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PAGE 1 MODELING AND RELIABILITY OF ELECTROTHERMAL MICROMIRRORS By SAGN IK 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 20 11 PAGE 2 2 2 2011 Sagnik Pal PAGE 3 3 3 To Baba, Ma, Eunj u Sageun and Tommy PAGE 4 4 4 ACKNOWLEDGMENTS I would like to thank my advisor Dr. Huikai Xie for his support and encourag ement in pursuing topics that a re 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 Las er, 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 providin g 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 Ghat u 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 optic al systems. Mingliang Wang, Jiping Li Sean R. Samuelson, Lin Liu Shuo Cheng, Jessica Meloy and Tiffany Re agan have trained me on several equipments. Al Ogden, David Hays and Bill Lewis at the PAGE 5 5 5 N anoscale R esearch F acility at U niversity of F lorida have given valuable suggestions on device fabrication and have provided clean room training Useful suggestions on p olyimide 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 o n 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, Car a Hall, Zhongyang Guo, and Dr. Hongzhi Jia. conference in Beijing a pleasant and memorable experience. This research was supported by grants from the National Science Foundation. SEM imag es 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 foremo st I thank my parents Prabir and Elonee Pal and my girlf riend 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. PAGE 6 6 6 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 vel ocity 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 (CTM s ) ................................ ........................... 44 2.2.4 Distributed Circuit Mode ls ................................ ................................ ........ 46 2.2.5 Numerical Model Order Reduction (MOR) ................................ ............... 46 2.3 Summary ................................ ................................ ................................ .......... 47 PAGE 7 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 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 Reduc tion ................................ ..... 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 Therm al 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 PAGE 8 8 8 5 TRANSMISSION LINE THERMAL MODEL WITH DISTRIBUTED H EAT 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 Back ground ................................ ................................ ................................ ..... 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 ................................ ................................ ......... 12 6 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 G 2 ( 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 PAGE 9 9 9 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 Contin uity 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 Dimensiona l 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 Mirro r with 92 ............... 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 PAGE 10 10 10 10.2.3 F inite 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 Mirro r with 92 ............... 186 10.4 A 400 wide Circular Mirror Actuated by a Semicircular Multimorph ........ 188 10.5 Su mmary 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 Is sues ................................ ................................ ........ 204 11.3.3.1 Creep ................................ ................................ ......................... 204 11.3.3.2 Fatigue ................................ ................................ ....................... 204 11.3.3.3 Enviro nmental factors ................................ ................................ 204 11.4 Summary and Future Work ................................ ................................ ........... 204 12 A PROCESS FOR FABRICATING ROBUST ELECTROTHERMAL MEMS WITH CUSTOMIZABLE T HERMAL 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 PAGE 11 11 11 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 Potenti al 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 PAGE 12 12 12 LIST OF TABLES Table page 3 1 Symbol s used in lumped element model ................................ ............................ 66 4 1 Material properties f or thermomechanical simulations ................................ ........ 73 4 2 Thermal cond uctivity 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 t o thermal diffusion through air ............... 172 11 1 Heater resistance before a nd after burn in for 12 devices ................................ 193 11 2 Candidate materials for fabricat ing electrothermal micromirrors ....................... 208 12 1 Scan angle per unit dc po wer input for 1D mirror designs ................................ 220 13 1 Comparison of Designs 1 and 2 ................................ ................................ ....... 235 PAGE 13 13 13 LIST OF FIGURES Figure page 1 1 Schematic of thermal bimorph ................................ ................................ ............ 28 1 2 SEMs of electrothermal m icromirrors ................................ ................................ 32 1 3 Schemati c of comprehensive mirror model ................................ ........................ 33 1 4 Top view of scan angle measur ement setup on an optical bench ...................... 34 3 1 SEM of a 1D micromirror ................................ ................................ .................... 49 3 2 Schematic o f 1D micromirror ................................ ................................ .............. 50 3 3 A rectang ular finite element. ................................ ................................ ............... 52 3 4 Simulated temperature distribution in a section of the device. ............................ 53 3 5 FE model of micromirror.. ................................ ................................ ................... 54 3 6 Temperat ure distribution in micromirror ................................ .............................. 55 3 7 Mirror rotation a ngle vs. input electrical power ................................ ................... 57 3 8 Tra nsfer 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 Tra nsfer 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 Compari son between two bimorph designs ................................ ........................ 72 4 4 Simulated temperature distribu tion in a section of the device ............................. 76 4 5 FE model for estimating heat loss coeff icient due to thermal diffusion ............... 77 4 6 FE thermal model of micromirror ................................ ................................ ........ 79 PAGE 14 14 14 4 7 Exaggerated schematic of bimorph subs trate junction ................................ ....... 81 4 8 Comparis on of two mi rrors with different etch times ................................ ........... 82 4 9 Passive transmission line mo del ................................ ................................ ........ 84 4 10 Equivalent circuit representation of thermal model ................................ ............. 89 4 11 Static electrothermal model ba sed on transmission line theory .......................... 91 4 12 Static electrothermal model based on lumped element approximation .............. 93 4 13 Dynamic elec trothermal model of micromirror ................................ .................... 95 4 14 Electrother momechanical model of 1D mi rror ................................ .................... 96 4 15 Comparison between model and experimental results ................................ ....... 97 4 16 Dependence of e rror in LEM result s on bimorph length ................................ ..... 98 4 17 Comparison between transmission line model and LEM ................................ .. 100 4 18 Fr equency 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 mi rror placed in air ................................ ...... 108 5 5 Equivalent circuit of an element of length x of a one dimensional heat flow region ................................ ................................ ................................ ............... 110 6 1 Schematic showing the three degrees of freedom of a 3D micromirror ............ 113 6 2 Sche matic of a multimorph ................................ ................................ ............... 114 6 3 Thin film structure of non inverted and inverted multimorphs ........................... 115 6 4 Series connection of a non in verted and an inverted actuator .......................... 117 6 5 Schematic of mechanical model for 1D mirror ................................ .................. 118 6 6 Free body diagram of actuator and mi rror plate ................................ ............... 119 PAGE 15 15 15 6 7 S tress distribution in a bimorph ................................ ................................ ........ 121 7 1 Schematic of a complete model of an electrothermally actuated microm irror .. 127 7 2 Static characteristic ................................ ................................ .......................... 129 7 3 Micromirror freq uency 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 mirro r 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 cr oss section of a multimorph .................. 147 8 4 FE simulation results for curved thermal multimorphs ................................ ...... 153 8 5 Deformation of curved multimorp h ................................ ................................ .... 154 8 6 Curved multimorph test structure ................................ ................................ ...... 15 5 8 7 Mirror plate tilt vs. chip temperature for test structure shown in Figure 8 6 A .... 156 8 8 Large deformation of a curved multimorph ................................ ....................... 157 9 1 Straight multimorph based 1D micromirror design at two different posi tions 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 PAGE 16 16 16 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 measu red 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 .................... 181 10 2 Static characteristic of device shown in Figu re 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 dimens ional scan pattern ................................ ................................ .......... 184 10 7 Simulated temperature distribution for an a pplied voltage of 400 mV ............... 185 10 8 Temperature distributio n 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 PAGE 17 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 finit e 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 m irror with no thermal isolation ................................ ...................... 212 12 4 SEM of 1D mirror with beam type thermal isolation at both ends of the actuator s ................................ ................................ ................................ ........... 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 bet ween 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 rob ust 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 f illed thermal isolation between the actuators and the mirror plate ................................ ................................ ........... 217 PAGE 18 18 18 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 cha racteristic 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 Optimize d D esign 1 f or l total 2 R m ................................ ................................ ... 230 13 4 Optimized D esign 1 for total transducer len gth, l total 2 R m .............................. 230 13 5 Top view of propose d transducer for achievi ng large in plane displacement ... 231 13 6 SEM of fabricat ed in plane transducer Design 2 ................................ .............. 233 13 7 Optical microscope image of fabricated in plane transduce r 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 PAGE 19 19 19 LIST OF ABBREVIATION S BCI Boundary Condition Independent CMOS Complementary Metal Oxide Semiconductor CTE Coefficient of Thermal Expansion CTM Compact Thermal Model DLC Diamond l ike 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 Electr on Devi ce 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 PAGE 20 20 20 RC Resistor Capacitor SEM Scanning Electron Microscope/Micrograph TCR Temperature Coefficient of Resistance TLM Transmission Line Matrix PAGE 21 21 21 COMMONLY USED SYMBOLS temperature coeffic ient of resistance strain propagation constant for thermal transm ission line deflection produces by bimorph/multimorph i CTE of i th layer of multimorph mirror rotation angle per unit bimorph temperature rise density 0 resisti vity of embedded heater time frequency in radians per second A i cross sectional area of i th layer of multimorph c thermal capacitance per unit length c p heat capacity per unit mass C bimorph C mirror thermal capaci tance of bimorphs, mirror plate, respe ctively C m curvature of straight multimorph upon deformation d in plane in plane displacement d design 1 d design 2 in plane displaceme nts produced by designs 1 and 2, respectively E i i th layer of multimorph E KE kinetic energy E P E potentia l energy f frequency g thermal conductance per unit length h h b h m heat loss co efficent; 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 repr esenting i ( x ) I cm moment of inertia of mirror plate about center of mass I E current flowing through embedded resistor R E PAGE 22 22 22 I i 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 power; power per unit length q power per unit volume r thermal resistance per unit length r M order of reduced model R A R B etc. resistances in thermal circuit model R c radius of curvature of undeformed curved multimorph heater resistance at temperature s T h and T 0 respectively R m radius of curvature of straight multimorph upon deformation R 0 dc value of characteristic impedance of thermal transmi ssion line s complex frequency s distance along curved multimorph S surface area t t b t i thickness; thickness of bimorph and i th layer of multimorph, respectively T T a T h T b temperature; reference temperature; temperat u re of ambient, embedded heater and bimorph respectively U out of plane deflection of curved multimorph v voltage, represents temperature rise in thermal circuit model v cm velocity of mirror plate center of mass V amplitude of phasor representing v ( x ) voltage applied to embedded heater Z Z A Z B etc. complex impedance s in thermal circuit model Z 0 characteristic impedance of thermal transmission line PAGE 23 23 23 Abstract of Dissertation Presented to the Graduate School of the Universi ty 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 multi morph actuator consists of two or more layers with different coefficients of thermal expansion (CTE). Micromirrors actuated by thermal multimorph s provide large scan range at low driving voltage. Up to 600 m o ut of pla ne displacement of mirror plate and f ull 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 i s 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 develo ped 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 PAGE 24 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 investiga ted 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, piezoelect ric, 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 SiO 2 thin film for thermal isolation. This makes the devices highly susceptible to impact failure during handling and packaging. SiO 2 is also used as one of the multimorph layers in several devices. The low thermal diffusivity of SiO 2 makes the thermal response sluggish. A novel process for fabr icating robust electrothermal MEMS with customizable thermal resp onse and power consumption is developed The process employs Al and W for forming the active layers of the multimorph structure. High temperatur e polyimide is used for thermal isolation The mirrors fabricated by the proposed process have improved robustne ss compared to PAGE 25 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 prop osed designs can produce 100s microns displacement along the substrate surface, which is an orde r of magnitude greater than previously reported designs. Possible applications include integrated Michelson interferome ter, movable MEMS stage and movable micro needles for biomedical applications. PAGE 26 26 26 CHAPTER 1 INTRODUCTION 1.1 Background MEMS scanning micromirrors have been widely used in displays [ 1 ] op tical communications [ 2 ] and biomedical imaging [ 3 ] Scanning may be achieved by piezoelectric [ 4 ] e lectrostatic [ 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 a n electrothermal multimorph MEMS mirror that rotated 124 at only 12.5 V [ 8 ] The main drawbacks of electrothermal micr o mirro r s 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 developme nt of electrothermomechanical (ETM) models of micromirrors. Many micromirror designs utilize SiO 2 thin film thermal isolation for confining most of the heat energy to the actuators, thereby minimizing power consumption The brittle nature of SiO 2 makes suc h devices susceptible to impact failure. Consequently, such devices cannot survive drop test from a height of a f ew centimeters on a vinyl floor However, practical applications such as hand held endoscopes may involve frequent drops from a height of sever al 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 gen eration micromirrors during handling and packaging need ed urgent attention. A major contribution of this thesis is a PAGE 27 27 27 novel process for fabricating robust micromirrors with customizable thermal response speed and power consumption requirements. Straight mu ltimorphs 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 foun d 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 displacement s as high as several hundred microns. This is an o rder 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 desig ns. Section 1.3 enumerates the key objectives of t his thesis. Section 1 .4 summarizes the significan ce 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 Principl e 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 mul timorph transducers have been widely reported. principle of thermal bimorph s [ 10 ] Wei nberg [ 11 ] and Devoe e t al. [ 12 ] discuss the equations governing multimorph transducers. Figure 1 1 shows a schematic of a thermal bimorph. PAGE 28 28 28 Figure 1 1 Schemat ic of thermal bimorph A thermal bimorph consists of two materials with different coefficients of the r mal expansion (CTE). Let the CTE of the top and bottom layers be 1 and 2 respectively. Let t 1 and t 2 represent the thickness and E 1 and E 2 represen t modulus of the top and bottom layers respectively. Let l b and t b represent the length and thickness of the bimorph. When the average temperature along the length of the bimorph is T 0 the tangential angle at the end of the bimorph is 0 When the average temperature along the bimorph length changes to T b the tangential angle ( T b ) is given by [ 10 ] ( 1 1 ) where, ( 1 2 ) ( 1 3 ) ( 1 4 ) Material 1 Material 2 ( T b ) d in plane Undeformed state Deformed bimorph PAGE 29 29 29 If the total bimorph thickness is fixed, maximum deflection is achieved when the ratio of thickness es of the two layers satisfy the condition [ 13 ] ( 1 5 ) If the bimorph bea m width is much greater than its thickness it can be assumed to be in plane strain. In this case replaced by their biaxial modulus in Equation s 1 4 and 1 5. As shown in Figure 1 1 the bimorph tip undergoes in plane displacement, d in plane along with out of plane displacement. In this thesis, two different actuators ar e proposed that amplify d in 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 stra ight bimorph beams to achieve z ero out of plane displacement. The bimorph shown in Figure 1 1 has zero curvature in the undeformed s tate. In this thesis, the analysis and fabrication of curved multimorphs which have a non zero curvature in the plane of the sub strate is reported The distinguishing feature of such actuators is that they undergo both bending and twisting deformations. Mirror designs actuated by curve d multimorphs are also reported. 1.2.2 Overview of Thermal Bimorph Actuated Micromirrors Reithml ler et al. suggested the thermal actuation of micromirrors in 1988 [ 14 ] Since then various e lectrothermal micromirrors actuated by straight multimorphs have been demonstrated with different fabrication pro cesses and different materials [ 15 21 ] For example, Buser et al. proposed an IC compatible fabrication process with Al Si bimorphs [ 15 ] Bhler et al. [ 16 ] and Tuantranont et al. [ 17 ] report CMOS fabrication PAGE 30 30 30 process es for making thermal micromirrors. A MUMPs polysilicon surface micromachining process for micro mirror arrays is discussed in [ 18 ] Lammel et al. report microscanners based on Cr SiO 2 bimorphs [ 6 ] The chromium thin film acts a s 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 SiO 2 bimorphs [ 22 ] A digital micromirror device from Texas Ins truments 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 thereb y producing a twisting action [ 23 ] The large displacement produced by thermal multimo rphs at low voltages is especially suited for micromirrors used in b iomedical imaging applications [ 24 26 ] Other applications inclu de image acquisition systems [ 27 ] lase r output control [ 28 ] and microprojector s [ 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 a pplications 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 SEM s of 1D, 2D and 3D micromirrors respectively previously reported by our group A CMOS based large vertica l displacement (LVD) electrothermal micromirror has been reported in [ 32 33 ] The LVD micromirror provides an out of plane disp lacement of 0.2 mm at 6 V and can scan 15 Heating is achieved by an embedded polysilicon resistive heater. Significant hysteresis PAGE 31 31 31 is observed in the current voltage characteristics of the polysilicon resistor. Also, the mirror plate center suffers signi ficant lateral shift while executing vertical motion. The center shift problem is overcome by the LSF LVD (lateral shift free large vertical displacement) 3D micro mirror reported in [ 34 ] Additionally replacing the polysilicon heater with platinum (Pt) makes the device repeatable with negligibl e hysteresis. The LSF LVD design utilizes four LSF actua tors at each edge of the mirror plate. When all four actuators are operated in phase, out of plane motion up to 600 m may be achieved. Two dimensional angular scanning may be ac hieved 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 2 C f our 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 m. Prior to this thesis, mirrors reported by our group utilized Al SiO 2 bimorphs with SiO 2 thin film thermal isolation at the bimorph ends. The brit tle nature of SiO 2 made these devices highly susceptible to impact failure. Repeatability and reliability studies on micromirrors actuated by Al SiO 2 bimorphs with integrated Pt heater have been included in this thesis [ 19 ] A major drawback of using SiO 2 as an active bimorph layer i s that the low thermal diffusivity of SiO 2 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 c hange thereby leading to actuation. PAGE 32 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) PAGE 33 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. = time, V E = applied voltage, p = power dissipated by Joule heating, T b = bimorph actuator temperature, T h = temperature of embedded resistive heater and R E ( T h ) = resistance of embedded heater. The degrees of freedom (DOF) of the mirror plate are angles x and y 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 be nch. 