Mechanics of Microsystems

Mechanics of Microsystems Alberto Corigliano, Raffaele Ardito, Claudia Comi, Attilio Frangi, Aldo Ghisi and Stefano Mariani, Politecnico di Milano, Italy   A mechanical approach to microsystems, covering fundamental concepts including MEMS design, modelling and reliability   Mechanics of Microsystems takes a mechanical approach to microsystems and covers fundamental concepts including MEMS design, modelling and reliability. The book examines the mechanical behaviour of microsystems from a design for reliability point of view and includes examples of applications in industry. Mechanics of Microsystems is divided into two main parts. The first part recalls basic knowledge related to the microsystems behaviour and offers an overview on microsystems and fundamental design and modelling tools from a mechanical point of view, together with many practical examples of real microsystems. The second part covers the mechanical characterization of materials at the micro–scale and considers the most important reliability issues (fracture, fatigue, stiction, damping phenomena, etc) which are fundamental to fabricate a real working device.     Key features: Provides an overview of MEMS, with special focus on mechanical–based Microsystems and reliability issues. Includes examples of applications in industry. Accompanied by a website hosting supplementary material.     The book provides essential reading for researchers and practitioners working with MEMS, as well as graduate students in mechanical, materials and electrical engineering. INDICE: Preface xv .Acknowledgements xvii .Notation xix .About the Companion Website xxiii .1 Introduction 1 .1.1 Microsystems 1 .1.2 Microsystems Fabrication 3 .1.3 Mechanics in Microsystems 5 .1.4 Book Contents 6 .References 7 .Part I Fundamentals 9 .2 Fundamentals of Mechanics and Coupled Problems 11 .2.1 Introduction 11 .2.2 Kinematics and Dynamics of Material Points and Rigid Bodies 12 .2.2.1 Basic Notions of Kinematics and Motion Composition 12 .2.2.1.1 Summary 15 .2.2.2 Basic Notions of Dynamics and Relative Dynamics 15 .2.2.2.1 Summary 17 .2.2.3 One–Degree–of–Freedom Oscillator 17 .2.2.3.1 Summary 21 .2.2.4 Rigid–Body Kinematics and Dynamics 22 .2.2.4.1 Rigid–Body Kinematics 22 .2.2.4.2 Rigid–Body Dynamics 23 .2.2.4.3 Summary 25 .2.3 Solid Mechanics 25 .2.3.1 Linear Elastic Problem for Deformable Solids 26 .2.3.1.1 Cubic 33 .2.3.1.2 Transversely Isotropic 34 .2.3.1.3 Orthotropic 34 .2.3.1.4 Summary 35 .2.3.2 Linear Elastic Problem for Beams 35 .2.3.2.1 Examples 39 .2.3.2.2 Summary 42 .2.4 Fluid Mechanics 43 .2.4.1 Navier Stokes Equations 43 .2.4.1.1 Newtonian Fluids 44 .2.4.1.2 Incompressible Fluids 46 .2.4.1.3 Perfect Fluids 46 .2.4.1.4 Stokes Flow 47 .2.4.1.5 Initial and Boundary Conditions 47 .2.4.1.6 Summary 48 .2.4.2 Fluid Structure Interaction 48 .2.5 Electrostatics and Electromechanics 49 .2.5.1 Basic Notions of Electrostatics 49 .2.5.1.1 Examples 52 .2.5.1.2 Summary 54 .2.5.2 Simple Electromechanical Problem 54 .2.5.2.1 Static Equilibrium 54 .2.5.2.2 Dynamic Response 56 .2.5.2.3 