To Baba, Ma, Eunju, Sageun and Tommyufdcimages.uflib.ufl.edu/UF/E0/04/33/28/00001/pal_s.pdf ·...

MODELING AND RELIABILITY OF ELECTROTHERMAL MICROMIRRORS

By

SAGNIK PAL

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

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© 2011 Sagnik Pal

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To Baba, Ma, Eunju, Sageun and Tommy

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Huikai Xie, for his support and

encouragement in pursuing topics that are of interest to me and for being accessible for

discussion at all times. This work would not have been possible without his insightful

comments. I am indebted to Dr. David Hahn, Dr. Toshikazu Nishida and Dr. Jenshan

Lin for kindly agreeing to be part of the supervisory committee.

I extend my deep gratitude to Doug Hamilton and Anh Phong Ngyuen at Lantis

Laser, Inc. for building several experimental setups including the MEMS video analysis

system, vibration table, drop-test system and vacuum chamber which have been very

useful for my research. I thank Sarah Dooley at Air Force Research Lab., Ohio for

providing thermal images of MEMS devices.

The foundation in basic sciences provided by my pre-undergraduate coaches, Mr.

Pratyush Singh and Dr. Tapan Battacharya, has stood by me till this day and enables

me to navigate interdisciplinary research areas. I thank Dr. David Hahn and Dr. Bhavani

Sankar at the University of Florida, Dr. Evgenii B. Rudnyi at the University of Freiburg

and Shane Todd at the University of California (Santa Barbara) for their insights on

thermal modeling. I am indebted to Dr. Subhash Ghatu who taught a course on failure

mechanisms and to Ying Zhou for useful discussions on device reliability. The

fabrication of in-plane actuators was done by Sean R. Samuelson. Mariugenia Salas

and Anupama Ramprasad assisted with mirror testing experiments. I am grateful to

Hongzhi Sun for useful suggestions on circuit simulation. I thank Dr. Shuguang Guo for

sharing his knowledge and expertise in optical systems. Mingliang Wang, Jiping Li,

Sean R. Samuelson, Lin Liu, Shuo Cheng, Jessica Meloy and Tiffany Reagan have

trained me on several equipments. Al Ogden, David Hays and Bill Lewis at the

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Nanoscale Research Facility at University of Florida have given valuable suggestions

on device fabrication and have provided clean-room training. Useful suggestions on

polyimide processing were provided by Vishwanath Sankar, Erin Patrick and Justin Zito.

I am indebted to Dr. Marc S. Weinberg, MIT; Todd Christenson, HT MicroAnalytical,

Inc.; and Jayanth Gobbalipur-Ranganathan, Ohio State University for useful discussions

on beam analysis. I thank Xuesong Liu for useful suggestions on MEMS layout. I am

grateful to Matt Williams for useful tips on ABAQUS and to Erin Patrick and Tai-An

Chen for their help in taking pictures of MEMS devices.

My life at graduate school has been an enriching experience, thanks to my

colleagues at the Biophotonics and Microsystems Lab.—Stephen Reid, Dr. Yiping Zhu,

Yi Lin, Anuj Virendrapal, Anirban Basu, Andrea Pais, Wenjing Liu, Wenjun Liao, Victor

Farm-Guoo Tseng, Jingjing Sun, Xiaoxing Feng, Cara Hall, Zhongyang Guo, and Dr.

Hongzhi Jia. Dr. Yingtao Ding’s kind hospitality made the trip to Transducers’11

conference in Beijing a pleasant and memorable experience.

This research was supported by grants from the National Science Foundation.

SEM images were obtained from the Major Analytical Instrumentation Center, University

of Florida. All fabrications were done at the Nanoscale Research Facility at the

University of Florida.

Last but foremost, I thank my parents Prabir and Elonee Pal, and my girlfriend

Eunju Kim, for their unconditional love and support. Throughout my growing years, my

father encouraged the maverick within me and my mother instilled the discipline and

perseverance that have enabled me as a researcher.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES .......................................................................................................... 12

LIST OF FIGURES ........................................................................................................ 13

LIST OF ABBREVIATIONS ........................................................................................... 19

COMMONLY USED SYMBOLS .................................................................................... 21

CHAPTER

1 INTRODUCTION .................................................................................................... 26

1.1 Background ....................................................................................................... 26

1.2 Micromirrors Actuated by Thermal Bimorphs .................................................... 27

1.2.1 Principle of Thermal Bimorph Actuation .................................................. 27

1.2.2 Overview of Thermal Bimorph Actuated Micromirrors ............................. 29

1.2.3 Mirror Testing .......................................................................................... 33

1.2.3.1 Static characterization .................................................................... 33

1.2.3.2 Frequency response ...................................................................... 34

1.2.3.3 Scan angle velocity for periodic actuation ...................................... 35

1.3 Research Objectives ......................................................................................... 35

1.3.1 Modeling .................................................................................................. 35

1.3.2 Novel Transducer Designs ...................................................................... 35

1.3.3 Reliability ................................................................................................. 36

1.4 Research Significance ...................................................................................... 36

1.5 Chapter Organization ........................................................................................ 38

2 REVIEW ON THERMAL MODELING ..................................................................... 39

2.1 Background ....................................................................................................... 39

2.2 Literature Review .............................................................................................. 42

2.2.1 Analytical Methods .................................................................................. 42

2.2.2 Numerical Methods.................................................................................. 42

2.2.2.1 Finite element method.................................................................... 42

2.2.2.2 Finite difference method (FDM) ..................................................... 43

2.2.2.3 Transmission line matrix (TLM) method ......................................... 44

2.2.3 Compact Thermal Models (CTMs)........................................................... 44

2.2.4 Distributed Circuit Models ........................................................................ 46

2.2.5 Numerical Model Order Reduction (MOR) ............................................... 46

2.3 Summary .......................................................................................................... 47

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3 DYNAMIC COMPACT THERMAL MODELING BY MODEL ORDER REDUCTION .......................................................................................................... 48

3.1 Background ....................................................................................................... 48

3.2 Electrothermally Actuated Micromirror .............................................................. 48

3.2.1 Device Description................................................................................... 48

3.2.2 Micromirror Modeling ............................................................................... 50

3.3 Finite Element Modeling ................................................................................... 51

3.3.1 The Heat Equation and Boundary Conditions ......................................... 51

3.3.2 Finite Element (FE) Formulation by Galerkin’s Weighted Residual Method .......................................................................................................... 51

3.3.3 FEM of the Micromirror ............................................................................ 52

3.3.4 FEM Simulation Results .......................................................................... 54

3.4 Model Order Reduction ..................................................................................... 57

3.4.1 Introduction .............................................................................................. 57

3.4.2 The Arnoldi Process for Model Order Reduction ..................................... 58

3.4.3 Results Obtained from Reduced Order Model ......................................... 60

3.5 Equivalent Circuit Model ................................................................................... 63

3.5.1 Discretization of the One-dimensional Heat Equation ............................. 63

3.5.2 Equivalent Thermal Model of Micromirror ................................................ 64

3.5.3 Results Obtained from Lumped Element Model ...................................... 66

3.6 Summary .......................................................................................................... 68

4 TRANSMISSION LINE THERMAL MODEL OF ELECTROTHERMAL MICROMIRRORS ................................................................................................... 69

4.1 Background ....................................................................................................... 69

4.2. 1D Electrothermally Actuated Micromirror ....................................................... 70

4.2.1 Device Description................................................................................... 70

4.2.2 Thermal Bimorph Actuation ..................................................................... 73

4.2.3 Electrothermal Model ............................................................................... 74

4.3 Thermal Model .................................................................................................. 75

4.3.1 Estimation of Heat Loss Coefficient ......................................................... 76

4.3.2 FE Thermal Model ................................................................................... 78

4.3.3 Effect of Process Variations on Thermal Model ....................................... 80

4.3.4 Transmission-line Model for 1D Heat Flow .............................................. 82

4.3.5 Equivalent Circuit Representation of Thermal Model .............................. 88

4.4 Electrothermal Model ........................................................................................ 91

4.4.1 Static Model ............................................................................................. 91

4.4.2 Dynamic Electrothermal Model ................................................................ 93

4.5 Static Electrothermomechanical Model ............................................................. 94

4.6 Comparison with Experimental Results ............................................................ 95

4.6.1 Static Electrothermomechanical Model ................................................... 95

4.6.2 Dynamic Thermal Model .......................................................................... 98

4.6 Summary and Discussion ................................................................................. 99

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5 TRANSMISSION-LINE THERMAL MODEL WITH DISTRIBUTED HEAT SOURCE .............................................................................................................. 102

5.1 Background ..................................................................................................... 102

5.2 Transmission-line Model for Uniformly Distributed Heat Source ..................... 102

5.2.1 Governing Equations for 1D Heat Flow ................................................. 102

5.2.2 Application of Transmission-line Model to Electrothermal Micromirrors 105

5.2.3 Simulation Results ................................................................................. 106

5.3 Distributed Temperature Dependent Resistive Heater in One-dimensional Heat Flow Region .............................................................................................. 109

5.4 Summary and Discussion ............................................................................... 111

6 MECHANICAL MODEL OF ELECTROTHERMAL MICROMIRRORS .................. 112

6.1 Background ..................................................................................................... 112

6.2 Mechanics of Bimorph Actuators .................................................................... 113

6.3 Optimization of the ISC Multimorph Actuators ................................................ 115

6.4 Mechanical Model of Micromirror .................................................................... 117

6.4.1 Newtonian Method................................................................................. 118

6.4.2 Energy Method ...................................................................................... 119

6.4.2.1 Evaluation of kinetic energy ......................................................... 120

6.4.2.2 Evaluation of potential energy ...................................................... 120

6.5 Summary ........................................................................................................ 123

7 COMPREHENSIVE ELECTROTHERMOMECHANICAL MODEL OF MICROMIRRORS ................................................................................................. 124

7.1 Background ..................................................................................................... 124

7.2 Model-based Open-loop Control ..................................................................... 124

7.2.1 Theoretical Background ......................................................................... 126

7.2.2 Linear Scanning by Open-loop Control.................................................. 128

7.2.2.1 Static characterization .................................................................. 128

7.2.2.2 Dynamic characterization ............................................................. 129

7.2.2.3 Determination of G2(s) .................................................................. 130

7.2.2.4 Fourier series expansion of desired output .................................. 132

7.2.2.5 Evaluation of voltage input ........................................................... 132

7.2.2.6 Pulse width modulation ................................................................ 133

7.2.3 Experimental Results ............................................................................. 134

7.2.3.1 Constant linear velocity scan ....................................................... 134

7.2.3.2 Constant angular velocity scan .................................................... 135

7.3 Electrothermomechanical Model Implemented in Simulink ............................. 137

7.3.1 Evaluation of Fourier Series Coefficients in MATLAB/Simulink ............. 137

7.3.2 Simulink Model ...................................................................................... 138



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8 ANALYSIS AND FABRICATION OF CURVED MULTIMORPH TRANSDUCERS THAT UNDERGO BENDING AND TWISTING ..................................................... 143

8.1 Background ..................................................................................................... 143

8.2 Curved Multimorph Analysis ........................................................................... 145

8.2.1 Deformation of Curved Beams .............................................................. 146

8.2.2 Strain Continuity between Adjacent Layers ........................................... 147

8.2.3 Force and Moment Balance .................................................................. 149

8.2.4 Curved Multimorph Deformation ............................................................ 150

8.2.5 Variation of Induced Strain along Multimorph Length ............................ 151

8.3 Results ............................................................................................................ 152

8.3.1 Analysis vs. FE Simulations .................................................................. 153


8.3.3 Large Deformation of Curved Multimorphs ............................................ 156


9 A 1MM-WIDE CIRCULAR MICROMIRROR ACTUATED BY A SEMICIRCULAR ELECTROTHERMAL MULTIMORPH ................................................................... 160

9.1 Background ..................................................................................................... 160

9.2 A 1 mm-wide Micromirror Actuated by Curved Multimorph ............................. 162

9.2.1 Device Description................................................................................. 162

9.2.2 Fabrication Process ............................................................................... 163

9.2.2.1 Material selection ......................................................................... 163

9.2.2.2 Process flow ................................................................................. 164

9.2.2.3 Thickness selection ...................................................................... 165

9.2.3 Device Characterization ........................................................................ 167

9.2.3.1 Static response ............................................................................ 167

9.2.3.2 Frequency response .................................................................... 168

9.2.4 Two Dimensional Scanning ................................................................... 170

9.3 Finite Element Model ...................................................................................... 170

9.3.1 Harmonic Analysis ................................................................................. 170

9.3.2 Estimation of Heat Loss Coefficient ....................................................... 171

9.3.3 Electrothermal Model ............................................................................. 172

9.3.4 Mechanical Model.................................................................................. 174

9.4 Comparison with Mirrors Actuated by Straight Multimorphs ........................... 175


10 ELECTROTHERMAL MICROMIRRORS ACTUATED BY CURVED MULTIMORPHS ................................................................................................... 180

10.1 Introduction ................................................................................................... 180

10.2 An Elliptical Mirror with 92 μm Minor Axis and 142 μm Major Axis ............... 180

10.2.1 Device Description ............................................................................... 181

10.2.2 Device Characterization ...................................................................... 181

10.2.2.1 Static characterization ................................................................ 181

10.2.2.2 Frequency response .................................................................. 182

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10.2.3 Finite Element Model ........................................................................... 184

10.2.3.1 Harmonic analysis ...................................................................... 184

10.2.3.2 Estimation of heat loss coefficient .............................................. 184

10.2.3.3 Electrothermal model ................................................................. 184

10.2.3.4 Mechanical model ...................................................................... 186

10.3 An Elliptical Mirror with 92 μm Minor Axis and 192 μm Major Axis ............... 186

10.4 A 400 μm-wide Circular Mirror Actuated by a Semicircular Multimorph ........ 188

10.5 Summary and Future Work ........................................................................... 190

11 BURN-IN, REPEATABILITY AND RELIABILITY OF ELECTROTHERMAL MICROMIRRORS ................................................................................................. 191

11.1 Background ................................................................................................... 191

11.2 Burn-in and Repeatability .............................................................................. 192

11.2.1 Embedded Heater Burn-in ................................................................... 192

11.2.2 Scan Angle Repeatability .................................................................... 195

11.2.3 Initial Tilt of Mirror-plate ....................................................................... 196

11.3 Device Failure ............................................................................................... 199

11.3.1 Failure Due to Overvoltage .................................................................. 199

11.3.2 Impact Failure ...................................................................................... 203

11.3.3 Other Reliability Issues ........................................................................ 204

11.3.3.1 Creep ......................................................................................... 204

11.3.3.2 Fatigue ....................................................................................... 204

11.3.3.3 Environmental factors ................................................................ 204

11.4 Summary and Future Work ........................................................................... 204

12 A PROCESS FOR FABRICATING ROBUST ELECTROTHERMAL MEMS WITH CUSTOMIZABLE THERMAL RESPONSE TIME AND POWER CONSUMPTION REQUIREMENTS ..................................................................... 206

12.1 Background ................................................................................................... 206

12.2 MEMS Materials for Thermal Multimorphs .................................................... 207

12.3 Fabrication Process ...................................................................................... 210

12.4 Device Characterization ................................................................................ 217

12.5 Device Robustness ....................................................................................... 224

12.5.1 Impact Testing with Two-Ball Setup .................................................... 224

12.5.2 Drop Tests ........................................................................................... 224

12.6 Summary and Conclusions ........................................................................... 225

13 NOVEL MULTIMORPH-BASED IN-PLANE TRANSDUCERS .............................. 226

13.1 Background ................................................................................................... 226

13.2 In-Plane Transducer Design 1 ...................................................................... 227

13.2.1 Topology of Design 1 ........................................................................... 227

13.2.2 Simulations .......................................................................................... 228

13.2.3 Analysis ............................................................................................... 228

13.2.4 Optimization ........................................................................................ 229

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13.3 In-Plane Transducer Design 2 ...................................................................... 230

13.3.1 Design Topology.................................................................................. 230

13.3.2 Device Fabrication ............................................................................... 232

13.3.3 Experimental Results ........................................................................... 233

13.4 Comparison of Designs 1 and 2 .................................................................... 235

13.5 Potential Applications .................................................................................... 235

13.6 Summary ...................................................................................................... 237

14 CONCLUSIONS AND FUTURE WORK ............................................................... 238

14.1 Summary of Work Done ................................................................................ 238

14.1.1 Device Modeling .................................................................................. 238

14.1.2 Curved Multimorph Actuators .............................................................. 240

14.1.3 Device Pre-conditioning and Repeatability .......................................... 240

14.1.4 Fabrication of Robust Micromirrors...................................................... 241

14.1.5 Novel In-plane Transducer Designs .................................................... 241

14.2 Future Work .................................................................................................. 242

LIST OF REFERENCES ............................................................................................. 243

BIOGRAPHICAL SKETCH .......................................................................................... 254

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LIST OF TABLES

Table page 3-1 Symbols used in lumped element model ............................................................ 66

4-1 Material properties for thermomechanical simulations ........................................ 73

4-2 Thermal conductivity values for simulations ....................................................... 79

4-3 Circuit model parameters for a mirror with 12 min release time .......................... 96

9-1 Simulated heat loss coefficients due to thermal diffusion through air ............... 172

11-1 Heater resistance before and after burn-in for 12 devices ................................ 193

11-2 Candidate materials for fabricating electrothermal micromirrors....................... 208

12-1 Scan angle per unit dc power input for 1D mirror designs ................................ 220

13-1 Comparison of Designs 1 and 2 ....................................................................... 235

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LIST OF FIGURES

Figure page 1-1 Schematic of thermal bimorph ............................................................................ 28

1-2 SEMs of electrothermal micromirrors ................................................................. 32

1-3 Schematic of comprehensive mirror model ........................................................ 33

1-4 Top view of scan angle measurement setup on an optical bench ...................... 34

3-1 SEM of a 1D micromirror .................................................................................... 49

3-2 Schematic of 1D micromirror .............................................................................. 50

3-3 A rectangular finite element. ............................................................................... 52

3-4 Simulated temperature distribution in a section of the device. ............................ 53

3-5 FE model of micromirror.. ................................................................................... 54

3-6 Temperature distribution in micromirror .............................................................. 55

3-7 Mirror rotation angle vs. input electrical power ................................................... 57

3-8 Transfer function of thermal model ..................................................................... 61

3-9 Experimentally obtained device response .......................................................... 61

3-10 LEM for one-dimensional heat flow. ................................................................... 64

3-11 Lumped element circuit model. ........................................................................... 65

3-12 Transfer function of thermal model ..................................................................... 67

3-13 Comparison of circuit model with experimental results ....................................... 68

4-1 SEM of electrothermal micromirror ..................................................................... 70

4-2 Schematic of electrothermal micromirror ............................................................ 70

4-3 Comparison between two bimorph designs ........................................................ 72

4-4 Simulated temperature distribution in a section of the device ............................. 76

4-5 FE model for estimating heat loss coefficient due to thermal diffusion ............... 77

4-6 FE thermal model of micromirror ........................................................................ 79

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4-7 Exaggerated schematic of bimorph substrate junction ....................................... 81

4-8 Comparison of two mirrors with different etch times ........................................... 82

4-9 Passive transmission-line model ........................................................................ 84

4-10 Equivalent circuit representation of thermal model ............................................. 89

4-11 Static electrothermal model based on transmission-line theory .......................... 91

4-12 Static electrothermal model based on lumped-element approximation .............. 93

4-13 Dynamic electrothermal model of micromirror .................................................... 95

4-14 Electrothermomechanical model of 1D mirror .................................................... 96

4-15 Comparison between model and experimental results ....................................... 97

4-16 Dependence of error in LEM results on bimorph length ..................................... 98

4-17 Comparison between transmission-line model and LEM .................................. 100

4-18 Frequency response of micromirror .................................................................. 100

5-1 Transmission-line model for a geometry with uniformly distributed heat source ............................................................................................................... 103

5-2 Thermal impedances at either ends of the transmission-line model of a thermal bimorph ................................................................................................ 106

5-3 Average bimorph temperature for mirror placed in vacuum .............................. 107

5-4 Average bimorph temperature for mirror placed in air ...................................... 108

5-5 Equivalent circuit of an element of length Δx of a one-dimensional heat flow region ............................................................................................................... 110

6-1 Schematic showing the three degrees of freedom of a 3D micromirror ............ 113

6-2 Schematic of a multimorph ............................................................................... 114

6-3 Thin film structure of non-inverted and inverted multimorphs ........................... 115

6-4 Series connection of a non-inverted and an inverted actuator .......................... 117

6-5 Schematic of mechanical model for 1D mirror .................................................. 118

6-6 Free body diagram of actuator and mirror-plate ............................................... 119

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6-7 Stress distribution in a bimorph ........................................................................ 121

7-1 Schematic of a complete model of an electrothermally-actuated micromirror .. 127

7-2 Static characteristic .......................................................................................... 129

7-3 Micromirror frequency response and fitted model ............................................. 130

7-4 Temperature of embedded heater .................................................................... 131

7-5 One period of optical scan angle vs. time ......................................................... 132

7-6 One period of the evaluated input waveform .................................................... 133

7-7 PWM representation of continuous waveform .................................................. 133

7-8 Constant linear velocity scan ............................................................................ 135

7-9 Constant angular velocity scan ......................................................................... 136

7-10 Constant angular velocity scan by PWM actuation ........................................... 137

7-11 Dynamic ETM mirror model implemented in Simulink ...................................... 139

7-12 Verification of Simulink mirror model ................................................................ 141

8-1 Schematics of straight and curved multimorphs ............................................... 144

8-2 Curved beam .................................................................................................... 146

8-3 Force and moment distribution on a cross-section of a multimorph .................. 147

8-4 FE simulation results for curved thermal multimorphs ...................................... 153

8-5 Deformation of curved multimorph .................................................................... 154

8-6 Curved multimorph test structure ...................................................................... 155

8-7 Mirror-plate tilt vs. chip temperature for test structure shown in Figure 8-6A .... 156

8-8 Large deformation of a curved multimorph ....................................................... 157

9-1 Straight multimorph based 1D micromirror design at two different positions during a scan cycle ........................................................................................... 160

9-2 Semicircular actuator based mirror design ....................................................... 161

9-3 A 1 mm-wide circular mirror .............................................................................. 163

9-4 Fabrication process flow on an SOI wafer ........................................................ 165

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9-5 Optimal thicknesses of multimorph layers ........................................................ 167

9-6 Experimentally obtained static characteristic of micromirror along with FE simulation data ................................................................................................. 168

9-7 Frequency response of the 1mm-wide micromirror .......................................... 169

9-8 Simulated resonant modes of the 1mm-wide micromirror ................................ 169

9-9 Two dimensional scan patterns ........................................................................ 171

9-10 Temperature distribution for an applied voltage of 0.7 V .................................. 173

9-11 Simulated temperature distribution along the length of the semicircular actuator ............................................................................................................ 173

9-12 Comparison of experimentally measured current with simulated data.............. 174

9-13 Simulated center-shift of the micromirror depicted in Figure 9-3 ...................... 177

10-1 An elliptical mirror with 92 μm major axis and 142 μm minor axis .................... 181

10-2 Static characteristic of device shown in Figure 10-1 ......................................... 182

10-3 Mirror center-shift obtained by observing the device shown in Figure 10-1 under a microscope .......................................................................................... 182

10-4 Frequency response of device shown in Figure 10-1 ....................................... 183

10-5 Simulated resonant modes of device depicted in Figure 10-1 .......................... 183

10-6 Two-dimensional scan pattern .......................................................................... 184

10-7 Simulated temperature distribution for an applied voltage of 400 mV ............... 185

10-8 Temperature distribution along actuator length ................................................ 185

10-9 Maximum actuator temperature ........................................................................ 186

10-10 SEM of elliptical micromirror ............................................................................. 187

10-11 Static characteristic of device shown in Figure 10-10 ....................................... 187

10-12 Frequency response of mirror shown in Figure 10-10 ...................................... 188

10-13 A circular micromirror actuated by a semicircular electrothermal multimorph ... 188

10-14 Static characteristic of device shown in Figure 10-13 ....................................... 189

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10-15 Frequency response of mirror shown in Figure 10-13 ...................................... 189

11-1 Embedded heater characteristic ....................................................................... 193

11-2 Burn-in characteristic of an unreleased device ................................................. 194

11-3 Scan angle vs. voltage for a released micromirror ............................................ 195

11-4 Mirror scan angle .............................................................................................. 197

11-5 Optical angle of a newly released micromirror placed on a hot-plate ............... 199

11-6 Deteriorated embedded heater characteristic at high voltage .......................... 200

11-7 Failure at high voltage ...................................................................................... 200

11-8 Damaged end of embedded heater .................................................................. 201

11-9 Current density obtained from a finite element model of the heater ................. 202

11-10 Current density distribution after inner Pt segment fails ................................... 202

11-11 SEM images of failed mirror ............................................................................. 203

12-1 Fabrication process for robust mirrors .............................................................. 210

12-2 Modified fabrication process for robust mirrors with trench isolation ................ 211

12-3 SEM of 1D mirror with no thermal isolation ...................................................... 212

12-4 SEM of 1D mirror with beam-type thermal isolation at both ends of the actuators ........................................................................................................... 213

12-5 SEM of mirror with polyimide-beam isolation at both ends of the actuators ..... 214

12-6 SEM of robust 1D mirror with beam-type thermal isolation between actuators and mirror-plate ................................................................................................ 215

12-7 SEM of robust 1D mirror with trench-type thermal isolation between actuators and mirror-plate only ......................................................................................... 215

12-8 SEM of robust 3D mirror with no thermal isolation ............................................ 216

12-9 SEM of robust 3D mirror with thermal isolation beams between the actuators and the mirror-plate .......................................................................................... 216

12-10 SEM of robust 3D mirror with trench-filled thermal isolation between the actuators and the mirror-plate........................................................................... 217

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12-11 Static characteristic of 1D mirror with no thermal isolation ............................... 218

12-12 Static characteristic of 1D mirror with beam-type thermal isolation at both ends of the actuators ........................................................................................ 218

12-13 Static characteristic of 1D mirror depicted in Figure 12-5 ................................. 219

12-14 Static characteristic of 1D mirror with beam-type thermal isolation between actuators and mirror-plate ................................................................................ 219

12-15 Static characteristic of 1D mirror with trench-type thermal isolation between actuators and mirror-plate ................................................................................ 220

12-16 Frequency response of the mirror shown in Figure 12-3 .................................. 221





12-21 The impact test setup consists of two steel balls .............................................. 224

13-1 Top view of proposed Design 1 for achieving large in-plane displacement. ..... 227

13-2 Deformed shape of Design 1 for a uniform temperature change of 400 K ........ 228

13-3 Optimized Design 1 for ltotal 2Rm ................................................................... 230

13-4 Optimized Design 1 for total transducer length, ltotal 2Rm .............................. 230

13-5 Top view of proposed transducer for achieving large in-plane displacement ... 231

13-6 SEM of fabricated in-plane transducer Design 2 .............................................. 233

13-7 Optical microscope image of fabricated in-plane transducer Design 2 ............. 233

13-8 In-plane displacement produced by Design 2 ................................................... 234

13-9 Out-of-plane displacement produced by Design 2 ............................................ 234

13-10 Schematic of a Michelson interferometer ......................................................... 236

13-11 Two opposing transducers can be used to form a MEMS tweezer or micro gripper .............................................................................................................. 237

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LIST OF ABBREVIATIONS

BCI Boundary Condition Independent

CMOS Complementary Metal-Oxide-Semiconductor

CTE Coefficient of Thermal Expansion

CTM Compact Thermal Model

DLC Diamond-like Carbon

DMD Digital Micromirror Device

DOF Degree of Freedom

ETM Electrothermomechanical

FDM Finite Difference Method

FE Finite Element

FEM Finite Element Method

IC Integrated Circuit

ISC Inverted Series Connected

IR Infrared

JEDEC Joint Electron Device Engineering Council

LEM Lumped Element Model

LSF Lateral Shift Free

LVD Large Vertical Displacement

MEMS Microelectromechanical Systems

MOR Model Order Reduction

MUMPs Multi-User MEMS Processes

PECVD Plasma-enhanced Chemical Vapor Deposition

PSGA Polymer Stud Grid Array

PSM Position Sensitive Module

20 20

RC Resistor-Capacitor

SEM Scanning Electron Microscope/Micrograph

TCR Temperature Coefficient of Resistance

TLM Transmission Line Matrix

21 21

COMMONLY USED SYMBOLS

α temperature coefficient of resistance

ε strain

γ propagation constant for thermal transmission-line

θ deflection produces by bimorph/multimorph

λi CTE of i th layer of multimorph

Λ mirror rotation angle per unit bimorph temperature rise

ν Poisson’s ratio

ρ density

ρ0 resistivity of embedded heater

η time

ω frequency in radians per second

Ai cross-sectional area of i th layer of multimorph

c thermal capacitance per unit length

cp heat capacity per unit mass

Cbimorph, Cmirror thermal capacitance of bimorphs, mirror-plate, respectively

Cm curvature of straight multimorph upon deformation

din-plane in-plane displacement

ddesign1, ddesign2 in-plane displacements produced by designs 1 and 2, respectively

Ei Young’s modulus of i th layer of multimorph

EKE kinetic energy

EPE potential energy

f frequency

g thermal conductance per unit length

h, hb, hm heat loss coefficent; heat loss coefficient on bimorph and mirror-

plate, respectively

i current, represents heat flow in thermal circuit model

I identity matrix

I amplitude of phasor representing i(x,η)

Icm moment of inertia of mirror-plate about center of mass

IE current flowing through embedded resistor RE

22 22

Ii area moment of inertia of i th layer of multimorph

j imaginary unit

J current density

k thermal conductivity

l length

L Lagrangian

m mass of mirror-plate

p, p power; power per unit length

q power per unit volume

r thermal resistance per unit length

rM order of reduced model

RA, RB etc. resistances in thermal circuit model

Rc radius of curvature of undeformed curved multimorph

ER , 0ER heater resistance at temperatures Th and T0 , respectively

Rm radius of curvature of straight multimorph upon deformation

R0 dc value of characteristic impedance of thermal transmission-line

s complex frequency

s distance along curved multimorph

S surface area

t, tb, ti thickness; thickness of bimorph and i th layer of multimorph,

respectively

T, 0T , Ta, Th, Tb temperature; reference temperature; temperature of ambient,

embedded heater and bimorph, respectively

U out-of-plane deflection of curved multimorph

v voltage, represents temperature rise in thermal circuit model

vcm velocity of mirror-plate center of mass

V amplitude of phasor representing v(x,η)

EV voltage applied to embedded heater

Z, ZA, ZB etc. complex impedances in thermal circuit model

Z0 characteristic impedance of thermal transmission-line

23 23

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

MODELING AND RELIABILITY OF ELECTROTHERMAL MICROMIRRORS

By

Sagnik Pal

December 2011

Chair: Huikai Xie Major: Electrical Engineering

Difference in strains in the layers of a multimorph causes it to curl, thereby leading

to transduction. Thermal, piezoelectric, shape-memory alloy and electroactive polymer

based multimorph transducers that undergo out-of-plane bending have been widely

reported. A thermal multimorph actuator consists of two or more layers with different

coefficients of thermal expansion (CTE). Micromirrors actuated by thermal multimorphs

provide large scan range at low driving voltage. Up to 600 μm out-of-plane displacement

of mirror-plate and full-circumferential scan angle have been reported in literature.

A major contribution of this thesis is the modeling of electrothermal micromirrors

for design, optimization and control. Procedure for building compact

electrothermomechanical (ETM) models is established and validated against

experiments. A key component of an ETM model is the thermal model. Thermal models

based on finite element (FE) simulations, lumped element method, model order

reduction (MOR) and transmission line theory have been developed. The mechanical

behavior of a micromirror may be modeled as a mass-spring-damper system. A

comprehensive ETM model was implemented in Simulink. Model-based open-loop

mirror control for bio-imaging systems has been demonstrated. Another contribution of

24 24

this thesis is the optimization of the inverted-series-connected (ISC) structure which

consists of a series connection of two different multimorph structures. Optimization

resulted in ten-fold increase in the scan angle of ISC actuator based micromirrors.

Most thermal multimorph-actuated MEMS devices reported in literature utilize

straight actuator beams which undergo bending deformation. On the other hand, curved

multimorph actuators that undergo combined bending and twisting have not been widely

investigated prior to this thesis. The small deformation analysis of curved multimorphs is

reported for the first time and validated against experiments. Analytical expressions

governing curved multimorphs can serve as design equations for novel thermal,

piezoelectric, shape-memory alloy, and electroactive polymer based devices. Mirrors

actuated by curved multimorphs are fabricated. The unique properties of curved

multimorphs are utilized to achieve lower power consumption, higher fill-factor, and

lower center-shift compared to previously reported designs.

The major drawbacks of thermal MEMS are high power consumption and slow

speed. Several micromirrors utilize SiO2 thin-film for thermal isolation. This makes the

devices highly susceptible to impact failure during handling and packaging. SiO2 is also

used as one of the multimorph layers in several devices. The low thermal diffusivity of

SiO2 makes the thermal response sluggish. A novel process for fabricating robust

electrothermal MEMS with customizable thermal response and power consumption is

developed. The process employs Al and W for forming the active layers of the

multimorph structure. High temperature polyimide is used for thermal isolation. The

mirrors fabricated by the proposed process have improved robustness compared to

25 25

previous designs and can withstand typical drop heights encountered in hand-held

applications.

Another contribution of this thesis is the development of two novel in-plane

transducers based on straight thermal multimorph actuators. The proposed designs can

produce 100s microns displacement along the substrate surface, which is an order of

magnitude greater than previously reported designs. Possible applications include

integrated Michelson interferometer, movable MEMS stage and movable micro needles

for biomedical applications.

26 26

CHAPTER 1 INTRODUCTION

1.1 Background

MEMS scanning micromirrors have been widely used in displays [1], optical

communications [2] and biomedical imaging [3]. Scanning may be achieved by

piezoelectric [4], electrostatic [5], electrothermal [6] or electromagnetic [7] actuation

mechanisms. Among them, multimorph-based electrothermal actuation provides the

largest scan range at low voltage and this makes them suitable for biomedical imaging

applications. For instance, Wu et al. demonstrated an electrothermal multimorph MEMS

mirror that rotated 124 at only 12.5 V [8]. The main drawbacks of electrothermal

micromirrors are high power consumption (~100 mW) and slow thermal response

(~ms) [2]. Consequently, device modeling is essential for design, optimization and

control. A major focus area of this thesis is the development of

electrothermomechanical (ETM) models of micromirrors.

Many micromirror designs utilize SiO2 thin-film thermal isolation for confining most

of the heat energy to the actuators, thereby minimizing power consumption. The brittle

nature of SiO2 makes such devices susceptible to impact failure. Consequently, such

devices cannot survive drop test from a height of a few centimeters on a vinyl floor.

However, practical applications such as hand-held endoscopes may involve frequent

drops from a height of several feet. Successful commercialization requires a thorough

investigation into reliability issues. During a project on hand-held dental imaging probes

with Lantis Laser Inc. [9], impact failure of previous generation micromirrors during

handling and packaging needed urgent attention. A major contribution of this thesis is a

27 27

novel process for fabricating robust micromirrors with customizable thermal response

speed and power consumption requirements.

Straight multimorphs that undergo out-of-plane bending have been widely reported

in literature. In this thesis, curved multimorphs that bend and twist upon deformation

have been analyzed for the first time. Novel mirror designs actuated by curved

multimorphs are found to have significantly better characteristics compared to

previously reported designs.

Another contribution of this thesis is the development of two novel in-plane

transducers that can produce displacements as high as several hundred microns. This

is an order of magnitude improvement over previously reported in-plane actuators. Such

actuators can be used in integrated Michelson interferometers and movable MEMS

stages.

The next section provides an overview of thermal bimorph actuation and

micromirror designs. Section 1.3 enumerates the key objectives of this thesis.

Section 1.4 summarizes the significance of this research. Section 1.5 details the

organization of the chapters in this dissertation.

1.2 Micromirrors Actuated by Thermal Bimorphs

1.2.1 Principle of Thermal Bimorph Actuation

Difference in strains in the layers of a multimorph causes it to curl, thereby leading

to transduction. A multimorph with two layers is a bimorph. Thermal, piezoelectric,

shape-memory alloy and electroactive polymer based multimorph transducers have

been widely reported. Timoshenko’s classic work establishes the principle of thermal

bimorphs [10]. Weinberg [11] and Devoe et al. [12] discuss the equations governing

multimorph transducers. Figure 1-1 shows a schematic of a thermal bimorph.

28 28

Figure 1-1. Schematic of thermal bimorph.

A thermal bimorph consists of two materials with different coefficients of thermal

expansion (CTE). Let the CTE of the top and bottom layers be λ1 and λ2, respectively.

Let t1 and t2 represent the thickness, and E1 and E2 represent the Young’s modulus of

the top and bottom layers, respectively. Let lb and tb represent the length and thickness

of the bimorph. When the average temperature along the length of the bimorph is T0,

the tangential angle at the end of the bimorph is θ0. When the average temperature

along the bimorph length changes to Tb, the tangential angle θ(Tb ) is given by [10],

0 2 1 0( ) ( )( )bb b

b

lθ T θ β λ λ T T

t (1-1)

where,

2

3 2

(1 )6

12(2 3 2)

ξβ

χξ ξ ξχξ

(1-2)

1

2

tξ

t (1-3)

1

2

Eχ

E (1-4)

Material 1

Material 2

θ(Tb)

din-plane

Undeformed state

Deformed bimorph

29 29

If the total bimorph thickness is fixed, maximum deflection is achieved when the

ratio of thicknesses of the two layers satisfy the condition [13],

1 2

2 1

t E

t E (1-5)

If the bimorph beam width is much greater than its thickness, it can be assumed to

be in plane strain. In this case, the Young’s modulus of the two materials must be

replaced by their biaxial modulus in Equations 1-4 and 1-5.

As shown in Figure 1-1, the bimorph tip undergoes in-plane displacement, din-plane,

along with out-of-plane displacement. In this thesis, two different actuators are proposed

that amplify din-plane to achieve 100s microns in-plane displacement. This corresponds to

a ten-fold improvement compared to previously reported in-plane displacement values.

These designs use a combination of straight bimorph beams to achieve zero out-of-

plane displacement.

The bimorph shown in Figure 1-1 has zero curvature in the undeformed state. In

this thesis, the analysis and fabrication of curved multimorphs which have a non-zero

curvature in the plane of the substrate is reported. The distinguishing feature of such

actuators is that they undergo both bending and twisting deformations. Mirror designs

actuated by curved multimorphs are also reported.

