the leakage currents to values that guarantee hold times for the charges of minutes to hours. For most practical applications this is fully sufficient.

A model of a piezoelectric sensor In Fig. 2.15(a) an electrical model of the piezoelectric sensor is shown, where QP are the charges generated by the piezoelectric effect, CP is the shunt capacitance between the collecting electrodes complemented by possible parasitic capacitances, and RP is the unavoidable shunt resistance, which is responsible for the dissipation of static charge. To avoid confusion with a current source, the charge generator is represented by a triple circle. Figure 2.15(a) is the so-called charge model, which is equivalent to the voltage model shown in Fig. 2.15(b). A typical example for measuring strain within a bent beam using a piezoelectric sensor is demonstrated in Fig. 2.16. Assuming that the sensor is small, thin, and tightly attached to the beam so that the strain in the x-direction, ε1 , is transferred to the sensor and can be assumed constant along all sensor dimensions, the charge generated is, according to Eq. (2.97), QP = −d31 σ1 BL = −d31 EP BLε1 .

(2.104)

2.4 Capacitive transducers

47

EP is here Young’s modulus of the piezoelectric material, which for simplicity is assumed to be mechanically isotropic (e.g. PZT). The sensor with length L, width B, and thickness H forms the capacitance ǫ33 LB , H

(2.105)

QP d31 EP H = ε1 . CP ǫ33

(2.106)

CP = the voltage across which is VP =

Piezoelectric actuation The converse piezoelectric effect is often characterized by the matrix of piezoelectric voltage constants g = {gij }, which is defined as the electrical field generated by unit mechanical stress, or as the strain generated by unit electrical displacement field: ¯ ¯ X ∂εi ¯¯ ∂Ei ¯¯ gij = = − (2.107) dk i ǫ−1 = ik . ∂Dj ¯ ∂σj ¯ ¯ σ=0 ¯

D =0

k

ǫ−1 ik

are the coefficients of the inverse matrix ǫ−1 . The coefficients The use of the inverse piezoelectric effect in a MEMS environment requires relatively high voltages. The reason is the low value of the voltage constant, which for applied voltages in the range up to some volts causes deformations in the nanometer range only. This is often not sufficient for efficient actuation in the micrometer range. However, piezoelectric MEMS actuators are well suited for nano and sub-nano technologies like scanning probe microscopy. High-voltage processes allowing one to realize driving voltages on the order of dozens of volts are normally a precondition for the implementation of piezoelectric actuators in inertial MEMS.

2.4

Capacitive transducers Capacitive transducers are easily customizable interlinks between inertial sensors and electronic signal processing. Unlike piezoresistors or piezoelectric sensors, they do not require additional technology steps. They enjoy wide popularity due to their insensitivity to temperature changes and low drift, which allows one to design high-performance products with very small changes in offset and sensitivity with temperature. The size of the capacitances can be adapted to the necessary transfer characteristics in a broad range. In-plane and out-of-plane capacitance sensing and actuation is possible in all MEMS technologies. A great variety of electronic input stages, such as charge amplifiers, transconductance stages, switch-capacitor devices and embedded sigma–delta converters, is at the disposal of the system designer for various sensitivity levels and various requirements for power consumption and size [Kraft et al. 1998, Petkov and Boser 2005,

48

Transducers

(a)

(b)

Flexures

Figure 2.17 Plate capacitance. (a) A generic plate capacitance. (b) A capacitance with

a movable plat.

Wu et al. 2004, Yazdi et al. 2004, Kulah et al. 2006, Lei et al. 2005]. Resolutions down to the sub-atto-farad region (< 10−18 F) are typical for modern inertial MEMS. Capacitive transducers comprise capacitive sensors and electrostatic actuators. The basis of all capacitive transducers is the plate capacitor as shown in Fig. 2.17(a), which usually consists of two in-plane electrodes or is arranged as a connected and interdigitated vertical comb structure. The capacitance of a plate capacitor is given by the well-known expression CP = εε0

A LB = εε0 . D D

(2.108)

ε0 = 8.854 × 10−12 F/m is the electrical permittivity in vacuum and ε is the relative permittivity of the medium between the plates. Since the capacitances in inertial sensors are usually located in cavities filled with inert gas, the relative permittivity is ε = 1. To simplify matters, this assumption is applied in what follows. In inertial MEMS four basic arrangements are used:

r plate capacitor with parallel, in-plane electrodes; one electrode movable in the z-direction (see Fig. 2.17(b));

r plate capacitor with one tilting electrode (see Fig. 2.19); r linear comb capacitor with one comb movable in plane (see Section 2.4.4 and Fig. 2.21);

r radial comb capacitor with one comb pivotable about the z-axis (see Fig. 2.22). Plate capacitors are characterized by small distances between the electrodes compared with the lateral plate dimensions. This makes them very sensitive with respect to tiny changes of distance. To measure or to generate such distance changes, one of the electrodes is typically fixed to the substrate and the other one is part of the moving structure of the inertial sensor. The simple parallelplate capacitor is used, if the sensed or actuated motion is out-of-plane, i.e. if one of the electrodes is moving in the z-direction as indicated in Fig. 2.17(b).

2.4 Capacitive transducers

49

The parallel-plate capacitor is invoked here as the introductory example for sensing and actuating mechanisms. It must be pointed out that any capacitor with a fixed electrode and a movable electrode acts as sensor and actuator at the same time. In particular, on applying a voltage to the electrodes of a sensing capacitance, a charge is created that is proportional to the capacitance value. This charge can be sensed by electronic means. Simultaneously the applied voltage creates a force, which may displace the movable electrode and consequently falsify the initial capacitance value. Capacitive sensing requires careful measures in order to avoid or to compensate for such effects.

2.4.1

Electrostatic forces In order to derive the forces acting on the electrodes of any capacitor, the combination “capacitance plus voltage generator” has to be considered as a whole. The potential energy of both is U , 1 U = UC + UV = − CV 2 , 2 where the potential energy UC stored in a capacitor with charge Q is

(2.109)

1 1 QV = CV 2 , (2.110) 2 2 and the potential energy UV of the voltage generator – with a voltage that is independent of the status of the capacitor – is UC =

UV = −QV = −CV 2 .

(2.111)

The negative sign indicates that the voltage generator is discharged on charging a capacitor. The force field of the charges is conservative. Thus, the force acting on the electrodes is the gradient of the potential energy of the system, where the gradients are taken with respect to the coordinates of admissible movement q1 , q2 , q3 : 1 F¯ = V 2 ∇C(q1 , q2 , q3 ). (2.112) 2 If, for instance, the upper plate electrode in Fig. 2.17(b) is movable in the z-direction, the capacitance according to (2.108) is CP = ε0

A , D+z

(2.113)

where z is the deviation from the nominal distance D. The corresponding electrostatic force becomes 1 ∂C(D + z) 1 A FPz = V 2 =− V2 (2.114) 2 ∂z 2 (D + z)2 and acts in the negative z-direction, creating an attractive force between the two electrodes. If the dimensions of the capacitance are scaled down in all three

50

Transducers

dimensions, Eq. (2.114) implies that the force does not change as long as the fringe fields do not corrupt the validity of the model (2.108). More generally speaking, the force vector is · ¸ 1 ∂C(¯ q) ∂C(¯ q) ∂C(¯ q) F¯ = V 2 e¯q 1 + e¯q 2 + (2.115) e¯q 3 , 2 ∂q1 ∂q2 ∂q3 where the unit vectors are denoted by e¯q i . For the sake of clarity the Cartesian coordinates are used, · ¸ 1 ∂C(x, y, z) ∂C(x, y, z) ∂C(x, y, z) F¯ = V 2 e¯x + e¯y + e¯z 2 ∂x ∂y ∂z = Fx e¯x + Fy e¯y + Fz e¯z ,

(2.116)

so that after linearization around the point x, y, z (for instance around the rest position x = 0, y = 0, z = 0) the force changes ∆F¯ can be described by    ∆x kxx kxy kxz r. (2.117) ∆F¯ = −  kxy ky y ky z   ∆y  = −K ∆¯ ∆z kxz ky z kz z

K = {kx i x j } is the symmetric spring-rate matrix kx 1 x 2 = −

V 2 ∂2 C = kx 2 x 1 . 2 ∂x1 ∂x2

(2.118)

As can be seen from Eq. (2.114), the spring rate for the plate capacitor kz z = − 21 V 2 A/(D + z)3 is negative. This effect is typical for movable-plate capacitances and is called “spring softening,” because the electrostatic attractive forces act against the mechanical spring. Sometimes the forces generated by induced or intentionally applied charges are of interest. If, for instance, the device is switched off, all voltage generators have zero energy, but the plates of different capacitances may still be charged or subjected to different charge-redistribution processes. Charges may have been induced also by external charge-separation processes. In all these cases the force is given by 1 Q2 F = −∇UC = −∇ QV = −∇ . 2 2C

(2.119)

As an example, a charged plate capacitor movable in the z-direction is subjected to the force FC z = −∇

(D + z)Q2 Q2 =− , 2εA 2εA

(2.120)

which is again an attractive force. It may cause undesirably large movements up to the point of mechanical contact between the electrodes, and subsequent sticking. Care must be taken in order to avoid undesired charging effects (e.g. ESD protection).

2.4 Capacitive transducers

51

Figure 2.18 The principle of differential sensing.

2.4.2

Parallel-plate capacitors The plate capacitor with parallel plates and one electrode movable in the zdirection is used for sensing small deviations of the movable electrode from some zero position z0 = 0. The capacitance change ∆CP for small deflections z < D can be expressed as a Taylor series, ∆CP = ε0

AX zk (−1)k k = SP1 z + SP2 z 2 + SP3 z 3 · · · . D D

(2.121)

k=1

A is the value of the overlapping area of the two parallel electrodes. As can be seen from Eq. (2.121), the distance measurement via ∆CP is not linear in z, but contains nonlinear distortions. The coefficients SPk are the sensitivities with respect to the kth power of z. The relative magnitude of the kth distortion is ¯ ¯ µ ¶k −1 ¯ SPk z k ¯ ¯ = |z| γk = ¯¯ . (2.122) SP1 z ¯ D

For example, if it is desired that the quadratic distortion for a singular plate measurement be less than 1% over the entire operating range |z| < zop m ax , the operating range should not exceed 1% of the distance D.

Capacitance sensing In order to detect a deflection z, an excitation voltage V must be applied to the capacitor in order to generate charge Q. Capacitance measurements are usually performed as relative measurements or differential measurements, where the capacitance is measured with respect to a reference. An idealized block diagram of a charge-amplifier measurement is presented in Fig. 2.18. It is assumed that the operational amplifier has infinite gain, bandwidth, and input impedance, so that both the input voltage and the input current are negligible.8 The noise 8

In practice the potential of the operational amplifier is not at ground, but shifted by some reference voltage V A G N D , so that the voltage V A G N D has to be added to all voltages V i and to U o u t . Additionally, the feedback capacitance in CMOS implementations must be

52

Transducers

of the amplifier is represented by the input noise voltage VNin . All parasitic capacitances of the movable plate to ground, including electrically connected parts of the sensor, wiring connections, and so on, are summarized under CPar . Thus, the current generated by the two voltages applied to the capacitances C1 and C2 , namely voltages V1 and V2 , flowing through the feedback capacitance of the charge amplifier CFB , is i = Q˙ 1 + Q˙ 2 =

d (C1 V1 + C2 V2 ). dt

Under the idealizations described above the output voltage Uout is Z 1 1 1 Uout = i(t)dt = (Q1 + Q2 ) = (C1 V1 + C2 V2 ). CFB CFB CFB

(2.123)

(2.124)

Let C1 = CP0 + ∆CP be the capacitance to be measured, and let the voltages be set in anti-phase, V2 = −V1 = −V ; then Uout =

V V (C1 − C2 ) = (∆CP + CP0 − C2 ). CFB CFB

(2.125)

The reference measurement allows one to compensate for the large contribution of the rest capacitance CP0 (CP0 − C2 → 0) and to get a much higher dynamic range and resolution. However, optimal conditions are achieved when the reference capacitance C2 is part of the mechanical sensor and arranged in such a way that it changes in anti-phase with C1 : C2 = CP0 − ∆CP ;

⇒ Uout = 2V

∆CP Az = −2ε0 V . CFB CFB D2

(2.126)

Compliance with the condition C1,2 = CP0 ± ∆CP

(2.127)

is part of nearly all designs of inertial sensors.

Noise The resolution of a sensor is strongly determined by the electronic noise component, which is represented here by the noise voltage VNin . The noise voltage creates fluctuating charges in all capacitances, including the parasitic capacitance CPar , QNin = VNin (C1 + C2 + CPar + CFB ),

(2.128)

which is transferred to the output noise voltage UNout , µ ¶ C1 + C2 + CPar + CFB C1 + C2 + CPar UNout = VNin = VNin 1 + . (2.129) CFB CFB shunted by some resistance in order to limit the impact of uncontrollable leakage currents etc. These deficiencies do not change the general relation presented in what follows, but must be considered in the real design.

2.4 Capacitive transducers

53

This equation illustrates the utmost importance of small parasitic capacitances. Furthermore, it shows that the signal-to-noise ratio depends only weakly on the value of the feedback capacitance, which should be chosen to be as small as possible but large enough to avoid the impact of leakage currents on the integrating capability of the operational amplifier. Typical values in sub-micrometer-CMOS technologies are in the range of 1 pF.

Excitation voltages The excitation voltages V1 and V2 can be DC voltages or periodic functions of time. The different approaches to choosing the sensing voltages and the kind of charge transfer and processing of the charges (not necessarily by a charge amplifier) have spawned a great variety of electronic solutions. Some of the principles will be discussed later (Chapter 6). Here it should be noted that periodic high-frequency excitation voltages, for instance, in the form of harmonic voltages V = VDC ± V0 sin(ωt) offer a lot of benefits.

r The charge flow ∆CP V is transferred by modulation into the high-frequency domain that is not affected by the impact of low-frequency noise of the input amplifier (1/f noise). r According to Eq. (2.114), the generated force has a DC component and highfrequency components with frequencies ω and 2ω that can be adjusted in order to have little impact on the mechanical behavior of the single plate. The impact of the DC component can be taken into account by considering the corresponding deflections or bending effects, or can be mitigated by using stiff constructions in the case of differential arrangements (see, for example, Section 2.4.3).

The pull-in effect As can be seen from Eqs. (2.124) and (2.126), the sensitivity of the capacitive transducer increases with the applied voltages. Apart from the nonlinear distortions of ∆CP (z) there are, however, two more limits: first, the limited voltage level in CMOS circuits; and second, the intrinsic pull-in effect of capacitors with a movable plate. The last effect should be kept in mind for any capacitive interface.

Stability limits The pull-in effect is the result of instability in the system “plate plus spring plus voltage source feeding the capacitance plates.” A movable plate according to Fig. 2.17(b) is subjected to two forces: the first is the elastic force Fz = −kz, created by the suspending springs and acting on the plate as part of the inertial sensor; the second one is the electrostatic force according to (2.114). In an equilibrium state they balance each other: FΣ = Fz + FPz = 0;

1 V2 ⇒ FΣ = − ε0 A − kz = 0. 2 (D + z)2

(2.130)

54

Transducers

The equilibrium state is stable when the total potential energy of the system “spring plus capacitance plus voltage generator” has a minimum. Hence, the second derivative should be positive or, according to Eq. (2.112), the first derivative of the force should be negative: ∂FΣ < 0; ∂z

⇒ ε0 A

V2 − k < 0. (D + z)3

(2.131)

The last expression can easily be interpreted by rewriting it using the definition of the spring rates according to Eq. (2.117): −kz z < k.

(2.132)

The negative electrostatic spring rate must be smaller than the (positive) mechanical spring rate in order to keep the equilibrium state z according to (2.130) stable. Since −kz z < 0, there exist displacements z for which this condition will be violated. With Eq. (2.130) it follows that the displacement in the negative z-direction (towards the fixed plate) must be smaller than D/3 in order to keep the equilibrium point z stable: a stable state exists for D . (2.133) 3 For larger deviations the movable plate becomes unstable, moving beyond the point of unstable static equilibrium towards the fixed plate, where it is stopped. This undesired contact may cause sticking effects, which have to be avoided. In any case, the mechanical operating range of the sensor should be much less than the stability limit −z < −zm ax =

D . (2.134) 3 If for a given elastic spring rate k the voltage V is increased, the electrostatic force grows until it can no longer be balanced by the spring force for any z. The corresponding pull-in voltage Vpull-in can be obtained from Eq. (2.130). Since µ ¶ ε0 AV 2 z z2 =− 1+ 2 , (2.135) 2kD3 D D |zop

m ax |

≪

the maximum of the right side is reached at z/D = − 31 , and, correspondingly, a stable dynamics bound to the domain |z| ≤ |zm ax | is possible only as long as s 8kD3 V ≤ Vpull-in = . (2.136) 27ε0 A According to Eq. (2.135), V 2 (z) is a cubic function of z. Such nonlinear dependencies, embedded in dynamic systems, typically cause a hysteresis behavior, which here manifests itself in an increase of |z| with V 2 until the instability point is reached, and in a subsequent jump of z into the position −D, i.e. into a contact position of both electrodes. Decreasing the voltage does not release the moving electrode (even in the absence of contact forces, causing the usual

2.4 Capacitive transducers

55

sticking). The capacitor must be completely switched off in order to release the moving plate.

