MICROSYSTEM DESIGN

Stephen D. Senturia Massachusetts Institute of Technology

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KLUWER ACÄDEMIC PUBLISHERS Boston / Dordrecht / London

Contents

Foreword xvii Preface xxi Acknowledgments xxv

Part I GETTING STARTED

1. INTRODUCTION 3 1.1 Microsystems vs. MEMS 3

1.1.1 Whatarethey? 3 1.1.2 How are they made? 5 1.1.3 What are they made of? 6 1.1.4 How are they designed? 7

1.2 Markets for Microsystems and MEMS 8 1.3 CaseStudies 9 1.4 LookingAhead 12

2. AN APPROACH TO MEMS DESIGN 15 2.1 Design: The Big Picture 15

2.1.1 Device Categories 15 2.1.2 High-Level Design Issues 16 2.1.3 The Design Process 17

2.2 Modeling Levels 19 2.2.1 Analytical or Numerical? 21 2.2.2 A Closer Look 22

2.3 Example: A Position-Control System 24 2.4 Going Forward From Here 26

3. MICROFABRICATION 29 3.1 Overview 29 3.2 Wafer-Level Processes 30

3.2.1 Substrates 30 3.2.2 WaferCleaning 34

viii MICROSYSTEM DESIGN

3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

Oxidation of Silicon Local Oxidation Doping Thin-Film Deposition Wafer Bonding

3.3 Pattern Transfer 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7

Optical Lithography Design Rules Mask Making Wet Etching Dry Etching Additive Processes: Lift-Off Planarization

3.4 Conclusion

34 37 38 42 47 50 50 54 55 57 67 71 74 77

PROCESS INTEGRATION 79 4.1 Developing a Process 79

4.1.1 A Simple Process Flow 79 4.1.2 The Self-Aligned Gate: A Paradigm-Shifting Process 83

4.2 Basic Principles of Process Design 85 4.2.1 From Shape to Process and Back Again 85 4.2.2 Process Design Issues 86

4.3 Sample Process Flows 91 4.3.1 A Bulk-Micromachined Diaphragm Pressure Sensor 92 4.3.2 A Surface-Micromachined Suspended Filament 97

4.4 Moving On 98

Part II MODELING STRATEGIES

5. LUMPED MODELING 103 5.1 Introduction 103 5.2 Conjugate Power Variables 104 5.3 One-Port Elements 106

5.3.1 Ports 106 5.3.2 The Variable-Assignment Conventions 106 5.3.3 One-Port Source Elements 108 5.3.4 One-Port Circuit Elements 109

5.4 Circuit Connections in the e -> V Convention 114 5.4.1 Kirchhoff's Laws 114

5.5 Formulation of Dynamic Equations 116 5.5.1 Complex Impedances 116 5.5.2 State Equations 117

5.6 Transformers and Gyrators 118 5.6.1 Impedance Transformations 119 5.6.2 The Electrical Inductor 120

6. ENERGY-CONSERVING TRANSDUCERS 125

Contents ix

6.1 Introduction 125 6.2 The Parallel-Plate Capacitor 125

6.2.1 Charging the Capacitor at Fixed Gap 126 6.2.2 Charging the Capacitor at Zero Gap, then Lifting 127

6.3 The Two-Port Capacitor 129 6.4 Electrostatic Actuator 130

6.4.1 Charge Control 131 6.4.2 Voltage Control 132 6.4.3 Pull-In 134 6.4.4 Adding Dynamics to the Actuator Model 137

6.5 The Magnetic Actuator 139 6.6 Equivalent Circuits for Linear Transducers 142 6.7 The Position Control System - Revisited 145

7. DYNAMICS 149 7.1 Introduction 149 7.2 Linear System Dynamics 150

7.2.1 Direct Integration 151 7.2.2 System Functions 152 7.2.3 Fourier Transform 157 7.2.4 Sinusoidal Steady State 158 7.2.5 Eigenfunction Analysis 160

7.3 Nonlinear Dynamics 164 7.3.1 Fixed Points of Nonlinear Systems 164 7.3.2 Linearization About an Operating Point 165 7.3.3 Linearization of the Electrostatic Actuator 166 7.3.4 Transducer Model for the Linearized Actuator 169 7.3.5 Direct Integration of State Equations 173 7.3.6 Resonators and Oscillators 178 7.3.7 And Then There's Chaos... 178

Part m DOMAIN-SPECIFIC DETAILS

8. ELASTICITY 183 8.1 Introduction 183 8.2 Constitutive Equations of Linear Elasticity 184

8.2.1 Stress 184 8.2.2 Strain 185 8.2.3 Elastic Constants for Isotropie Materials 186 8.2.4 Other Elastic Constants 188 8.2.5 Isotropie Elasticity in Three Dimensions 189 8.2.6 Plane Stress 190 8.2.7 Elastic Constants for Anisotropie Materials 191