1.2.2 Overview of Thermal Bimorph Actuated Micromirrors

Reithmüller et al. suggested the thermal actuation of micromirrors in 1988 [14].

Since then, various electrothermal micromirrors actuated by straight multimorphs have

been demonstrated with different fabrication processes and different materials [15-21].

For example, Buser et al. proposed an IC-compatible fabrication process with Al-Si

bimorphs [15]. Bühler et al. [16] and Tuantranont et al. [17] report CMOS fabrication

30 30

processes for making thermal micromirrors. A MUMPs polysilicon surface

micromachining process for micromirror arrays is discussed in [18]. Lammel et al. report

microscanners based on Cr-SiO2 bimorphs [6]. The chromium thin-film acts as a

resistive heater as well as one of the layers of the bimorphs. Singh et al. report a

micromirror for bio-imaging applications based on Al-SiO2 bimorphs [22]. A digital

micromirror device from Texas Instruments may utilize an electrothermal actuator to

overcome stiction [21]. Kim et al. report a thermal micromirror actuated by two parallel

bimorphs bending in opposite directions thereby producing a twisting action [23]. The

large displacement produced by thermal multimorphs at low voltages is especially

suited for micromirrors used in biomedical imaging applications [24-26]. Other

applications include image acquisition systems [27], laser output control [28] and

microprojectors [28].

Most micromirrors can be classified as 1D [29], 2D [30] or 3D [31]. A 1D

micromirror can scan about one axis. A 2D micromirror has scan capability about two

axes. A 3D mirror can generate rotation about two axes and can also undergo out-of-

plane displacement. Angular scanning finds applications in biomedical imaging, optical

displays and bar code readers [6]. Out-of-plane motion which is also known as piston

motion is useful in interferometric systems [32].

Figures 1-2A through 1-2C show SEMs of 1D, 2D and 3D micromirrors,

respectively, previously reported by our group. A CMOS based large-vertical-

displacement (LVD) electrothermal micromirror has been reported in [32, 33]. The LVD

micromirror provides an out-of-plane displacement of 0.2 mm at 6 V and can scan 15.

Heating is achieved by an embedded polysilicon resistive heater. Significant hysteresis

31 31

is observed in the current-voltage characteristics of the polysilicon resistor. Also, the

mirror-plate center suffers significant lateral shift while executing vertical motion. The

center-shift problem is overcome by the LSF-LVD (lateral shift free-large vertical

displacement) 3D micromirror reported in [34]. Additionally, replacing the polysilicon

heater with platinum (Pt) makes the device repeatable with negligible hysteresis. The

LSF-LVD design utilizes four LSF actuators at each edge of the mirror-plate. When all

four actuators are operated in-phase, out-of-plane motion up to 600 μm may be

achieved. Two-dimensional angular scanning may be achieved by operating opposite

actuators with a phase-shift with respect to each other. Another 3D micromirror design

is based on the inverted-series-connected (ISC) bimorph actuator [35]. As shown in

Figure 1-2C, four actuators are located at the four edges of the mirror-plate [36]. The

optical scan range is 30 and the out-of-plane scan range is 480 μm.

Prior to this thesis, mirrors reported by our group utilized Al-SiO2 bimorphs with

SiO2 thin-film thermal isolation at the bimorph ends. The brittle nature of SiO2 made

these devices highly susceptible to impact failure. Repeatability and reliability studies on

micromirrors actuated by Al-SiO2 bimorphs with integrated Pt heater have been

included in this thesis [19]. A major drawback of using SiO2 as an active bimorph layer

is that the low thermal diffusivity of SiO2 makes the thermal response sluggish. A novel

process for fabricating robust mirrors with improved thermal response is a major

contribution of this research [37]. An electrothermal micromirror may be represented by

the schematic shown in Figure 1-3. An applied voltage results in Joule heating in the

embedded heater. This causes the temperature of the bimorphs to change thereby

leading to actuation.

32 32

Figure 1-2. SEMs of electrothermal micromirrors. (A) 1D mirror [3]. (B) 2D mirror [3]. (C) 3D mirror based on inverted series connected (ISC) actuator [36, 38].

(A)

(B)

(C)

33 33

The three components of the comprehensive dynamic mirror model are the

electrical, thermal and mechanical models. Mirror modeling [20, 39] and model-based

open loop control [40, 41] have been addressed in this dissertation. Mirror testing is

discussed next.

Figure 1-3. Schematic of comprehensive mirror model. η = time, VE = applied voltage,

p = power dissipated by Joule heating, Tb = bimorph actuator temperature,

Th = temperature of embedded resistive heater and RE (Th ) = resistance of

embedded heater. The degrees of freedom (DOF) of the mirror-plate are angles θx and θy

, and out-of-plane displacement z.

1.2.3 Mirror Testing

Mirror testing typically involves applying an actuation voltage to the device and

tracking the position of the mirror-plate optically.

1.2.3.1 Static characterization

Figure 1-4 shows a typical setup used for scan angle measurement. The setup is

assembled on an optical bench. A dc voltage source is used to actuate the device. The

laser beam reflected by the mirror-plate is tracked on a screen. The laser spot position

on the screen is then used to obtain the scan angle. The measured current and voltage

values are noted at each data point.

Some mirrors are designed to execute out-of-plane motion, also known as piston

motion. One such device is shown in Figure 1-2C. Out-of-plane displacement can be

Mechanical Model

Thermal Model

( )E hR T = Temperature

dependent resistance of embedded heater

θx, θy, z

2( )

( )E

E h

V ηp η

R T

Electrical Model

( )EV η( )p η ( )bT η

( )hT η

34 34

measured by using an optical microscope with low depth-of-focus. The measured

current and dc voltage values are noted at each data point along with the elevation of

the mirror-plate from the substrate.

Figure 1-4. Top view of scan angle measurement setup on an optical bench.

1.2.3.2 Frequency response

For measuring the frequency response corresponding to angular scanning, the

mirror is first biased in the linear region of the scan angle vs. voltage characteristic. An

ac voltage superimposed on the dc bias is then applied. Typically, the ac amplitude is

an order of magnitude less than the dc bias voltage. The frequency of the ac actuation

voltage is varied and the scan range is monitored on a screen using a laser beam

reflected from the mirror-plate. Alternatively, the frequency response may be obtained

by tracking the light beam reflected from the mirror-plate using a position sensitive

module (PSM) [42].

Frequency response corresponding to piston motion is typically obtained using an

interferometric system known as a laser vibrometer. Similar to angular scanning, a small

ac voltage superimposed on a dc bias is used as the actuation signal.

Screen

Light beam from laser

Light reflected by mirror-plate

Breadboard mounted on micro-positioner

Packaged micromirror

35 35

1.2.3.3 Scan angle velocity for periodic actuation

A periodic actuation waveform can be used to excite mirror scanning. Mirror-scan

velocity may be determined by using a laser-diode driven by a pulse waveform that is

synchronized with the periodic actuation signal. The light reflected by the mirror-plate is

tracked on a screen. Only those points on the screen are illuminated which correspond

to the laser-diode being turned on. By varying the phase between the actuation signal

and the laser-diode drive, the scan angles corresponding to different points on the

actuation waveform can be determined. The derivative of scan angle with respect to

time gives the scan angle velocity. Scan velocity may also be determined by tracking a

continuous wave laser beam reflected from the mirror-plate using a position sensitive

module (PSM) [42].

The next section lists the key goals of this thesis.

1.3 Research Objectives

1.3.1 Modeling

The objectives of device modeling have been enumerated below:

Developing generic procedures for building compact, parametric thermal models of electrothermal micromirrors. Such models will represent the thermal behavior in the form of an equivalent circuit with few elements.

Demonstrating comprehensive ETM model that takes actuation voltage waveform as input and predicts mirror motion. ETM model was implemented in SPICE and Simulink.

Demonstrating model-based open-loop control.

1.3.2 Novel Transducer Designs

Research on novel transducers deals with the following topics:

Small-deformation analysis of curved multimorphs that have a non-zero curvature in the plane of the substrate in the undeformed state. The analysis will be validated against experiments and FE simulations.

36 36

Qualitative study of large deformation of curved multimorphs by experiments and FE simulations.

Design and fabrication of mirrors actuated by curved multimorphs.

Design of two novel in-plane transducers that utilize straight multimorphs to achieve 100s microns displacement.

1.3.3 Reliability

The main goals of reliability study have been enumerated below:

Experiments on preconditioning and repeatability of micromirrors.

Experiments on device failure under impact.

Fabrication of robust micromirrors using a novel process that allows customization of thermal response speed and power consumption.

Comparison of robustness and performance of current generation micromirrors with previous designs.

1.4 Research Significance

This thesis addresses several important issues and proposes novel electrothermal

micromirror designs. The models developed in this thesis are compact and therefore

computationally efficient. They can be implemented as a circuit model in SPICE or as a

Simulink model. Additionally, the models are parametric, i.e., device parameters are

treated as variables. Consequently, the models can be used for design and

optimization. Another application of the ETM models is open-loop device control.

Constant angular velocity scanning for biomedical imaging applications has been

demonstrated. The optimization reported in [36] resulted in a ten-fold improvement in

mirror scan range.

A key contribution of this thesis is the improvement in mirror robustness. Unlike

previous generation devices that would fail drop tests from a few centimeters height, the

robust mirrors can withstand drop tests from a height of several feet. This improvement

37 37

addresses a major hurdle in micromirror commercialization. The robust mirrors can be

used for hand-held clinical applications. The fabricated mirrors have significantly lower

voltage requirements than previous designs, making them suitable for in vivo

applications. Additionally, the novel fabrication process allows the design engineer to

customize power consumption and speed requirements.

The curved multimorphs analyzed in this thesis bend and twist upon deformation.

The reported analysis greatly expands the design space for MEMS engineers and

paves the way for novel devices. Micromirrors actuated by curved multimorphs were

designed and fabricated. These mirrors have higher fill-factor, lower mirror-plate center-

shift and lower power consumption than previously reported designs. Additionally, they

can achieve 2D scanning by using a single signal line. The improved fill-factor and 2D

scan capability can be utilized for miniaturizing micromirror based endoscopic probes.

The in-plane actuators proposed in this thesis are capable of achieving 100s

microns displacement, which is a ten-fold improvement over previously reported design.

The novel actuators can be used for actuating mirrors that are vertical to the substrate

and execute to and fro in-plane motion. The vertical mirror can be part of an integrated

Michelson interferometer. Such interferometers can lead to overall miniaturization of

several bio-imaging systems. Another potential application area is a movable MEMS

stage. The device shown in Figure 1-2C can be used as a 3 degrees-of-freedom

movable stage. Without incurring additional fabrication steps, the proposed actuators

can add two more degrees-of-freedom along mutually perpendicular in-plane directions.

Therefore, a 5 degrees-of-freedom movable stage can be realized.

38 38

1.5 Chapter Organization

This dissertation is organized as follows. Chapter 2 provides an introduction on

thermal modeling. Chapter 3 deals with numerical model order reduction. Chapter 4

outlines a thermal modeling procedure that draws analogy between signal flow in a

passive electrical transmission line and heat flow in a source free medium. In Chapter 5,

the transmission line method is extended to include distributed, temperature-dependent

heat sources. Chapter 6 discusses mechanical modeling. Comprehensive ETM model is

described in Chapter 7. Curved multimorph analysis is presented in Chapter 8.

Micromirrors actuated by curved multimorphs are discussed in Chapters 9 and 10.

Experiments on repeatability and reliability are presented in Chapter 11. Chapter 12

details the fabrication of robust electrothermal mirrors. In-plane actuator designs are

discussed in Chapter 13. Chapter 14 summarizes the dissertation.

39 39

CHAPTER 2 REVIEW ON THERMAL MODELING

2.1 Background

As discussed in Chapter 1, one of the main goals of this thesis is to develop

compact parametric models. This dissertation will focus on the modeling of

electrothermal micromirrors. However, the approach adopted in this thesis will be

generic and applicable to a wide range of electrothermal devices. The schematic of a

micromirror model has been shown in the form of a block diagram in Figure 1-3.

This chapter focuses on thermal model which is one of the main components of

the complete ETM model. Thermal modeling is essential for predicting device response,

reducing power consumption, preventing overheating and optimizing performance. The

most general equation governing the motion of electrothermal MEMS is given by the

equation of thermoelasticity [43],

11 22 330

( )3 2p

ε ε εTρc k T q λ ψ μ T

η η (2-1)

where, ρ = density, cp = heat capacity per unit mass, k = thermal conductivity,

T = temperature, q = power input per unit volume, λ = coefficient of thermal expansion

(CTE), T0 = reference temperature, and η = time. The symbols μ and ψ are the Lamé

constants. The Lamé constants are related to the Young’s modulus, E, and the

Poisson’s ratio, ν, by [43],

(2 3 )μ μ ψE

ψ μ (2-2)

2( )

ψ

ψ μ

ν (2-3)

40 40

The strain εij is defined by [43],

1

2

jiij

j i

uuε

x x (2-4)

where, ui is the i th component of the displacement vector and the partial derivatives are

taken with respect to directions x1, x2 and x3. Newton’s second law gives [43],

2

2

ijii

j

ζuρ

η x (2-5)

where, ζij are the components of the stress tensor and i is the body force per unit

volume.

The constitutive equation for thermoelasticity is given by [43],

02 (3 2 )( )ij ij ij kk ijζ με ψδ ε δ λ ψ μ T T (2-6)

where δij is the Kronecker delta.

Nowacki provides a detailed treatment of thermoelasticity [44]. Equations 2-1

through 2-3 rigorously describe the behavior of electrothermal MEMS devices and have

been included here for completeness. For most engineering materials, the CTE is in the

range 0.36–120 μm m-1 K-1 [45] which is very small. Therefore the third term on the right

hand side of Equation 2-1 is usually neglected [43] and the temperature distribution is

obtained by solving the general heat equation [43],

p

Tρc k T q

η (2-7)

Equation 2-7 must be solved subject to boundary conditions. Let n denote the

outward unit normal vector to a certain boundary. Boundary conditions commonly

encountered in thermal problems are listed below [46]:

41 41

Constant temperature or heat sink condition

Input power p through an area Ap: This condition is mathematically expressed as,

p

T pk T k

n An (2-8)

Heat loss due to convection: Let Ta be the ambient temperature and h be the

convection coefficient. Then the convective heat loss is given by,

( )a

Tk T k h T T

nn (2-9)

At microscale, heat loss due to thermal diffusion usually dominates over heat loss due to buoyancy driven air flow. Therefore, in subsequent chapters, h will be used to represent the total heat loss coefficient due to convection and thermal diffusion.

Thermal Insulation: Substituting p = 0 in Equation 2-8 gives the thermal insulation condition.

The low CTE of solid engineering materials implies that the efficiency, i.e.,

mechanical work done per unit power input by solid state electrothermal devices, is

typically less than 1% [47]. Such low efficiency is primarily responsible for the relatively

high power consumption in electrothermal MEMS (~mW) [2] compared to electrostatic

and piezoelectric actuators.

A second challenge in electrothermal design is that the thermal response time is

typically slow (~ms) [2]. The low thermal diffusivity, dT = k / (ρcp), of engineering

materials limits the minimum attainable thermal response time [48]. Hence, careful

modeling and optimization is required to meet design goals. Understanding the thermal

behavior is critical for minimizing power consumption and achieving fast response.

In spite of the challenges involved in thermal design, the large force (~mN) and

displacement (~102 μm) [34] produced by electrothermal actuators makes it attractive

42 42

for several applications such as micromirrors for biomedical imaging [49] and

micromanipulators [50].

Owing to the integration and miniaturization of electronic circuits, thermal

management in IC chips has been an active area of research for the past three

decades. As a result, thermal modeling has received a lot of attention and several

approaches have been reported in literature [51, 52]. These studies provide a starting

point for the development of thermal models for electrothermal MEMS. The next section

will provide a synopsis of key developments in thermal modeling. Thereafter, the

various approaches investigated in this thesis will be described in detail.

2.2 Literature Review

Solution of the heat equation has been widely investigated. This section will

provide a summary of key approaches available in literature.

2.2.1 Analytical Methods

Özişik describes analytical methods for solving the heat equation [53]. Analytical

approaches based on Green’s function, Fourier series [54] and Fourier transform [55]

are usually feasible for simple geometries [56].

2.2.2 Numerical Methods

Closed-form solution of thermal problems is limited to simple geometries only [53].

For most practical problems, numerical models such as finite element (FE) models and

finite difference models are usually used [29]. This section deals with three commonly

used numerical techniques.

2.2.2.1 Finite element method

Lewis et al. provide a detailed background on thermal FE models [57]. Several

commercial FE software tools such as COMSOL [58], Ansys [59] and IntelliSuite [60]

43 43

are available. After drawing the geometry, the model is meshed. Meshing involves

dividing the model into small elements. An FE model numerically evaluates the

temperature at discrete points of the elements. These discrete points are known as

nodes. The FE method approximately represents Equation 2-7 as n ordinary differential

equations, where n is the number of nodes. Let T1(η ), T2(η ),…,Tn (η ) represent the

temperature at the n nodes. Then the n ordinary differential equations are given by,

1 1

1

( )( ) ( )

n nn n n nn

T ηC K T η F η

η (2-10)

In Equation 2-10, C, K and F(η ) are the heat capacitance matrix, thermal

conductivity matrix and forcing vector, respectively. The vector 1

( )n

T η contains the

temperature values at the n nodes [39].

Hsu et al. report equivalent electrical network representation for finite elements

that constitute the complete model [46]. The networks representing the individual

elements may be combined to form a circuit representation for the complete model. The

complete circuit model may be solved using simulators such as SPICE. Hsu et al. report

network reduction techniques for increasing the efficiency of the complete circuit

model [61].

2.2.2.2 Finite difference method (FDM)

Equation 2-7 is a differential equation governing temperature distribution. In FDM,

this differential equation is approximated using a set of difference equations. The

resulting algebraic equations may be solved numerically using MATLAB or SPICE [62]

to obtain the temperature at discrete points in the model.

44 44

2.2.2.3 Transmission line matrix (TLM) method

Heat flow in a structure is analogous to current flow in a circuit. Temperature

difference that drives the heat flow is analogous to voltage difference in an electrical

circuit. In circuit models, resistors may be used to represent the thermal resistances and

capacitors are used to account for thermal capacitances. In such models, nodal

voltages represent temperature. The TLM method draws analogy between heat flow

and the flow of electrical signal in a transmission line. The model is first meshed. The

time step is chosen such that during each step a pulse can propagate from any node to

its nearest neighboring nodes. The flow of pulses in the 3D transmission line is

simulated, and reflection and scattering of pulses at each node are accounted for in the

simulation algorithm [63]. Gui et al. and Mimouni et al. report the application of TLM for

modeling semiconductor devices [52, 64]. An analytical method based on the analogy

between transmission line and heat flow will be in introduced in Chapter 4.

In spite of their widespread application and commercial success, numerical

methods are computationally expensive. For instance, FEM for practical problems may

have as many as ~105 nodes [56] making them computationally inefficient. Therefore,

they do not provide a cost-effective solution for device design and optimization.

Moreover, numerical methods obfuscate the relationship between device response and

device parameters, making optimization difficult. To overcome this limitation several

attempts have been made to develop simple circuit models in the last three decades.

2.2.3 Compact Thermal Models (CTMs)

Christiaens et al. report a method for synthesizing a compact boundary condition-

independent (BCI) RC network for simulating the thermal behavior of a PSGA (Polymer

Stud Grid Array) package [51]. This method involves choosing a suitable RC network

45 45

based on engineering intuition. The values of the resistors and capacitors are chosen

such that the difference between circuit model output and FEM simulations are

minimized. As few network elements are used to build such models, they are known as

CTMs (compact thermal models). This method can be used to determine compact

models that can be simulated without using excessive computational resources.

Moreover, it is desired that the model demonstrates a certain degree of boundary

condition independence. A BCI (boundary condition independent) model may be used

as a component in a complete system level simulation. The major drawback of the

method described in [51] is that the extraction of the compact model requires a large

amount of time and resource-intensive FEM simulations. Moreover, there are no

rigorous guidelines for selecting the topology of the RC network model. Consequently,

the model extraction procedure cannot be automated. Also, the model is not parametric.

The method reported in [51] depends on engineering intuition and user intervention for

selecting the CTM network topology. Palacin et al. propose the selection of CTM

network topology by using genetic algorithm [65]. Hence, genetic algorithms may be

used to make the extraction of CTM automated. However, the model reported in [65] is

not parametric. Lasance et al. report the BCI CTM of an electronic package [66].

BCI CTMs have received considerable interest from the industry. For instance, the

European project DELPHI [67, 68] was a major step in getting component

manufacturers to supply validated BCI CTMs to their end users. The intent was to

enable equipment manufacturers to build system level thermal models by integrating the

individual component models. The DELPHI project was followed up by the SEED and

PROFIT projects [69]. The DELPHI, SEED and PROFIT projects culminated in the Joint

46 46

Electron Device Engineering Council (JEDEC) standard JESD15-1 on compact thermal

models [70].

2.2.4 Distributed Circuit Models

In a real thermal system, thermal resistances and capacitances are distributed. A

one dimensional heat flow path is represented as a Foster or Cauer network [51].

Lumped models reported in literature are approximations to these distributed networks.

Székely reports the analysis of a one-dimensional distributed RC network [71]. A

distributed network has infinite number of capacitors and is therefore associated with

infinite number of time constants [72]. Székely et al. define a time constant density

function [71-73]. For a 3D model, the time constant density function may be evaluated

by using solvers such as SUNRED [74]. The dominant time constants may be chosen

and used to construct a thermal lumped element model (LEM) with only a few circuit

elements [72]. The major disadvantage of this approach is that it is difficult to associate

any physical significance with the elements of the LEM.

2.2.5 Numerical Model Order Reduction (MOR)

The need for automatic extraction of compact models has resulted in keen interest

in numerical MOR algorithms [75]. Numerical MOR algorithms project a set of equations

from a higher dimensional space to a lower dimensional space without incurring

significant error [56]. For example, let us consider an electrothermal micromirror. As

discussed in Chapter 1, the heater and bimorph temperatures are of interest to the

design engineer. Equation 2-10 will provide the temperature distribution in the device.

An MOR may be used to approximate the system of n equations in Equation 2-10 by a

set of rM equations, rM << n. These rM equations may then be used to calculate the

heater and bimorph temperatures. MOR algorithms can achieve a high degree of

47 47

computational efficiency by significantly reducing the model order [76]. However,

reduced order models are not parametric and therefore it is difficult to assign any

physical significance to its elements. More details and a practical MOR model will be

discussed in Chapter 3.

2.3 Summary

Temperature distribution may be evaluated by solving the heat equation along with

boundary conditions. This chapter reviews key methods for solving the heat equation.

Analytical methods are limited to simple geometries only. Numerical methods such as

FE method, finite difference method and TLM method may be used for complex

geometries. Numerical models are computationally expensive and do not provide

explicit relationship between device parameters and response. Therefore, there has

been a growing interest in CTMs. MOR algorithms may be used to automate CTM

extraction from FE models. The next chapter will introduce a practical MOR application.

48 48

CHAPTER 3 DYNAMIC COMPACT THERMAL MODELING BY MODEL ORDER REDUCTION

3.1 Background

Several algorithms are available in literature that can extract a compact thermal

model (CTM) from the complete FE model automatically [77]. For practical problems,

FE models may have 103–105 nodes [56]. Typically, the order of the compact model

obtained by automatic model order extraction is less than 10 [56]. Such a drastic

increase in computational efficiency with minimal effect on accuracy and the prospect of

automatic compact model extraction makes model order reduction (MOR) valuable for

several applications. The mor4ansys program takes ANSYS FE models as input and

generates the corresponding reduced order model [78]. This chapter deals with the

reduced order thermal model of a 1D micromirror.

Section 3.2 describes the electrothermal micromirror. Section 3.3 presents an FE

model. Extraction of a reduced order model from the FE model is discussed in

Section 3.4. The insight provided by the reduced order model is used to construct a

lumped element dynamic thermal model in Section 3.5.

3.2 Electrothermally Actuated Micromirror

3.2.1 Device Description

Figure 3-1 shows an SEM of a 1D electrothermally actuated micromirror device.

The 1 mm1.2 mm mirror-plate is attached to the substrate by an array of 72 thermal

bimorph actuators. Figure 3-2 shows a schematic of the 1D electrothermally actuated

micromirror device. Each thermal bimorph actuator consists of a 1 μm-thick PECVD

SiO2 layer at the bottom and 1 μm-thick aluminum (Al) layer on the top. A 0.2 μm-thick

platinum (Pt) heater layer is sandwiched between the top Al and bottom SiO2 layers.

49 49

The Pt and Al layers are isolated electrically by a 500 Å thick layer of SiO2. Each

bimorph beam is 8.5 μm wide and 135 μm long. The 40 μm-thick single-crystal-silicon

(SCS) layer below the mirror-plate serves to improve the flatness of the mirror surface.

When a current is passed through the embedded Pt heater, it results in Joule heating.

The difference in the thermal expansion coefficients of the SiO2 and Al layers causes

the bimorphs to bend, thereby causing the mirror to tilt. When no actuation is applied,

the bimorphs are curled up due to internal stresses in the SiO2 and Al layers. The mirror

is fabricated by a process similar to that described by Wu et al. [8]. The SiO2

connections at either end of the bimorphs provide thermal isolation, thereby reducing

power consumption (Figure 3-2).

The resonance frequency of the mirror is 328 Hz. The mirror scans as much as

32 at an applied dc voltage of 8 V. For scanning applications, the actuation voltage

consists of an ac voltage superimposed on a dc bias. The dc voltage is used to bias the

device at a certain angle. The ac voltage excites the mirror to scan about the bias angle.

Figure 3-1. SEM of a 1D micromirror [29].

1 mm

Bimorphs

Mirror-plate

50 50

Figure 3-2. Schematic of 1D micromirror (not to scale).

3.2.2 Micromirror Modeling

Figure 1-3 shows a schematic of the complete model of an electrothermally

actuated micromirror device. An applied voltage, VE, across the embedded Pt heater

causes Joule heating, thereby producing a power, p. This raises the temperature, Tb , of

the bimorph actuators and the mirror-plate rotates by an angle, θ. The change in

temperature also determines the resistance, RE(Th ), of the Pt heater. A complete model

of the form shown in Figure 1-3 can be used for simulating the mirror rotation for an

arbitrary voltage input. In this chapter, thermal modeling of the micromirror device is

discussed. In the following sections, an FEM thermal model of the micromirror is

developed. The FEM model is reduced to a model of order 2 by using an order-

Substrate Thermal isolation

Bimorph

Thermal isolation

Mirror- plate

TOP VIEW

Pt

SiO2

Si

Al

SIDE VIEW

51 51

reduction method. Finally, a simple lumped-element circuit is developed based on the

reduced-order model.

3.3 Finite Element Modeling

A 3D FEM simulation of the device shown in Figure 3-1 is very time-consuming

due to the large number of bimorph beams. Therefore, any variation in temperature

across the bimorph array is neglected and a 2D finite element model for the micromirror

is developed. This assumption will be verified in Section 3.3.3. This simplification leads

to a significant saving in computational resources.

3.3.1 The Heat Equation and Boundary Conditions

The temperature distribution is described by Equation 2-7. The micromirror shown

in Figure 3-1 is attached to a metallic package which can be approximated by a perfect

heat sink. Therefore, constant temperature boundary condition can be applied to the

surface of the substrate that is in contact with the package. All other surfaces of the

device are in contact with the surrounding air. Consequently, heat loss due to

convection and diffusion may be applied as boundary condition at these surfaces.

3.3.2 Finite Element (FE) Formulation by Galerkin’s Weighted Residual Method

The Galerkin’s method [57] may be used to approximate Equation 2-7 with a set of

equations which can be solved numerically. The first step is to divide the geometry of

interest into a number of small elements. Rectangular finite elements have been used

for the present work. Figure 3-3 shows a schematic of a rectangular finite element with

length 2b1 and width 2b2. Nodes 1 through 4 are numbered in a counterclockwise

fashion and Гe represents the boundary of the element.

Finite element formulation for the heat equation based on the Galerkin’s method

gives [57],

52 52

1 1

1

( )( ) ( )

n nn n n nn

T ηC K T η F u η

η (3-1a)

Τ

1 1{ ( )} [ ] { ( )}m m n nY η X T η (3-1b)

where, n and η denote the number of nodes in the FE model and time, respectively. C

and K are the heat capacitance matrix and thermal conductivity matrix, respectively.

The term 1{ } ( )nF u η accounts for heat source and boundary conditions. T is a vector

containing the temperature at all n nodes. Typically, in thermal problems, the

temperatures at m nodes are of interest to the engineer, where m << n. For instance,

the rotation angle of the micromirror device described in Section 3.2 depends on the

average temperature along the length of the thermal bimorph actuators [79]. Let ni, i = 1

to k, be k nodes equispaced along the bimorph length. Let us define X such that the nith

(i = 1 to k) elements of X are 1 / k and all other elements are zero. Then, Τ( ) ( )Y η X T η

is the average temperature along the length of the bimorph actuators.

Figure 3-3. A rectangular finite element.

3.3.3 FEM of the Micromirror

The electrothermal mirror shown in Figure 3-1 is actuated by an array of 72

bimorphs. From the device topology, it appears that if end effects at the two extremes of

the array are neglected, the individual bimorphs have a nearly identical temperature

distribution. In order to verify this claim, a 3D thermal model of a section of the device

was built in IntelliSuite 8.2 MEMS simulation package [60]. The temperature distribution

node1 node2

node3 node4

Гe

2b1

2b2

53 53

obtained from the model has been shown in Figure 3-4. It was found that the bimorph

beams indeed have a nearly identical temperature distribution. Therefore, in order to

conserve computational resources, any temperature gradient across the array is

ignored and a 2D model is built.

Figure 3-4. Simulated temperature distribution in a section of the device. The section

has eight bimorphs. The input power is 22.5mW and substrate temperature is 300K.

The 2D finite element thermal model is shown in Figure 3-5A. An enlarged view of

the bimorph region is shown in Figure 3-5B. The dimensional parameters are the same

as those given in Section 3.2.1. The boundary condition at the bottom and left side of

the substrate is assumed to be a fixed temperature. Heat loss by convection and

diffusion is assumed at all other boundaries and Equation 2-9 is applied. The boundary

condition expressed by Equation 2-8 is applied to the heater region. During device

operation, the temperature is well below the melting point of the bimorph materials.

Mirror-plate

Bimorphs

347.0

342.7

338.5

334.2

329.9

325.6

321.4

317.1

312.8

308.5

304.3

300.0

Temperature (K)

54 54

Hence, heat loss due to radiation may be neglected [32]. The finite element formulation

has been implemented in MATLAB. All the elements in the FEM model are rectangles

as shown in Figure 3-3. The variations in the thermal conductivity of the bimorph

materials with temperature change are neglected. This assumption has been verified

with simulations in Section 3.3.4. The effect of temperature dependence of heat

capacitance values has been addressed in Section 3.4.3.

Figure 3-5. FE model of micromirror. (A) Schematic of the 2D finite element model

(length scales in x and y direction are different). (B) Enlarged view of the bimorph region.

3.3.4 FEM Simulation Results

The simulated temperature distribution along the bimorph is shown in Figure 3-6A.

The thermal image of an actual device, acquired using a high-resolution IR (infrared)

camera, is shown in Figure 3-6B. The simulated and experimental data are also plotted

in Figure 3-6C. Both the simulation and experiment were done for 22.55 mW power

input and 323.15 K substrate temperature. The elevated substrate temperature is

required by the IR imager to reduce environmental noises during measurement. As

shown in Figure 3-6C, the FEM simulation matches the experiment well. It also can be

seen that the simulated temperature distribution along the bimorph has a maxima at

(A)

(B) x

y

Mirror Bimorph

Substrate

Pt Al

Si SiO2

55 55

about 80 µm, but no maxima is observed in the thermal imaging. This discrepancy

arises because the actual device has an array of bimorphs and the embedded Pt heater

has right-angled corners at the ends of the bimorphs. The presence of right-angled

corners causes current crowding [80] and creates local hot spots at the bimorph ends.

The presence of these local hot spots has not been accounted for in the FE model.

Since the temperature variation along the bimorph length is less than 5%, neglecting the

local hot spots in the FE model does not cause significant error.

Figure 3-6. Temperature distribution in micromirror. (A) Simulated temperature

distribution for power input = 22.55 mW and substrate temperature = 323 K (length scales used in x and y directions are different). (B) Thermal imaging data. (C) Comparison of simulation results with thermal imaging data.

0 40 80 120

361.4

360.6

360.2 Min:

323.1K Max: 361.4K (A)

Min: 322.08K

Max: 372.93K

A

B

363.15

353.15

343.15

333.15

0 20 40 60 80 100

Pixel number along AB

Bimorph

Length along bimorph (μm)

Sim

ula

ted

tem

pera

ture

(K

) Im

ag

ing d

ata

(K

)

(B)

(C)

56 56

Figure 3-7 shows the simulated and experimental data of rotation angle versus

power. The thermomechanical simulation was done using IntelliSuite and is in close

agreement with the experimental data. The slight discrepancy is possibly due to process

variations. Also, the rotation angle is linearly dependent on the electrical power input.

Since the rotation angle is directly proportional to the temperature change of the

bimorphs [10], it may be inferred that the steady state temperature distribution is linearly

proportional to the input power.

In order to study the effect of temperature dependence of thermal conductivity, a

non-linear FEM model with temperature dependent material properties was built using

COMSOL. The temperature-dependent thermal conductivity for Si and SiO2 were

obtained from literature [81, 82]. The Wiedemann-Franz law was used to estimate the

thermal conductivity of aluminum and platinum [83]. The average bimorph temperature

obtained from the non-linear model was compared with that obtained from the model

with constant material properties. It was found that the difference in the results is less

that 2% for input powers up to 100 mW. Consequently, a model with constant thermal

conductivity values can be used for the entire range of device operation depicted in

Figure 3-7.

A complete finite element model can accurately represent device behavior.

However, the number of nodes for a practical problem can easily exceed 105. Repetitive

dynamic simulations for arbitrary inputs can be time and resource consuming.

Therefore, in the next section a compact thermal model of the micromirror device is

developed.

57 57

3.4 Model Order Reduction

3.4.1 Introduction

Model order reduction algorithms provide a formal and rigorous way of obtaining a

reduced order model. Reduced order models consume minimal computational

resources and do not affect the accuracy of the final output significantly. Equation 3-1

may be rewritten as [56],

1 1

1

( )( ) ( )

n nn nn

T ηW T η b u η

η (3-2a)

Τ

1 1( ) [ ] { ( )}b n nT η M T η (3-2b)

where, W = -K -1C, b = -K -1F and Tb is the average temperature along the bimorph

length. The matrix M is defined such that it extracts useful information from the

temperature vector, 1{ ( )}nT η . In this case, M is used to obtain the average of

temperature values corresponding to equispaced nodes along the bimorph length.

Figure 3-7. Mirror rotation angle vs. input electrical power.

The mirror rotation angle is directly proportional to the average bimorph

temperature [10, 79] and hence Tb is of interest to the device engineer. Model order

reduction seeks to find a system [56],

25

20

15 10 5

0

Input electrical power (mW)

0 20 40 60 80 100

Mirro

r ro

tatio

n (

de

gre

es)

Simulation result

Experimental data

2.7°

58 58

1 11

( )( ) ( )

M MMM M M

M

r rrr r rr

T ηW T η b u η

η (3-3a)

Τ

1 1( ) [ ] { ( )}M M Mb r r rT η M T η (3-3b)

such that, rM << n and the error ( ) ( )b bT η T η is less than an upper bound.

Several algorithms for order reduction such as balanced truncation approximation,

singular perturbation approximation and Hankel norm approximation are based on

rigorous control theoretic approach. These methods preserve the stability and passivity

of the original system. Furthermore, they provide a rigorous upper bound for the error in

the reduced model output [84]. However, computational complexity of these methods is

O 3( )n , which is prohibitively large for most practical problems. The Arnoldi process,

which has been described below, has been reported to be suitable for practical model

order reduction problems [56].

3.4.2 The Arnoldi Process for Model Order Reduction

For the Arnoldi process, the computational complexity is O 2( 2 ( ))M M zr n r N M ,

where Nz (M ) is the number of non-zero elements in the matrix M and rM is the order of

the reduced model [56]. Hence, it requires lesser computational resources as compared

to control theory based methods. Additionally, it preserves the stability and passivity of

the original system. The biggest disadvantage of the Arnoldi process is that it does not

provide a rigorous upper bound for the error in the reduced model output. However,

several heuristic methods can be used for estimating the error [56]. Hence, the Arnoldi

process has been used for obtaining a reduced order model from Equation 3-2. The

transfer function of the original discretized Equation 3-2 is given by [77],

59 59

Τ( ) (Ι )G s M sW b (3-4)

where, Ι is an identity matrix. W, b and M have been previously introduced in

Equation 3-2. The Taylor series expansion of G(s ) about the point s = 0 is given by,

2 2

0

( ) (Ι )T i

i

i

G s M sW s W b m s (3-5)

where, T i

im M W b for i = 0,1,2,…, are called the moments of G(s ) about s = 0 [77].

The transfer function of the reduced order system described by Equation 3-3 is

given by,

( ) (Ι )M M M M

T

r r r rG s M sW b (3-6)

The Arnoldi process computes a reduced order model, i.e., Equation 3-3, such that

the first rM moments of the transfer function of the reduced system and the first rM

moments of the original system, i.e., Equation 3-2, are equal. Order reduction is

achieved by projecting the original discretized equation to a Krylov-subspace. The

output of the algorithm is a matrix Q such that [77],

Τ

MrW Q WQ (3-7a)

Τ

Mrb Q b (3-7b)

Τ

M Mr rM Q M (3-7c)

The algorithm for the Arnoldi process was implemented in MATLAB and the

discretized heat equation for the electrothermally actuated micromirror device was

reduced to a lower order system.

60 60

3.4.3 Results Obtained from Reduced Order Model

Figure 3-8A and Figure 3-8B show the magnitude and phase plot, respectively, of

the transfer function obtained from a reduced order model of order rM = 2. It was found

that increasing rM (not shown in Figure 3-8) does not change the model output

significantly up to a frequency of 104 Hz. This implies that a second order transfer

function can be used to approximate the relationship between input power and average

bimorph temperature up to a frequency of 104 Hz. Since 104 Hz is more than one order

of magnitude greater than the mechanical resonant frequency of the device, the mirror

scan angle is negligible beyond this frequency. Hence, a second order model was

chosen. The plot shows the change in the average bimorph temperature from the

ambient temperature, i.e., 300 K. Hence, for a dc power input of 22.55 mW, the average

temperature along the bimorph length is 300 K+51 K = 351 K, as shown in Figure 3-8A.

