Understanding MEMS
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Preface xiii
About the Companion Website xv
1 Scaling of Forces 1
1.1 Scaling of Forces Model 1
1.2 Weight 2
1.2.1 Example: MEMS Accelerometer 2
1.3 Elastic Force 3
1.3.1 Example: AFM Cantilever 4
1.4 Electrostatic Force 4
1.4.1 Example: MEMS RF Switch 6
1.5 Capillary Force 6
1.5.1 Example: Wet Etching Force 8
1.6 Piezoelectric Force 8
1.6.1 Example: Force in Film Embossing 9
1.7 Magnetic Force 10
1.7.1 Example: Compass Magnetometer 10
1.8 Dielectrophoretic Force 11
1.8.1 Example: Nanoparticle in a Spherical Symmetry Electric Field 12
1.9 Summary 13
Problems 13
2 Elasticity 15
2.1 Stress 15
2.2 Strain 18
2.3 Stress-strain Relationship 20
2.3.1 Example: Plane Stress 21
2.4 Strain-stress Relationship in Anisotropic Materials 22
2.5 Miller Indices 23
2.5.1 Example: Miller Indices of Typical Planes 24
2.6 Angles of Crystallographic Planes 25
2.6.1 Example 25
2.7 Compliance and Stiffness Matrices for Single-Crystal Silicon 26
2.7.1 Example: Young's Modulus and Poisson Ratio for (100) Silicon 27
2.8 Orthogonal Transformation 29
2.9 Transformation of the Stress State 31
2.9.1 Example: Rotation of the Stress State 31
2.9.2 Example: Matrix Notation for the Rotation of the Stress State 32
2.10 Orthogonal Transformation of the Stiffness Matrix 32
2.10.1 Example: C11 Coefficient in Rotated Axes 33
2.10.2 Example: Young's Modulus and Poisson Ratio in the (111) Direction 34
2.11 Elastic Properties of Selected MEMS Materials 36
Problems 36
3 Bending of Microstructures 37
3.1 Static Equilibrium 37
3.2 Free Body Diagram 38
3.3 Neutral Plane and Curvature 39
3.4 Pure Bending 40
3.4.1 Example: Neutral Plane for a Rectangular Cross-section 41
3.4.2 Example: Cantilever with Point Force at the Tip 42
3.5 Moment of Inertia and Bending Moment 43
3.5.1 Example: Moment of Inertia of a Rectangular Cross-section 43
3.6 Beam Equation 44
3.7 End-loaded Cantilever 45
3.8 Equivalent Stiffness 47
3.9 Beam Equation for Point Load and Distributed Load 48
3.10 Castigliano's Second Theorem 48
3.10.1 Strain Energy in an Elastic Body Subject to Pure Bending 50
3.11 Flexures 51
3.11.1 Fixed-fixed Flexure 51
3.11.2 Example: Comparison of Stiffness Constants 53
3.11.3 Example: Folded Flexure 53
3.12 Rectangular Membrane 54
3.13 Simplified Model for a Rectangular Membrane Under Pressure 55
3.13.1 Example: Thin Membrane Subject to Pressure 57
3.14 Edge-clamped Circular Membrane 58
Problems 60
4 Piezoresistance and Piezoelectricity 65
4.1 Electrical Resistance 65
4.1.1 Example: Resistance Value 66
4.2 One-dimensional Piezoresistance Model 67
4.2.1 Example: Gauge Factors 68
4.3 Piezoresistance in Anisotropic Materials 69
4.4 Orthogonal Transformation of Ohm's Law 70
4.5 Piezoresistance Coefficients Transformation 71
4.5.1 Example: Calculation of Rotated Piezoresistive Components ¿¿¿¿' 11, ¿¿¿¿' 12 and ¿¿¿¿' 16 for unit axes X' [110], Y' [ ¯110] and Z' [001] 72
4.5.2 Analytical Expressions for Some Rotated Piezoresistive Components 74
4.6 Two-dimensional Piezoresistors 74
4.6.1 Example: Accelerometer with Cantilever and Piezoresistive Sensing 76
4.7 Pressure Sensing with Rectangular Membranes 79
4.7.1 Example: Single-resistor Pressure Sensor 82
4.7.2 Example: Pressure Sensors Comparison 85
4.8 Piezoelectricity 86
4.8.1 Relevant Data for Some Piezoelectric Materials 88