8.3 Thermal Expansion and Thin-Film Stress 193 8.3.1 Other Sources of Residual Thin-Film Stress 195

8.4 Selected Mechanical Property Data 196 8.5 Material Behavior at Large Strains 196

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8.5.1 Plastic Deformation 197

9. STRUCTURES 201 9.1 Overview 201 9.2 Axially Loaded Beams 201

9.2.1 Beams With Varying Cross-section 203 9.2.2 Statically Indeterminate Beams 203 9.2.3 Stresses on Inclined Sections 205

9.3 Bending of Beams 207 9.3.1 Typesof Support 207 9.3.2 Typesof Loads 207 9.3.3 Reaction Forces and Moments 208 9.3.4 Pure Bending of a Transversely Loaded Beam 211 9.3.5 Differential Equation for Beam Bending 213 9.3.6 Elementary Solutions of the Beam Equation 216

9.4 Anticlastic Curvature 218 9.5 Bending ofPlates 219

9.5.1 Plate in Pure Bending 220 9.6 Effects of Residual Stresses and Stress Gradients 222

9.6.1 Stress Gradients in Cantilevers 222 9.6.2 Residual Stresses in Doubly-Supported Beams 226 9.6.3 Bückling of Beams 231

9.7 Plates With In-Plane Stress 235 9.8 What about large deflections? 237

10. ENERGY METHODS 239 10.1 Elastic Energy 240 10.2 The Principleof Virtual Work 243 10.3 Variational Methods 244

10.3.1 Properties of the Variational Solution 247 10.4 Large Deflections of Elastic Structures 249

10.4.1 A Center-Loaded Doubly-Clamped Beam 249 10.4.2 Combining Variational Results with Simulations 253 10.4.3 The Uniformly Loaded Doubly-Clamped Beam 254 10.4.4 Residual Stress in Clamped Structures 255 10.4.5 Elastic Energy in Plates and Membranes 256 10.4.6 Uniformly Loaded Plates and Membranes 257 10.4.7 Membrane Load-Deflection Behavior 259

10.5 Rayleigh-Ritz Methods 260 10.5.1 Estimating Resonant Frequencies 260 10.5.2 Extracting Lumped-Element Masses 263

11. DISSIPATION AND THE THERMAL ENERGY DOMAIN 267 11.1 Dissipation is Everywhere 267 11.2 Electrical Resistance 267 11.3 Charging a Capacitor r 269 11.4 Dissipative Processes 271

Contents xi

11.5 The Thermal Energy Domain 272 11.5.1 The Heat-Flow Equation 275 11.5.2 Basic Thermodynamic Ideas 275 11.5.3 Lumped Modeling in the Thermal Domain 277

11.6 Self-Heating of a Resistor 278 11.6.1 Temperature Coefficient of Resistance 279 11.6.2 Current-source drive 279 11.6.3 Voltage-source drive 281 11.6.4 A Self-Heated Silicon Resistor 282

11.7 Other Dissipation Mechanisms 286 11.7.1 Contact Friction 286 11.7.2 Dielectric losses 287 11.7.3 Viscoelastic losses 288 11.7.4 Magnetic Losses 289 11.7.5 Diffusion 290

11.8 Irreversible Thermodynamics: Coupled Flows 291 11.8.1 Thermoelectric Power and Thermocouples 293 11.8.2 Thermoelectric Heating and Cooling 295 11.8.3 Other Coupled-Flow Problems 296

11.9 Modeling Time-Dependent Dissipative Processes 296

12. LUMPED MODELING OF DISSIPATTVE PROCESSES 299 12.1 Overview 299 12.2 The Generalized Heat-Flow Equation 299 12.3 The DC Steady State: The Poisson Equation 300 12.4 Finite-Difference Solution of the Poisson Equation 301

12.4.1 Temperature Distribution in a Self-Heated Resistor 303 12.5 Eigenfunction Solution of the Poisson Equation 305 12.6 Transient Response: Finite-Difference Approach 307 12.7 Transient Response: Eigenfunction Method 307 12.8 One-Dimensional Example 308 12.9 Equivalent Circuit for a Single Mode 309 12.10 Equivalent Circuit Including All Modes 311

13. FLUIDS 317 13.1 What Makes Fluids Difficult? 317 13.2 Basic Fluid Concepts 318

13.2.1 Viscosity 318 13.2.2 Thermophysical Properties 319 13.2.3 Surface Tension 320 13.2.4 ConservationofMass 322 13.2.5 Time Rate of Change of Momentum 323 13.2.6 The Navier-Stokes Equation 324 13.2.7 Energy Conservation 324 13.2.8 Reynolds Number and Mach Number 325