Pull-in during switch-on If the capacitor is driven with a constant voltage V0 < Vpull-in , the corresponding deflection z can be calculated by solving Eq. (2.130). However, keeping the amplitudes of stationary plate motions within the indicated stability region does not yet guarantee stable dynamics since during the switch-on process an overshoot appears that may drive the movable plate into the unstable condition. Neglecting the damping effects and assuming a step-like switch-on process, the overshoot distance can be found from the fact that, at the overshoot distance, the velocity vP is zero and, consequently, the accumulated kinetic energy EP is zero. The kinetic energy is the work performed by the total force, thus Z zm in 1 EP = mP vP2 = (Fz + FP z )dz = 0. (2.137) 2 0

The integration of the forces results in

AV0 2 . (2.138) kD However, even if the condition −zm in ≤ D/3 is also fulfilled, the stability condition may still be violated due to additional forces caused by external shocks and vibration. A large safety margin is needed, at least for failure-critical applications. A hard switch-on effect with large overshoot should be avoided by implementing a soft electronic ramp-up procedure. zm in (D + zm in ) = −ε0

Increased stability region Theoretically, the stability region can be increased significantly by adding a capacitor connected in series to the plate capacitance and by compensating the resulting voltage loss by enlarged drive voltages [Puers and Lapadatu 1996, Bao 2000]. The two serially connected capacitors can be envisaged as an equivalent capacitor with the same plate area, but an increased distance between moving and fixed plate. A serial capacitor CS < 21 CP0 corresponds to a distance that is increased by a factor of more than three, in which case the tendency for instability disappears for all voltages. The moving plate may now contact the fixed plate of the initial capacitor, remaining in a stable state. However, due to the limitations on the drive voltages for cost-efficient CMOS circuits, this method is usually regarded as unfavorable.

2.4.3

Tilting-plate capacitors Paired capacitances with tilting plates are often used to sense or actuate rotary motions about an in-plane axis. A typical arrangement is shown in Fig. 2.19. Two movable plates are stiffly connected with a torsional spring suspension, which permits a tilting movement of both plates about the y-axis. The tilting

56

Transducers

(a)

(b)

θ

Figure 2.19 A tilting-plate capacitance. (a) Top view. (b) Side view.

angle of the right plate in Fig. 2.19(b) is θ, while the left plate is rotated by −θ. The capacitance of the right plate CT1 is Z y U (x) Z L2 dy dx CT1 = ε0 , (2.139) D + z(x, y) y L (x) L1 where yL (x) is the lower border line in the x–y-plane of the plate capacitance, and, correspondingly, yU (x) is the upper border. L1,2 are the minimal and maximal dimensions along the x-axis, respectively. The z-deflection at a given point x, y does not depend on y: z(x, y) = x tan θ.

(2.140)

The plates are usually symmetric with respect to the x-axis, yL (x) = −yU (x), so that Z L2 dx yU (x) CT1 = 2ε0 . (2.141) D + x tan θ L1 Since the second capacitance plate is stiffly connected to the first one, and since θ2 = −θ1 , the second capacitance changes in the opposite direction: Z L2 dx yU (x) CT2 = 2ε0 . (2.142) D − x tan θ L1

Nonlinear distortions In MEMS implementations the gap between the movable plate and the fixed electrode D is much smaller than the lateral dimension L2 . Consequently, the maximum tilting angle θm ax is very small, µ ¶ D θm ax = arctan ≪ 1, (2.143) L2 and the tangent can be approximated by the angle θ. Typical values are L2 > 0.5 mm and D < 2 µm. Thus θm ax < 0.2◦ ; however, θm ax values on the order of 1◦ are not unusual. Owing to the pull-in effect, the operating angles must be chosen to be even significantly smaller than θm ax . Typical values for capacitive sensors are in the

2.4 Capacitive transducers

57

range of milli-degrees and less. Thus, the actual capacitance changes can be calculated assuming |xθ| ≪ D,

(2.144)

which allows one to represent the capacitance as a Taylor series with only a few terms: X (−1)k θk Z L 2 CT1 (θ) = 2ε0 yU (x)xk dx (2.145) Dk + 1 L1 k=0

or

∆CT1 = ST1 θ + ST2 θ2 + ST3 θ3 + · · ·

(2.146)

with STk = 2ε0

(−1)k Dk + 1

Z

L2

yU (x)xk dx.

(2.147)

L1

The sensitivity integrals STk can be easily derived for rectangular or trapezoidal plates, for which yU (x) = B/2 or y√ U (x) = χx. For disk-shaped plates with radius R and upper boundary yU (x) = R2 − x2 analytical expressions can also be derived using the substitution x/R = sin ϕ and integrating over ϕ. In the case of harmonic high-frequency, differential sensing, the excitation voltages are V1 = VDC + V0 sin(ωt) and V2 = VDC − V0 sin(ωt), and the highfrequency component of the total charge flow ∆QΣ = V1 ∆CT1 + V2 ∆CT2 = 2V0 sin(ωt)(ST1 θ + ST3 θ3 ) + 2VDC ST2 θ2 · · · (2.148) exhibits a suppressed quadratic nonlinearity, i.e. the total capacitance change is linearized. The “DC”-component 2VDC ST2 θ2 varies slowly with θ and is proportional to the second-order sensitivity ST2 . It can be suppressed by the signalprocessing electronics. For rectangular plates with areas (L2 − L1 )B the sensitivities are STk = ε0 B

(−1)k 1 − β k +1 . k +1 k + 1 θm ax

(2.149)

β is the length ratio, β = L1 /L2 . The corresponding relative distortions µ ¶k −1 STk θk 2 1 − βk + 1 θ γTk = = . (2.150) ST1 θ k+1 1−β θm ax reach their theoretical maximum for β = 1: µ ¶k −1 k θ γTk m ax = . 2 θm ax

(2.151)

In the case of differential sensing, according to (2.148), the first distortion of interest is γT3 . If, for instance, a cubic distortion of 1% is allowed, the operating range should be limited to θ < 0.08θm ax .

Transducers

-0.32

-0.34

θ / θmax Pull-in

58

-0.36

-0.38

Rectangular Plate Trapezoidal Plate

-0.4

-0.42

-0.44 0

0.2

0.4

0.6

0.8

1

L1 / L2

Figure 2.20 Stability limits of rectangular and trapezoidal plates.

Instabilities of a singular tilting plate As in the case of parallel-plate capacitances, the tilting capacitor may be subjected to the pull-in effect. For a singular tilting plate the spring torque Mθ of the torsional suspension is Mθ = −kθ θ, and the electrostatic moment of a single plate is Z L2 1 ∂CP1 2 yU (x)x dx MC θ = V = −ε0 V 2 (1 + θ2 ) . 2 ∂θ (D + xθ)2 L1

(2.152)

(2.153)

The total torque MΣ = MC θ + Mθ = 0 is zero, and stability is guaranteed by the condition ∂MΣ /∂θ < 0. Performing the same calculations as in Section 2.4.2, the stability condition can be represented as Z 1 L 2 yU (x)x(D + 3xθ) dx < 0, θ 6= 0. (2.154) θ L1 (D + xθ)3 Even for a rectangular plate with yU (x) = B/2, the integral must be evaluated numerically, and the stability limit must be determined iteratively. In Fig. 2.20 the stability limits of rectangular and trapezoidal plates are presented for various plate dimensions. L1 /L2 is the ratio of the shortest to the largest distance of the plate edges according to Fig. 2.19(b). The shape of the trapezoidal plates is here determined by boundaries orientated radially with respect to the pivotal point, whereby the radial edges enclose an angle of 90◦ (i.e. yU (x) = x; yL = −x).

Instabilities of the tilting capacitance pair For differential sensing as introduced in Section 2.4.2, paired or balanced plates with central suspension as in Fig. 2.19(a) are preferably used. They are excited by voltages V1 = VDC + VAC and V2 = VDC − VAC . VDC is supposed to be constant and VAC may be taken as a time-varying periodic or harmonic

2.4 Capacitive transducers

59

excitation, applied in anti-phase. The plate pair is then subject to the total moment MBPθ = 21 [V12 ∂CP1 /∂θ + V22 ∂CP2 /∂θ]: MBPθ = 2ε0

Z

L2

yU (x)x

L1

2 2 ) − 2VDC VAC (D2 + x2 θ2 ) + VAC Dxθ(VDC Dx dx. (D2 − x2 θ2 )2 (2.155)

If θ 6= 0, a moment is created even in the case of equal voltages (VAC = 0); it can be represented by a negative spring force. The spring rate is easily derived from kBPθ = −∂MBPθ /∂θ, yielding kBPθ = Z −2ε0

L2

yU (x)x2

L1

2 2 ) − 4xθ(3D2 + x2 θ2 )VDC VAC + VAC D(D2 + 3x2 θ2 )(VDC dx. 2 (D − x2 θ2 )3

For small angles θ the spring rate is approximated by kBPθ = −2ε0

2 2 VDC + VAC D3

Z

L2

yU (x)x2 dx.

(2.156)

L1

Spring softening The last expression reflects again the spring-softening effect. Since kBPθ < 0, the total spring rate kΣθ = kθ + kBPθ of the suspended plate pair is smaller than kθ , and the resulting suspension is “softer” than in the absence of electrostatic forces. This effect is often used for electrostatic trimming of resonance frequencies. If the mass of the plate pair including stiffly associated bodies is denoted by mBP , the corresponding modal frequency of the system is 2 ωBP =

kΣθ . mBP

(2.157)

Increasing the voltages reduces the resonance frequency. This effect has to be taken into account during the design flow. It allows also a fine tuning of the resonance frequency. In the case of high-frequency, harmonic sensing the AC voltage is not constant but a high-frequency sinusoidal function VAC = V0 sin(ωt). The expression 1 1 2 2 2 VDC + VAC = VDC + V02 − V02 cos(2ωt) 2 2

(2.158)

contains a high-frequency component. Usually the mechanical system is designed 2 + 21 V02 not to respond to such high-frequency excitations. Only the term VDC remains effective. The spring softening as discussed above pertains to small excitation voltages and small tilting angles for which system stability is guaranteed. However, care has to be taken to remain within the stability regime.

60

Transducers

Stability limits For the plate pair there are two stability limits: one on the right side and one on the left. When these limits are exceeded, the right or left plate will be pulled in and will contact the substrate. Since different voltages may be applied to the right and the left plate, pull-in will occur on the side with the higher voltage. In order to get a rough estimate of the stability limits, a simplification is made using Eq. (2.145), and truncating the series after the fourth power of θi , ∆CTi = ST1 θi + ST2 θi2 + ST3 θi3 + ST4 θi4 ,

i = 1, 2.

(2.159)

The moments of both plates, 21 V1,2 ∂C1,2 /∂θ, have to be added, keeping in mind that, for the second plate, the tilting angle is θ2 = −θ1 = −θ. The total moment is balanced by the torque −kθ θ, 1 2 V [(1 − α2 )ST1 + 2(1 + α2 )ST2 θ + 3(1 − α2 )ST3 θ2 + 4(1 + α2 )ST4 θ3 ] − kθ θ = 0. 2 1 (2.160) Here α2 = V22 /V12 is the squared ratio of the driving voltages. The stability regime is determined – as above – by the negative slope of the total torque: ∂MΣ = V12 [(1 + α2 )ST2 + 3(1 − α2 )ST3 θ + 6(1 + α2 )ST4 θ2 ] − kθ < 0. (2.161) ∂θ If equal absolute voltages are applied, then α2 = 1, and the solutions of Eq. (2.160) can be found analytically: s kθ − 2V 2 ST2 θ1 = 0, θ2,3 = ± . (2.162) 4V 2 ST4 According to Eq. (2.161), the corresponding slopes of the moments obey the equations ∂MΣ ¯¯ θ1 = 0 : (2.163) ¯ = 2V 2 ST2 − kθ ; ∂θ θ 1 ¯ ∂MΣ ¯ θ2,3 : = −2(2V 2 ST2 − kθ ). (2.164) ¯ ∂θ θ 2 , 3

Consequently, as long as 2V 2 ST2 < kθ , the equilibrium point θ1 = 0 is stable and the points θ2,3 are unstable. If the voltage is increased to exceed the stability limit V 2 = kθ /(2ST2 ), the equilibrium point θ1 = 0 becomes unstable, and the two additional states θ2,3 disappear (become imaginary values). In this case, tiny deviations from the zero position result in a collapse. Under these circumstances the tilting-plate pair can no longer be operated. It is important to check this worst-case condition for any design. A plausibility check can be based on the fact that ST2 is proportional to the second moment of inertia Iθ of the plate pair, Z L2 x2 yU (x)dx. (2.165) Iθ = 2ρH0 L1

2.4 Capacitive transducers

61

ρ is the density of the plate material – for silicon, ρ = 2300 kg/m3 . H0 is the thickness of the plate. As is well known, the tilting pair has a resonance frequency of 9 kθ ωT2 = . (2.166) Iθ The stability condition for θ1 according to (2.163) can now be expressed as ρ V 2 < ωT2 H0 D3 . (2.167) ε0 For a typical gap between the electrodes of D = 1.75 µm, a plate thickness of H0 = 10 µm, and a resonance frequency fT = 5 kHz, the voltage must be less than 3.7 V. This example shows how critical this condition may be, especially for small gaps and low resonance frequencies. The case of equal voltages as considered above is, of course, not very representative for the tilting plates. In order to drive or sense the pair of capacitances, different time-varying voltages are applied. For a simplified analysis they are considered as acting so slowly in comparison with the transfer processes which start once an unstable point has been reached that they can be approximated by constant values. If different absolute voltages are applied, the central stationary state θ1 will be shifted from the zero position. In a crude approximation, the equilibrium point θ1 can be determined by neglecting the higher-order terms in Eq. (2.160), θ1 ≈

(V12 − V22 )ST1 . 2[kθ − (V12 + V22 )ST2 ]

The moment’s slope is calculated to the same level of accuracy: ∂MΣ ¯¯ −[kθ − V12 (1 + α2 )ST2 ]2 + 1.5(1 − α2 )2 V14 ST1 ST3 . ¯ ≈ ∂θ θ 1 kθ − V12 (1 + α2 )ST2

(2.168)

(2.169)

If the system has been stable for V12 = V22 (i.e. according to (2.163) kθ − (V12 + V22 )ST2 > 0), and V1 is kept constant while V2 is decreased, then the denominator remains positive since α2 ≤ 1. In this case the stability criterion ∂MΣ /∂θ|θ 1 < 0 can be simplified to √ · ¸ √ ST1 ST3 V12 ST2 1 + α2 + 1.5(1 − α2 ) < kθ . (2.170) ST2 √ The actual change of the pull-in voltage depends on the ratio ST1 ST3 /ST2 , which for various plate geometries is on the order of 1. Consequently, the pull-in voltage remains on the same order as for the symmetric voltage condition. If, on the other hand, the voltage V2 is increased, stability loss happens if the denominator becomes negative. To preserve stability the condition 9

Usually the tilting plates are connected to some additional co-moving masses, so that the actual moment of inertia is larger. Consequently, the sensitivity S T 2 is overestimated, and the stability limit for the voltage in reality is higher than that estimated here.

62

Transducers

kθ − (V12 + V22 )ST2 > 0 must be satisfied, which again leads to the same order of average pull-in voltages. In summary, the stability check for symmetric voltages is a qualified starting point for an estimate of the pull-in point. The unstable states θ2,3 as well as their existence limits have to be determined numerically for any given capacitance configuration and spring torque kθ . The limits beyond which stability is no longer possible can be derived analogously to Eq. (2.154) from 1 θ

Z

L2

L1

·

D + 3xθ D − 3xθ yU (x)x − α2 (D + xθ)3 (D − xθ)3

¸

< 0,

θ 6= 0.

(2.171)

It is sufficient to investigate the stability limit θpull-in only for 0 < α2 < 1 (corresponding to V22 < V12 ), because for α2 > 1 the absolute value of the corresponding pull-in angle is the same as for 1/α2 , but with opposite sign. In other words, for V22 < V12 the right side of the tilting plate pair is going to collapse, whereas for V22 > V12 the left plate loses stability for angles outside the stability region.

2.4.4

Comb capacitors To sense or actuate in-plane movements of inertial proof masses, capacitors with plates orientated normal to the plane of the proof mass are needed. These socalled vertical plates have the drawback that the extent of the electrodes in the z-direction (the direction normal to the sensor plane) is strongly limited by constraints of the MEMS technologies to some 10 to 100 µm if surface micromachining (SMM) is used. Thus, for the design of sufficiently large capacitors in SMM technologies, multiple structures are required.

