Understanding MEMS

Luis Castañer, Universitat Politecnica de Cataluña, Barcelona, SpainLuis Castañer is a Professor at Universitat Politecnica de Cataluña, where he teaches courses focusing on semiconductor devices, analog circuits, photovoltaic systems, solar cells and MEMS.

• Preface xiiiAbout the Companion Website xv1 Scaling of Forces 11.1 Scaling of Forces Model 11.2 Weight 21.2.1 Example: MEMS Accelerometer 21.3 Elastic Force 31.3.1 Example: AFM Cantilever 41.4 Electrostatic Force 41.4.1 Example: MEMS RF Switch 61.5 Capillary Force 61.5.1 Example: Wet Etching Force 81.6 Piezoelectric Force 81.6.1 Example: Force in Film Embossing 91.7 Magnetic Force 101.7.1 Example: Compass Magnetometer 101.8 Dielectrophoretic Force 111.8.1 Example: Nanoparticle in a Spherical Symmetry Electric Field 121.9 Summary 13Problems 132 Elasticity 152.1 Stress 152.2 Strain 182.3 Stress–strain Relationship 202.3.1 Example: Plane Stress 212.4 Strain–stress Relationship in Anisotropic Materials 222.5 Miller Indices 232.5.1 Example: Miller Indices of Typical Planes 242.6 Angles of Crystallographic Planes 252.6.1 Example 252.7 Compliance and Stiffness Matrices for Single-Crystal Silicon 262.7.1 Example: Young’s Modulus and Poisson Ratio for (100) Silicon 272.8 Orthogonal Transformation 292.9 Transformation of the Stress State 312.9.1 Example: Rotation of the Stress State 312.9.2 Example: Matrix Notation for the Rotation of the Stress State 322.10 Orthogonal Transformation of the Stiffness Matrix 322.10.1 Example: C11 Coefficient in Rotated Axes 332.10.2 Example: Young’s Modulus and Poisson Ratio in the (111) Direction 342.11 Elastic Properties of Selected MEMS Materials 36Problems 363 Bending of Microstructures 373.1 Static Equilibrium 373.2 Free Body Diagram 383.3 Neutral Plane and Curvature 393.4 Pure Bending 403.4.1 Example: Neutral Plane for a Rectangular Cross-section 413.4.2 Example: Cantilever with Point Force at the Tip 423.5 Moment of Inertia and Bending Moment 433.5.1 Example: Moment of Inertia of a Rectangular Cross-section 433.6 Beam Equation 443.7 End-loaded Cantilever 453.8 Equivalent Stiffness 473.9 Beam Equation for Point Load and Distributed Load 483.10 Castigliano’s Second Theorem 483.10.1 Strain Energy in an Elastic Body Subject to Pure Bending 503.11 Flexures 513.11.1 Fixed–fixed Flexure 513.11.2 Example: Comparison of Stiffness Constants 533.11.3 Example: Folded Flexure 533.12 Rectangular Membrane 543.13 Simplified Model for a Rectangular Membrane Under Pressure 553.13.1 Example: Thin Membrane Subject to Pressure 573.14 Edge-clamped Circular Membrane 58Problems 604 Piezoresistance and Piezoelectricity 654.1 Electrical Resistance 654.1.1 Example: Resistance Value 664.2 One-dimensional Piezoresistance Model 674.2.1 Example: Gauge Factors 684.3 Piezoresistance in Anisotropic Materials 694.4 Orthogonal Transformation of Ohm’s Law 704.5 Piezoresistance Coefficients Transformation 714.5.1 Example: Calculation of Rotated Piezoresistive Components 𝜋′ 11, 𝜋′ 12 and 𝜋′ 16 for unit axes X′ [110], Y′ [ ̄110] and Z′ [001] 724.5.2 Analytical Expressions for Some Rotated Piezoresistive Components 744.6 Two-dimensional Piezoresistors 744.6.1 Example: Accelerometer with Cantilever and Piezoresistive Sensing 764.7 Pressure Sensing with Rectangular Membranes 794.7.1 Example: Single-resistor Pressure Sensor 824.7.2 Example: Pressure Sensors