Figure 3-8B compares the phase response obtained from reduced order model and the

FEM. Some higher order effects are observed in the FEM output above 1 kHz. These

effects may be neglected as the device is operated below 1 kHz only. Consequently, it

is sufficient to use a reduced model of order 2.

In order to validate the model results, the frequency response of the mirror was

obtained by applying a 0.54 V peak-to-peak sine wave at a dc offset of 6.01 V. A laser

beam reflected from the mirror was tracked by a position sensitive module (PSM).

Figure 3-9 shows the frequency response of the mirror. The peak at 328 Hz

corresponds to the mechanical resonant frequency of the mirror (Figure 3-9A). The low

frequency response in Figure 3-9B shows two distinct cutoff frequencies in the mirror

response. Since the first mechanical resonance is observed at 328 Hz, the low

61 61

frequency drops may be attributed to the thermal response of the device. This is in

agreement with the reduced order model output. Moreover, a comparison of Figure 3-

8A and Figure 3-9B shows that the reduced order model accurately predicts the range

of frequencies over which the mirror response decays.

Figure 3-8. Transfer function of thermal model. (A) Transfer function magnitude plot from finite element model and reduced order models (rM = 2). The two plots

practically overlap. The plot shows the change in temperature at the center of the bimorph from the ambient temperature, i.e., 300 K. (B) Transfer function phase plot.

Figure 3-9. Experimentally obtained device response. (A) Mirror frequency response. (B) Response from dc to 200Hz.

0

-0.4

-0.8

-1.2

-1.6

Frequency (Hz)

60

50

40

30

20

10

0

Incre

ase in a

vera

ge

bim

orp

h t

em

pera

ture

(K

)

FEM

Reduced model of order 2

Phase (

radia

ns)

Reduced model of order 2

FEM

Frequency (Hz) 10-4 10-2 100 102 104 106

10-4 10-2 100 102 104 106

(A) (B)

1.4

1.2

1

0.8

0.6

PS

M O

utp

ut p

eak-p

eak (

V)

10-2 100 102 104

101

100

10-1

Frequency (Hz) Frequency (Hz)

10-2 100 102 104

PS

M O

utp

ut p

eak-p

eak (

V)

(A) (B)

62 62

In general, finite element models for many practical problems have more than105

nodes, so finite element simulations may take up too much time and resources.

Consequently, it may not always be feasible to verify the reduced order model against

finite element simulations. In such cases, several heuristic methods may be used to

validate the reduced order model. In general, if increasing the order of the reduced

model does not change the simulation results significantly, the model may be assumed

to be accurate [56]. Hence, reduced order modeling is essential for finding a lower order

approximation for many practical problems.

Constant thermal conductivity and heat capacitance values were used for

obtaining the simulation results in Figure 3-8. However, in general thermal conductivity

and heat capacitance values are temperature dependent. The effect of temperature

dependence of thermal conductivity on the static response of the mirror has been

discussed in Section 3.4. The dynamic response of the mirror depends on both thermal

conductivity and heat capacitance values. The temperature dependent thermal

conductivity was obtained from literature [81-83]. The temperature dependence of heat

capacitance values was estimated using the data available for bulk materials [85]. The

temperature range corresponding to the 0-100 mW input power range depicted in

Figure 3-7 was established and several values of thermal conductivity and heat

capacitance were chosen in this range. It was found that the first and second cutoff

frequencies of the transfer function can vary by ±0.01 Hz and ±12 Hz, respectively, due

to temperature dependent material properties. For most applications, a dc voltage is

used to bias the mirror at a particular tilt angle and an ac voltage superimposed on the

dc bias is used to produce scanning motion. The temperature distribution corresponding

63 63

to the dc bias can be used to estimate the thermal conductivity and heat capacitance

values for building the device model.

From Figure 3-8, it is evident that a reduced order model of order 2 closely

approximates the magnitude and phase response of the micromirror over a frequency

range that is more than 100 times greater than the micromirror resonant frequency.

Being a purely numerical model, it does not provide an explicit relationship between the

device dimensions and the mirror response. Hence, in the next section an equivalent

circuit model that treats the device parameters as variables is described. The second

order thermal response will be explained by incorporating two capacitors into the

equivalent circuit.

3.5 Equivalent Circuit Model

3.5.1 Discretization of the One-dimensional Heat Equation

The one-dimensional heat equation is given by,

2

2 p

T Tk q ρc

x η (3-8)

where x is the direction along which the temperature varies. Let us consider the

rectangular section shown in Figure 3-10A and assume that the temperature varies only

along the length of this section. Let T1, T2, and T3 denote the temperature at cross-

sections 1, 2 and 3, respectively. Replacing the second order derivative in Equation 3-8

by its central difference approximation,

1 3 2 2

2

22 2

(Δ ) / (2 )p

T T T dTAq Aρc

x kA dη (3-9)

Treating temperature as an across variable and heat flow as through variable,

Equation 3-9 may be equivalently represented by the circuit shown in Figure 3-10B, in

64 64

which heat loss to the atmosphere is ignored. Let h be the heat loss coefficient, Ta be

the ambient temperature, and S be the area from which heat loss to the atmosphere is

taking place. A lumped resistor (1 / (hS)) can be used to account for heat loss to the

atmosphere. Figure 3-10C shows the equivalent thermal circuit that takes heat loss to

the surrounding air into account.

Figure 3-10. LEM for one-dimensional heat flow. (A) A rectangular section with

temperature variation along the x-direction only. (B) Equivalent conductive thermal network (T2 satisfies Equation 3-9). (C) Equivalent thermal network

taking heat loss to atmosphere into account.

3.5.2 Equivalent Thermal Model of Micromirror

From Figure 3-4B, the temperature variation along the length of the bimorph is

less than 5%. Hence, the array of bimorphs may be represented by the equivalent

circuit shown in Figure 3-10C, with the impedances scaled by the number of bimorphs,

nb. Since the isolation region length (12 µm) is less that 10% of the length of the

bimorphs, the thermal capacitance of the isolation region may be neglected.

Furthermore, the mirror-plate is more than 40 µm thick. Hence, the conductive thermal

Δx/kA Δx/kA

q(2AΔx) ρcp(2AΔx)

T1

T2

T3

Δx

Δx

1 2 3

(A)

(B) (C)

T1 T2 T3

Δx/kA

Δx/kA

ρcp(2AΔx)

q(2AΔx)

1/hS

Ta

65 65

impedance of the mirror-plate may be neglected. The complete lumped element circuit

model has been shown in Figure 3-11.

Figure 3-11. Lumped element circuit model.

The symbols used in Figure 3-11 have been described in Table 3-1. The 1 / hbSb

resistance in series with a dc voltage source accounts for heat loss from the bimorphs

to the surrounding. In general, the heat loss coefficient is different on different surfaces

of the bimorphs. Since, the width and thickness of the actuators is at least an order of

magnitude less than the length, temperature variation perpendicular to the length is

neglected. Accordingly, an average heat loss coefficient is used to account for the

combined heat loss through all bimorph surfaces. The 1 / hmSm resistance in series with

a dc voltage source accounts for the heat loss from the mirror-plate. Since, the

temperature distribution on the mirror-plate is not required for predicting the mirror

rotation angle, an average heat loss coefficient value is used to account for heat loss

from all surfaces of the mirror-plate. The capacitors Cbimorph and Cmirror account for the

thermal capacitances of the bimorphs and mirror-plate, respectively. Hence, they

1

b bh S

2

b

b b b

l

k A n

2b

b b b

l

k A n

i

i i i

l

k A n

1

m mh S

Cmirror Cbimorph p

Ta

Ts

i

i i i

l

k A n

Isolation Bimorph Mirror

Tb

Isolation

Ta

66 66

provide explicit relationship between the second order response shown in Figure 3-8,

Figure 3-9; and dimensions of the device and the properties of its constituent materials.

Table 3-1. Symbols used in lumped element model.

Symbol Description

Tb Bimorph temperature

Ts Substrate temperature

Ta Ambient temperature

Ii, Ai, ki, ni Length, cross-sectional area, average thermal conductivity and number of beams in the thermal isolation region, respectively

lb, Ab, kb, nb Length, cross-sectional area, average thermal conductivity and number of beams in the bimorph array, respectively

p Electrical power input Cbimorph Total thermal capacitance of nb bimorphs

Cmirror Total thermal capacitance of mirror-plate

hb Heat loss coefficient on bimorphs

hm Heat loss coefficient on mirror-plate

Sb Total surface area of nb bimorphs

Sm Total surface area of mirror-plate

3.5.3 Results Obtained from Lumped Element Model

Figure 3-12 compares the magnitude and phase response of the lumped element

circuit model with the complete finite element model. Clearly, the lumped element model

is in good agreement with the finite element model up to a frequency of 104 Hz and

explains the two cutoff frequencies in Figure 3-8A and Figure 3-9B. From Figure 3-12A,

the magnitude response obtained from the circuit model has an error of 10% in the low

frequency range. This may be attributed to the fact that the circuit model does not

account for the thermal resistance of the connecting region between the thermal

isolation and the bimorph array. Additionally, the thermal resistance of the mirror-plate

has been neglected in the lumped element model. No such assumptions are made in

the finite element model. For frequencies up to 104 Hz, the error in the phase plots

shown in Figure 3-12B and Figure 3-8B is less than 10%. However, significant error is

67 67

observed at higher frequencies. This is because the two lumped capacitances used in

Figure 3-11 cannot account for higher order effects produced by the distributed nature

of thermal capacitances in a real device.

Figure 3-12. Transfer function of thermal model. (A) Transfer function magnitude plot from finite element model and lumped element circuit model. The plot shows the rise in bimorph temperature. (B) Transfer function phase plot.

A comparison of Figure 3-9B and Figure 3-12A shows that the lumped element

circuit model accounts for the two low-frequency cutoffs observed in the device

response. Figure 3-13 compares the normalized PSM output of Figure 3-9B with the

normalized circuit model output. At frequencies well below mechanical resonance, the

PSM output is expected to be proportional to alternating thermal stresses. The

difference between the two plots is possibly due to the change in heat loss coefficient

value with frequency and angular displacement. The current work does not account for

this effect. Future work will involve the investigation of variation in heat loss coefficient

with mirror tilt angle. A feedback path from the angular output, θ, to the thermal model

can possibly be used to account for this phenomenon.

Incre

ase in a

vera

ge

bim

orp

h t

em

pera

ture

(K

)

0

-0.5

-1

-1.5

-2

Phase (

radia

ns)

Frequency (Hz) Frequency (Hz)

(A) (B)

FEM

Circuit Model

FEM

Circuit model

100 105

10-4 10-2 100 102 104 106

60

40

20

0

68 68

Figure 3-13. Comparison of normalized output of circuit model with normalized experimental results.

3.6 Summary

A dynamic compact thermal model of an electrothermally actuated micromirror is

reported. The micromirror is actuated by an array of thermal bimorph actuators with an

embedded platinum heater. A 2D thermal finite element model of the device is built. The

model is validated by comparing the simulation results with thermal imaging data. The

discretized heat equation is reduced to a lower order equation by a Krylov-subspace

based model order reduction algorithm. It is found that a reduced order model of order 2

closely approximates the complete finite element simulation output. In order to explain

the thermal response, a parametric circuit model is proposed. Two capacitors are

incorporated into the model to account for the second order response. The lumped

model is validated with finite element simulations and experimental results. The model

predicts the thermal response well.

1.1

1.0

0.9

0.8

0.7

0.6 0.5 0.4

0.3

0.2

Frequency (Hz)

Norm

aliz

ed

outp

ut

Normalized PSM output Normalized circuit model output

10-4 10-2 100 102 104

69

CHAPTER 4 TRANSMISSION LINE THERMAL MODEL OF ELECTROTHERMAL MICROMIRRORS

4.1 Background

A distributed RC network can rigorously represent the heat flow path in a

packaged IC chip [72]. Such distributed networks have the same topology as a

transmission line [64]. Therefore, equations that govern transmission lines can be used

for analyzing thermal models. The main focus of this chapter is a transmission line

based thermal model of a micromirror that is actuated by thermal bimorph actuators [20,

86]. The thermal model is simplified to obtain a simple LEM with only a few circuit

elements. The approach used in this chapter provides a rigorous strategy for deriving a

thermal model that saves computation time and provides explicit relationship between

device response and physical parameters. An electrical model is used to determine the

power dissipated, p, in the embedded resistor, for an applied voltage, VE. A mechanical

model predicts the mirror rotation for a change in bimorph temperature. The electrical,

thermal and mechanical models are then integrated to obtain the complete model, as

shown schematically in Figure 1-3. As discussed in Chapter 1, the average temperature

of the bimorph actuator, Tb, determines the mirror rotation angle, θ [10]. The resistance

of the heater is dependent on its temperature, Th.

This chapter is organized as follows. Section 4.2 introduces the micromirror and

presents the equations governing electrical, thermal and mechanical behavior. In

Section 4.3, a dynamic thermal model of the micromirror is developed. Simple lumped

element thermal models are proposed for both static and dynamic case. Electrothermal

modeling is discussed in Section 4.4. In Section 4.5, the thermal model is integrated into

70

a static electrothermomechanical model. Experimental results are used to validate the

model in Section 4.6.

4.2. 1D Electrothermally Actuated Micromirror


Figure 4-1 and Figure 4-2 show the SEM and schematic of the micromirror,

respectively.

Figure 4-1. SEM of electrothermal micromirror [19].

Figure 4-2. Schematic of electrothermal micromirror [19].

SiO2

Si

Al

Pt

SIDE VIEW

TOP VIEW

Mirror- plate Thermal

isolation Bimorphs

Heater

Substrate

Bond pads

Mirror- plate

Bimorph array

71

The 1.1 mm1.2 mm mirror-plate is attached to the substrate by an array of 72

thermal bimorph actuators. The thermal bimorph actuators consist of a 1.1 μm thick

PECVD SiO2 layer and a 0.83 μm thick e-beam evaporated aluminum (Al) layer. Each

actuator is 8.5 μm wide and 173.2 μm long. The single-crystal-silicon (SCS) layer

underneath the mirror-plate serves to improve the flatness of the mirror surface. The

thickness of the SCS layer is in the range 26–37 μm and depends on process

parameters. A current passing through the 1.2 mm long embedded platinum (Pt) heater

results in Joule heating. The difference in the coefficients of thermal expansion (CTE) of

the SiO2 and Al layers causes the bimorphs to bend, thereby causing the mirror to tilt.

When no actuation is applied, the bimorphs are curled up due to internal stresses in the

thin films. At either ends of the bimorphs, there is an SiO2 thermal isolation region. The

low thermal conductivity of SiO2 serves to limit the flow of heat from the bimorphs to the

mirror-plate and the substrate. The device is attached to a dual in-line package (DIP)

with H20E silver-epoxy adhesive from Epoxy Technology [87]. The package acts as a

good heat sink and aids thermal dissipation.

The devices shown in Figure 4-1 and Figure 3-1 are similar. The key difference

lies in the embedded heater design. For the device in Figure 3-1 [39], the heater runs

along the length of the bimorph; the heater in the design shown in Figure 4-1 is located

only at the substrate side of the bimorphs. This also leads to different bimorph layer

structures. Figure 4-3 compares the thin film layers that constitute the bimorph structure

along with typical thickness values. In Figure 4-3A, the thin SiO2 layer isolates the

heater from Al electrically; the bimorph shown in Figure 4-3B has no heater layer. In

both cases, the mirror-plate rotation is proportional to the average temperature along

72

the bimorph length [19, 39]. Simulations show that for the same rise in the average

bimorph temperature, the deflection produced by the structure shown in Figure 4-3B is

greater than that produced by the structure shown in Figure 4-3A by 14%. Also in case

of Figure 4-3B, the heater lies along the side of the bimorph array and undergoes much

less deformation during actuation as compared to the heater corresponding to Figure 4-

3A. Therefore, when large deflections are involved, the design corresponding to Figure

4-3B is used. For instance, Wu et al. report a micromirror that can scan as much as

124° by using the bimorph structure shown in Figure 4-3B [8].

The key advantage of the design shown in Figure 4-3A is that the bimorph is

heated almost uniformly. Typically, the maximum allowable temperature is limited by the

melting point of Al. Therefore, if the heater is along the bimorph length, all parts of the

bimorph almost reach maximum temperature simultaneously [39]. Thus the structure in

Figure 4-3A maximizes bimorph utilization. It will be shown in Section 4.3 that there is a

significant temperature gradient along the bimorph length for the design corresponding

to Figure 4-3B.

Figure 4-3. Comparison between two bimorph designs. (A) Thin films in bimorphs corresponding to Figure 3-1 [39]. (B) Thin films in bimorphs of the device shown in Figure 4-2. Typical thickness values have been indicated beside the schematics.

1 µm

0.83 µm

0.25 µm

0.1 µm

1.1 µm

0.83 µm

(A) (B)

Al

SiO2

Pt

73

4.2.2 Thermal Bimorph Actuation

The principle of operation of thermal bimorphs is well established [10, 79] and has

been outline in Chapter 1. From Equation 1-1, the change in tangential angle at the tip

of a bimorph, θ(Tb), is directly proportional to the change in the average temperature

along the bimorph length. Consequently, for the device shown in Figure 4-1, the change

in the mirror tilt angle varies linearly with the average bimorph temperature. If the

material properties of the bimorph layers do not vary significantly over the operation

range, the tilt angle change of the mirror-plate is proportional to the electrical power

input to the embedded heater [29, 39]. Equation 1-1 gives the mechanical rotation angle

of a mirror-plate attached to the bimorphs. The optical angle scanned by a light beam

reflected from the mirror is twice the mechanical rotation angle.

Equation 1-1 may be rewritten as,

0 0( ) Λ ( )b bθ T θ T T (4-1)

The coefficient of the (Tb-T0) term in Equation 4-1 is Λ; it represents the change in

deflection angle per unit rise in bimorph temperature. T0 denotes a reference

temperature, typically room temperature, i.e., 298 K. The material properties of the two

bimorph layers, i.e., Al and SiO2, are listed in Table 4-1. Plugging the values from

Table 4-1 into Equation 1-1 yields Λ = 0.08833.

Table 4-1. Material properties for thermomechanical simulations [58].

Material Young’s modulus (GPa)

Poisson’s ratio CTE (K-1)

Al 70 0.35 23.1×10-6

SiO2 70 0.17 0.5×10-6

From FEM simulation using COMSOL [58], it was found that Λ = 0.08673 [19].

Therefore, the value of Λ obtained using Equation 4-1 has less than 2% error.

74

Equation 4-1 is valid for the temperature range in which material properties do not vary

significantly.

4.2.3 Electrothermal Model

Neglecting edge effects at either ends of the bimorph array, the embedded heater

has a nearly uniform temperature distribution. When a voltage, VE, is applied, the power,

p, dissipated due to Joule heating is given by,

2

( )E

E h

Vp

R T (4-2)

where, Th is the average temperature of the embedded heater and RE (Th ) is the heater

resistance. If the TCR of the heater is α, the reference temperature is T0, and the heater

resistance at T0 is RE0, then RE (Th ) can be expressed as

0 0( ) (1 ( ))E h E hR T R α T T (4-3)

2

0 0(1 ( ))E

E h

Vp

R α T T (4-4)

To measure α, the reference temperature T0 was set as the room temperature, i.e.,

298 K. A mirror was placed in an oven and the resistance of the heater was measured

in the 298– 498 K temperature range. The TCR was determined to be 0.0025 K-1 by

using a linear fit for the resistance vs. temperature plot.

As discussed above, both mechanical and electrical behavior can be readily

described by the simple models given by Equations 4-1 and 4-4, respectively. What is

missing in the complete model shown in Figure 1-3 is the thermal model, which is the

focus of this chapter. A dynamic thermal model will be developed in the next section.

After that, simple lumped element electrothermal circuit models will be proposed for

75

both static and dynamic cases. A static electrothermomechanical model will be

discussed in Section 4.5.

4.3 Thermal Model

The electrothermal mirror shown in Figure 4-1 is actuated by an array of 72

bimorphs. From the device topology, it appears that if edge effects at the two extremes

of the array are neglected, the individual bimorphs have a nearly identical temperature

distribution [39]. In order to verify this claim, a 3D thermal model of a section of the

device was built using the IntelliSuite 8.2 MEMS simulation package [60]. For the 10

bimorphs in Figure 4-4, the variation in the average temperature rise and maximum

bimorph temperatures are within 1%. Thus, the bimorph beams have nearly identical

temperature distribution. Therefore, in order to conserve computational resources, any

temperature gradient across the array may be ignored. The temperature distribution

obtained from the model has been shown in Figure 4-4. Thermal transfer from the

device to the surrounding air has been accounted for by employing a heat loss

coefficient. The heat loss coefficient may be defined as thermal power transfer from a

surface to the surrounding atmosphere per unit area per unit rise in surface

temperature. Details on the value of heat loss coefficient from the device to the

atmosphere have been given in Section 4.3.1. The heat loss coefficient accounts for

thermal transfer due to conduction and convection. Since the maximum temperature is

limited by the melting point of Al, heat loss due to radiation is neglected [29]. The next

subsection will provide details on the estimated heat loss coefficient from the device to

the surrounding atmosphere. After that, a simple FE model consisting of a single

bimorph and a section of the mirror-plate will be presented.

76

Figure 4-4. Simulated temperature distribution in a section of the device. The section has ten bimorphs. The input power to the device is 10 mW and the substrate temperature is 298 K.

4.3.1 Estimation of Heat Loss Coefficient

Both convection and thermal diffusion will result in heat loss from the device to the

surrounding atmosphere. Convection occurs due to buoyancy driven motion of the

surrounding air whereas thermal diffusion is a result of heat conduction through the

surrounding atmosphere. In microscale, thermal diffusion dominates and buoyant forces

are weak compared to viscous forces [43]. Hence, to estimate the heat loss due to

thermal diffusion alone, the FE model shown in Figure 4-5 was built using

COMSOL [58]. The model consists of the air region surrounding a single bimorph and a

section of the mirror-plate. Since the device is kept at room temperature, i.e., 298 K,

constant temperature boundary condition is imposed on the substrate and package as

Mirror-plate

Bimorphs

363.59

357.23

351.55 345.70

339.74 333.78

327.81 321.85

315.89

309.93 303.96

298

Temperature (K)

77

well as the outer boundary of the air region. The bimorph and mirror section define the

internal boundary of the air region. A uniform elevated temperature of 310 K was

applied to the surfaces of the bimorph and mirror-plate that are in direct contact with air.

The thermal conductivity of air, kair , was assumed to be 0.026 Wm-1K-1 [58]. From the

solution of the steady state heat equation obtained using COMSOL [58], the average

heat loss coefficient on the bimorph and mirror-plate were found to be 188 Wm-2K-1 and

47 Wm-2K-1, respectively. These values are significantly greater than typical values of

free-convection heat loss coefficient for macroscopic bodies in air ranging from 1 to

10 Wm-2K-1 [88]. Hence, the simulations suggest that the heat loss from the device to

the atmosphere may be primarily attributed to thermal diffusion.

Figure 4-5. Simple FE model for estimating heat loss coefficient due to thermal diffusion. Constant temperature condition is imposed on the bimorph and the section of the mirror-plate. Air at a distance of 2 mm from the mirror (not shown) is assumed to be at room temperature.

To further substantiate this assertion, coupled thermal-fluidic simulations were

performed using COMSOL [58]. For fluidic simulations, the no-slip boundary condition

was imposed on the substrate, package, mirror-plate and bimorph surfaces. Since the

bimorphs have a nearly identical temperature distribution, the symmetry boundary

x

y

z Package (room temperature)

Substrate (room temperature)

Mirror (elevated temperature)

Bimorph (elevated temperature)

Air

Air

78

condition was used at the faces common to adjacent mirror sections. The open

boundary condition was imposed on all other boundaries. The Boussinesq

approximation was used to account for the buoyancy driven air flow [58]. The model

was simulated for several package orientations. In all cases, the difference in the total

heat loss coefficient and the heat loss coefficient due to diffusion alone is found to be

less than 5%. Hence, diffusion heat loss coefficients have been used for all thermal

simulations.

The estimated values of heat loss coefficients given in this section are valid for a

stationary micromirror only. When the mirror is in motion, these values may change due

to the forced motion of air around the device. For the mirror-plate and the bimorph, the

heat loss coefficients are denoted by hm and hb , respectively. Based on simulation

results, hm and hb are chosen to be 188 Wm-2K-1 and 47 Wm-2K-1, respectively.

4.3.2 FE Thermal Model

Neglecting temperature variation across the bimorph array, the FE model shown in

Figure 4-6 was built using COMSOL [58]. The thermal conductivity values of the

bimorph materials are listed in Table 4-2. The thickness of the silicon substrate to which

the bimorphs are attached is 550 µm, which is more than two orders of magnitude

thicker than the bimorphs. The substrate is attached to a metallic heat sink with a

thermally conductive H20E [87] silver-epoxy adhesive. Hence, the substrate is

considered to be a nearly perfect heat sink. Constant temperature boundary condition is

imposed at the end of the bimorph connected to the substrate. Heat loss coefficient

obtained from the simulation described in Section 4.3.1 is specified for all boundaries

exposed to air. Since the temperature distribution in all the bimorphs in the device is

79

nearly identical, symmetry condition is imposed on all other boundaries. Figure 4-6A

shows the simulation results for a 10 mW power input to the device.

Figure 4-6. FE thermal model of micromirror. (A) Simulated temperature distribution for 0.01 W power input. (B) Temperature distribution along bimorph.

Table 4-2. Thermal conductivity values for simulations [20].

Material Thermal Conductivity (W/(m-K)

Al 94

SiO2 1.4 Si 130 Pt 71.6

Figure 4-6B shows the temperature distribution along the bimorph length. The

temperature is highest at the point where the heater is located, i.e., Th = 363.65 K. The

average temperature, Tb , along the bimorph length is found to be 356.55 K.

298 K

363.65 K

364

360

356

352

348 0 40 80 120 160

Distance along bimorph (µm)

Tem

pe

ratu

re (

K)

(A)

(B)

80

Due to process variations, the temperature distribution in a real device may vary

significantly from the simulation results shown in Figure 4-6. The next subsection will

address the impact of process variations on the thermal model.

4.3.3 Effect of Process Variations on Thermal Model

The micromirrors were fabricated by a process similar to the one described in [8].

All the fabrication steps, except the final release step, are done at wafer level. The final

release step involves Si isotropic etch using DRIE. Typically, 4-6 devices are released

at a time. The etch time determines the amount of Si etched underneath the thermal

isolation regions at the ends of the bimorph actuators. A longer etch time results in

better thermal isolation. On the other hand, a short etch time may not remove the Si

under the thermal isolation region completely. The schematics shown in Figure 4-7

illustrate the effect of etch time on the thermal isolation region at the bimorph ends. For

illustration, the substrate end of the bimorph has been shown. As shown in Figure 4-7A,

for low etch times, the Si below the isolation region is not removed completely, resulting

in lower thermal resistance at the bimorph-substrate junction. So, in this case the actual

temperature will be lower than that depicted in Figure 4-6. Figure 4-7B shows that for

longer etch time, the Si below the isolation region is completely removed and some Si

may also get etched from the substrate, resulting in higher thermal resistance. Hence,

for longer etch times, the actual temperature may be greater than the one depicted in

Figure 4-6. The etch time also determines the thickness of Si below the mirror-plate.

Longer the etch time, thinner the Si layer.

In order to elucidate the effect of process variations experimentally, two mirrors

with release etch times of 5 min and 12 min were compared. A dc source was used for

actuation and the optical angle was tracked by using a laser beam reflected from the

81

mirror surface. Measured values of applied voltage and current flowing through the

heater were used to determine the power input. Figure 4-8 compares the responses of

these devices. Clearly, the optical angle per unit power input may change by more than

5 times due to process variations. This may be incorporated into the thermal model by

scaling the thermal conductivity of the thermal isolation region by a suitable factor. The

isolation region consists of SiO2 which is transparent to visible light. Hence, the thermal

resistance of the isolation region may be estimated by measuring the silicon undercut

using a microscope.

Figure 4-7. Exaggerated schematic of bimorph substrate junction [19]. (A) Short etch time for final device release. (B) Long etch time for final device release.

For both the plots shown in Figure 4-8, the difference between measured data and

a linear fit is less than 1%. The linear device response suggests that the effect of

variation in material properties and heat loss coefficients is negligible. Hence, models

with constant parameters may be used. Moreover, Figure 4-6 shows that temperature

gradient is mainly present along the length of the model, i.e., the x direction. Therefore,

most of the heat flow takes place along the x direction. Based on these observations, a

simple transmission line-based thermal model will be derived in the following sections.

(A)

(B)

Pt

SiO2

Si

Al

82

Figure 4-8. Comparison of two mirrors with different etch times. A longer etch time results in better thermal isolation and consequently a larger optical angle per unit power input.

4.3.4 Transmission-line Model for 1D Heat Flow

Let us consider the 1D geometry shown in Figure 4-9A. Let the thermal

conductivity be k and the heat capacitance per unit length be c. If the 1D geometry

consists of layers of different materials, the equivalent thermal conductivity is obtained

by taking the weighted average of thermal conductivities of all the layers, with the

weights as the layer thicknesses [6]. The equivalent thermal resistance per unit length is

r = 1 / (kA), where A is the area of cross-section. Let g represent the thermal

conductance per unit length. The thermal conductance accounts for heat loss due to

convection and diffusion to the surroundings. The thermal model may be represented by

the two-port distributed network [72] shown in Figure 4-9B. An element of length Δx0

is represented by resistors r Δx, 1 / (gΔx), and the capacitor cΔx; a series of such

infinitesimal elements represent thermal impedances in the complete length l of the 1D

geometry. The voltage, v, denotes the increase in temperature above ambient. The

current, i, represents heat flow. Principles governing transmission lines are well

0 10 20 30 40 50 60

O

ptica

l an

gle

(de

gre

es)

Input electrical power (mW)

Etch time = 12 min

Etch time = 5 min

40

30

20

10

0

83

established [89] and may be used to analyze the circuit shown in Figure 4-9B. At x, let

the voltage be v(x,η) and the current be i(x,η).

Kirchhoff’s voltage law gives,

( , ) ( Δ ) ( , ) ( Δ , )v x η r x i x η v x x η

( , )( , )

v x ηri x η

x (4-5)

Kirchhoff’s current law at node N gives,

( Δ , )( , ) ( Δ , ) ( Δ ) ( Δ , ) Δ

v x x ηi x η i x x η g x v x x η c x

η

( , ) ( , )( , )

i x η v x ηgv x η c

x η (4-6)

For harmonic variation,

( , ) Re( ( ) )jωηv x η V x e (4-7)

( , ) Re( ( ) )jωηi x η x eI (4-8)

Let,

( )γ r g jωc (4-9)

From Equations 4-5 and 4-7,

( ) ( )γx γxV x e e 1 2A A (4-10)

From Equations 4-5, 4-6, 4-7, 4-8 and 4-10,

( ) ( )γx γxγx e e

r

1 2A AI (4-11)

It will be shown in Section 4.5.2 that A1 and A

2 can be determined by the heat

sources and thermal impedances connected at the two ends of the 1D geometry.

84

Figure 4-9. Passive transmission-line model. (A) Geometry with 1D heat flow. (B) Transmission-line model representing 1D heat flow. (C) Equivalent circuit model.

Let a voltage V(0) be applied at x = 0 and let the other end be open circuit, i.e.,

I( l ) = 0.

1 2A A( ) ( ) 0γl γlγ

l e er

I

1

2

A

A

2γ le (4-12)

rΔx

+

-

+

-

v(l ) v(0)

i(0) i(l )

(B)

ZA

ZA

ZB v(0) v(l )

i(l )

i(0)

+

-

+

- (C)

N

i(x)

i(x+Δx) v(x)

v(x+Δx)

1

Δg x

cΔx

l

t

Direction of heat flow

(A)

Network representation of an

element of length Δx, Δx0

w

rl / 2

rl / 2

cl v(0)

i(0)

+

-

v(l )

i(l )

+

- (D)

1

gl

85

1 2

1 2

A A

A A

( )(0)

(0) ( / )( )A B

VZ Z

γ rI (4-13)

Using Equations 4-12 and 4-13,

2

2

( 1)

( 1)

γ l

A B γ l

r eZ Z

γ e (4-14)


1

2 2

2

AA A

A

2(0) ( 1) (1 )γ lV e (4-15)

2A

2

(0)

(1 )γ l

V

e (4-16)

From Equations 4-10, 4-12 and 4-16,

2A

2

( ) 2( ) (2 )

(0) (1 )

γlγl

γ l

V l eV l e

V e (4-17)

But from Figure 4-9C,

( )

(0)B

A B

ZV l

V Z Z (4-18)

So from Equations 4-17 and 4-18,

2

2

(1 )

γl

B

γ l

A B

Z e

Z Z e (4-19)

Equations 4-14 and 4-19 are solved to give,

0

1tanh( / 2)

1

γl

A γ l

r eZ Z γl

γ e (4-20)

0

2

2

1 sinh( )

γ l

B γl

Zr eZ

γ e γl (4-21)

where, Z0 = r / γ.

86

The circuit model shown in Figure 4-9C has been rigorously derived and accounts

for the distributed nature of thermal resistances. Since MEMS devices involve small

length scales, it may be possible to approximate the hyperbolic functions in Equations

4-20 and 4-21 by using Taylor series expansion. Expanding Equations 4-20 and 4-21,

3 5

00

( )( / 2) 2( / 2)( / 2 )

3 15 2 2 2A

r g jωc lZ γlγl γl r rlZ Z γl

g jωc (4-22)

0 0

3 5

/ ( ) 1

( ) ( ) ( )( )

6 120

B

r g jωcZ ZZ

γl γl γl gl jωclr g jωc lγl

(4-23)

The approximations used in Equations 4-22 and 4-23 are valid if,

3Re( ) Re( )γl γl (4-24a)

and,

3Im( ) Im( )γl γl (4-24b)

Equations 4-22 and 4-23 elucidate the physical significance of ZA and ZB,

respectively. From Equation 4-22, ZA ≈ rl / 2 represents half of the conductive resistance

of the 1D geometry shown in Figure 4-9A. Similarly, from Equation 4-23, ZB represents

the parallel connection of a resistor (1/(gl )) and a capacitor cl. Hence, the heat loss to

the atmosphere and the total capacitance get lumped together as ZB. Therefore, if the

Inequalities 4-24a and 4-24b are satisfied, the thermal LEM shown in Figure 4-9D can

be used without incurring large error.

Equation 4-9 may be equivalently expressed as,

1 2γ γ jγ (4-25a)

87

2 2

1

( ) ( )

2

rg ωrc rgγ (4-25b)

2 2

2

( ) ( )

2

rg ωrc rgγ (4-25c)

where, γ1 and γ2 are the real and imaginary parts of γ.

Substituting Equation 4-25 into Inequality 4-24a gives,

2 2 2 2 2( ) ( ) 2 1rgl ωrcl rgl (4-26)

Substituting Equation 4-25 into Inequality 4-24b gives,

2 2 2 2 2( ) ( ) 2 1rgl ωrcl rgl (4-27)

The conditions necessary for the LEM shown in Figure 4-9D to be valid may be

obtained by simplifying Inequalities 4-26 and 4-27,

( )( ) 1 1/rl gl l rg (4-28a)

and,

( )( ) 1/ 1/rl cl ω l ωrc (4-28b)

Inequalities 4-28a and 4-28b illustrate the physical conditions necessary for the

lumped-element approximation to hold. Inequality 4-28a requires that the total

resistance associated with heat loss to the surrounding, i.e., 1 / (gl ), be significantly

greater than the conductive resistance, rl. Inequality 4-28b requires that the time

constant, (rl )(cl ), be significantly less than the reciprocal of the frequency, ω. If the 1D

geometry satisfies Inequality 4-28a but not Inequality 4-28b, the lumped-element

approximation may still be used for the static case.

88

If Inequalities 4-28a and 4-28b do not hold good, the 1D geometry may be

partitioned into smaller segments such that Inequalities 4-28a and 4-28b are satisfied by

each segment. The circuit models corresponding to these segments may be cascaded

in series to obtain the thermal model of the entire geometry. Alternatively, the rigorous

equivalent circuit shown in Figure 4-9C must be used.

For the static case, i.e., ω = 0, Equations 4-9, 4-20 and 4-21 give,

00tanh( / 2)A A ω

R Z R rgl (4-29a)

0

0sinh( )

B B ω

RR Z

rgl (4-29b)

where, R0 = (r / g)1/2

denotes the dc value of Z0.

If (rl )(gl ) 1, Equations 4-22 and 4-23 yield,

/ 2AR rl (4-30a)

1/ ( )BR gl (4-30b)

These results will be applied to develop a thermal model of the micromirror in the

next subsection.

4.3.5 Equivalent Circuit Representation of Thermal Model

From the FE model in Figure 4-6, it was found that temperature gradient primarily

occurs along the length of the bimorph and mirror-plate section. Temperature variation

along any cross-section perpendicular to the bimorph length is negligible. Hence, the

heat flow is mainly one-dimensional and the results derived in Section 4.3.4 are

applicable. Figure 4-10 shows the equivalent circuit representation of the thermal model

of the 1D mirror. The equivalent thermal impedances for the bimorph and the mirror-

plate are denoted by subscripts b and m, respectively, and may be calculated using

89

Equations 4-22 and 4-23. Since the lengths of the thermal isolation regions are an order

of magnitude smaller than the bimorph length, heat loss from the isolation regions to the

atmosphere has been neglected and they have been replaced by a single resistor to

represent the conductive thermal resistance. These resistors are Riso1 and Riso2 in Figure

4-10. The power input is p and the number of bimorphs is nb. So, the current source with

magnitude p / nb represents the power input per bimorph.

Figure 4-10. Equivalent circuit representation of thermal model.