Unidirectional linear combs The interdigitated comb structure as shown in Fig. 2.21 is the most common building block of such multiple arrangements. It consists of two half-combs carrying finger electrodes, which – together with the counter-electrodes of the second half-comb – form the capacitance. The operating motion is in the y-direction and can be employed for actuation or for sensing (operating motions in the x-direction will be discussed in Section 2.4.4). However, in contrast to the tilting-plate capacitor, where a high stiffness against orthogonal movements can be assumed, the suspending springs for the comb usually yield in more than one direction. The limited stiffness, especially in the z-direction, has to be taken into consideration. As long as the linear dimensions of the interleaving fingers are much larger than the distance between them, the expression for the plate capacitance can be used. In a first approximation the capacitances of the end-faces of the fingers to the opposite walls are neglected. If the movable half-comb in Fig. 2.21 is displaced in the x-direction from the zero position by the value x, in the y-direction by the value y and in the z-direction by the value z, the total capacitance of the

2.4 Capacitive transducers

63

Figure 2.21 Comb structure.

comb is CC = ε0 N (H0 − f (z))(Y0 + y)

µ

1 1 + D0 + x D0 − x

¶

.

(2.172)

N is the number of fingers in the movable half-comb. The number of fingers in the fixed half-comb is usually N + 1. The expression (2.172) has to be interpreted with care: the overlapping area of the movable fingers with the fixed electrodes is simply (H0 − |z|)(Y0 + y). However, here the parallel-plate capacitor model is too crude, because a displacement in the z-direction will change the fringe capacitances provided by the upper and lower finger surfaces. A better-suited model should reflect the fact that, around the zero position z = 0, the capacitance change should be approximated by a function like f (z) = κz 2 , which, for increasing |z|, quickly approaches the linear region f (z) = |z|. The exact shape of the function f (z) around z = 0 is not relevant for the following discussion of the stability limits.

Forces in unidirectional combs According to Eq. (2.115), the force vector acting on the movable comb is given by · (H0 − f (z))(Y0 + y)2x 2 N D0 ¯ F C = ε0 V e¯x D02 − x2 D02 − x2 ¸ ∂f (z) + (H0 − f (z))¯ ey − (Y0 + y) e¯z ∂z = F¯Cx e¯x + F¯Cy e¯y + F¯Cz e¯z . (2.173) The attractive forces between the individual fingers in the x-direction correspond to face-to-face forces as in the plate capacitances considered above. They balance each other, so that the combs – if movable in the x-direction – take the average

64

Transducers

position x = 0 between the fingers of the non-movable half-comb. However, again, the stability condition must be considered. The movable half-comb is subject to the sum of the spring force Fx = −kx x and the electrostatic force FCx in the x direction, which balance each other at the equilibrium point x x FΣx = 2V 2 ε0 N D0 HY − kx x = 0. (2.174) 2 (D0 − x2 )2 H = H0 − f (z) and Y = Y0 + y are the actual overlapping dimensions. Stability holds for ∂FΣx /∂x < 0, or, in more detail, for 2V 2 ε0 N D0 HY

D02 + 3x2 − kx < 0. (D02 − x2 )3

(2.175)

As in the case of tilting plates in the previous paragraph, the first equilibrium point is x = 0, which is stable for 2V 2 ε0 N

HY < kx . D03

(2.176)

As the comb is moving preferably in the y-direction, the spring is stiff in the x-direction and the spring constant kx is very large, so this condition usually holds for standard designs.

Driving forces At the stable equilibrium point x = 0, the force in the y-direction, acting on the movable half-comb, is, according to Eq. (2.173), equal to FCy =

1 2 ∂C(Y0 + y) H 2 V = ε0 N V . 2 ∂y D0

(2.177)

It is orientated in the positive y-direction and, thus, is also an attractive force towards the fixed half-comb. ε0 N (H/D0 )V is the voltage-to-force gain of the driving comb. As for the horizontal plate capacitors, the independency of the generated force of dimensional scaling remains intact. Remarkably, the force is independent of y as long as the comb capacitance can be approximated by a collection of individual plate capacitances. Thus, the comb is an excellent actuator with quadratic voltage-to-force transformation.

Out-of-plane forces According to Eq. (2.173), the force in the z-direction FCz = −ε0 V 2

N Y D0 ∂f (z) D02 − x2 ∂z

(2.178)

follows the slope ∂f (z)/∂z. It is balanced by the spring force of the comb, Fz = −kz z, or −ε0 V 2

N Y D0 ∂f (z) − kz z = 0, D02 − x2 ∂z

(2.179)

2.4 Capacitive transducers

65

Figure 2.22 A schematic view of a radial comb structure.

and does not cause any instability. Indeed, the stability condition ∂FΣz /∂z < 0 or −ε0 V 2

N Y D0 ∂ 2 f (z) < kz D02 − x2 ∂z 2

(2.180)

is always satisfied because, in accordance with the properties of the function f (z) introduced above, the relation ∂ 2 f (z)/∂z 2 > 0 holds for all z.

Bidirectional actuation Applied voltages on plates or unidirectional combs create unidirectional attractive forces. In order to drive an object in both directions, a bidirectional force is needed. To create it, at least two plate capacitors or two combs are required. For instance, two combs can be arranged as a common, movable beam with rows of fingers on both sides, complemented by two corresponding interdigitated finger rows fixed on the substrate. The design principle is shown in Fig. 2.22, which at the same time illustrates the radial comb structure. For practical applications the mechanical cross-coupling and stability conditions are of interest.

The spring matrix of a bidirectional comb If the two identical linear combs with capacitances µ ¶ 1 1 CC1(2) = ε0 N H(Y0 + (−)y) + D0 + x D0 − x

(2.181)

are driven by two different voltages V1 = VDC + VAC and V2 = VDC − VAC , the resulting force of the balanced comb can be represented as ( N D0 4xH 2 2 ¯ FBC = ε0 2 [Y0 (VDC + VAC ) + 2yVDC VAC ]¯ ex + 4HVDC VAC e¯y 2 D0 − x D02 − x2 ) ∂f (z) 2 2 −2 (2.182) [Y0 (VDC + VAC ) + 2yVDC VAC ]¯ ez . ∂z

66

Transducers

The spring-rate matrix ki,j = ∂FBCi /∂xj becomes  2H(D02 + 3x2 )[Y V 2 ] 4HxVDC VAC − − 2 − x2 )2  (D D02 − x2 0    4HxVDC VAC N D0   − 0 KBC = 2ε0 2 D02 − x2 D0 − x2      2f ′ x[Y V 2 ] − 2 2f ′ VDC VAC D0 − x2

 2f ′ x[Y V 2 ] − 2 D0 − x2      ′ 2f VDC VAC  ,     ′′ 2 f [Y V ]

(2.183)

2 2 )+ + VAC where the following abbreviations were introduced: [Y V 2 ] = Y0 (VDC ′ ′′ 2 2 2yVDC VAC , f = ∂f (z)/∂z, and f = ∂ f (z)/∂z . Balanced combs are usually suspended by elastic beam-shaped springs. The beams are yielding in the y-direction, but also inevitably exhibit some elasticity in the z-direction. As mentioned above, the stiffness in the x-direction is much larger, so that, for an estimate of cross-coupling terms, x can be set to zero and the rates kxy and kxz vanish. The spring rate kBCxx leads to a light softening of the comb suspension in the x-direction as described in Section 2.4.1 and can usually be neglected. Remarkable is the cross-coupling term ky z = 2f ′ VDC VAC . As discussed above, the slope f ′ should be set to zero for small z-deflections, i.e. under normal operating conditions of the comb. Thus, this term is usually negligible. However, for larger z, caused, for instance, by shock or vibrations, the cross-coupling term may become relevant and induce an additional parasitic modulation in the z-direction. The spring rate kBCz z = f ′′ [Y V 2 ] is proportional to f ′′ , which is practically zero outside a very small interval around the zero position. However, within this interval, f ′′ and the corresponding spring rate become very large, keeping the movable comb close to the position z = 0.

The driving force of a bidirectional comb The desired driving force in the y-direction depends linearly on the excitation voltage VAC : FBCy = 4ε0

N H0 VDC VAC . D0

(2.184)

The voltage-to-force gain is now independent of the applied driving function VAC . If the AC voltage is a sinusoid, the resulting force creates vibrational oscillations of the mass linked to the driving combs. This is the basic principle for the driving of vibrational gyroscopes.

Radial combs Radial combs as shown in Figs. 2.22 and 2.23 are used in order to create torques. The finger lengths are proportional to the distance to the pivotal point so that

67

2.4 Capacitive transducers

Figure 2.23 A radial comb structure manufactured in a surface micromachining

process.

the overlap area of an individual finger with its neighbors is proportional to the common overlap angle δk = δ defined in Fig. 2.22. The capacitance of one comb, rotated from the zero position by an angle θ < 21 δ, is µ ¶ X H0 θ CCR = 2ε0 rk = CCR0 1 + (δ + θ) , (2.185) D0 δ k

where CCR0 is the comb capacitance in the zero position. Since the fingers are positioned equidistantly as shown in Fig. 2.22, rk = r0 + k ∆r, the sum over rk can be easily calculated, and one · H0 H0 CCR0 = 2ε0 δN Ravg = 2ε0 δN r0 + D0 D0

(2.186) obtains ¸ 1 (N − 1)∆r , 2

(2.187)

where Ravg is the average distance between fingers and pivot. On the basis of expression (2.185), the torque Mθ generated about the z-axis is 1 ∂CCR 2 1 Mθ = V = CCR0 V 2 . (2.188) 2 ∂θ 2δ As in the case of linear combs, the combination of two radial comb structures according to Fig. 2.22 with voltages applied in anti-phase creates a driving moment 2 MΣθ = CCR0 VDC VAC , (2.189) δ which for VAC = V sin(ωt) is a clean sinusoidal excitation.

Frame-based capacitors As mentioned above, in-plane sensing and actuation require interdigitated structures, where the opposite faces of the movable fingers see electrodes with different voltages. Thus, between two movable fingers there must be placed two fixed fingers that carry two different excitation voltages. Each group of fixed fingers with

Transducers

(a)

Top view

(b) Fixed Plates

Moving Frame

Moving Box Shield

x

Fixed Plates

Wiring

Contacts

...

68

Side view Figure 2.24 A sensing box formed by vertical walls of a frame. (a) A bird’s-eye view of a sensing box. (b) A separated sensing box.

one of the applied excitation voltages constitutes a half-comb. In summary, three half-combs – two fixed and one movable – are necessary in order to create a comb sensitive with respect to in-plane movements. Usually such an arrangement is mirrored: one movable comb with fingers on both sides and two fixed combs on each side of the movable comb. The electrical interconnection of such multiple structures is not a simple task. A more advantageous solution is presented in Fig. 2.24. Here, the electrodes of the moving capacitances are formed by the vertical walls of the moving frame. The counter-electrodes are arranged as pairs and can be supplied with different voltages. They are fixed mechanically on the substrate, but are electrically wired to different voltage sources V1 and V2 . Additional shields with the same potential as the moving box may be implemented. They shield the working capacitances from possible horizontal box deflections which may take place in gyroscopes. The opposite horizontal walls of an individual box see electrodes with different voltage loads. If the deflection of the frame from the zero position upwards in Fig. 2.24(a) is x, then the capacitances CF1,2 between frame and upper fixed electrodes and between frame and lower fixed electrodes, respectively, are CF1,2 = ε0

H0 L . D0 ∓ x

L is the length of the fixed, vertical electrode plates. The total force of both capacitances is · ¸ V12 V22 FFx = ε0 H0 L − . (D0 − x)2 (D0 + x)2

(2.190)

(2.191)

The stability considerations are identical to the explanation in Section 2.4.4 and result in a formula equivalent to (2.176): H0 L < kx . (2.192) D3 The unidirectional combs, considered in the section “Unidirectional linear combs,” are preferentially sensitive in the direction of the fingers (denoted by y) and stiff in the in-plane direction normal to the faces of the fingers (x). For 2V 2 ε0 N

2.4 Capacitive transducers

(a)

69

(b)

te

Figure 2.25 (a) A cross-section of comb fingers and electrical field lines.

(b) A simplified model of fringe impact.

frame-based combs the sensitivity direction is orientated perpendicular to the faces of the moving walls. Consequently, the spring rate kx is much lower by design, which increases the susceptibility to a pull-in effect. Thus, a careful check of tolerable voltages is mandatory during the design. The frame-based capacitance shown in Fig. 2.24 can be divided in two halfparts by cutting it along the x-axis. The resulting construction of one part is the mentioned three-subcomb design.

2.4.5

Levitation In microelectronics, fringe capacitances, i.e. capacitances associated with the edges or outer surfaces of comb- or frame-based capacitances, increase the nominal value of a capacitance but do not change its basic properties. This is not always the case for MEMS capacitances, because the movability of one of the electrodes may cause unexpected movements due to forces originating in the fringe capacitances. This is true in particular for comb capacitances, which were described in the preceding section.

Comb levitation Comb levitation in MEMS is the result of an asymmetric field configuration around the comb fingers. Let’s consider a comb with N fingers per movable halfcomb and with an overlapping area Hy Y = H0 (Y0 + y) as shown in Fig. 2.21. A cut across three fingers is shown in Fig. 2.25(a). For driving combs the applied voltages V1 and V2 are, of course, equal to each other: V = V1 = V2 .10 The movable comb is kept at a potential VAGND , commonly called analog ground. This potential is higher than the potential of the substrate, because 10

The case of different voltages is relevant for sensing combs or frames, where the movable finger is encased between the fingers of two different half-combs.

70

Transducers

modern integrated circuits are powered by one voltage only and therefore are not able to efficiently deliver positive and negative voltages around some external zero potential that could be connected to the substrate. Thus, the neutral potential of the integrated circuit has to be shifted by some voltage that is called the “analog ground”; all positive and negative signals are referenced to this potential. As a consequence, the moving structure is biased by VAGND with respect to the substrate. Since VAGND is on the order of volts, the attractive forces may become so high that the moving structure may collapse and contact the substrate. A common countermeasure is therefore the insertion of a conducting shield layer between the moving structure and the substrate, which is supplied by VAGND , so that the potential difference between moving structure and underlying surface becomes zero. This layer is electrically insulated with respect to the substrate. This grounded “shield” prevents the field lines between the fingers in the upper and in the lower space closing symmetrically. In a very crude approximation, the impact of the fringe field can be represented by some additional capacitances CF1 and CF2 between the top of the movable fingers and the fixed neighbors as shown in Fig. 2.25(b), and correspondingly between the bottom and the neighbors (not shown due to limited drawing space). These fringe capacitors represent the impact of field lines that are directed out of the upper or lower plane of the movable structure. In the zero position, z = 0, the bottom fringe capacitors are small, because the ground plane in close proximity to the bottom of the moving structure blocks the creation of field lines between the bottoms of the moving and fixed structures. Thus, the larger top capacitors create attractive forces in the positive z-direction – the levitation forces. With increasing displacement in the positive direction the bottom fringe capacitances increase, because more and more field lines starting at the bottom of the moving structure end at the fixed comb, until a point z0 > 0 where the upwards and downwards forces balance each other is reached. The exact analysis of the levitation forces requires the calculation of the charge densities at the finger surfaces and the integration of charges according to Coulomb’s law in order to derive the force distribution. Only a little work on analytical expressions for the resulting forces is available (e.g. Johnson and Warne [1995] and Yeh et al. [2000a]), because the inevitable idealization of the model entails extremely large inaccuracies. Most of the work is based on FEM and BEM simulations that allow a remarkably good insight into the consequences of the levitation effect. The starting point was the fundamental work of W. C. Tang and others [Tang et al. 1989, 1992, Tang 1990], who found a nearly linear dependency of the levitation force on the z-displacement. On the basis of the recent work of Chyuan [Chyuan 2008, Chyuana et al. 2004], the dependency of the force density per micrometer length on the overlapping area was verified for a test example with comb fingers of width 4 µm and height 2 µm (i.e. h = layer thickness) and with 2-µm gaps between combs and ground layer. The result is presented in Fig. 2.26. It shows that the levitation force FLz can be modeled by

71

2.4 Capacitive transducers

250 200

Force density

150 100 50 0 –50 –100 –150 0

2

4

6

8

10

12

14

16

18

20

z

Figure 2.26 Levitation force density fL z in ε0 /µm. Adapted from Chyuana et al.

[2004].

the equation FLz = fLz N Y,

fLz = χz ε0 V 2

z0 − z . z0

(2.193)

fLz is the force density per unit length of the overlapping area. N Y denotes the total length of the overlapping area and z0 the zero-levitation point. ε0 χz expresses the vertical drive capacity per unit length in micrometers. For the example presented, the value of χz is about 0.15 (≈ 0.19 according to Tang et al. [1992]). The zero-levitation point is at z0 ∼ 1.2 µm. The proportionality factor χz decreases with decreasing finger width and increases with decreasing fingerto-finger gaps. For a given comb configuration the exact values should be derived by an appropriate FEM simulation. The electrostatic force FLz corresponds to a positive spring constant kLz = −

∂FLz χz ε 0 = NY V 2 ∂z z0

(2.194)

and does not cause stability problems. It shifts the equilibrium point of a comb, which is governed by a mechanical spring rate Fz = −kz z, according to the force balance FLz + Fz = 0 from z = 0 to a state between z = 0 and z = z0 , z=

N Y χz ε0 V 2 z0 . N Y χz ε0 V 2 + kz z0

(2.195)

The z-shift reduces the capacitance between the fingers. As long as the voltage is constant, stable, and not too large, the effect is not really troublesome, particularly for combs with z0 < (0.1–0.2)H0 and for voltages in the range of some volts only. However, the situation is more critical for variable voltages.