Comparison 854.8 Piezoelectricity 864.8.1 Relevant Data for Some Piezoelectric Materials 884.8.2 Example: Piezoelectric Generator 89Problems 915 Electrostatic Driving and Sensing 935.1 Energy and Co-energy 935.2 Voltage Drive 975.3 Pull-in Voltage 975.3.1 Example: Forces in a Parallel-plate Actuator 995.4 Electrostatic Pressure 1015.5 Contact Resistance in Parallel-plate Switches 1015.6 Hold-down Voltage 1015.6.1 Example: Calculation of Hold-down Voltage 1025.7 Dynamic Response of Pull-in-based Actuators 1025.7.1 Example: Switching Transient 1035.8 Charge Drive 1055.9 Extending the Stable Range 1055.10 Lateral Electrostatic Force 1065.11 Comb Actuators 1065.12 Capacitive Accelerometer 1085.13 Differential Capacitive Sensing 1085.14 Torsional Actuator 110Problems 1116 Resonators 1156.1 Free Vibration: Lumped-element Model 1156.2 Damped Vibration 1166.3 Forced Vibration 1176.3.1 Example: Vibration Amplitude as a Function of the Damping Factor 1206.4 Small Signal Equivalent Circuit of Resonators 1216.4.1 Example: Series and Parallel Resonances 1256.4.2 Example: Spring Softening 1256.5 Rayleigh–Ritz Method 1266.5.1 Example: Vibration of a Cantilever 1286.5.2 Example: Gravimetric Chemical Sensor 1296.6 Resonant Gyroscope 1306.7 Tuning Fork Gyroscope 1336.7.1 Example: Calculation of Sensitivity in a Tuning Fork Gyroscope 134Problems 1357 Microfluidics and Electrokinetics 1377.1 Viscous Flow 1377.2 Flow in a Cylindrical Pipe 1407.2.1 Example: Pressure Gradient Required to Sustain a Flow 1417.3 Electrical Double Layer 1427.3.1 Example: Debye Length and Surface Charge 1447.4 Electro-osmotic Flow 1447.5 Electrowetting 1467.5.1 Example: Droplet Change by Electrowetting 1487.5.2 Example: Full Substrate Contacts 1497.6 Electrowetting Dynamics 1517.6.1 Example: Contact-angle Dynamics 1537.7 Dielectrophoresis 1537.7.1 Electric Potential Created by a Constant Electric Field 1547.7.2 Potential Created by an Electrical Dipole 1557.7.3 Superposition 156Problems 1578 Thermal Devices 1598.1 Steady-state Heat Equation 1598.2 Thermal Resistance 1618.2.1 Example: Temperature Profile in a Heated Wire 1628.2.2 Example: Resistor Suspended in a Bridge 1658.3 Platinum Resistors 1668.4 Flow Measurement Based on Thermal Sensors 1668.4.1 Example: Micromachined Flow Sensor 1698.5 Dynamic Thermal Equivalent Circuit 1718.6 Thermally Actuated Bimorph 1728.6.1 Example: Bimorph Actuator 1748.7 Thermocouples and Thermopiles 1758.7.1 Example: IR Detector 175Problems 1769 Fabrication 1819.1 Introduction 1819.2 Photolithography 1829.3 Patterning 1839.4 Lift-off 1849.5 Bulk Micromachining 1849.5.1 Example: Angle of Walls in Silicon (100) Etching 1859.6 Silicon Etch Stop When Using Alkaline Solutions 1869.6.1 Example: Boron drive-in at 1050◦C 1869.7 Surface Micromachining 1869.7.1 Example: Cantilever Fabrication by Surface Micromachining 1879.8 Dry Etching 1889.9 CMOS-compatible MEMS Processing 1889.9.1 Example: Bimorph Actuator Compatible with CMOS Process 1899.10 Wafer Bonding 1909.11 PolyMUMPs Foundry Process 1909.11.1 Example: PolyMUMPs Cantilever for a Fabry–Perot Pressure Sensor 191Problems 192APPENDICES 195A Chapter 1 Solutions 197B Chapter 2 Solutions 207C Chapter 3 Solutions 221D Chapter 4 Solutions 239E Chapter 5 Solutions 249F Chapter 6 Solutions 267G Chapter 7 Solutions 277H Chapter 8 Solutions 285I Chapter 9 Solutions 299References 307Index 311