Summary 58 .2.5.3 General Electromechanical Coupled Problem 58 .2.5.3.1 Summary 60 .2.6 Piezoelectric Materials in Microsystems 60 .2.6.1 Piezoelectric Materials 60 .2.6.2 PiezoelectricModelling 62 .2.6.2.1 Summary 64 .2.7 Heat Conduction and Thermomechanics 64 .2.7.1 Heat Problem 64 .2.7.1.1 Example: UnidimensionalThermal Problem 66 .2.7.1.2 Summary 67 .2.7.2 Thermomechanical Coupled Problem 67 .2.7.2.1 Summary 70 .References 70 .3 Modelling of Linear and NonlinearMechanical Response 73 .3.1 Introduction 73 .3.2 Fundamental Principles 74 .3.2.1 Principle of Virtual Power 74 .3.2.2 Total Potential Energy Principle 74 .3.2.3 Hamilton s Principle 75 .3.2.4 Specialization of the Principle of Virtual Powers to Beams 76 .3.3 Approximation Techniques andWeighted Residuals Approach 76 .3.4 Exact and Approximate Solutions for Dynamic Problems 79 .3.4.1 Free Flexural Linear Vibrations of a Single–span Beam 79 .3.4.2 Nonlinear Vibration of an Axially Loaded Beam 80 .3.5 Example of Application: Bistable Elements 84 .References 90 .Part II Devices 91 .4 Accelerometers 93 .4.1 Introduction 93 .4.2 Capacitive Accelerometers 94 .4.2.1 In–Plane Sensing 94 .4.2.2 Out–of–Plane Sensing 96 .4.3 Resonant Accelerometers 98 .4.3.1 Resonating Proof Mass 98 .4.3.2 Resonating Elements Coupled to the Proof Mass 99 .4.4 Examples 101 .4.4.1 Three–Axis Capacitive Accelerometer 101 .4.4.2 Out–of–Plane Resonant Accelerometer 104 .4.4.3 In–Plane Resonant Accelerometer 105 .4.5 Design Problems and Reliability Issues 107 .References 107 .5 Coriolis–Based Gyroscopes 109 .5.1 Introduction 109 .5.2 BasicWorking Principle 109 .5.2.1 Sensitivity of Coriolis Vibratory Gyroscopes 112 .5.3 Lumped–Mass Gyroscopes 113 .5.3.1 Symmetric and Decoupled Gyroscope 113 .5.3.2 Tuning–Fork Gyroscope 114 .5.3.3 Three–Axis Gyroscope 115 .5.3.4 Gyroscopes with Resonant Sensing 115 .5.4 Disc and Ring Gyroscopes 118 .5.5 Design Problems and Reliability Issues 118 .References 119 .6 Resonators 121 .6.1 Introduction 121 .6.2 Electrostatically Actuated Resonators 123 .6.3 Piezoelectric Resonators 125 .6.4 Nonlinearity Issues 126 .References 128 .7 Micromirrors and Parametric Resonance 131 .7.1 Introduction 131 .7.2 Electrostatic Resonant Micromirror 132 .7.2.1 Numerical Simulations with a Continuation Approach 136 .7.2.1.1 Computation of the Electrostatic Torque and its Derivative via Direct Finite Element Method 138 .7.2.2 Experimental Set–Up 140 .7.2.2.1 Sinusoidal Excitation 142 .7.2.2.2 SquareWave Excitation 143 .References 145 .8 Vibrating Lorentz ForceMagnetometers 147 .8.1 Introduction 147 .8.2 Vibrating Lorentz Force Magnetometers 148 .8.2.1 Classical Devices 148 .8.2.2 Improved Design 151 .8.2.3 Further Improvements 155 .8.3 Topology or Geometry Optimization 156 .References 159 .9 Mechanical Energy Harvesters 161 .9.1 Introduction 161 .9.2 Inertial Energy Harvesters 162 .9.2.1 Classification of Resonant Energy Harvesters 162 .9.2.2 Mechanical Model of a Simple Piezoelectric Harvester 165 .9.2.2.1 Piezoelectric Constitutive Law for Beams 165 .9.2.2.2 The Principle of Virtual Power for