Levitation in drive combs A drive comb, consisting of a movable central comb and two fixed half-combs, is presented in Fig. 2.27. With γz = N Y χz ε0 , the upper fingers in the figure

72

Transducers

Figure 2.27 A comb drive with turnable half-combs.

generate the levitation force FLz 1 = γz V12

z0 − z1 z0

(2.196)

and, correspondingly, the lower fingers provide FLz 2 = γz V22

z0 − z2 . z0

(2.197)

As long as the movable beam is yielding in the z-direction, but is very stiff with respect to rotations about the x-axis (i.e. z1 = z2 ), the forces can be added and create a levitation force proportional to V12 + V22 . For sinusoidal excitation the voltages are given, as usual, by V1,2 = VDC ± VAC with VAC = V0 sin(ωt); in this case the total force contains no component at frequency ω, but one at the double frequency 2ω. This component, together with the sinusoidal force in the y-direction, may create deflections in the z-direction at frequency ω, which, for instance, disturb the correct operation of vibratory gyroscopes. It can be suppressed by choosing a non-sinusoidal alternating voltage with the shape of a meander: V = V0 h(t), where ( µ ¶ X 1, for 0 ≤ t < T /2, T k (−1) 10 t − k h(t) = (2.198) , 10 (t) = 2 0, otherwise, k so that V 2 and the corresponding levitation force are constant. The limited stiffness with respect to torsional movements is another root cause for parasitic movements of the comb. It causes a time-varying reaction on the total torque M = (FLz 1 − FLz 2 )r. The torque is proportional to VDC V0 sin(ωt): µ · ¶ ¸ ¡ 2 ¢ z1 + z2 2 z1 − z2 2 M = rγz 4VDC V0 1 − sin(ωt) − VDC + V0 sin (ωt) . 2z0 z0 (2.199)

73

2.4 Capacitive transducers

(a)

V

(b)

-V

-V

V

Figure 2.28 (a) Differential drive with a biased shield. (b) Differential drive strips.

r denotes the average distance from the top or the bottom fingers to the xaxis, and z1 and z2 are the average deflections of the upper and lower fingers, respectively, in Fig. 2.27 during torsional movements. The tilt angle about the x-axis, θx , is θx =

z1 − z2 . 2r

(2.200)

The last term in Eq. (2.199) lies outside the operating frequency range of an oscillating drive. However, the first term causes tilting movements of the drive comb, which change the capacitances of the upper and lower half-combs. Since in resonant structures the torsional resonance is usually far away from the drive frequency, the driving excitation is transferred by the mechanical transfer function into oscillatory tilt movements that are approximately in phase with the excitation voltages. Being transferred mechanically to sensing combs, such inphase movements may cause distortions of the sensing capacitors and create contributions to the so-called quadrature error in gyroscopes.

Reduction of levitation forces The reduction of levitation forces is desirable, particularly for high-voltage drives. There are various techniques to reduce the field asymmetries or to compensate for levitation by mechanisms attracting the moving structure towards the shield. Tang et al. [1992] proposed differential biasing. This idea is based on the fact that the driving forces between two capacitor fingers are proportional to the square of the voltages. On applying on one of the fixed fingers the voltage V and on the adjacent fixed finger −V , the field configuration of the finger triple changes as shown in Fig. 2.28(a). The upper field lines of a first fixed finger no longer terminate on the top face of the movable finger, but do so on the adjacent fixed finger. Therefore, the upper fringe capacitance is split into two

74

Transducers

Fixed half-comb

Movable half-comb Figure 2.29 A comb drive with a partially biased shield.

parts: a main contribution between the fixed fingers and a significantly reduced part between fixed and movable fingers. The fringe fields, which are still present at the lower finger faces, can be reduced further by dividing the former ground shield into stripes that are alternately grounded and biased with V and −V , respectively. The fact that all lower surfaces on the fingers now are facing shield stripes loaded with the same potential completely eliminates direct finger-to-shield field lines in the intermediate space. There remains a weak residual force directed towards the substrate that is generated by field lines that connect lower surfaces of adjacent fingers; see Fig. 2.28(b). The last solution is technologically difficult, if not impossible, because it requires very narrow stripes and stripe-wiring. Another solution is based on the balance between levitation and attractive forces [Geen and Carow 2000], which is schematically presented in Fig. 2.29. The former ground shield is subdivided into areas that are grounded and some smaller areas that are biased with the driving voltage V and consequently attract the moving fingers. If the comb is stiff enough, the areas can be chosen in such a way as to balance the forces. Remarkably, the force balance is valid for any applied voltage. This method, obviously, was used in some industrial gyroscopes. A dynamic suppression of the drive modulation due to levitation forces is also possible. If the impact of the levitation on the transversal forces FCi , particularly due to the loss of overlap in the z-direction, is taken into account, the drive force for a half-comb can be approximated by FCy i = N ε0 Vi2

H0 − χz , D0

(2.201)

where χ should be determined experimentally. The drive-comb motion is balanced by the spring forces acting in the y-direction (ky y) and in the z-direction (kz z). Thus, voltages V1 and V2 can be determined for which a predetermined constant z-level (0 < z < zm ax ) and a desired motion y(t) can be established without distortion of the intended drive motion! (See, for instance, Timpe et al. [2008].)

References

75

It should be mentioned that fringe fields are not always undesirable. They may be used, for instance, in order to build asymmetric combs. In this case, one half-comb consists of fixed fingers with height H0 , whereas the fingers of the opposite row have only a fraction of this height. If the center of gravity of the second half-comb is additionally shifted a little bit upwards and the halfcomb is suspended in a way that allows one to perform small rotations about the lateral axis, the fringe fields create a large torque, which can be used to create considerable deflection angles. Such approaches were discussed for optical switches and similar applications [Yeh et al. 2000b]. For inertial sensors this technique is applied for some types of gyroscopes with out-of-plane excitation (see Chapter 8 and Adams et al. [2003]).

References Abbaspour-Sani, E., Huang, R. S., and Kwok, C. Y. (1994). A novel electromagnetic accelerometer. IEEE Electron Device Letters, 15(8):272–273. Adams, S. G., Groves, J., Cardarelli, D., Caroll, R., and Dauwalter, C. R. (2003). Electrically decoupled silcon gyroscope. US Patent 6,626,039 B1. Bao, M. H. (2000). Handbook of Sensors and Actuators, Micro Mechanical Transducers, volume 8. Amsterdam: Elsevier. Bassiri-Gharb, N., Fujii, I., Hong, E. et al. (2007). Domain wall contributions to the properties of piezoelectric thin films. Journal of Electroceramics, 19(1):47– 65. Beeby, S., Ensell, G., Kraft, M., and White, N. (2004). MEMS Mechanical Sensors. Boston: Artech House, Inc. Bohua, S. and Rui, Z. (2005). MEMS accelerometer with two thin film piezoelectric read-out, in International Conference on MEMS, NANO, and Smart Systems, p. 318. Brantley, W. A. (1973). Calculated elastic constants for stress problems associated with semiconductor devices. Journal of Applied Physics, 44:534–535. Chou, P. C. and Pagano, N. J. (1967). Elasticity: Tensor, Dyadic, and Engineering Approaches. New York: Dover Publications, Inc. Chyuan, S. W. (2008). Computational simulation for MEMS combdrive levitation using FEM. Journal of Electrostatics, 66:361–365. Chyuana, S. W., Liaoa, Y. S., and Chen, J. T. (2004). Computational study of variations in gap size for the electrostatic levitating force of MEMS device using dual BEM. Microelectronics Journal, 35:739–748. Dauderstadt, U. A., de Vries, P. H. S., Hiratsuka, R., and Sarro, P. M. (1995). Silicon accelerometer based on thermopiles. Sensors and Actuators A, 46/ 47(1–3):201–204. DeVoe, D. and Pisano, A. (2001). Surface micromachined piezoelectric accelerometers (PiXLs). Journal of MicroElectroMechanical Systems, 10(2):180–186.

76

Transducers

Draganovic, D. (1998). Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Reports on Progress in Physics, 61(9):1267– 1324. Franssila, S. (2004). Introduction to Microfabrication. New York: Wiley. Gad-el-Hak, M. (2002). The MEMS Handbook. Boca Raton, FL: CRC Press. Geen, J. and Carow, D. (2000). Micromachined gyros. US Patent 6,122,961. (Assignee: Analog Devices, Inc.) Hall, N. A., Okandan, M., Littrell, R. et al. (2008). Micromachined accelerometers with optical interferometric read-out and intgrated electrostatic actuation. Journal of MicroElectroMechanical Systems, 17(1):37–44. Haronian, D. (1999). Direct integration (DI) of solid state stress sensors with single crystal micro-electro-mechanical systems for integrated displacement sensing, in Technical Digest IEEE Workshop on Micro Electro Mechanical Systems (MEMS ’99). New York: IEEE, pp. 88–93. Huang, P. S., Ren, T. L., Lou, Q. W. et al. (2003). Design of a triaxial piezoelectric accelerometer. Integrated Ferroelectrics, 56(1):1115–1122. Iula, A., Lamberti, N., and Pappalardo, M. (1999). Analysis and experimental evaluation of a new planar piezoelectric accelerometer. IEEE/ASME Transactions on Mechatronics, 2:207–212. Johnson, W. and Warne, L. (1995). Electrophysics of micromechanical comb actuators. Journal of MicroElectroMechanical Systems, 4(1):49–59. Kagawa, Y., Wakatsuki, N., Tsuchiya, T., and Terada, Y. (2006). A tubular piezoelectric vibrator gyroscope. IEEE Sensors Journal, 6(2):325– 330. Kniffin, M. L., Wiegele, T. G., Masquelier, M. P., Fu, H., and Whitfield, J. D. (1998). Modeling and characterization of an integrated FET accelerometer, in Technical Digest IEE International Conference on Modeling and Simulation of Microsystems, Sensors and Actuators. Cambridge, MA: Computational Publications, pp. 546–551. Kovacs, G. T. A. (1998). Micromachine Transducers Sourcebook. New York: McGraw-Hill. Kraft, M., Lewis, C., Hesketh, T., and Szymkoviak, S. (1998). A novel micromachined accelerometer capacitive interface. Sensors and Actuators A, 68:466– 473. Kubena, R. L., Atkinsin, G. M., Robinson, W. P., and Stratton, F. P. (1996). A new miniaturized surface micromachined tunneling accelerometer. IEEE Electron Device Letters, 17(6):306–308. Kulah, H., Chae, J., Yazdi, N., and Najafi, K. (2006). Noise analysis and characterization of Σ∆ capacitive silicon micro-accelerometers. IEEE Journal of Solid-State Circuits, 41(2):352–361. Lei, S., Zorman, C. A., and Garverick, S. L. (2005). An oversampled capacitanceto-voltage converter IC with application to time-domain characterization of MEMS resonators. IEEE Sensors Journal, 5(6):1353–1361.

References

77

Li Yang, S. Z. (2009). Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature. Composite Structures, 87:257– 264. Macke, W. (1962). Mechanik der Teilchen – Systeme und Kontinua. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. Madni, A., Costlow, L., and Knowles, S. (2003). Common design techniques for BEI GyroChip quartz rate sensors for both automotive and aerospace/defense markets. IEEE Sensors Journal, 3(5):569–578. McCullom, B. and Peters, O. S. (1924). A new electric telemeter. Technology Papers of the National Bureau of Standards, 17(247). Middelhoek, S. and Audet, S.A. (1989). Silicon Sensors. New York: Academic Press. Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Oxford University Press. Pereyma, M. (2008). Mathematical model of piezoelectric MEMS accelerometer. MEMSTECH 2008, Polyana, Ukraine, 87:257–264. Petkov, V. P. and Boser, B. E. (2005). A fourth-order Σ∆ interface for micromachined inertial sensors. IEEE Journal of Solid-State Circuits, 40(8):1602–1609. Popov, E. P. (1968). Introduction to Mechanics of Solids. Englewood Cliffs, NJ: Prentice-Hall, Inc. (1999). Engineering Mechanics of Solids. Upper Saddle River, NJ: PrenticeHall Inc. Puers, R. and Lapadatu, D. (1996). Electrostatic forces and their effects on capacitive mechanical sensors. Sensors and Actuators A, 56(3):203–210. Sirohi, J. and Chopra, I. (2000). Fundamental understanding of piezoelectric strain sensors. Journal of Intelligent Material Systems and Structures, 11:246– 257. Smith, C. S. (1954). Piezoresistive effect in germanium and silicon. Physical Review, 94:42–49. (1958). Macroscopic symmetry and properties of crystals. Solid State Physics, Advances in Research and Applications, 6:175–249. Tang, W. (1990). Electrostatic comb drive for resonant sensor and actuator applications. Ph.D. thesis, Dept. EECS, University of California, Berkeley. Tang, W., Lim, M., and Howe, R. (1992). Electrostatic comb drive levitation and control method. Journal of MicroElectroMechanical Systems, 1(4):170–178. Tang, W., Nguyen, T. C., and Howe, R. (1989). Laterally driven polysilicon resonant microstructures. Sensors and Actuators, 20:25–32. Timpe, S., Hook, D., Dugger, M., and Komvopoulos, K. (2008). Levitation compensation method for dynamic electrostatic comb-drive actuators. Sensors and Actuators A, 143:383–389. Wakatsuki, N., Kagawa, Y., and Haba, M. (2004). Tri-axial sensors and actuators made of a single piezoelectric cylindrical shell. IEEE Sensors Journal, 4(1):102–107.

78

Transducers

Wang, Q. M., Yang, Z., Li, F., and Smolinski, P. (2004). Analysis of thin film piezoelectric microaccelerometer using analytical and finite element modeling. Sensors and Actuators A, 113:1–11. Weinberg, M. S. (1999). Working equations for piezoelectric actuators and sensors. Journal of MicroElectroMechanical Systems, 8(4):529–533. Wittenburg, J. (2008). Dynamics of Multibody Systems. 2nd edn. Berlin: Springer. Wu, J., Fedder, G. K., and Carley, √L. R. (2004). A low-noise low-offset capacitive sensing amplifier for a 50-µg/ Hz monolithic CMOS MEMS accelerometer. IEEE Journal of Solid-State Circuits, 39(5):722–730. Yazdi, N., Kulah, H., and Najafi, K. (2004). Precision readout circuits for capacitive microaccelerometers, in IEEE Sensors 2004 (Proceedings). Yeh, C. and Najafi, K. (1997). Micromachined tunneling accelerometer with a low-voltage CMOS interface circuit. Technical Digest IEEE International Conference on Solid-State Sensors and Actuators (Transducers ’97), pp. 1213– 1216. Yeh, J. L. A., Hui, C. Y., and Tien, N. C. (2000a). Electrostatic model for an asymmetric combdrive. Journal of MicroElectroMechanical Systems, 9(1):49– 59. (2000b). Electrostatic model for an asymmetric combdrive. Journal of MicroElectroMechanical Systems, 9(1):126–135. Yu, H., Zou, L., Deng, K., Wolf, R., Tadigadapa, S., and Trolier-McKinstry, S. (2003). Lead zirconate titanate MEMS accelerometer using interdigitated electrodes. Sensors and Actuators A, 107(1):26–35. Yu, J. and Lan, C. (2001). System modeling of microaccelerometer using piezoelectric thin films. Sensors and Actuators A, 88(2):178–186. Zou, Q., Tan, W., Kim, E. S., and Loeb, G. E. (2008). Single- and triaxis piezoelectric-bimorph accelerometers. Journal of MicroElectroMechanical Systems, 17(1):45–57.

3

Non-inertial forces

Force compensation is the basic principle of inertial MEMS. For instance, accelerations acting on a movable mass create deflections that are limited by counter forces. Without restoring forces the inertial mass would bump into the walls of the sparingly calculated measurement cavity. Hence, the exploitation of restoring forces is the key principle of all inertial sensors. For stationary accelerations or rate signals they allow the system to reach a spatially stationary equilibrium state of moving masses within the measuring room. Restoring forces typically are generated by elastic deformations.1 If the movable mass is suspended by cantilever beams, tethers, hinges or other elastic members with negligible masses, and if the deformation of the inertial object itself can be neglected, then the system can be described by a lumped-element approximation consisting of stiff members and elastic springs or hinges, which are described in terms of their spring rates. In most cases this approximation holds for the operating range of applied inertial forces. For overloads like shocks and large vibrations the deformation of all elements of the mechanical system – including the inertial masses – should be taken into account. Furthermore, for some systems, such as an elastic beam under acceleration, the mass of the beam itself creates inertial forces, so a lumped-element approximation is impossible. Some such systems will be considered later (see, for example, Section 7.4). Besides the intended restoring forces, inevitable damping forces caused by interaction of movable parts with a gaseous environment or by mechanical friction within the moving parts complement the following triad of forces acting within inertial sensors: inertial forces, proportional to the acceleration of the body; damping forces, typically proportional to the velocity of movable parts; and restoring forces, proportional to the mass deflection.

