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THERMOELASTICALLY ACTUATED ACOU STIC PROXIMITY SENSOR WITH INTEGRATED ELECTRICAL THRO UGH-WAFER INTERCONNECTS By VENKATARAMAN CHANDRASEKARAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004
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Copyright 2004 by Venkataraman Chandrasekaran
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iii To my parents and my wife, Anu.
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iv ACKNOWLEDGEMENTS Financial support for the project was pr ovided by the Office of Naval Research (contract #N00014-00-1-0343) monitored by Dr. Kam Ng. I would like to express my sincere gratitude to my advisor, Mark Sheplak, for giving me the opportunity to work at the Interdisciplinary Microsystems Group. His guidance over the years has been invaluable. His penchant for good quality research and his aggressive approach to it have had a ve ry positive influence on me. I would like to thank Professors Toshikazu Nishida, Louis N. Cattafesta, Bhavani V. Sankar and Wei Shyy for their help and valuable insights on different aspects of this multidisciplinary project and for serving on my committee. I would also like to thank Professor David Hahn for his help with the thermal modeling. My thanks go to all my colleagues at IM G over the past years, especially Anthony Cain, David Arnold, Stephen Horowitz, Sunil Bhardwaj, Karthik Ka dirvel and Anurag Kasyap. It has been a pleasure to have worked with them. Their company made the innumerable hours spent at the lab a lot more interesting, memories of which I will carry for the rest of my life. I would like to thank Mr. Ken Reed from TMR Engineering for his excellent and prompt services. His help in fabricating th e experimental set-up and the sensor package is greatly appreciated. I would like to thank Professor Thomas Ke nny and his students, especially Eugene Chow, for all their help and insights du ring the fabrication of the sensors.
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v Most of all, I would like to thank my fam ily. My parents have always wanted the best for me and their endless support a nd guidance throughout my life are beyond words and will always be cherished. I hope to fulfill all their dreams. I am forever grateful to my wife, Anu, for her love and understanding du ring my years as a graduate student. She continues to bring out the best in me.
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vi TABLE OF CONTENTS ACKNOWLEDGEMENTS............................................................................................... iv LIST OF TABLES............................................................................................................. ix LIST OF FIGURES .............................................................................................................x LIST OF SYMBOLS.........................................................................................................xv ABSTRACT.................................................................................................................... xvi i CHAPTER 1 INTRODUCTION........................................................................................................1 Cavity Physics ..............................................................................................................1 Cavity Sensing Issues ...................................................................................................3 Scope.......................................................................................................................... ...5 Research Contributions.................................................................................................5 Dissertation Organization .............................................................................................5 2 LITERATURE REVIEW.............................................................................................7 Conventional Ultrasonic Transducers...........................................................................7 Micromachined Ultrasonic Transducers.......................................................................9 3 ACOUSTIC PROXIMITY SENSOR.........................................................................14 Operating Principles ...................................................................................................17 Thermoelastic Actuation .....................................................................................17 Piezoresistive Detection ......................................................................................18 Device Fabrication......................................................................................................19 Electrical Through-Wafer Interconnect...............................................................20 Acoustic Sensor/Actuator....................................................................................24 4 DEVICE MODELING ...............................................................................................27 Thermoelastic Actuation.............................................................................................27 2-D Thermomechanical Model...................................................................................29 Heat Conduction Model ......................................................................................31 Plate Equations ....................................................................................................33
