Contents iii
Preface xi
Symbols and units xiii
1 Introduction 1
1.1 History of MEMS 2
1.2 MEMS applications are diverse 2
1.3 MEMS fabrication is based on batch processing 4
1.3.1 Surface micromachining makes thin structures 5
1.3.2 Bulk micromachining makes thick structures 7
1.4 Introduction to the Practical MEMS Practical MEMS book
2 Noise in micromechanical systems 13
2.1 Noise as a statistical quantity 13
2.2 Noise in frequency domain 15
2.2.1 White noise 15
2.2.2 1/f-noise 18
2.3 Equipartition theorem and noise 20
2.3.1 Thermal noise in electrical systems 20
2.3.2 Thermal noise in mechanical systems 23
2.4 Signal-to-noise ratio 26
2.5 Input referred noise 27
2.6 Averaging signals 29
3 Accelerometers 33
3.1 Operation principle 34
3.2 Accelerometer equation 35
3.2.1 Low-frequency response 36
3.2.2 High-frequency response 37
3.2.3 Time domain response 38
3.3 Damping 40
3.4 Mechanical noise in accelerometers 41
3.5 Commercial devices 44
3.5.1 Case study: A surface micromachined accelerometer 44
3.5.2 Case study: A bulk micromachined accelerometer 44
4 Beams as micromechanical springs 49
4.1 Hooke's law for parallel and serial springs 50
4.2 Material properties and theory of elasticity 51
4.2.1 Normal stress and strain 52
4.2.2 Shear stress and strain 54
4.2.3 Material properties 54
4.2.4 General definition of 3D strain 54
4.3 Spring design equations 56
4.3.1 Rod extension 57
4.3.2 Cantilever bending 58
4.3.3 Torsional springs 62
4.3.4 Guided beams 64
4.4 Computer simulations 65
5 Piezoresistive sensing 73
5.1 Piezoresistive effect 74
5.2 Piezoresistive properties of silicon 75
5.3 Piezoresistance measurement 78
5.3.1 Single-ended ratiometric measurement 79
5.3.2 Differential ratiometric measurement 80
5.4 Noise in piezoresistors 84
Thermal noise 84
Shot noise 85
Flicker noise (1/f-noise) 85
6 Capacitive sensing 91
6.1 Capacitance measurement 91
6.1.1 Rate-of-change measurement 92
6.1.2 Displacement measurement 94
6.2 Minimizing the effect of parasitic capacitances 97
6.2.1 Single-chip integration 97
6.2.2 Physical separation 98
6.2.3 Shielding and grounding 98
6.2.4 Bootstrapping 100
6.2.5 Current measurement 102
6.3 Temperature dependency 104
6.4 Demodulation 105
7 Piezoelectric sensing 109
7.1 Piezoelectric effect 110
7.1.1 Longitudinal transducer 111
7.1.2 Transverse transducer 113
7.2 Sensing circuits 114
7.2.1 Current measurement 114
7.2.2 Voltage measurement 115
7.3 Case study: A micromechanical accelerometer 117
8 Signal amplification 121
8.1 Operation amplifiers 121
8.1.1 Inverting amplifier 123
8.1.2 Non-inverting amplifier 124
8.1.3 Transimpedance amplifier 125
8.1.4 Differential amplifier 127
8.1.5 Instrumentation amplifier 128
8.2 Transistor amplifiers 129
8.2.1 Common source amplifier 130
8.2.2 Differential pair 133
9 Amplifier noise 137
9.1 Noise in transistors 137
9.1.1 Noise in common source amplifier 139
9.1.2 Noise in a differential pair 142
9.2 Operational amplifier noise 143
9.3 Amplifier noise in microsystems 149
9.3.1 Case study: A piezoresistive accelerometer 149
9.3.2 Case study: A capacitive accelerometer 150
10 Switched capacitor circuits 157
10.1 Switched capacitor amplifier 157
10.2 Noise in switched capacitor amplifiers 160
10.3 Case study: A switched capacitor accelerometer circuit 162
11 Sensor specifications 167
11.1 System specifications 167
11.2 Element specifications 170
11.3 Commercial accelerometer comparison 172
12 Damping 175
12.1 Damping and quality factor 175
12.2 Damping mechanisms 176
12.2.1 Material losses 176
12.2.2 Anchor losses 178
12.2.3 Surface related losses 180
12.2.4 Mode conversion losses 180
12.2.5 Air damping 181
12.3 Models for the air damping 182
12.3.1 Mean free path and Knudsen number 183
12.3.2 Squeeze film damping 184
12.3.3 Lateral damping 190
12.3.4 Air damping in complex geometries 191
Analytical modeling of approximate nature 191
Semi-analytical simulation 193
Computational flow simulation 194
13 Pressure sensors 197
13.1 Pressure sensing with micromechnical diaphragms 197
13.1.1 Circular diaphragm 198
13.1.2 Square diaphragm 200
13.2 Electromechanical transduction 202
13.2.1 Piezoresistive pressure sensors 202
13.2.2 Capacitive pressure sensors 204
13.3 Large deformation effects 207
13.4 Packaging and specifications 207