3.1

Springs There exists a great variety of elastic members used within micromechanical sensors. A comprehensive description can be found, for instance, in Lobontiu 1

In levitating disks, electrostatic compensating forces are used (e.g. Houlihan et al. [2001] and Kraft and Evans [2000]).

80

Non-inertial forces

and Garcia [2004]. The general theory of compliant mechanisms is presented in Howell [2001] (see also Timoschenko and Goodier [1970]). The most important compliant members in inertial MEMS are springs. Plates and membranes are also very popular for many MEMS, especially for pressure sensors, but they are rarely used for inertial MEMS. An ideal spring should have high compliance for one degree of freedom and infinite stiffness in the remaining five degrees of freedom (DOF). For instance, a beam should be compliant to forces along one of the axes perpendicular to the beam axis and should not respond to other forces and moments. It is desirable that a linear relation between force and displacement or between moment and torsion angle holds: Fz = kz z ∆z

Mθ x = kθ x θx ,

or

(3.1)

where kz z is the bending spring rate for deflections in the z-direction and kθ x is the torsional spring constant for a rotation about the x-axis. The ratios ∆z/Fz = 1/kz z and θx /Mθ x = 1/kθ x are called compliances.

3.1.1

Beams Since spiral-shaped or z-folded springs are difficult to manufacture, the basic spring types in inertial MEMS are made up of beams, normally with uniform cross-section. The planar nature of most of the MEMS technologies limits the z-dimensions of beams. They can be long and wide, but not high. The length may be limited by the action of electrostatic forces that attract the beam towards the substrate. The minimal width is limited by technological restrictions. A wide beam can be made quite insensitive against in-plane forces and torques about the z-axis. In general, the limited aspect ratios and the necessarily small widths of in-plane bending beams make them susceptible mainly to moments about the beam axis and to forces in the out-of-plane direction. Therefore, multiple beam structures based on singular beams or beam-chains are necessary in order to limit the compliance of a suspension in undesired directions. However, it must be noted that cross-coupling effects in beams are generally unavoidable. For instance, the relation between forces and moments acting on a beam on the one side, and linear deflections and rotations on the other, is illustrated in Fig. 3.1. Assuming that the beam bends and twists in the x−zplane, only Fz and My are different from zero. Accordingly, Fz = kz z ∆z + kz θ y ∆θy

and

My = kz θ y ∆z + kθ y ∆θy ,

(3.2)

or µ

Fz My

¶

=

µ

kz kz θ y kz θ y kθ y

¶µ

∆z ∆θy

¶ .

(3.3)

3.1 Springs

(a)

81

(b)

Figure 3.1 Beams under different loads. (a) A beam under a force load. (b) A beam

under a moment load.

One sees that the same deformation can be generated by a force (Fig. 3.1(a)) or by a moment (Fig. 3.1(b)). kz reflects the desired effect of linear deflection under force, whereas kθ y represents the intended rotation under an applied moment. The rate kz θ y characterizes the cross-coupling effect: an applied force generates not only a deflection but also a rotation; and vice versa, an applied torque creates besides the rotation also a displacement. In any case, a singular beam is a far cry from a one-DOF idealization. At best, it can approximate it for kz z ∆z ≫ kz θ y ∆θy or for kθ y ∆θy ≫ kz θ y ∆z. The acceptability of the inherent cross-coupling has to be checked for any spring design.

3.1.2

The stiffness matrix For more complicated suspensions consisting of many springs, movements in one or more directions may be intended. Consequently, the full stiffness matrix is required. This consists of the intentionally non-zero spring rates as well as the cross-coupling terms: µ ¶ µ ¶ ∆¯ r F¯ (3.4) ¯ = KF ,M ∆θ¯ , M       ∆x kxx kxy kxy kxθ x kxθ y kxθ z Fx    F  k  y   xy ky y ky z ky θ x ky θ y ky θ z   ∆y         Fz   kxz ky z kz z kz θ x kz θ y kz θ z   ∆z  (3.5) . = =   Mx   kxθ x ky θ x kz θ x kθ x kθ x θ y kθ x θ z   ∆θx         My   kxθ y ky θ y kz θ y kθ x θ y kθ y kθ y θ z   ∆θy  ∆θz kxθ z ky θ z kz θ z kθ x θ z kθ y θ z kθ z Mz

In accordance with the theorem of Maxwell and Betti, the matrix is symmetric. It should be noted that the coefficients kij have different dimensions, depending on whether they are related to bending or to torsion, or to both. For practical applications two cases are of special interest: movement of a body within the x–y-plane that include rotations about the z-axis; and out-ofplane movements with rotations about both in-plane axes and deflection in the z-direction. The corresponding spring matrixes are given by   kxx kxy kxθ z (3.6) Kin-plane =  kxy ky y ky θ z , kxθ z ky θ z kθ z

82

Non-inertial forces

(a)

(b)

(c)

Figure 3.2 (a) A fixed–guided beam. (b) The deformation principle. (c) A symbolic

representation.

Kout-of-plane

 kz z kz θ x kz θ y =  kz θ x kθ x kθ x θ z . kz θ y kθ x θ z kθ y 

(3.7)

The off-diagonal elements of the stiffness matrix play a crucial role, especially in gyroscopes, where they are responsible for some very important parasitic effects. In the case of parallel or serial beam connections their emergence is related to imperfections in forming a constant, rectangular cross-section. Manufacturing flaws generate axis misalignment leading to skewed beam bending, or sidewall tilt with a similar effect. Often analytical models are missing and careful FEM simulations must be carried out for in-depth investigations of the most important secondary effects.

3.1.3

The bending equation for beams An elastic beam can be exploited under different load conditions. It can be clamped2 at one or both ends and can be subjected to distributed and concentrated point loads. Hence, different boundary conditions and load applications must be used. In inertial MEMS the typical applications are

r a cantilever with a free end r a cantilever with a guided end (fixed-guided beam) r a bridge (beam, clamped at both ends). The cantilever beam is clamped only at one end, whereas the fixed–guided beam, as shown in Fig. 3.2, is clamped at one end and guided at another. The guidance changes the boundary condition by forcing the displacement slope to be zero not only at the clamped end but also at the guided end. 2

Clamping fixes the position and the slope of the beam at the clamped end, whereas a support limits only the position.

83

3.1 Springs

(a)

(b)

Figure 3.3 Differential relations along a beam. (a) Definitions of cutting forces and loads. (b) Differential relations.

Internal forces and moments In order to derive a differential equation governing beam bending, the internal forces and moments have to be considered first. Figure 3.3(a) illustrates their definition. A virtual cut creates a left positive and a right negative cutting edge. For the positive cutting edge the coordinates correspond to (positive) coordinate axes. At the negative cutting edge the coordinates are inverted. Positive forces and moments – the normal force, transverse (shear) force, and bending moment – are orientated in line with the corresponding coordinate axes. Forces and moments at the cutting edges, together with load forces and reaction forces in the supports, keep the two parts in static equilibrium. As usual in technical mechanics the z-axis is orientated “down,” i.e. in the direction of a reference fiber, which is drawn on the bottom of the framework under consideration.3 The bending moment is defined positive if the reference fiber is strained. Correspondingly, the bending torque on a positive cutting edge acts counterclockwise (right-handed about the y-axis) and that on a negative cutting edge acts clockwise.

Differential relations of a bent beams The differential relations along the beam can be easily understood considering an infinitesimally small slice as shown in Fig. 3.3(b). The force and moment balances deliver

FS ∆x +

q(x)∆x + ∆FS = 0

⇒

1 ∆FS ∆x − ∆Mby = 0 2

⇒

∂FS = −q(x), ∂x ∂Mby FS = , ∂x

(3.8) (3.9)

where q(x) is the distributed load. The index y in the notation Mby is added to emphasize the moment orientation, which acts about the y-axis. 3

For lateral bending in the y-direction the indices “z” and “y” must be interchanged.

84

Non-inertial forces

(a)

(c)

(b)

Figure 3.4 (a) A cross-section of a beam. (b) An unbent beam. (c) A bent beam.

Bending moment and internal stress A bending moment Mby about the y-axis can be created only by normal forces within the beam’s cross-section, not by shear forces. In Fig. 3.4 a bent beam is presented. The bending moment is the sum of all partial moments dMby = σxx z dS = σz dS generated by the stress distribution over the cross-section S shown in Fig. 3.4(a): Z zσ(z)dS. (3.10) Mby = S

The bending moment is equivalent to a pair of forces within the upper and lower parts of the cross-section. Hence, a positive bending moment produces compressive stress in the upper part and tensile stress in the lower part. Between these regions there exists a neutral line where the normal stress remains zero. In an unloaded case, the neutral line is located at z = 0 if the origin of the z-axis is located at the cross-section’s center of gravity, i.e. if Z z dS = 0. (3.11) S

The stress distribution σ(z) is found by assuming the validity of Bernoulli’s hypothesis. Bernoulli argued that all of the points which initially are located in a planar cross-section perpendicular to the longitudinal beam axis x will, after pure bending (no axial torsion), remain in a plane normal to the deformed beam axis as indicated in Figs. 3.4(b) and (c). In other words, cross-sectional planes behave like rigid planes that rotate when a bending moment about the y-axis is applied. Therefore, the elongation of longitudinal fibers in the beam is linearly proportional to the distance from the neutral line. Since strain and stress are linked by the linear form of Hooke’s law, the stress also changes linearly with z: σ(z) = constant × z.

(3.12)

85

3.1 Springs

The constant can be found by considering Eq. (3.10): Z Mby constant = z 2 dS. , Iy y = Iy = Iy y S

(3.13)

Iy y = Iy is the geometric moment of inertia of the beam’s cross-section (i.e. R 2 z dS, I one of the area moments defined in the y–z-plane with I = z = y S R R 2 yz dS). y dS, and I = − yz S S The stress distribution can now be written as σ(z) =

Mby z. Iy

(3.14)

It has to be noted that this equation is valid under the following conditions:

r the origins of the y- and z-axes are located at the center of gravity of the beam’s cross-section; and

r the cross area moment is zero: Iy z = − R yz dS = 0. S

Iy z is the second-order biaxial moment (centrifugal moment). The last condition defines the main axes of the cross-section and simply states that bending stress does not create a moment about the z-axis. If this condition is not fulfilled, the bending moment is skewed and can be projected onto the main axes: σ(z) =

Mby Mbz z+ y. Iy y Iz z

(3.15)

In the linear approximation, the bending in the z- and y-directions can be calculated as a superposition of two orthogonal bending processes. Here the case of bending about the y-axis is considered first.

The Bernoulli equation To analyze the beam displacement, pure bending is assumed (with a constant bending moment). According to Bernoulli’s hypothesis, an arbitrary longitudinal fiber with distance z from the neutral axis and initial length ∆x = |R|∆ϕ elongates under bending by the increment ∆l (see Figs. 3.4(b) and (c)): ∆l = (|R| + z)∆ϕ − ∆x = z ∆ϕ.

(3.16)

while the neutral fiber retains the initial length ∆x = |R|∆ϕ. R is the radius of curvature of the neutral axis under bending. The strain of fiber elongation is4 ε=

z ∆ϕ z =− . ∆x R

(3.17)

On the other hand, with Eq. (3.14) and Hooke’s law the following is obvious: σ=

4

Mby z z = Eε = −E , Iy R

Note that with the axis convention specified above dx = −R dϕ.

(3.18)

86

Non-inertial forces

which yields the relationship between bending moment and deformation Mby = −EIy

1 . R

(3.19)

The curvature of the neutral line is proportional to the bending moment. The proportionality factor is the bending stiffness, which is the product of the modulus of elasticity and the moment of inertia: EIy . For practical applications it is more convenient to use displacements of beams w(x) instead of curvatures. The mathematical relationship between the displacement curve w(x) and the radius of curvature R is given by the formula 1 w′′ = 3 . R [1 + (w′ )2 ] 2

(3.20)

Substitution into Eq. (3.19) delivers a nonlinear differential equation for the beam displacement, w′′ (x) +

3 Mby [1 + (w′ )2 ] 2 = 0. EIy

(3.21)

In inertial MEMS spring displacements are small, and so is the slope w′ (x). This justifies the simplification EIy (x)w′′ (x) = −Mby (x),

(3.22)

which is the well-known Bernoulli equation (often called the Euler–Bernoulli equation). This differential equation is normally used in the context of MEMS. It is common to allow for variable moments of inertia in this equation, although this is not fully consistent with Bernoulli’s hypothesis. However, the longer the beam is relative to the lateral beam extensions, the smaller the warping of the cross-sectional planes becomes. Thus, Bernoulli’s hypothesis is the more accurate the longer the beam is in comparison with the typical deflection. The deflection error exceeds 10% if the deflection becomes on the order of 30% of the length [Gere and Timoshenko 1984]. ′′ = FS′ = −q(x)), the second-order differential equaIn light of Eq. (3.8) (Mby tion can be transformed into [EIy (x)w′′ (x)]′ = FS , and by a further differentiation into [EIy (x)w′′ (x)]′′ = q(x).

(3.23)

It is a matter of convenience which of Eqs. (3.22) and (3.23) is used for solving a given problem. Equation (3.23) requires the determination of four boundary conditions, whereas Eq. (3.22) needs a-priori determination of the bending moment for all x. From Eqs. (3.14) and (3.22) the relation between stress and displacement follows: σ(z) = −Ezw′′ (x).

(3.24)

87

3.1 Springs

If the acting force has to be estimated by stress measurement, e.g. by using piezoresistors at the beam’s surface, this relation allows one to link the measured variable with the acting load.

3.1.4

Cantilever beams Cantilevers under different loads A cantilever beam of length L can be subjected to a distributed load q(x) and a point load. As illustrated in Fig. 3.3(a), point loads usually are applied at the free end at x = L as a force Fz or as a bending moment about the y-axis.

Boundary conditions Assuming that the beam’s cross-section does not depend on x (Iy = constant), differentiation of Eq. (3.22) yields ∂Mby = −FS (x), ∂x ∂FS EIy w′′′′ (x) = − = q(x). ∂x EIy w′′′ (x) = −

(3.25) (3.26)

These equations allow one to derive boundary conditions for a point-force or point-torque load, for instance, at the free end. In general, the differential equation (3.23) has to be complemented by boundary conditions at the clamped end, w(0) = 0,

w′ (0) = 0,

(3.27)

and at the free end, w′′ (L) = 0, ′

w (L) = 0, ′′

EIy w (L) = −M, ′′

w (L) = 0,

EIy w′′′ (L) = −Fz

for force loading of the cantilever,

′′′

for force loading at the guided end,

′′′

for moment loading of the cantilever,

′′′

for the bridge (double-clamped beam). (3.28)

EIy w (L) = −Fz w (L) = 0 w (L) = 0

In the case of a distributed load the boundary conditions are homogeneous, w(0) = w′ (0) = w′′ (L) = w′′′ (L) = 0. Bent bridges must obey symmetric boundary conditions, w(0) = w′ (0) = w(L) = w′ (L) = 0. The bridge load is usually a distributed force or a point load at position x0 , which can be represented by a delta function q(x) = Fz δ(x − x0 ).

Load–deflection relations R If f1 (x) is the integral function of q(x) (f1 (x) = q(x)dx), f2 (x) the integral function of f1 (x), and f3 (x) the integral function of f2 (x), integration of

88

Non-inertial forces

Eq. (3.26) generates a set of equations EIy w′′′ (x) = f1 (x) + C1 , EIy w′′ (x) = f2 (x) + xC1 + C2 , x2 C1 + xC2 + C3 , (3.29) 2 x3 x2 EIy w(x) = f4 (x) + C1 + C2 + xC3 + C4 , 6 2 where the integration constants have to be determined using boundary conditions (3.27) and (3.28). The corresponding solutions are summarized in Table 3.1. For the bridge the integral functions fi (x) have a discontinuity in order to fulfill Eq. (3.26): µ ¶ L (x − L/2)i−1 fi (x) = Fz h x − , i ≥ 1, 2 (i − 1)! EIy w′ (x) = f3 (x) +

where h(x) is the unit-step (Heaviside) function ( 0 for x < 0, h(x) = 1 for x > 0.

(3.30)

The spring rate for the guided beam is four times higher than for a free-end cantilever and, as is intuitively clear, is highest for a doubly clamped bridge. Where desired, the cross-coupling stiffness according to Eq. (3.2) can be easily derived by calculating the ratio between the slope at the beam’s tip and the applied load.