13.3 Incompressible Laminar Flow 326 13.3.1 CouetteFlow 327

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13.3.2 Poiseuille Flow 328 13.3.3 Development Lengths and Boundary Layers 331 13.3.4 StokesFlow 332

13.4 Squeezed-Film Damping 332 13.4.1 Rigid Parallel-Plate Small-Amplitude Motion 334

13.5 Electrolytes and Electrokinetic Effects 339 13.5.1 Ionic Double Layers 340 13.5.2 Electroosmotic Flow 343 13.5.3 Electrophoresis 344 13.5.4 Diffusion Effects 347 13.5.5 Pressure Effects 348 13.5.6 Mixing 348 13.5.7 Modelingof Electrokinetic Systems 349

PartlV CIRCUIT AND SYSTEM ISSUES

14. ELECTRONICS 353 14.1 Introduction 353 14.2 Elements of Semiconductor Physics 353

14.2.1 Equilibrium Carrier Concentrations 354 14.2.2 Excess Carriers 355

14.3 The Semiconductor Diode 357 14.4 The Diffused Resistor 363 14.5 The Photodiode 364 14.6 The Bipolar .Function Transistor 365 14.7 The MOSFET 365

14.7.1 Large-Signal Characteristics of the MOSFET 367 14.7.2 MOSFET Capacitances 371 14.7.3 Small-Signal Model of the MOSFET 371

14.8 MOSFET Amplifiers 372 14.8.1 The CMOS Inverter 373 14.8.2 Large-Signal Switching Speed 376 14.8.3 The Linear-Gain Region 379 14.8.4 Other Amplifier Configurations 381

14.9 Operational Amplifiers 381 14.10 Dynamic Effects 383 14.11 Basic Op-Amp Circuits 384

14.11.1 inverting Amplifier 3 84 14.11.2 Short Method for Analyzing Op-Amp Circuits 387 14.11.3 Noninverting Amplifier 3 87 14.11.4 Transimpedance Amplifier 388 14.11.5 Integrator 389 14.11.6 Differentiator 390

14.12 Charge-Measuring Circuits 391 14.12.1 Differential Charge Measurement 391 14.12.2 Switched-Capacitor Circuits 393

Contents xiu

15. FEEDBACK SYSTEMS 397 15.1 Introduction 397 15.2 Basic Feedback Concepts 397 15.3 Feedback in Linear Systems 398

15.3.1 Feedback Amplifiers 399 15.3.2 Example: The Position Controller 400 15.3.3 PIDControl 405 15.3.4 The Effect of AmplifierBandwidth 407 15.3.5 Phase Margin 408 15.3.6 Noise and Disturbances 409 15.3.7 Stabilization of Unstable Systems 410 15.3.8 ControUability and Observability Revisited 411

15.4 Feedback in Nonlinear Systems 411 15.4.1 Quasi-static Nonlinear Feedback Systems 412

15.5 Resonators and Oscillators 413 15.5.1 Simulink Model 417 15.5.2 The (Almost) Sinusoidal Oscillator 418 15.5.3 Relaxation Oscillation 420

16. NOISE 425 16.1 Introduction 425 16.2 The Interference Problem 426

16.2.1 Shields 427 16.2.2 GroundLoops 428 16.2.3 Guards 429

16.3 Characterization of Signals 430 16.3.1 Amplitude-Modulated Signals 431

16.4 Characterization of Random Noise 433 16.4.1 Mean-Square and Root-Mean-Square Noise 434 16.4.2 Addition of Uncorrelated Sources 434 16.4.3 Signal-to-Noise Ratio 435 16.4.4 Spectral Density Function 435 16.4.5 Noise in Linear Systems 436

16.5 Noise Sources 436 16.5.1 Thermal Noise 436 16.5.2 Noise Bandwidth 438 16.5.3 Shot Noise 439 16.5.4 Flicker Noise 440 16.5.5 Amputier Noise 441

16.6 Example: A Resistance Thermometer 442 16.6.1 Using a DC source 445 16.6.2 Modulation of an AC Carrier 446 16.6.3 CAUTION: Modulation Does Not Always Work 447

16.7 Drifts 447

xiv MICROSYSTEM DESIGN

PartV CASESTUDIES

17. PACKAGING 453 17.1 Introduction to the Case Studies 453 17.2 Packaging, Test, and Calibration 454 17.3 An Approach to Packaging 455 17.4 A Commercial Pressure-Sensor Case Study 459

17.4.1 Device Concept 461 17.4.2 System Partitioning 461 17.4.3 Interfaces 462 17.4.4 Details 463 17.4.5 A Final Comment 467

18. A PIEZORESISTIVE PRESSURE SENSOR 469 18.1 Sensing Pressure 469 18.2 Piezoresistance 470

18.2.1 Analytic Formulation in Cubic Materials 471 18.2.2 Longitudinal and Transverse Piezoresistance 472 18.2.3 Piezoresistive Coefficients of Silicon 473 18.2.4 Structural Examples 474 18.2.5 Averaging over Stress and Doping Variations 477 18.2.6 A Numerical Example 480