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 e ach 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 2 C. Out of plane displacement can be Mechanical Model Thermal Model = Temperature dependent resistance of embedded heater x y z Electrical Model PAGE 34 34 34 measured by using an optical microscope with low dept h 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 s can angle measurement se tup 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 sc an angle vs. voltage characteristic An ac voltage superimposed on the dc b ias 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. Alt ernatively, 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 correspondi ng 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 a ctuation signal. Screen Light beam from laser Light reflected by mirror plate Breadboard mounted on micro positioner Packaged micr omirror PAGE 35 35 35 1.2.3. 3 Scan angle velocity for periodic act uation 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 angle s corresponding to different point s on the actuation wav eform 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 s ensitive module (PSM) [ 42 ] The next section lists the key goals of this thesis. 1.3 Research Objectives 1.3 .1 Modeling The ob jectives of device modeling have been enumerated below: Dev eloping 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 ta kes 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 transducer s 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. PAGE 36 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 ther mal response speed and power consumption. Comparison of robustness and performance of current generation micromirrors with previous designs. 1.4 Research Significance T his thesis addresses several important issue s and proposes novel electrothermal micromi rror 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 optimizatio n 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 dro p tests from a few centimeters height, the robust mirrors can withstand drop tests from a height of several feet. This improvement PAGE 37 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 requirement s 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 req uirements. 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 des igned 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 an d 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 wh ich is a ten fold improvement over previous ly 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 miniaturiza tion of several bio imaging systems. Another potential application area is a movable MEMS stage. The device shown in Figure 1 2 C can be used as a 3 degrees of freedom movable stage. W ithout 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. PAGE 38 38 38 1.5 Chapter Organization This dissertation is organized as follows. Chapte r 2 pro vides an introduction on thermal modelin g 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 incl ude 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 Chapter s 9 and 10 Experiments on repeatability and reliabi lity are presented in Chapter 11 Chapter 12 details the fabrication of robust electrothermal mirrors. In plane actuator d es igns are discussed in Chapter 13. Chapter 14 summarizes the dissertation. PAGE 39 39 39 CHAPTE R 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 par ametric models. This dissertation will focus on the modeling of electrothermal micromirrors. However, the approach adopte d 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 o f a block diagram in Figure 1 3 This chapter focuses on thermal model whi ch 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 elect rothermal MEMS is given by the equation of thermoelasticity [ 43 ] ( 2 1 ) where = density, c p = heat capacity per unit mass, k = thermal conductivity, T = temperature, q = power input per unit volume, = coefficient of thermal expansion (CTE), T 0 = reference temperature and = time The symbols and are the Lam constants. The Lam E, and the by [ 43 ] ( 2 2 ) ( 2 3 ) PAGE 40 40 40 The strain ij is defined by [ 43 ] ( 2 4 ) where u i is the i th component of the displacement vector and the partial derivatives are taken with respect to directions x 1 x 2 and x 3 law gives [ 43 ] ( 2 5 ) where ij are the components of the stress tensor a nd is the body force per unit volume The constitutive equation for thermoelasticity is given by [ 43 ] ( 2 6 ) where ij 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 i ncluded here for completeness. For most engineering materials the CTE is in the range 0.36 m m 1 K 1 [ 45 ] which is very sma ll. Therefore the third term on the right hand side of Equation 2 1 is us ually neglected [ 43 ] and the temperature distribution is obtained by solving the general heat equation [ 43 ] ( 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 comm only encountered in thermal problems are listed below [ 46 ] : PAGE 41 41 41 Constant tem perature or heat sink condition Input power p through an area A p : This condition is mathematically expressed as, ( 2 8 ) Heat loss due to convection : Let T a be the ambient temperature and h be the convection coefficient. Then the convective heat loss is given by ( 2 9 ) At micro scale, 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 Insu lation: 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, d T = k / ( p ), 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 a nd achieving fa st response In spite of the challenges invo lved in thermal design, the large force (~mN) and displacement (~10 2 [ 34 ] produced by electrothermal act uators makes it attractive PAGE 42 42 42 for several applications such as micr omirrors 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 starti ng 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 inv estigated in this thesis will be described in detail. 2.2 Litera ture 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 zi ik describes analyti cal methods for solving the heat equation [ 53 ] Analytical appro Fourier series [ 54 ] and Fourier transform [ 55 ] are usually feas ible 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 u sed [ 29 ] This section deals with three commonly used numerical tech niques. 2.2.2.1 Finite element m ethod Lewis et al. provide a detailed background on thermal FE models [ 57 ] Several commercial FE software tools such as COMSOL [ 58 ] Ansys [ 59 ] and IntelliS uite [ 60 ] PAGE 43 43 43 are available. After drawing the geometry, the model is meshed. Meshing involves dividing the model into small elements. A n FE model num erically evaluates the tempe rature at discrete points of the elements These discrete points are known as nodes. The FE method approximately represents Equation 2 7 as n ordinary differe ntial equations, where n is the number of nodes. Let T 1 ( ), T 2 ( T n ( ) represent the temperature at the n nodes. Then the n ordinary differential equations are given by ( 2 10 ) In Equation 2 10 C K and F ( ) are the heat capacitance matrix, thermal conductivity matrix and for cing vector respectively. The vector 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 combi ned to form a circuit representation for the complete model. The complet e 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 m etho d (FDM) Equation 2 7 is a differential equation governing temperature distribution. In FDM, this differential equation is approximated using a set of differen ce equations. The resulting algebraic equations may be solved numerically using MATLAB or SPICE [ 62 ] to obtain the temperat u re at discrete points in the model. PAGE 44 44 44 2.2.2.3 Transmission line m atrix (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 circ uit. In circuit models resistors may be used to represent the thermal resistance s and capacitors are used to account for thermal capacitance s In such models, nodal voltages represent temperature. The TLM me thod draws analogy between heat flow and the flo w 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 r eflection 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 wi ll be in introduced in Chapter 4 In spite of their widespread application a nd commercial success, numerical methods are computatio nally expensive For instance, FEM for practical problems may have as many as ~10 5 nodes [ 56 ] making them computatio nally 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 s everal attempts have been made to develop simple circuit models in the last three decades. 2.2.3 Compact Thermal Models (CTM s ) 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 PAGE 45 45 45 based on engineering intuition. The values of the resistors and capacitors are chosen such that the difference between circuit model out put and FEM simulations are minimized. As few network elements are used to build such models, they are known as CTM s (compact thermal models). This method can be used to determine compact models that can be simulated without using excessive computational r esources. Moreover, it is desired that the model demonstrate s a certain degree of bound ary condition independence A BCI (boundary condition independent) mode l may be used as a component in a com plete system level simulation. The major drawback of the meth od 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 procedu re 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 th e BCI CTM of an electronic package [ 66 ] BCI CTM s have receive d 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 u p by the SEED and PROFIT projects [ 69 ] The DELPHI, SEED and PROFIT projects culminated in the Joint PAGE 46 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 approxi mations to these distributed networks. Szkely reports the analysis of a one dimensional distributed RC network [ 71 ] A distributed network has infinite nu mber of capacitors and is therefore associated with infinite number of time constants [ 72 ] Szkely et al. define a time constant de nsity 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 l umped element model (L EM ) with only a few circuit element s [ 72 ] The major disadvantage of this appro ach is that it is difficult to associate any physi cal signifi cance 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 o f equations from a higher dimensional space to a lower dimensional space withou t incurring significant error [ 56 ] F or example, let us consider an electrothermal micromirror As discussed in Chapter 1, the heater and bimorph temperature s are of interest to the design engine er. Equation 2 10 will provide the temperat ure distribution in the device. An MOR may be used to approximate the system of n equation s in Equation 2 10 by a s et of r M equations, r M << n These r M equations may then be used to calculate the heater and bimorph temperature s MOR algorithms c an achieve a high degree of PAGE 47 47 47 computational efficiency by significantly reducing the model order [ 76 ] H owever, 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 mod el 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 rev iews key methods for solving the heat equation. Analytical method s are limited to simple geometries only. Numerica l methods such as FE method, finite difference method and TLM method may be used for complex geome tries. 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 practic al MOR application. PAGE 48 48 48 CHAPTER 3 DYNAMIC COMPACT THER MAL MODELING BY MODE L ORDER RED UCTION 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 10 3 10 5 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 com pact 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 chapte r 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 provide d by the reduced order model is used to construct a lumped element d ynamic thermal model in Section 3.5. 3.2 E lectrothermally Actuated M icromirror 3.2.1 Device D escription Figure 3 1 shows an SEM of a 1D electrothermally actuated micromirror device. The 1 mm 1.2 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 a ctuator consists of a 1 thick PECVD SiO 2 layer at the bottom and 1 thick aluminum (Al) layer on the top. A 0.2 thick platinum (Pt) heater layer is sandwiched between the top Al and bottom SiO 2 layers. PAGE 49 49 49 T he Pt and Al layers are isolated electrically by a 500 thick layer of SiO 2 Each bimorph beam is 8.5 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 i n J oule heating. The difference in the thermal expansion coefficients of the SiO 2 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 SiO 2 a nd Al layers. The mirror is fabricated by a process s imilar to that described by Wu et al. [ 8 ] The SiO 2 connections at either end of the bimorphs provide thermal isolation, thereby reduci ng power consumption ( Figure 3 2 ). The resonance frequency of the mirror is 328 Hz. The mirror scans as much as 32 at a n app lied dc voltage of 8 V For scanning applications, the actuation voltage consists of an ac volt age 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 PAGE 50 50 50 Figure 3 2 Schematic of 1 D micromirror (not to scale) 3. 2.2 Micromirror M odeling Figure 1 3 shows a schematic of the complete model of an electrothermally actuated micromirror device. An applied voltage V E across the embedded Pt heater causes Joule heating, thereby producing a power p This raises the temperature T b of the b imorph actuators and the mirror plate rotates by a n angle The change in temperature also determines the resistance R E ( T h ), of the Pt heater. A complete m odel 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 m icromirror device is discussed In the following sections a n FEM thermal model of the micromirror is developed The FEM model is reduced to a model of order 2 by using a n order Substrate Thermal isolation Bimorph Thermal isolation Mirror plate TOP VIEW Pt SiO 2 Si Al SIDE VIEW PAGE 51 51 51 reduction method. Finally a simpl e lumped element circuit i s developed based on the reduced order model 3.3 F inite E lement M odeling A 3D FEM simulati on of the device shown in Figure 3 1 is very time consuming due to the large number of bimor ph 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 S ection 3. 3.3. This simplification leads to a significant saving in computational resour ces. 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 surroundi ng air. Consequently, heat loss due to convection and diffusion may be applied as boundary condition at these surfaces. 3. 3. 2 Finite Element (FE) F o ethod s method [ 57 ] ma y be used to approxi mate 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 num ber of small elements. Rectangular finite elements have been used for the present wor k. Figure 3 3 shows a schematic of a rectangular finite element with length 2 b 1 and width 2 b 2 Nodes 1 through 4 are numbered in a counterclockwi se fashio e represents the boundary of the element. gives [ 57 ] PAGE 52 52 52 ( 3 1 a) (3 1b) where, n and 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 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 averag e temperature along the length of the thermal bimorph actuators [ 79 ] Let n i i = 1 to k be k nodes equispaced along the bimorph length. Let us define X such that the n i th ( i = 1 to k ) elements of X are 1 / k and all other elements are zero. Then, is the average temperature alon g the length of the bimorph actuators. Figure 3 3 A rectangular finite element 3. 3.3 FEM of the Micromirror The ele ct rothermal mirror shown in Figure 3 1 is actuated by a n array of 72 bimorphs. From the device topolog y 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 e 2 b 1 2 b 2 PAGE 53 53 53 obtained from th e model has been shown in Figure 3 4 It was fou nd that the bimorph beams indeed have a nearly identical temperature distribution. Therefore, in order to conserve com putational 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 5 A An enlarged view of the b imorph region is shown in Figure 3 5 B The dimensional parameters are the sa me as those given in S ection 3. 2.1. The boundary condition at the bottom and left side of the substrate is ass umed to be a fixed temperature. H eat loss by convection and diffusion is assumed at all other boundaries and Equation 2 9 is applied. The boundary condition e xpressed 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) PAGE 54 54 54 H ence, heat loss due to radiation may be neglected [ 32 ] The finite element formulation has been implemented in MATLAB. All the elements in the FEM m odel a re rectangles as shown in Figure 3 3 T he variation s in the thermal conductivity of the bimorp h materials with temperature change are neglected. This assumption has bee n verified with simulations in S ection 3. 3.4. The effect of temperature dependence of heat capacitanc e values has been addressed in S ection 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 alon g the bimorph is shown in Figure 3 6 A The thermal image of an actual device, acquired usi ng a high resoluti on IR (infrared) camera, is shown in Figure 3 6 B The simulated and experimental data are also plotted in Figure 3 6 C Both the simulation and experiment were done for 22.55 mW power in put 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 6 C the FEM simulation matches the experiment well. It also can be seen that the simulated temperature distribution along the b imorph has a maxima at (A) (B) x y Mirror Bimorph Substrate Pt Al Si SiO 2 PAGE 55 55 55 about 80 m, 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 he ater 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 leng th 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 p ower 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 Simulated temperature (K) Imaging data (K) (B ) (C ) PAGE 56 56 56 Figure 3 7 shows the simu lated 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, t he rotation a ngle is li nearly 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 temperatu re dependent material properties was built using COMSOL. The temperature dependent thermal conductivity for Si and SiO 2 were obtained from literature [ 81 82 ] The Wiedemann Franz law was used to estimate the ther mal conductivity of aluminum and platinum [ 83 ] The average bimorph temperature obtained from the non linear model was compared with that obtained from the mo del 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 dev ice operation depic ted 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 10 5 R epetitive dynamic simulations for arbitrary inputs can be ti me and resource consuming. Therefore, in the next section a compact thermal model of the micromirror device is developed. PAGE 57 57 57 3.4 Model Order Reduction 3.4.1 Introduction Model order reduction algorithms provide a formal and rigorous way of obtaining a reduce d 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 ] ( 3 2 a) (3 2b) where, W = K 1 C b = K 1 F and T b is the average temperature along the bimorph length. The matrix M is defined such that it extracts useful information from the tem perature vector 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 henc e T b i s of interest to the device engineer. Model order reduc tion seeks to find a system [ 56 ] 25 20 15 10 5 0 Input electrical power (mW) 0 20 40 60 80 100 Mirror rotat ion (degrees) Simulation result Experimental data 2.7 PAGE 58 58 58 ( 3 3 a) (3 3b) such that r M << n and the error 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 out put [ 84 ] However, computational complexity of these methods is O which is prohibitively large for most practical problems. The Arnoldi process which has been described below, has been reported to be suitable for practical mod el order reduction problems [ 56 ] 3. 4.2 The Arnoldi Process for Model Order R eduction For the Arnoldi process, the computational complexity is O where N z ( M ) is the number of non zero elements in the matrix M and r M 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 rig orous upper bound for the error in the reduced model output. However, several he uristic 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 d iscretiz ed Equation 3 2 is given by [ 77 ] PAGE 59 59 59 ( 3 4 ) where 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, ( 3 5 ) where for i = G ( s ) about s = 0 [ 77 ] The transfer function of the reduced order system described by Equation 3 3 is given by, ( 3 6 ) The Arnoldi process computes a r educed order model, i.e., Equation 3 3 such that the first r M moments of the transfer function of the reduced system and the first r M moments of the orig inal 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 ] ( 3 7 a) (3 7b) (3 7c) The algorithm for the Arnoldi process wa s implemented in MATLAB and the discretized heat equation for the electrothermally actuated micromirror device was reduced to a lower order system. PAGE 60 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 r M = 2. It was found that increasing r M (not shown in Fi gure 3 8 ) does not change the model output significantly up to a frequency of 10 4 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 10 4 Hz. Since 10 4 Hz is more than one order of magnitude great er 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 chan ge 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 Fi gure 3 8 A Fi gure 3 8 B compares the phase response obtained from reduced order model and the FEM. Some higher order effects are obs erved 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 vali date 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 (PS M ). 