For the bimorph, Equation 4-10 may be rewritten as,

( ) ( )b bγ x γ x

b bV x e e

1 2A A (4-31)

( )b b b bγ r g jωc (4-32)

The average bimorph temperature increase can be obtained by integrating

Equation 4-31 over the bimorph length, lb , and is given by,

1 2A A0

( )( )(1 )

b

b b b b

l

γ l γ l

b baverage

b b b

V x dxe e

Vl γ l

(4-33a)

Δ Re[ ]jωη

b averageT V e (4-33b)

p / nb

Riso1

ZAb

ZBb

ZAm

ZBm

ZAb +

-

+

-

M

V = (Th-T0)

Riso2

( )V l(0)V

I( l ) I(0)

90

Node M in Figure 4-10 represents the embedded heater. Therefore the rise in

embedded heater temperature is given by,

1 2A AΔ Re[ (0) ] Re[( ) ]jωη jωη

h b bT V e e (4-34)

Let, ZL = Riso1 represent the impedance to the left of the bimorph. The current

through ZL is ((p / nb ) - (γb / rb)(A1b - A2b )). Therefore,

1 2

1

A A

A -A1

2( / ) ( / )( )b b

L iso

b b b b b

Z Rp n γ r

(4-35)

Similarly, let ZR represent the impedance to the right of the bimorph. Then ZR is

given by,

1 2

1 2

A A

A A2

( / )( )

b b b b

b b b b

γ l γ l

b bR iso Am Bm γ l γ l

b b b b

e eZ R Z Z

γ r e e (4-36)

Equations 4-35 and 4-36 can be solved to find A1b and A

2b,

1A

2

2

( / ) ( )

( )( ) ( )( )

b b

b b

l γ

b b L b R bb l γ

b L b b R b b L b b R b

e p n r Z r Z γ

r Z γ r Z γ e r Z γ r Z γ (4-37a)

2A

2

( / ) ( )

( )( ) ( )( )b b

b b L b R bb l γ

b L b b R b b L b b R b

p n r Z r Z γ

r Z γ r Z γ e r Z γ r Z γ (4-37b)

Equations 4-33, 4-34 and 4-37 can be used to evaluate the average bimorph

temperature rise and the embedded heater temperature rise for a particular frequency

ω. If the power dissipated in the embedded heater has several frequency components,

the total temperature rise may be obtained by summing the contribution from each

frequency component ,i.e., for m components,

m

,

1

Δ Re[ ]ijω η

b average i

i

T V e (4-38)

where, the subscript i denotes the i th component.

91

The results derived in this section are based on rigorous transmission-line theory.

Simple electrothermal models will be developed in the next section.

4.4 Electrothermal Model

4.4.1 Static Model

Equation 4-29 may be used to evaluate the thermal resistances for the static

model shown in Figure 4-11. As described by Equation 4-2, the applied voltage, VE ,

across the embedded heater resistance, RE , causes power dissipation. Both the heater

temperature, Th, and the bimorph temperature, Tb , may be obtained from the thermal

model.

Figure 4-11. Static electrothermal model based on transmission-line theory. The voltages at M and Q represent the rise in heater temperature and average bimorph temperature rise, respectively.

From Figure 1-3, the thermal model must determine the average bimorph

temperature increase and the embedded heater temperature. Comparing Figure 4-10

and Figure 4-11 we conclude that the voltage at node M gives the rise in embedded

heater temperature. For determining the average bimorph temperature, the impedance

RAb is partitioned into ηRAb and (1-η)RAb, where η is a partition factor. The value of η is

+

V(0)

-

+

-

V(l ) zdcR0b

V = ΔTb

Q

M

VE

RE0αΔTh

RE0

IE

V = (Th-T0)

I(0) I( l )

ηRAb

(1-η)RAb

RAb

RBb Riso1 2

E E

b

R

n

I

92

chosen such that the voltage at node Q in Figure 4-11 gives the average bimorph

temperature increase,

Δ (0) (0) ( )b AbT V R ηI (4-39)

Let,

b b br g lδ e (4-40)

0 /b b bR r g (4-41)

1 0( ) /dc iso Am Bm bz R R R R (4-42)

where, zdc represents the normalized thermal resistance to the right of the bimorph.

Let, a = A1b / A2b for ω = 0. Equation 4-37 gives,

1

2

A

A

2

0

0

( ) 1

( ) 1b b dc

dc b

b b dcω

V l zz R a δ

l zI (4-43)

Substituting V(0), I(0) , ΔTb from Equations 4-10, 4-11 and 4-33 into Equation 4-43

and using Equation 4-42,

( 1)( ) ( 1)( 1)

( 1)ln( ) ( 1)( 1)

δ a δ δ a

a δ δ a

η (4-44)

An equation to explicitly represent η in terms of device parameters may be

obtained by substituting Equation 4-43 into Equation 4-44,

2 2 2

2 2

(1 )((1 ) ( (1 )) ) (1 )((1 ) (1 ))

(1 )(1 ) ((1 ) (1 ))ln( )dc dc dc dc

dc dc dc

δ z δ z δ δ z δ z

δ z z δ z δ

η (4-45)

where, δ and zdc are defined by Equations 4-40 and 4-42, respectively. For the device

shown in Figure 4-1, the value of η is 0.81.

Equation 4-45 has been rigorously derived using transmission-line theory. Next, let

us check if the condition for lumped-element approximation, i.e., Inequality 4-28a is

93

satisfied by the bimorphs and the mirror-plate section. For the bimorph,

(rblb)(gblb) = 1.410-7 1 and for the mirror-plate (rmlm)(gmlm) = 0.04 1. Therefore,

Inequality 4-28a holds good for the bimorph and the mirror-plate section. Hence, RAb,

RBb, RAm and RBm may be replaced by the lumped element approximation given by

Equation 4-30. Figure 4-12 shows the lumped-element static electro-thermal model.

Since (rblb) << 1/(gblb), the current through the resistor 1/(gblb) is negligible compared to

the current through the (rblb) / 2 resistors. Consequently, I1 ≈ I2 and voltage drops across

both the (rblb) / 2 resistors are approximately equal. Therefore, voltage at node Y in

Figure 4-12 represents the average bimorph temperature.

Figure 4-12. Static electrothermal model based on lumped-element approximation. The voltages at M and Y represent the rise in heater temperature and average bimorph temperature rise, respectively.

4.4.2 Dynamic Electrothermal Model

The experimentally obtained mechanical resonant frequency of the mirror is

fresonant = 173 Hz. For frequencies greater than flimit = 265 Hz, i.e., ωlimit = 1665 rad/s, the

I1

I2

M

Y

VE

RE0αΔTh

RE0

IE

Riso1

2

E E

b

R

n

I

2

1

2m m

iso

m m

r lR

g l

1

b bg l

V = ΔTb

2b br l

2b br l

V = ΔTh

Electrical domain

Thermal domain

94

mirror deflection is found to be negligible. Therefore, it is desirable to have a thermal

model that is accurate in the 0–265 Hz frequency range. Inequality 4-28b may be used

to evaluate the length of a segment of a bimorph, lbs, that can be represented by a

model of the form shown in Figure 4-9D, i.e.,

limit1/ 115.2μmbs b bl ω r c (4-46)

Let us partition the bimorph into 8 segments, each 21.6 μm long. Also, since

the thermal resistance of the isolation between the bimorph and mirror-plate is an order

of magnitude larger than the conductive resistance of the mirror-plate, the resistance

corresponding to heat conduction through the mirror-plate may be neglected. Figure 4-

13 shows the schematic of the lumped-element dynamic model.

The rise in average bimorph temperature is obtained by taking the average voltage

corresponding to nodes 1– 8. This can be easily obtained from the ammeter output in

Figure 4-13, i.e.,

8 8

1

1 1 8Δ

8 8total

b i i

i ib b b b

T Vg l g l

II (4-47)

4.5 Static Electrothermomechanical Model

The mirror deflection may be evaluated using Equation 4-1. The average bimorph

temperature (Tb) and the embedded heater temperature (Th) can be determined from

the electrothermal model discussed in Section 4.4. Therefore, the mechanical, electrical

and thermal models for the micromirror can be integrated into a complete

electrothermomechanical model as shown in Figure 4-14. Circuit elements for

implementing Equation 4-1 have been added to the electrothermal model in Figure 4-11

to obtain the schematic shown in Figure 4-14. Figure 4-14 provides a circuit

95

representation for the schematic described in Figure 1-3. The next section will compare

the model against experimentally obtained results.

Figure 4-13. Dynamic electrothermal model of micromirror. (A) LEM with bimorph divided into 8 segments. (B) Circuit representing the thermal model of one-eighth of a bimorph actuator corresponding to node i (i = 1 to 8) in (A).

4.6 Comparison with Experimental Results

4.6.1 Static Electrothermomechanical Model

As described in Section 4.3.3, the thermal resistance of the isolation region at

either ends of the bimorphs may be estimated by observing the device under a

microscope. The device parameters for a mirror that was released using an etch time of

12 min has been listed in Table 4-3. From the etch time, the Si layer thickness below

(B)

i

Ii

RE0

Riso1

VE

A

Itotal

Riso2

V = ΔTh

Bimorph divided into 8 segments

1 2 3 4 5 6 7 8

(A)

Thermal domain Electrical domain

RE0αΔTh

2

E E

b

R

n

I 1

m mg l cmlm

8b bc l

16b br l

16b br l

8

b bg l

96

the mirror-plate was estimated to be 28.6 µm. From Equation 4-45, the value of the

partition factor, η, for the thermal model was found to be 0.81. Figure 4-15A and Figure

4-15B compare the FE, circuit model (Figure 4-11) and LEM (Figure 4-12) results with

experimentally obtained data. The distinction between the circuit model and the LEM is

that the circuit model is based on Equation 4-29, whereas the LEM is based on

Equation 4-30. Hence, the circuit model is based on rigorous transmission-line theory.

Figure 4-14. Electrothermomechanical model of 1D mirror. The voltage at node H gives the mechanical rotation angle of the micromirror.

Table 4-3. Circuit model parameters for a mirror with 12 min release time.

Impedance (mirror section with 1 bimorph)

Value

Riso1 2.1647 MΩ

Riso2 0.91137 MΩ

RAb 125.2 kΩ

RBb 1190.7 kΩ

RAm 10.2 kΩ

RBm 458 kΩ

The error in FE model results arises because the material properties in the actual

device may be different from the one used for simulations. Even though the circuit

RE0

VE

RE0αΔTh

IE

Riso1 RBb

RAb

Z

θ0

ΛΔTb

v = ΔTh

v = ΔTb

M

Q

H

Electrical Domain

Thermal Domain

Mechanical Domain

2 /E E bR nI

AbRη (1 ) AbRη

( )bθ THV

97

model has been rigorously derived using transmission-line theory, Figure 4-15 shows

that there is a difference between the circuit-model and FEM results. This is because

the circuit model has been derived by assuming one-dimensional heat flow, which is not

strictly true for the real device. For instance, an accurate evaluation of the thermal

resistance of the isolation regions must take into account the three-dimensional heat

flow in the SiO2 thin film.

Figure 4-15. Comparison between model and experimental results. The optical angle is

02( ( ) )bθ T θ . (A) Optical angle vs. input power. (B) Optical angle vs. applied

voltage.

Figure 4-15A and Figure 4-15B reveal that the circuit model and LEM are in close

agreement. For instance, in Figure 4-15A the slopes of the lines corresponding to circuit

model and LEM differ by 1.4% only. Therefore, the assumptions for lumped element

modeling hold good for the current device. However, the LEM shown in Figure 4-12 may

not always be an accurate representation of a real device. To illustrate, let us vary the

bimorph length, lb, while keeping all others device parameters fixed. Figure 4-16 shows

the error in the optical angle per unit input power evaluated using the LEM shown in

Experiment

FE model

Circuit model

LEM

0 5 10 15 20 25 30

Input power (mW)

40

35

30

25

20

15

10

5

0

Op

tica

l an

gle

(d

eg

ree

s)

35

30

25

20

15

10

5

0

Op

tica

l an

gle

(d

eg

ree

s)

0 0.5 1 1.5 2 2.5

Voltage (V)

Experiment

FE model

Circuit model

LEM

(A) (B)

98

Figure 4-12. The circuit model has been used as a reference for estimating the error. As

lb is increased, the accuracy of the LEM diminishes. This is because the approximations

used in Equation 4-30 are valid for small length scales. As the length is scaled up, the

higher order terms in the Taylor expansion in Equations 4-22 and 4-23 become

significant. LEM accuracy may be increased by partitioning the bimorph into smaller

segments, such that Inequality 4-28a is satisfied by each segment.

Figure 4-16. Dependence of error in LEM results on bimorph length. All other device parameters are same as the device shown in Figure 4-2.

4.6.2 Dynamic Thermal Model

The rigorous dynamic transmission-line model is described in Section 4.3.5. The

approximate dynamic electrothermal LEM has been discussed in Section 4.4.2. In order

to validate the dynamic LEM, let us consider the part of the circuit corresponding to the

thermal domain shown in Figure 4-13. A 0.01 W sinusoidal power source is assumed

and the results are compared with the transmission-line model. Figure 4-17A and Figure

4-17B show the magnitude and phase response, respectively. Clearly, the plots

corresponding to the transmission-line model and LEM practically coincide.

50 150 250 350 450

Bimorph length (µm) %

Err

or

in o

ptica

l a

ng

le e

valu

ate

d b

y L

EM

30

25

20

15

10

5 0

99

In order to validate the thermal model, the frequency response of the mirror was

obtained by applying a 0.22 V peak-to-peak sine wave at a dc offset of 1.6V. The dc

offset serves to bias the mirror in the linear region of the device characteristics shown in

Figure 4-15B. The frequency response has been shown in Figure 4-18A. The decay in

response observed at frequencies below the mirror resonant frequency, i.e., 173 Hz,

may be attributed to the thermal response time of the micromirror. Figure 4-18B

compares the normalized bimorph temperature rise predicted by the transmission-line

model with the normalized experimentally obtained scan angle at low frequencies.

Clearly, the transmission-line model predicts the range of frequencies over which the

frequency response of the micromirror decays. This proves the feasibility of the

transmission-line approach for modeling both static and dynamic response.

4.6 Summary and Discussion

This chapter reports a general procedure for modeling bimorph based

electrothermal MEMS and demonstrates it with a micromirror device. Electrical, thermal

and mechanical models are developed and integrated into a complete static

electrothermomechanical model. The electrical model provides the output power for an

applied voltage. It takes the temperature dependent resistance of the embedded heater

into account. The mechanical model gives the device response for a certain

temperature change in the bimorph actuators. A thermal FE model is built. A circuit

model that predicts the thermal behavior is then developed by drawing analogy between

heat transfer and signal flow in an electrical transmission line. A simplification of the

circuit model into an LEM is proposed when small length scales are involved. The FE,

circuit and lumped element models show good agreement with experimental results.

Furthermore, it has been shown that the error in LEM results increases as device

100

dimensions are scaled up. When the bimorphs are 173.2 µm long, the error in LEM was

less than 1.4%. However, when the bimorph beam length was scaled up to 450 µm, the

error in the LEM was greater than 20%. Hence, caution must be exercised while using

LEM. LEM accuracy can be increased by partitioning the geometry into smaller

segments. Rigorous condition for choosing the number of partitions is derived.

Figure 4-17. Comparison between transmission-line model and LEM in the 0– 300 Hz range. (A) Magnitude response. (B) Phase response.

Figure 4-18. Frequency response of micromirror. (A) Experimentally obtained frequency response. (B) Comparison between normalized frequency response data and the normalized value of rise in bimorph temperature predicted by the thermal model in the 0.005-50Hz frequency range.

140

120

100

80

60

40

20

0

10-3 10-2 10-1 100 101 102

Frequency (Hz)

Frequency (Hz)

Ma

gn

itud

e o

f Δ

Tb (

K)

Transmission-line model

LEM

10-3 10-2 10-1 100 101 102

0

-0.5

-1

-1.5

Ph

ase o

f Δ

Tb (

rad

ian

s)


LEM

(A) (B)

10-2 100 102

101

100

10-2 10-1 100 101

1.1

1

0.9

0.8

0.7

0.6

0.5

Frequency (Hz)

Frequency (Hz)

Op

tica

l an

gle

(d

eg

ree

s)

Norm

aliz

ed

da

ta

(A)

(B)

Normalized frequency response

Normalized bimorph temperature rise

101

In this chapter, the complete electrothermomechanical model for the static case

has been reported. However, it has been demonstrated that the transmission-line

thermal model is suitable for modeling both static and dynamic response. Transmission-

line thermal modeling provides a generic approach for a wide range of electrothermal

problems. The model presented in this chapter has a single heat source. If a part of a

device has an embedded distributed heat source, the theory of active transmission lines

may be utilized [90]. The transmission-line models for the different parts of the device

may be cascaded together to obtain the complete thermal model.

Since, [39, 79] also deal with LEM thermal models of a micromirror, a comparison

with the present work is in order. The LEM models reported in [39, 79] are easy to use.

However, they do not provide any rigorous bounds on the error and their error increases

as the device dimensions are scaled up. So, LEM models offer simplicity at the cost of

accuracy. On the other hand, FE models offer a high degree of accuracy at the cost of

computational resources. Moreover, they do not provide a direct relation between

device parameters and response. The transmission-line based thermal model reported

in this chapter provides the best of both worlds and offers rigor and computational

efficiency. The models developed in this chapter are parametric, i.e., the device

parameters can be varied. Therefore, it is useful for design and optimization. In this

chapter, the resistive heater was modeled as a localized heater at one end of the

actuator beams. In the next chapter, the transmission-line model will be extended to

account for distributed heat sources, such as a resistive heater embedded along the

entire bimorph length.

102

CHAPTER 5 TRANSMISSION-LINE THERMAL MODEL WITH DISTRIBUTED HEAT SOURCE

5.1 Background

In Chapter 4, heat flow is modeled as electrical signal flow through a passive

transmission-line. The resistive heater is treated as a localized heat source. In general,

the heat source may be distributed along the thermal transmission-line. For example,

the 1D mirror discussed in Chapter 3 has an embedded resistive heater along the

length of the bimorphs.

Distributed heat source for which the power density is uniform along the bimorph

length will be considered in Section 5.1 and validated using FE simulations. In the most

general case, the power density may vary along the length of the actuator due to

temperature dependence of the embedded resistive heater. Temperature-dependent

distributed resistive heater will be discussed in Section 5.2.

5.2 Transmission-line Model for Uniformly Distributed Heat Source

5.2.1 Governing Equations for 1D Heat Flow

Let us consider the 1D heat flow depicted in Figure 4-9A. Let the power dissipated

per unit length be ( )p η . Figure 5-1A and Figure 5-1B show the equivalent circuit model

of an element of length Δx0 and the corresponding ac model, respectively. In Figure

5-1A, the ambient temperature is denoted by Ta and is represented by a dc source.

Heat dissipation has been incorporated by adding a current source, ( )Δp η x . For the

circuit in Figure 5-1A, voltage and current represent temperature and heat flow,

respectively. In the circuit in Figure 5-1B, voltage represents temperature rise above the

ambient temperature. The other symbols have been defined in Section 4.3.4. The

analysis proceeds similar to that outlined in Chapter 4. In thermal transmission line

103

models, voltage change represents temperature change and current represents heat

flow.

Figure 5-1. Transmission-line model for a geometry with uniformly distributed heat source. (A) Circuit model for element of length Δx. (B) ac model.

Kirchhoff’s voltage law gives,

( , ) ( Δ ) ( , ) ( Δ , )v x η r x i x η v x x η

( , )

( , )v x η

ri x ηx

(5-1)

Kirchhoff’s current law at node N in Figure 5-1B gives,

rΔx rΔx

cΔx

1/(gΔx)

+

-

Ta

N

(A)

N

+

-

+

-

(B)

( )Δp η x

( )Δp η x cΔx 1/(gΔx)

rΔx rΔx

( , )v x η

( , )i x η

( Δ , )v x x η

( Δ , )i x x η

104

( Δ , )

( , ) ( Δ , ) ( )Δ ( Δ ) ( Δ , ) Δv x x η

i x η i x x η p η x g x v x x η c xη

( , ) ( , )

( , ) ( )i x η v x η

gv x η p η cx η

(5-2)

For harmonic variation,

( , ) Re( ( ) )jωηv x η V x e (5-3)

( , ) Re( ( ) )jωηi x η x e I (5-4)

( ) Re( )jωηp t Pe (5-5)

Equations 5-1 and 5-2 may be re-written as,

( )

( )dV x

r xdx

I (5-6)

( )

( ) ( )d x

g jωc V x Pdx

I

(5-7)


2

2

( )( ) ( )

d V xr g jωc V x rP

dx (5-8)

2

2

( )( ) ( ) 0

d xg jωc r x

dx

II (5-9)

The propagation constant is given by,

( )γ r g jωc (5-10)

From Equation 5-8,

2

( ) ( )γx γx PrV x e e

γ

1 2A A (5-11)

From Equations 5-1, 5-2, 5-3, 5-4, 5-5, 5-10 and 5-11,

( ) ( )γx γxγx e e

r

1 2A AI (5-12)

105

As discussed in Chapter 4, A1 and A2 can be determined from boundary conditions

at the ends of the 1D geometry. A1 and γ A1 / r are the magnitude of voltage and current

waves, respectively, propagating in the +ive x-direction. A2 and γ A2 / r are the

magnitude of voltage and current waves, respectively, propagating in the -ive x-

direction. The term 2/Pr γ in Equation 5-11 accounts for power dissipated in the

geometry. Let the length of the geometry be l.

0

2

( )(1 )( )

l

γl γl

average

V x dxlPr e e γ

Vl lγ

1 2A A

(5-13)

The average temperature rise along the 1D geometry is given by,

Δ Re[ ]jωη

average averageT V e (5-14)

Equation 5-11 can be used to determine the temperature rise at any point of a 1D

heat flow region. Equation 5-14 may be used to evaluate the rise in average actuator

temperature above the ambient temperature, Ta.

5.2.2 Application of Transmission-line Model to Electrothermal Micromirrors

The schematic shown in Figure 5-2 could represent a 1D micromirror, in which Z1

and Z2 correspond to thermal impedances at the mirror and substrate ends,

respectively. Voltage and current values have been indicated in Figure 5-2 based on

Equations 5-11 and 5-12.

From the voltage, current and impedances indicated in Figure 5-2,

2

1

/Pr γrZ

γ

1 2

1 2

A A

A A (5-15)

2

2

/b b

b b

γl γl

γl γl

e e Pr γrZ

γ e e

1 2

1 2

A A

A A (5-16)

106


2

22

( )

( )

b b

b

γl γl

γl

Pr e e

γ e

2 3

1

2 4 1 3

B BA

B B B B (5-17)

2

22

( )

( )

b

b

γl

γl

Pr e

γ e

4 1

2

2 4 1 3

B BA

B B B B (5-18)

Where,

1r Z γ 1B (5-19)

1r Z γ 2B (5-20)

2r Z γ 3B (5-21)

2r Z γ 4B (5-22)

Figure 5-2. Thermal impedances Z1 and Z2 at either ends of the transmission-line model

of a thermal bimorph. The bimorph length is lb. The voltages and currents at

both ends of the bimorph are also shown.

5.2.3 Simulation Results

In order to verify the transmission-line thermal model, a 1D electrothermal mirror

similar to that described in Chapter 3 is considered. It is assumed that there is no

thermal isolation between the mirror-plate and the actuators. All other device

Z1 Z2

Transmission-line model of bimorph

1M 2( )

PrV

γ

1 2A A

2M 2( )b bγl γl Pr

V e eγ

1 2A A

M1 M2

( )b bγl γlγe e

r

1 2A A( )

γ

r

1 2A A

107

parameters are chosen to be the same as those of the device shown in Figure 3-1.

Figure 5-3 and Figure 5-4 compare transmission-line model results and FE simulations

for the mirror kept in vacuum and air, respectively.

Figure 5-3. Average bimorph temperature for mirror placed in vacuum. A 28 mW sinusoidally varying heat source is uniformly distributed along the length of the bimorphs. The mirror topology and dimensions are the same as that of the device described in Chapter 3. The only difference is that the simulated design does not have any SiO2 thermal isolation between the actuators and the mirror-plate. (A) Magnitude plot. (B) Phase plot.

At low frequencies, Figure 5-3 and Figure 5-4 show good agreement between FE

and transmission-line models. The error at low frequencies may be attributed to the fact

that the transmission-line model assumes one-dimensional heat flow, but in practice the

200

150

100

50

0 10-4 10-2 100 102 104 106 108

Frequency (Hz)

FE model


Tem

pe

ratu

re r

ise (

K)

(A)

0

-0.5

-1.0

-1.5

-2.0

10-4 10-2 100 102 104 106 108

FE model


Frequency (Hz)

Ph

ase

(ra

dia

ns)

(B)

108

heat flow is three-dimensional. Beyond 104 Hz, there is a significant discrepancy

between transmission-line model and FE simulations in the phase-plots shown in Figure

5-3B and Figure 5-4B. This discrepancy may be attributed to the fact that the FE

simulation accuracy is limited by its mesh density. However, discrepancies at high

frequencies do not pose a problem as device response is practically zero beyond

104 Hz.

Figure 5-4. Average bimorph temperature for mirror placed in air. A 28 mW sinusoidally varying heat source is uniformly distributed along the length of the bimorphs. The heat loss coefficients on the actuators and mirror-plate are assumed to be 200 Wm-2K-1 and 50 Wm-2K-1, respectively. The mirror topology and dimensions are the same as that of the device described in Chapter 3. The only difference is that the simulated design does not have any SiO2 thermal isolation between the actuators and the mirror-plate. (A) Magnitude plot. (B) Phase plot.

10-4 10-2 100 102 104 106 108

100

80

60

40

20

0

Frequency (Hz)

FE model


Tem

pe

ratu

re r

ise (

K)

10-4 10-2 100 102 104 106 108

Frequency (Hz)

FE model


0

-0.5

-1

-1.5

-2

Pha

se

(ra

dia

ns)

(A)

(B)

109

The method outlined in this section is applicable only when the temperature

dependence of the embedded resistive heater is negligible. Temperature dependence

of resistance may be significant in many practical problems. Pt, which is used as an

embedded resistive heater in several mirror designs, has one of the highest TCR values

among MEMS materials [91]. The next section will address the dependence of

embedded heater resistance on temperature.

5.3 Distributed Temperature Dependent Resistive Heater in One-dimensional Heat Flow Region

Let us consider the one-dimensional heat flow region shown in Figure 4-9A. Let a

distributed temperature dependent resistor be embedded along this geometry. Let the

power per unit length be,

2

0( , ) ( ) (1 Δ ( , ))hq x η A J η α T x η ρ (5-23)

where, Ah is the heater cross-section area; ( )J η is the time dependent electric current

density; ρ0 and α are the resistivity and temperature coefficient of resistance of the

embedded heater, respectively, at temperature Ta ; and ΔT is the rise in temperature

above the ambient temperature, Ta. The distance along the geometry and time are

denoted by x and η, respectively.

From (5-23),

( , ) ( ) ( ( )( ( , ) ))aq x η p η α p η T x η T (5-24)

where,

2

0( ) ( ( ))hp η A J η ρ (5-25)

The first term in Equation 5-24 is represented by a current source in the equivalent

circuit model shown in Figure 5-5. The second term in Equation 5-24 can be

110

represented by a time dependent resistor as shown in Figure 5-5A. The ac equivalent

circuit is shown in Figure 5-5B. For the circuit in Figure 5-5A, voltage and current

represent temperature and heat flow, respectively. In the circuit in Figure 5-5B, voltage

represents temperature rise above the ambient.

Figure 5-5. Equivalent circuit of an element of length Δx of a one-dimensional heat flow region. The TCR of the distributed embedded resistor is α. (A) Circuit model in which the ambient temperature, Ta, is modeled as a dc source. (B) ac

model.

rΔx rΔx

1/(gΔx)

cΔx

+

-

Ta

N

rΔx rΔx

cΔx

1/(gΔx)

1

( )Δαp η x

N

+

-

+

-

(A) (B)

( )Δp η x

( , )v x η

( , )i x η

( Δ , )v x x η

( Δ , )i x x η

1

( )Δαp η x

( )Δp η x

111

The time dependent resistor in Figure 5-5 obscures an analytical solution for

voltage and current. Such time varying circuit elements lead to generation of harmonics

in a circuit [92]. Therefore, the analysis procedure used in Chapter 4 and Section 5.2

cannot be readily adapted. However, advanced SPICE programs that support current-

controlled resistors may be used to obtain numerical solutions. For obtaining a

numerical solution, the geometry may be divided into a finite number of segments and

each segment may be represented as a circuit of the form shown in Figure 5-5. The

time dependent resistor can be expressed in terms of the ( )Δp η x current source.


This chapter extends the transmission-line thermal model discussed in Chapter 4

to account for distributed resistive heater embedded along the geometry. If the

temperature dependence of the embedded resistor is negligible, it is possible to obtain

closed-form expression for temperature distribution. The analytical expressions have

been validated against FE simulations. If the temperature dependence of the embedded

heater cannot be neglected, it is possible to use the transmission-line approach to

obtain numerical solutions. However, the presence of a time-dependent resistor in the

transmission line model obscures a closed-form solution in this case. Mechanical

modeling will be discussed in the next chapter.

112

CHAPTER 6 MECHANICAL MODEL OF ELECTROTHERMAL MICROMIRRORS

6.1 Background

The mechanical model of the micromirror is an essential component of the

complete ETM model shown in Figure 1-3. The mechanical model takes the actuator

temperature predicted by the thermal model as input and evaluates the motion of the

mirror-plate. As discussed in Section 4.2.2, in the case of 1D mirrors, the output of the

mechanical model is the mirror rotation angle. The output of a 2D mirror includes

rotation about two orthogonal axes. In case of mirrors reported in [34, 36], the output of

the mechanical model consists of three quantities — the displacement of the mirror-

plate perpendicular to the substrate (z) and the mirror rotation angle along two

orthogonal axes (θx and θy), as schematically represented in Figure 6-1. Due to their

three degrees of freedom, such mirrors are also called tip-tilt-piston (TTP) micromirrors.

In general, the mirror-plate of a micromirror device has some lateral shift in the x and y

directions. However, the lateral shift is negligible in case of the designs reported in [34,

36]. For instance, the micromirror design based on lateral-shift-free (LSF) actuators has

a lateral shift of 10 μm for a vertical displacement of 0.62 mm. Low lateral shift greatly

simplifies optical system design.

The mechanics of thermal bimorph actuators is central to the development of the

complete mechanical model of the micromirror. Static bimorph mechanics has been

briefly discussed in Section 1.2.1. The next section will deal with bimorph actuation.

Section 6.3 will discuss the optimization of ISC multimorph actuators. Section 6.4

outlines the development of mechanical models based on Newtonian method and

energy method.

113

Figure 6-1. Schematic showing the three degrees of freedom of a 3D micromirror. (A) displacement along z-axis. (B) rotation about x axis. (C) rotation about y axis.

6.2 Mechanics of Bimorph Actuators

Timoshenko’s Analysis of bi-metal thermostats is a classic text on the mechanics

of singly and doubly clamped bimetal strips [10]. Since bimetal strips and thermal

bimorph actuators work on the same principle, the equations derived in [10] are valid for

thermal bimorph actuators as well. Several papers have extended the theory of

bimorphs to multimorph actuators [93, 94]. The theory of multimorphs is useful to MEMS

engineers for two reasons. Firstly, most fabrication processes involve several layers of

thin film depositions and one often ends up with multimorphs after the fabrication

process. For instance, the structure shown in Figure 4-3B consists of 4 distinct thin-film

layers. Secondly, the general theory of multimorphs is applicable to a wide range of

devices such as electrothermal, piezoelectric and SMA (Shape Memory Alloy) actuated

MEMS [11]. Weinberg provides a comprehensive treatment of multimorphs and the key

results have been reproduced in the remainder of this section [11]. Figure 6-2 shows the

x

z

y

x

z

y

x y

z

(A) (B) (C)

Substrate

Mirror-plate

Displaced mirror-plate

Δθx

Δθy

Δz

114

schematic of a multimorph. A moment Mm, a transverse force F1m and an axial force F2m

have been shown at the free end. The bending moment may be external or generated

internally due to residual stresses in the thin films. The micromirrors shown in Figure 3-1

and Figure 4-1 are curled up on release due to the moment produced by residual

stresses. The position of the neutral axis of the i th layer with respect to an arbitrary

reference is zi. The CTE, Young’s modulus, area, and area moment of inertia of the i th

layer are λi, Ei, Ai and Ii , respectively.

Figure 6-2. Schematic of a multimorph (adapted from [11]).

Let the beam width be comparable to its thickness, so that the plane-stress

approximation is valid. If the multimorph temperature changes by ΔT, the curvature of

the multimorph, Cm, is given by [11],

2

22

( Δ ) ( Δ )1

( )

m i i i i i i m i i i i i ii i i im

mi i i i i i i i ii i i

z E A λ T E A E A λ T z E A

E A E A z z E A

M FC

R I

(6-1)

In Equation 6-1, Rm denotes the radius of curvature of the multimorph.

6281.72

zi

i th layer

θm

F1m F

2m

Mm

z

x

115

6.3 Optimization of the ISC Multimorph Actuators

The optimization and fabrication of the inverted-series-connected (ISC) actuator,

and micromirrors actuated by ISC actuators are described in [36, 95, 96]. A major

contribution of this thesis has been the successful optimization of an ISC actuator based

micromirror design [36]. The key component of the ISC actuator is a series connection

of two multimorphs. These two multimorphs will be referred to as non-inverted and

inverted multimorphs. The thin-film structures of the two multimorphs have been shown

in Figure 6-3.

Figure 6-3. Thin film structure of (A) non-inverted and (B) inverted multimorphs. The numbers on the side represent the thickness of the thin films in μm.

Upon heating, the non-inverted multimorph curls down and the inverted

multimorph curls up. Consequently, a series connection of the two multimorphs may be

1

0.1

0.05 0.2

1.1

1

0.05

1

0.1

0.05

0.2

0.05 0.1

Al

SiO2

Cr

Pt

(A) (B)

116

used to achieve tilt-free motion. This has been schematically shown in Figure 6-4. Let

the lengths of the inverted and non-inverted multimorphs be denoted by linverted and

lnon-inverted , respectively. The optimal ratio of linverted and lnon-inverted, roptimal, that achieves tilt

free motion has been determined by FE simulations. As shown in Figure 6-4A, when the

ratio of linverted and lnon-inverted is less than roptimal, the inverted multimorph is unable to

compensate the tilt completely. On the other hand, Figure 6-4B shows that when the

ratio of linverted and lnon-inverted is greater than roptimal, the inverted multimorph

overcompensates the tilt. The optimal configuration corresponding to zero tilt at the

actuator tip has been shown in Figure 6-4C. Assuming static actuation and neglecting

the effect of gravity on the mirror, the terms Mm and F2m are set to zero in Equation 6-1.

Let the multimorphs be initially flat at a reference temperature. Let the temperature of

the inverted and non-inverted multimorphs change uniformly by ΔT. From FE

simulations,

2.2invertedoptimal

non inverted

l

l

r (6-2)

The optimal ratio in Equation 6-2 was verified by using the analytical expression of

multimorph curvature given by Equation 6-1. The uniform temperature assumption is not

strictly valid as the temperature may be different at different points on the beams.

However, since the embedded heater runs along the length of the ISC actuator and

there is SiO2 thermal isolation at either ends, the uniform temperature approximation

may be used. Let the curvature of the inverted and non-inverted beams be Rm-inverted and

Rm-non-inverted, respectively. The bending of the inverted and non-inverted beams cancel

each other if,

117

inverted non inverted

m inverted m non inverted

l l

R R

(6-3)

The effect of residual stresses and non-uniform temperature distribution requires

further investigation. Optimization of the ISC actuator design has resulted in more than

ten-fold improvement in mirror scan angle and vertical displacement range [36, 96].

Figure 6-4. Series connection of a non-inverted and an inverted actuator. The detailed thin film structure has been shown in Figure 6-3. (A) The tilt is undercompensated. (B) The tilt is overcompensated. (C) When the ratio of lengths of the inverted and non-inverted multimorphs is optimal, the actuator tip executes tilt free motion.

6.4 Mechanical Model of Micromirror

In this section, the mechanical model of an idealized 1D mirror structure, in which

the bimorph has a uniform temperature, is derived. This assumption is approximately

true for the 1D mirror shown in Figure 3-1 and may be verified from the thermal image

shown in Figure 3-6B. In general, the temperature may vary along a bimorph actuator

and must be taken into account. However, the procedure for building the model will be

essentially the same as that outlined in this section. The next subsection will present

Non-inverted multimorph

Inverted multimorph

(A)

Actuator tip is parallel to substrate

(C)

(B)

118

mirror modeling by Newtonian method. The energy method will be introduced in

Section 6.4.2.

6.4.1 Newtonian Method

Figure 6-5 shows an idealized schematic of a 1D mirror. The output of the

mechanical model must determine the angle θ.

Figure 6-5. Schematic of mechanical model for 1D mirror. The mirror-plate has been approximated as a 1D geometry of mass m and length lm. The bimorph of

length lb is assumed to act like an ideal spring.

The typical thickness of a multimorph actuator is ~2 μm. The thickness of the

mirror-plate is much larger than that of the actuators, typically 20–40 μm. Therefore, the

mirror-plate may be regarded as a rigid body. Also, the mirror-plate thickness is much

less than its length. Therefore, as shown in Figure 6-5, the mirror-plate has been

represented by a line. The lengths of the actuator and mirror-plate have been denoted

by lb and lm, respectively. The radius of curvature of the actuator is Rm. The point

(xcm,ycm) denotes the center of mass of the mirror-plate.

lm

lb

θ

Rm (xcm,ycm)

x

y

119

Figure 6-6 shows the free body diagrams of the actuator and the mirror-plate. The

actuator mechanics is described by Equation 6-1. The motion of the mirror-plate may be

determined by rigid body mechanics [97].

Figure 6-6. Free body diagram of (A) actuator and (B) mirror-plate. Simulations show that the effect of gravity is negligible for a mirror-plate size of ~1 mm. Therefore, gravitational force has been neglected in the free body diagrams. For heavier mirror-plates, the initial elevation angle may shift due to the orientation dependent gravitational force. However, the device response will not be affected significantly.

6.4.2 Energy Method

The energy method described in [98] may be applied for building the mechanical

model. Since the mirror-plate is significantly thicker than the actuator, it will be assumed

that the kinetic energy associated with actuator motion is negligible. The actuator acts

like a spring and therefore stores potential energy in the deformed state.