18.3 The Motorola MAP Sensor 481 18.3.1 ProcessFlow 481 18.3.2 Details of the Diaphragm and Piezoresistor 483 18.3.3 Stress Analysis 485 18.3.4 Signal-Conditioning and Calibration 488 18.3.5 Device Noise 492 18.3.6 Recent Design Changes 493 18.3.7 Higher-Order Effects 494

19. A CAPACinVE ACCELEROMETER 497 19.1 Introduction 497 19.2 Fundamentals of Quasi-Static Accelerometers 498 19.3 Position Measurement With Capacitance 500

19.3.1 Circuits for Capacitance Measurement 502 19.3.2 Demodulation Methods 507 19.3.3 Chopper-Stabilized Amplifiers 510 19.3.4 Correlated Double Sampling 511 19.3.5 Signal-to-Noise Issues 512

19.4 A Capacitive Accelerometer Case Study 513 19.4.1 Specifications 516 19.4.2 Sensor Design and Modeling 518 19.4.3 Fabrication and Packaging 520 19.4.4 Noise and Accuracy 523

19.5 Position Measurement With Tunneling Tips 525

20. ELECTROSTATIC PROJECTION DISPLAYS 531

Contents xv

20.1 Introduction 531 20.2 Electromechanics of the DMD Device 536

20.2.1 Electrode Structure 536 20.2.2 Torsional Pull-in 537

20.3 Electromechanics of Electrostatically Actuated Beams 541 20.3.1 M-Test 544

20.4 The Grating-Light-Valve Display 544 20.4.1 Diffraction Theory 544 20.4.2 Device Fabrication and Packaging 548 20.4.3 Quantitative Estimates of GLV Device Performance 550

20.5 AComparison 558

21. A PIEZOELECTRIC RATE GYROSCOPE 561 21.1 Introduction 561 21.2 Kinematics of Rotation 561 21.3 The Coriolis Rate Gyroscope 563

21.3.1 Sinusoidal Response Function 565 21.3.2 Steady Rotation 566 21.3.3 Response to Angular Accelerations 567 21.3.4 Generalized Gyroscopic Modes 567

21.4 Piezoelectricity 570 21.4.1 The Originof Piezoelectricity 570 21.4.2 Analytical Formulation of Piezoelectricity 571 21.4.3 Piezoelectric Materials 573 21.4.4 Piezoelectric Actuation 575 21.4.5 Sensing with Piezoelectricity 577

21.5 A Quartz Rate Gyroscope Case Study 578 21.5.1 Electrode Structures 579 21.5.2 Lumped-Element Modeling of Piezoelectric Devices 582 21.5.3 QRS Specifications and Performance 592 21.5.4 A Quantitative Device Model 594 21.5.5 The Drive Mode 595 21.5.6 Sense-Mode Displacement of the Drive Tines 598 21.5.7 Coupling to the Sense Tines 599 21.5.8 Noise and Accuracy Considerations 602 21.5.9 Closing Comments 602

22. DNA AMPLMCATION 605 22.1 Introduction 605 22.2 Polymerase Chain Reaction (PCR) 606

22.2.1 Elements of PCR 606 22.2.2 Specifications for a PCR System 610

22.3 Microsystem Approaches to PCR 611 22.3.1 Batch System 611 22.3.2 PCR Flow System 614

22.4 Thermal Model of the Batch Reactor 616 22.4.1 Control Circuit and Transient Behavior 618

xvi MICROSYSTEM DESIGN

22.5 Thermal Model of the Continuous Flow Reactor 621 22.6 AComparison 625

23. A MICROBRIDGE GAS SENSOR 629 23.1 Overview 629 23.2 System-Level Issues 630 23.3 First-Order Device and System Models 632

23.3.1 Filament Characteristics 632 23.3.2 Resistance-Control System 634

23.4 A Practical Device and Fabrication Process 639 23.4.1 Creating the Filament 639 23.4.2 High-Temperature Bond Pads 641 23.4.3 Catalyst Coating 642

23.5 Sensor Performance 643 23.5.1 Demonstration of Hydrogen Detection 643 23.5.2 Mass-Transport-Limited Operation 644 23.5.3 Reaction-Rate-Limited Operation 645

23.6 Advanced Modeling 646 23.7 Epilogue 648

Appendices 650 A-Glossary of Notation 651 B - Electromagnetic Fields 657

B.l Introduction 657 B.2 Quasistatic Fields 657 B.3 Elementary Laws 657 B.4 Electroquasistatic Systems 658 B.5 Magnetoquasistatic Systems 659

C- Elastic Constants in Cubic Material 663

References 665

Index 677