Rectangular beams The derived expressions have to be specified for the most common cross-sections. For a cantilever beam with height h and z-dependent width b(z), the moment of inertia is Z h 2 Iy = z 2 b(z)dz. (3.31) − h2

A rectangular beam of width b possesses the moments of inertia Iy = bh3 /12 and Iz = b3 h/12 and the corresponding stiffness values along the three axes 1 bh3 1 b3 h bh E 3, ky = E 3 , kx = E . (3.32) 4 L 4 L L Therefore, the ratio between the spring rates for z- and y-deflections of a freeend or a guided beam is kz /ky = h2 /b2 . To make the beam much stiffer in the y-direction than in the z-direction one requires b ≫ h. The axial stiffnes, kx = 4 (L2 /h2 ) kz , is orders of magnitude larger then the lateral stiffness (L ≫ h, L ≫ b). kz =

Bridge with point load Fz at x = L/2

Moment M at the free end

Force Fz at the guided end

Force Fz at the free end

Constant distributed load q(x) = q0

Load

Table 3.1. A beam under different loads

x2 (3L − 2x) 12EIy

x2 (3L − x) 6EIy

M

x2 2EIy " ¶3 # ¶µ µ L L Fz 2 3 x− 3Lx − 4x + 8h x − 2 2 48EIy

Fz

Fz

Deflection w(x) x2 (6L2 − 4xL + x2 ) q0 24EIy

w

L3 3EIy

µ ¶ L = Fz 1 2

L2 2EIy

L3 12EI M

Fz

Fz

Tip deflection L q0 EIy 8E

90

Non-inertial forces

(a)

(b)

(c)

Figure 3.5 Different cross-sections of MEMS beams: (a) rectangular, (b) trapezoidal,

and (c) with tilted sidewalls.

Non-rectangular beams The specific features of isotropic etching of silicon bulk material entail that crosssections of beams that originally were designed to be rectangular emerge as trapezoids or with tilted sidewalls, as illustrated in√Fig. 3.5. The parameter d of the bulk-micromachined trapeze is given by d = ( 2/2)h, and θ is the sidewall inclination. To keep the notation of the axes in accordance with the previously used indices, the cross-section of the real manufactured beam with a small inclination θ of the vertical sidewalls is rotated by 90◦ , resulting in a horizontally inclined sidewall. The area moments of the trapezoidal beam can be easily calculated taking into consideration that the center of gravity is shifted by z0 = −dh/6b, µ ¶ µ ¶ bh3 d2 hb3 d2 Iy ,trap ez = 1− 2 , Iz ,trap ez = 1+ 2 , Iy z ,trap ez = 0, 12 3b 12 b (3.33) where b = (b1 + b2 )/2 and d = (b1 − b2 )/2. For the with tilted sidewall beam the main area moments are Iy ,sidewall tilt =

bh 2 bh3 (h + tan2 θ b2 ) ∼ , = 12 12

Iz ,sidewall tilt =

b3 h 12

(3.34)

and Iy z b2 b2 ∼ = tan θ 2 θ . = 2 Iy h + tan θ b2 h2

(3.35)

The mixed moment of inertia of a beam with height b = 10 µm, thickness 2 µm and sidewall tilt of 0.1◦ amounts to 4.4% of the moment Iy . This causes a coupling of 4.4% of the lateral beam’s deflection into an out-of-plane movement. Summarizing, symmetric cross-sections may cause a shift of the central beam axis, whereas asymmetric cross-sections cause a mixed moment of inertia that is the root cause for off-diagonal elements in the stiffness matrix.

Skew beam bending and asymmetric suspensions Skew beam bending and the more general case of skewed deflections of complex suspensions is one of the most dangerous imperfections in designing highperformance gyroscopes. To derive some quantitative relations the cross-coupling

3.1 Springs

91

Figure 3.6 The cross-section and main axes of a skewed beam.

of the intended motion into a parasitic motion can be analyzed on the singular beam level, or – in the best case – on the level of simple spring chains according to Fig. 3.12. For complex suspensions few general relations are available. Skew beam bending is an effect where deflections along two orthogonal axes are caused by only one force or torque component along one of these axes. Let’s consider first the case of two orthogonal forces. The equations in Table 3.1, derived for deflections in the z-direction, can be applied to the orthogonal direction y by substituting Iy by Iz . Since superposition holds, arbitrary beam bending can be calculated, projecting the applied force onto the main axes y and z and adding the corresponding responses. No cross-effects arise.

Principal axes and principal moments of inertia However, for asymmetric cross-sections the biaxial moment Iy z is not zero, and the main axes do not coincide with the coordinate axes. In this case, the main axes y ′ and z ′ of an asymmetric cross-section can be determined by rotating the coordinate system by the angle ϕ as illustrated in Fig. 3.6. The coordinates of a point in the rotated coordinate system are given by y ′ = y cos ϕ + z sin ϕ,

z ′ = −y sin ϕ + z cos ϕ.

(3.36)

Using this fact, moments of inertia in the rotated coordinate system Iy ′ , Iz ′ , and Iy ′ z ′ , Z Z Z y ′ z ′ dS, (3.37) y ′2 dS, Iy ′ z ′ = − z ′2 dS, Iz ′ = Iy ′ = dS

dS

dS

can be expressed in terms of the corresponding moments in the initial system. Simple calculations show that 1 (Iy + Iz ) + 2 1 = (Iy + Iz ) − 2

Iy ′ = Iy ′ y ′ = Iz ′ = Iy ′ y ′

1 (Iy − Iz )cos(2ϕ) + Iy z sin(2ϕ), 2 1 (Iy − Iz )cos(2ϕ) − Iy z sin(2ϕ), 2

1 Iy ′ z ′ = − (Iy − Iz )sin(2ϕ) + Iy z cos(2ϕ). 2

(3.38) (3.39) (3.40)

92

Non-inertial forces

To define the main axes, ϕ has to be chosen to bring the biaxial moment to zero, Iy ′ z ′ = 0, tan(2ϕ) =

2Iy z . Iy − Iz

(3.41)

The inertial moments with respect to the main axes are correspondingly " # q 1 2 2 Iy ′ (z ′ ) = Iy + Iz ± (Iy − Iz ) + 4Iy z . (3.42) 2 The sign “−” corresponds to the moment Iz ′ . The non-principal moments can be expressed in terms of the principal moments and the tilt angle ϕ of the principal axes: Iy =

Iy ′ + Iz ′ Iy ′ − Iz ′ Iy ′ + Iz ′ Iy ′ − Iz ′ + cos(2ϕ), Iz = − cos(2ϕ), 2 2 2 2 Iy ′ − Iz ′ Iy z = sin(2ϕ). (3.43) 2

The same relation holds for the spring rates.

The stiffness matrix for rotated principal axes Indeed, for forces and displacements of an arbitrary suspension that act along the principal axes, the relation ¶ µ µ ¶ µ ¶ kz ′ 0 ∆z Fz = = Fy 0 ky ′ ∆y holds. No cross-coupling terms exist. Rotating first the acting forces by the angle ϕ on the principal axes and re-projecting the reactions onto the actual coordinates yields the real stiffness matrix, µ ¶µ ¶µ ¶ c −s kz ′ 0 c s K= , s c 0 ky ′ −s c where c = cos ϕ and s = sin ϕ. On performing the matrix multiplications one gets   kz ′ + ky ′ kz ′ − ky ′ kz ′ − ky ′ − cos(2ϕ) − sin(2ϕ)   2 2 2 K=  kz ′ − ky ′ kz ′ + ky ′ kz ′ − ky ′ − sin(2ϕ) + cos(2ϕ) 2 2 2 ¶ µ 3E Iz −Iy z = 3 . (3.44) −Iy z Iy L The first relation of Eq. (3.44) between principal stiffness and real stiffness is valid for any suspension because it was derived without using any information on the nature of the suspension. It can be used for investigating the impact of cross-coupling stiffness elements (anisoelasticity) on parasitic effects in gyroscopes.

3.1 Springs

93

Figure 3.7 The moment Mσ0 created by residual stress.

If applied to a single beam, the second equation of (3.44) shows that the cross-coupling terms are proportional to the cross moment of inertia Iy ,z , i.e. to the asymmetries in the beam’s cross-section with respect to the force–deflection axes. For well-designed, complex suspensions consisting of multiple springs or springchains, the dominant reason for the emergence of off-diagonal stiffness elements is differences in the constituting springs, caused by manufacturing imperfections, including varying cross-sections over the length of the individual beams, sidewall tilts etc.

Residual stress in bending beams If elastic members are formed out of stressed materials like polysilicon or thinfilm silicon, the “frozen” residual stress may significantly change the compliance. The origins of residual stress include thermal mismatch during production, lattice mismatch between layers or between the material and its sacrificial layer, substitutional or interstitial impurities disturbing the homogeneous structure, non-uniform plastic deformation and so on. The stress may vary from MPa up to some hundreds of MPa and – if balanced by the reaction forces of supports – may lead to buckling or cracking. Here, structures below the buckling limit will be considered. Besides frozen stress, stress gradients within a structural layer may exist. Stress gradients lead, for instance, to static bending of free-standing beams. Apart from some exceptions such as processes based on active layers with embedded metal stripes, modern MEMS processes used for manufacturing of elastic members exhibit small stress gradients below 0.05–0.5 MPa/µm. Hence, the impact of stress gradients is not considered here.

The beam equation with residual stress A prestressed cantilever beam is shown in Fig. 3.7. The residual stress in the axial direction is represented by the force arrow Fσ0 : Fσ0 = σ0 S,

(3.45)

94

Non-inertial forces

where σ0 is the uniform residual stress and S the area of the cross-section. σ0 is positive in the case of tensile stress and negative in the case of residual compression. The force Fσ0 generates an additional bending moment Mσ0 = Fσ0x w(x), whereas Fσ0x = Fσ0 cos(arctan w′ (x)) ∼ = Fσ0 is the projection of the residual stress force onto the x-axis. Since the bending slope is very small, the approximation Mσ0 = Fσ0x w(x) holds. It should be noted that, in the case of large residual stress, the full nonlinear relation must be used. The bending moment Mσ0 is acting against the moment Mby and, consequently, has to be taken with negative sign. At a given position the local moment Mby is MbΣ = Mby − σ0 Sw(x).

(3.46)

Since for a local bending moment the Euler equation (3.22) remains in force, the resulting expression has the form EIy (x)w′′ (x) = −Mby + σ0 Sw(x)

(3.47)

or w′′ (x) − κ2 w(x) = −

Mby , EIy

κ2 =

|σ0 |S . EIy

(3.48)

For a homogeneous beam the fourth-order differential equation yields w′′′′ (x) − κ2 w′′ (x) =

q . EIy

(3.49)

(See also, for example, Landau and Lifschitz [1986], Clark [1997], Kuehnel [1995], and Osterberg and Senturia [1997].)

The impact of residual stress First, the case of residual tensile stress is considered: σ0 > 0. Thereby the parameter κ is a real number and the solution of the fourth-order homogeneous equation can be written as w′′ (x) = C1 sinh(κx) + C2 cosh(κx). Further integration leads to the demanded function w(x) =

C1 C2 sinh(κx) + 2 cosh(κx) + C3 x + C4 . κ2 κ

(3.50)

In the case of residual compressive stress σ0 < 0, and κ has to be substituted by jκ. The coefficients of the total solution should be determined in accordance with boundary conditions (3.27) and (3.28) (the first two equations). For example, the solution for a fixed guided beam is given by  1 − cosh(κL)   (1 − cosh(κx)) − sinh(κx) + κx, σ0 ≥ 0 Fz  sinh(κL) . (3.51) w(x) = 3 1 − cos(κL) κ EIy    (1 − cos(κx)) + sin(κx) − κx, σ0 ≤ 0 sin(κL)

3.1 Springs

The maximal deflection at x = L is  Fz 1 − cosh(κL)    κ3 EI × 2 sinh(κL) + κL, y w(L) =  F 1 − cos(κL) z   3 ×2 − κL, κ EIy sin(κL)

95

σ0 > 0, (3.52) σ0 < 0.

and the spring rate results with kz = Fz /w(L). On substituting Iy by Iz one gets the spring constant ky . Serial expansion reveals the order of magnitude particularly for small amounts of stress. The first three members of the maximal deflection at x = L are " # µ ¶ 2 2 2 F σ SL σ SL z 0 0 3 w(L) ∼ 1 − 0.1 + 0.0101 − ··· . (3.53) =L 12EIy EIy EIy If the residual stress is zero, the equation is in accordance with the third equation in Table 3.1. Equation (3.53) is valid for positive and negative stress values. Thus, for tensile stress the spring stiffness increases, whereas for compressive stress the spring becomes weaker. For compressive stress the limits of validity of Eq. (3.52) are given by κL ≤

π 2

or

|σ0 |S ≤

π2 EIy . 4L2

(3.54)

Limits of the linear approximation For larger compressive stresses the denominator of the second equation in (3.52) decreases and the deflection starts to grow, eventually approaching instability limits; this indicates that the limits of validity for a linear approximation have been reached and that the onset of buckling is incipient (see Section 3.1.6). The limit at which buckling takes place is four times larger than the indicated buckling onset value and is called the Euler buckling limit. It corresponds to the value κL = π at which reaction forces can no longer balance the stress associated with a cosine-shaped deflection. The tendency for instability is the result of reaction forces at the second end of the guided beam and is also present for bridges. Thus, not only residual stress may cause such behavior. The same effect takes place if a compressive axial force is applied. Such a force is equivalent to the stress-induced force Fσ0 = σ0 S and causes the same bending moments. In contrast, tensile stress does not cause any instability or buckling problems. An example should demonstrate the impact of residual stress. Assuming a rectangular beam’s cross-section with length-to-height ratio L/h ∼ 50, the expression for κL becomes r σ0 L2 κL = 12 . (3.55) E h2 In standard MEMS processes residual stress on the order of 50 MPa could be expected. With the elasticity of polysilicon E ∼ 160 GPa and σ0 ∼ 50 MPa,

96

Non-inertial forces

Figure 3.8 A slice of a torsion bar.

the value of κL is approximately 3.0. Compared with an unstressed beam, the corresponding displacement reduction is 47%. This example shows that residual stress may increase the spring rate remarkably. The improvements of polysilicon deposition and annealing processes in the 1990s, however, have further reduced the residual stress, pushing the impact on the spring rate down to the percent level. The impact of residual stress can be mitigated by structural measures. For instance, they can aim at avoiding reaction forces at the second end of a spring construction. So stress relief is facilitated by using folded beams instead of fixed guided ones.

3.1.5

Torsion springs Cylindrical torsion bars The simplest torsional flexure would be a full cylinder similar to the hollow cylinder used for the shear test described in Chapter 2, in the section on “Shear stresses,” Fig. 2.2(a). It can be assumed that, under an applied external moment, the cross-sectional planes of the bar remain planar and are only rotated with respect to each other, creating the constant shear angle γ. Thus, no bowing exists. As illustrated in Fig. 3.8, the torsion angle θ along the bar slice with radius r changes by dθ according to the relation γ dx = r dθ

⇒

γ=r

dθ . dx

(3.56)

The shear stress is subject to Hooke’s law, σt = Gγ, and, since it is constant over the cross-section, it generates the moment Z Z dθ rGr dS = GIP θ′ . rσt dS ⇒ Mt = Mt = (3.57) dx S S Here IP is the polar geometric moment of inertia, Z r2 dS = Iy y + Iz z , IP = S

(3.58)

3.1 Springs

97

and GIP denotes the torsional stiffness of the cylindrical bar. The polar moment of a full cylinder is π IPcyl = R4 2 and that of an annular cylinder is π IPannul = (R24 − R14 ), (3.59) 2 where R2,1 are the outer and inner radii, respectively. One obtains the torsion angle of the bar with length L by integration over x, Mt =

GIP θ, L

(3.60)

which leads to a torsional spring rate ∂Mt GIP = . (3.61) ∂θ L The cylindrical bar presented here is a nice example for introducing torsional flexures but is of little interest for implementation in MEMS technologies. Technologically, cylindrical cross-sections are possible only within the plane of a wafer and, thus, have lengths that are too short in the out-of-wafer-plane direction. Therefore, most torsional springs that are designed for twisting about in-plane axes have a rectangular or near-rectangular cross-section. In addition, the presented results on the cylindrical bar are based on the assumption that there is no warping of cross-sectional planes. In the next section this assumption will be weakened and substituted by presupposing constant warping along the beam according to Saint-Venant’s theory. Only full cylinders remain without bulging, and IPcyl delivers correct results. For hollow cylinders an improved relation will be given in the section on “Cylindrical bars.” kθ =

Torsion bars with arbitrary cross-section The cross-sectional planes in torsion bars without central symmetry undergo bowing. Ideally, the bow is identical for any cross-section. However, if one or two ends of the flexure are clamped, the bulge is disturbed or eliminated at the clamped ends. In this case the full three-dimensional deformation has to be found, by solving the exact equations of elasticity. For long flexures, disablement of the end bulge can be neglected, assuming uniform torsion according to the theory of Saint-Venant. Thus, the introduction of an external moment does not disturb the bulge shape. Consequently, the twisting angle γ is constant, which leads to a torsion angle that increases in proportion to x, dθ(x) x = θ′ x. (3.62) dx The analysis is based on consideration of warped cross-sections that are independent of x, and basically follows Wittenburg and Pestel [2001]. θ(x) =

98

Non-inertial forces

(a)

(b)

Figure 3.9 A segmental cut through a torsion bar and a corresponding slice.

After torsion, the point P(x, y, z) is shifted to P(x + u, y + ν, z + w). This is shown in Fig. 3.9(a). Taking into account the similarity of the triangle 0– A–P(x, y, z) to the triangle P(x, y, z)–B–P(x + u, y + ν, z + w) constituted by the deflections w(x, y, z), −ν(x, y, z), and rθ(x), displacements w and ν can be related to (small) torsion angles θ by the equations −ν z = , rθ(x) r

w y = , rθ(x) r

(3.63)

which, by comparison with Eq. (3.62), yields ν = −θ′ xz,

w = θ′ xy.