4.8.2 Example: Piezoelectric Generator 89
Problems 91
5 Electrostatic Driving and Sensing 93
5.1 Energy and Co-energy 93
5.2 Voltage Drive 97
5.3 Pull-in Voltage 97
5.3.1 Example: Forces in a Parallel-plate Actuator 99
5.4 Electrostatic Pressure 101
5.5 Contact Resistance in Parallel-plate Switches 101
5.6 Hold-down Voltage 101
5.6.1 Example: Calculation of Hold-down Voltage 102
5.7 Dynamic Response of Pull-in-based Actuators 102
5.7.1 Example: Switching Transient 103
5.8 Charge Drive 105
5.9 Extending the Stable Range 105
5.10 Lateral Electrostatic Force 106
5.11 Comb Actuators 106
5.12 Capacitive Accelerometer 108
5.13 Differential Capacitive Sensing 108
5.14 Torsional Actuator 110
Problems 111
6 Resonators 115
6.1 Free Vibration: Lumped-element Model 115
6.2 Damped Vibration 116
6.3 Forced Vibration 117
6.3.1 Example: Vibration Amplitude as a Function of the Damping Factor 120
6.4 Small Signal Equivalent Circuit of Resonators 121
6.4.1 Example: Series and Parallel Resonances 125
6.4.2 Example: Spring Softening 125
6.5 Rayleigh-Ritz Method 126
6.5.1 Example: Vibration of a Cantilever 128
6.5.2 Example: Gravimetric Chemical Sensor 129
6.6 Resonant Gyroscope 130
6.7 Tuning Fork Gyroscope 133
6.7.1 Example: Calculation of Sensitivity in a Tuning Fork Gyroscope 134
Problems 135
7 Microfluidics and Electrokinetics 137
7.1 Viscous Flow 137
7.2 Flow in a Cylindrical Pipe 140
7.2.1 Example: Pressure Gradient Required to Sustain a Flow 141
7.3 Electrical Double Layer 142
7.3.1 Example: Debye Length and Surface Charge 144
7.4 Electro-osmotic Flow 144
7.5 Electrowetting 146
7.5.1 Example: Droplet Change by Electrowetting 148
7.5.2 Example: Full Substrate Contacts 149
7.6 Electrowetting Dynamics 151
7.6.1 Example: Contact-angle Dynamics 153
7.7 Dielectrophoresis 153
7.7.1 Electric Potential Created by a Constant Electric Field 154
7.7.2 Potential Created by an Electrical Dipole 155
7.7.3 Superposition 156
Problems 157
8 Thermal Devices 159
8.1 Steady-state Heat Equation 159
8.2 Thermal Resistance 161
8.2.1 Example: Temperature Profile in a Heated Wire 162
8.2.2 Example: Resistor Suspended in a Bridge 165
8.3 Platinum Resistors 166
8.4 Flow Measurement Based on Thermal Sensors 166
8.4.1 Example: Micromachined Flow Sensor 169
8.5 Dynamic Thermal Equivalent Circuit 171
8.6 Thermally Actuated Bimorph 172
8.6.1 Example: Bimorph Actuator 174
8.7 Thermocouples and Thermopiles 175
8.7.1 Example: IR Detector 175
Problems 176
9 Fabrication 181
9.1 Introduction 181
9.2 Photolithography 182
9.3 Patterning 183
9.4 Lift-off 184
9.5 Bulk Micromachining 184
9.5.1 Example: Angle of Walls in Silicon (100) Etching 185
9.6 Silicon Etch Stop When Using Alkaline Solutions 186
9.6.1 Example: Boron drive-in at 1050¿C 186
9.7 Surface Micromachining 186
9.7.1 Example: Cantilever Fabrication by Surface Micromachining 187
9.8 Dry Etching 188
9.9 CMOS-compatible MEMS Processing 188
9.9.1 Example: Bimorph Actuator Compatible with CMOS Process 189
9.10 Wafer Bonding 190
9.11 PolyMUMPs Foundry Process 190
9.11.1 Example: PolyMUMPs Cantilever for a Fabry-Perot Pressure Sensor 191
Problems 192
APPENDICES 195
A Chapter 1 Solutions 197
B Chapter 2 Solutions 207
C Chapter 3 Solutions 221
D Chapter 4 Solutions 239
E Chapter 5 Solutions 249
F Chapter 6 Solutions 267
G Chapter 7 Solutions 277
H Chapter 8 Solutions 285
I Chapter 9 Solutions 299
References 307
Index 311