Mm

F1m F2m

Mm F2m

(A) (B)

(xcm,ycm)

F1m

120

6.4.2.1 Evaluation of kinetic energy

Let the mass of the mirror-plate be m. The moment of inertia of mirror-plate about

its center of mass is given by,

2

12m

cm

lm

I (6-4)

The coordinates of the center of mass of the mirror-plate are given by,

( , ) sin cos , (1 cos ) sin2 2

b bm mcm cm

l ll lx y θ θ θ θ

θ θ (6-5)

The magnitude of the velocity of the center of mass is given by,

2 2

cm cm cmx y v (6-6)

When the mirror is in motion, its kinetic energy is given by [97],

2 21 1

2 2KE cm cmθ E mvI (6-7)

From Equations 6-4 through 6-7,

2( )KE θ θ FE (6-8)

where,

2

2 2 2 4 2

4( ) 8 4 ( ) 4 ((2 )cos 2 sin )

8 24m

b b b m m b b m b

lθ l l l l θ l θ l l l θ θ l θ θ

θ

Fmm

(6-9)

6.4.2.2 Evaluation of potential energy

The potential stored in the actuator may be evaluated by the stress distribution

given in [10]. In this section, it will be assumed that the actuator consists of 2 layers of

thin films. The radius of curvature is Rm. Figure 6-7 shows the stress distribution in layer

1, i.e., the top layer. The diagram has been drawn assuming that the top layer is under

compressive stress. The thicknesses of the top and bottom layers have been denoted

by t1 and t2, respectively.

121

Figure 6-7. Stress distribution in a bimorph.

The stresses max,1 and min,1 are given by [10],

3 3 2

1 1 2 2 1 1 1 2,1

1 1 2

3 ( )1

6 ( )max

m

E t E t t E t tζ

t t t

R (6-10)

3 3 2

1 1 2 2 1 1 1 2,1

1 1 2

3 ( )1

6 ( )min

m

E t E t t E t tζ

t t t

R (6-11)

Similar expressions can be obtained for the stresses in the bottom layer, i.e., max,2 and

min,2,

3 3 2

1 1 2 2 2 2 1 2,2

2 1 2

3 ( )1

6 ( )max

m

E t E t t E t tζ

t t t

R (6-12)

3 3 2

1 1 2 2 2 2 1 2,2

2 1 2

3 ( )1

6 ( )min

m

E t E t t E t tζ

t t t

R (6-13)

The strain distribution for the top layer is given by,

11( )

m m

Kyε y

R R (6-14)

where,

3 3 2

1 1 2 2 1 1 1 21

1 1 1 2

3 ( )

6 ( )

E t E t t E t tK

E t t t (6-15)

Similarly, the strain distribution in layer 2 may also be evaluated.

t1

t2

|max,1|

|min,1|

x

y

122

The elastic energy in the top layer of the bimorph is given by,

2 2

1 1 1 1 11 1 2

( 3 3 )1

2 3PE b

m

t t t K KE wl

E

R (6-16)

A similar expression may be found for the elastic energy stored in the bottom layer. Let,

3 3 2

1 1 2 2 2 2 1 22

2 2 1 2

3 ( )

6 ( )

E t E t t E t tK

E t t t (6-17)

The total potential energy is given by the sum of elastic energy stored in the top and

bottom layers,

2

PE PEθ FE (6-18)

where,

2 2 2 2

1 1 1 1 1 2 2 2 2 21 2

( 3 3 ) ( 3 3 )1 1

2 3 2 3PE

b b

t t t K K t t t K KE w E w

l l

F (6-19)

The Lagrangian is given by [98],

2 2( )KE PE PEL θ θ θ F FE E (6-20)

If the mirror oscillates with a small amplitude about θbias,

( )KE biasθF F (6-21)

where, F(θ) is given by Equation 6-9.

2 2

KE PEL θ θ F F (6-22)

For free oscillations Lagrange’s equation gives,

,

10 0

2PE

KE PE res undamped

KE

d L Lθ θ f

dη θ πθ

FF F

F (6-23)

Let FThermal and Fdamping denote the thermal and damping forces. For forced oscillations,

the equation governing mirror motion is given by,

123

thermal damping

d L L

dη θθ

F F (6-24)

For small oscillations about a bias point,

ΔKE damping PE Thermalθ θ θ T F C F C (6-25)

The constant CThermal may be evaluated from the steady state device characteristic.

6.5 Summary

The mechanics of multimorph actuators is well documented in literature and has

been reviewed in this chapter. Optimization of the multimorph structure in ISC actuators

has resulted in a ten-fold improvement in the mirror scan angle and vertical

displacement range. The mechanical model of electrothermal micromirrors is a key

component of the complete ETM schematic shown in Figure 1-3. Newtonian and

Lagrangian methods for deriving mechanical models have been outlined. It has been

shown that for small oscillations, the mechanical model of a 1D mirror can be

represented as a second order spring-mass-damper system. Analytical expression for

mirror resonant frequency has been derived.

124

CHAPTER 7 COMPREHENSIVE ELECTROTHERMOMECHANICAL MODEL OF MICROMIRRORS

7.1 Background

Several electrothermal bimorph based designs have been reported in literature.

However, complete dynamic models have not been reported. This chapter builds upon

the material presented in Chapters 1–6 and discusses the development of dynamic

electro-thermo-mechanical (ETM) models of thermal bimorph based 1D micromirrors.

The schematic of an electrothermal mirror model has been shown in Figure 1-3. Model

based open-loop mirror control is discussed in Section 7.2. This work was done as part

of a project on mirror-based dental imaging with Lantis Laser Inc. [9] and has been

reported in [40, 41]. Section 7.3 presents a dynamic Simulink model of a 1D

micromirror.

7.2 Model-based Open-loop Control

This section deals with open-loop drive methods that minimize nonlinearity by

using special input waveforms. A procedure for open-loop control of electrothermal

MEMS is established and demonstrated by using a thermal bimorph based one-

dimensional (1D) electrothermal micromirror.

In many applications, a constant angular velocity or constant linear velocity scan

range is desirable. In case of constant angular velocity scanning, the angle of the light

beam reflected from the mirror traverses equal angles in equal intervals of time. In case

of constant linear velocity scanning, the reflected light traverses equal distances in

equal intervals of time over a target surface. Constant velocity scanning has two major

advantages. Firstly, it greatly simplifies signal acquisition in imaging systems. Secondly,

it ensures that the pixels in the constant velocity range have a uniform resolution. Zara

125

et al. suggest that the nonlinear portion of the scan range of a micromirror may be

truncated, but this greatly reduces the scan range [99]. Another solution is to correct for

the nonlinearity in the output by signal processing, but this makes the system more

complex. Also, any nonlinearity in mirror motion leads to non-uniform pixel resolution.

Both open-loop and closed-loop MEMS control for achieving linear response have

been reported in literature [100]. The advantages of closed-loop control such as

reduction in system error and improvement of stability and sensitivity are well

known [101]. Unlike macroscopic systems, the closed-loop control of MEMS is often

complicated due to limited availability of sensor data, fast actuator dynamics, and

noise [100]. Additionally, fabrication variations often culminate into significant

performance variations among MEMS devices. For instance, the scan angle of an

electrothermal micromirror may vary by a factor of ten due to process variations

alone [19]. Closed-loop control can be used to compensate large variations. In some

cases, feedback may be implemented without considerably increasing system

complexity. Blecke et al. report a closed-loop system whose feedback only consists of

an on-off switch [102]. However, in most cases feedback increases system complexity,

size and cost significantly. For instance, bulky optical systems may be used for

feedback [100-102]. Messenger et al. report the fabrication of a thermomechanical

actuator integrated with a piezoresistive sensor for feedback [103]. The inclusion of the

piezoresistor and associated bond-pads trades off miniaturization for performance.

The key advantage of open-loop drive is simplicity. Open-loop control may be

implemented without increasing system cost and size. Borovic et al. report the open-

loop control of a MEMS variable optical attenuator [100]. Since a large class of MEMS

126

devices are underdamped, the input shaping procedure reported in [104] may be used

for reducing vibrations. Input shaping has been applied to the open-loop control of

electrostatic torsional micromirrors [105] and hot-arm cold-arm type thermal

actuators [106]. In [40, 107], constant linear and angular scanning have been reported.

Empirically derived custom waveforms were used to compensate for device nonlinearity

and achieve the desired scanning profile.

In this section, a formal procedure is developed for deriving custom actuation

waveforms for achieving desired performance. Next, it is demonstrated that the

continuous custom waveform is equivalent to a pulse width modulated (PWM) drive. In

case of electrothermal micromirror arrays which may have as many as 64 signal

lines [108], PWM can greatly reduce system cost and size.

The topology of the electrothermal mirror used for the experiments in this section

is the same as that depicted in Figure 4-2. For the device under test, the static mirror-

deflection angle can be predicted by setting Λ = 0.17/K in Equation 4-1.

Section 7.2.1 provides theoretical background. The procedure for custom

waveform generation is outlined in Section 7.2.2. Section 7.2.3 presents experimental

results on continuous wave and PWM actuation.

7.2.1 Theoretical Background

Figure 7-1 shows a schematic of a 1D electrothermal micromirror model. The

applied voltage, VE , results in power dissipation, p, in the embedded heater with

resistance, RE. The thermal model predicts the heater temperature, Th, and average

bimorph temperature, Tb. The mechanical model predicts the mirror rotation, θ.

127

Figure 7-1. Schematic of a complete model of an electrothermally-actuated micromirror.

η = time, VE(η) = applied voltage, p(η) = power dissipated by Joule heating,

Tb(η) = bimorph actuator temperature, θ = mirror rotation angle,

Th(η) = temperature of embedded resistive heater and RE(Th) = resistance of

embedded heater.

Let P(s), Tb(s), Th(s) and Θ(s) denote the power, average bimorph temperature,

heater temperature and mirror rotation in the frequency domain. From Chapters 3–5, it

can be inferred that a typical micromirror thermal model consists of a few poles and

zeros. Therefore, the relationship between temperature and power, p, can be

represented by a transfer function. Also, as discussed in Chapter 6, the bimorph and the

mirror-plate are analogous to a spring and mass, respectively. Therefore, the

mechanical model may also be represented by a simple transfer function. Let us define

the transfer functions G1(s), G2(s) and G3(s) as,

1

( )( )

( )bT s

G sP s

(7-1)

2

( )( )

( )hT s

G sP s

(7-2)

3

Θ( )( )

( )b

sG s

T s (7-3)


2( )( )

( )E

E h

V ηp t

R T

Thermal Model

( )E hR T =Temperature

dependent resistance of Pt heater

( )p η( )EV η ( )bT η

( )hT η

Mechanical Model ( )θ η

128

1 3

Θ( )( )

( ) ( )

sP s

G s G s (7-4)

Therefore, for a desired scan angle profile, the input power may be determined by

using Equation 7-4. Typically, θ(η) for a scanning mirror is a periodic function which can

be easily expanded in a Fourier series. Equation 7-4 may then be used to evaluate the

power signal corresponding to each frequency component. The components may be

added to obtain the input power signal. Equation 7-2 may be used to evaluate the

heater temperature, Th(η), for a certain power input, p(η). The heater resistance, RE, is

given by Equation 4-3. For the device under test, the TCR, α = 0.0025 K-1 and the

reference temperature,T0, is chosen to be 298 K.

Finally, the desired voltage signal is given by,

( ) ( ) ( )E EV η p η R η (7-5)

Since, most electrothermal devices can be represented by a schematic similar to

the one shown in Figure 7-1, the procedure outlined in this section is widely applicable.

Next, the principles described in this subsection will be applied to micromirror control.

7.2.2 Linear Scanning by Open-loop Control

The key steps involved in determining the custom actuation signal have been

outlined below.


In order to determine the angular range and establish the safe voltage limit, the

static device characteristic was experimentally obtained by using a dc source. The

voltage, current and scan angle were monitored. Figure 7-2A and Figure 7-2B show the

optical angle vs. voltage and optical angle vs. input power, respectively. From Equation

129

7-4, G1(0)G3(0)=Θ(0)/P(0). Therefore, the linear angle-power relationship in Figure 7-2B

is in agreement with the use of linear transfer functions G1(s) and G3(s).

Figure 7-2. Static characteristic. (A) Optical angle vs. applied dc voltage. (B) Optical

angle vs. input power.

7.2.2.2 Dynamic characterization

A dc voltage of 5.01 V was used to bias the mirror in the linear region of the angle

vs. voltage plot shown in Figure 7-2A. A 150 mV amplitude sine wave superimposed on

the dc bias was used to obtain the mirror response. This approximately corresponds to

a sinusoidal input power wave of 8.545 mW amplitude. Figure 7-3 shows the frequency

response. Figure 7-3 may be used to evaluate the transfer function with power wave as

input and angular motion as output, i.e., G1(s)G3(s).

The fitted model shown in Figure 7-3 corresponds to the transfer function given by,

11 3 1 2 2

1

( ) 1( ) ( )

( ) ( 2 )z

p n n

sG s G s K

s s ςω s ω

r

r (7-6)

where, K1 = 7.56/ W, rz1 = 31.4 rad/s, rp1 = 11.3 rad/s, ωn = 1451.4 rad/s and

ς = 0.0084.

25

20

15

10

5

0

0 2 4 6 8 10 0 100 200 300 400

Optical an

gle

(degre

es)

25

20

15

10

5

0

Voltage (V) Input power (mW)

(A) (B)

Optical an

gle

(degre

es)

130

Figure 7-3. Micromirror frequency response and fitted model.

The zero, rz1, and pole, rp1, correspond to the thermal response of the device.

From the reduced order model of an electrothermal micromirror [39], it is expected that

there will be two poles and a zero. In fact, harmonic FE thermal simulations of the

device indeed show that there are two poles and a zero. However, one of the poles is

close to the mechanical resonant frequency of the mirror and is therefore not apparent

in Figure 7-3. Mechanical FE simulations show that the mirror has only one resonant

mode in the operation frequency range. The constants ς and ωn are the damping ratio

and the natural resonance frequency of the mechanical system.

7.2.2.3 Determination of G2(s)

A thermal FE model was built and simulated in COMSOL [58]. A harmonic input

power of 0.01 W was used for simulations. Figure 7-4 shows the simulated heater

temperature Th(s) along with a fitted model.

Experimental results

Fitted model

10-1 100 101 102 103

Frequency (Hz)

Optical an

gle

scan r

an

ge (

degre

es)

102

101

100

10-1

10-2

131

Figure 7-4. Temperature of embedded heater. (A) Magnitude and (B) Phase of Th for an

input power of 0.01 W.

From the fitted model,

22 2

2 3

( )( )( )

( ) ( )( )z

p p

sT sG s K

P s s s

r

r r (7-7)

where, K2 = 2.93106 K / W, rz2 = 4.83 rad/s, rp2 = 4.65 rad/s and rp3 = 13823 rad/s.

2.3

2.2

2.1

2

1.9

1.8

FE results

Fitted model

10-2 10-1 100 101 102 103 104

Frequency (Hz)

Magn

itu

de o

f T

h (

K)

FE results

Fitted model

10-2 10-1 100 101 102 103 104

Frequency (Hz)

0

-0.05

-0.1

-0.15

-0.2

-0.25

Phase o

f T

h (

radia

ns)

(A) (B)

132

7.2.2.4 Fourier series expansion of desired output

Figure 7-5A shows an ideal periodic ramp output. The mirror-plate has some

inertia and therefore requires a finite time interval to switch directions. Thus, in practice

the mirror output will be rounded at the corners. Figure 7-5B shows the Fourier series

approximation of the ramp waveform. All frequency components greater than 50 Hz

have been truncated. This eliminates frequencies close to mirror resonance which may

cause unwanted oscillations. Truncating the Fourier series also serves to round off the

corners of the ramp waveform.

Figure 7-5. One period of optical scan angle vs. time. (A) Ideal ramp waveform, and (B)

Fourier series approximation.

7.2.2.5 Evaluation of voltage input

The frequency domain representation of the ramp output can be used along with

Equation 7-4 to evaluate the input power waveform. Thereafter, Equations 7-2, 4-3 and

7-5 may be used to obtain the voltage input. Figure 7-6A shows the input power. Figure

7-6B shows the voltage input corresponding to constant angular velocity actuation.

In the next subsection, the principle of PWM actuation will be presented.

0 0.02 0.04 0.06 0.08 1 0 0.02 0.04 0.06 0.08 1

10

9

8

7

6

5

4

10

9

8

7

6

5

4

(A) (B)

Optical an

gle

(degre

es)

Optical an

gle

(degre

es)

Time (s) Time (s)

133

7.2.2.6 Pulse width modulation

Let us consider the analog voltage actuation signal shown in Figure 7-7A. Let the

PWM waveform shown in Figure 7-7B be equivalent to this analog signal in the interval

Δη.

Figure 7-6. One period of the evaluated input waveform. (A) Power, and (B) Voltage

waveforms for achieving linear scan.

Figure 7-7. PWM representation of continuous waveform. (A) Continuous waveform. (B)

Equivalent PWM.

In the interval Δη, the PWM signal delivers the same amount of energy to the

micromirror as the analog signal. Moreover, since the two waveforms are equivalent,

they result in identical temperature distribution in the embedded Pt heater. It is assumed

vavg

vpulse

Δη

Δηon

Δηoff

time

time

(A) (B)

0 0.02 0.04 0.06 0.08 1 0 0.02 0.04 0.06 0.08 1

200

150

100

50

0

7

6

5

4

3

2

1

0

Time (s) Time (s)

Input

pow

er

(mW

)

(degre

es)

Voltag

e (

V)

(A) (B)

134

that Δη is sufficiently small, so that the analog voltage and the embedded heater

resistance are approximately constant over Δη. Equating the energy delivered to the

device by the analog waveform and its equivalent PWM representation,

2 2 2

ΔΔ Δ

( ) ( ) ( )on

ηE h E h E h

v v vdη η η

R T R T R T

analog avg pulse (7-8)

where, vanalog is the analog voltage, vavg is the average voltage of the analog waveform in

the interval Δη, vpulse is the voltage amplitude of the pulse waveform, Th is the embedded

heater temperature, RE(Th) is the temperature dependent resistance of the embedded

heater. Hence,

2

Δ Δavg

on

pulse

vη η

v

(7-9)

Equation 7-9 may be used to evaluate the equivalent PWM waveform for any

continuous signal. It must be ensured that the maximum value of Δηon is less than the

device response time. Experimental results will be discussed in the following section.

7.2.3 Experimental Results

7.2.3.1 Constant linear velocity scan

The 1D micromirror reported in [39] was used for obtaining a constant linear scan

profile. Figure 7-8A shows the 10 Hz actuation waveform that was generated by using a

Tektronix AFG3102 arbitrary function generator. The reflected laser beam was tracked

by using the ON-TRACK PSM 2-10 position sensing module (PSM) [42]. Figure 7-8B

shows the constant linear velocity scan profile of the light spot incident on the PSM

screen. The linear scan range is around 90% [107]. The profile shown in Figure 7-8B

corresponds to an optical scan range of ±3º.

135

Figure 7-8. Constant linear velocity scan. (A) 10 Hz custom actuation waveform. (B)

Constant linear velocity scan profile obtained using a PSM.

7.2.3.2 Constant angular velocity scan

Figure 7-9A and Figure 7-9B show a 10 Hz actuation waveform and the

corresponding optical scan angle, respectively. Clearly, the optical scan angle is linear

over a wide range, i.e., the velocity is constant in that range. Over 75% of a time period,

the optical angle is linear to within 6%. For simplicity of implementation, a symmetric

voltage waveform was used. Further improvement in linearity may be obtained by using

an unsymmetrical waveform similar to the one shown in Figure 7-6B.

In order to obtain the equivalent PWM representation of the waveform shown in

Figure 7-9A, a single period of the waveform was partitioned into 800 subintervals. The

on-time of the equivalent pulses was determined using Equation 7-9 for vpulse = 7 V.

Figure 7-10A shows a section of the PWM waveform. Figure 7-10B shows the scan

profile obtained by using the PWM actuation signal. For an actuation signal of 10 Hz,

0 0.05 0.1 0.15 0.2 0.25 0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

10

9

8

7

6

5

4

Time (s)

Time (s)

Actu

ation

voltage

(V

)

6

4

2

0

-2

-4

-6

-8

PS

M o

utp

ut

(V)

(A) (B)

136

each subinterval is 0.125 ms long, i.e., Δηon+ Δηoff = 0.125 ms for each pulse. Since the

resonant frequency of the mirror is 231 Hz, its response time is estimated to be

(1/231) s, i.e., 4.3 ms. Hence, the pulse widths are at least one order of magnitude

smaller than the device response time. For the experiments, the pulses were generated

by using the Tektronix AFG3102 function generator followed by an emitter-follower

stage. The emitter-follower stage ensures that change in load resistance does not affect

the pulse amplitude. Since the mirror is underdamped, pulsed excitation leads to ripples

in the mirror output as depicted in Figure 7-10. It is possible to minimize the ripples by

using notch filters to eliminate frequencies that excite mirror resonance.

Figure 7-9. Constant angular velocity scan. (A) 10Hz Actuation voltage for obtaining

constant angular velocity scan profile. (B) Optical scan angle versus time. The linear portion corresponds to the constant scan velocity range.

In this section, experiments, FE modeling and experience gathered from mirror

modeling have been used to obtain a simple dynamic model of an electrothermal

micromirror. Voltage waveforms that achieve constant linear and angular velocity scan

were then derived from the device model. The next section discusses a Simulink model

of a 1D micromirror.

0 0.05 0.1 0.15 0.2 0.25

0 0.05 0.1 0.15 0.2 0.25

V

olta

ge (

V)

Optical an

gle

(degre

es)

6

4

2

0

12

8

4

Time (s)

Time (s)

(A) (B)

137

Figure 7-10. Constant angular velocity scan by PWM actuation. (A) A section of the

PWM waveform. (B) Large constant angular scan velocity range obtained using PWM actuation.

7.3 Electrothermomechanical Model Implemented in Simulink

In this section, the dynamic ETM Simulink model of a 1D micromirror is discussed.

The model takes a periodic voltage waveform as input and predicts the device response

as output. The transmission-line approach discussed in Chapter 4 is used to build the

thermal model. The mechanical model is represented as a second order system as

discussed in Section 7.2. Before presenting the ETM model, Fourier series will be

reviewed briefly.

7.3.1 Evaluation of Fourier Series Coefficients in MATLAB/Simulink

A periodic function can be represented in the form of a Fourier series [109]. Let us

consider a periodic function f p(η) of period p. Let the Fourier series representation of f p(η)

be,

0

1

2 2( ) sin cosn n

n

πnη πnηη

a a b

pf

p p (7-10)

0 0.05 0.1 0.15 0.2 0.25

21.54 21.74 21.94 22.14 22.34 22.54 22.74 Time (ms)

Time (s)

(A)

(B)

Optical an

gle

(degre

es)

Voltag

e (

V)

7

0

12

8

4

138

where, a0, an and bn are the Fourier expansion coefficients. Let the vector f pv contain the

values of f p(η) at η = {0, p/2m, 2p/2m, ... , (2m-1)p/2m}, where m is an integer. Applying

the fft function in MATLAB to f pv gives the vector,

pv

f

c c c c c c c0 1 2 ( 1) ( 2) 1fft( ) { , , , , , , , }m m m (7-11)

The vector on the right-hand side of Equation 7-11 can be used to evaluate the

Fourier coefficients of f p(η) as follows [110],

00

2c

am

(7-12)

1

Im( ) , 1,2,3 ,n n n a c mm

(7-13)

1

Re( ) , 1,2,3 ,n n n b c mm

(7-14)

The above results are useful for building the Simulink model discussed in the next

subsection.

7.3.2 Simulink Model

The dynamic ETM Simulink model of a 1D mirror is depicted in Figure 7-11. The

mirror has the same topology as the mirror discussed in Chapter 4. The actuation

voltage is assumed to be periodic, which is typical for mirror scanning applications. The

Simulink model takes two inputs—a vector containing actuation voltage values at 500

equispaced points in time in one period of the waveform, and the frequency of actuation

voltage which is set to 82 Hz in the model depicted in Figure 7-11. The MATLAB

function labeled ‘Power Spectrum’, expands the input power waveform in a Fourier

series by using the procedure discussed in Section 7.3.1. Similarly, ‘Tbimorph

Spectrum’ expands the bimorph temperature waveform in a Fourier series. The

‘Thermal Model’ is based on the transmission-line approach discussed in Chapter 4.

Since the thermal model can be represented using resistors and capacitors, it is a linear

139

Figure 7-11. Dynamic ETM mirror model implemented in Simulink.

140

time invariant (LTI) model. Therefore, it handles each component of the input power

spectrum separately. The final temperature is obtained by linear superposition. The

output of the ‘Thermal Model’ is a vector containing the average bimorph temperature,

Tb, and embedded heater temperature, Th, at 500 equispaced points in one time period.

The functions ‘Tbimorph’ and ‘Theater’ extract the bimorph and heater temperatures,

respectively, from the ‘Thermal Model’ output. The ‘Mechanical Model’ function is

implemented as a second order LTI system. It computes mirror scan angle for each

component of the bimorph temperature waveform and uses linear superposition to

evaluate the final result. The function ‘R(T)’ evaluates the heater resistance, which is

then used for calculating the power input.


An 82 Hz sinusoidal voltage waveform was applied to the mirror using a signal

generator with 50 source resistance. The voltage across the mirror was measured

using an oscilloscope and has been shown in Figure 7-12A. The mirror position was

tracked by using a laser-diode. Light from the laser-diode was reflected by the mirror-

plate onto a screen. The laser diode was driven by a pulse waveform that is

synchronized with the mirror actuation voltage. As a result, a single point on the screen

is illuminated when the mirror executes periodic scanning motion. By varying the phase

between the mirror actuation waveform and the laser-diode drive waveform, the scan

angle can be determined at any instant of time. Figure 7-12B shows good agreement

between experimental data and the scan angle predicted by the Simulink model.

141

Figure 7-12. Verification of Simulink mirror model. (A) One period of an 82 Hz voltage waveform applied to the mirror. (B) Mirror-scan angle produced by the applied voltage.


In this chapter, experiments, FE modeling and experience gathered from reduced

order modeling have been used to achieve model-based open-loop control of an

electrothermal micromirror. Continuous voltage waveforms that achieve constant linear

Time (ms) 0 2 4 6 8 10 12

1.6

1.2

0.8

0.4

0

Vo

lta

ge

(V

)

Simulink model

Experimental data

0 2 4 6 8 10 12 Time (ms)

Op

tica

l an

gle

(d

eg

ree

s)

30

25

20

15

10

5

0

-5

(A) (B)

142

and angular velocity scanning are then evaluated. Such scan profiles are useful in

biomedical imaging and optical display applications. Constant linear and angular

velocity over 90% and 75% of scan period, respectively, are demonstrated. It is shown

that an equivalent PWM may be used to achieve the same response as the continuous

signal. PWM actuation can lead to significant saving is system size and cost in case of

large micromirror arrays.

The dynamic ETM model of a micromirror was implemented in Simulink. The

thermal response was modeled based on the transmission-line approach discussed in

Chapter 4. The thermal model predicts the bimorph temperature as well as the

temperature of the embedded heater. The mechanical response is represented as a

second order LTI system as discussed in Chapter 6. Good agreement between Simulink

model and experimental results is observed.

Future work will involve the modeling and control of 2D and 3D micromirrors, and

mirror arrays.

143

CHAPTER 8 ANALYSIS AND FABRICATION OF CURVED MULTIMORPH TRANSDUCERS THAT

UNDERGO BENDING AND TWISTING

8.1 Background

A multimorph consists of two or more layers of different materials. Difference in

strains produced in the constituent layers causes a multimorph to deform, thereby

producing transduction. A multimorph with two layers is a bimorph. Figure 8-1A shows

the schematic of a straight multimorph that bends upon deformation. Various devices

based on straight multimorphs have been reported in literature. Kim et al. report an

electrothermal micromirror actuated by two multimorphs that bend in opposite directions

to produce twisting [23]. Lee et al. report a piezoelectric multimorph based MEMS

generator [111]. Ho et al. report the use of polypyrrole-gold actuators for micro-mixing

applications [112]. Polypyrrole is an electroactive polymer that undergoes swelling and

shrinking during redox cycling [112]. Kniknie et al. investigate the dynamic response of

silicon micro-cantilevers coated with shape-memory alloy [113]. The analysis of straight

multimorphs has been investigated by several researchers. Timoshenko’s classic work

provides closed-form expressions for the deflection of singly and doubly clamped

straight thermal bimorphs [10]. DeVoe and Pisano report a model for predicting the

static behavior of straight piezoelectric multimorphs [12]. Weinberg provides a generic

derivation that is applicable to thermal, piezoelectric and shape-memory alloy

multimorphs [11].

Unlike straight multimorphs that undergo bending, curved multimorphs bend and

twist upon deformation. Figure 8-1B shows the schematic of a curved multimorph. It

must be emphasized that in this dissertation, the term ‘curved multimorph’ refers to a

multimorph with non-zero curvature in the plane of the substrate.

144

Figure 8-1. Schematics of straight and curved multimorphs. (A) A straight multimorph bends upon deformation. (B) A curved multimorph bends and twists upon deformation. The radius of curvature in the undeformed state is Rc. The twist

angle and out-of-plane displacement are denoted by θ and U, respectively.

There is limited literature on curved multimorphs. Manalis et al. report an array of

spiral shaped curved bimorphs for detecting thermal radiation [114]. A thermal bimorph

spiral not only exhibits a shape-altering response to thermal radiation, but can also have

a focusing effect on visible light by acting as a quasi-Fresnel element. Such bimorphs

may be used for uncooled photothermal spectroscopy [115]. Xu et al. report a

micromirror actuated by curved multimorphs, but do not give insight into the differences

between curved and straight multimorphs [116]. An elliptical micromirror actuated by a

curved thermal multimorph that is concentric to the mirror-plate [37] is discussed in

Chapter 9. Mirror designs based on curved actuators offer several advantages over

previously reported straight actuator based designs. Some of the advantages include

low mirror-plate center-shift, low power consumption, high resonant frequencies and

compact layout. We have previously reported the small deformation analysis and

simulation of curved bimorphs [117]. Curved bimorph analysis has limited practical use

as many processes involve more than two thin film layers.

DEFORMED

UNDEFORMED

DEFORMED

Clamped end

Free end

Undeformed state

UNDEFORMED Rc

Undeformed state

θ(l) U(l)

Clamped end

(s = 0)

Free end

(A) (B)

145

This chapter presents the small deformation analysis of curved multimorphs [118].

This analysis will greatly expand the design space for MEMS engineers. Closed-form

expressions for the out-of-plane displacement, U, and actuator twist angle, θ, shown in

Figure 8-1B are derived. The derivation is experimentally validated by monitoring a

curved thermal multimorph test structure in an oven. Numerical techniques are required

for studying large deformation. Large deformation is investigated experimentally and by

using finite element (FE) simulations.

This chapter is organized as follows. Section 8.2 describes the analysis of curved

multimorphs. Validation through experiments and simulations are provided in

Section 8.3. Results on large deformation of curved multimorphs are given in

Section 8.4.

8.2 Curved Multimorph Analysis

As in the case of straight multimorphs [11], the key components of curved

multimorph analysis are the beam deformation equations [119], the force and moment

balance equations, and strain continuity at the interface between adjacent layers. The

subsequent subsections will establish the theory governing the bending and twisting of

curved multimorphs. First, curved multimorphs in which the induced axial strain in the

constituent layers due thermal, piezoelectric, SMA, electroactive effect etc. and axial

strain due to residual stresses is constant along the multimorph length are considered.

A typical example in which the induced axial strain is constant along the length is a

thermal multimorph at a uniform temperature. The effect of variation of induced axial

strain along multimorph length will be discussed in Section 8.2.4.

146

8.2.1 Deformation of Curved Beams

Let us consider the bending of a curved beam. Figure 8-2A shows a section of a

curved beam of in-plane radius of curvature, Rc, subjected to a bending moment, Mi. The

thickness, width, area of cross-section and Young’s modulus are ti, wi, Ai and Ei ,

respectively. Figure 8-2B shows a cross-section of the curved beam. The cross-

sectional moment of inertia about the centroidal axis is Ii = 3( ) /12i iw t . In subsequent

sections, the subscript i will be used to denote the i th layer of a curved multimorph. The

distance along the beam is s. The beam is clamped at s = 0. The vertical deflection and

twist angle are denoted by U(s) and θ(s), respectively.

Figure 8-2. Curved beam. (A) Portion of a curved beam (B) Beam cross-section. The

origin of the local ui -ri coordinate system coincides with the center of the

cross-section.

The deformation of a curved beam is governed by the following equations [119],

2

2

( ) ( )0i

i i c

θ d U

E d

M s s

I R s (8-1)

( ) 1 ( )

0c

dθ dU

d d

s s

s R s (8-2)

If the beam is clamped at s = 0, the boundary conditions are U(0) = 0, U’(0) = 0

and θ(0) = 0. Solving Equations 8-1 and 8-2, and imposing boundary conditions,

2

( ) 1 cosi c

i i c

UE

M R ss

I R (8-3)

(0,0)

ui = -ti / 2

ri = -wi / 2 r

θ(s)

Mi

Mi

Rc

(A) (B)

s

U(s)

ri = wi / 2

ui = ti / 2 u

147

( )

( )c

Uθ

ss

R (8-4)

8.2.2 Strain Continuity between Adjacent Layers

The strains produced in adjacent layers must match at the interface between the

two. Figure 8-3A shows the forces and moments on a cross-section of a multimorph

with n layers. The forces and moments on the i th layer are denoted by Fi and Mi ,

respectively. The total axial strain, εT(i ), in the i th layer of the multimorph consists of

three components—strain due to axial force, strain due to bending moment and strain

due to thermal, piezoelectric, SMA effects, electroactivity, residual stresses etc. These

three strain components are denoted by εF(i ), εM(i ) and εi , respectively.

Figure 8-3. Force and moment distribution on a cross-section of a multimorph. (A) The

force and moment on the i th layer are denoted by Fi and Mi , respectively. (B)

Equivalent representation of forces and moments.

The strain due to axial force is [120],

( )i

F i

i i

εE A

F

(8-5)

F1

F2

F3

Fn

M1

M2

M3

Mn

M1

M2

M3

Mn

F1

F1 F1+F2

F1+F2

F1+ F2+F3

F1+F2+F3+…Fn-1

F1+ F2+F3+…Fn=0

(A) (B)

148

The axial strain due to bending moment is [120],

( )

1

i iM i

ii i

c

εE

M u

rJR

(8-6)

where, ui and ri are defined as shown in Figure 8-2B and Ji is defined by the area

integral,

32 2

ln12 21i

i c c ii ii

i c iA

c

t wdA

w

R RuJ

r RR

(8-7)

The dependence of εM(i ) on ri in Equation 8-6 implies that εM(i ) varies along the

width of the cross-section, wi. If wi << Rc, Equation 8-6 may be approximated as,

( )i i

M i

i i

εE

M u

J (8-8)

Strain continuity between the (i -1)th and i th layers gives,

( ) ( ) ( 1) ( 1) 1;2F i M i i F i M i iε ε ε ε ε ε i n (8-9)

Equations 8-5, 8-8 and 8-9 give,

1 1 11

1 1 1 1

;22 2

i i i i i ii i

i i i i i i i i

t tε ε i n

E A E E A E

F M F M

J J (8-10)

Since Equation 8-1 holds for 1 i n,

1 2

1 1 2 2

n

n nE E E

I

M M Mξ

I I (8-11)

where, the quantity ξ may be evaluated from Equation 8-3,

2

( )

(1 cos( / ))c c

U

sξ

R s R (8-12)

149

Using Equation 8-11, the second term on the left hand side of Equation 8-10 may be

written as,

2 2

i i i ii

i i i

t tB

E

M Iξ ξ

J J (8-13)

where,

2

i ii

i

tB

I

J (8-14)


11 1

1 1

( ) ( ) ( ) ;2ii i i i i i i

i i

E A B B ε ε i nE A

FF ξ (8-15)

Equation 8-15 may be used to evaluate Fi in terms of Fi-1 ,i.e., the force in the i th

layer can be expressed in terms of the force in the (i -1)th layer. Using Equation 8-15

successively, all forces may be expressed in terms of F1,

1

11 1

21 1

( ) 2 ;2i

i i i k i i

k

E A B B B ε ε i nE A

FF ξ (8-16)

8.2.3 Force and Moment Balance

Let us consider the force and moment distributions shown in Figure 8-3A. Force

balance gives,

1

0n

i

i

F (8-17)

In order to formulate the moment balance equation, the force and moment

distributions shown in Figure 8-3A may be equivalently represented by the distribution

depicted in Figure 8-3B. Equating the total moment produced by the axial forces to the

total bending moment gives,

150

2 31 2 11 1 2 1 2 3 1 1 2

2 2 2n n

n n

t tt t t t

F F F F F F F M M M (8-18)

8.2.4 Curved Multimorph Deformation

Equations 8-11, 8-16, 8-17 and 8-18 provide a set of 2n equations in Fi and Mi,

1 i n. These equations along with Equation 8-12 will be utilized to determine the

axial forces, bending moments and beam deflection.


i i i C DF ξ (8-19)

Ci and Di are given by,

1

112 2

1

2

1

2

( ) 2

n i

i i k iii k

i i i k i nk

i i

i

E A B B B

E A B B B

E A

C (8-20)

1

21

1

( )

n

i i i

ii i i i n

i i

i

ε ε E A

E A ε ε

E A

D (8-21)

Equation 8-11 may be rewritten as,

i i iE M ξ I (8-22)

Substituting force and moment expressions from Equations 8-19 and 8-22 into

Equation 8-18 and solving for ξ,

1 1

1 1

1 1

1 1 1

2 2

2 2

n ni n

i j

i j i

n n ni n

i i i j

i i j i

t tt

t tE t

C

ξ

ξ

I

(8-23)


151

1 1

1 1 2

1 1

1 1 1

2 2( ) 1 cos( / )

2 2

n ni n

i j

i j i

c cn n n

i ni i i j

i i j i

t tt

Ut t

E t

D

C

s R s R

I

(8-24)

The twist angle can be evaluated by substituting the expression for U(s) into

Equation 8-4. Equations 8-4 and 8-24 represent the small deformation of curved

multimorphs when the induced strains, εi, are constant along the multimorph length.

At large Rc, the deflection predicted by Equation 8-24 must approach the deflection

of a straight multimorph. Let us consider a straight multimorph of length ls that is initially

undeformed. Let the out-of-plane radius of curvature of the straight multimorph in the

deformed state be Rm. If small deformation is assumed, the out-of-plane deflection of the

tip of the straight multimorph is,

2

2( ) 1 cos sin2 2

s s smstraight s m

m m m

l l lU l

RR

R R R (8-25)

The expression for Rm was obtained from [11]. It has been verified using

MATHEMATICA [121] that at large Rc the deflections predicted by Equations 8-24 and

8-25 are identical,

Limit ( ) ( )c

s straight sU l U l

R

(8-26)

The effect of variation in εi along the multimorph length will be addressed in the

next subsection.