(3.64)

Now, shear strains εxy and εxz in the y–z-plane can be determined and related to shear stresses σxy and σxz according to Hooke’s law: µ ¶ µ ¶ 1 ∂ν ∂u 1 ∂u σxy ′ εxy = + = −θ z + = , 2 ∂x ∂y 2 ∂y 2G (3.65) µ ¶ µ ¶ 1 ∂w ∂u 1 ∂u σxz ′ εxz = + = θy+ = . 2 ∂x ∂z 2 ∂z 2G The dislocation u(y, z) represents the cross-sectional bulge. It changes the torsional stiffness in comparison with that of a non-warped bar. For further calculation the axial deformation u has to be either determined or excluded. To do so, a natural equilibrium condition for shear stress along an arbitrary smooth curve that cuts the cross-section is used. In Fig. 3.9(b) the shear stress σxn denotes stress orientated perpendicular to the cutting curve L(l) which cuts free the lower part of the dx-bar slice. Since the cut-free part is in equilibrium, the stress σxn along the x-axis is equal to the shear stress in the x-plane as indicated in Fig. 3.9 (no resulting moment). The total force in the x-direction must be zero: µZ C ¶ Z C σxn (l)dl = 0. (3.66) σxn (l)dl dx = 0, ⇒ A

A

3.1 Springs

99

The integral from point A to point P, shown in Fig. 3.9(b), represents the socalled stress function. More precisely, the function Φ, defined as Z P(y ,z ) 1 Φ=− σxn (l)dl, (3.67) 2Gθ′ A is the stress function which fulfills the following Poisson equation: ∂2 Φ ∂2 Φ + =1 ∂y 2 ∂z 2

(3.68)

Φ(boundary) = 0.

(3.69)

with boundary condition

Indeed, it first has to be noted that Φ depends not on the form of the cutting curve L(l), in particular on the starting point A, but only on the coordinates of the final point P(y, z). This follows from the fact that a substitution of curve L(l)A→P by L(l)B→P does not change the balancing integral over L(l)P→C : Z P(y ,z ) Z C Z P(y ,z ) σxn (l)dl. (3.70) σxn (l)dl = σxn (l)dl = − P(y ,z )

A

B

Secondly, it is obvious by definition that Φ(boundary) = 0, where the boundary is defined by arbitrary endpoints A and C. Thirdly, differentiating Eq. (3.67) with respect to the curve parameter l gives ∂Φ 1 =− σxn (l) dl 2Gθ′

⇒

∂Φ 1 = σxz , dy 2Gθ′

−

∂Φ 1 = σxy . (3.71) dz 2Gθ′

Here the direction of the normal of a curve extending along the positive ydirection is downwards so that σxn = −σxz . For a curve extending along the positive z-direction, the normal is pointed in the positive y-direction so that σxn = σxy . If now Eqs. (3.71) and (3.65) (the last right-hand equations) are combined, one gets ∂u ∂Φ = −2θ′ , ∂y ∂z ∂u ∂Φ θ′ y + = 2θ′ . ∂z ∂y

−θ′ z +

(3.72) (3.73)

Differentiation of the first equation with respect to z and of the second with respect to y, and subtraction of one of the results from the other immediately leads to the Poisson equation (3.68). Now the interrelation between shear stresses and the applied moment Mt must be determined. If Eq. (3.71) is taken into consideration, the balance of moments can be presented as follows: ¶ Z Z µ ∂Φ ∂Φ y Mt = (yσxz − zσxy )dS = 2Gθ′ +z dS. (3.74) ∂y ∂z S S

100

Non-inertial forces

It is easy to show by partial integration with boundary conditions Φ(boundary) = 0 that Z Z Z ∂Φ ∂Φ y z dS = (3.75) dS = − Φ dS. ∂y ∂z S S S Substitution into Eq. (3.74) leads to the final equation for torsion of flexures with arbitrary but constant cross-sections in Saint-Venant’s approximation Z Mt θ′ = (3.76) , It = −4 Φ dS. GIt S It is called the areal moment and plays the same role as the polar moment for annular bars. For a bar of length L the torsion spring rate is then kθ =

∂Mt GIt = . ∂θ L

(3.77)

Rectangular bars The Poisson equation (3.68) with homogeneous boundary conditions can be analytically solved for a rectangular bar with height h and width b (b ≤ h), using a known Green function G(y, z, η, ζ) Z b Z h dζ G(y, z, η, ζ) dη Φ=− 0

0

with G(y, z, η, ζ) =

∞ 2 X sin(pn y) sin(pn η) Hn (z, ζ), b n = 1 pn sinh(pn h)

where Hn (z, ζ) =

( sinh(pn ζ)sinh[pn (h − z)]

sinh(pn z)sinh[pn (h − ζ)]

for for

0 ≤ ζ < z < h,

0 ≤ z < ζ ≤ h,

with pn = πn/b. The integration is straightforward and must be completed by a second integration in order to obtain the areal moment It . Assuming b ≤ h the final result is " # ∞ hb3 192 b X 1 − cosh[(2n + 1)πb/h] It = 1+ 5 3 π h 0 (2n + 1)5 sinh[(2n + 1)πb/h] " # µ ¶5 hb3 b b = 1 − 0.63 + 0.052 + ··· . (3.78) 3 h h For b ≥ h the rectangle edges should be renamed. If the applied moment is skewed with respect to the bar axis, it has to be projected onto the three axes of the coordinate system. The moments which are orthogonal to the bar axis create a slope of the beam tip, which can be calculated according to Section 3.1.4.

3.1 Springs

101

Cylindrical bars In the case of hollow cylindrical bars with inner and outer radii R1 and R2 the Poisson equation (3.68) can be presented more conveniently in cylindrical coordinates, accounting for the fact that axial symmetry entails the condition ∂Φ/∂θ = 0: µ ¶ 1 ∂ ∂Φ r = 1. (3.79) r ∂r ∂r The solution pertaining to zero-boundary conditions takes the form · µ ¶ ¸ R22 r2 R12 ln(r/R2 ) Φ(r) = −1+ 1− 2 . 4 R22 R2 ln(R1 /R2 ) RR The areal moment Itcyl = R 12 2πrΦ dr of an annular bar is therefore · ¸ π (R22 − R12 )2 4 4 Itcyl = R2 − R1 + 2 ln(R1 /R2 )

(3.80)

(3.81)

and will be used later for the calculation of squeeze damping forces in Section 3.2.3. If the thickness of the cylinder wall is small, (R2 − R1 )/R2 = t/R2 ≪ 1, the areal moment becomes Itcyl = 2πtR23 , and substitution into (3.77) shows that the result fits with the initial defining Eqs. (2.9) and (2.10) in Chapter 2. A full cylinder has the areal moment Itcyl full = (π/2)R24 .

3.1.6

Stress concentration The maximal stress calculated for bent beams or torsional bars does not reflect the worst-case stress conditions which usually emerge at the corners of clamping supports. However, such calculations are useful in order to estimate the operational stress to be measured by piezoresistors or piezoelectric sensors. Furthermore, they give a first orientation for estimating worst-case values. The maximal stress in a rectangular beam follows from Eq. (3.14) on setting zm ax = h/2: σb,m ax =

Mby Mby zm ax = 6 2 . Iy bh

(3.82)

The maximal shear stress for a rectangular torsion bar has to be determined using Eq. (3.71): ¯ ¯ ¯ ¯¸ · ¯ ¯ ¯ ¯ ′ ∂Φ ¯ ′ ∂Φ ¯ ¯ ¯ σt,m ax = max max ¯2Gθ , max ¯2Gθ . (3.83) dy ¯ dz ¯

It can be shown that

· ¸ 0.65(t/h)3 hMt σt,m ax = 1 − 1 + (t/h)3 It

(3.84)

102

Non-inertial forces

(a)

(b)

Figure 3.10 (a) Stress relief by fillets. (b) Stress relief by forking.

At clamped ends of beams the real maximum stress values can reach multiples of values calculated for homogeneous members. Over the lifetime of a sensor, either fractures or, in cases of overload, complete immediate breaks may occur. In practical MEMS processes, the sharp corners at the clamped ends become slightly smoothed out by fillets even without additional design measures, since these fillets arise naturally during the etch process. A typical natural shape is shown in Fig. 3.10(a). The curvature of fillets can be purposely decreased in order to further reduce stress concentration at the corners. For critically stressed beams, shapes formed in accord with archetypes of natural evolution, such as forks or multiple forks, should be used (see Fig. 3.10(b)). The exact stress loads and spring constants for such complicated elastic members must be determined by FEM simulations.

Buckling Inertial MEMS usually are based on mature technologies with low residual stress and well-matched temperature behavior of different substructures. However, situations with high loads in the lateral direction of beams may occur. The result of axial loads may be not only reduced spring stiffness but also buckling. This is an indication of high stress concentration and should be avoided at all cost.5 The theory of buckling is related to the nonlinear analysis of deformation and presented for instance in Timoschenko and Gere [1961] and Fang and Wickert [1994]. To check the onset of buckling for a fixed guided beam the following equation, which has already been mentioned in Section 3.1.4, can be used: π 2 EI (3.85) 4 L2 where for I the smaller of the two area moments should be taken. As indicated in Section 3.1.4 the limit at which buckling takes place is four times larger than the value at which the onset of buckling occurs. Fx,onset =

5

Owing to fatigue risk, applications of buckling beams, for instance, in the form of mechanical switches are not considered in this book.

3.1 Springs

(a)

103

(b) (c)

Figure 3.11 (a) Serial, (b) anti-parallel, and (c) parallel connection of springs.

For a doubly clamped beam (bridge) it can be calculated by solving Eq. (3.49) for a point load using an eigenfunction expansion. The Euler buckling limit for a bridge is four times that for a fixed guided beam: Fx,buckling ≤ 4

3.1.7

π2 EI. L2

(3.86)

Suspensions In inertial MEMS the suspension usually should constrain the motion to rectilinear directions and, thus, can be created by straight beams. The suspension may be formed by connecting various springs with each other and/or with the rigid subject of suspension. Such spring combinations allow one to increase compliance in desired directions and stiffness for all other motions.

Parallel and serial spring connections Basic spring combinations are serial and parallel connections as illustrated in Fig. 3.11. According to Fig. 3.11(a), in a serial spring connection the forces acting on the two springs are equal, F1 = F2 = F . Consequently, the partial displacements of both springs add up to ∆yΣ = ∆y1 + ∆y2 =

F F + k1 k2

(3.87)

and the resulting spring rate for serial spring connection is given by 1 1 1 = + . kΣ k1 k2

(3.88)

For parallel and anti-parallel connections the two deflections are equal, and the spring forces add up to F = F1 + F2 = k1 ∆yΣ + k2 ∆yΣ ,

(3.89)

kΣ = k1 + k2

(3.90)

resulting in

for parallel spring connection.

104

Non-inertial forces

Figure 3.12 Different types of rectilinear beam chains: from folded beam to serpentine.

Beam chains Rectilinear connections of beams are used in order to implement comparable compliances in two directions or to mitigate the impact of compressive stress and axial loads. The most common types of beam connections are shown in Fig. 3.12. The exact calculation of spring constants requires the consideration of force/torque propagation through the chain. A more convenient approach is based on Castigliano’s theorems [Lobontiu and Garcia 2004, Wittenburg and Pestel 2001], which state that ∂UF = ∆i ∂Fi

(first theorem),

∂U∆ = Fi ∂∆i

(second theorem),

(3.91)

where UF is the elastic energy expressed as a function of the generalized forces, Fi , and U∆ is the same energy expressed as a function of the generalized deflections, ∆i . In particular, the second theorem allows one to calculate the forces and moments Fi at the connecting points of a beam chain. Since the calculations should be performed by a formula-manipulation program and lead to long expressions, the results are cited only for a folded beam and a crab-leg structure.

Folded beam For the estimation of the main stiffness elements the folded beam according to Fig. 3.12 is considered simply as a serial connection of three cantilever beams. If the x-axis is along the anchored beam axis and the z-axis is the in-plane axis, the spring rate kz ,FB is given by 1 kz ,FB

=

L31 L2 L33 + + , 3EIy 1 ES2 3EIy ,3

(3.92)

whereby the axial stiffness of the linking member with length L2 has been added. In many practical designs the intermediate beam is made very short and stiff, and can be neglected. If both beams have the same length, L1 = L3 = L, and identical cross-sections, the total spring rate kz ,FB = 23 EIz /L3 is eight times smaller than that of a guided beam with the same length. In order to get the same spring rate, two springs each, with length 2L, have to be used. Nevertheless, folded beams are preferable if residual stress or temperature-dependent displacements of load

3.1 Springs

105

points (axial stress) give rise to concern. A folded beam mitigates such stress considerably because the beam ends are free to move. Folded beams have also a smaller nonlinearity than fixed guided or doubly clamped beams. The reason is the smaller stress concentration. Therefore, tolerable deflections are significantly smaller in fixed guided beams than in folded beams. If the folded beam features different lengths L1 and L3 , cross-coupling between the two in-plane axes (x and z; y is the out-of-plane axis) emerges as a result of the transfer of torques at the connecting points of the sub-beams. For L1 ≃ L3 the off-diagonal stiffness becomes (derived as a special case of U-shaped springs from Iyer [2003]) kxz ,FB = 3E

Iy ,2 (L1 − L3 ) L2 L31

⇒

kxz ,FB ∼ Iy ,2 L1 − L3 . =2 kz ,FB Iy ,1 L2

(3.93)

Crab-leg spring Crab-leg springs can also reduce the maximal stress and buckling tendency by virtue of the added “thigh” fraction. Iyer [2003] gives the following expression for the cross-coupling stiffness: kxz ,CL = 9E

Iy ,1 Iy ,2 . L1 L2 (L1 Iy ,2 + L2 Iy ,1 )

(3.94)

The basic stiffness values are, according to Fedder [1994], kz ,CL = 3E

Iy ,1 4L1 Iy ,2 + L2 Iy ,1 , L31 L1 Iy ,2 + L2 Iy ,1

kx,CL = 3E

Iy ,2 L1 Iy ,2 + 4L2 Iy ,1 . (3.95) L32 L1 Iy ,2 + L2 Iy ,1

As may be expected, the in-plane cross-coupling is incomparably higher then for folded springs, for which it can be brought to zero.

Plate suspension Plate or disk suspensions have to facilitate linear or rotatory motions in one (accelerometers with one DOF) or two directions (gyroscopes or gyroscope members).

Linear suspensions Plate suspensions supporting linear motions are usually designed as combinations of four beams or four beam-chains as shown in Fig. 3.13. The beams are connected individually to the plate and anchored to the substrate at four points. The anchors may be located outside or inside the plate (frame). The springs may extend along the plate to save footprint area rather than diverging as in the figure. Since the springs are connected in parallel, the resulting main stiffness values of the suspension are four times larger then the individual spring rates, provided that identical springs are used, kx(y )(z ),susp ension = 4kx(y )(z ),spring .

(3.96)

106

Non-inertial forces

Figure 3.13 Standard suspensions for supporting orthogonal linear motions.

Small differences in the main stiffness values and asymmetries in the individual springs lead to different deflections at the plate-suspension points. The suspension deviates from a parallel-spring connection, which makes an exact analysis very difficult. The main characteristics of the different suspension types are as follows.

r The simplest four-beam suspension according to Fig. 3.13(a) consists of four fixed guided beams. It is susceptible to residual stress or external axial loads that may lead to buckling. Since the lateral endpoints have a fixed distance, it features also an increased nonlinearity. With growing deflections there emerges a nonlinearly increasing axial load that requires increasing additional deflection forces. Thus, this kind of suspension is not suited for large deflections as are necessary, for instance, for driving a gyroscope. r An excellent countermeasure is the use of folded beams as illustrated in Fig. 3.13(b). Owing to the increased overall length and the possibility of shifting freely one of the beam ends, it provides good linearity, albeit at the price of increased sensitivity to accelerations in the out-of-plane direction. For large deflections the pendulous effect increases the compliance in the axial direction. Using doubly folded beams avoids this effect. The U-shaped suspension is basically a folded beam suspension with an elongated middle fraction of the folded beam. It has similar properties to the foldedbeam suspension.

r The crab-leg suspension used for a one-DOF system (with short “thigh”) may tolerate residual stress, but the main advantage is a better symmetry of the overall construction, because cross-couplings in one direction are to some extent absorbed by the compliance in the orthogonal direction. A crab-leg

3.1 Springs

107

Figure 3.14 Typical suspensions for rotating plates or disks.

suspension may be designed to support comparable sensitivity in the x- and y-directions (two-DOF movement). In this case an improved symmetrization may be achieved by connecting the break points by relatively stiff bars as shown in Fig. 3.13(e), forming an H-type suspension. r The five-beam serpentine suspension (Fig. 3.13(d)) exists in two versions: the intrinsically serpentine suspension and the hairpin suspension. In a hairpin suspension the serpentine length LS is considerably smaller than the outer length of the spring chain members and often located at one of their ends. Owing to small asymmetries any motion in one direction in crab-leg and serpentine suspensions causes deflections of the orthogonal beams that constitute an additional potential root cause of increased cross-coupling. This is why folded beams or U-shaped suspensions are preferred in mode-couplingsensitive applications, such as gyroscopes.