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 9 A ). The low frequency response in Figure 3 9 B shows t wo distinct cutoff frequencies in the mirror response. Since the first mechanical resonance is observed at 328 Hz, the low PAGE 61 61 61 frequency drops may be attributed to the thermal response of the device. This is in agreement with the reduced order model output. Mo reover, a comparison of Fi gure 3 8 A and Figure 3 9 B shows that the reduced order model accurately predicts the range of frequencies over which the mirror response decays. Fi gure 3 8 Transfer function of thermal model. (A ) Transfer function magnitude plot from finite element model and reduced order models ( r M = 2). The two plots practically overlap. The plot shows the change in temperature at the c enter of the bimor ph 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 Increase in average bimorph temperature (K) FEM Reduced model of order 2 Phase (radians) Reduced model of order 2 FEM Frequency (Hz) 10 4 10 2 10 0 10 2 10 4 10 6 10 4 10 2 10 0 10 2 10 4 10 6 (A ) (B) 1.4 1.2 1 0.8 0.6 PSM Output peak peak (V) 10 2 10 0 10 2 10 4 10 1 10 0 10 1 Frequency (Hz) Frequency (Hz) 10 2 10 0 10 2 10 4 PSM Output peak peak (V) (A ) (B) PAGE 62 62 62 In general, finite element models for many practical problems have more than 10 5 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 agains t 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 pra ctical problems. Constant thermal conductivity and heat capacitance values were used for obtaining the simulation res ults in Fi gure 3 8 However, in general thermal conductivity and heat capacitance values are temperature dependen t. The effect of temperature dependence of thermal conductivity on the static response of th e mirror has been discussed in S ection 3.4. The dynamic response of the mirror depends on both thermal conductivity and heat capacitance values. The temperature dep endent thermal conductivity was obtained from literature [ 81 83 ] The temperature dependence of heat capacitance values was estimated using the data available fo r bulk materials [ 85 ] The temperature range corresponding to the 0 100 mW in put power range depi cted 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 scan ning motion The temp erature distribution corresponding PAGE 63 63 63 to the dc bias can be used to estimate the thermal conductivity and heat capacitance values for building the device model. From Fi gure 3 8 it is evident that a reduced order model of order 2 cl osely approximates the magnitude and phase response of the micromirror over a frequency range that is more than 100 time s greater than the microm irror 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 E quation Th e one dimensional heat equation is given by, ( 3 8 ) where x is the direction along which the temperature varies. Let us consider the rec t angular section shown in Figure 3 10 A and assume that the temperature varies only along the length of this section. Let T 1 T 2 and T 3 denote the temperature at cross sections 1, 2 and 3 respectively. Replacing the second order derivative in Equation 3 8 by its central differ ence approximation ( 3 9 ) Treating temperature as an across variable and heat flow as through variable Eq uation 3 9 may be equivalently represen ted by the ci rcuit shown in Figure 3 10 B in PAGE 64 64 64 w hich heat loss to the atmosphere is ignored. Let h be the heat loss coefficient T a be the ambient temperature and S be the area from which heat loss to the atmosphere is taking place A lumped re sistor (1 / ( hS )) can be used to account for heat loss to the atmosphere Figure 3 10 C shows the e quivalent 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 ne twork ( T 2 satisfies Equation 3 9 ). (C ) Equivalent thermal network taking heat loss to atmosphere into account. 3. 5.2 Equivalent Thermal Model of M icromirror From Figure 3 4 B the temperature variation along the le ngth of the bim orph is less than 5%. Hence, the array of bimorphs may be represented by the eq uivalent circuit shown in Figure 3 10 C with the impedances scaled by the number of bimorphs n b Since the isolation region length (12 m) is less that 10% of the length of the bimorphs, the thermal capacitance of the isolation region may be neg lected. Furthermore, the mirror plate is more than 40 m thick. Hence, the conductive thermal x / kA x / kA q (2 A x ) p (2 A x ) T 1 T 2 T 3 x x 1 2 3 (A ) (B ) (C) T 1 T 2 T 3 x / kA x / kA p ( 2A x ) q (2 A x ) 1/ hS T a PAGE 65 65 65 impedance of the mirror plate may be neglected. The complete lumped element circui t model has been shown in Figure 3 11 Figure 3 11 Lumped element circuit model The symbols used in Figure 3 11 ha ve been described in Table 3 1. The 1 / h b S b resistance in series with a dc voltage source acco unts 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 wid th and thickness of the actuators is at least an order of magnitude less than the length, temperature variation perpendicular to the length is neglected. Ac cordingly, an average heat loss coefficient is used to account for the combined heat loss through al l bimorph surfaces. The 1 / h m S m 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 rotat ion angle, an average he at loss coefficient value is used to account for heat loss from all surfaces of the mirror plate. The capacitors C bimorph and C mirror account for the thermal capacita nces of the bimorphs and mirror plate respectively. Hence, they C mirror C bimorph p T a T s Is olation Bimorph Mirror T b Isolation T a PAGE 66 66 66 provide explicit relatio nship between the seco nd order response shown in Fi gure 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 Sy mbol Description T b Bimorph t emperature T s Substrate temperature T a Ambient temperature I i A i k i n i Length, cross sectional area, average thermal conductivity and number of beams in the thermal isolation region respectively l b A b k b n b Length cross sectional area, average thermal conductivity and number of beams in the bimorph array respectively p Electrical power input C bimorph Total thermal capacitance of n b bimorphs C mirror Tota l thermal capacitance of mirror plate h b Heat loss coeffi cient on bimorphs h m Heat loss coefficient on mirror plate S b Total surface area of n b bimorphs S m Total surface area of mirror plate 3. 5.3 Results Obtained from Lumped Element M odel Figure 3 12 compare s the magnitude and pha se 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 10 4 Hz and explains the two cutof f frequencies in Fi gure 3 8 A and Figure 3 9 B From Figure 3 12 A 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 t hermal resistance of the mirror plate has been neglected in the lumped element model. No such as sumptions are made in the finite element model. For frequencies up to 10 4 Hz, the error in the phase plots shown in Figure 3 12 B and Fi gure 3 8 B is less than 10%. However, significant error is PAGE 67 67 67 observed at higher frequencies. This is because the two lum ped 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 mode l. The plot shows the rise in bimorph temperature. (B ) Transfer function phase plo t A comparison of Figure 3 9 B and Figure 3 12 A shows that the lumped element circuit model accounts for the two low frequency cutoffs obser ved in the device response. Figure 3 13 compares the normalized PSM output of Figure 3 9 B 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 diff erence 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 investig ation of variation in heat loss coeff icient with mirror til t angle. A feedback path from the angular output to the thermal model can possibly be used to account for this phenomenon. Increase in average bimorph temperature (K) 0 0.5 1 1.5 2 Phase (radians) Frequency (Hz) Frequency (Hz) (A ) (B) FEM Circuit Model FEM Circuit model 10 0 10 5 10 4 10 2 10 0 10 2 10 4 10 6 60 40 20 0 PAGE 68 68 68 Figure 3 13 Comparison of normalized output of ci rcuit 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 reductio n 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 mo del 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) Normalized output Normalized PSM output Normalized circuit model output 10 4 10 2 10 0 10 2 10 4 PAGE 69 69 CHAPTER 4 TRANSMISSION LINE THERMAL MODEL OF ELECTROTHERMAL MICROMIRRORS 4.1 Background A d istributed RC ne twork can rigorously represent the heat flow path in a packaged IC chip [ 72 ] Such distributed networks have the same topo logy as a transmission line [ 64 ] Therefore, equations that govern transmission lines can be used for analyzing thermal mode ls. The main focus of this chapter is a transmission line based thermal model of a micromirror that is actuated by thermal bimorph actuator s [ 20 86 ] The thermal model is simplified to obt ain a simple LEM with only a few circuit elements. The approach used in t his chapter provides a rigorous strategy for deriving a thermal model that saves computation time and provides explicit relationship between device response and physical parameters. A n el ectrical model is used to determine the power dissipated, p in the embedded resistor for an applied voltage, V E A mechanical model predicts the mirror rotation for a change in bimorph temperature. The electrical thermal and mechanical models are th en integrated to obtain the complete model, as sh own schematically in Figure 1 3 As discussed in Chapter 1 t he average temperature of the bimorph actuator, T b determines the mirror rotation angle, [ 10 ] The resistance of the heater is dependent on its temperature, T h This chapter is organized as follows. Section 4. 2 int roduces the micromirror and pre sents the equations governing electrical, thermal and mechanical behavior In S ection 4. 3, a dynamic thermal model of the micromirror is developed. S imple lumped element thermal model s are pr oposed for both static and dynamic case. Electrothermal modeling is discussed in Section 4.4. In S ection 4. 5 the thermal model is integrated into PAGE 70 70 a static electrothermomechanical model Experimental results are used to validate the model in Section 4. 6 4. 2. 1 D Electrothermally Actuated Micromirror 4. 2.1 Device D escrip tion Figure 4 1 and F igure 4 2 show the SEM and schematic of the micromirror respectively. Figure 4 1 SEM of electrothermal micromirror [ 19 ] F igure 4 2 Schematic of electrothermal micromirror [ 19 ] SiO 2 Si Al Pt SIDE VIEW TOP VIEW Mirror plate Thermal isolation Bimorphs Heater Substrate Bond pads Mirror plate B imorph array PAGE 71 71 The 1.1 mm 1.2 mm mirror plate is attached to the substrate by an array of 72 thermal bimorph actuators. The thermal bim orph actuators consist of a 1.1 PECVD SiO 2 beam evaporated aluminum (Al) layer. Each crystal silicon (S CS) layer underneath the mirror plate serves to improve the flatness of the mirror surface. Th e thickness of t he SCS layer is in the range 26 37 parameters. A current passing through the 1.2 mm long embedded platinum (Pt) heater results in Joule heating. The difference in the coefficient s of thermal expansion ( CTE ) of the SiO 2 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 SiO 2 thermal isolation regi on. The low thermal conductivity of SiO 2 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 adh esive 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 simil ar The key difference lies in the embedded heat er design. For the device in Figure 3 1 [ 39 ] the heater r uns along t he length of the bimorph; the heat er 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 c ompares the thin film layers that constitute the bimorph structure along with typ ical thickness values. In Figure 4 3 A the thin SiO 2 layer isolates the heater from Al electrica lly; the bimorph shown in Figure 4 3 B has no heater layer. In both cases, the mirror plate rotation is proportional to the average temperat ure along PAGE 72 72 the bimorph length [ 19 39 ] Simulations show that for the same ris e in the average bimorph temperature, the deflection produced by the structure shown in Figure 4 3 B is grea ter than that produced by the structure shown in Figure 4 3 A by 14%. Also in case of Figure 4 3 B the heater lies along the side of the bimorph array and undergoes much less deformation during actuation as compared to th e heater corresponding to Figure 4 3 A Therefore, when large deflections are invo lved, th e design corresponding to Figure 4 3 B 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 3 B [ 8 ] The key advanta ge of the design shown in Figure 4 3 A is that the bimorph i s 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 maximu m temperature simultaneously [ 39 ] Thus the structure in Figure 4 3 A maximizes bimorp h utilization. It will be shown i n S ection 4. 3 that there is a significant temperature gradient along the bimorph length for the design corresponding to Figure 4 3 B Figure 4 3 Comparison between two bimorph designs. (A ) Thin f ilms in bimorph s corresponding to Figure 3 1 [ 39 ] (B ) Thin films in bimo rphs of the device shown in F igure 4 2 Typical thickness values have been indicated beside the schematics. 1 m 0.83 m 0.25 m 0.1 m 1.1 m 0.83 m (A ) (B ) Al SiO 2 Pt PAGE 73 73 4. 2.2 T hermal Bimorph A ctuation The principle of operation of thermal b imorphs is well established [ 10 79 ] and has been outline in Cha pter 1. From Equation 1 1 t he change in tangential angle at the tip of a bimorph ( T b ) is directly proportional to the change in the average temperature along the bimorph length. Consequ ently 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 ran ge, 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 bimorph s 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, ( 4 1 ) T he coefficient of the ( T b T 0 ) term in Equation 4 1 is ; it represents the change in defl ection angle per unit rise in bimorph temperature. T 0 denotes a reference temperature, typically room temperature, i.e., 298 K. The material properties of the two bimorph layers, i.e., Al and SiO 2 are listed in Table 4 1 Plugging the values from Table 4 1 into Equation 1 1 yields = 0.08833. Table 4 1 Material properties for thermomecha nical simulations [ 58 ] Material modulus (GPa) C TE (K 1 ) Al 70 0.35 23.110 6 SiO 2 70 0.17 0.510 6 From FEM simulation using COMSOL [ 58 ] it was found that = 0.08673 [ 19 ] Therefore, the value of obtained using Equation 4 1 has less than 2% error. PAGE 74 74 Equatio n 4 1 is v alid for the temperature range in which material properties do not vary significantly. 4. 2.3 Electrothermal M odel Neglecting edge effects at either ends of the bimorph array the embedded h eater has a nearly uniform temperature distribution. W hen a voltage, V E is applied the power, p dissipated due to Joule heating is given by, ( 4 2 ) where T h is the average temperature of the embedded heater and R E ( T h ) is the heater resistance. If the TCR of the heater is the refer ence temperature is T 0 and the heater resistance at T 0 is R E 0 then R E ( T h ) can be expressed as ( 4 3 ) ( 4 4 ) To measure the reference temperature T 0 was set as the room temperature i.e. 298 K. A mirror was placed in an oven and the resistance of the h eater was measured in the 298 498 K temperature range. The TCR was determined to be 0.0025 K 1 by using a linear fit fo r the resistance vs. temperature plot. As disc ussed above, both mechanical and electrical behavior can be readily described by the si mple models given by Equations 4 1 and 4 4 respectively. What is missing in the com plete model shown in Figure 1 3 is the thermal model, whi ch 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 PAGE 75 75 both static and dynamic cases. A static electrothermomechanical model will be discussed in Section 4.5. 4.3 Thermal Model The elect rothermal mirror shown in Figure 4 1 is actuated by an array of 72 bimorphs. From the device topolog y it appears that if edge effects at the two extremes of the array are n eglected the individual bimorphs have a nearly identi cal temperature distribution [ 39 ] In orde r 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 v ariation 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 computa tional resources, any temperature gradient across the ar ray 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 l oss 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 S ec tion 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 d ue to radiation is neglected [ 29 ] The next subsection will provide details on the estimated heat loss coefficient from the device to the surr ounding atmosph ere. After that a simple FE model consisting of a single bimo rph and a section of the mirror plate will be presented PAGE 76 76 Figure 4 4 Simulated temperature distribution in a section of th e 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 C oefficient Both convection and thermal diffusion will result in heat loss from the device to the surround ing 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 compa r ed 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) PAGE 77 77 well as the outer boundary of the air region. The bimorph and mi rror section define the internal boundary of the air region. A uniform elevated temperature of 310 K was applied to the sur faces of the bimorph and mirror plate that are in direct contact with air. The thermal conductivity of air, k air was assumed to be 0.026 Wm 1 K 1 [ 58 ] From the solution of the steady state heat equ ation obtained using COMSOL [ 58 ] the average heat loss coefficient on the bimorph and mirror plate were found to be 188 Wm 2 K 1 and 47 Wm 2 K 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 2 K 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 bimorp h an d 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 COMSO L [ 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 te mperature) Mirror (elevated temperature) Bimorph (elevated temperature) Air Air PAGE 78 78 condition w as used at the f aces common to adjacent mirror sections. The open boundary condition was imposed on all other boundaries. The Boussinesq approximation was used to account for t he 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 coef ficient 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 coefficient s 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 h m and h b respectively. Based on simulation results, h m and h b are chosen to be 188 Wm 2 K 1 and 47 Wm 2 K 1 respectively. 4. 3.2 FE Thermal M odel Neglecting temperature variation ac ross the bimorph arr ay, the FE model shown in Figure 4 6 was bu ilt 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 m, 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 subst rate. Heat loss coefficien t 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 PAGE 79 79 nearly identical, symmetry condition is imposed on all other boundaries. Figure 4 6 A shows the simulation results for a 10 mW power input to the device. Figure 4 6 FE thermal model of micromirror (A ) Simulated temperature distributi on 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 SiO 2 1.4 Si 130 Pt 71.6 Figure 4 6 B shows the temperature distribution along the bimorph length. The temperature is highest at the point where the heater is located, i.e., T h = 363.65 K. The average temperature, T b along the bimor ph 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 (m) Temperature (K) (A) (B) PAGE 80 80 Due to process variations, the temperature distribution in a real device may vary significantly from the si mulation results shown in Figure 4 6 The next sub section will address the impact of proc ess variations on the thermal model. 4. 3.3 Effect of Process Variations on T herma l M odel The micromirrors were fabricated by a process sim ilar 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 re move 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 b een shown As shown in Figure 4 7 A for low etch times th e 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 tempera ture will be lower than that depicted in Figure 4 6 Figure 4 7 B shows that for longer etch time the Si below the isolation region is co mpletely 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 t hickness of Si below the mirror plate. Longer the etch time, thinner the Si layer. In order to elucidate the effect of process variations exper imentally, two mirrors with release etch times of 5 min and 12 min were compa red. A dc source was used for actuation and the optical angle was tracked by using a laser beam reflected from the PAGE 81 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 the rmal conductivity of the thermal isolation region by a suitable factor. The isolation region consists of SiO 2 which is transparent to visible light. Hence, the thermal resistance of the isolation region may be estimated by meas uring the silicon undercut us ing 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 c onstant 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 w ill be derived in the following sections. (A ) (B ) Pt SiO 2 Si Al PAGE 82 82 Figure 4 8 Comparison of two mirrors with different etch times. A lo nger etch time results in better thermal isolation and consequently a larger optical angle per unit power input. 4. 