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.

Alberto Corigliano, Raffaele Ardito, Claudia Comi, Attilio Frangi, Aldo Ghisi, and Stefano Mariani – Politecnico di Milano, Italy

Alberto Corigliano is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Corigliano has authored and co-authored more than 240 scientific publications in fields related to solid and structural mechanics at various scales, including 2 book chapters in Microsystems area, and 7 patents on Microsystems. During his research activity, A. Corigliano covered a wide range of subjects in the fields of structural and materials mechanics, with particular reference to theoretical and computational problems relevant to non-linear material responses.

Raffaele Ardito is an Associate Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. He graduated in 2000 (cum laude) at the Politecnico di Milano in Civil Engineering and he received the Ph.D. degree, cum laude, in 2004. From 2004 to 2006 he was a research fellow at the National Institute for Nuclear Physics, joining an international research group with focus on solid mechanics in cryogenic conditions. He spent, in 2008 and 2010, two periods of research at the Research Laboratory of Electronics, Massachusetts Institute of Technology, as visiting scientist. His scientific contributions to the field of MEMS focus on theoretical and computational aspects of adhesion and multi-physics behavior.

Claudia Comi is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. C. Comi has authored and co-authored more than 140 scientific publications in various fields of solid and structural mechanics and 4 patents on Microsystems. Her main research interests concern theoretical and computational mechanics of materials and structures. Her research activities focus on damage and quasi-brittle fracture modelling, on instability phenomena and nonlocal models for elastoplastic and damaging one-phase and multi-phase materials, including functionally graded materials, and on design and reliability of MEMS.

Attilio Frangi is a Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Frangi has authored and co-authored more than 150 scientific publications on issues of computational mechanics and micromechanics and 5 patents on Microsystems. He has co-edited one scientific monograph on the multi-physics simulation of MEMS and NEMS. The research interests of A. Frangi in the field of MEMS include: the design of new devices; the theoretical and numerical analysis of multi-physics phenomena; the analysis of non-linear phenomena in the dynamical response of MOEMS.

Aldo Ghisi is an Assistant Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. A. Ghisi has authored and co-authored more than 70 scientific publications on various subjects related to materials and structural mechanics. His research areas include multi-physics phenomena in micro/nano structures, particularly related to mechanical simulation of drop impacts, fatigue in polysilicon, gas-solid interaction, study of wafer-to-wafer bonding. Besides microsystems, he is also involved in the numerical and experimental study of metallic alloys for cryogenic applications and in dam engineering.

Stefano Mariani is an Associate Professor of Solid and Structural Mechanics at the Department of Civil and Environmental Engineering of Politecnico di Milano, Italy. S. Mariani has authored and co-authored about 170 scientific publications. His main research interests are: numerical simulations of ductile fracture in metals and quasi-brittle fracture in heterogeneous and functionally graded materials; extended finite element methods; calibration of constitutive models via extended and sigma-point Kalman filters; multi-scale solution methods for dynamic delamination in layered composites; reliability of MEMS subject to shocks and drops; structural health monitoring of composite structures through MEMS sensors.

Series Preface xiii

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

2.2.2 Basic Notions of Dynamics and Relative Dynamics 15

2.2.3 One-Degree-of-Freedom Oscillator 17

2.2.4 Rigid-Body Kinematics and Dynamics 22

2.3 Solid Mechanics 25

2.3.1 Linear Elastic Problem for Deformable Solids 26

2.3.2 Linear Elastic Problem for Beams 35

2.4 Fluid Mechanics 43

2.4.1 Navier–Stokes Equations 43

2.4.2 Fluid–Structure Interaction 48

2.5 Electrostatics and Electromechanics 49

2.5.1 Basic Notions of Electrostatics 49

2.5.2 Simple Electromechanical Problem 54

2.5.3 General Electromechanical Coupled Problem 58

2.6 Piezoelectric Materials in Microsystems 60

2.6.1 Piezoelectric Materials 60

2.6.2 PiezoelectricModelling 62

2.7 Heat Conduction and Thermomechanics 64

2.7.1 Heat Problem 64

2.7.2 Thermomechanical Coupled Problem 67

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

7.2.2 Experimental Set-Up 140

References 145

8 Vibrating Lorentz Force Magnetometers 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

9.3 Frequency Upconversion and Bistability 174

9.4 Fluid–Structure Interaction Energy Harvesters 176

9.4.1 Synopsis of Aeroelastic Phenomena 177

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

10.3.2 Reliability Issues 192

10.4 MultiphysicsModel of an Electrostatic Micropump 196

10.5 Piezoelectric Micropumps 198

10.5.1 Modelling of the Actuator 198

10.5.2 Complete Multiphysics Model 201

References 202

Part III Reliability and Dissipative Phenomena 205

11 Mechanical Characterization at theMicroscale 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 TestingMethodologies 210

11.2.3 Quasi-Static Testing 211

11.2.4 High-Frequency Testing 214

11.3 Weibull Approach 215

11.4 On-Chip TestingMethodology 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

11.4.2 On-Chip Bending Test through a Parallel-Plate Electrostatic Actuator 225

11.4.3 On-Chip Tensile Test through an Electrothermomechanical Actuator 229

11.4.4 On-Chip Test forThick Polysilicon Films 233

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

15.6.2 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

16.6.2 Accelerated Numerical Techniques 383

References 387

Index 393

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