a Piezoelectric Cantilever Beam 167 .9.2.2.3 Governing Equations via theWeighted Residuals Approach 168 .9.2.2.4 Solution in the Frequency Domain 171 .9.3 Frequency Upconversion and Bistability 174 .9.4 Fluid Structure Interaction Energy Harvesters 176 .9.4.1 Synopsis of Aeroelastic Phenomena 177 .9.4.1.1 Vortex–Induced Vibration 177 .9.4.1.2 Flutter Instability 178 .9.4.2 Energy Harvesting through Vortex–Induced Vibration 179 .9.4.3 Energy Harvesting through Flutter Instability 180 .References 181 .10 Micropumps 185 .10.1 Introduction 185 .10.2 Modelling Issues for Diaphragm Micropumps 186 .10.3 Modelling of Electrostatic Actuator 188 .10.3.1 Simplified Electromechanical Model 188 .10.3.1.1 The Principle of VirtualWork for an Axisymmetric Plate 189 .10.3.1.2 Electrostatic Forces 190 .10.3.1.3 Governing Equations via theWeighted Residuals Approach 190 .10.3.2 Reliability Issues 192 .10.3.2.1 Electrostatic Pull–In 192 .10.3.2.2 Adhesion 194 .10.3.2.3 Actuator Control 195 .10.4 MultiphysicsModel of an Electrostatic Micropump 196 .10.5 Piezoelectric Micropumps 198 .10.5.1 Modelling of the Actuator 198 .10.5.2 CompleteMultiphysicsModel 201 .References 202 .Part III Reliability and Dissipative Phenomena 205 .11 Mechanical Characterization at the Microscale 207 .11.1 Introduction 207 .11.2 Mechanical Characterization of Polysilicon as a Structural Material for Microsystems 209 .11.2.1 Polysilicon as a Structural Material for Microsystems 209 .11.2.2 Testing Methodologies 210 .11.2.3 Quasi–Static Testing 211 .11.2.3.1 Off–Chip Tension Test 212 .11.2.3.2 Off–Chip and On–Chip Bending Test 212 .11.2.3.3 Test on Membranes (Bulge Test) 213 .11.2.3.4 Nanoindenter–Driven Test 214 .11.2.4 High–Frequency Testing 214 .11.2.4.1 Fatigue Mechanisms 215 .11.3 Weibull Approach 215 .11.4 On–Chip Testing Methodology for Experimental Determination of Elastic .Stiffness and Nominal Strength 219 .11.4.1 On–Chip Bending Test through a Comb–Finger Rotational Electrostatic Actuator 220 .11.4.1.1 General Description 220 .11.4.1.2 Data Reduction Procedure 220 .11.4.1.3 Experimental Results 224 .11.4.2 On–Chip Bending Test through a Parallel–Plate Electrostatic Actuator 225 .11.4.2.1 General Description 225 .11.4.2.2 Data Reduction Procedure 227 .11.4.2.3 Experimental Results 227 .11.4.3 On–Chip Tensile Test through an Electrothermomechanical Actuator 229 .11.4.3.1 General Description 229 .11.4.3.2 Data Reduction Procedure 230 .11.4.3.3 Experimental Results 232 .11.4.4 On–Chip Test forThick Polysilicon Films 233 .11.4.4.1 General Description 233 .11.4.4.2 Data Reduction Procedure 237 .11.4.4.3 Experimental Results 237 .References 240 .12 Fracture and Fatigue in Microsystems 245 .12.1 Introduction 245 .12.2 Fracture Mechanics: An Overview 245 .12.3 MEMS Failure Modes due to Cracking 249 .12.3.1 Cracking and Delamination at Package Level 249 .12.3.2 Cracking at Silicon Film Level 250 .12.4 Fatigue in Microsystems 256 .12.4.1 An Introduction to Fatigue in Mechanics 256 .12.4.2 Polysilicon Fatigue 259 .12.4.3 Fatigue in Metals at the Microscale 261 .12.4.4 Fatigue Testing