3.4 Transmission l ine M odel for 1 D Heat Flow Let us consider t he 1 D geometry shown in Figure 4 9 A Let the thermal conductivity be k and the heat capacitance per unit length be c. If the 1 D 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 weig hts 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 9 B An element of length x 0 is represented by resistors r x 1 / ( g x ) and the capacitor c x ; 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. Th e current, i represents heat flow. Principles governing transmission lines are well 0 10 20 30 40 50 60 Optical angle (degrees) Input electrical power (mW) Etch time = 12 min Etch time = 5 min 40 30 20 10 0 PAGE 83 83 established [ 89 ] and may be used to anal yze the circuit shown in Figure 4 9 B At x, let the voltage be v ( x ) and the current be i ( x ) Kirch ( 4 5 ) Kirch h N gives, ( 4 6 ) For harmonic variation, ( 4 7 ) ( 4 8 ) Let, ( 4 9 ) From Equation s 4 5 and 4 7 ( 4 10 ) From Equations 4 5 4 6 4 7 4 8 and 4 10 ( 4 11 ) It will b e shown in Section 4.5.2 that A 1 and A 2 can be determined by the heat sources and thermal impedances connected at the two ends of the 1D geometry. PAGE 84 84 Figure 4 9 Passive transmission line model. (A ) G eometry 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. ( 4 12 ) r x + + v ( l ) v ( 0 ) i ( 0 ) i ( l ) (B) Z A Z A Z B v ( 0 ) v ( l ) i ( l ) i ( 0 ) + + (C) N i ( x ) i ( x+ x ) v ( x ) v ( x+ x ) l t Direct ion of heat flow (A) Network representation of an element of length x x 0 w rl / 2 rl / 2 cl v (0) i (0) + v ( l ) i ( l ) + (D) PAGE 85 85 ( 4 13 ) Using Equations 4 12 and 4 13 ( 4 14 ) From Equations 4 10 and 4 12 ( 4 15 ) ( 4 16 ) From Equations 4 10 4 12 and 4 16 ( 4 17 ) But from Figure 4 9 C ( 4 18 ) So from Equations 4 17 an d 4 18 ( 4 19 ) Equations 4 14 and 4 19 are solved to give, ( 4 20 ) ( 4 21 ) where, Z 0 = r / PAGE 86 86 The circuit model shown in Figure 4 9 C has been rigoro usly 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 ( 4 22 ) ( 4 23 ) The appr oximations used in Equations 4 22 and 4 23 are valid if, ( 4 24 a) and, (4 24b ) Equations 4 22 and 4 23 elucidat e the physical significance of Z A and Z B respectively. From Equation 4 22 Z A rl / 2 represents half of the conductive resistance of the 1 D geometry shown in Figure 4 9 A Similarly, from Equation 4 23 Z B rep resents 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 Z B Therefore, if the I nequalities 4 24 a and 4 24 b are satisfied, the thermal LEM shown in Figure 4 9 D can be used without inc urring large error. Equation 4 9 may be equivalently expressed as, ( 4 25 a) PAGE 87 87 (4 25b) (4 25c) where, 1 and 2 are the real and imaginary parts of Substituting Equation 4 25 into Inequality 4 24 a gives, ( 4 26 ) Substituting Equat ion 4 25 into Inequality 4 24 b gives, ( 4 27 ) The conditions necessary for the LEM shown in Figure 4 9 D t o be valid may be obtained by simplifying Inequalities 4 26 and 4 27 ( 4 28 a) and, (4 28b) Inequalities 4 28 a and 4 28 b illustrate the physical conditions necessary for the lumped element approx imation to hold. Inequality 4 28 a 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 28 b requires that the time const ant ( rl ) ( cl ) be significantly less than the reciprocal of the frequency, If the 1D geometry satisfies Inequality 4 28 a but not Inequality 4 28 b, the lumped element approximation may still be used for the static case. PAGE 88 88 If Inequalities 4 28 a and 4 28 b do not hold good, the 1 D geometry may be partitioned into smaller segments such that Inequalities 4 28 a and 4 28 b 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 equiva lent circuit shown in Figure 4 9 C must be used. For the static case, i.e., = 0, Equations 4 9 4 20 and 4 21 give, ( 4 29 a ) ( 4 29b ) where, R 0 = ( r / g ) 1/2 denotes the dc value of Z 0 If ( rl )( gl ) 1 Equations 4 22 and 4 23 yield ( 4 30 a ) (4 30 b ) These results will be applied to develop a thermal model of the micromirror in the next subsection. 4. 3.5 Equivalent C ircuit Representation of Thermal M odel 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 dimensio nal and the results derived in S ection 4.3.4 are applicable. Figure 4 10 shows the equivalent circuit representatio n of the thermal model of the 1 D mirror. Th e equivalen t thermal impedances for the bimorph and the mirror plate are denoted by subscripts b and m respectively and ma y be calculated using PAGE 89 89 Equations 4 22 and 4 23 Since the length s of the thermal isolation regions are an order of magnitude smaller than the bimorph length heat loss from the isolation regions to the at mosphere has been neglected and they have been replaced by a single resistor to represent the conductive thermal resistance. These resistors are R iso 1 and R iso 2 in Figure 4 10 The power input is p and the number o f bimorphs is n b So, the current source with magnitude p / n b 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, ( 4 31 ) ( 4 32 ) The a verage bimorph temperature increase can be obtained by integrating Equation 4 31 over the bimorph length l b and is given by, ( 4 33 a ) (4 33 b) p / n b R iso 1 Z Ab Z Bb Z Am Z Bm Z Ab + + M V = ( T h T 0 ) R iso 2 I ( l ) I (0 ) PAGE 90 90 N ode M in Figure 4 10 represents the embedded heater. Therefore the rise in embedded heater temperature is given by, ( 4 34 ) Let, Z L = R iso 1 represent the impedance to the left of the bimorph. The current through Z L is (( p / n b ) ( b / r b )( A 1 b A 2 b )) Therefore, ( 4 35 ) Similarly, let Z R represent t he impedance to the right of the bimorph. Then Z R is given by, ( 4 36 ) Equations 4 35 and 4 36 can be solved to find A 1 b and A 2 b ( 4 37 a ) ( 4 37 b) 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 If the power dissipated in the embedded heater has several frequency components, t he total temperature rise may be obtained by summing the contribution from each frequency component i.e., for m components, ( 4 38 ) where, the subscript i denotes the i th component. PAGE 91 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 M odel 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 V E across the embedded heater resistance R E causes power dissipation. Both t he heater temperature T h and the bimorph temperature T b 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 temp erature. 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 R Ab is p artitioned into R Ab and (1 ) R Ab where is a partition factor. The value of is + V (0) + V ( l ) z dc R 0 b V = T b Q M V E R E 0 h R E 0 I E V = ( T h T 0 ) I (0 ) I ( l ) R Ab (1 ) R Ab R Ab R Bb R iso 1 PAGE 92 92 chosen such that the voltage at node Q in Figure 4 11 gives the avera ge bimorph temperature increase, ( 4 39 ) Let, ( 4 40 ) ( 4 41 ) ( 4 42 ) where z dc represents the normalized thermal resistance to the right of the bimorph. Let a = A 1 b / A 2 b for = 0 Equation 4 37 gives, ( 4 43 ) Substituting V (0) I (0) T b from Equations 4 10 4 11 and 4 33 into Equation 4 43 and using Equation 4 42 ( 4 44 ) An equation to explicitly represent in terms of device parameters may be obtained by substituting Equation 4 43 into Equation 4 44 ( 4 45 ) where, and z dc are defined by Equations 4 40 and 4 42 respective ly. For the device shown in Figure 4 1 the value of is 0.81. Equation 4 45 has been rigorously derived using transmission line th eory. Next, let us check if the condition for lumped element approximation, i.e., Inequality 4 28 a is PAGE 93 93 satisfied by the bimorphs and the mirror plate section. For the bimorph, ( r b l b )( g b l b ) = 1.4 10 7 1 and for the mirror plate ( r m l m )( g m l m ) = 0.04 1. Therefore, Inequality 4 28 a holds good for the bimorph and the mirror plate section. Hence, R Ab R Bb R Am and R Bm may be replaced by the lumped ele ment approximation given by Equation 4 30 Figure 4 12 shows the lumped element stat ic electro thermal model. Since ( r b l b ) << 1/( g b l b ), the current through the resistor 1/( g b l b ) is negligible comp ared to the current through the ( r b l b ) / 2 resistors. Consequently, I 1 I 2 and voltage drops across both the ( r b l b ) / 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 ob tained mechanical resonant frequency of the mirror is f resonant = 173 Hz. For frequencies greater than f limit = 265 Hz, i.e. limit = 1665 rad/s, the I 1 I 2 M Y V E R E 0 T h R E 0 I E R iso 1 V = T b V = T h Electrical domain Thermal domain PAGE 94 94 mirror deflection is found to be negligible. Therefore, it is desirable to have a thermal mode l that is accurate in the 0 265 Hz frequency range. Inequality 4 28 b may be used to evaluate the length of a segment of a bimorph, l b s that can be represented by a model of the form shown in Figure 4 9 D i.e., ( 4 46 ) Let us partition the bimorph into 8 segments, each 21.6 Also, since the thermal resistance of the isolation between the bimorph and mirror plate is an order of magnitude larger than the cond uctive resistance of the mirror plate, the resistance cor responding 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 corre sponding to nodes 1 8. This can be easily obtained from the ammeter output in Figure 4 13 i.e., ( 4 47 ) 4. 5 Static Electrothermomechanical Model The mirror deflection may be evaluated using Equation 4 1 The average bimorph temperature ( T b ) and the embedded heater temperature ( T h ) 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 t he electrothermal model in Figure 4 11 to obtain the schematic shown in Figure 4 14 Figure 4 14 provides a circuit PAGE 95 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 s egments. (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 Co mparison with Experimental R esults 4. 6 .1 Static Electrothermomechanical Model As described in Section 4.3.3, the the rmal 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. F rom the et ch time t he Si layer thickness below (B) i I i R E 0 R iso 1 V E A I total R iso 2 V = T h Bimorph divided into 8 segments 1 2 3 4 5 6 7 8 (A) Thermal domain Electrical domain R E 0 T h c m l m PAGE 96 96 the mirror plate was estimated to be 28.6 m. From Equation 4 45 the value of the partition factor for the thermal model was found to be 0.81. Figure 4 15 A and Figure 4 15 B 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 cir cuit model is based on Equation 4 29 where as the LEM is based on Equation 4 30 Hence, the cir cuit 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 (mirro r section with 1 bimorph) Value R iso 1 R iso 2 R Ab 125.2 R Bb 1190.7 R Am 10.2 R Bm 458 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 R E 0 V E R E 0 T h I E R iso 1 R Bb R Ab Z 0 T b v = T h v = T b M Q H Electrical Domain Thermal Domain Mechanical Domain PAGE 97 97 mode l 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 SiO 2 thin film. Figure 4 15 Comparison between model and experimental results. The optical angle is (A) Optical angle vs. input power. (B) Optical angle vs. applied voltage. Figure 4 15 A and Figure 4 15 B reveal that the circuit model and LEM are in close agreement. For instance, in Figure 4 15 A the slopes of the lines correspondin g to circ uit model and LEM differ by 1.4 % only. Th erefore, 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 l b 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 Optical angle ( degrees ) 35 30 25 20 15 10 5 0 Optical angle ( degrees ) 0 0.5 1 1.5 2 2.5 Voltage (V) Experiment FE model Circuit model LEM (A) (B) PAGE 98 98 Figure 4 12 The circuit model has been used as a reference for estimating the error. As l b is increased, the accuracy of the LEM diminishes. This is because the approximations used in Equation 4 30 are val id 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 Ine quality 4 28 a is satisfied by each segment. Figure 4 16 Dependence of e r ror in LEM results on bimor ph length All other device parameters are same as the device shown in F igure 4 2 4.6 .2 Dynamic Thermal Model The rigorous dynamic transmission line model is described in Section 4. 3.5. The approximate dynamic ele ctro thermal 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 assum ed and the results are compared with the transmission line model. Figure 4 17 A and Figure 4 17 B show the magnitude and phase response respectively. Clearly, th e plots corresponding to the transmission li ne model and LEM practically coincide. 50 150 250 350 450 Bimorph length (m) % E rror in optical angle evaluated by LEM 30 25 20 15 10 5 0 PAGE 99 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 regio n of the device characteristics shown in Figure 4 15 B. The frequency response has been shown in Figure 4 18 A. The decay in response observed at frequencies below the mirror resonant frequency, i.e. 173 H z may be attributed to the thermal response time of the micromirror. Figure 4 18 B 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 respons e. 4.6 Summary and D iscussion This chapter reports a general procedure for modeling bimorph based electrothermal MEMS and demonstrates it with a mi cromirror device. Electrical, thermal and mechanical models are developed and integrated into a complete st atic electrothermomech anical 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 tempe rature 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 c ircuit 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 PAGE 100 100 dimensions are scaled up. When the bimorphs are 173.2 m long, the error in LEM was less than 1.4%. However, when the bimorph beam length was scaled up to 450 m, the error in the LEM was greater than 20%. Hence, caution must be exercised while using LEM. LEM accura cy ca n be increased by partitioning the geometry into small er segments. Rigorous condition for choosing the number of partitions is derived. Figure 4 17 Comparison between tr ansmission line model an d LEM in the 0 300 Hz range (A) Magnitude response. (B) Phase response. Figure 4 18 Freq uency response of micromirror. (A) Experimentally obtained frequency response. (B ) Comparison between norm alized 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 10 0 10 1 10 2 Frequency (Hz) Frequency (Hz) Magnitude of T b ( K ) Transmission line model LEM 10 3 10 2 10 1 10 0 10 1 10 2 0 0.5 1 1.5 Phase of T b ( radians ) Transmission line model LEM (A) (B) 1 0 2 10 0 10 2 10 1 10 0 10 2 10 1 10 0 10 1 1.1 1 0.9 0.8 0.7 0.6 0.5 Frequency (Hz) Frequency (Hz) Optical angle ( degrees ) Normalized data (A) (B) Normalized frequency response Normalized bimorph temperature rise PAGE 101 101 In this chapter the complete electrothermomechanical model for the static case has been reported Howe ver, it has been demonstrate d that the transmission line thermal model is suitable for modeling both static and dy namic response. T ransmission line thermal modeling provides a generic approach for a wide range of electrothermal problems. T he model presente d in this chapter has a single heat source. If a part of a device has an embedded distributed heat source, the theory of active transmi ssion 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 ther mal models of a micromirror, a comparison with the present work is in order The LEM models reported in [ 39 79 ] a re 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 acc uracy at the cost of computational resources. Moreover, they do not provide a direct relation between device parameters and response. The transmission line based ther mal model reported in this chapter provides the best of both worlds and offers rigor and c omputational 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. PAGE 102 102 CHAPTER 5 TRANSMISSION LINE THERMAL MODEL WITH DISTRIBUTED HEAT SOURCE 5.1 Background In Chapter 4, heat flow is modeled as electrical s ignal 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 embedde d 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 m ay 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 Sou rce 5.2.1 Governing Equations for 1D Heat Flow Let us consider the 1D heat flow depicted in Figure 4 9 A Let the power dissipated per unit length be Figure 5 1 A and Figure 5 1 B show the equivalent ci rcuit model of an x 0 and the corresponding ac model respectively. I n Figure 5 1 A, t he ambient temperature is denoted by T a and is represented b y a dc source. Heat dissipation has been incorporated by adding a current source For the circuit in Figure 5 1 A voltage and cur rent represent temperature and heat flow respectively. In the circuit in Figure 5 1 B 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 I n thermal transmission line PAGE 103 103 models, voltage change represents temperature change and current represents heat flow. Figure 5 1 Transmission line model for a geometry with unifo rmly distributed heat source. (A ) Circuit model for element of length x (B ) ac model. voltage law gives, ( 5 1 ) Kirc h N in Figure 5 1 B gives, r x r x c x 1 / ( g x ) + T a N (A) N + + (B) c x 1 / ( g x ) r x r x PAGE 104 104 ( 5 2 ) For harmonic variation, ( 5 3 ) ( 5 4 ) ( 5 5 ) Equations 5 1 and 5 2 may be re written as, ( 5 6 ) ( 5 7 ) From Equations 5 6 and 5 7 ( 5 8 ) ( 5 9 ) The propagation constant is given by, ( 5 10 ) From Equation 5 8 ( 5 11 ) From Equation s 5 1 5 2 5 3 5 4 5 5 5 10 and 5 11 ( 5 12 ) PAGE 105 105 As discussed in Chapter 4, A 1 and A 2 can be determined from boundary conditions at the ends of the 1D geometry. A 1 and A 1 / r are the magnitude of voltage and c urrent waves respectively, propa gating in the +ive x direction. A 2 and A 2 / r are the magnitude of voltage and current waves respectively, propagating in the ive x direction. The term in Equation 5 11 accounts for pow er dissipated in the geometry Let the length of the geometry be l ( 5 13 ) The avera ge temperature rise along the 1D geometry is given by, ( 5 14 ) Equation 5 11 can be used to determine the temperature rise at any point of a 1D heat flow r egion. Equation 5 14 may be used to evaluate the rise in average actuator temperature above the ambient temperature, T a 5.2.2 Application of Transmission lin e Model to Electrothermal Micromirrors The schematic shown in Figure 5 2 could represent a 1D micromirror in which Z 1 and Z 2 correspond to thermal impedances at the mirror and substrate ends respectively. Voltage and current va lues 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 ( 5 15 ) ( 5 16 ) PAGE 106 106 From Equations 5 15 and 5 16 ( 5 17 ) ( 5 18 ) Where, ( 5 19 ) ( 5 20 ) ( 5 21 ) ( 5 22 ) Figure 5 2 Thermal impedances Z 1 and Z 2 at either ends of the tra nsmission line model of a thermal bimorph The bimorph length is l b 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 Z 1 Z 2 Transmission line model of bimorph M 1 M 2 PAGE 107 107 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 sinus o i dally 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 Cha pter 3. The only difference is that the simulated design does not have any SiO 2 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 g ood 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 10 0 10 2 10 4 10 6 10 8 Frequen cy (Hz) FE model Transmission line model Temperature rise (K) (A) 0 0.5 1.0 1.5 2.0 10 4 10 2 10 0 10 2 10 4 10 6 10 8 FE model Transmission line model Fre quency (Hz) Phase (radians) (B) PAGE 108 108 heat flow is three dimension al. Beyond 10 4 Hz, there is a significant discrepancy between transmission line model and FE simulations in the phase plots shown in Figure 5 3 B and Figure 5 4 B 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 10 4 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 coefficient s on the actuators and mirro r plate are assumed to be 200 W m 2 K 1 and 50 W m 2 K 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 SiO 2 thermal isolation between the actuators and the mirror plate. (A) Ma gnitude plot. (B) Phase plot. 10 4 10 2 10 0 10 2 10 4 10 6 10 8 100 80 60 40 20 0 Frequency (Hz) FE model Transmission line model Temperature rise ( K) 10 4 10 2 10 0 10 2 10 4 10 6 10 8 Frequency (Hz) FE model Transmission line model 0 0.5 1 1.5 2 Phase (radians) (A) (B) PAGE 109 109 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 whi ch 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 Dist ributed Temperature Dependent Resistive Heater in One dimensional Heat Flow Region Let us consider the on e dimensional heat flow region shown in Figure 4 9 A. Let a distributed temperature dependent resistor be embedded along this geometry. Let the power per unit length be, ( 5 23 ) w here A h is the he ater cross section area; is the time dependent electric current density; 0 and are the resistivity and temperature coefficient of resistance of the embedded heater respectively, at temperature T a ; and T is the rise in temp erature above the ambient temperature, T a The distance along the geometry and time are denoted by x and respectively. From ( 5 23 ), ( 5 24 ) where, ( 5 25 ) The first term in Equation 5 24 is represen ted by a current source in the equivalent circuit model shown in Figure 5 5 The second term in Equation 5 24 can be PAGE 110 110 represented by a time dependent resistor as shown in Figure 5 5 A. The ac equivalent circuit is shown in Figure 5 5 B. For the circuit in Figure 5 5 A, voltage and current repr esent temperature an d heat flow, respectively. In the circuit in Figure 5 5 B, voltage represents temperature ri se above the ambient Figure 5 5 Equivalent circuit of an element of length x of a one dimensional heat flow region. The TCR of the distributed embedded resistor is (A) Circuit model in which the ambient temperature, T a is modeled as a dc source. (B) ac model. r x r x 1 / ( g x ) c x + T a N r x r x c x 1 / ( g x ) N + + (A ) (B) PAGE 111 111 The time dependent resistor in Figure 5 5 ob scures an analytical solution for voltage and current. Such time varying circuit elements lead to generation of harmonics in a circuit [ 92 ] T herefore 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 numbe r 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 current source. 5.4 Summary and Discussion T his 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 fo rm 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 nume rical 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. PAGE 112 112 CHAPTER 6 MECHANICAL MODEL OF ELECTROTHERMAL MICROMIRRORS 6 .1 Background The mechanical model of the micromirror is an essential compo nent of the complet e 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 ca se 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 o utput of the mechanical model consist s of three quantities the displacement of the mirror plate perpendicular to the substrate ( z ) and the mirror rotation angle along two orthogonal ax e s ( x and y ) as schem atically represented in Figure 6 1 Due to their three degrees of freedom, such mirrors are also called tip tilt piston (TTP) micromirrors. In ge neral, the mirror plate of a micromirror device has some lateral shift in the x and y direction s However, the lateral shift is negligible in case of the designs reported in [ 34 36 ] For in stance, the micromirror design based on lateral shift free ( LSF ) actuators has a lateral shift of 10 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 mic romirror. Static bimorph mechanics has been b riefly discussed in Section 1.2.1 The next section will deal w ith bimorph actuation. Section 6 .3 will discuss the optimization of ISC multimorph actuators. Section 6.4 outlines the development of mechanical mod el s base d on Newtonian method and energy method. PAGE 113 113 Figure 6 1 Schematic showing the three deg rees of freedom of a 3 D micromirror (A ) displacement along z axis (B ) rotation about x axis (C ) rotati on about y axis 6.2 Mechanics of Bimorph A ctuators Timoshenko Analysis of bi metal thermostats is a classic text on the mechanics of singly and doubly clamped bimetal strip s [ 10 ] S ince 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 multimorph s is useful to MEMS engineers for two reasons. Firstly, most fabrication process es 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 3 B consists of 4 distinct thin film layers. Secondly, the general theory of multimor phs is applicable to a wide range of devices such as electrothermal, piezoelectric and SMA (Shape Memory Alloy) ac tuated 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 x y z PAGE 114 114 schema tic of a multimorph. A moment M m a transverse force F 1 m and an axial force F 2 m 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 z i T he CTE Young area and area moment of inertia of the i th layer are i E i A i and I i respectively. Figure 6 2 Schematic of a multimorph (adapted from [ 11 ] ) Let the beam width be comparable to its thickness, so that th e plane stress approximation is valid. If the multimorph temperature changes by T the curvature of the multimorph C m is given by [ 11 ] ( 6 1 ) In Equation 6 1 R m denotes the radius of curvature of the multimorph. z i i th layer m F 1 m F 2 m M m z x PAGE 115 115 6 .3 Optimization of the ISC Mult imorph A ctuators 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 multimorph s The thin film structures of the two multim orphs have been sho wn in Figure 6 3 Figure 6 3 Thin film structure of (A) n on inverted and (B ) inverted m ultimorphs. The numbers on the side represent the thickness of the thin films in m. Upo n 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 SiO 2 Cr Pt (A) (B) PAGE 116 116 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 l inverted and l non inverted respectively. The optimal ra tio of l inverted and l non inverted r optimal that achieves tilt free motion has been determined by FE simula tions. As shown in Figure 6 4 A when the ratio of l inverted and l non inverted is less than r optimal the inverted multimorph is unable to compensate the tilt completely. On the other hand, Figure 6 4 B sho ws that when the ratio of l inverted and l non inverted is greater than r optimal the inverted multimorph overcompensates the tilt. The optimal configuration corresponding to zero tilt at the actuator tip has been shown in Figure 6 4 C Assuming static actuation and neglecting the effect of gravity on the mirror, the terms M m and F 2 m 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 T Fr om FE simulations ( 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 SiO 2 ther mal isolation at either ends, the uniform temperature approximation may be used. Let the curvature of the inverted and non inverted beams be R m inverted and R m non inverted respectively. The bending of the inverte d and non inverted beams cancel each othe r if, PAGE 117 117 ( 6 3 ) The effect of residual stresses and non uniform temperatu re distribution requires further investigation. Optimizatio n of the ISC actuator design has resulted in more than ten fold impr ovement 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 bee n shown in Figure 6 3 (A ) The tilt is undercompensated. (B ) The tilt is overco mpensated. (C ) When the ratio of lengths of the inverted and non inverted multimorphs is optimal, the actuator tip executes tilt free motion 6 .4 Mecha nical Model of M icromirror In this section, t he 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 6 B In ge neral the temperature may vary along a bimorph actuator and must be taken into account However, the procedure for building the model will be essential ly the same as that outlined in this section. The next sub section will present Non inverted multimorph Inverted multimo rph (A) Actuator tip is parallel to substrate (C) (B) PAGE 118 118 mirror modeling by Newtonian method. The energy method will be introduced in Section 6 .4 .2. 