8.2.5 Variation of Induced Strain along Multimorph Length

In many practical situations, εi, may vary along the multimorph length. A typical

example is a thermal multimorph whose temperature varies along its length. The

152

quantities Fi, Mi, ξ, εi, εF(i ), εM(i ) and Di are now functions of s. In this case, Equations 8-1

through 8-23, except Equations 8-3, 8-4 and 8-12, hold true. Equation 8-1 may be

rewritten as,

2

2

( ) ( )( ) 0

c

θ d U

d

s sξ s

R s (8-27)

If a closed-form solution for Equations 8-27, 8-2 does not exist, ξ(s) may be

expressed as a Fourier expansion. Solutions corresponding to the sine and cosine

terms may be obtained separately and added to give the complete solution.

For ξ(s) = Q1 cos(Q2s), Equations 8-27 and 8-2 may be solved and boundary

conditions at the clamped end, s = 0, may be imposed to give,

2

1 2

2 2

2

cos / cos( )( )

1

c c

c

U

Q Q

Q

R s R ss

R (8-28)

Similarly, for ξ(s) = Q1 sin(Q2s), Equations 8-27 and 8-2 may be solved to give,

2

1 2 2

2 2

2

sin / sin( )( )

1

c c c

c

U

Q Q Q

Q

R R s R ss

R (8-29)

Once the solution for Equations 8-27, 8-2 are obtained, the rest of the analysis

proceeds similar to that described in Sections 8.2.1-8.2.4. The next section provides

validation of the analysis using FE simulations and experiments.

8.3 Results

The next two subsections discuss the validation of the analysis with simulations

and experiments. For simulations and calculations, material properties were obtained

from COMSOL’s materials library [58]. All FE simulations are done using the COMSOL

simulation software [58]. Section 8.3.3 deals with large deformation of curved

multimorphs.

153

8.3.1 Analysis vs. FE Simulations

Let us consider a 50 μm long, 6 μm wide thermal multimorph with four layers. Let

the constituent layers from bottom to top be SiO2, W, SiO2 and Al, respectively. Let the

thicknesses of the layers from bottom to top be 0.15 μm, 0.6 μm, 0.25 μm and 0.7 μm,

respectively. These values have been chosen based on typical film thicknesses

reported in literature [37]. Let the multimorph be subjected to a 100 K uniform

temperature rise from a reference temperature. At the reference temperature, the

multimorph is assumed to be flat. FE simulations were done for Rc in the range 25 μm to

1000 μm. Figures 8-4A through 8-4C show the simulated deformation for in-plane radius

of curvature equal to 25 μm, 100 μm and 300 μm, respectively.

Figure 8-4. FE simulation results for curved thermal multimorphs with length 50 μm and

width 6 μm, subjected to a uniform temperature change of 100 K. The radius of curvature Rc in the undeformed state are (A) 25 μm (B) 100 μm (C) 300 μm.

The layers from top to bottom are Al, SiO2, W, SiO2 with thicknesses 0.7 μm, 0.15 μm, 0.6 μm and 0.25 μm, respectively. The shading represents the out-of-plane displacement.

z

x y

z

x y

z

x y

0

-0.4

-0.8

-1.2

-1.52

0

-0.4

-0.8

-1.2

-1.6

-1.97

z displacement (μm) z displacement (μm)

0

-0.4

-0.8

-1.2

-1.6

-1.96

z displacement (μm)

(A) (B)

(C)

154

Figure 8-5 compares FE simulation and analytical results. Figure 8-5A shows that

the magnitude of out-of-plane deflection increases with Rc. At large Rc, the out-of-plane

deflection approaches the deflection of a straight multimorph. Figure 8-5B shows that

the beam twist reduces monotonically with increasing Rc. From Figure 8-5, it may be

concluded that the analysis is in good agreement with FE simulations. The error in the

closed-form expressions may be attributed to the assumptions involved in small

deformation analysis.

Figure 8-5. Deformation of curved multimorph. (A) Tip-deflection. (B) Beam twist of a

50 μm long, 6 μm wide singly-clamped multimorph with the following layers—0.15 μm SiO2 - 0.6 μm W - 0.25 μm SiO2 - 0.7 μm Al— for a temperature change of 100 K.


Figure 8-6A shows a test structure that was fabricated on an SOI wafer by a

process similar to that described in [37]. The 10 μm wide curved multimorph is attached

to a 245 μm wide mirror-plate. Figure 8-6B shows a cross-section of the curved

multimorph. The multimorph thin-film layers from bottom to top are 0.14 μm PECVD

SiO2, 0.6 μm sputtered W, 0.29 μm PECVD SiO2 and 0.58 μm sputter-deposited Al,

respectively. Figure 8-6C shows a simplified cross-section that will be used for analysis

Analytical Finite-element simulations

In-plane radius of curvature, Rc (μm) 0 200 400 600 800 1000

-1.4

-1.6

-1.8

-2.0

Bea

m tw

ist (r

adia

ns)

0 200 400 600 800 1000

In-plane radius of curvature, Rc (μm)

Analytical

Finite-element simulations

(A) (B)

0.06

0.04

0.02

0

tip-d

eflectio

n (

μm

)

155

and simulations. The idealized cross-section in Figure 8-6C simplifies calculations and

modeling. The mirror-plate consists of 20 μm-thick single-crystal Si coated with 0.58 μm

Al. At 300.8 K, the mirror-plate is tilted at 12.4 to the substrate due to residual stresses.

An embedded tungsten resistor is incorporated in the same chip to monitor the

temperature. The resistance and temperature coefficient of resistance of the resistor are

201 Ω and 0.0012/K, respectively, at 298 K.

Figure 8-6. Curved multimorph test structure. (A) SEM (B) Schematic of multimorph

cross-section (not to scale). (C) Idealized cross-section used for analysis and simulations.

The test structure shown in Figure 8-6A was placed in an oven with a glass

window. The temperature was monitored using the on-chip W resistor. A laser beam

reflected from the mirror-plate was used to measure the mirror-plate tilt with respect to

the substrate. Figure 8-7 shows the mechanical tilt angle vs. chip temperature. From

experimental results, it is found that mirror-plate tilt is zero at 379 K. Therefore, 379 K is

the reference temperature used for analysis and simulations. At zero tilt, the slope of the

200 μm

Multimorph

Mirror-plate

Substrate

0.14 μm

0.58 μm

0.6 μm

0.29 μm

7 μm

10 μm

Sputtered Al

Sputtered W

PECVD SiO2

(A) (B)

0.14 μm

0.58 μm

0.6 μm

0.29 μm

10 μm

(C)

7 μm

156

experimentally obtained curve shown in Figure 8-7 is 0.26/K. This is in close

agreement with the analytically obtained sensitivity of 0.24/K. Since Equations 8-1 and

8-2 are based on small deformation assumptions, at large tilt angles the difference

between experimental and analytical sensitivities increases. Close agreement between

experiment and analytical results is observed. Discrepancy in the experimental and

simulated values may be attributed to the difference in material properties of thin-films

with those used in calculations and errors in the measurement of thin-film thicknesses.

Large deformation of curved multimorphs will be discussed next.

Figure 8-7. Mirror-plate tilt vs. chip temperature for test structure shown in Figure 8-6A.

8.3.3 Large Deformation of Curved Multimorphs

A curved multimorph may undergo large deformation if the residual stresses are

high, the actuation signal is large, if the multimorph is significantly long, the thickness of

the multimorph is significantly low, or a combination of the aforementioned factors.

Figure 8-8A shows the schematic of a curved multimorph test structure layout. The

semicircular Al-SiO2 multimorph is anchored to a circular mirror-plate. The thicknesses

of the Al and SiO2 layers are 0.58 μm and 0.43 μm, respectively. The mirror-plate

consists of a 20 μm thick single crystal silicon coated with 0.58 μm Al.

300 320 340 360 380 400 420

Experiment

Analytical

Finite Element Simulation

20

15

10

5

0

-5

-10

Mirro

r tilt ()

Chip temperature (K)

157

Figure 8-8. Large deformation of a curved multimorph. (A) Schematic of a curved

multimorph test structure layout. The 546 μm wide mirror-plate is anchored to a semicircular multimorph. The top and bottom layers of the multimorph are Al and SiO2 , respectively, with thicknesses 0.58 μm and 0.43 μm, respectively. The mirror-plate consists of a 20 μm thick single crystal silicon layer coated with a 0.58 μm Al layer. (B) Optical microscope image of a released structure. (C) SEM of a released test structure. (D) Simulated deformation for a uniform temperature of -250 K. The color bar represents total displacement.

The process for fabricating the test structure shown in Figure 8-8 is similar to that

for the structure shown in Figure 8-6A. Device release involves Si etch using DRIE [37].

The optical microscope image and SEM of the released structure are shown in Figure 8-

546 μm

Clamped-end of multimorph

Undeformed semicircular multimorph

Mirror-plate

500 μm

(A)

(C)

Clamped-end of multimorph

Deformed semicircular multimorph

Mirror-plate

Deformed semicircular multimorph

(B)

Undeformed state

0 (D)

620 μm

158

6B and Figure 8-6C, respectively. The multimorph is found to undergo large

deformation due to residual stresses.

Large deformation of curved multimorphs results in large in-plane displacement

along with out-of-plane displacement and twisting. Numerical techniques are required

for investigating large deformations [119]. The FE model shown in Figure 8-8D provides

a qualitative understanding of the large deformations observed in Figure 8-8B and

Figure 8-8C. In the FE simulation, the residual stress in the thin-films is modeled by

applying a uniform temperature change of -250 K to the undeformed multimorph. As the

mirror-plate undergoes negligible deformation compared to the multimorph, it has not

been included in the FE model. Clearly, the simulated deformed shape is qualitatively

similar to that observed using a microscope. It is evident from the SEM shown in Figure

8-8C that the mirror-plate is constrained by the surrounding substrate. This leads to

some discrepancy between the simulated shape and the fabricated device.


Difference in strains in the layers of a multimorph causes it to deform. Straight

multimorphs that undergo out-of-plane bending have been widely investigated. In this

chapter, the small deformation analysis of multimorphs that have a non-zero curvature

in the plane of the substrate is reported. Curved multimorphs undergo both out-of-plane

bending and twisting deformations. The analysis involves the deflection equations for

curved beams, static equilibrium equations and strain continuity condition between

adjacent layers. Closed-form expressions are derived for out-of-plane displacement and

beam twist angle. The analysis is validated against FE simulations and experimental

results. At large deformations, significant in-plane displacement is produced along with

159

out-of-plane bending and twisting. The investigation of large deformations requires

numerical techniques.

Previously, curved multimorphs have been used for thermal sensing and

micromirror actuation only. The analysis reported in this dissertation will lead to greater

understanding of curved multimorphs and enable MEMS engineers to conceive novel

devices. This chapter has focused on curved transducers that have a constant in-plane

radius of curvature along their length. However, the procedure reported in this chapter

can be adapted to multimorphs of arbitrary shape. Future work will involve the design of

curved multimorph based devices, investigation into large deformation of curved

multimorphs, multimorphs of arbitrary shape and multimorphs subjected to external

loading. The next two chapters present micromirrors actuated by curved multimorphs.

160

CHAPTER 9 A 1MM-WIDE CIRCULAR MICROMIRROR ACTUATED BY A SEMICIRCULAR

ELECTROTHERMAL MULTIMORPH

9.1 Background

As discussed in Chapters 1 and 8, most multimorph based MEMS designs

reported in literature utilize straight multimorph beams. This chapter will present a novel

electrothermal micromirror design actuated by a curved multimorph. Unlike straight

multimorphs that bend upon deformation, curved multimorphs undergo bending and

twisting. This unique feature of curved multimorphs may be used to address several

drawbacks of mirrors actuated by straight multimorphs. As depicted in Figure 9-1, the

large mirror-plate center-shift in a straight multimorph-based design hampers optical

alignment. As shown in Figure 9-2, low mirror-plate center-shift can be achieved by

using a curved multimorph actuator. The center-shift produced by bending and that

produced by twisting partially cancel each other and this leads to an overall low center-

shift. Other advantages of micromirrors actuated by curved multimorphs include

compact layout, high resonant frequency and low power requirements.

Figure 9-1. Straight multimorph based 1D micromirror [20, 39] design at two different positions during a scan cycle. The mirror-plate center suffers significant shift during actuation. 1D mirrors actuated by straight multimorphs have been discussed in Chapters 3-4.

Mirror-plate

Straight multimorph

Shift in mirror-plate

center

161

Figure 9-2. Semicircular actuator based mirror design. (A) Schematic of a 50 μm wide mirror-plate with an Al-SiO2 semicircular actuator (Rc = 35 μm, w = 10 μm) (B)

FE simulation shows that the mirror-plate tilts by 13.5 for a temperature change of 200 K. The mirror-plate center shifts by 0.4 μm only.

In this chapter, a 1 mm-wide mirror actuated by a semicircular multimorph [122]

will be used to elucidate the advantages of curved multimorph-based designs. The

micromirror has an optical scan range of 60 at 0.68 V applied voltage and 11 mW

power input. The mirror-plate size is comparable to older designs discussed in Chapters

3 and 4. Therefore, the 1 mm-wide mirror provides a suitable baseline for evaluating the

improvement in performance due to the incorporation of curved multimorphs. Mirror

center-shift produced by actuator bending and twisting partially compensate each other

and this results in 1.6 times lower center-shift compared to previously reported straight

multimorph based designs. The curved actuator design not only maximizes the

efficiency of chip area usage, but also achieves high resonant frequency due to

torsional stiffness encountered during beam deformation. The first three resonant

modes of the micromirror are at 104 Hz, 400 Hz and 416 Hz, respectively. Two-

dimensional (2D) optical scanning is demonstrated by using the second resonant mode.

Mirror-plate

0

9.4 μm

(A)

(B)

Clamped end

Semicircular actuator

Out-of-plane displacement

162

The performance and robustness of the device exceeds previously reported designs

actuated by straight multimorphs. Furthermore, since most laser spots are circular or

elliptical, a circular mirror-plate actuated by a concentric curved actuator utilizes chip

area more efficiently than a square mirror-plate actuated by straight multimorphs.

This chapter is organized as follows. Section 9.2 describes the device design,

fabrication process and experimental results. An electrothermomechanical finite

element (FE) model is presented in Section 9.3. Section 9.4 underscores the efficacy of

the present design by comparing it with older micromirrors actuated by straight

multimorphs.

9.2 A 1 mm-wide Micromirror Actuated by Curved Multimorph


Figure 9-3A shows an SEM of a fabricated circular micromirror with a semicircular

multimorph actuator. The diameter of the mirror-plate is 1 mm. The initial tilt of the

mirror-plate is caused by residual stresses in the multimorph, which matches the

simulation, as shown in Figure 9-3B. More finite-element (FE) modeling will be

discussed in Section 9.3. Figure 9-3C shows a cross-section of the 20 μm wide curved

multimorph. Multimorph deformation can be mainly attributed to differential strain

between the Al and W layers. W also acts as a resistive heater. The mirror-plate

consists of a 15-20 μm single-crystal-silicon (SCS) layer coated with a 0.1-0.2 μm

PECVD SiO2 layer and a 0.6 μm Al layer on top. The thick SCS layer serves to improve

mirror flatness. Mirror curvature can be obtained from the spot size of a reflected laser

beam and the distance from the mirror at which the spot size is measured. For a

measured SCS thickness of 180.5 μm, the radius of curvature of the mirror is 6 cm.

Equations given in [2, 11] can be used to show that the radius of curvature of the mirror-

163

plate varies approximately as square of the SCS thickness. After release, the mirror-

plate is tilted at an angle of 24 with respect to the substrate. The first three resonant

modes are observed at 104 Hz, 400 Hz and 416 Hz, respectively.

Figure 9-3. A 1 mm-wide circular mirror. (A) SEM of a circular micromirror actuated by a semicircular electrothermal multimorph. (B) Simulated initial tilt of micromirror upon release. The color bar shows the out-of-plane displacement from the unreleased position. A uniform actuator temperature change of -67 K is used to simulate the tilt from the unreleased flat position. (C) Schematic of a cross-section AA’ (Figure 9-3A) of the multimorph actuator (not to scale).

9.2.2 Fabrication Process

9.2.2.1 Material selection

For large deflection, material pairs with widely different CTEs are commonly used

for thermal multimorphs. Several designs utilize metal-SiO2 [19] or metal-polymer [48]

pairs for generating large differential strain. However, the low thermal diffusivities of

SiO2 [48, 58] and MEMS polymers [48, 58] result in slow thermal response of metal-

Mirror- plate

Curved multimorph

Mirror position before release

-268 μm

123 μm

0.14 μm

0.6 μm

0.6 μm

0.29 μm

15.6 μm

20 μm

Sputtered Al

Sputtered W

PECVD SiO2

(A) (B)

(C)

y x

z

A

A’

200 μm 200 μm

164

SiO2 and metal-polymer based designs. Additionally, designs that utilize SiO2 beams for

thermal isolation are susceptible to impact failure due to the brittle nature of SiO2 [19].

Therefore, common MEMS materials are surveyed [48] and three criteria for choosing

candidate material pairs are established:

1. Large difference in CTE values for achieving large deflection 2. High thermal diffusivities of both materials for achieving fast response 3. Avoiding materials like SiO2 that are susceptible to brittle failure

Based on the above criteria and the material properties listed in [46, 121], three

candidate pairs stand out—Al-DLC (Diamond-like Carbon), Al-Invar and Al-W. Due to

the wide availability of W deposition facilities and recipes, Al-W was chosen to form the

active layers of the multimorph. As shown in Figure 9-3C, a thin layer of SiO2

encapsulates W. The SiO2 acts as electrical isolation between Al and W, and protects W

from fluoride based dry etch recipes.

9.2.2.2 Process flow

The micromirror fabrication process is illustrated in Figure 9-4. SOI wafers are

used to ensure the flatness of mirror-plates with single-crystal-silicon microstructures.

The handle-layer and device-layer thicknesses of the SOI wafer are 500 μm and 20 μm,

respectively. A 0.1-0.2 μm thick PECVD oxide deposited in Figure 9-4A acts as

electrical isolation between W and silicon. Both Al and W are fabricated by sputtering

and lift-off as shown in Figure 9-4B and Figure 9-4D, respectively. The 0.3 μm thick

PECVD oxide deposited in Figure 9-4C electrically isolates W and Al. The SiO2

depositions in Figure 9-4A and Figure 9-4C protect the W layer from fluoride based etch

recipes. The device is released by backside DRIE silicon etching (Figure 9-4E), buried

oxide etch (Figure 9-4F), and front-side silicon etch (Figure 9-4G).

165

Figure 9-4. Fabrication process flow on an SOI wafer. (A) PECVD SiO2 deposition, patterning, and etching by RIE. (B) W sputtering and lift-off. (C) PECVD SiO2 deposition, patterning, and etching by RIE. (D) Al sputtering and lift-off. (E) Backside lithography and DRIE Si etch. (F) Buried oxide etch by RIE. (G) Front-side anisotropic and isotropic Si etch for device release.

9.2.2.3 Thickness selection

The dimensions of a cross-section of the multimorph are shown in Figure 9-3C.

The W beam is narrower than Al and SiO2 layers to account for potential misalignment

during fabrication. From past experience, the etch selectivity of Si over SiO2 during

DRIE is in excess of 100:1. During the isotropic Si etch shown in Figure 9-4G, 10 μm of

Si must be undercut from either edge of the 20 μm wide multimorph to release the

device. Therefore, the SiO2 protecting the W layer must have thickness in excess of

0.1 μm. Additionally, from past experience, a PECVD SiO2 layer with thickness greater

than 0.1 μm provides satisfactory electrical insulation between two metal layers [20].

Si SiO2

W Al

(A) (E)

(B) (F)

(C) (G)

(D)

166

Therefore, the 0.1-0.2 μm and 0.3 μm thick PECVD SiO2 layers corresponding to Figure

9-4A and Figure 9-4C, respectively, meet the fabrication requirements.

Two criteria are used for choosing the thicknesses of W and Al:

1. Maximum deflection criteria: For a certain thickness of W, the Al thickness is

chosen to produce maximum deflection. Let us consider a curved beam with in-

plane radius of curvature Rc and thickness t. If Rc >> t, the components of stress

on its cross-section may be approximated by assuming that the beam is

straight [117, 120]. As the radius of curvature of the multimorph actuator is two

orders of magnitude greater than its thickness, straight multimorph equations

may be used for choosing optimum thickness values of the thin-films. Material

properties are obtained from [58]. Figure 9-5A shows the tangential angle at the

free-end of a singly-clamped 100 μm long straight multimorph for a uniform

temperature change of 100 K. Optimum Al thickness corresponds to the maxima

of the plots shown in Figure 9-5A. Figure 9-5B shows the optimum Al thickness

as a function of the thickness of the W layer. If the W layer is 0.6 μm thick, the

optimum Al thickness is 0.7 μm. As shown in Figure 9-3C, the measured value of

Al thickness in 0.6 μm.

2. Stiffness criteria: The thicknesses of Al and W are chosen to produce similar

actuator stiffness as older designs based on Al-SiO2-Pt-SiO2 multimorphs [20].

The stiffness criteria ensures that for the same device dimensions, mirrors

fabricated using the new process shown in Figure 9-4 will have similar resonant

frequencies as the older designs. Equations provided in [11] were used to verify

that the bending stiffness of multimorphs fabricated using the process shown in

167

Figure 9-4 is similar to the stiffness of multimorphs fabricated using previously

reported fabrication process [20].

Figure 9-5. Optimal thicknesses of multimorph layers. (A) Change in tangential angle at the free-end of a singly-clamped 100 μm long straight multimorph for a uniform temperature change of 100 K. The thicknesses of the SiO2 layers are 0.14 μm and 0.29 μm (Figure 9-3C). The thickness of W is varied as a parameter from 0.2 to 1 μm. For a certain W thickness, the optimum Al thickness corresponds to the maxima. (B) Optimum Al thickness values obtained from Figure 9-5A.

9.2.3 Device Characterization

9.2.3.1 Static response

The static response was obtained by applying a dc voltage to the device. A laser

beam reflected from the mirror-plate was used to monitor the scan angle. The voltage

was increased from 0 to 0.68 V and then decreased back to 0. Figure 9-6A and Figure

9-6B show the experimentally obtained scan angle along with applied voltage and input

power, respectively. It is found that the mirror can scan as much as 60 at an applied

voltage of 0.68 V. The corresponding power consumption is 11 mW. As shown in

Figure 9-6, the device characteristic is repeatable with negligible hysteresis. Figure 9-6

0 0.5 1.0 1.5 2.0 2.5 3.0

15

10

5

0

Aluminum thickness (microns)

Chan

ge in t

ang

entia

l an

gle

at fr

ee e

nd

of str

aig

ht

multim

orp

h (

degre

es)

Decreasing W thickness

1μm

0.2μm

0.8μm

0.6μm

0.4μm

0.2 0.4 0.6 0.8 1.0

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

Tungsten thickness (microns) O

ptim

um

alu

min

um

th

ickne

ss (

mic

rons)

(A) (B)

168

also provides a comparison between FE simulations and experimental data. Details on

the FE model will be provided in Section 9.3.

Figure 9-6. Experimentally obtained static characteristic of micromirror along with FE simulation data. (A) Optical scan angle vs. applied voltage. (B) Optical scan angle vs. input power.


In order to obtain the frequency response, the mirror is biased in the linear region

of the scan-angle vs. voltage characteristic shown in Figure 9-6A and a sine-wave

voltage superimposed on the dc bias is used [20]. The dc bias is 608 mV and the

amplitude of the sine voltage is 17.5 mV. The frequency response is shown in Figure 9-

7. The high thermal diffusivities of Al and W ensure a nearly flat response at low

frequencies.

Figure 9-8 shows the simulated resonant modes. Two one-dimensional scanning

modes about the x-axis are observed at 104 Hz and 416 Hz and are depicted in Figure

9-8A and Figure 9-8C, respectively. The modes at 104 Hz and 416 Hz correspond to

the two peaks in the frequency response shown in Figure 9-7. A transverse mode about

0 2 4 6 8 10 12

70

60

50

40

30

20

10

0

Increasing voltage

Decreasing voltage

FE Simulation

Power consumption (mW)

Optical scan a

ngle

(d

egre

es)

Voltage (V) 0 0.2 0.4 0.6 0.8

70

60

50

40

30

20

10

0

Increasing voltage

Decreasing voltage

FE Simulation

Optical scan a

ngle

(d

egre

es)

(B) (A)

169

the y-axis is observed at 400 Hz and has been shown in Figure 9-8B. The simulated

resonant frequencies match the experimentally measured frequencies to within 8%.

Figure 9-7. Frequency response of the 1mm-wide micromirror. The actuation voltage is a sinusoid of amplitude 17.5 mV at a dc bias of 608 mV. The optical scan angle depicted in this plot is the scan angle about the y-axis shown in Figure 9-3B.

Figure 9-8. Simulated resonant modes of the 1mm-wide micromirror are at (A) 96 Hz, (B) 376 Hz, and (C) 391 Hz. The corresponding experimentally obtained resonance frequencies are 104 Hz, 400 Hz and 416 Hz, respectively. The shading represents the out-of-plane displacement along z-axis.

10-2 10-1 100 101 102 103

Optical scan r

an

ge (

de

gre

es)

15

10

5

0

Frequency (Hz)

(A) (B)

(C)

y

x

z

y

x

z

x

y

z

170

9.2.4 Two Dimensional Scanning

As shown in Figure 9-8, the mirror can scan about two mutually-perpendicular

axes. The Tektronix AFG 3022B signal generator, which has a 50 Ω source resistance,

was used for two dimensional scanning experiments. The voltage applied by the signal

generator appears across the mirror and the source resistance in series. A 400 Hz, 0-

1.14 V sine wave was applied, which excites the transverse mode shown in Figure 9-

8B. In accordance with the frequency response plot in Figure 9-7, scanning about the y-

axis is also produced. Figure 9-9A shows the 2D pattern scanned on a screen by a

laser beam reflected from the mirror-plate. By modulating the amplitude of the applied

voltage, it is possible to achieve 2D scanning. The scan pattern shown in Figure 9-9B

was obtained by applying a 400 Hz sine wave amplitude modulated using a 10 Hz

sinusoid. The corresponding voltage waveform across the mirror and source resistance

in series is (570+285cos(2400η)(1+cos(210η))) mV. This expression represents an

amplitude modulated waveform superimposed on a dc bias. A signal synchronized with

the actuation signal was used to drive a laser diode and the smiley face pattern shown

in Figure 9-9C was generated.

9.3 Finite Element Model

9.3.1 Harmonic Analysis

The device layer of the SOI wafer is 20 μm thick. The exact silicon thickness, tSi,

below the mirror-plate depends on the etch recipe used for device release in Figure 9-

4G. The mirror-plate of the device was broken and the thickness of the SCS layer was

measured using a 150X magnification optical microscope. The measured value of tSi is

171

180.5 μm. Harmonic simulation was performed using COMSOL [58] by setting

tSi = 18 μm. The resonant modes have been shown in Figure 9-8.

Figure 9-9. Two dimensional scan patterns generated by actuating the mirror using a signal generator with 50 Ω source resistance. The applied voltage appears across the series connection of the mirror and the source resistance (A) Scan pattern generated by a 400 Hz, 0-1.14 V sinusoidal actuation voltage. (B) Scan pattern generated by the amplitude modulated signal (570+285cos(2400η)(1+cos210η)) mV. (C) Smiley face pattern generated by driving the laser diode with a signal synchronized with the mirror actuation waveform.

9.3.2 Estimation of Heat Loss Coefficient

The heat loss coefficient is the heat lost per unit area per unit temperature rise

above ambient temperature. It accounts for both conductive and convective heat loss

through air. At the length scales under consideration, heat loss due to convection is

negligible compared to conductive heat loss through air [20]. The procedure outlined in

[20] is used to estimate the heat loss coefficient. A thermal FE model consisting of the

air region surrounding the device was built using COMSOL [58] to estimate the heat

loss due to thermal diffusion alone. The boundaries that represent the package and the

5

(A)

(B) (C)

3.5

Mirror bonded on a DIP package mounted on a bread-board

Two-dimensional scan pattern

5.5

5

172

handle layer silicon of the device are set to 300 K. At a distance of 4 mm from the

device, the air temperature is set to 300 K. The temperature of the air adjacent to the

mirror-plate and multimorph is set to 310 K. The thermal conductivity of air is set to

0.026 Wm-1K-1 [20]. The simulated heat loss coefficients have been listed in Table 9-1

and will be used in the electrothermal model discussed in Section 9.3.3.

Table 9-1. Simulated heat loss coefficients due to thermal diffusion through air.

Region of model Simulated average heat loss coefficient (Wm-2K-1)

Top surface of mirror-plate 48 Bottom surface of mirror-plate 68.4

Edge of mirror-plate 236 Top surface of actuator 407

Bottom surface of actuator 541 Edge of actuator 335

9.3.3 Electrothermal Model

The electrothermal model predicts the device temperature distribution for an

applied voltage. The temperature dependent resistivity of the W heater is first

determined experimentally by monitoring the resistance of a test structure in an oven. It

is found that the resistivity, ρ0, and temperature coefficient of resistivity, α, at 297 K are

2.1310-7 Ω-m and 0.00118 K-1, respectively. Thermal conductivities are obtained

from [58]. Figure 9-10 shows the simulated temperature distribution for an applied

voltage of 0.7 V. Figure 9-11 depicts the simulated temperature along the actuator

length at applied voltages of 0.3 V, 0.5 V and 0.7 V.

Figure 9-12 compares the heater current predicted by the electrothermal model

with the experimentally measured heater current. Good agreement between simulation

and experiment is observed. The error in the simulated value of the current is less than

173

3.7%. Next, the simulated temperature is used by the mechanical model to predict the

mirror scan angle.

Figure 9-10. Temperature distribution for an applied voltage of 0.7 V simulated using an electrothermal FE model.

Figure 9-11. Simulated temperature distribution along the length of the semicircular actuator at applied voltages of 0.3 V, 0.5 V and 0.7 V, respectively.

420 K

400 K

380 K

360 K

340 K

320 K

300 K

Max: 432.4 K

Min: 297 K

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

440

420

400

380

360

340

320

300

280

Tem

pera

ture

(K

)

0.7 V

0.5 V

0.3 V

Distance along semi-circular actuator (mm)

174

Figure 9-12. Comparison of experimentally measured current with simulated data.

9.3.4 Mechanical Model

Material properties for the mechanical model are obtained from [58], which are

also verified in literature [20, 122]. The initial tilt of the mirror-plate is determined by the

residual stresses in the thin films. It was found that, a uniform actuator temperature

change of - 64 K can be used to mimic the initial tilt. Figure 9-3B shows the simulated

initial tilt. The actuator temperature evaluated by the electrothermal model described in

Section 9.3.3 is then used to evaluate the scan angle as a function of the applied

voltage. Figure 9-6A and Figure 9-6B compare the simulated angle-voltage and angle-

power curves with the experimental data. Good agreement between simulation and

experiments is observed. The error in simulation results may be attributed to the

difference between the material properties used for simulations and the actual material

properties of the thin-films.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

18

16

14

12

10

8

6

4

2

0

Experimental

Electrothermal simulation

Voltage (V)

H

eate

r C

urr

en

t (m

A)

175

9.4 Comparison with Mirrors Actuated by Straight Multimorphs

Since straight multimorph-based designs with comparable mirror-plate size and

scan angle have been reported previously [19, 20], a comparison with the new design

shown in Figure 9-3 has been given below:

1. Resonant frequency: The design reported in [19] consists of a 1.1 mm x 1.2 mm

mirror-plate actuated by an array of 72 straight multimorphs and has a resonant

frequency of 200 Hz. In contrast, the present design achieves comparable

resonant frequencies of 104 Hz, 400 Hz and 416 Hz by using a single curved

actuator only. The high resonant frequency may be attributed to the high

torsional stiffness encountered during beam twisting. Higher resonant

frequencies may be obtained by increasing the width of the curved multimorph.

2. Scan angle and power consumption: The design reported in [19] utilizes a large

number of straight multimorphs and therefore has a high power consumption of

355 mW at 36 scan angle. By using large SiO2 thermal isolations at the mirror

and substrate ends of the straight actuators, it is possible to achieve 22 scan

angle at 23 mW [19]. In contrast, the design presented in this chapter consumes

only 11 mW at 60 scan angle as it utilizes a single actuator only. Therefore, the

design presented in this chapter scans larger angle and has lower power

consumption compared to older designs [19].

3. Voltage requirements: The design reported in [19] achieves a 29 scan angle at

an applied voltage of 11 V. In contrast, the present design has significantly lower

voltage requirements and can scan 60 at 0.68 V. The low voltage requirement

may be attributed to the choice of resistive heater material as well as the

thickness of the heater layer. For a certain applied voltage, the power dissipated

176

is inversely proportional to the heater resistance. The W heater used in the

present design is three times thicker than the Pt heater used in [19].

Furthermore, W has lower electrical resistivity than Pt [58]. Therefore, the W

heater can dissipate higher power at a certain applied voltage.

4. Robustness: The designs reported in [19, 20] utilize SiO2 for thermal isolation

which makes them prone to impact failure. In contrast, the present design does

not require thermal isolation to minimize power consumption. Therefore, it has

significantly better robustness than previously reported designs.

5. Two-dimensional scanning capability: As described in Section 9.2.4, the present

device can achieve 2D scanning by utilizing a transverse resonant mode. The

ability to achieve 2D scan using a single actuator can help miniaturize

micromirror-based imaging systems. Devices reported in [19, 20] are capable of

1D scan only. Schweizer et al. report an L-shaped actuator consisting of two

mutually perpendicular straight multimorphs that can execute 2D scanning [123].

However, the mirror-plate, which is attached to one end of the L-shaped actuator,

shifts significantly during actuation thereby hampering optical alignment.

Furthermore, unlike curved actuators that undergo bending and twisting, L-

shaped actuators undergo bending only. Therefore, for a comparable mirror-plate

size, L-shaped designs will have lower resonant frequencies than designs based

on curved actuators.

6. Mirror center-shift: A low mirror center-shift is desirable for good optical

alignment during system design. Figure 9-13 compares the simulated center-shift

of the present design with the design reported in [19]. The present design

177

undergoes 1.6 times lower center-shift and therefore provides better optical

alignment. The low center-shift may be attributed to the fact that bending and

twisting of the actuator result in center-shifts in opposite directions.

Figure 9-13. Simulated center-shift of the micromirror depicted in Figure 9-3 and the mirror reported in [19] that is actuated by straight multimorphs. The mirror depicted in Figure 9-3 undergoes lower center-shift and therefore provides better optical alignment.

7. Thermal response time: Al and W, that constitute the bulk of the curved

multimorph, have high thermal diffusivities. Thus, the present fabrication process

can achieve significantly faster thermal response than those reported in [19, 20].

Based on thermal diffusivities of the multimorph materials [48, 58], it is estimated

that designs fabricated by the current process have 1.4 times faster thermal

response time than similar designs with Al-SiO2 based multimorphs reported

in [19, 20].

8. System miniaturization and fill-factor: Unlike the device reported in [19, 20] that

requires an array of 72 straight multimorphs, the present design utilizes a single

curved actuator only. The 72 straight multimorphs require more than 3 times the

area of the semicircular actuator. Furthermore, since most laser spots are

0 10 20 30 40 50 60 70

350

300

250

200

150

100

50

0

Optical Scan Angle (Degrees)

Shift in

mir

ror-

pla

te c

en

ter

(μm

) Mirror actuated by semicircular multimorph

Mirror actuated by straight multimorphs

178

circular or elliptical, a circular mirror-plate actuated by a concentric curved

actuator can lead to greater system miniaturization than a rectangular mirror-

plate actuated by straight multimorphs.


This chapter describes a 1 mm wide circular micromirror actuated by a

semicircular electrothermal multimorph. Al and W constitute the active layers of the

multimorph. W also acts as a resistive heater. The unique feature of the device is the

combined bending and twisting deformations of the curved multimorph actuator. This

enables the device to exceed the performance of previously reported designs actuated

by straight multimorphs. At an applied voltage as low as 0.68 V, the mirror scans 60

and consumes 11 mW power only. The torsional stiffness encountered during

multimorph deformation ensures high resonant frequencies in excess of 100 Hz. The

device has scanning modes about two mutually perpendicular axes and can therefore

achieve two-dimensional scanning. A possible application is in hand-held devices for

biomedical imaging applications such as dental optical coherence tomography [124].

Higher resonant frequencies can be achieved by increasing the actuator width. The

mirror-plate center-shift produced by actuator bending partially compensates the shift

produced by actuator twisting. Therefore, the present design has 1.6 times lower center-

shift compared to straight multimorph based designs. Furthermore, the concentric layout

of the curved actuator along the periphery of the mirror-plate utilizes chip area more

efficiently than straight multimorph based designs of comparable size.

In this chapter, a curved electrothermal multimorph for micromirror actuation is

described. The mirror-plate size is comparable to previously reported devices that utilize

straight multimorphs [20]. This enables us to quantify the improvement in performance

179

due to the use of a curved multimorph actuator. The next chapter will present test

results on other micromirror designs actuated by curved actuators.

180

CHAPTER 10 ELECTROTHERMAL MICROMIRRORS ACTUATED BY CURVED MULTIMORPHS

10.1 Introduction

The previous chapter described a 1mm-wide micromirror actuated by a

semicircular multimorph. Detailed comparison to previously reported devices actuated

by straight multimorphs was provided. Advantages of curved actuator-based mirrors

including compact layout, low power requirements and high resonant frequency were

illustrated. The subsequent sections provide a compilation of test results for several

mirror designs actuated by curved multimorphs. All the designs reported in this chapter

share the fabrication process depicted in Figure 9-4. The thin-film thicknesses are

depicted in Figure 9-3C.

10.2 An Elliptical Mirror with 92 μm Minor Axis and 142 μm Major Axis

This section describes an elliptical micromirror actuated by a curved

electrothermal multimorph. The major and minor axes of the mirror-plate are 142 μm

and 92 μm, respectively. The micromirror has an optical scan angle of 22 at 0.37 V

applied voltage or 9 mW power input. Mirror center-shift produced by multimorph

bending and twisting compensate each other and is only 14.5 μm at an optical scan

angle of 22.The curved actuator shape maximizes chip-area utilization and ensures a

high resonant frequency. The first three resonant modes are at 3.9 kHz, 8.6 kHz and

17 kHz. Two-dimensional (2D) optical scanning is demonstrated using the second and

third resonant modes. The subsequent sections provide detailed device description and

test results.