14 Actuation 211
14.1 Scaling laws 211
14.2 Scaling of actuation forces 215
14.2.1 Electrostatic forces (capacitive actuation) 215
14.2.2 Magnetic forces 216
14.2.3 Thermal forces 217
14.2.4 Piezoelectric forces 218
15 Capacitive actuation 221
15.1 Force acting on a capacitor 221
15.2 Parallel plate transducer 223
15.2.1 Equilibrium and pull-in point 224
15.2.2 Capacitive spring forces 228
15.3 Longitudinal comb finger capacitor (comb drive) 229
15.4 Capacitive actuation with ac voltages 232
15.4.1 Time harmonic actuation with dc bias 232
15.4.2 w0/2-actuation 234
16 Piezoelectric actuation 239
16.1 Actuation force 239
16.1.1 Longitudinal actuator 240
16.1.2 Transverse actuator 244
17 Thermal actuation 249
17.1 Principle of operation 249
17.2 Leverage for large displacement 251
17.3 Transient analysis 253
17.3.1 Steady state 254
17.3.2 Heating 254
17.3.3 Cooling 256
17.4 Higher order models 257
17.5 Bi-stable actuators 258
18 Micro-optical devices 261
18.1 Huygens' principle 261
18.2 Gaussian beam optics 263
18.3 Micromirrors 266
18.3.1 Optical scanners 266
18.3.2 Displays 268
18.4 MEMS for fiber optical communications 270
18.4.1 Attenuators 271
18.4.2 Optical switches 273
18.4.3 Vanishing optical MEMS 276
18.5 Interferometry 277
18.6 Exercises 280
19 RF MEMS 283
19.1 Solid-state switches and varactors 284
19.1.1 Solid-state switches 285
19.1.2 Solid-state varactors 288
19.2 Micromechanical varactors 289
19.3 Micromechanical switches 291
19.3.1 Capacitive switches 293
19.3.2 Ohmic switches 296
19.3.3 Switching speed 299
19.3.4 Cost and reliability 300
19.4 RF inductors 301
20 Modeling microresonators 305
20.1 Lumped model for mechanical vibrations 305
20.1.1 Vibration mode 306
20.1.2 Effective mass and spring for the lumped resonator 308
20.1.3 General calculation of the lumped model parameters 310
20.2 Electromechanical transduction 312
20.2.1 Transduction factor 312
20.2.2 Transduction in distributed resonators 312
20.2.3 Capacitive transduction 315
20.2.4 Piezoelectric transduction 316
20.3 Electrical equivalent circuit 320
20.4 Nonlinear effects in microresonators 322
20.4.1 Case study: Nonlinearity in a clamped-clamped beam 324
21 Microresonator applications 329
21.1 Clock oscillator 329
21.2 Reference oscillators 332
21.3 RF filters 337
21.3.1 FBAR filters 338
21.3.2 Silicon MEMS filters 340
22 Gyroscopes 343
22.1 Coriolis force 344
22.2 Vibrating two-mode gyroscope 346
22.2.1 Drive-mode vibrations 347
22.2.2 Sense-mode vibrations 347
Matched modes
Separated modes
22.3 Capacitive gyroscopes 349
22.4 Quadrature error 352
22.5 Measurement circuitry 355
22.6 Commercial gyroscopes 357
22.6.1 Case study: Quartz tuning fork gyroscope 357
22.6.2 Case study: Piezoelectric metal/ceramic gyroscope 358
22.6.3 Case study: Surface micromachined gyroscope 359
23 Microfluidics 365
23.1 Flow in microchannels 366
23.2 Mixing 369
23.3 Microfluidic systems: valves and pumps 371
23.3.1 Microvalves 371
23.3.2 Micropumps 372
23.4 Nonmechanical pumps 375
23.5 Minimum sample volume 378
24 Economics of microfabrication 383
24.1 Yield analysis 384
24.2 Cost analysis 386
24.2.1 Cost case study: MEMS integration 389
24.3 Profit analysis 391
24.3.1 Profit case study: Fabless start-up 392
24.3.2 Profit case study: VTI Technologies 394
24.4 Beyond the high cost manufacturing 395
A Laplace transform 399
B Mechanical harmonic resonators 403
B.1 Frequency response 404
B.1.1 Amplitude response 405
B.1.2 Phase response 406
B.2 Impulse response 406
B.2.1 Under damped 407
B.2.2 Critically damped 408
B.2.3 Over damped 408
B.3 Step response 408
B.3.1 Under damped 409
B.3.2 Critically damped 409
B.3.3 Over damped 410
B.4 Transient response of forced vibrations 410
C Nonlinear vibrations in resonators 413
C.1 Nonlinear spring forces 413
C.2 Unforced vibrations 414
C.3 Forced vibrations 416
D Thermal noise generator 419
D.1 Derivation of noise voltage generator 419
D.2 Derivation of mechanical noise force generator 421
E Anisotropic elasticity of silicon 423
F Anisotropic piezoresistivity of silicon 429
G Often used formulas 431
G.1 Constants 431
G.2 Decibel (dB) units 431
G.3 Mode shapes and resonant frequencies for beams with different boundary conditions 432
G.4 Second moment of inertias 432
Bibliography 435
Index 455