6 .4 .1 Newtonian M ethod Figure 6 5 shows an idealized schema tic of a 1D mirror. The output of the mechanical model must determine the angle Figure 6 5 Schematic of mechanical model for 1D mirror. The mi rror plate has been approximated as a 1D geometry of mass m and length l m The bimorph of length l b is assumed to act like an ideal spring. The typical thickness of a multimorph actuator is ~2 T he thi ckness of the mirror plate is much larger than that of the actuators, typically 20 40 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 mirro r plate have been denoted by l b and l m respectively. The radius of curvature of the actuator is R m The point ( x cm y cm ) denotes t he center of mass of the mirror plate. l m l b R m ( x cm ,y cm ) x y PAGE 119 119 Figure 6 6 shows the free body diagrams of the actuator and the mi rror plate. The actuator mechanics is described by Equation 6 1 The moti on of the mirror plate may be determined by rigid body mechanics [ 97 ] Figure 6 6 Free body diagram of (A) a ctuator and ( B ) m irror 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 t o the orientation dependent gravitational force. However, the device response will not be affected significantly. 6 .4 .2 Energy M ethod The e nergy method described in [ 98 ] may be applied for building the mechanical model. Si nce 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. M m F 1 m F 2 m M m F 2 m (A) (B) ( x cm ,y cm ) F 1 m PAGE 120 1 20 6.4.2.1 Evaluation of kinetic e nergy Let the mass of the mirror plate be m The moment of inertia of mirror plate about its center of mass is given by, ( 6 4 ) The c oor dinates of the center of mass of the mirror plate are given by, ( 6 5 ) The m agnitud e of the velocity of the center of mass is given by, ( 6 6 ) Whe n the mirror is in motion, its kinetic energy is given by [ 97 ] ( 6 7 ) From Equations 6 4 through 6 7 ( 6 8 ) where, ( 6 9 ) 6.4.2.2 Evaluation of p otential e nergy The potential stored in the actuator may be evaluated by the stress distribution given in [ 10 ] In this section, it will be assume d that the actuator consists of 2 layers of thin films. The radius of curvature is R m 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 t 1 and t 2 respectively. PAGE 121 121 Figure 6 7 Stress distribution in a bimorph The stresses max, 1 and min, 1 are given by [ 10 ] ( 6 10 ) ( 6 11 ) Similar expressions can be obtained for the stresses in the bottom layer, i.e., max ,2 and min ,2 ( 6 12 ) ( 6 13 ) The strain distribution for the top layer is given by, ( 6 1 4 ) where, ( 6 15 ) Similarly the strain distribution in layer 2 may also be evaluated t 1 t 2  max ,1   min 1  x y PAGE 122 122 The elastic energy in the top layer of the bimorph is given by, ( 6 16 ) A similar expression may be found for the elastic energy stored in the bottom layer. Let, ( 6 17 ) The total potential energy is given by the sum of elastic energy stored in the top and bottom layers, ( 6 18 ) where, ( 6 19 ) The Lagrangian is given by [ 98 ] ( 6 20 ) If the mi rror oscillates with a small amplitude about bias ( 6 21 ) where, F ( ) is given by Equation 6 9 ( 6 22 ) ( 6 23 ) Let F Thermal and F damping denote the thermal and damping force s Fo r forced oscillations the equation governing mirror motion is given by PAGE 123 123 ( 6 24 ) For small oscillations about a bias point, ( 6 25 ) The constant C Thermal may be evaluated from the ste ady state device characteristic 6 .5 Summary The mechanics of multimorph actuators is well documente d 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 micromi rrors is a key component of the complete ET M 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 mec hanical 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. PAGE 124 124 CHAPTER 7 COMPREHENSIVE ELECTROTHERMOMECHANICAL MODEL OF MICROMIRRORS 7 .1 Background Several electrothermal bimorph based designs have been reported in literature. However, co mplete 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) model s of thermal bimorph based 1D micromirrors. The schematic of an elec trothermal 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 micro mirror. 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 the rmal 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 advant ages. 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 PAGE 125 125 et al. suggest that the nonlinear portion of the scan range of a micromirror may be truncated, but this greatly reduc es the scan range [ 99 ] Another solution is to correct for the nonlinearity in the output by signal processing but this make s 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 culmina te 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 b e 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 a nd 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 feedb ack [ 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 PAGE 126 126 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 a nd angular scanning have been reported Empirically derived custom waveforms were used to compensat e for device nonlinearity and achieve the desired scanning profile. In this section, a formal procedure is developed for deriving custom actuation waveforms for achievi ng desired performance. Next, it is demonstrate d 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. T he topology of the electrothermal mirror used for the experiments in this section is th e same as that depicted in F igure 4 2 For the device under test, the static mirror deflection angle can be predicted by setting = 0.17 /K in Equation 4 1 Section 7.2.1 provides theoretical background. The procedure for custom waveform generat ion is outlined in Section 7.2.2 Section 7.2.3 presents experimental results on continuous wave and PWM actuation. 7 .2 .1 The oretical Background Figure 7 1 shows a schematic of a 1D electrothermal micromirror model The applied voltage V E results in power dissipation p in the embedded heater with resistance, R E The thermal model predicts the heat er temperature, T h and average bimorph temperature, T b The mechanical model predicts the mirror rotation PAGE 127 127 Figure 7 1 Schematic of a complete model of an electrothermally actuated micromi rror. = time, V E ( ) = applied voltage, p ( ) = power dissipated by Joule heating, T b ( ) = bimorph actuator temperature, = mirror rotation angle, T h ( ) = temperature of embedded resistive heater and R E ( T h ) = resistance of embedded heater. Let P ( s ) T b ( s ) T h ( s ) and ( s ) denote the power, average bimorph temperature, heater temperature and mirror rotation in the frequency domain. From Chapters 3 5, it can be infer red 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 res pectively. Therefore, the mechanical model may also be repre sented by a simple transfer function. Let us define the transfer functions G 1 ( s ), G 2 ( s ) and G 3 ( s ) as, ( 7 1 ) ( 7 2 ) ( 7 3 ) From Equations 7 1 and 7 3 Thermal Model =Temperature dependent resistance of Pt heater Mechanical Model PAGE 128 128 ( 7 4 ) Therefore, for a desired scan angle profile, the input powe r may be determined by using Equation 7 4 Typically, ( ) for a scanning mirror is a periodic function which can be easily expanded i n 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. E quation 7 2 may be used t o evaluate the heater temperature T h ( ) for a certain power input p ( ). The heater resistance R E is given by Equation 4 3 For the device under test, the TCR, = 0.0025 K 1 and the reference temperature, T 0 is chosen to be 298 K. Finally, the desired voltage signal is given by ( 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 thi s sub section 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 c haracterization In order to determine the ang ular range and establish the safe voltage limit, t he static device characteristic was experimentally obtained by using a dc source. The voltage, current an d scan angle were monitored. Figure 7 2 A and Figure 7 2 B show the optical angle vs. voltage and optical angle vs. input power respectively. From Equation PAGE 129 129 7 4 G 1 (0) G 3 (0) = (0)/ P (0) Therefore, the linear ang le power relationship in Figure 7 2 B is in agreement with the use of linear transfer functions G 1 ( s ) and G 3 ( s ) Figure 7 2 Static characteristic (A ) Optical angle vs. app lied dc voltage (B ) Optical angle vs. input power 7.2.2.2 Dynamic c haracterization 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 2 A A 150 mV amplitude si ne wave super im posed on the dc bias was used to obtain the mirror response. This approximately corresponds to a sinusoidal input power wa ve of 8.545 mW amplitude. Figure 7 3 show s 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. G 1 ( s ) G 3 ( s ). The fitted model shown in Figure 7 3 corresponds to the transfer function given by, ( 7 6 ) where, K 1 = 7.56 / W, r z 1 = 31.4 rad/s, r p 1 = 11.3 rad/s, n = 1451.4 rad/s and = 0.0084. 25 20 15 10 5 0 0 2 4 6 8 10 0 100 200 300 400 Optical angle (degrees) 25 20 15 10 5 0 Voltage (V) Input power (mW) (A ) (B) Optical angle (degrees) PAGE 130 130 Figure 7 3 Mic romirror frequency response and fitted model The zero, r z 1 and pole, r p 1 correspond to the thermal response of the device. From the reduced order model of an electrothe rmal micromirror [ 39 ] it is expected that there will be two poles and a zero. In fact, harmonic FE thermal simulation s 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 i n Figure 7 3 Mechanical FE simulations show that the mirror has only one resonant mode in the operation frequency range. The constants and n are the damping ratio and the natural resonance frequency of the mechanical system. 7.2.2.3 Determination of G 2 ( 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 T h ( s ) along with a fitted model. Experimental results Fitted model 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) Optical angle scan range (degrees) 10 2 10 1 1 0 0 10 1 10 2 PAGE 131 131 Figure 7 4 Temperature of embedded heater. (A) Magnitude and (B ) Phase of T h for an input power of 0.01 W. From the fitted model, ( 7 7 ) where, K 2 = 2.93 10 6 K / W, r z 2 = 4.83 rad/s, r p 2 = 4.65 rad/s and r p 3 = 13823 rad/s. 2.3 2.2 2.1 2 1.9 1.8 FE results Fitted model 10 2 10 1 10 0 10 1 10 2 10 3 10 4 Frequency (Hz) Magnitude of T h ( K ) FE results Fitted model 10 2 10 1 10 0 10 1 10 2 10 3 10 4 F requency (Hz) 0 0.05 0.1 0.15 0.2 0.25 Phase of T h ( radians ) (A ) (B) PAGE 132 132 7.2.2.4 Fourier s eries e xpansion of d esired o utput Fi gure 7 5 A shows an ideal periodic ramp output. The mirror plate has some inertia a nd therefore requires a finite time interval to switch directions. Thus in practice the mirror output will b e rounded at the corners. Fi gure 7 5 B shows the Fourier series approximation of the ramp waveform. All frequency componen ts 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. Fi gure 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 v oltage i nput The frequency domain representation of the ramp o utput can be used along wit h Equation 7 4 to evaluate the input power waveform. Thereafter, Equations 7 2 4 3 and 7 5 may be used to obtain the vo ltage input. Figure 7 6 A shows the input power. Figure 7 6 B shows the voltage input corresponding to constant angular velocity actuation. In the next subsection, the principle of PWM actuation will be pr esented. 0 0.02 0.04 0.06 0.08 1 0 0.02 0.04 0.0 6 0.08 1 10 9 8 7 6 5 4 10 9 8 7 6 5 4 (A ) (B) Optical angle (degrees) Optical angle (degrees) Time (s) Time (s) PAGE 133 133 7.2.2.6 Pulse width modulation Let us consider the analog voltage actuation signal shown in Figure 7 7 A Let the PWM waveform shown in Figure 7 7 B be equivalent to this analog signal in the inte rval Figure 7 6 One period of the evaluated input waveform. (A ) P ower and (B ) V oltage waveforms for achieving linear scan Figure 7 7 PWM representation of continuous waveform. (A ) Continuous waveform (B ) Equivalent PWM In the interval the PWM signal delivers the same amount of energy to the micromirror as the analog signal Moreover, since the two waveforms are equivalent, they re sult in identical temperature distributi on in the embedded Pt heater. It is assume d v avg v pulse on off 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 power (mW) (degrees) Voltage (V) (A ) (B) PAGE 134 134 is sufficiently small, so that the analog voltage and the embedded heater Equating the energy delivered to the device by the analog waveform and its equivalent PWM representation, ( 7 8 ) w here, v analog is the analog voltage, v avg is the average voltage of the analog waveform in v pulse is the voltage amplitude of the pulse waveform, T h is the embedded heater temperature, R E ( T h ) is the temperature dependent resistance of the embedded heater. Hence, ( 7 9 ) Equation 7 9 may be used to evaluate the equivalent PWM waveform for any continuous signal. on 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 const ant linear scan profile. Figure 7 8 A sh ows 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 8 B shows the constant linear velocity scan profile of th e light spot incident on the PSM screen. The linear scan range is around 90% [ 107 ] The profile shown in Figure 7 8 B corresponds to an optical scan range of 3. PAGE 135 135 Figure 7 8 Constant linear velocity scan. (A ) 10 Hz custom actuation wave form (B ) Constant linear velocity scan profile obtain ed using a PSM 7.2.3.2 Constant angular velocity scan Figure 7 9 A and Figure 7 9 B 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 wave form was used. Further improvement in linearity may be obtained by using an unsymmetrical waveform similar to the one shown in Figure 7 6 B In order to obtain the equivalent PWM representation of the waveform shown in Figure 7 9 A, 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 v pulse = 7 V. Figure 7 10 A shows a section of the PWM waveform. Figure 7 10 B 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) Actuation voltage (V) 6 4 2 0 2 4 6 8 PSM output (V) (A ) (B) PAGE 136 136 each subint erval is 0.125 ms long, i.e. on + off = 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. F or 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 i s 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 t o eliminate frequencies that excite mirror resonance. Figure 7 9 Constant angular velocity scan. (A ) 10Hz Actuation voltage for obtaining constant a ngular 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 sc an 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 Voltage (V) Optical angle (degrees) 6 4 2 0 12 8 4 Time (s) Time (s) (A) (B) PAGE 137 137 Figure 7 10 Constant angular velocity scan by PWM actuation. (A ) A section of the PWM wav eform. ( B ) Large c onstant 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 volta ge waveform as input and predicts the device response as output. The t ransmission 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 pres enting 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 ( ) of period p Let the Fourier series representation of f p ( ) be, ( 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 angle (degrees) Voltage (V) 7 0 12 8 4 PAGE 138 138 where, a 0 a n and b n are the Fourier expansion coefficients. Let the vector f pv contain the values of f p ( ) at = {0, p / 2 m 2 p / 2 m ... (2 m 1) p / 2 m }, where m is an integer. Applying the fft function in MATLAB to f pv gives the vector, ( 7 11 ) The vecto r on the right hand side of Equation 7 11 can be used to evaluate the Fourier coefficients of f p ( ) as follows [ 110 ] ( 7 12 ) ( 7 13 ) ( 7 14 ) The above results are useful for building the Simulink model discussed in the next sub section. 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 rier line approach discussed in Chapter 4. Since the the rmal model can be represented using resistors and capacitors, it is a linear PAGE 139 139 Figure 7 11 Dynamic ETM mirror model implemented in Simulink. PAGE 140 140 time invariant (LTI) model. Therefore, it ha ndles each component of the input power spectrum separately. The final temperature is obtained by linear superposition. The T b and embedded heater temperature, T h at 50 0 equispaced points in one time period. respectively implemented as a second order LTI system. It compu tes mirror scan angle for each component of the bimorph temperature waveform and uses linear superposition to which is then used for calculating the power input. 7.3.3 Experim ental 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 12 A 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 sing le 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 12 B shows good agreement between experimental data and the scan angle p redicted by the Simulink model. PAGE 141 141 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 base d 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 Voltage (V) Simulink model Experimental data 0 2 4 6 8 10 12 Time (ms) Optical angle (degrees) 30 25 20 15 10 5 0 5 (A) (B) PAGE 142 142 and angular velocity scan ning 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 temper ature 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 th e modeling and control of 2D and 3D micromirrors, and mirror arrays. PAGE 143 143 CHAPTER 8 ANALYSIS AND FABRICATION OF CURVED MULTIMORPH TRANSDUCERS THAT UNDERGO BENDING AND TWISTING 8 .1 Background A multimorph co nsists of two or more layers of different materials Difference in strains produced in the constituent layers causes a multimorph to d eform, thereby producing transduction. A multimorph with two layers is a bimorph. Figure 8 1 A shows the schematic of a straight multimorph that bends upon deformation. Various devices based on straight multimorphs have been repor ted 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 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 p iezoelectric 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 1 B shows the schematic of a curved multimorph. It must be emphasized that in this dissertation, curved multimorph refers to a multimorph with non zero curvature in the plane of the substrate. PAGE 144 144 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 i s R c The twist angle and out of plane displacement are denoted by and U respectively. There is limited literature on curved mult imorphs. 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 ] A n elliptical micromirror actuated by a curved thermal multimorph that is concentric to the mirror plate [ 37 ] is discussed in Chapter 9 Mirror desig ns 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 previ ously 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 R c Undeformed state (l) U(l) Clamped end ( s = 0 ) Free end (A) (B) PAGE 145 145 This chapter presents the small deformation analysis of curved mul timorphs [ 118 ] T his analysis will greatly expand the design space for MEMS engineers. Closed form expressions for the out of plane displacement, U and actuator twist angle, shown in Figure 8 1 B 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 deformati on 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 simu lations are provided in Section 8. 3. Re sults 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 adjacen t layers. The subsequent subsections will establish the theory governing the bending and twisting of curved multi morphs. First, curved multimorphs in which the induced axial strain in the constituent layers due thermal, piezoelectric, SMA, electroactive ef fect 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 v ar iation of induced axial strain along multimorph length will be discussed in Section 8. 2.4. PAGE 146 146 8. 