181


Figure 10-1A shows an SEM of the elliptical electrothermal micromirror. The major

and minor axes of the mirror-plate are 142 μm and 92 μm, respectively. Upon release,

the mirror-plate is tilted at 2.5 with respect to the substrate due to residual stresses in

the multimorph actuator. The FE model in Figure 10-1B shows the simulated initial tilt.

Details on FE modeling will be provided in Section 10.2.3. The cross-section of the 9 μm

wide elliptical multimorph is shown in Figure 10-1C.

Figure 10-1. An elliptical mirror with 92 μm major axis and 142 μm minor axis. (A) SEM.

(B) Simulated initial displacement upon release. (C) Multimorph cross-section (not to scale).

10.2.2 Device Characterization

The device was characterized by applying a voltage to the W heater and

monitoring the position of a laser-spot reflected from the mirror-plate on a screen.


Figure 10-2A and Figure 10-2B illustrate the optical scan angle as a function of

applied dc voltage and input power, respectively. FE simulation results are also shown

Mirror plate

(A) (B) 200 μm

Curved multimorph

0

33 μm

0.14 μm

0.6 μm

0.6 μm

0.29 μm

6 μm

Sputtered Al

Sputtered W

PECVD SiO2

(C)

9 μm

182

and will be described in detail in Section 10.2.3. The mirror scans 22 at 0.37 V applied

voltage. The corresponding input power is 9 mW.

Figure 10-2. Static characteristic of device shown in Figure 10-1. (A) Optical scan angle

vs. voltage (B) Optical scan angle vs. power.

A unique feature of the proposed design is the low mirror center-shift during

scanning. Figure 10-3 plots the center-shift vs. optical scan angle. At 22 scan angle,

the center-shift is only 14.5 μm.

Figure 10-3. Mirror center-shift obtained by observing the device shown in Figure 10-1

under a microscope.


A 16 mV amplitude sine wave offset at 334 mV was used to obtain the frequency

response shown in Figure 10-4. The thermal time constant is only about 3 ms and may

be attributed to the high thermal diffusivities of Al and W. The first three resonant modes

0 5 10 15 20 25

14

12

10

8

6

4

2

0

Optical Angle (degrees)

Mirro

r cente

r-shift (μ

m)

0 100 200 300 400

25

20

15

10

5

0

25

20

15

10

5

0

0 2 4 6 8 10

Increasing voltage

Decreasing voltage

FE model

Voltage (mV) Power consumption (mW)

Optical an

gle

(degre

es)

Optical an

gle

(degre

es)

(A) (B)

Increasing voltage

Decreasing voltage

FE model

183

are very high, which are at 3.9 kHz, 8.6 kHz and 17 kHz, respectively. The resonant

frequencies from FE simulations are 4.1 kHz, 8.6 kHz and 17.4 kHz, respectively.

Figure 10-4. Frequency response of device shown in Figure 10-1 obtained using a sine

wave of 16 mV amplitude at 334 mV dc offset.

The simulated resonant modes have been shown in Figure 10-5. The high

resonant frequencies may be attributed to the large torsional stiffness encountered

during multimorph twisting. Interestingly, the first two modes correspond to scanning

about the y-axis and the third mode corresponds to scanning about the x-axis. This

unique feature may be attributed to the curved actuator geometry and enables the

mirror to achieve 2D scanning.

Figure 10-5. Simulated resonant modes of device depicted in Figure 10-1 at (A)

3.9 kHz, (B) 8.6 kHz, and (C) 17 kHz.

Figure 10-6 shows a 2D scan pattern generated by a reflected laser beam on a

screen. A 0-312 mV, 17 kHz sinusoidal signal amplitude-modulated at 8.04 kHz is used

for actuation. The 17 kHz signal excites the third resonant mode and produces an

(A) (B) (C) y

x

z

Frequency (Hz)

Optical an

gle

()

101

100

10

-1

10-2

100 102 104

184

optical scan range of 6.1. The 8.04 kHz modulation signal is close to the second

resonant mode and produces a scan range of 17.4.

Figure 10-6. Two-dimensional scan pattern generated on a screen using a laser beam

reflected from the mirror-plate of the device depicted in Figure 10-1. The mirror is actuated by a 0-312 mV, 17 kHz sinusoidal waveform amplitude modulated at 8.04 kHz.

10.2.3 Finite Element Model

10.2.3.1 Harmonic analysis

The simulated modes are show in Figure 10-5. The simulated frequencies of the

first and third modes are in agreement with experimental results to within 5%.

10.2.3.2 Estimation of heat loss coefficient

At the length scales involved, convection may be neglected and heat loss to the

surrounding air may be mainly attributed to thermal diffusion [19]. The heat loss

coefficient was obtained from a thermal FE model consisting of the air surrounding a

packaged device. The air adjacent to the package was assumed to be at room

temperature. The air adjacent to the device was at an elevated temperature. The

simulated heat loss coefficients at the actuator and the mirror-plate are 580 Wm-2K-1

and 200 Wm-2K-1, respectively. These coefficients were used in the electrothermal

model described in the next subsection.

10.2.3.3 Electrothermal model

An electrothermal model was used to simulate the temperature distribution for an

applied voltage of 0– 400 mV. The clamped ends of the actuator are fixed at room

17.4

6.1

185

temperature. Figure 10-7 shows the temperature distribution at 400 mV. The

temperature distribution along the actuator is symmetric about the connecting beam

attached to the mirror-plate. The thermal resistance in the mirror-plate region is low due

to the thick SCS layer. This results in a nearly uniform temperature distribution along the

mirror-plate. Figure 10-8 shows the temperature distribution along the actuator length.

The maximum temperature points are located close to the connecting beam. As shown

in Figure 10-9A, the maximum actuator temperature has a quadratic dependence on

applied voltage. Figure 10-9B shows linear dependence between the maximum actuator

temperature and input power.

Figure 10-7. Simulated temperature distribution for an applied voltage of 400 mV.

Figure 10-8. Temperature distribution along actuator length at applied voltages of 100 mV, 200 mV, 300 mV and 400 mV.

0 100 200 300 400 500 600

500

450

400

350

300

Tem

pera

ture

(K

)

100 mV

200 mV

300 mV

400 mV

Length along actuator (μm)

522 K

294 K

186

Figure 10-9. Maximum actuator temperature. (A) Maximum actuator temperature has a

quadratic dependence on applied voltage. (B) Maximum actuator temperature varies linearly with input power.

10.2.3.4 Mechanical model

The temperature distribution obtained from the electrothermal model was used to

simulate the scan angle and the results have been depicted in Figure 10-2. Good

agreement with experimental results is observed. The error may be attributed to the

differences between the material properties used for simulation and the actual

properties of the thin-films.

10.3 An Elliptical Mirror with 92 μm Minor Axis and 192 μm Major Axis

The previous section describes an elliptical mirror whose mirror-plate eccentricity

is 0.76. For the device discussed in this section, the mirror-plate eccentricity is 0.88.

The SEM of the device has been shown in Figure 10-10. The multimorph cross-section

is the same as that shown in Figure 10-1C. The gap between the mirror-plate edge and

the concentric actuator is 19 μm. The scan angle vs. voltage plot for dc excitation has

been shown in Figure 10-11A. Figure 10-11B depicts the scan angle vs. input power.

0 100 200 300 400 0 2 4 6 8 10

550

500

450

400

350

300

Power consumption (mW) Voltage (V)

550

500

450

400

350

300

M

axim

um

te

mp

era

ture

(K

)

M

axim

um

te

mp

era

ture

(K

)

(A) (B)

187

The mirror can scan 19.5 at an applied voltage of 300 mV. The corresponding power

consumption is 10.3 mW.

Figure 10-10. SEM of elliptical micromirror. The minor and major axes are 92 μm and 192 μm, respectively.

Figure 10-11. Static characteristic of device shown in Figure 10-10. (A) Optical scan angle vs. voltage (B) Optical scan angle vs. power.

The frequency response was obtained by applying a 5.5 mV amplitude sine wave

at 344 mV dc offset and has been shown in Figure 10-12. The first two resonant modes

are observed at 3.7 kHz and 6.1 kHz, respectively, and are similar to the modes shown

in Figure 10-5A and Figure 10-5B, respectively. The third mode corresponding to

transverse scanning is not observed because the corresponding frequency is very high

and the device response decays to zero at high frequencies.

The next section describes a circular micromirror with 400 μm optical aperture.

0 100 200 300 400 Voltage (mV)

20

15

10

5

0

Optical an

gle

(degre

es)

increasing voltage

decreasing voltage

20

15

10

5

0

Optical an

gle

(degre

es)

0 2 4 6 8 10 Input power (mW)

increasing voltage

decreasing voltage

(A) (B)

100 μm

188

Figure 10-12. Frequency response of mirror shown in Figure 10-10. The actuation voltage is a sine wave with 5.5 mV amplitude and 344 mV dc offset.

10.4 A 400 μm-wide Circular Mirror Actuated by a Semicircular Multimorph

Figure 10-13A shows an SEM of a 400 μm-wide circular mirror. The gap between

the mirror-edge and the concentric semicircular multimorph is 20 μm. The cross-section

of the multimorph is depicted in Figure 10-13B.

Figure 10-13. A 400 μm-wide circular micromirror actuated by a semicircular electrothermal multimorph. (A) SEM. (B) Multimorph cross-section.

10-1 100 101 102 103 104

101

100

10-1

10-2

10-3

Frequency (Hz)

Optical scan r

an

ge

()

Semicircular actuator

Mirror-plate

(A)

(B) 0.14 μm

0.6 μm

0.6 μm

0.29 μm

Sputtered Al

Sputtered W

PECVD SiO2

10 μm

7 μm

200 μm

189

The scan angle vs. voltage plot for dc excitation has been shown in Figure 10-14A.

Figure 10-14B depicts the scan angle vs. input power. The mirror can scan 19.5 at an

applied voltage of 300 mV. The corresponding power consumption is 10.5 mW.

Figure 10-14. Static characteristic of device shown in Figure 10-13. (A) Optical scan angle vs. voltage. (B) Optical scan angle vs. power.

The frequency response was obtained by applying a 13.5 mV amplitude sine wave

at 278 mV dc offset and has been shown in Figure 10-15. A resonant mode is observed

at 514 Hz. The mode corresponding to transverse scanning is not observed because

the corresponding frequency is very high.

Figure 10-15. Frequency response of mirror shown in Figure 10-13. The actuation voltage is a sine wave with 13.5 mV amplitude and 278 mV dc offset.

0 100 200 300

20

16

12

8

4

0

Voltage (mV)

(A)

increasing voltage

decreasing voltage

20

16

12

8

4

0

Optical an

gle

(degre

es)

0 2 4 6 8 10 Input power (mW)

increasing voltage

decreasing voltage

(B)

10-1 100 101 102 103 Frequency (Hz)

101

100

10-1

10-2

10-3

Optical scan r

an

ge

()

Optical an

gle

(degre

es)

190

10.5 Summary and Future Work

Circular and elliptical mirrors actuated by curved electrothermal multimorphs have

been presented in Chapters 9 and 10. The active layers of the multimorphs are Al and

W. The high thermal diffusivity of Al and W leads to fast thermal response. W also acts

as a resistive heater and is used for Joule heating. The high resonant frequencies of the

reported devices may be attributed to the twisting deformation of curved multimorphs.

Other advantages include high-fill factor, low mirror-plate center-shift and low power

consumption.

Future work may include the fabrication of curved piezoelectric and SMA

multimorphs. For instance, piezoelectric micromirrors for display systems may utilize the

twisting deformation of curved multimorphs to achieve high resonant frequencies.

Furthermore, the combined bending and twisting deformations of curved multimorphs

may be applied to other devices as well and greatly expands the design space available

to the MEMS engineer.

191

CHAPTER 11 BURN-IN, REPEATABILITY AND RELIABILITY OF ELECTROTHERMAL

MICROMIRRORS

11.1 Background

Successful commercialization of a device requires a thorough investigation into its

burn-in, repeatability and reliability. Transistor reliability has been widely investigated

over the last few decades [125, 126]. Texas Instruments’ digital micromirror devices

(DMDs) owe their phenomenal success to in-depth investigation into failure

mechanisms as part of the design cycle [1]. Reliability studies on ink jet print heads,

inertial sensors, pressure sensors, micro-mirror arrays, and RF switches have been

reported [127]. Similar to electronic circuit elements, MEMS devices must go through a

burn-in phase at the beginning of their lifetime [128]. The burn-in process sieves out

devices that would otherwise fail in their infancy by stressing them [129]. Therefore,

burn-in acts as a screening process in which devices that fail during their early lifetime

are rejected [130]. Additionally, it makes the characteristics of the working devices

repeatable and stable [19, 131]. Hence, conditions for achieving successful burn-in

must be clearly identified. In this dissertation, the term burn-in refers to the pre-

conditioning process that makes newly released devices repeatable and stable. Failure

rate has not been addressed in this thesis and requires further investigation.

It is important to investigate the range of actuation signals that may be applied

without degrading the device performance. The study of device failure due to

overvoltage, creep, fatigue and impact provide valuable insights for design

improvement. Since SiO2 is brittle, the oxide thermal isolation region fails on accidental

impact. Current generation of Al-SiO2 bimorph based MEMS devices cannot withstand

192

drop test on a hard surface for drop heights greater than a few centimeters. A key goal

of this thesis is to design a process for fabricating robust mirrors.

As discussed in Chapter 1, the large scan range and low voltage requirements

make electrothermal mirrors suitable for a wide range of applications. Pal et al. report

preliminary results on the repeatability of electrothermal micromirrors [19]. The next

section deals with device burn-in and repeatability. Device failure modes are

enumerated in Section 11.3.

11.2 Burn-in and Repeatability

As discussed in Section 11.1, the burn-in process makes a device repeatable and

stable. The three important parameters of the micromirror shown in Figure 4-1 are

embedded heater resistance, initial mirror-plate tilt angle and mirror scan angle. In this

section, these parameters are discussed one by one.

11.2.1 Embedded Heater Burn-in

A newly released micromirror was used for studying heater burn-in. A dc voltage

source was used for actuation and the current passing through the heater was

monitored. The voltage was gradually increased from 0 to 4.5 V and then decreased to

0. This process was repeated for 6 V, 7.5 V, 9 V, 11 V and 13 V. As shown in Figure 11-

1A, significant hysteresis is observed in the heater characteristic. However, as shown in

Figure 11-1B, the heater resistance becomes repeatable once 7.5 V is applied.

Thereafter, the characteristic continues to be repeatable when the voltage is increased

to 9 V and brought back to zero. Repeatability is observed up to an applied voltage of

11 V (not shown). During subsequent actuation, the heater is found to operate along the

repeatable characteristic for applied voltages less than 11 V. From Figure 11-1, the

burn-in process causes a reduction in the heater resistance.

193

Figure 11-1. Embedded heater characteristic. (A) Heater characteristic of newly released device at low voltage. (B) Heater characteristic becomes repeatable at higher voltage.

Heater burn-in was studied for 12 micromirrors. The mean and standard deviation

of the resistance values have been summarized in Table 11-1. The post burn-in

resistance value is significantly less than that of a newly released device. Also, burn-in

causes a marked reduction in the standard deviation of the resistance values. So, burn-

in makes the device repeatable and reduces performance variations from one device to

another. It was found that 9 V was sufficient to achieve successful burn-in for all mirrors.

Table 11-1. Heater resistance before and after burn-in for 12 devices.

Before burn-in After burn-in Mean

Standar

d deviation

Mean Standard

deviation

356.7Ω 197.1Ω 153.9Ω 5.2Ω

Heater burn-in was also observed in unreleased devices. In an unreleased device,

the silicon below the bimorphs, heater and isolation region is still present and the silicon

etch step [8] has not been performed. Figure 11-2 shows the heater characteristic of an

unreleased device. Data 1–3 have been obtained in chronological order. The voltage is

first increased from 0 V (Data 1), then decreased back to 0 V (Data 2) and finally

0 1 2 3 4 5 6

0 2 4 6 8 10

0 – 4.5 V

4.5 V – 0

0 – 6 V

6 V – 0

0 – 7.5 V

7.5 V – 0

0 – 9 V

9 V – 0

Voltage (V)

Voltage (V)

35

30

25

20

15

10

5

0

50

40

30

20

10

0

C

urr

ent (m

A)

(A) (B) Curr

ent (m

A)

194

increased again (Data 3). As in the case of released devices, the post burn-in heater

characteristic was found to be repeatable. Hence, it is possible to do wafer level burn-in

of the embedded heater.

Figure 11-2. Burn-in characteristic of an unreleased device. Data 1–3 are in chronological order.

An unreleased device is one in which the silicon below the bimorphs has not been

removed. Therefore, for the same current, the heater temperature of an unreleased

device is expected to be lower than that of a released device. From Figure 11-1B, for a

released device, heater burn-in occurs at about 7.5 V, which corresponds to a current of

40 mA and heater temperature of 529 C. From Figure 11-2, in the case of an

unreleased device, burn-in occurs at about 10 V. The corresponding current and

estimated heater temperature are 55 mA and 503 C, respectively. Therefore, when the

temperature is lower, a higher current is required to achieve burn-in. Thus, both current

and temperature influence the burn-in of embedded heaters. Furthermore, when a

newly released device was heated up to 547 C, no heater burn-in was observed.

Therefore, current is essential for achieving successful heater burn-in.

Voltage (V)

0 5 10 15

80

60

40

20

0

Curr

ent (m

A)

Data 1 Data 2

Data 3

195

11.2.2 Scan Angle Repeatability

A newly released mirror was used for studying device burn-in. A dc voltage source

was used for actuation. A laser beam reflected from the mirror-plate was tracked on a

screen. The position of the laser spot on the screen was used to evaluate the optical

angle scanned by the mirror. The finite size of the mirror-plate leads to an uncertainty in

the measured optical angle data. For an optical angle of 40, the error is estimated to be

less than 0.15º. The results have been plotted in Figure 11-3. Data 1–3 were obtained

in chronological order. The actuation voltage was first increased from 0 to 11 V (data 1),

then decreased to zero (data 2) and thereafter increased again (data 3). After this

process, the scan angle of the mirror was found to be repeatable for voltages less than

11 V.

Figure 11-3. Scan angle vs. voltage for a released micromirror.

In Figure 11-3, the vertical intercepts of the plots corresponding to data 2 and 3

differ from that corresponding to data 1. This indicates a change in the mirror-plate tilt

angle during the burn-in process. The effect of burn-in on the initial mirror-plate tilt is the

subject of the next subsection.

0 2 4 6 8 10 12

30

25

20

15

10

5

0

-5

Optical an

gle

(degre

es)

Data1

Data2

Data3

dc voltage (V)

196

11.2.3 Initial Tilt of Mirror-plate

As shown in Figure 4-1, the mirror-plate is tilted at an angle to the substrate due to

the residual stresses in the bimorph thin films. The tilt angle is an important parameter

for designing optical systems. The initial tilt of the mirror-plate with respect to the

substrate was about 20º for a newly released micromirror. This parameter was found to

increase by about 0.5º–5º during the burn-in process.

In Figure 11-3, the vertical intercept of the plots corresponding to data 2 and 3 is

-2.8. This indicates that the post burn-in mechanical tilt of the mirror-plate in the

unactuated state increased by 2.8/2, i.e., 1.4.

To further investigate the change in the initial tilt angle, a newly released mirror

was actuated by a dc source. Before actuation, the mirror-plate tilt was found to be

20.5º. The optical scan angle, applied voltage and device current were monitored. The

optical angle has been plotted against the applied voltage in Figure 11-4A. Data 1

through data 5 were obtained in chronological order. Data 1–2 in Figure 11-4A

correspond to applied voltage in the range 0–10 V and are similar to the plots shown in

Figure 11-3. After the mirror is subjected to 10 V for the first time, its characteristic is

found to be repeatable in the range 0–10 V. As long as the mirror is operated below

10 V, it will continue to operate along the curve corresponding to data 2. At this point the

initial mirror tilt has increased by 1.5º. Next, the applied voltage was increased to 13.4 V

(data 3) and then decreased to 0 (data 4). This shifts the device characteristic and the

initial tilt of the mirror-plate is found to be 24. Thus, the total increase in initial mirror tilt

is 3.5º. Plasticity effects such as slip are believed to cause the change in the tilt angle.

As long as the applied voltage is maintained below 13.4 V, the device continues to

operate along the characteristic corresponding to data 4 and 5.

197

It may be concluded from Figure 11-3 and Figure 11-4A that for making the device

characteristic repeatable, it must first be operated at the maximum voltage required by

the specific application.

Figure 11-4. Mirror scan angle. (A) Optical angle vs. voltage plots for a newly released mirror. Data 1–5 are obtained in chronological order. (B) Optical angle vs. input power plots corresponding to the data shown in (A).

The optical angle corresponding to Figure 11-4A has been plotted against the

electrical power input in Figure 11-4B. Some non-linearity is observed for power inputs

greater than 0.65 W. The optical angle varies linearly over the range of temperatures for

which the material properties do not vary significantly [29]. Therefore, the temperature

0 2 4 6 8 10 12 14

dc voltage (V)

Data1 010V

Data2 10V0

Data3 013.4V

Data4 13.4V0

Data5 012.5V

50

40

30

20

10

0

-10

O

ptical an

gle

(degre

es)

(A)

O

ptical an

gle

(degre

es)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Input electrical power (W)

50

40

30

20

10

0

-10

Data1 010V

Data2 10V0

Data3 013.4V

Data4 13.4V0

Data5 012.5V

(B)

198

dependence of material properties becomes significant for power inputs greater than

0.65 W. The embedded heater resistance can be evaluated using the measured values

of voltage and current at each data point. The highest temperature point on the bimorph

corresponds to the embedded heater, which is located at the edge of the bimorph close

to the substrate. Equation 4-3 may be rewritten as,

0

0

11E

h

E

RT T

α R

(11-1)

Using Equation 11-1, the heater temperature is estimated to be 613 C for 0.65 W

power input. From data available in literature [131], it has been estimated that the

Young’s modulus of Al drops by about 30% from room temperature to 613 C. The drop

in Young’s modulus of Al is believed to be the main reason for the nonlinearity observed

for high power input.

At low power input, the plots corresponding to data 2 and 3 are parallel to the plots

corresponding to data 4 and 5. Therefore, a higher burn-in voltage does not alter the

optical angle scanned per unit power input. It only changes the initial tilt of the mirror-

plate.

Repeatability of the mirror tilt angle may also be achieved by heating the

micromirror to a high temperature. A newly released device was heated using a hot

plate and the optical angle was tracked. The experimental data are plotted in Figure 11-

5. Data 1–3 were obtained in chronological order. The temperature was first gradually

increased (data 1), then decreased (data 2) and finally increased again (data 3). The

plots corresponding to data 2 and 3 were found to coincide.

199

Figure 11-5. Optical angle of a laser beam reflected from a newly released micromirror placed on a hot-plate.

11.3 Device Failure

11.3.1 Failure Due to Overvoltage

A device was actuated by 9 V dc and monitored over a period of three hours. No

significant change in the heater current and optical angle was observed. The measured

heater current was 46 mA. Using Equation 11-1, the heater temperature is estimated to

be 562 C for an applied voltage of 9 V. Next, the voltage was increased from 0 to 11 V

and brought back to 0 in steps of 0.5 V. For voltages in the range 0–11 V, the

embedded heater characteristic was found to be repeatable. However, when the voltage

was increased to 13 V for the very first time, the heater resistance increased as shown

in Figure 11-6. On subsequent actuation, the embedded heater was found to operate

along the characteristic corresponding to data 2 in Figure 11-6. The increased heater

resistance is the result of physical damage to the heater structure. Thus, 13 V defines

an upper limit to the safe actuation voltage range.

In order to study the failure at high voltage, 16.2 V was applied to a micromirror

and the heater current and optical angle were monitored. Figure 11-7A shows the

heater current and Figure 11-7B shows the optical angle. The heater became open

0 50 100 150 200 250

Hot plate temperature (C)

Data1

Data2

Data3

50

40

30

20

10

0

-10

Optical an

gle

(degre

es)

200

circuit after 14,700 minutes. As shown in Figure 11-7, the entire duration may be

roughly divided into two stages. During stage I, the rates of change of the heater current

and optical angle are high. Stage I roughly spans the first 5,000 minutes. During stage

II, the drift in heater current and optical angle is slow.

Figure 11-6. Deteriorated embedded heater characteristic at high voltage.

Figure 11-7. Failure at high voltage. (A) Heater current and (B) Optical angle for an applied voltage of 16.2 V

In order to investigate the damage caused at high voltage, the failed device was

observed under a microscope. As shown in Figure 11-8, damage was observed at the

platinum segment connecting the embedded heater to the bond pad. Points on the

heater that have high temperature and current density are potential locations of failure.

0 2 4 6 8 10 12 14

dc voltage (V)

dc c

urr

ent (A

)

013 V (data 1)

13 V0 (data 2)

0.06

0.05

0.04

0.03

0.02

0.01

0

Stage II Stage I

Heater fails at this point

dc c

urr

ent

(A)

Time (minutes) Time (minutes)

70

60

50

40

30

20

10

0.06

0.058

0.056

0.054

0.052

0.05

0.048

O

ptica

l a

ng

le (

degre

es)

0 5,000 10,000 15,000

0 5,000 10,000 15,000

(A) (B)

Stage I Stage II

201

Figure 11-8. Damaged end of embedded heater.

The electrical finite-element (FE) model of the heater shown in Figure 11-9 may be

used to understand the failure mechanism. Figure 11-9 shows the current density in the

embedded heater. The model ignores the temperature dependence of the heater

resistance. Therefore, it will only be used to provide a qualitative understanding of the

failure mechanism. From Figure 11-9, it is observed that there is a high current density

at the corner of the inner segment that connects the heater to the bonding pad. The

current crowding effect [80] is responsible for the uneven distribution in current density.

The high current density and power dissipation cause the inner Pt segment to be a local

hot spot. The Pt-SiO2 bi-layer experiences bending stresses due to the difference in

their thermal expansion coefficients. This results in yielding of the inner Pt segment due

to thermomechanical creep. This explanation is in agreement with the image of the

failed device shown in Figure 11-8. It is posited that the failure of the inner Pt segment

corresponds to stage I shown in Figure 11-7.

24µm

Bond pad Damaged Pt segment between heater an bond-pad

202

Figure 11-9. Current density obtained from an electrical finite element model of the embedded heater for an applied voltage of 1 V.

After the inner Pt segment fails, the current density distribution is qualitatively

represented by the FE model shown in Figure 11-10.

Figure 11-10. Current density distribution after inner Pt segment fails. Applied voltage = 1 V.

As in the case of the FE model shown in Figure 11-9, the temperature

dependence of resistivity is ignored. Therefore, Figure 11-10 provides a qualitative

explanation only. As shown in Figure 11-10, a large current density exists in the Pt

segment connecting the heater to the bond pad. This results in a local hot spot at one

end of the heater. A significant temperature gradient now exists across the bimorph

array. Therefore, different bimorphs tend to curl at different angles thereby exerting

torsion on the embedded heater. It is proposed that the torsion produced due to the

temperature gradient across the bimorph array results in heater failure due to shear.

Min: 0

Max: 12.210

9 A/m

2

High current density

Min: 0

Max: 10109 A/m

2

High current density

203

This corresponds to stage II in Figure 11-7. Microscopic examination of the failed device

shows that the Al layer near the damaged end of the heater melted during the device

failure. The Al far away from the local hot spot did not melt. This confirms that there was

a large temperature gradient across the bimorph array.

11.3.2 Impact Failure

A major limitation faced by several electrothermal mirrors arises due to the brittle

nature of SiO2. The thermal isolation region consists of thin film SiO2 only and is

especially susceptible to failure. Brittle fracture is often encountered during device

handling and packaging.

A micromirror was attached inside a plastic box using double sided tape and

dropped from a height of 40 cm on a vinyl floor. Figure 11-11 shows SEM images of the

failed device. It is found that the mirror fails at the thermal isolation connecting the

bimorphs to the mirror-plate. A major contribution of this thesis is a novel process for

fabricating robust micromirrors, which is discussed in the next chapter.

Figure 11-11. SEM images of failed mirror. The mirror was dropped from a height of 40 cm. (A) Substrate. (B) Zoomed-in view of the mirror-plate.

1 mm 100 μm

(A) (B)

204

11.3.3 Other Reliability Issues

11.3.3.1 Creep

A mirror was placed on a hot plate at 240 ºC and the position of a laser beam

reflected from the mirror-plate was tracked. No significant change in the mirror rotation

angle was observed over a period of 4 hours. However, thermomechanical creep may

become significant at higher temperatures [132].

11.3.3.2 Fatigue

When a specimen is subjected to alternating stress cycles, damage accrues over

several stress cycles. This phenomenon is known as fatigue [132]. It may manifest in

the form of change in scan angle and resonant frequency.

11.3.3.3 Environmental factors

Both Al and SiO2 are resistant to atmospheric corrosion. On exposure to the

atmosphere, a thin oxide layer forms on the Al surface [133]. This protects Al from

environmental corrosion. The device may also be enclosed to avoid the deposition of

particulate matter.

11.4 Summary and Future Work

The three key parameters of a 1D bimorph based electrothermally-actuated

micromirror are the embedded heater resistance, initial mirror tilt and mirror scan angle.

The conditions necessary for making these three parameters repeatable and stable are

established. It is found that device preconditioning can be achieved by using an

appropriate actuation voltage. A voltage of 7.5 V is sufficient for preconditioning the

embedded heater. The burn-in process reduces the standard deviation in the resistance

values of the embedded heater. The heater resistance decreases during burn-in. The

initial mirror-plate tilt is found to increase by a few degrees during the burn-in process.

205

In order to make the initial tilt and mirror scan angle repeatable, the device must first be

operated at the maximum voltage required by the particular application.

Device failure observed at high voltage is investigated. Based on finite element

simulations, it is posited that one of the two Pt segments connecting the bond pad to the

heater has a high current density and fails first. Failure of the Pt segment is also

confirmed by microscopic examination of the failed device. Failure of one of the Pt

segments results in a high current density in the other Pt segment, thereby creating a

hot spot. This hot spot establishes a large temperature gradient across the bimorph

array. The resulting torsion causes the heater to shear. The presence of the hot spot is

confirmed by microscopic examination of the failed device and by finite element

simulations.

For practical drive circuit design, it is essential to quantify device repeatability, as

well as expected variations in response for a particular design. Furthermore, it is

important to identify the safe actuation signal range to minimize degradation in device

performance. Therefore, investigation into device repeatability and reliability takes

electrothermal micromirrors a step closer to real world applications such as biomedical

imaging and micromirror arrays.

Micromirrors with SiO2 thermal isolation are highly susceptible to impact failure

due to the brittle nature of SiO2. A novel process for fabricating robust mirrors will be

discussed in the next chapter. Failure mechanisms such as creep and fatigue require

long term testing and further investigation.

Future work will involve long term device testing and further investigation into

mirror failure mechanisms.

206

CHAPTER 12 A PROCESS FOR FABRICATING ROBUST ELECTROTHERMAL MEMS WITH CUSTOMIZABLE THERMAL RESPONSE TIME AND POWER CONSUMPTION

REQUIREMENTS

12.1 Background

As discussed in Chapter 1, several bimorph actuated MEMS devices utilize SiO2

as an active layer of the bimorph and for thermal isolation. Due to its use in CMOS

processes, deposition and etch recipes for SiO2 are well developed and this has lead to

widespread use of SiO2 in MEMS devices. Also, the low CTE of SiO2 [48] enables

metal-SiO2 bimorphs to produce large deflections. Additionally, the low thermal

conductivity of SiO2 [134] makes it suitable for thermal isolation. However, micromirrors

actuated by Al-SiO2 bimorphs have two major drawbacks. Firstly, the low thermal

diffusivity of SiO2 makes the response of metal-SiO2 bimorph devices sluggish.

Secondly, devices employing SiO2 thermal isolation are susceptible to impact failure

due to its brittle nature. As discussed in Chapter 11, micromirrors with SiO2 thermal

isolation cannot withstand drop tests from more than a few centimeters height. This

makes them unsuitable for hand-held applications that may involve frequent drops from

up to a height of few feet. During a project on hand-held dental imaging probes with

Lantis Laser Inc. [124], the susceptibility of the micromirrors to impact failure proved to

be a major stumbling block. Another major drawback of SiO2 is that films thicker than

~1μm tend to crack due to large residual stresses.

Several metal-polymer bimorphs have also been reported [48]. Since polymers

have the lowest Young’s modulus among MEMS materials, the polymer layer is much

thicker than the metal layer [48]. Therefore, the thermal response is mainly determined

207

by the diffusivity of the polymer. Typically, polymers have very low thermal diffusivity

(~10-7m2/s [46, 135]) which makes the overall response slow.

As a solution to the aforementioned problems, a novel process for fabricating

robust micromirrors [135] is discussed in this chapter. The process allows the design

engineer to customize device speed and power requirements depending on the

application. A simple test-setup based on Newton’s cradle [136] is used for quantifying

device robustness. Drop tests are used for simulating real-world impact events. This

chapter focuses on micromirror fabrication, but the proposed process can be adapted to

a wide range of electrothermal MEMS devices.

This chapter is organized as follows. The next section surveys candidate materials

for mirror fabrication. Section 12.3 discusses the fabrication process. Static

characterization and frequency response of fabricated devices are presented in Section

12.4. Experimental results on improvement in device robustness are presented in

Section 12.5.

12.2 MEMS Materials for Thermal Multimorphs

A wealth of information on the properties of candidate materials for thermal

multimorphs is available in literature [45, 48]. The goal of this research effort is to allow

design engineers to customize thermal response time and power consumption

requirements. In order to increase the thermal response time, materials with large

thermal diffusivity must be employed. Metals usually have higher thermal diffusivity than

non-metals [48]. Certain non-metals such as diamond-like-carbon (DLC) have very high

thermal diffusivity as well [46, 121]. Also, according to Equation 1-1, it must be ensured

that the two active layers of the thermal bimorphs have widely different CTEs.

208

The thermal isolation region must have low thermal conductivity. Among metals,

invar has one of the lowest thermal conductivities. Polymers may also be used for

thermal isolation as they typically have low thermal conductivity [48]. High temperature

polymers [134] are especially suitable as they can withstand high temperatures

encountered during fabrication as well as device operation.

Several materials were surveyed as candidate materials for the new fabrication

process. These have been listed in Table 11-2.

Table 11-2. Candidate materials for fabricating electrothermal micromirrors.

Material

Young's Modulus

GPa CTE

(microns/m/K)

Thermal Conductivity

(W/m/K)

Thermal diffusivity

(m2/s)

Electrical conductivity

(S/m) comments

Al [134] 70 23.1 237 9.710-5

35.5106

High CTE, Fast thermal response

SiO2 [134] 70 0.5 1.4 8.710-7

Insulator Brittle, slow thermal response

Pt [134] 168 8.8 71.6 2.510-5

8.9106

Slow thermal response

W [134] 411 4.5 174 6.810-5

20106

Can be used as heater and active bimorph layer

Invar 145 [48] 0.36 [48] 13 [48] 3.110-6

[48]

One of the least thermally conductive metal

DLC [137] 700 1.18 1100 6.0610-4

Doping

dependent High thermal speed

Polyimide 2.3 [134] 20 [134] 0.15 [134] 1.0410-7

[134] Insulator Can achieve good thermal isolation

The material combinations that were considered for forming the active layers of

the actuators have been listed below:

Polymer-metal bimorphs: Polymers have a very high CTE among MEMS materials [45]. However, according to Equation 1-5, the polymer layer must have a

209

large thickness. Since polymers have very low thermal diffusivities [48], their thermal response time is very slow. Therefore, a thick polymer layer will adversely affect the thermal response time of a bimorph electrothermal actuator. Additionally, the low thermal conductivity of a polymer may result in a large thermal gradient across its thickness. The thermal gradient effect tends to counter the bimorph actuation effect.

DLC based bimorphs: Certain types of DLC have very high thermal diffusivity [137] which makes their thermal response very fast. The low CTE of DLC makes an Al-DLC bimorph feasible. The fracture strength of diamond thin films is in the 2.8–4.1 GPa range, which is one of the highest among MEMS materials [137]. However, as shown in Table 11-2, DLC has a very high Young’s modulus. Therefore, according to Equation 1-5, the thickness of the DLC film is much less than the other layer in the bimorph structure. Thus, the thermal speed of the bimorph is mainly determined by the thermal diffusivity of the latter. Consequently, the bimorph structure is significantly slower than DLC itself. Another problem is that it is difficult to grow good quality DLC film with low thickness.

Metal-metal bimorphs: Since Al and W have reasonably high thermal diffusivity, Al-W bimorph is a viable option. Also, W has a very high Young’s modulus. Therefore according to Equation 1-5, a thin layer of W may produce optimum results. Another advantage of using W as a bimorph material is that it can double up as a resistive heater.

Based on high diffusivities and large difference in CTEs, three possible pairs stand

out — Al-DLC, Al-Invar, and Al-W. The newly fabricated micromirrors are actuated by

Al-W bimorphs with W also acting as a resistive heater. The thin-film thickness values

are same as that depicted in Figure 9-3C. As discussed in Section 9.2.2.3, the thickness

values are chosen to maximize multimorph deflection. The PECVD SiO2 that

encapsulates W, provides electrical isolation between Al and W. SiO2 also serves to

protect W from fluoride based etch recipes.

Other than SiO2, invar [48] and polyimide [46, 135] can potentially be used for

forming the thermal isolation region at one or both ends of the multimorph actuators.

The newly fabricated micromirrors utilize high temperature polyimide, PI 2574 from HD

Microsystems [134], for thermal isolation. Both 1D and 2D scanning micromirrors were

fabricated. The next section describes the process flow in detail.

210

12.3 Fabrication Process

The micromirror fabrication process is illustrated in Figure 12-1. SOI wafers are

used to ensure flatness of mirror-plates with single-crystal-silicon microstructures.

Figure 12-1. Fabrication process for robust mirrors. (A) Oxide deposition, patterning, etching on SOI wafer. (B) W sputtering and lift-off. (C) Oxide deposition, patterning, etching. (D) Al sputtering and lift-off. (E) Polyimide spin coating and baking, PECVD oxide deposition, patterning, oxide etching and polyimide etching. (F) Backside lithography, DRIE Si etch and buried oxide etch. (G) Front-side isotropic Si etch for device release.