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vii Results and Discussion........................................................................................36 Effect of in-plane heat conduction ...............................................................41 Effect of static heating..................................................................................43 Generation of Acoustic Waves ...................................................................................46 Piezoresistive Sensing ................................................................................................50 5 EXPERIMENTAL CHARACTERIZATION ............................................................61 Electrical Characterization..........................................................................................61 Current vs. Voltage Characteristics.....................................................................61 Capacitance vs. Voltage Characteristics..............................................................63 Noise Floor Spectra.............................................................................................63 Mechanical Characterization ......................................................................................65 Acoustic Characterization-Receiver ...........................................................................67 Acoustic Characteri zation-Transmitter.......................................................................68 Proximity Sensing.......................................................................................................69 6 RESULTS AND DISCUSSION.................................................................................70 Electrical Characterization..........................................................................................70 Current vs. Voltage Characteristics.....................................................................70 Capacitance vs. Voltage Characteristics..............................................................73 Noise Floor Spectra.............................................................................................74 Mechanical Characterization ......................................................................................75 Acoustic Characterization-Receiver ...........................................................................82 Linearity ..............................................................................................................82 Frequency Response............................................................................................83 Acoustic Characteri zation-Transmitter.......................................................................84 Proximity Sensing.......................................................................................................86 7 CONCLUSIONS AND FUTURE WORK.................................................................88 Conclusions.................................................................................................................88 Future Work................................................................................................................90 APPENDIX A PROCESS TRAVELER.............................................................................................92 B PIEZORESISTOR DESIGN ......................................................................................98 FLOOPS Input Files .................................................................................................98 Resistance Calculation..............................................................................................100 C THERMAL ACTUATION OF A COMPOSITE DIAPHRAGM............................103 Heat Conduction Model............................................................................................103
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viii Green’s Function Solution Technique...............................................................105 Finite Hankel Transform ...................................................................................109 Plate Analysis ...........................................................................................................115 D SENSOR PACKAGE...............................................................................................123 LIST OF REFERENCES.................................................................................................124 BIOGRAPHICAL SKETCH ...........................................................................................132
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ix LIST OF TABLES Table Page 4-1: Material properties and geometry used in the analytical and finite element models (radius of the diaphragm: 500 !m , radius and thickness of the heater: 30 !m , 0.5 !m , input power: 20 mW at 50 kHz ).........................................................................36 4-2: Comparison of the effective piezoresistiv e coefficients with that of a uniformly low-doped p-type silicon at room temperature.........................................................57 6-1: List of devices used for the characte rization (The thickness of the silicon dioxide and silicon nitride layers are 0.7 !m and 0.3 !m respectively)................................70 6-2: Resistance of the diffused resistors (in k")..............................................................72 6-3: Mechanical characteristics of the thermoelastic resonators. ....................................77 6-4: Critical buckling loads. ............................................................................................78 6-5: Comparison of the MEMS-based acoustic transducer with a c onventional ultrasonic transducer. ................................................................................................................86