2.1 Deformation of Curved Beams Let us consider the bending of a curved beam. Figure 8 2 A shows a section of a curved beam of in plane ra dius of curvature R c subjected to a bending moment M i The thickness, width, area of cross section and s modulus are t i w i A i and E i respectively. Figure 8 2 B shows a cross section of the curved beam. The cross sectio nal moment of inertia about the centroidal axis is I i = 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 ( s ), respectively. Figure 8 2 Curved beam. (A) Portion of a curved beam (B ) Beam cross section The origin of the local u i r i coordinate system coincides with the center of the cross section. The deformation of a curved beam is governed by the following equations [ 119 ] ( 8 1 ) ( 8 2 ) If the beam is clamped at s = 0, the boundary condi tions are U (0) = 0, (0) = 0 and (0) = 0. Solving Equations 8 1 and 8 2 and imposing bound ary conditions, ( 8 3 ) (0,0) u i = t i / 2 r i = w i / 2 r ( s ) M i M i R c (A) (B) s U ( s ) r i = w i / 2 u i = t i / 2 u PAGE 147 147 ( 8 4 ) 8. 2.2 Strain Continuity between Adjacent Layers The strain s produced in adjacent layers must match at the interf ace between the two. Figure 8 3 A 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 F i and M i respectively. The total axial strain, T ( i ) in t he 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 F ( i ) M ( i ) and i 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 F i and M i respectively (B ) Equivalent representation of forces and moments. The strain due to axial force is [ 120 ] ( 8 5 ) F 1 F 2 F 3 F n M 1 M 2 M 3 M n M 1 M 2 M 3 M n F 1 F 1 F 1 + F 2 F 1 + F 2 F 1 + F 2 + F 3 F 1 + F 2 + F 3 F n 1 F 1 + F 2 + F 3 F n =0 (A) (B) PAGE 148 148 The axial strain due to bending moment is [ 120 ] ( 8 6 ) where, u i and r i are defined as shown in Figure 8 2 B and J i is defined by the area integral, ( 8 7 ) The dependence of M ( i ) on r i in Equation 8 6 implies that M ( i ) varies along the width of the cross section, w i If w i << R c Equation 8 6 may be approximated as, ( 8 8 ) Strain continuity between the ( i 1)th and i th layer s gives, ( 8 9 ) Equations 8 5 8 8 and 8 9 give, ( 8 10 ) Since Equation 8 1 holds for 1 i n ( 8 11 ) where, the quantity may be evaluated from Equation 8 3 ( 8 12 ) PAGE 149 149 Usin g Equation 8 11 the second te rm on the left hand side of Equation 8 10 may be written as, ( 8 13 ) where, ( 8 14 ) From Equat ions 8 10 and 8 13 ( 8 15 ) Equation 8 15 may be used to evaluate F i in terms of F i 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 F 1 ( 8 16 ) 8. 2.3 Force and Moment Balance Let us consider the force and mome nt distributions shown in Figure 8 3 A Force balance gives, ( 8 17 ) In order to formulate the moment balance equation, the force and mome nt distributions shown in Figure 8 3 A may be equivalently represen ted by the distribution depicted in Figure 8 3 B Equating the total moment produced by the axial forces to the total bending moment gives, PAGE 150 150 ( 8 18 ) 8. 2.4 Curved Multimorph Deformation Equations 8 11 8 16 8 17 and 8 18 provide a set of 2 n equati ons in F i and M i 1 i n T hese 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 ( 8 19 ) C i and D i are g iven by, ( 8 20 ) ( 8 21 ) Equation 8 11 may b e rewritten as, ( 8 22 ) Substituting force and mome nt expressions from Equations 8 19 and 8 22 into Equation 8 18 and solving for ( 8 23 ) From Equations 8 12 and 8 23 PAGE 151 151 ( 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, i are constant along the multimorph length. At large R c the deflection predicted by Equation 8 24 must approach the deflection of a straight multimorph. L et us consider a straight multimorph of length l s that is initially undeformed. Let the out of plane radius of curvature of the straight multimorph in the deformed state be R m If small deformation is assumed, the out of plane deflection of the tip of the straight multimorph is, ( 8 25 ) The expression for R m was obtained from [ 11 ] It has been ve rified using MATHEMATICA [ 121 ] that at large R c the deflection s pred icted by Eq uations 8 24 and 8 25 are identical, ( 8 26 ) The effect of variation in i along the multimorph length will be addressed in the nex t subsection. 8. 2.5 Variation of Induced Strain along Multimorph Length In many practical situations, i may vary along the multimorph length. A typical example is a thermal multimorph whose temperature varies along its length. The PAGE 152 152 quantities F i M i i F ( i ) M ( i ) and D i 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, ( 8 27 ) If a closed form solution for Equations 8 27 8 2 does not exist, ( 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 ( s ) = Q 1 cos( Q 2 s ), Equations 8 27 and 8 2 may be solved and boundary conditions at the clamped end, s = 0, may be imposed to give, ( 8 28 ) Similarly, for ( s ) = Q 1 sin( Q 2 s ) Equations 8 27 and 8 2 may be solved to give, ( 8 29 ) Once the solution for Equations 8 27 8 2 are obtained, the rest of the analysis proceeds similar to that described i n 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 calculatio ns, material properties were obtained [ 58 ] All FE simulations are done using the COMSOL simulation software [ 58 ] Section 8. 3.3 deals wit h large deformation of curved multimorphs. PAGE 153 153 8. 3.1 Analysis vs. FE Simulations Let us consider a 50 6 multimorph with four layers. Let the constituent layers from bottom to top be SiO 2 W, SiO 2 and Al respectively. Let the thickness es of the layers from bottom to top be 0.15 and 0.7 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 R c in the range 25 1000 Figures 8 4A through 8 4C show the simulated deformation for in plane radius of curvature eq ual to 25 respectively. Figure 8 4 FE simulation results for curved thermal multimorphs with length 50 width 6 K. The radius of curvature R c in the undeformed state are (A) 25 (C ) 300 The layers from top to bottom are Al, SiO 2 W, SiO 2 with thicknesses 0.7 0.15 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 z displacement 0 0.4 0.8 1.2 1.6 1.96 z displacement (A) (B) (C) PAGE 154 154 Figure 8 5 compares FE simulatio n and analyt ical results. Figure 8 5 A shows that the magnitude of out of plane deflection increases with R c At large R c the out of plane deflection approaches the deflection of a straight multimorph. Figure 8 5 B sh ows that the beam twist reduces monotonically with increasing R c 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 a ss umptions involved in small deformation analysis. Figure 8 5 Deformation of curved multimorph. (A) Tip deflection. (B ) Beam twist of a 50 clamped multimorph with the foll owing layers 2 0.2 5 2 0.7 for a temperature change of 100 K. 8. 3.2 Experimental Results Figure 8 6 A shows a test structure that was fabricated on an SOI wafer by a process similar to that de scribed in [ 37 ] wide curved multimorph is a ttached to a 245 plate. Figure 8 6 B shows a cross section of the curved multimorph. The multimorph thin SiO 2 2 a deposited Al, respectively. Figure 8 6 C shows a simplified cross section that will be used for analysis Analytical Finite element simulations In plane radius of curvature, R c 0 200 400 600 800 1000 1.4 1.6 1.8 2.0 Beam twist (radians) 0 200 400 600 800 1000 In plane radius of curvature, R c Analytical Finite element simulations (A ) (B) 0.06 0.04 0.02 0 tip PAGE 155 155 and simulations. The i dealized cross section in Figure 8 6 C simplifies calculations and modeling. The mirror thick single 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 incorpor ated in the same chip to monitor the temperature. The resistance and temperature coefficient of resistance of the resistor are nd 0.0012/K, respectively at 298 K. Figure 8 6 Curved multimor ph 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 6 A 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 mirr or 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 Multimorph Mirror plate Substrate 0.14 0. 58 0.6 0.29 7 10 Sputtered Al Sputtered W PECVD SiO 2 (A) (B) 0.14 0. 58 0.6 0.29 10 (C) 7 PAGE 156 156 experimentall y 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 exp eriment and analytical results is observed. Discre pancy in the experimental and simulated values may be attributed to the difference in material properties of thin films wit h those used in calculations and errors in the measur ement of thin film thicknesses Large deformation of curved multimorphs will be discussed next. Figure 8 7 Mirror plate tilt vs. chip temperature fo r test structure shown in Figure 8 6 A 8. 3.3 Large Def ormation of Curved Multimorph s 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 thi ckness of the multimorph is significantly low or a combinati on of th e aforementioned factors. Figure 8 8 A shows the schematic of a curved multimorph test structure layout. The semicircular Al SiO 2 multimorph i s anchored to a circular mirror plate. The thicknesses of the Al and SiO 2 layers are 0.5 8 m and 0.43 respectively. The mirror plate consists of a 20 thick single crystal silicon coated with 0.58 300 320 340 360 380 400 420 Experiment Analytical Finite Element Simulation 20 15 10 5 0 5 1 0 Mirror tilt ( ) Chip temperature (K) PAGE 157 157 Figure 8 8 Large deformation of a curved multimorph. (A ) Schematic of a curv ed multimorph test structure layout. The 546 plate is anchored to a semicircular multimorph. The top and bottom layers of the multimorph are Al and SiO 2 respectively with thicknesses 0.58 respectively. The mirror plate con sists of a 20 ayer coated with a 0.58 ) Optical microscope image of a released structure. (C ) SE M of a released test structure. (D ) Simulated deformation for a uniform temperature of 250 K. The color bar rep resents 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 6 A Device release inv olves Si etch using DRIE [ 37 ] The optical microscope image and SEM of the released structure are shown in Figure 8 546 Clamped end of multimorph Undeformed semicircular multimorph Mirror plate 500 (A) (C) Clamped end of multimorph Deformed semicircular multimorph Mirror plate Deformed semi circular multimorph (B) Undeformed state 0 (D) PAGE 158 158 6 B and Figure 8 6 C 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 8 D provides a qualitative understanding of the large deformations observed in Figure 8 8 B and Figure 8 8 C In the FE s imulation, t he 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 evid ent from the SEM shown in Figure 8 8 C that the mirror plate is constrained by the surrounding substrate. This le ads 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 inve stigated. 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 def lection 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 t wist angle. The analysis is validated against FE simulatio ns and experimental results. At large deformations, significant in plane displacement is produced along with PAGE 159 159 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 transducer s 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 chapter s present micro mirror s actuated by curved multimorphs. PAGE 160 160 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 el ectrothermal micromirror design actuated by a curved multimorph Unlike straight multimorphs that bend upon deformation, c urved 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 ce nter shift in a st raight 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 multi morph based 1D micromirror [ 20 39 ] design at two different positions during a scan cycle. The mirror plate center suffers significant shift during a ctuation. 1D mirrors actuated by straight multimorphs h ave been discussed in Chapters 3 4. Mirror plate Straight multi morph Shift in mirror plate center PAGE 161 161 Figure 9 2 Semicircular actuator based mirror design. (A) S chematic of a 50 mirror plate with an Al SiO 2 semicircular actuator ( R c = 35 w = 10 ) 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 In this chapter a 1 mm wide mirr or actuated by a semicircula r 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 ch ip 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 (A ) (B ) Clamped end Semicircular actuator Out of pl ane displacement PAGE 162 162 T he performance and robustness of the device exceeds previously reported designs actuated by straight multimorphs. Furthermore, since most laser spots are circular o r elliptical, a circular mirror plate a ctuated by a concentric curved actuator utilizes chip area more e fficiently than a square mirror plate actuated by straight multimorphs. This c hapter is organized as follows. Section 9. 2 describes the device design fabrication process and experimental res ults. 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 Actuate d b y Curved Multimorph 9. 2.1 Device Description Figu re 9 3 A show s an SEM of a fabricated circular micromirror with a semicircular multimorph ac t ua tor. 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 Figu re 9 3 B More finite element ( FE ) modeling will be discussed in Section 9. 3. Figu re 9 3 C shows a cross sect ion of the 20 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 crystal sil i con (SCS) layer coate d with a 0.1 0.2 PECVD SiO 2 layer and a 0.6 thic k 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 18 0.5 cm. Equations given in [ 2 11 ] can be used to show that the radius of curvature of the mirror PAGE 163 163 plate varies appr oximately 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. Figu re 9 3 A 1 mm wide circular mirror. (A ) SEM of a circular micromirror actuated by a semicircular electrotherm al multimorph. (B ) Simulated initial tilt of micromirror upon release. The color bar shows the out of plane displaceme nt from the unreleased position. A uniform actuator temperature change of 67 K is used to simulate the tilt from the unreleased flat posi tion. (C ) Schematic of a cross section ( Figu re 9 3 A ) of the multimorph actuator (not to scale). 9. 2.2 Fabrication Process 9. 2.2.1 Material s election For large deflection, material pairs with widely different CTEs are commonly used for thermal multim orphs. Several designs utilize m etal SiO 2 [ 19 ] or m etal polymer [ 48 ] pairs for generating large differential strain. However, the low thermal diffusivities of SiO 2 [ 48 58 ] and MEMS polymers [ 48 58 ] resu lt in slow thermal response of m etal Mirror plate Curved multimorph Mirror position before release 123 0.14 0. 6 0.6 0.29 15.6 20 Sputtered Al Sputtered W PECVD SiO 2 (A) (B) (C) y x z A A 200 m 200 PAGE 164 164 SiO 2 and m etal polymer based designs. Additionall y, designs that utilize SiO 2 beams for thermal isolation are susceptible to impact failure due to the brittle nature of SiO 2 [ 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. Avoid ing materials l ike SiO 2 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 t he wide availability of W deposition facilities and recipes, Al W was chosen to form the active layers of th e multimorph. As shown in Figu re 9 3 C a thin layer of SiO 2 encapsulates W. The SiO 2 acts as electrical isolation between Al and W and protects W from fluoride based dry etch recipes. 9. 2.2.2 Process f low The micromirror fabrication process is illustrated in Figure 9 4 SOI wafers are used t o ensure the flatness of mirror plates with single crystal silicon microstructures. The handle layer and device layer thicknesses of the SOI wafer are 500 respectively. A 0.1 0.2 PECVD oxide deposited in Figure 9 4 A acts as electrical isolation between W and silicon. Both Al and W are fabricated by sputtering and lift off as shown in Figure 9 4 B and Figure 9 4 D respectively. The 0. 3 PECVD oxide deposited in Figure 9 4 C electrically isolates W and Al. The SiO 2 depositions in Figure 9 4 A and Figure 9 4 C protect the W layer from fluoride based etch recipes. The device is released by bac kside DRIE silicon etching ( Figure 9 4 E ), buried oxide etch ( Figure 9 4 F ) and front side silicon etch ( Figure 9 4 G ). PAGE 165 165 Figure 9 4 Fabrication process flow on an SOI wafer (A ) PECVD SiO 2 deposition patter ning, and etching by RIE (B ) W sputtering and lift off (C ) PECVD SiO 2 deposition patterning, and etching by RIE (D ) Al sputtering and lift off (E ) Backsi de 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 s election The dimensions of a cross section of th e multimorph are shown in Figu re 9 3 C The W beam is narrower than Al and SiO 2 layers to account for potential misalignment during fabrication. F rom past experience, the etch selectivity of Si over SiO 2 during DRIE is in excess of 100:1. During the i sotropic Si etch shown in Figure 9 4 G 10 Si must be undercut from either edge of the 20 the device. Therefore, the SiO 2 protecting the W layer must have thickness in excess of 0.1 a PECVD SiO 2 layer with thickness greater than 0.1 [ 20 ] Si SiO 2 W Al (A) (E) (B) (F) (C) (G) (D) PAGE 166 166 Therefore, the 0.1 0.2 3 2 layers corresponding to Figure 9 4 A and Figure 9 4 C respectively meet the fabrication requirements. Two criteria are used for choosing the thicknesses of W and Al: 1. Maxi mum 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 R c and thickness t If R c >> t the components of stress on its cross section m ay 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 i ts thickness, straight multimorph equations may be used for choosing optimum thickness values of the thin films. Material properties are obtained from [ 58 ] Figure 9 5 A shows the tangenti al angle at the free end of a singly clamped 100 a uniform temperature change of 100 K. Optimum Al thickness corresponds to the max ima of the plots shown in Figure 9 5 A Figure 9 5 B shows the optimum Al thickness as a function of the thickness of the W layer. If the W layer is 0.6 optimum Al thickness is 0.7 Figu re 9 3 C the measured value of Al thickness in 0.6 2. St iffness criteria: The thicknesses of Al and W are chosen to produce similar actuator stiffness as older designs based on Al SiO 2 Pt SiO 2 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 u sing the process shown in PAGE 167 167 Figure 9 4 is similar to the stiffness of multimorphs fabricat ed 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 uniform temperature change of 100 K. The thicknesses of the SiO 2 layers are 0.14 ( Figu re 9 3 C) The thickness of W is varied as a parameter from 0.2 to 1 thickness corresponds to the maxima. (B) Optimum Al thickness values obtained from Figure 9 5 A. 9. 2.3 Device Characterization 9. 2 .3.1 Static r esponse The static response was obtained by applying a dc voltage to the device. A laser beam reflected from the mirror plate was used to m onitor the scan angle. The voltage was increased from 0 to 0.68 V a nd then decreased back to 0. Figure 9 6 A and Figure 9 6 B 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 consumpti on is 11 mW. As shown in Figure 9 6 the device characteristic is repeatable wi th negligi ble hysteresis. Figure 9 6 0 0.5 1.0 1.5 2.0 2.5 3.0 15 10 5 0 Aluminum thickness (microns) Change in t angential angle at free end of straight multimorph (degrees) Decreasing W thickness 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) Optimum aluminum thickness (microns) (A) (B) PAGE 168 168 also provides a comparison between FE simulations a nd experimental data. Details on the FE model will be provided in Section 9. 3. Figure 9 6 Exp erimentally obtained static characteristic of micromirror along with FE simulation data ( A) Optical scan angle vs. applied v oltage ( B) Optical scan angle vs. i nput power 9. 2.3.2 Freque ncy r esponse In order to obtain the frequency response, the mirror is biased in the linear region of the scan angle vs. vol tage characteristic shown in Figure 9 6 A and a sine wave voltage superimp osed 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 frequ ency response is shown in F igure 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 8 A and Figure 9 8 C respectively. The modes at 104 Hz and 416 Hz correspond to the two peaks in the fr equency response shown in F igure 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 angle (d egrees) 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 angle (d egrees) (B ) (A ) PAGE 169 169 the y axis is observed at 400 Hz and has been shown in Figure 9 8 B The simulated resonant frequencies match the experimentally measured frequencies to within 8%. F igure 9 7 Frequency response of the 1mm wide micro mirror. 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 Figu re 9 3 B 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 resona nce 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 10 0 10 1 10 2 10 3 Optical scan range (degrees) 15 10 5 0 Frequency (Hz) (A) (B) (C) y x z y x z x y z PAGE 170 170 9. 2.4 Two Dimensional Scanning As shown in Figure 9 8 the mirror can scan about two mutually perpendi cular axes. The Tektronix AFG 3022B signal generator, which has a 50 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 s hown in Figure 9 8 B In accordance with the frequency response plot in F igure 9 7 scanning about the y axis is also produced. Figure 9 9 A 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 9 B 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(2 400 ) (1+cos(2 10 ))) mV. This expression repre sents 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 smi ley face pattern shown in Figure 9 9 C was generated. 9.3 Finite Element M odel 9. 3.1 Harmonic Analysis The device layer of the SOI wafer is 20 t Si below the mirror plate depends on the etch recipe used for device release in Figure 9 4 G The mirror plate of the device was broken and the thickn ess of the SCS layer was measured using a 150X magnification optical microscope. The measured value of t Si is PAGE 171 171 18 0.5 [ 58 ] by setting t Si = 18 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 e resistance. The applied voltage appears across the series connection of the mir ror 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(2 400 ) (1+cos2 10 )) 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 consi sting 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 PAGE 172 172 handle layer silicon of the de vice 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 conductiv ity of air is set to 0.026 Wm 1 K 1 [ 20 ] The simulated heat loss coefficients have been listed in Ta ble 9 1 and will be used in the electrothermal m odel discussed in Section 9. 3.3 Table 9 1. Simulated heat loss coefficient s due to thermal diffusion through air Region of m odel Simulated average heat loss coefficient (Wm 2 K 1 ) Top surface of mirror pla te 48 Botto m 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 depend ent 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, 0 and temperature coefficient of resistivity, at 297 K are 2.13 10 7 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 PAGE 173 173 3.7%. Next, the simulated tempe rature 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 perature (K) 0.7 V 0.5 V 0.3 V Distance along semi circular actuator (mm) PAGE 174 174 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 film s It was found that, a uniform actuator temperature change of 64 K can be used to mimic the initial tilt. Figu re 9 3 B 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 6 A and Figure 9 6 B 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 attribu ted 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) Heater Current (mA) PAGE 175 175 9.4 Comparison w ith Mirrors Actuated b y Straight Multimorphs Since straight multimorph based designs with comparable mirror p late size and scan angle have been reported previously [ 19 20 ] a comparison wit h the new design shown in Figu re 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 mul timorph. 