at the Microscale 263 .References 266 .13 Accidental Drop Impact 271 .13.1 Introduction 271 .13.2 Single–Degree–of–Freedom Response to Drops 272 .13.3 Estimation of the Acceleration Peak Induced by an Accidental Drop 276 .13.4 A Multiscale Approach to Drop Impact Events 277 .13.4.1 Macroscale Level 277 .13.4.2 Mesoscale Level 279 .13.4.3 Microscale Level 279 .13.5 Results: Drop–Induced Failure of Inertial MEMS 280 .References 287 .14 Fabrication–Induced Residual Stresses and Relevant Failures 291 .14.1 Main Sources of Residual Stresses in Microsystems 291 .14.2 The Stoney Formula and its Modifications 292 .14.3 ExperimentalMethods for the Evaluation of Residual Stresses 299 .14.4 Delamination, Buckling and Cracks inThin Films due to Residual .Stresses 304 .References 310 .15 Damping in Microsystems 313 .15.1 Introduction 313 .15.2 Gas Damping in the Continuum Regime with Slip Boundary Conditions 314 .15.2.1 Experimental Validation at Ambient Pressure 317 .15.2.2 Effects of DecreasingWorking Pressure 318 .15.3 Gas Damping in the Rarefied Regime 320 .15.3.1 Evaluation of Damping at Low Pressure using KineticModels 321 .15.3.2 Linearization of the BGK Model 323 .15.3.3 Numerical Implementation 324 .15.3.4 Application to MEMS 325 .15.4 Gas Damping in the Free–Molecule Regime 328 .15.4.1 Boundary Integral Equation Approach 328 .15.4.2 Experimental Validations 330 .15.5 Solid Damping: Thermoelasticity 335 .15.6 Solid Damping: Anchor Losses 338 .15.6.1 Analytical Estimation of Dissipation 339 .15.6.1.1 Applications: Axial and BendingModes 340 .15.6.2 Numerical Estimation of Anchor Losses 342 .15.7 Solid Damping: Additional unknown Sources Surface Losses 346 .15.7.1 Solid Damping: Deviations from Thermoelasticity 346 .15.7.2 Solid Damping: Losses in Piezoresonators 346 .References 348 .16 Surface Interactions 351 .16.1 Introduction 351 .16.2 Spontaneous Adhesion or Stiction 352 .16.3 Adhesion Sources 353 .16.3.1 Capillary Attraction 353 .16.3.2 Van derWaals Interactions 356 .16.3.3 Casimir Forces 358 .16.3.4 Hydrogen Bonds 359 .16.3.5 Electrostatic Forces 360 .16.4 Experimental Characterization 361 .16.4.1 Experiments by Mastrangelo and Hsu 361 .16.4.2 Experiments by the Sandia Group 362 .16.4.3 Experiments by the Virginia Group 365 .16.4.4 Peel Experiments 367 .16.4.5 Pull–in Experiments 368 .16.4.6 Tests for Sidewall Adhesion 372 .16.5 Modelling and Simulation 374 .16.5.1 Lennard–Jones Potential 374 .16.5.2 Tribological Models: Hertz, JKR, DMT 375 .16.5.3 Computation of Adhesion Energy 377 .16.6 Recent Advances 380 .16.6.1 Finite Element Analysis of Adhesion between Rough Surfaces 380 .16.6.1.1 Artificial Rough Surfaces 380 .16.6.1.2 Details of the Finite Element Model and Results 381 .16.6.2 Accelerated Numerical Techniques 383 .16.6.2.1 Rough Surface Represented by Spherical Caps 383 .16.6.2.2 Numerical Outcomes and Comparison with Experiments 384 .References 387 .Index 393

• ISBN: 978-1-119-05383-5
• Editorial: Wiley–Blackwell
• Encuadernacion: Cartoné
• Páginas: 464
• Fecha Publicación: 19/01/2018
• Nº Volúmenes: 1
• Idioma: Inglés