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x LIST OF FIGURES Figure page 1-1: Schematic of a bump-b onded sensor package placed on the hull of an underwater supercavitating vehicle for monitoring th e state and thickness of the gas/water interface...................................................................................................................... 3 2-1: Cross-sectiona l schematic of a 1 mm x 1 mm thermally actuated membrane resonator.....................................................................................................................9 2-2: Cross-section of a capacitive micromachined ultras onic transducer element..........10 2-3: Cross-section of a ferroelectric transducer...............................................................11 2-4: Cross-sectional sche matic of the P(VDF-TrFE)/si licon-based transducer. .............12 3-1: Top view microscopic (diffraction in terference contrast) image of the acoustic proximity sensor with integrat ed through-wafer interconnects................................15 3-2: Cross-sectional schematic of the acoustic proximity sensor....................................15 3-3: Traditional wire-bonde d sensor package versus the more compact and rugged bump-bonded package..............................................................................................16 3-4: Schematic of the arc and tapered piez oresistors arranged in a Wheatstone bridge configuration. ...........................................................................................................19 3-5: A schematic of the fabr ication process illustrating th e creation of the through-wafer vias. .......................................................................................................................... 21 3-6: SEM cross-section after plas ma etching vias through the wafer..............................22 3-7: Fabrication sequence: Thermal oxide growth for el ectrical isolation of the interconnects from the silicon substrate...................................................................22 3-8: Fabrication sequence: Deposition and boron diffusion doping of polysilicon layers for electrical conduction...........................................................................................23 3-9: Fabrication sequence: Patterning the interconnects. ................................................23
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xi 3-10: SEM illustrating the oxide insulation a nd the polysilicon conduction layers of the ETWI........................................................................................................................24 3-11: Through-wafer interconnect integrated with a metal bond pad. ..............................24 3-12: Fabrication sequence: Resistor implantation and thermal oxide growth. ................25 3-13: Fabrication sequence: Sputtering and patterning alumin um (1%-Si), silicon nitride deposition and DRIE to release the diaphragm........................................................25 3-14: 3-D schematic of the in tegrated acoustic sensor/ETWI...........................................26 4-1: Axisymmetric model of the composite diaphragm consisting of three transversely isotropic layers and a diffused resistive heater (H1, H2, H3 and HHeat represent axial distances from the reference plane. The piezoresistors are not represented in the analytical model)......................................................................................................30 4-2: A flow chart of the solution procedure.....................................................................35 4-3: Plot illustrating the convergence of the analytical series solution for temperature (calculated at r = 40 m , z = 4 m and t = 12/5 .................................................37 4-4: Temperature distribution in the composite diaphragm (at 0 r and 0 H ) as a function of time. .......................................................................................................37 4-5: Non-uniform temperature profile thr ough the thickness of the composite diaphragm (at 0 r ). .................................................................................................................39 4-6: Radial temperature di stribution in the composite diaphragm along the center of silicon layer. .............................................................................................................39 4-7: Vibration amplitude of the composite diaphragm (at 0 r and 0 H ) as a function of time. .......................................................................................................40 4-8: Radial variation of the vibration amplitude of the composite diaphragm................40 4-9: Comparison of the analytical model with an experimental measurement of the diaphragm vibration amplitude for an input power of 39 mW at 40 kHz .................41 4-10: Normalized radial te mperature distribution at the center of a homogenous silicon diaphragm (500 m -diameter and 10 m -thick) as a function of driving frequency.42 4-11: Axisymmetric finite element model of a released diaphragm (500 m -radius, 10 m -thick) with the surrounding substrate (500 m -thick), illustrating the temperature distribution due to a combined dc (20 mW ) and ac (20 mW at 50 kHz ) input.......................................................................................................................... 44
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xii 4-12: Plot of temperature as a function of time at the cent er of the diaphragm due to a combined dc (20 mW ) and ac (20 mW at 50 kHz ) input...........................................44 4-13: Radiation from a circular piston...............................................................................47 4-14: Comparison of the normal velocity di stribution of the thermoelastic resonator operated at resonance an d a clamped radiator..........................................................48 4-15: Polar plot of the amplitude directivit y factor of a clamped circular radiator for different values of ka ................................................................................................49 4-16: Comparison of the directional characteristics of two transducers ( ka = 5 ): (a) radiator with uniform vibration amplitude , (b) radiator clamped at the edges.........49 4-17: Room temperature piezor esistive coefficients in the ( 100) plane of p-type silicon (10-11 Pa-1)................................................................................................................52 4-18: Plot of the piezoresistance factor P(N,T) as a function of impurity concentration for p-type silicon. The line is based on a theoretical model by Kanda (1982) and the points are experimental data.....................................................................................55 4-19: Piezoresistor doping profile with a junction