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 SiO 2 thermal isolations at the mirror and substrate ends of the straight actua tors, it is possible to achieve 22 scan angle at 23 mW [ 19 ] In contrast the design presented in t his chapter consumes only 11 mW at 60 scan angle as it utilizes a single actuator only. Therefore, th e design presented in this chapter scans larger angle and has lower power consumpt ion 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 PAGE 176 176 is inversely proportional to the heater resistance. The W heater used in the present design is three times thicker than the Pt heate r used in [ 19 ] Fur thermore, 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 SiO 2 for thermal isolation which makes them prone to impact failure. In contra s t the present design does not require thermal isolation to minimize power consumption. Ther efore, 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 miniatur ize micromirror based imaging systems. Devices reported in [ 19 20 ] are capable of 1D sca n 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 c omparable mirror plate size, L shaped design s will have lower resonant frequenc ies than designs based on curved actuators. 6. Mirror center shift: A low mirror center shift is desirable for good optical alignm ent during system design. Figure 9 13 compares the simulated center shift of the present des ign with the design reported in [ 19 ] The present design PAGE 177 177 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 twi sting of the actuator result in center shifts in opposite directions. Figure 9 13 Simulated center shift of the micromirror depicted in Figu re 9 3 and the mirror reported in [ 19 ] that is actuated by straight multimorph s. The mirror depicted in Figu re 9 3 undergoes lower center shift and therefore provides better optical alignment. 7. Thermal response time: Al and W, that constitut e 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 SiO 2 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 Mirror actuated by semicircular multimorph Mirror actuated by straight multimorphs PAGE 178 178 circular o r 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 layer s 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 a ctuated 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 b ending partially compensates the shift produ ced 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 mor e 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 m ultimorphs [ 20 ] This enables us to quantify the imp rovement in performance PAGE 179 179 due to the use of a curved multimorph actuator. The next chapter will present test results on other micromirror designs actuated by curved actuators. PAGE 180 180 CHAPTER 10 ELECTROTHERMAL MICROMIRRORS ACTUATED BY CURVED MULTIMORPH S 10.1 Introduction The previous chapter described a 1mm wide micromirror actuated by a semicir cular 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. T he 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 t hicknesses are depicted in Figu re 9 3 C 10. 2 An E lliptical Mirror with 92 Axis This section describes an elliptical micromirror actuated by a curved electrothermal multimorph. The major and minor axe s of the mirror plate are 142 and 92 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 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 us ing the second and third resonant modes. The subsequent sections provide detailed device description and test results. PAGE 181 181 10.2 .1 Device Description Figure 10 1 A shows an SEM of the elliptical electrothermal micromirro r. The major and minor axes of the mirror plate are 142 m and 9 2 m respect ively. 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 1 B shows the simulated initi al tilt. Details on FE modelin g will be provided in Section 10.2 .3 The cross wide elliptical multimorph is shown in Figure 10 1 C Figure 10 1 An e lliptical m irror with 92 xis. ( 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 volt age 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 2 A and Figure 10 2 B illustrate the optical scan ang le as a function of applied dc voltage and input power respectively. FE simulation results are also shown Mirror plate (A ) (B ) 200 m Curved multimorph 0 0.14 0. 6 0.6 0.29 6 Sputtered Al Sputtered W PECVD SiO 2 (C) 9 PAGE 182 182 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. p ower. 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 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 amp litude sine wave offset at 334 mV was used to obtain the fr equency response shown in Figure 10 4 T he thermal time constant is only about 3 ms and may be attributed to the high thermal diffusivities of Al and W The first three re sonant modes 0 5 10 15 20 25 14 12 10 8 6 4 2 0 Optical Angle (degrees) Mirror center 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 angle (degrees) Optical angle (degrees) (A) (B) Increasing voltage Decreasing voltage FE model PAGE 183 183 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 Frequ ency 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 m ay 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 att ributed to the curved actuator geometry and enables the mirro r to achieve 2D scanning. Figure 10 5 Simulated resonant modes of device depicted in Figure 10 1 at (A) 3.9 k Hz (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 ex cites the third resonant mode and produces an (A) (B) (C) y x z Frequency (Hz) Optical angle ( ) 10 1 10 0 10 1 10 2 10 0 10 2 10 4 PAGE 184 184 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 dim ensional 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 Estimat ion of heat loss coefficient At the length scales involved, convection may be neglected and heat loss to the surrounding air may be mainly a ttributed 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 a re 580 W m 2 K 1 and 200 W m 2 K 1 respec tively. These coefficients were used in the electrothermal model described in the next sub section. 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 en ds of the actuator are fi xed at room 17.4 6.1 PAGE 185 185 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 t hermal 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 le ngth. The maximum temperature points are located close to the connecting beam. As shown in Figure 10 9 A the maximum actuator temperature has a quadratic depend ence on applied voltage. Figure 10 9 B 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 Temperature (K) 100 mV 200 mV 300 mV 400 mV 522 K 294 K PAGE 186 186 Figure 10 9 Maximum actuator temperature. ( A ) Maximum actuat or 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 ang le and the resu lts 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 properti es of the thin films. 10. 3 An E lliptical Mirror with 92 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 1 C The gap between the mirror plate edge and the concentric actuator is 19 m. The scan angle vs. voltage plot for dc excitation has been shown in Figure 10 11 A. Figure 10 11 B 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 Maximum temperature (K) Maximum temperature (K) (A) (B) PAGE 187 187 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 m and 192 m respectively. Figure 10 11 Static characteristic of device shown in Figure 10 10 ( A ) O ptical scan angle vs. voltage (B) Optical scan angle vs. p ower. The frequency response was obtained by applying a 5.5 mV amplitude sine wave at 34 4 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 5 A and Figure 10 5 B respectively. The third mode corresponding to transverse scanning is not observed because the corresponding frequency is very high and the device response decays to zer o at high frequencies The next section describes a circular micro mirror with 400 m optical aperture. 0 100 200 300 400 Voltage (mV) 20 15 10 5 0 Optical angle (degrees) increasing voltage decreasing voltage 20 15 10 5 0 Optical angle (degrees) 0 2 4 6 8 10 Input power (mW) increasing voltage decreasing voltage (A) (B) PAGE 188 188 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 wide Circular Mirror Actuated by a Semicircular Multimorph Figure 10 13 A shows an SEM of a 400 wide circular mirror. The gap between the mirror edge and the concentric semicircular multimorph is 20 m The cross section of the multimorph is depicted in Figure 10 13 B Figure 10 13 A 400 m wide circular micromirror actuated by a semicircular electrothermal multimorph ( A ) SEM ( B ) M ultimorph cross section. 10 1 10 0 10 1 10 2 10 3 10 4 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) Optical scan range ( ) Semicircular actuator Mirror plate (A) (B ) 0.14 0.6 0.6 0.29 Sputtered Al Sputtered W PECVD SiO 2 10 7 200 PAGE 189 189 The scan angle vs. voltage plot for dc excitation has been shown in Figure 10 14 A. Figure 10 14 B 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 c orresponding 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 angle (degrees) 0 2 4 6 8 10 Input power (mW ) increasing voltage decreasing voltage (B) 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) 10 1 10 0 10 1 10 2 10 3 Optical scan range ( ) Optical angle (degrees ) PAGE 190 190 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 mul timorphs 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 deforma tion of curved multimorphs. Other advantages inclu de high fill factor, low mirror plate center shift and low power consumption. Future work may include the fabrication of c urved piezoelectric and SMA mu ltimorphs For instance, piezoelectric micromirrors f or 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 expand s the design space available to the MEMS engineer PAGE 191 191 CHAPTER 11 BURN IN, REPEATABIL ITY 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 ] (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 sw itches 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 the ir 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, t he 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. I t is important to investigate the range of actuation sig nals that may be applied without degradi ng the device performance. The study of device failure due to overvoltage, creep, fatigue and impact provide valuable insight s for design improvement. Since SiO 2 is brittle, the oxide thermal isolation region fails o n accidental impact. Current generation of Al SiO 2 bimorph based MEMS devices cannot withstand PAGE 192 192 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 m odes 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 i n A newly released micromirror was used for s tudying 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 an d 13 V. As shown in Figure 11 1 A significant hysteresis is observed in the heater characteristic. However, as shown in Figure 11 1 B the heater resistance becomes repeatable once 7.5 V is applied. Therea fter, 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. PAGE 193 193 Figure 11 1 Embedded hea ter 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 repea table 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 M ean Standar d deviation Mean Standar d deviation 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 Current (mA) (A) (B) Current (mA) PAGE 194 194 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 unrele ased 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 1 B 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 heat ers. 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 Current (mA) Data 1 Data 2 Data 3 PAGE 195 195 11 .2 .2 Scan Angle R epeatability 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. T he 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 0.15. The results have been plotted in Figure 11 3 Data 1 3 were obtained in chrono logical 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 cor responding 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 angle (degrees) Data1 Data2 Data3 dc voltage (V) PAGE 196 196 11 .2 .3 Initial Tilt of M irror 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 sub strate was about 20 for a newly released micromirror. This parameter was found to increase by about 0.5 5 during the burn in process In Figure 11 3 the vertical inter cept of the plots co rresponding to data 2 and 3 is 2.8 This indicates that the post burn i n 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 s ource. Before actuation, the mirror plate til t was foun d to be 20.5. The optical scan angle, applied voltage and device current were monitored. The optical angle has been plotted against the applied voltage in Figure 11 4 A Data 1 through data 5 were obtained in chronological order. Data 1 2 in Figure 11 4 A 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 1.5 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 t otal increase in initial mirror tilt is 3.5 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 correspond ing to data 4 and 5. PAGE 197 197 It may be concluded from Figure 11 3 and Figure 11 4 A 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 4 A has been plotted against the electrical power input in Figure 11 4 B 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 0 10V Data2 10V 0 Data3 0 13.4V Data4 13.4V 0 Data5 0 12.5V 50 40 30 20 10 0 10 Optical angle (degrees) (A) Optical angle (degrees) 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 0 10V Data2 10V 0 Data3 0 13.4V Data4 13.4V 0 Data5 0 12.5V (B ) PAGE 198 198 dependence of material properties becomes significant for power inputs g reater 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 bi morph close to the substrate. Equation 4 3 may be rewritten as, ( 11 1 ) U sing 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 out 30% from room temperature to 613 C. The drop for high power input. At low power input, the plots corresponding to data 2 and 3 are parallel to the plots correspon ding 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 micr omirror 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 temperatur e 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. PAGE 199 199 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 O vervoltage A device was actuated by 9 V dc and monitored over a period of three hours. No significant change in the heater current an d optical angle was observed. The measured heater current was 46 mA. Using Equatio n 11 1 th e heater temperature is estimated to be 562 C for an applied volt age 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 ver y 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 Th e 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 hea ter current and optical angle were monitored. Figure 11 7 A shows the heater current and Figure 11 7 B 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 angle (degrees) PAGE 200 200 circuit after 14,700 minutes. As shown in Figure 11 7 the entire duration may be roughly divi ded 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 dam age was observed at the platinum segment connecting the embedded heat er 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 current (A) 0 13 V (data 1) 13 V 0 (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 current (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 Optical angle (degrees) 0 5,000 1 0,000 15,000 0 5,000 10,000 15,000 (A) (B) Stage I Stage II PAGE 201 201 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 SiO 2 bi layer experiences bending stresses due to the difference in their thermal expansion coefficient s This results in yielding of the inner Pt segment due to thermomechanical creep. This explanation is in agreement with the image of the failed devic e shown in Figure 11 8 It is posited that the failure of the inne r Pt segment corresponds to stage I shown in Figure 11 7 24m Bond pad Damaged Pt segment between heater an bond pad PAGE 202 202 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 te mperature dependence of resistivity is ignored. Therefore, Figure 11 10 provide s a qualitative explanation only. As shown in Figure 11 10 a large current density exists in the Pt segment connecting the h eater 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 embedde d heater. It is proposed that the torsion produced due to the temperature gradient across t he bimorph array results in heater failure due to shear. Min: 0 Max: 12.2 10 9 A/m 2 High current density Min: 0 Max: 10 10 9 A/m 2 High current density PAGE 203 203 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 Fa ilure A major limitatio n faced by several electrothermal mirrors arises due to the brittle nature of SiO 2 The thermal isolation region consists of thin film SiO 2 only and is especially susceptible to failure. Brittle fracture is often encountered during d evice 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 m irror 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 micro mirrors, 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 m (A) (B) PAGE 204 204 11 .3.3 Other Reliability Issues 11.3.3.1 Creep A mirror was placed on a hot plate at 240 C 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 fatigu e [ 132 ] It may manifest in the form of change in scan angle and resonant frequency. 11.3.3.3 Environmental factors Both Al and SiO 2 are resistant to atmospheric corrosion. On exposure to the atmosphere, a thin oxide layer forms on the Al su rface [ 133 ] This protects Al from environmental corrosion. The device may also be enclosed to avoid the deposition of particulate matter. 11 .4 Summa ry 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 dur ing burn in. The initial mirror plate tilt is found to increase by a few degrees during the burn in process. PAGE 205 20 5 In order to make the initial tilt and mirror scan angle repeatable, th e 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 microscopi c examination o f the failed device and by finit e 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 min imize 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 SiO 2 thermal isolation are highly susceptible to impact failure due to the brittle nature of SiO 2 A novel process for fabricating robust mirrors will be discussed in the next chapter. Failure mechanisms such as c reep and fatigue require long term testing and further investigation. Future work will involve long term device testing and further investigation into mirror failure mechanisms. PAGE 206 206 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 actuat ed MEMS devices utilize SiO 2 as an active layer of the bimorph and for thermal isolation. Due to its use in CMOS processes, deposition and etch recipes for SiO 2 are well developed and this has lead to widespread use of SiO 2 in MEMS devices. Also, the low C TE of SiO 2 [ 48 ] enables m etal SiO 2 bimorphs to produce large deflections. Additionally, the low thermal conductivity of SiO 2 [ 134 ] makes it suitable for thermal isolation. However, micromirrors actuated by Al SiO 2 bimorphs have two major drawbacks. Firstly, the low thermal diffusivity of SiO 2 makes the response of m etal SiO 2 bimorph devices sluggish. Secondly devices employing SiO 2 thermal isolation are susceptible to impact failure due to its brittle nature. As discussed in Chapter 11, micromirrors with SiO 2 t hermal 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 SiO 2 is th at films thicker than ~ tend to crack due to large residual stresses. Several metal polymer bimorphs have also been reported [ 48 ] Since polymers thicker than the metal layer [ 48 ] Therefore, the thermal response is mainly determined PAGE 207 207 by the diffusivity of the polymer. Typically, polymers have very low thermal diffusivity (~10 7 m 2 /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 [ 136 ] is used for quantifying device robustness. Drop tests are used for s imulating real world impact events This chapter focuses on micromirror fabrication but the proposed proce ss 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 cha racterization and frequency response of fabricated devices are presented in Section 12.4. Experimental results on improvement in device robustness are presented in S ection 12.5 12.2 MEMS Materials for Thermal Multimorphs A wealth of information on the pr operties of candidate mat erials 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 high er thermal diffusivity than non metals [ 48 ] Certain non metals such as diamond like carbon (DLC) have very high ther mal diffus ivity as well [46, 121] Also, according to Equation 1 1 it must be ensured that the two active layers of the therma l bimorphs have widely different CTEs. PAGE 208 208 The thermal isolation region must have low therm al conductivity. Among metals, i nvar has one of the lowest thermal conductivities. Polymers may also be used for thermal isolation as they typically have low thermal con ductivity [ 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 p rocess. 