Torsional suspensions For rotating plates the basic design consists of one- or two-anchor constructions as illustrated by Fig. 3.14. Inner suspensions as in Figs. 3.14(a) and (c) are one-anchor suspensions. Outer suspensions require two anchors. In reality anchor areas for inner suspensions are often divided into sub-anchors that are symmetrically arranged around the central anchor point. Torsional springs in inertial MEMS are typically shorter then linear beams. Therefore, according to Eq. (3.86) residual stress is more critical. For stress release the torsion springs should include compensating members, for instance, formed as tridents without the middle jag or as inserted loops as illustrated in Figs. 3.14(a) and (b). Torsional suspensions supporting rotations about the out-of-plane axis as in Fig. 3.14(c) are usually made from two or more bending beams. If the dimension of the anchor can be neglected, i.e. the beam lengths L are equal to the radius of rotation of the load points, L = ri , the torsional spring rate can be calculated as parallel connection of N beams with loads Mθ Z /ri at the free end: kθ = 3EIy /(N L). The real beams attached to an anchor with non-negligible dimensions are shorter: L < ri . A rotation of the outer load points causes not only a bending force but also axial stresses, and changes the boundary conditions at the loaded end.

108

Non-inertial forces

A suspension as in Fig. 3.14(a) is compliant with respect to torsion about the beam axis and about the out-of-plane axis. Correspondingly, a suspension with four beams linked in two orthogonal pairs can be used to design a given compliance about all three rotation axes. The design of an optimized suspension for a given application is a challenging task, especially for gyroscopes. It usually requires a careful refinement and validation of the analytical top design by extensive FEM simulations.

3.2

Damping forces

3.2.1

Fluid-flow models Damping forces acting on oscillating masses determine the quality factor of spring–mass systems according to Q = 2π

Utot . Udiss

(3.97)

Utot is the total system energy and Udiss is the average energy loss during one period at the resonance frequency. The energy dissipation is the combination of intrinsic energy losses Uint within the structure and the loss caused by interaction of surfaces with surrounding fluid Us : Q = 2π

Utot 1 = . Uint + Us 1/Qint + 1/Qs

(3.98)

It is well known that, as the device shrinks, surface phenomena become more and more important in comparison with volumetric effects, because the volumeto-surface ratio scales down with linear dimension. In MEMS devices the impact of surface effects such as damping increases accordingly. Moreover, since internal energy losses are usually related to friction in the most stressed device parts and can be reduced by proper design geometry, the internal losses often become small or negligible, particularly at not-too-low pressures. They can be determined by Q-factor measurements in vacuum. In general, the determination of damping forces in MEMS requires analysis of the interaction between a moving body and the surrounding fluid – usually air or an inert gas at various pressure levels. To some extent, the fluid flow around an object follows the object’s movement due to normal (pressure) and tangential (shear) forces acting on the fluid. Forces at the boundary are translated into volumes of fluid creating a velocity field. Pressure and density may change locally. Inner friction of the gas creates temperature changes, which will be transferred to the surrounding walls. A complete description of all effects requires that the velocity field of flow, together with its pressure and density as well as its temperature, must be analyzed. Even if tiny thermal effects are neglected, however, such an analysis may be quite complicated due to the numerous different interacting

3.2 Damping forces

109

Figure 3.15 Damping forces acting on a moving comb.

geometries and different surface-to-fluid interactions as well as fluid-flow models. For instance, bodies in continuum flow can steadily move through fluid far away from other objects and experience drag forces. Or, the body can interact with fluid that is bound by another, fixed surface. If the moving and fixed surfaces are parallel to the direction of motion, sliding forces occur. In a different scenario, two parallel surfaces move towards each other and squeeze out the gas between them. As a result, squeeze damping takes place. In Fig. 3.15 some of the different situations are illustrated. If the comb is moving in the x-direction, the indicated slide-film forces occur at all surfaces parallel to the x-axis and orthogonal to the z-axis. Drag forces occur at the front and back surfaces, which open and close the encasing flow, and mix with slide forces along the surfaces. If the comb is moving in the y-direction, squeeze-film forces arise between opposite fingers of moving and fixed combs. It is clear that substitution of the entire damping process by superposition of different damping mechanisms neglects complicated corner effects and leads to approximate solutions only. When this is not sufficient, numerical simulations of the Navier–Stokes equations should refine qualitative results. The classes of interaction models specified above must be adapted accordingly for non-continuum fluids.

Fluid flows and Knudsen number A careful differentiation of the nature of the fluid flow is necessary. Different regimes can be identified according to the nature of the gas–particle interaction. This nature is determined by the ratio between the mean free path λ of gas particles and a characteristic device length LK (including distances of interest to other surfaces), which is known as the Knudsen number Kn: Kn =

λ . LK

(3.99)

110

Non-inertial forces

It is important to recognize that there may be different Knudsen numbers for one and the same device depending on what fluid flow is modeled. If, for instance, the flow in a narrow gap between a moving plate and a substrate is of interest, the characteristic length is the gap height. If the flow on top of the plate is analyzed, the characteristic length is the distance between the plate and the top of the cavity (the upper gap). If λ ≪ LK ⇒ Kn ≪ 1, the flow can be considered as a continuum where the molecular nature of the fluid can be neglected. This regime is well described by the Navier–Stokes equation mentioned earlier. Vice versa, if λ ≫ LK ⇒ Kn ≫ 1, then collisions of gas particles with each other and with device walls determine the nature of the fluid. Here the general Boltzmann transport equation must be analyzed. The mean free path of molecules in air under standard conditions (atmospheric pressure P0 ) is about λ0 = 65 nm. For minimal characteristic dimensions of 2 µm this results in a Knudsen number of 0.033, where the continuum approximation holds. However, the mean free path is inversely proportional to pressure, λ = λ0

P0 . P

(3.100)

Since gyroscopes and resonators are often driven under low-pressure conditions, down to 0.1 mbar and below, Knudsen numbers of some hundreds may be encountered, indicating the regime of free molecular flow. Transition zones exist between the regimes of continuum and free molecular flow. They are characterized by an intermediate gas status where features of continuum behavior are present and must be completed by free molecular idiosyncrasies. The choice of the correct flow model is not trivial. To select a correct approach, the flow models are grouped into four regions [Beskok 2002, Bird 1996], according to the corresponding Knudsen numbers:

r r r r

Kn ≤ 0.01, 0.01 ≤ Kn ≤ 0.1, 0.1 ≤ Kn ≤ 10, 1 ≤ Kn,

continuum flow with no-slip boundary condition;6 continuum flow with slip boundary condition; transitional flow; free molecular flow.

Flow analysis in any of these regions requires further constraints. In inertial MEMS two classes of movement can be considered in order to simplify the models:

r slow motion like in accelerometers; and r fast harmonic oscillation like in vibratory gyroscopes or resonators. The general case of arbitrary motions with a broad spectrum is relevant, for instance, in cases of slowly or harmonically driven objects under the impact 6

No relative movement between the surface and adjacent fluid at the boundary.

3.2 Damping forces

(a)

111

(b)

Figure 3.16 (a) Viscosity and Couette flow. (b) Stokes flow.

of shock and vibration. The models usually are based on convolution of an impulse transfer function and an input excitation, assuming linearity of the damping equation. Nonlinear extensions, which are necessary for shock-induced large amplitudes of motion, are complicated. So far, they have attracted little attention in the literature and, thus, still represent a challenge for further investigations.

Continuous viscous flow A continuum flow is regarded as a collection of fluid volume elements that interact by shear and normal forces. To introduce basic properties of fluid a twodimensional flow model wherein a fluid is located between two infinite parallel plates (see Fig. 3.16(a)) is used. The upper plate is moving with velocity vxP ; the lower one is at rest. Friction between wall and fluid generates a (steady) shear force σz x that forces the fluid to flow. The fluid can be imagined as layers, which are moving with different velocities; the topmost layer has the highest velocity, equal to the plate velocity (vx (0) = vxP – no slip), while the lowermost remains at rest (vx (d) = 0). This corresponds to a changing strain with a strain rate −vx (0)/(2d). If the distance d between plates is small, the steady velocity distribution of layers is linear, ³ z´ vx (z) = vx (0) 1 − . (3.101) d

The ratio between the velocity change perpendicular to the flow, vx (0)/d, and the shear force is a material property called viscosity and is denoted by µ, σz x = −µ

vx (0) . d

(3.102)

Since the shear force of fluid must oppose plate motion, the sign is negative in order to have a positive viscosity constant. On decreasing the distance d one obtains the general expression for the shear stress of an infinitesimal thin layer, σz x = µ

∂vx (z) . ∂z

(3.103)

112

Non-inertial forces

For a large class of fluids – called Newtonian fluids – the viscosity µ (dimension: Pa · s)7 is constant. Non-Newtonian fluids like polymers, blood, starch and so on exhibit a nonlinear dependency between shear stress and strain rate ∂vx (z)/∂z.

Viscosity of gases The viscosity of gases can be estimated using the common formula µ=

1 ̺hviλ, 3

(3.104)

where ̺ is the density, hvi the average velocity of molecules, and λ the mean free path. It is obvious that the viscosity is proportional to density. The proportionality to velocity and free path can be understood as a result of interaction between molecules of adjacent layers, which are accelerated as they cross the border between layers in the direction of the faster layer and hence retard it; this interaction increases with hvi and λ. Molecules crossing the border in the direction of the slower layer are decelerated, transferring their impulse to the layer and speeding it up. p From the kinetic theory of gases the three relations ̺ = mn, hvi = 8kT /(πm) and λ = 1/(nσm ) follow, where n is the particle density, m the the mass of the molecule, k the Boltzmann constant, T the absolute temperature, and σm = π(2rm )2 the scattering cross-section of the molecule (rm is the radius of molecule). Substitution into Eq. (3.104) yields s 1 8kmT µ= . (3.105) 2 3 πσm Therefore, viscosity depends on temperature, but not on pressure. The inclusion of the temperature dependency of the scattering cross-section can be managed successfully by use of a semi-empirical relation – the Sutherland equation [White 1974]: µ = µ0

1 + TS /T0 T . 1 + TS /T T0

(3.106)

T0 = 273.16 K is room temperature and µ0 the viscosity at T0 . TS is the Sutherland temperature and T the actual temperature. For the most relevant fill gases, N2 , air, argon, and neon, the corresponding values (from Bao [2000], Gudeman et al. [1998], and Thornton and Baker [1962]) are shown in Table 3.2.

Slip at borders Viscosity was introduced on assuming that the velocity of the wall and of gas flow at the border are equal (no slip). This is true for small Knudsen numbers. 7

µ is the dynamic viscosity, in contrast to the kinematic viscosity µ ∗ = µ/̺, which is normalized by the density ̺.

3.2 Damping forces

113

Table 3.2. Viscosity at T0 and TS for N2 and air

−6

µ0 (10 TS K

Pa · s)

N2 16.6 104

Air 17.2 124

Argon 22.2

Neon 30.8

For larger mean free paths a slip between the wall and the adjacent fluid layer appears. In other words, the flow velocity vx (d) of a moving fluid layer is not zero at the border with a stationary wall at z = d. The flow velocity of the boundary layer is proportional to the velocity gradient in the z-direction at the wall and to the mean free path. The velocity gradient is responsible for the intensity of velocity transfer in adjacent layers, particularly between the boundary layer and the surface. For ∂vx (z)/∂z = 0 there is no transfer of wall velocity to fluid or vice versa. Slip means that particles approaching the wall have different velocities than particles leaving it. This behavior can be modeled by the following Maxwellian boundary conditions [Chapman and Cowling 1970, Arkilic et al. 2001, Sattler and Wachutka 2004]: ¯ 2 − σ ∂vx (z) ¯¯ vx (d) = −λ σ ∂z ¯z =d

or at a moving plate

vx (0) = vxP

¯ 2 − σ ∂vx (z) ¯¯ +λ , σ ∂z ¯z = 0

(3.107)

where 0 ≤ σ ≤ 1 is the fraction of diffusely reflected molecules and is called the tangential momentum accommodation coefficient. It represents the average tangential momentum exchange between impinging gas molecules and the solid surface. Experiments have shown that, for smooth silicon, σ is about 0.7 [Arkilic et al. 2001]. For engineering surfaces σ is close to 1, i.e. the particle loses all of its normal momentum. σ = 0 indicates an ideal reflection of particles without momentum transfer.

Effective viscosity Slip of a boundary layer can be accounted for by substituting the viscosity, µ, by a reduced effective viscosity, µeff , keeping the non-slip condition: µeff =

µ . 1 + f (Kn)

(3.108)

f (Kn) must be determined using the introduced slip condition. However, there is an urgent need to further extend the powerful models of continuum flow towards higher Knudsen numbers, even exceeding the value Kn = 0.1 of the slip area and entering into the region of transitional flow of

114

Non-inertial forces

rarefied gases. This is due to the complicated nature of transitional flow models that makes analysis very cumbersome. Different approaches to defining the function f (Kn) for high Knudsen numbers were summarized by the broadly accepted work of Veijola et al. [1995], who presented an empirical approximation of the effective viscosity, which is applicable also in the lower region of transitional flow, µeff =

µ . 1 + 9.658Kn1.159

(3.109)

Boltzmann’s transport equation provides a basis for similar approximations. Simulation results of Boltzmann’s equation can be fitted with the Navier–Stokes model. Such numerical calculations include the slip region and refine the function f (Kn), derived for the non-slip region, for higher Knudsen numbers. Strictly speaking, Eq. (3.109) was deduced in this way for squeeze-film damping and hence should be applied to other flows only with caution. Taking into account the described and other possible extensions, the continuum-flow model, which will be briefly recapitulated in the next section, may serve as a basis for damping analysis up to Knudsen numbers significantly exceeding unity.

Continuous-flow equations The two-dimensional fluid flow between two plates with constant velocity gradient which was described above and used to introduce viscosity is called Couette flow. Assuming that edge effects of the – finitely sized – plates can be neglected, the force acting on the plate and caused by a steady or slowly oscillating flow is A FCouette = −vx (0)µ , d

(3.110)

where A = bL denotes the area of the moving plate. This is one of the rare examples where the damping force can be directly estimated on the basis of strong assumptions on the flow’s nature. In more general cases, especially in the case of non-steady flows, the interrelation of velocity, pressure, and density must be found in order to calculate the stress distribution on the object’s surface.

Generalization of the Couette model Therefore, a generalization of the simple layer model is necessary. First, a strain (or deformation) rate for arbitrary fluid configurations is introduced. In analogy to the shear-strain definition (see Chapter 2) it is given by µ ¶ 1 ∂vi (¯ x) ∂vj (¯ x) ε˙ij = + (3.111) 2 ∂xj ∂xi so that with Eq. (3.103) the shear stress becomes σij = 2µε˙ij .

(3.112)

3.2 Damping forces

115

Comparison with elastic solid objects – where shear stress is generated by shear strain – reveals that in fluids shear strain is replaced by a shear-strain rate, which is responsible for the appearance of shear stress. Normal stress components also depend on the velocity field of fluid. Clearly, a velocity component, which is, for instance, normal to the surface x = constant, will increase the stress by 2µ ∂vx /∂x. A further stress component is caused by velocity sources or sinks at a given location of fluid. This contribution is related to the continuity equation, which states that the mass M of a fluid with local density ̺(¯ x, t), encased in an initial volume V , is constant even if the volume is changing.8 Z Z I dM d ∂̺ = ̺(¯ x, t)dV = ̺¯ v · dS¯ = 0. (3.113) dV + dt dt V (t) V (t) ∂t S (t) S is the surface of the volume V (t) which moves with the velocity of the flow v¯(¯ x, t). dS¯ is a vector that is normal to the surface and possesses area dS. Applying Gauss’ theorem and taking into account that the resulting volume integral must vanish for any volume, one obtains the continuity equation ∂̺ ∂̺ + div(̺¯ v) = + v¯ · grad ̺ + ̺ div v¯ = 0. (3.114) ∂t ∂t A fluid is incompressible if div v¯ = 0. In general, gases are compressible so that sinks and sources of velocity field emerge, which contribute to normal stresses. In the theory of fluid mechanics it has been shown that this contribution is η div v¯, where η is the so-called bulk viscosity of the fluid, which for gases is related to the shear viscosity according to 2 1 η=− µ ⇒ µ + η = µ. (3.115) 3 3 To summarize, the components of the stress tensor can be expressed as ∂vi σi = σii = −p + 2µ + η div v¯, ∂xi µ ¶ ∂vi ∂vj σij = µ + = 2µε˙ij , (3.116) ∂xj ∂xi 1 p = − (σ11 + σ22 + σ33 ). 3 p denotes the pressure of fluid, defined as the average of the negative normal stresses, directed into a volume element.

The Navier–Stokes equation The density, pressure, and velocity field are connected by the Navier–Stokes equation, which expresses Newton’s law applied to an elementary volume element. Indeed, the principle of linear momentum states that dP /dt = F . The 8