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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
7 7
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
7.3.3 Experimental Results ............................................................................. 140
7.4 Summary and Discussion ............................................................................... 141
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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
8.3.2 Experimental Results ............................................................................. 154
8.3.3 Large Deformation of Curved Multimorphs ............................................ 156
8.4 Summary and Discussion ............................................................................... 158
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
9.5 Summary and Discussion ............................................................................... 178
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
17 17
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
12-17 Frequency response of the mirror shown in Figure 12-4 .................................. 222
12-18 Frequency response of the mirror shown in Figure 12-5 .................................. 222
12-19 Frequency response of the mirror shown in Figure 12-6 .................................. 223
12-20 Frequency response of the mirror shown in Figure 12-7 .................................. 223
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
19 19
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
4.2.1 Device Description
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)
From Equations 4-10 and 4-12,
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)
Transmission-line model
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)
From Equations 5-6 and 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
From Equations 5-15 and 5-16,
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
Transmission-line 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
Transmission-line 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
Transmission-line model
Tem
pe
ratu
re r
ise (
K)
10-4 10-2 100 102 104 106 108
Frequency (Hz)
FE model
Transmission-line 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.
5.4 Summary and Discussion
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)
From Equations 7-1 and 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.
7.2.2.1 Static characterization
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.
7.3.3 Experimental Results
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.
7.4 Summary and Discussion
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)
From Equations 8-10 and 8-13,
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.
From Equations 8-16 and 8-17,
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)
From Equations 8-12 and 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.
8.3.2 Experimental Results
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.
8.4 Summary and Discussion
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
9.2.1 Device Description
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.
9.2.3.2 Frequency response
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.
9.5 Summary and Discussion
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
10.2.1 Device Description
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.
10.2.2.1 Static characterization
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.
10.2.2.2 Frequency response
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
Voltage (V) Input power (mW)
(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
()
Voltage (V) Input power (mW)
(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.
13.3.3 Experimental Results
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
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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.