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 (m 2 /s) Electrical conductivity (S/m) comments Al [ 134 ] 70 23.1 237 9.7 10 5 35.5 10 6 High CTE, Fast thermal response SiO 2 [ 134 ] 70 0.5 1.4 8.7 10 7 Insulator Brittle, slow thermal response Pt [ 134 ] 168 8.8 71.6 2.5 10 5 8.9 10 6 Slow thermal response W [ 134 ] 411 4.5 174 6.8 10 5 20 10 6 C an be used as heater and active bimorph layer Invar 145 [ 48 ] 0.36 [ 48 ] 13 [ 48 ] 3.1 10 6 [ 48 ] One of the least thermally conductive metal DLC [ 137 ] 700 1.18 1100 6.06 10 4 Doping dependent H igh thermal speed Polyimide 2.3 [ 134 ] 20 [ 134 ] 0.15 [ 134 ] 1.04 10 7 [ 134 ] Insulator Can achieve good thermal isolation The material comb inations that were considered for form ing 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 PAGE 209 209 large thickness. Since polymers have very low thermal diffusivities [ 48 ] their thermal response time is very slow. Therefore, a thick polymer layer will adver sely 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 actuat ion 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 dia mond 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 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 sign ificantly slower than DLC itself. Another problem is that it is difficul t 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 Therefore according to Equation 1 5 a thin layer of W may produce optimum resul ts. Another advantage of using W as a bimorph material i s that it can double up as a resistive heater. Based on high diffusiv ities and large difference in C TE s, three possible pairs stand out Al DLC, Al Invar, and Al W. The newly fabricated micromirrors are actu ated by Al W bimorphs with W also acting as a resistive heater. The thin film thickness values are same as that depicted in Figu re 9 3 C As discussed in Section 9.2.2.3, the thickness values are chosen to maximize multimorph deflection. The PECVD SiO 2 that encapsulates W, pro vides electrical isolation between Al and W SiO 2 also serves to protect W from fluoride based etch recipes. Other than SiO 2 i nvar [ 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 polyimid e, PI 2574 from HD Microsystems [ 134 ] for thermal isolation. Both 1D and 2D scanning micromirrors were fabricated. The next s ection describes the process flow in detail. PAGE 210 210 12.3 Fabrication Process The micromirror fabrication process is illustrated in Figure 12 1 SOI wafers are used t o ensure flatness of mirror plates with single crystal silicon microstru cture s. Figure 12 1 Fabrication process for robust mirrors. (A) Oxide deposition, pa tterning, etching on SOI wafer. (B) W sputtering and lift off. (C) Oxide deposition, patterning, etching. (D) Al sp uttering and lift off. (E) Polyimide spin coating and baking, PECVD oxide deposition, patterning, oxide etchin g and polyimide etching. (F) Backside lithography, DRIE Si etch and buried oxide etch. (G) Front side isotropic Si etch for device release. Th ough thermal isolation is shown at one end of the bimorph only, there may be isolation at both ends. The SiO 2 deposited in Figure 12 1 A and Figure 12 1 C protect W from fluoride based etch recipes. Additio nally, t he SiO 2 deposited in the step show n in Figure 12 1 C electrically isolates the Al and W layers. Both Al and W are fabricated b y sputtering and lift off ( Figure 12 1 B and Figure 12 1 D ). Polyimide is spin coated, baked and then covered with PECVD SiO 2 ( Figure 12 1 E ). This layer of SiO 2 acts as a mask for etching the 5 m thick polyimide. The device is released by back side DRIE silicon etching ( Figure 12 1 F ) buried oxide etch and frontside isotropic silicon etching ( Figure (A) (B) (C) (D) (E) (F) (G) Si SiO 2 W Al Polyimide Substrate Mirror plate Polyimide thermal isolation Bond pad Multimorph PAGE 211 211 12 1 G ). 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 show n in Figure 12 2 In this variation, trenches are created by isotropic Si RIE etch in Figure 12 2 E 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, pa tterning, etching on SOI wafer. (B) W sputtering and lift off. (C) Oxide deposition, patte rning, etching. (D) Al sp uttering 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 etchin g 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 SiO 2 W Al Polyimide (F ) (G) (H) Bond pad Trench filled thermal isolation Multimorph actuator Substrate Mirror plate PAGE 212 212 After driving out the air bubbles, the polyimide is spin coated and baked. Figure 12 2 H shows trench filled polyimide thermal isolation between multimorph actuator s 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 isol ation 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 actuator s Therefor e, this device has similar topology to the device discussed in Chapter 3, which employs SiO 2 thin film isolation. Mirror plate Array of b imorphs 1 mm PAGE 213 213 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 o ut 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 re sistance of the tungsten heater. 1 mm Beam type polyimide thermal isolation Mirror plate Array of Al W Bimorphs PAGE 214 214 Therefore, the fabrication process shown in Figure 12 2 requires modifications and further refinement. Figure 12 5 SEM of mirror with po lyimide 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 t his 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 5 0 m (B) (C) 100 m Mirror plate Array of bimorphs PAGE 215 215 Figure 12 6 SEM of robust 1D mir ror 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 3 D 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 PAGE 216 216 Figure 12 8 SEM of robust 3D lateral shift free (LSF) mirror with no thermal isola tion. 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 mirr or plate. (A) (B) 50 m 1 mm Mirror plate Bimorphs Rig id beam PAGE 217 217 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 ac tuator 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 fabricat ed 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 s ection. 12.4 Device Characterization The static char acteristic 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 11 B through Figure 12 15 B and have been listed in Table 12 1. 1 mm Mirror plate Bimorphs Trench filled thermal isolation 100 m Trenches (A) (B) PAGE 218 218 Figure 12 11 Static characteristic of 1D mirror with no t hermal 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 actuator s 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 angle ( ) 0 20 40 60 80 100 120 140 12 10 8 6 4 2 0 Increasing voltage Decreasing voltage Optical angle ( ) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Vol tage (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 Vol tage (V) Input power (mW) Optical angle ( ) Optical angle ( ) (A) (B) PAGE 219 219 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 t he 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 isolatio n 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 isolat ion between actuators and mirror plate. Th e 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 angle ( ) 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 ang le ( ) Increasing voltage Decreasing voltage Vol tage (V) Input power (mW) (A) (B) 0 10 20 30 40 50 60 70 Optical angle ( ) 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 angle ( ) Vol tage (V) Input power (mW) (A) (B) PAGE 220 220 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 a ngle 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 ( /mW) No thermal isolation Figure 12 3 0.098533 Beam type isola tion 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 show n 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 angle ( ) 10 8 6 4 2 0 Optical angle ( ) 0 0.2 0.4 0.6 0.8 Vol tage (V) (A) (B) PAGE 221 221 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 device s with no thermal iso lation. 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 n early 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 generat ion micromirrors that used SiO 2 thermal isolation. Figure 12 16 Frequency response of the mirror shown in Figure 12 3 The response is obtained by applying a sinusoidal v oltage of 55 mV amplitude at a dc bias of 1.475 V. 10 1 10 0 10 1 10 1 10 0 10 1 10 2 Frequency (Hz) Optical s can range ( ) PAGE 222 222 Figure 12 17 Frequency res ponse of the mirror shown in Figure 12 4 The response is obtained by applying a sinusoidal vo ltage of 25 mV amplitude at a dc bias of 870 m V. Figure 12 18 Frequency res ponse of the mirror shown in Figure 12 5 The response is obtained by applying a sinusoidal volt age of 32.5 mV amplitude at a dc bias of 820.5 m V. 10 1 10 0 10 1 10 2 Frequency (Hz) 10 1 10 0 10 1 Optical s can r ange ( ) 10 1 10 0 10 1 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) Optical s can range ( ) PAGE 223 223 Figure 12 19 Frequency res ponse of the mirror shown in Figure 12 6 The response is obtained by applying a sinusoidal v oltage of 46 .5 mV amplitude at a dc bias of 799 .5 m V. Figure 12 20 Frequenc y 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 m V. 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) 10 1 10 0 10 1 Optical s can range ( ) 10 1 10 0 10 1 10 2 10 3 Frequency (Hz) 10 1 10 0 10 1 Optical s can range ( ) PAGE 224 224 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 ava 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/s 2 experi enced by the mirror is given by [ 136 ] ( 12 1 ) where g is the acceleration due to gravity (m/s 2 ) and h is the drop height (m). It was found that mirrors desc r i bed in Chapter 3, which have SiO 2 thermal isolation, fail in the 1600g 2 000g range. In contrast, the new device s hown in Figure 12 5 can withst and accelerations greater than 8 000g 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 suspend ed from a height of 130 mm. ( A ) Before impact. ( B ) After imp act. 12.5.2 Drop Tests It is found that the 3D device shown in Figure 12 9 does not suffer catastr o phic 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 PAGE 225 225 on a vinyl floor. Older designs with Si O 2 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 mak es 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 polyimid e allow the optimization of device speed and power consumption. The process for fabricating robust mirrors brings them a step closer to real world applications. PAGE 226 226 C HAPTER 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 d in plane in the plane of the substrate. In this chapter, novel schemes for amplifying the in plane displace ment are proposed. The des igns can achieve 100s mi crons 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. Electrostati c comb drive actuators can produce up to 30 plane displacement [ 138 ] Cragun et al. report electrothe rmal transducers that produce in plane displacement s up to 20 [ 139 ] Waterfall et al. report a bent bent beam type thermal actuator that produces up to 5 plane displacement. Lee et al. report a n electrothermally actuated MEMS stage that has a maximum displacement of 40 [ 140 ] Noworolski et al. report electrothermal designs that produce large in plane displacements up to 200 but the device area is greater than 10 mm 2 which is very large [ 141 ] The reported values of in plane displacements produced by piezoelectric transducers are much less that those produced by elec trothermal transducers [ 138 ] Conway et al. report a piezoelect ric in plane actuator that produces a peak displacement of 1.18 [ 142 ] Two novel transducers are propo sed in this c hapter. These will be referred to as Design 1 and Design 2. Sections 13.2 and 13.3 discuss Designs 1 and 2 respectively. PAGE 227 227 Section 13.4 provides a comparison of the two designs. Potential applicatio ns are discussed in Section 13.5 13.2 In Plan e 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 beam s 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. l 1 l 2 l 1 l 2 2l 1 Multimorph Rigid Beam PAGE 228 228 13.2.2 Simulations Let u s consider Al W thermal bimorphs. Let the thickness of Al be 1 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 The lengths l 1 and l 2 ( Figure 13 1 ) are chosen to be 100 respectively. The transducer is simulated for a uniform temperature change of 400 K from an initially flat positi on Figure 13 2 shows the deformed shape The simulated in plane tip displacement is 238 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 fr om finite element simulations. The color bar shows the in plane displacement. The simulated in plane displacement is 238 13.2.3 Analysis Let the transducer be completely flat at a reference temperature. Let R m = 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 In plane displacement PAGE 229 229 ( 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 l 1 and l 2 to achieve m aximum displacement for constant total transducer length l total As show n in Figure 13 1 if l 1 l 2 l total = 2 l 2 In general l total is given by, ( 13 2 ) It can be shown that if the total transducer length, l total is less than 2 R m the condition l 1 = l 2 maximizes in plane displacement. Figure 13 3 shows the top view of the optimized transd ucer design for l total < 2 R m If l total > 2 R m and l 1 l 2 the partial derivative of d design 1 with respect to l 1 may be equated to zero to give l 1 = R m According to Equation 13 2 in this case l 2 is constant. If l total > 2 R m and l 1 l 2 the partial derivative of d design 1 with respect to l 2 may be equated to zero to give l 1 = R m According to Equation 13 2 in this case l 1 is constant. Therefore for l total > 2 R m l 1 = R m must be chosen to maximize in plane displacement. Figure 13 4 shows the side view of the optimized transducer fo r l total > 2 R m To summarize, if l total < 2 R m l 1 = l 2 maximizes in plane displacement. Otherwise, l 1 = R m maximizes in plane displacement. After optimizing the design shown in Figure 13 2 l 1 and l 2 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. PAGE 230 230 Figure 13 3 Optimized D esign 1 for l total 2 R m The top view of the undeformed state is depicted in this figure. Figure 13 4 Optimized D esign 1 for total transducer length, l total 2 R m The side view in the deformed state is sho wn. In this case l 1 = R m is chosen. The total in plane displacement is 4 l 2 13.3 In Plane Transducer Design 2 13.3.1 Design Topology Figure 13 5 A shows a schematic of the transducer topology. If the inverted multimorph is an exac t inversion of the non inverted multimorph as shown in Figure 13 5 B and Figure 13 5 C and the material thicknesses are the same in the inverted and non inverted multimorphs, l 2 may be chosen to be twice of l 1 The non inverted and inverted multimorphs bend in opposite directions and result in a net zero out of plane displacement. Rigid Beam Rigid Beam Multi morph Multimorph Multimorph Multimorph Multimorph Rigid Beam l total PAGE 231 231 Figure 13 5 Top view of proposed transducer for achieving large in pl ane 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, l 2 must be twice of l 1 ( 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 t ungsten (W) at the bottom. The invert ed multimorph consists of 1 a luminum (Al) at the bottom and 0.4 ungsten (W) on top. The lengths l 1 and l 2 are chosen to be 100 respectively. l 1 l 1 l 2 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 (D) In plane displacement PAGE 232 232 If the inverted mult imorph is an exact inversion of the non inverted multimorph, l 1 = 2 l 2 Let the radius of curvature of the multimorphs be R m in the deformed state. Let the multimorphs be flat, i.e., undeformed at a reference state. Then the total in plane displacement is, ( 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 m SiO 2 0 .2 m Pt 0 .2 m SiO 2 and 1 m Al. The inverted multimorph layers from bottom to top are 50 nm SiO 2 0 .2 m Pt 0 .2 m SiO 2 1 m Al, and 1.6 m SiO 2 The Pt layer acts as an embedded resistive heater. The SiO 2 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 thick silicon layer coated with Al. The thick silicon layer under the platform ensures its flatn ess. Fabrication was done on an SOI wafer with 80 thick device layer. The fabrication process is similar to the one described in [ 36 ] With reference to Figure 13 5 l 1 and l 2 are 100 m and 400 m respectively. The values of l 1 and l 2 ar e chosen in accordance with Equation 6 1 to ensure zero out of plane displacement Only therm al 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 m with respect to the substrate. PAGE 233 233 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 8 A, an in plane displacement of 11 m is produced at 1.65 V. As depicted in Figure 13 8 B, 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 SiO 2 thermal isolation 80 thick single crystal silicon Rigid platform Bond pads Rigid platform Actuators PAGE 234 234 platform is shown in Figure 13 9 The maximum out of plane displacement is 17 m in the downward direction, i.e., towards the substrate. Figure 13 8 In plane displacement produced by Design 2. (A) In plane displacement vs. v oltage. (B) In plane displacement vs. input power. Figure 13 9 Out of plane displacem ent produced by Design 2. (A) Out of plane displacement vs. voltage. (B) Out of plane displacement vs. input pow er. 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 u se 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 10 0 12 8 4 0 0 0.5 1 1.5 2 12 8 4 0 Voltage (V) Input power (mW) (A) (B) In plane displacement ( ) In plane displacement ( ) 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 Out of plane displacement ( ) Out of plane displacement ( ) (A) (B) PAGE 235 235 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 assumed. The multimorphs consist of 1 ungsten. For Design 1, optimum beam lengths are chosen as discussed in Section 13.2.4. Transduction is achieved by the rmal 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 Desi gn 1 1.6 mm Design 2 298 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 ha ve been listed below: 1. Movable MEMS s tage with 5 degrees of freedom : The design shown in Figure 1 2 C 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 p erpendicular axes. The in plane designs can ad d 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, additio nal masks are not required for fabrication. A possible application is the handling of small biological samples. PAGE 236 236 2. Integrated Michelson i nterferometer : Michelson interferometer s 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 u sing 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 m ay be fabricated on the sa me substrate as discussed in [ 2 ] V grooves may be etched on the substrate to hold t he optical fib er 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 op tical coherence tomography setup. 3. MEMS tweezer or micro gripper : By using two opposing tr ansducers, 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 11 A 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 actuati ng 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 inter fere Beam splitter PAGE 237 237 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 choosin g 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 trans ducers for achieving large in plane displacement of the order of 10 0s 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 e xpansion, 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 PAGE 238 238 CHAPTER 14 CONCLUSIONS AND FUTURE WORK 14 .1 Summary of Work Done This dissertation deals wit h 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 M odeling Device modeling is essential for design, optimization and co ntrol 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 computa tionally efficient and it saves time and reso urces. The key components of the complete device model of an electrothermal mi cromirror are the electrical thermal and mechanical models. The electrical model provides the power dissipated in the embedded heater for a certain actuation voltage. The therm al 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 be en surveyed in Chapter 2. As discussed in Chapter 3, n umerical 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 mod el satisfactorily represents the thermal response of the device. Based on intuition, a circuit model was built to explain the second order thermal response. Two PAGE 239 239 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 numer ic al, 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 he at 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, t he transmission line method has been demonstrated for a de vice in which the resi stive heater is embedded at one end of the bimorphs. The bimorphs themselves are treated as a passive transmission line. Active t hermal transmission line with distributed embedded heater is discussed in Chapter 5. The mechanical model predicts mirror plat e 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 base d on free body diagrams. The Lagrangian approach is based on energy equations. In th e Lagrangian method, the actuator beams are treated as springs that can store elastic potential energy. Since the mirror is significantly heavier than the ac tuators, kineti c 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 fol d increase in the scan angle of micromirrors employing ISC actuators and has been discussed in Chapter 6. PAGE 240 240 14.1.2 Curved Multimorph Actuators Prior to this thesis, several MEMS devices based on straight multimorphs have been reported by researchers. Howeve r, 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 expressi ons 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 electro thermal micromirrors. These micromirrors have been discussed in Chapters 9 and 10. Micromirrors actuated by curved multimorph actuators have several a ttractive features Compared to previously reported designs, the curved multimorph based designs un dergo l ower 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 R epeatability Device repeatability has been discussed in Chapter 11. Prel iminary investigation on the pre conditioning and repeatability of devices based on Al SiO 2 bimorphs and embedded Pt heater has been carried out Three key parameters initial eleva tion, 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 successfu l pre conditioning. The pre conditioning makes the device repeatable and reduce s performance variations among a certain population of devices. PAGE 241 241 14.1.4 Fabrication of Robust Micromirrors Commercial success of a device requires a th o rough investigation into its reliab ility issues. Devices actuated by Al SiO 2 bimorphs have two major drawbacks. Firstly, such devices utilize SiO 2 thin film thermal isolation which makes them susceptible to impact failure. As a result, such devices cannot withstand drop tests fr om more than a fe w centimeters height. Secondly, the low thermal diffusivity of SiO 2 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 utilize s 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 d evices. 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, t wo 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 previou sly 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. PAGE 242 242 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 co ntrol 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 inves tigation 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 ac hievement 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. 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"Miniaturized time scanning Fourier transform spectrometer based on silicon technology," Optics Letters vol. 24, pp. 1705 1707, 1999. PAGE 254 254 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 swit ch, 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 Biophotoni cs and Mi crosystems 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 f abrication process for robust, fast electrothermal micromirrors; novel micromirrors with low center shift, high fill factor, improved voltage and power requirements; comprehensive electrothermomechanical devic e models; and model based mirror control. He co mpleted 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. 