Though thermal isolation is shown at one end of the bimorph only, there may be

isolation at both ends. The SiO2 deposited in Figure 12-1A and Figure 12-1C protect W

from fluoride based etch recipes. Additionally, the SiO2 deposited in the step shown in

Figure 12-1C electrically isolates the Al and W layers. Both Al and W are fabricated by

sputtering and lift-off (Figure 12-1B and Figure 12-1D). Polyimide is spin coated, baked

and then covered with PECVD SiO2 (Figure 12-1E). This layer of SiO2 acts as a mask

for etching the 5 μm-thick polyimide. The device is released by backside DRIE silicon

etching (Figure 12-1F), buried oxide etch and frontside isotropic silicon etching (Figure

(A)

(B)

(C)

(D)

(E)

(F)

(G)

Si SiO2 W Al Polyimide

Substrate Mirror-plate

Polyimide thermal isolation

Bond-pad

Multimorph

211

12-1G). The process shown in Figure 12-1 can be used to fabricate polyimide beams for

thermal isolation. Therefore, such thermal isolation will be referred to as beam-type

thermal isolation.

A variation of the process depicted in Figure 12-1 is shown in Figure 12-2. In this

variation, trenches are created by isotropic Si RIE etch in Figure 12-2E. The polyimide

is dispensed and the wafer is placed in vacuum to drive air bubbles out of the trenches.

Figure 12-2. Modified fabrication process for robust mirrors with trench isolation. (A) Oxide deposition, patterning, etching on SOI wafer. (B) W sputtering and lift-off. (C) Oxide deposition, patterning, etching. (D) Al sputtering and lift-off. (E) Trench formation by Si isotropic etch using RIE. (F) Polyimide dispensation, vacuum pressure for trench filling, spin coating, baking, oxide deposition, patterning, oxide etching and polyimide etching. (G) Backside lithography, DRIE Si etch and buried oxide etch. (H) Front-side isotropic Si etch for device release.

(A)

(B)

(C)

(D)

(E)

Si SiO2 W Al Polyimide

(F)

(G)

(H)

Bond-pad

Trench-filled thermal isolation

Multimorph actuator

Substrate

Mirror-plate

212

After driving out the air bubbles, the polyimide is spin coated and baked. Figure 12-2H

shows trench filled polyimide thermal isolation between multimorph actuators and the

mirror-plate. The process shown in Figure 12-2 can be used to fabricate polyimide-filled

trenches for thermal isolation. Therefore, such thermal isolation will be referred to as

trench-type thermal isolation.

1D and 3D mirrors were fabricated using the processes shown in Figure 12-1 and

Figure 12-2. SEMs of fabricated 1D mirrors are shown in Figure 12-3 through Figure 12-

7.

Figure 12-3. SEM of 1D mirror with no thermal isolation.

The device shown in Figure 12-3 does not have any thermal isolation and can be

used to quantify the improvement in power consumption when thermal isolation is

incorporated in device design. Figure 12-4 shows a 1D mirror with beam-type thermal

isolation at both ends of the actuators. Therefore, this device has similar topology to the

device discussed in Chapter 3, which employs SiO2 thin film isolation.

Mirror-plate

Array of bimorphs

1 mm

213

Figure 12-4. SEM of 1D mirror with beam-type thermal isolation at both ends of the

actuators.

Figure 12-5 shows a device with beam-type thermal isolation at both mirror and

substrate ends. This design has larger openings around the polyimide beams compared

to the design shown in Figure 12-4. The large openings allow easy removal of Si from

under the polyimide during device release by Si etch. Figure 12-6 shows a device with

beam-type isolation between the actuators and mirror-plate only. Figure 12-7 shows a

device with polyimide filled trench-type isolation between the actuators and the mirror-

plate. Close examination of the trenches shows that they are not filled completely. This

is possibly because the uncured polyimide tends to flow out of the trenches during wafer

spinning. Another issue with trench filled designs is that the RIE Si etch process that

was used to define the trenches partially damages the tungsten layer. In some designs,

this damage manifests in the form of increased resistance of the tungsten heater.

1 mm

Beam-type polyimide thermal isolation

Mirror-plate

Array of Al-W Bimorphs

214

Therefore, the fabrication process shown in Figure 12-2 requires modifications and

further refinement.

Figure 12-5. SEM of mirror with polyimide-beam isolation at both ends of the actuators.

(A) 1D mirror. (B) Connection between actuators and substrate. (C) Connection between actuators and mirror-plate. Compared to the device shown in Figure 12-4, this device has larger openings around the polyimide beams. These openings allow easier removal of silicon from underneath the polyimide beams during device release.

1 mm

(A)

Polyimide thermal isolation

50 μm

(B) (C)

100 μm

Mirror-plate

Array of bimorphs

215

Figure 12-6. SEM of robust 1D mirror with beam-type thermal isolation between

actuators and mirror-plate only.

Figure 12-7. SEM of robust 1D mirror with trench-type thermal isolation between

actuators and mirror-plate only. It is observed that the trenches are not completely filled. This is possibly due to the outflow of polyimide from the trenches during wafer spinning.

SEMs of 3D lateral-shift-free (LSF) [31] mirrors are shown in Figure 12-8 through

Figure 12-10. The mirror shown in Figure 12-8 does not have any thermal isolation.

1 mm Trench-type isolation

Mirror-plate

Array of bimorphs

1 mm

Polyimide beam

Mirror-plate

Array of bimorphs

216

Figure 12-8. SEM of robust 3D lateral-shift-free (LSF) mirror with no thermal isolation.

Figure 12-9. SEM of robust 3D lateral-shift-free (LSF) mirror with thermal isolation

beams between the actuators and the mirror-plate. (A) Device. (B) Polyimide beams between actuator and mirror-plate.

(A) (B)

50 μm

1 mm

Mirror-plate

Bimorphs

Rigid beam

217

Figure 12-10. SEM of robust 3D lateral-shift-free (LSF) mirror with trench-filled thermal

isolation between the actuators and the mirror-plate. (A) Device. (B) Thermal isolation between actuator and mirror-plate.

The designs shown in Figure 12-9 and Figure 12-10 have thermal isolation

between mirror-plate and actuators. The mirror shown in Figure 12-9 was fabricated

using the process shown in Figure 12-1. The device shown in Figure 12-10 has trench-

type thermal isolation between the mirror-plate and the actuators.

Device characterization is discussed in the next section.

12.4 Device Characterization

The static characteristic of the 1D mirror shown in Figure 12-3 through Figure 12-7

are depicted in Figure 12-11 through Figure 12-15, respectively. By using a linear fit, the

scan angle per unit power input values were extracted from Figure 12-11B through

Figure 12-15B and have been listed in Table 12-1.

1 mm

Mirror-plate

Bimorphs

Trench-filled thermal isolation

100 μm

Trenches

(A)

(B)

218

Figure 12-11. Static characteristic of 1D mirror with no thermal isolation (Figure 12-3). (A) Optical scan angle vs. applied dc voltage. (B) Optical scan angle vs. input power.

Figure 12-12. Static characteristic of 1D mirror with beam-type thermal isolation at both ends of the actuators. The mirror has been depicted in Figure 12-4. (A) Optical scan angle vs. applied dc voltage. (B) Optical scan angle vs. input power.

Among all devices, the device with no thermal isolation consumes the highest

power for the same scan angle. The sparsely spaced beam-type isolation ensures that

the device in Figure 12-5 consumes the least amount of power. The devices shown in

Figure 12-4 and Figure 12-5 have similar topology; their power requirements are

Increasing voltage

Decreasing voltage

12

10

8

6

4

2

0

Optica

l an

gle

()

0 20 40 60 80 100 120 140

12

10

8

6

4

2

0

Increasing voltage

Decreasing voltage

Optical an

gle

()

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Voltage (V) Input power (mW)

(A) (B)

Increasing voltage Decreasing voltage

Increasing voltage Decreasing voltage

0 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1

30

25

20

15

10

5

0

30

25

20

15

10

5

0 Voltage (V) Input power (mW)

Optical an

gle

()

Optical an

gle

()

(A) (B)

219

comparable. The devices shown in Figure 12-6 and Figure 12-7 have thermal isolation

between the actuators and the mirror-plate; their power consumption is only slightly less

than the device with no thermal isolation. This may be attributed to the large heat loss

from the actuators to the substrate.

Figure 12-13. Static characteristic of 1D mirror depicted in Figure 12-5. The mirror has beam-type thermal isolation at both ends of the actuators. (A) Optical scan angle vs. applied dc voltage. (B) Optical scan angle vs. input power.

Figure 12-14. Static characteristic of 1D mirror with beam-type thermal isolation between actuators and mirror-plate. The mirror has been depicted in Figure 12-6. (A) Optical scan angle vs. applied dc voltage. (B) Optical scan angle vs. input power.

0 10 20 30 40

45

40

35

30

25

20

15

10

5

0

Optical an

gle

()

Increasing voltage

Decreasing voltage

0 0.2 0.4 0.6 0.8 1

45

40

35

30

25

20

15

10

5

0

Optical an

gle

()

Increasing voltage

Decreasing voltage


(A) (B)

0 10 20 30 40 50 60 70

Optical an

gle

()

0 0.2 0.4 0.6 0.8 1

Increasing voltage

Decreasing voltage

Increasing voltage

Decreasing voltage

7

6

5

4

3

2

1

0

7

6

5

4

3

2

1

0

Optical an

gle

()


(A) (B)

220

Figure 12-15. Static characteristic of 1D mirror with trench-type thermal isolation between actuators and mirror-plate. The mirror has been depicted in Figure 12-7. (A) Optical scan angle vs. applied dc voltage. (B) Optical scan angle vs. input power.

Table 12-1. Scan angle per unit dc power input for 1D mirror designs.

Device Description Scan angle per unit dc power input (º/mW)

No thermal isolation, Figure 12-3 0.098533

Beam-type isolation at both ends of actuator beams, Figure 12-4

0.71595

Beam-type isolation at both ends of actuator beams, Figure 12-5

1.0649

Beam-type isolation between actuator beams and mirror-plate, Figure 12-6

0.10056

Trench-filled isolation between actuator beams and mirror-plate, Figure 12-7

0.13492

The frequency response of the 1D mirrors shown in Figure 12-3 through Figure

12-7 are depicted in Figure 12-16 through Figure 12-20, respectively. The resonant

frequencies of all the devices are in excess of 200 Hz. The device with no thermal

isolation has the most sluggish response (Figure 12-16) as the thermal capacitances

associated with the mirror-plate and substrate make the device response slow.

0 10 20 30 40 50 60 70

Input power (mW)

Increasing voltage

Decreasing voltage

Increasing voltage

Decreasing voltage

9

8

7

6

5

4

3

2

1

0

Optical an

gle

()

10

8

6

4

2

0

Optical an

gle

()

0 0.2 0.4 0.6 0.8 Voltage (V)

(A) (B)

221

Comparing Figure 12-17 and Figure 12-18 to Figure 12-16, devices with beam-type

thermal isolation at both mirror and substrate ends have about 10 times faster response

than devices with no thermal isolation. The improved response may be attributed to the

isolation of mirror-plate and substrate thermal capacitances from the actuators. The

device with beam-type isolation at the mirror-plate junction only, has a fast thermal

response as evidenced by the nearly flat frequency response shown in Figure 12-19.

The frequency response of the device with trench-type isolation at mirror-plate junction

only (Figure 12-20), is similar to that of the device with no isolation (Figure 12-16). The

similarity suggests that the Si was not fully removed from the trenches resulting in poor

thermal isolation.

The next section will compare the device robustness of the new devices with older

generation micromirrors that used SiO2 thermal isolation.

Figure 12-16. Frequency response of the mirror shown in Figure 12-3. The response is obtained by applying a sinusoidal voltage of 55 mV amplitude at a dc bias of 1.475 V.

101

100

10-1

10-1 100 101 102 Frequency (Hz)

Optical scan r

an

ge ()

222

Figure 12-17. Frequency response of the mirror shown in Figure 12-4. The response is obtained by applying a sinusoidal voltage of 25 mV amplitude at a dc bias of 870 mV.

Figure 12-18. Frequency response of the mirror shown in Figure 12-5. The response is obtained by applying a sinusoidal voltage of 32.5 mV amplitude at a dc bias of 820.5 mV.

10-1 100 101 102 Frequency (Hz)

101

100

10-1

Optical scan r

an

ge ()

101

100

10-1

10-1 100

101 102

103

Frequency (Hz)

Optical scan r

an

ge ()

223

Figure 12-19. Frequency response of the mirror shown in Figure 12-6. The response is

obtained by applying a sinusoidal voltage of 46.5 mV amplitude at a dc bias of 799.5 mV.

Figure 12-20. Frequency response of the mirror shown in Figure 12-7. The response is

obtained by applying a sinusoidal voltage of 52 mV amplitude at a dc bias of 632 mV.

10-1 100 101 102 103

Frequency (Hz)

101

100

10-1

Optical scan r

an

ge ()

10-1

100 10

1 10

2 10

3

Frequency (Hz)

101

10

0

10

-1

Optical scan r

an

ge ()

224

12.5 Device Robustness

12.5.1 Impact Testing with Two-Ball Setup

Figure 12-21 illustrates the setup for mirror robustness test. The setup was

implemented using a commercially available Newton’s cradle. The mirror is attached to

Ball-2 which suffers an impact with Ball-1. After the first impact, Ball-1 is caught to

prevent successive collisions. The maximum acceleration in m/s2 experienced by the

mirror is given by [136],

3/528851 max

a h g (12-1)

where, g is the acceleration due to gravity (m/s2) and h is the drop height (m). It was

found that mirrors described in Chapter 3, which have SiO2 thermal isolation, fail in the

1600g–2000g range. In contrast, the new device shown in Figure 12-5 can withstand

accelerations greater than 8000g which is the maximum that can be generated by the

setup in Figure 12-21.

Figure 12-21. The impact test setup consists of two steel balls of diameter 22 mm suspended from a height of 130 mm. (A) Before impact. (B) After impact.

12.5.2 Drop Tests

It is found that the 3D device shown in Figure 12-9 does not suffer catastrophic

failure when bonded to a standard dual-in-line package and dropped from 4 feet height

(A) (B)

Ball-1 Ball-1 Ball-2

Ball-2

Micromirror Nylon line

225

on a vinyl floor. Older designs with SiO2 thermal isolation cannot withstand drops from

more than a few centimeters height.

12.6 Summary and Conclusions

In conclusion, electrothermal micromirrors actuated by Al/W bimorphs and

polyimide thermal isolation demonstrate improved robustness. This makes them

especially suitable for hand-held imaging probes that suffer frequent drops during

handling. The proposed process can be adapted to a wide range of electrothermal

MEMS. High thermal diffusivities of Al and W and low thermal conductivity of polyimide

allow the optimization of device speed and power consumption. The process for

fabricating robust mirrors brings them a step closer to real-world applications.

226

CHAPTER 13 NOVEL MULTIMORPH-BASED IN-PLANE TRANSDUCERS

13.1 Background

Multimorphs have been widely used for producing out-of-plane displacement.

Figure 1-1 shows the schematic of a multimorph. The tip of the multimorph undergoes a

displacement din-plane in the plane of the substrate. In this chapter, novel schemes for

amplifying the in-plane displacement are proposed. The designs can achieve 100s

microns to ~1 mm in-plane displacement, which is an order of magnitude greater than

the displacements produced by previously reported designs. Potential applications such

as miniature Michelson interferometer and movable MEMS stage are proposed.

Electrostatic comb-drive actuators can produce up to 30 μm in-plane

displacement [138]. Cragun et al. report electrothermal transducers that produce in-

plane displacements up to 20 μm [139]. Waterfall et al. report a bent-bent beam type

thermal actuator that produces up to 5 μm in-plane displacement. Lee et al. report an

electrothermally actuated MEMS stage that has a maximum displacement of

40 μm [140]. Noworolski et al. report electrothermal designs that produce large in-plane

displacements up to 200 μm, but the device area is greater than 10 mm2, which is very

large [141]. The reported values of in-plane displacements produced by piezoelectric

transducers are much less that those produced by electrothermal transducers [138].

Conway et al. report a piezoelectric in-plane actuator that produces a peak

displacement of 1.18 μm [142].

Two novel transducers are proposed in this chapter. These will be referred to as

Design 1 and Design 2. Sections 13.2 and 13.3 discuss Designs 1 and 2, respectively.

227

Section 13.4 provides a comparison of the two designs. Potential applications are

discussed in Section 13.5.

13.2 In-Plane Transducer Design 1

13.2.1 Topology of Design 1

Figure 13-1 shows the top view of the proposed transducer. The three multimorph

segments have identical cross-section and are capable of producing bending

deformation as discussed in Chapter 1. Possible ways to achieve transduction include

thermal expansion, piezoelectric effect, shape-memory effect, expansion/contraction of

electroactive polymer etc. The rigid beams undergo negligible deformation. Low

deformation of the rigid beams may be achieved by using a thick layer of commonly

available MEMS materials such as single crystal silicon. Alternatively, the rigid beam

may be implemented as a multilayer structure in which the materials and

layer-thicknesses are chosen to achieve very low deformation.

Figure 13-1. Top view of proposed Design 1 for achieving large in-plane displacement.

l1 l2

l1 l2

2l1

Multimorph

Rigid Beam

228

13.2.2 Simulations

Let us consider Al-W thermal bimorphs. Let the thickness of Al be 1 μm, which is a

typical thickness value encountered in literature [20]. In accordance with Equation 1-5,

the thickness of W is chosen to be 0.4 μm. The lengths l1 and l2 (Figure 13-1) are

chosen to be 100 μm and 400 μm, respectively. The transducer is simulated for a

uniform temperature change of 400 K from an initially flat position. Figure 13-2 shows

the deformed shape. The simulated in-plane tip displacement is 238 μm. In practice, a

nearly uniform temperature may be achieved by using large thermal isolation at either

ends of the transducer. Temperature change may be achieved by Joule effect.

Analytical expression for in-plane displacement is derived next.

Figure 13-2. Deformed shape of Design 1 for a uniform temperature change of 400 K

obtained from finite-element simulations. The color-bar shows the in-plane displacement. The simulated in-plane displacement is 238 μm.

13.2.3 Analysis

Let the transducer be completely flat at a reference temperature. Let, Rm = radius

of curvature of the multimorph in deformed state at the highest possible operation

temperature. The in-plane tip displacement is given by,

Clamped-end

Undeformed state

Deformed shape

0

238 μm

In-plane displacement

229

11 1 2 2( , ) 2 1 cosdesign

m

ld l l l

R (13-1)

Equation 13-1 will be used to optimize Design 1 in the next subsection.

13.2.4 Optimization

The goal of this optimization is to determine l1 and l2 to achieve maximum

displacement for constant total transducer length, ltotal . As shown in Figure 13-1, if l1 l2,

ltotal = 2l2. In general ltotal is given by,

2 1 2

1 1 2

2 ,

2 ,total

l l ll

l l l (13-2)

It can be shown that if the total transducer length, ltotal , is less than 2Rm, the

condition l1 = l2 maximizes in-plane displacement. Figure 13-3 shows the top-view of the

optimized transducer design for ltotal < 2Rm.

If ltotal > 2Rm and l1 l2, the partial derivative of ddesign1 with respect to l1 may be

equated to zero to give l1 = Rm. According to Equation 13-2, in this case l2 is constant. If

ltotal > 2Rm and l1 l2, the partial derivative of ddesign1 with respect to l2 may be equated to

zero to give l1 = Rm. According to Equation 13-2, in this case l1 is constant. Therefore

for ltotal > 2Rm, l1 = Rm must be chosen to maximize in-plane displacement. Figure 13-4

shows the side-view of the optimized transducer for ltotal > 2Rm.

To summarize, if ltotal < 2Rm, l1 = l2 maximizes in-plane displacement. Otherwise,

l1 = Rm maximizes in-plane displacement.

After optimizing the design shown in Figure 13-2, l1 and l2 are chosen to be 400

microns. This gives an in-plane displacement as high as 1.6 mm for a temperature

change of 400 K.

230

Figure 13-3. Optimized Design 1 for ltotal 2Rm. The top-view of the undeformed state

is depicted in this figure.

Figure 13-4. Optimized Design 1 for total transducer length, ltotal 2Rm. The side-view

in the deformed state is shown. In this case l1 = Rm is chosen. The total in-

plane displacement is 4l2.

13.3 In-Plane Transducer Design 2

13.3.1 Design Topology

Figure 13-5A shows a schematic of the transducer topology. If the inverted

multimorph is an exact inversion of the non-inverted multimorph as shown in Figure 13-

5B and Figure 13-5C and the material thicknesses are the same in the inverted and

non-inverted multimorphs, l2 may be chosen to be twice of l1. The non-inverted and

inverted multimorphs bend in opposite directions and result in a net zero out-of-plane

displacement.

Rigid Beam Rigid Beam

Multimorph Multimorph Multimorph Multimorph

Multimorph

Rigid Beam

ltotal

231

Figure 13-5. Top view of proposed transducer for achieving large in-plane displacement.

Possible cross-sections of non-inverted and inverted multimorphs are shown in (B) and (C), respectively. The non-inverted and inverted multimorphs bend in opposite directions thereby resulting in zero out-of-plane displacement at the free-end of the transducer. If materials with the same thicknesses are used to form the inverted and non-inverted multimorphs, l2 must be twice of l1.

(D) Finite element simulation of a transducer that employs thermal multimorphs and is subjected to a uniform temperature change of 400 K. The non-inverted multimorph consists of 1 μm aluminum (Al) on top and 0.4 μm tungsten (W) at the bottom. The inverted multimorph consists of 1 μm aluminum (Al) at the bottom and 0.4 μm tungsten (W) on top. The lengths l1

and l2 are chosen to be 100 μm and 200 μm, respectively.

l1 l1 l2

Non-inverted Multimorph

Inverted Multimorph

Clamped end

Free end

Material 1

Material 2

Material n Material 1

Material 2

Material n

(A)

(B) (C)

Clamped end

Undeformed state

Deformed state

0

38 μm

(D)

In-plane displacement

232

If the inverted multimorph is an exact inversion of the non-inverted multimorph,

l1 = 2l2. Let the radius of curvature of the multimorphs be Rm in the deformed state. Let

the multimorphs be flat, i.e., undeformed, at a reference state. Then the total in-plane

displacement is,

12 14 sindesign m

m

ld l

R

R (13-3)

13.3.2 Device Fabrication

Figure 13-6 and Figure 13-7 show the SEM and optical microscope image of a

fabricated structure. The device consists of an array of 32 actuators connected to a

platform. The non-inverted multimorph layers from bottom to top are 1 μm SiO2, 0.2 μm

Pt , 0.2 μm SiO2, and 1 μm Al. The inverted multimorph layers from bottom to top are

50 nm SiO2, 0.2 μm Pt , 0.2 μm SiO2, 1 μm Al, and 1.6 μm SiO2. The Pt layer acts as an

embedded resistive heater. The SiO2 thermal isolation at either ends of the actuator

beams minimizes power consumption and ensures a nearly uniform temperature

distribution along the actuators. The platform consists of an 80 μm thick silicon layer

coated with Al. The thick silicon layer under the platform ensures its flatness.

Fabrication was done on an SOI wafer with 80 μm thick device layer. The fabrication

process is similar to the one described in [36].

With reference to Figure 13-5, l1 and l2 are 100 μm and 400 μm, respectively. The

values of l1 and l2 are chosen in accordance with Equation 6-1 to ensure zero out-of-

plane displacement. Only thermal stresses are considered. However, due to residual

stresses in the thin films, the end of the actuators connected to the platform is initially

elevated by 150 μm with respect to the substrate.

233

Figure 13-6. SEM of fabricated in-plane transducer Design 2.

Figure 13-7. Optical microscope image of fabricated in-plane transducer Design 2.


Figure 13-8 shows the in-plane displacement produced by the device shown in

Figure 13-6. As shown in Figure 13-8A, an in-plane displacement of 11 μm is produced

at 1.65 V. As depicted in Figure 13-8B, the corresponding power consumption is

94 mW. The out-of-plane displacement at the end of the actuators connected to the

Non-inverted multimorph

1 mm

Inverted multimorph

SiO2 thermal isolation

80 μm thick single-crystal silicon

Rigid platform

Bond-pads

Rigid platform

Actuators

234

platform is shown in Figure 13-9. The maximum out-of-plane displacement is 17 μm in

the downward direction, i.e., towards the substrate.

Figure 13-8. In-plane displacement produced by Design 2. (A) In-plane displacement vs. voltage. (B) In-plane displacement vs. input power.

Figure 13-9. Out-of-plane displacement produced by Design 2. (A) Out-of-plane displacement vs. voltage. (B) Out-of-plane displacement vs. input power.

In the design shown in Figure 13-6, all actuator beams are identical. As a result, it

is difficult to compensate for the out-of-plane displacement due to residual stresses and

thermal stresses completely. Future designs will use two different actuators beam

designs that undergo out-of-plane displacement in opposite directions. The two types of

actuators will be controlled by separate actuation signals to ensure zero net out-of-plane

displacement.

0 20 40 60 80 100

12

8

4

0 0 0.5 1 1.5 2

12

8

4

0 Voltage (V) Input power (mW)

(A) (B)

In-p

lane d

isp

lace

ment

(μm

)

In-p

lane d

isp

lace

ment

(μm

)

0 20 40 60 80 100 Input power (mW)

0 0.4 0.8 1.2 1.6 Voltage (V)

0

-4

-8

-12

-16

-20

0

-4

-8

-12

-16

-20 O

ut-

of-

pla

ne d

isp

lace

ment

(μm

)

Out-

of-

pla

ne d

isp

lace

ment

(μm

)

(A) (B)

235

The next section compares the displacements generated by Designs 1 and 2.

13.4 Comparison of Designs 1 and 2

In order to compare Designs 1 and 2, a total transducer length of 800 μm is

assumed. The multimorphs consist of 1 μm aluminum and 0.4 μm tungsten. For Design

1, optimum beam lengths are chosen as discussed in Section 13.2.4. Transduction is

achieved by thermal expansion. At reference temperature, the transducer is assumed to

be flat. In-plane displacement is evaluated for uniform temperature change of 400 K.

Table 13-1 compares the in-plane displacements obtained using Equations 13-1 and

13-3. Clearly, for the dimensions chosen, the displacement produced by Design1 is

more that 5 times greater than that produced by Design 2. The next section will discuss

possible applications of the in-plane transducer designs.

Table 13-1. Comparison of Designs 1 and 2.

In-plane displacement produced by 800 μm long transducer.

Design 1 1.6 mm

Design 2 298 μm

13.5 Potential Applications

Multimorphs based on thermal, piezoelectric, shape memory effect, electroactive

polymer expansion etc. can be employed in the novel designs presented in this chapter.

Potential application areas have been listed below:

1. Movable MEMS stage with 5 degrees-of-freedom: The design shown in Figure 1-2C and those reported in [34, 36] utilize multimorphs to achieve motion along 3 degrees-of-freedom. These designs can produce out-of-plane displacement and rotation about two mutually perpendicular axes. The in-plane designs can add two more degrees-of-freedom, i.e., in-plane displacement about two mutually perpendicular axes. Hence, a 5 degrees-of-freedom MEMS stage can be realized. Furthermore, motion along all 5 degrees-of-freedom can be achieved using multimorphs. Hence, additional masks are not required for fabrication. A possible application is the handling of small biological samples.

236

2. Integrated Michelson interferometer: Michelson interferometers are used in several optical systems such as optical coherence tomography [143], Fourier transform spectroscopy [144] etc. Figure 13-10 shows the schematic of a Michelson interferometer. Previously, miniaturization of the interferometric setup has been attempted by using discrete MEMS components. For instance, a MEMS mirror that undergoes out-of-plane displacement may be used to implement the movable mirror [32] and the beam splitter and fixed mirror may be assembled to form the complete setup. Additional components for optical fiber alignment, light collimation etc. may also be required. Handling the discrete optical components is very challenging and achieving good optical alignment requires manual fine-tuning. The proposed in-plane transducers may be used to actuate a mirror that is vertical to the substrate. This movable mirror can be part of an integrated Michelson interferometer. Various optical components may be fabricated on the same substrate as discussed in [2]. V-grooves may be etched on the substrate to hold the optical fiber in position [2]. An integrated Michelson interferometer will lead to a significantly miniaturized design and good optical alignment is automatically achieved during fabrication. The large displacement produced by the reported designs is especially suited for achieving large depth of scanning in a miniature optical coherence tomography setup.

3. MEMS tweezer or micro gripper: By using two opposing transducers, it may be possible to realize a MEMS tweezer as shown in Figure 13-11. The small out-of-plane displacement shown in Figure 13-11A may be achieved in several ways. One possible way is to choose the transducer dimensions such that the out-of-plane displacement does not get cancelled out completely. Another option is to utilize a non-uniform temperature distribution along the length of thermal multimorphs or a non-uniform electric field for actuating piezoelectric multimorphs.

4. Temperature sensor: The large displacement produced by transducers based on thermal multimorphs can be used for sensing temperature.

Figure 13-10. Schematic of a Michelson interferometer

Movable mirror executes axial scanning

Fixed mirror

Incident laser beam

Beams reflected from the fixed and movable mirrors

interfere

Beam splitter

237

Figure 13-11. Two opposing transducers can be used to form a MEMS tweezer or micro

gripper. This schematic shows a modified version of the design shown in Figure 13-5. A small out-of-plane displacement may be achieved by choosing suitable design parameters. (A) Micro gripper in the released state. (B) Micro gripper holding the sample at a fixed position.

13.6 Summary

Two multimorph based MEMS transducers for achieving large in-plane

displacement of the order of 100s microns to 1 mm have been proposed. The

displacement produced is at least an order of magnitude greater that previously

reported designs. Possible means of transduction include thermal expansion,

piezoelectric effect, shape-memory effect, expansion/contraction of electroactive

polymers etc. Possible applications include 5 degrees-of-freedom MEMS stage,

integrated Michelson interferometer, MEMS tweezer, temperature sensor etc.

(A) (B)

Clamped-end Sample

238

CHAPTER 14 CONCLUSIONS AND FUTURE WORK

14.1 Summary of Work Done

This dissertation deals with the modeling, reliability and design of scanning

micromirrors actuated by electrothermal multimorphs. The subsequent sections

summarize the work done in this thesis.

14.1.1 Device Modeling

Device modeling is essential for design, optimization and control. It is desirable to

build parametric, compact models. A parametric model predicts device response in

terms of device parameters. Therefore, it is suitable for optimization and design. A

compact model is computationally efficient and it saves time and resources.

The key components of the complete device model of an electrothermal

micromirror are the electrical, thermal and mechanical models. The electrical model

provides the power dissipated in the embedded heater for a certain actuation voltage.

The thermal model predicts the actuator and heater temperatures for a certain power

input. The mechanical model takes the actuator temperature as input and predicts

mirror-motion.

Chapters 2–5 deal with thermal modeling. Several thermal modeling techniques

have been surveyed in Chapter 2. As discussed in Chapter 3, numerical model order

reduction may be used to extract a compact thermal model from a complete FE model.

Model order reduction has been applied to a 1D mirror. It is found that a second order

reduced model satisfactorily represents the thermal response of the device. Based on

intuition, a circuit model was built to explain the second order thermal response. Two

239

capacitors were used to represent the heat capacitances of the actuators and the

mirror-plate. The second order response was attributed to these two capacitances.

Being purely numerical, a reduced order model is not parametric. Another

approach well suited for MEMS modeling is the transmission line based method. This

method draws analogy between heat flow in a MEMS structure and signal flow in a

transmission line. The transmission line model is then simplified to obtain a parametric

compact thermal model. In Chapter 4, the transmission line method has been

demonstrated for a device in which the resistive heater is embedded at one end of the

bimorphs. The bimorphs themselves are treated as a passive transmission line. Active

thermal transmission line with distributed embedded heater is discussed in Chapter 5.

The mechanical model predicts mirror-plate motion for a certain temperature

change in the bimorph actuators. Two possible approaches for building the mechanical

model—the Newtonian method and Lagrangian method have been discussed in

Chapter 6. The Newtonian method involves classical analysis based on free body

diagrams. The Lagrangian approach is based on energy equations. In the Lagrangian

method, the actuator beams are treated as springs that can store elastic potential

energy. Since the mirror is significantly heavier than the actuators, kinetic energy is

attributed to mirror motion only. A comprehensive electrothermomechanical model of a

1D mirror is discussed in Chapter 7.

Another contribution of this thesis is the optimization of ISC actuators. This

optimization resulted in more than ten-fold increase in the scan angle of micromirrors

employing ISC actuators and has been discussed in Chapter 6.

240

14.1.2 Curved Multimorph Actuators

Prior to this thesis, several MEMS devices based on straight multimorphs have

been reported by researchers. However, curved multimorphs that bend and twist upon

deformation have not been widely reported. In this thesis, the small-deformation

analysis of curved multimorphs has been reported for the first time and has been

discussed in Chapter 8. The analytical expressions have been verified by comparing

them to simulation and experimental results. Large deformation has been qualitatively

studied by experiments and simulations.

The unique properties of curved multimorphs have been utilized in the design of

novel electrothermal micromirrors. These micromirrors have been discussed in

Chapters 9 and 10. Micromirrors actuated by curved multimorph actuators have several

attractive features. Compared to previously reported designs, the curved multimorph-

based designs undergo lower mirror-plate center-shift, consume less power and utilize

chip-area more efficiently. Furthermore, mirrors actuated by curved multimorphs can

achieve two-dimensional scanning by using a single electrical signal line.

14.1.3 Device Pre-conditioning and Repeatability

Device repeatability has been discussed in Chapter 11. Preliminary investigation

on the pre-conditioning and repeatability of devices based on Al-SiO2 bimorphs and

embedded Pt heater has been carried out. Three key parameters—initial elevation,

mirror scan angle and embedded heater resistance have been identified for studying the

initial burn-in phase and characterizing device repeatability. It is found that a certain

voltage must be applied to newly released devices for achieving successful pre-

conditioning. The pre-conditioning makes the device repeatable and reduces

performance variations among a certain population of devices.

241

14.1.4 Fabrication of Robust Micromirrors

Commercial success of a device requires a thorough investigation into its reliability

issues. Devices actuated by Al-SiO2 bimorphs have two major drawbacks. Firstly, such

devices utilize SiO2 thin-film thermal isolation, which makes them susceptible to impact

failure. As a result, such devices cannot withstand drop-tests from more than a few

centimeters height. Secondly, the low thermal diffusivity of SiO2 hampers the thermal

speed of the device. These challenges have been addressed using a novel fabrication

process that has been discussed in Chapter 12. This process utilizes Al and W as

bimorph materials and high temperature polyimide for thermal isolation. The W thin-film

also doubles up as a resistive heater. Mirrors fabricated by the proposed process have

significantly improved robustness compared to previous generation devices. The

devices can withstand drops from a height of several feet. Furthermore, the new

fabrication process allows the layout engineer to customize power consumption and

thermal response speed.

14.1.5 Novel In-plane Transducer Designs

In Chapter 13, two multimorph-based in-plane transducers have been proposed.

Simulations show that these transducers can produce 100s microns to a few millimeters

displacement along the substrate. The displacement produced is an order of magnitude

improvement over previously reported designs. A possible application is an integrated

Michelson interferometer in which the proposed transducer actuates a mirror-plate that

is vertical to the substrate. Such interferometers can significantly miniaturize biomedical

imaging systems. Other possible applications include movable MEMS stage,

temperature sensor and micro-gripper.

242

14.2 Future Work

Several device modeling approaches have been explored in this thesis.

Electrothermomechanical model of a 1D mirror and model based-open loop control

have been demonstrated. Future work may involve the modeling of 2D and 3D MEMS

mirrors. The understanding developed during the course of this thesis may be further

extended to enable design synthesis and automatic layout generation.

Preliminary investigation of device pre-conditioning and repeatability has been

reported in this dissertation. Future work may involve thorough investigation into device

burn-in, long term testing and failure modes such as creep and fatigue.

A major achievement of this thesis is the small-deformation analysis of curved

multimorphs. In future, the large deformation behavior of curved multimorphs may be

investigated in detail. Several micromirrors actuated by curved multimorphs have been

discussed in this dissertation. Hereafter, the unique features of curved multimorphs may

be utilized to design other novel MEMS devices.

Mirrors fabricated using the novel process discussed in Chapter 12 have

significantly improved robustness compared to older designs. Future work may involve

the fine-tuning of process parameters. The isotropic Si etch recipe that is used for

fabricating polyimide trench-filled thermal isolation needs further refinement. Candidate

mirror-fabrication materials discussed in Chapter 12, such as DLC and invar, may also

be explored.

The transducers presented in Chapter 13 can produce 100s microns in-plane

displacement. Future work may involve the fabrication of novel MEMS devices that

utilize the large in-plane displacements produced by these transducers.

243

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BIOGRAPHICAL SKETCH

Sagnik Pal was born in 1982 in Kolkata, India. As a student at Delhi Public School,

Ranchi, he received several certificates of merit in mathematics, physics, chemistry and

English. In 2000, he enrolled in the undergraduate program in electrical engineering at

Indian Institute of Technology, Kharagpur. He developed keen interest in

Electromagnetics and Control Theory. His B.Tech thesis involved the analysis of

electrical machines using numerical techniques such as finite element and finite

difference methods. After receiving his B.Tech (Hons.) in 2004, he pursued M.Tech in

electrical engineering with specialization in Microwave and Photonics at Indian Institute

of Technology, Kanpur. In his M.Tech dissertation on optically actuated MEMS

structures, he proposed several novel devices including optical switch, photodetector

cum beam profiler, and diffraction grating based switch. Thereafter, he joined the

doctoral program at the Department of Electrical and Computer Engineering, University

of Florida in August 2006. His innovations at the Biophotonics and Microsystems Lab

include novel curved multimorph transducers; in-plane MEMS transducers that produce

order of magnitude greater displacement compared to existing designs; mirror design

optimization that resulted in ten-fold improvement in scan-range; novel fabrication

process for robust, fast electrothermal micromirrors; novel micromirrors with low center-

shift, high fill-factor, improved voltage and power requirements; comprehensive

electrothermomechanical device models; and model-based mirror control. He

completed his PhD in December 2011. He is interested in interdisciplinary research,

especially the design, modeling, reliability and fabrication of MEMS and nano devices.

Besides authoring 6 journal papers and 17 conference papers, he has 3 patents

pending.

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