depth of 0.46 !m obtained using FLOOPS &.................................................................................................................56 4-20: Plot of the piezoresistance factor as function of depth of the resistor......................56 4-21: Conductivity variation through th e thickness of the piezoresistor...........................57 4-22: Radial and circumferent ial stress distribution in the composite diaphragm subjected to in-plane (300 MPa -compressive) and transverse (200 Pa ) loading.....................60 5-1: Chain of 6, 22, 62 and 100 ETWI connect ed in series via surface polysilicon lines (dark and bright lines indicate the to p and the bottom surface of the wafer respectively). ............................................................................................................62 5-2: Schematic representation of the device noise measurement setup...........................64 6-1: Forward and reverse bias characteristics of the pn junction, indicating negligible leakage current (< 14 pA at -10 V ) (Device C3).......................................................71 6-2: Reverse bias breakdown voltage of the pn junction for devices C1 and C9............71 6-3: I-V characteristics fo r chains of 6, 22 and 62 ETWI. ..............................................72 6-4: Plot illustrating the linear variation in the resist ance of the ETWI with (a) 4 !m thick and (b) 6 !m -thick doped polysilicon layers (s lope indicates average ETWI resistance).................................................................................................................73
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xiii 6-5: High-frequency (1 MHz ) capacitance vs. voltage characteristics for a chain of 22 interconnects.............................................................................................................74 6-6: Noise power spectral density of the sensor at a bridge bias of 9 V (devices tested C1 and C3). ....................................................................................................................75 6-7: Visualization of the fi rst three vibration modes of th e thermoelastic resonator using the scanning laser vibrometer...................................................................................76 6-8: Plot of the vibration amplitude meas ured at the center of the diaphragm as a function of excitation frequency for vary ing thickness of the silicon layer.............76 6-9: Vibration amplitude of a 1 mm -diameter diaphragm calculated at the center as function of the diaphragm thickness. .......................................................................79 6-10: Resonant frequency of the thermoelastic resonators as a function of their buckled height measured at the center (data represen ts 7 devices)........................................80 6-11: Vibration amplitude of the thermoelas tic resonator as a function of the buckled height both measured at the center of th e diaphragm (data represents 7 devices)...80 6-12: Deflection mode shape as a function of static power...............................................81 6-13: Change in the buckled height of th e diaphragm as a function of static power. .......81 6-14: Resonant frequency of the diaphr agm as a function of static power. ......................82 6-15: Plot illustrating the de vice linearity in sensing acoustic pressure perturbations (up to 140 dB ) at a frequency of 1 kHz (Device C3). .....................................................82 6-16: Magnitude of the sens or frequency response function to a constant sound pressure level of 110 dB (Device C3).....................................................................................83 6-17: Directivity of the generated acoustic field at a frequency of 60 kHz (Device C3) ( 0.55 ka # ). ..............................................................................................................84 6-18: Sound pressure level of generated acoustic field as a function of radial distance, at a frequency of 60 kHz (Device C3 and C9). ...............................................................85 6-19: Plot of the true distance versus th e measured distance obtained using a CW phaseshift technique. .........................................................................................................86 B-1: A schematic of the discretized arc piezo resistor and the equiva lent resistor model illustrating the series connection of the elements in the circumferential direction and the parallel connection of the se gments in the radial direction. ......................100 B-2: Piezoresistor geometry a nd layout (all dimensions are in mm )..............................102
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xiv C-1: Axisymmetric model of the composite diaphragm consisting of three transversely isotropic layers and a diffused resistive heater (H1, H2, H3 and HHeat represent distances from the reference plane)........................................................................103 C-2: Force and moment resultants on an element of the circular plate..........................116 D-1: Lucite package for the acoustic proximity sensor to enable testing of the integrated sensor/ETWI (all dimensions are in mm )...............................................................123
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xv LIST OF SYMBOLS a radius of the composite diaphragm b radius of the diffused resistive heater 'iH axial distance from reference axis of the composite diaphragm $ angular frequency V dc voltage (V) V ( ac voltage (V) ,arctaperRR total resistance of the arc and tapered piezoresistors under zero load conditions (") ,arctaperRR )) change in resistance of the arc and tapered piezoresistors (") ij E elastic moduli ( N/m2) ijQ elements of the material stiffness matrix ( N/m2) * Poissons ratio + degree of anisotropy k thermal conductivity ( W/mK ) pk C, -# thermal diffusivity ( m2/sec ) 2. axisymmetric Laplacian operator m / radial-direction spatial eigenvalues ( m -1) nm 0 temporal eigenvalues ( s -1) og internal heat source ( W/m3) 1,oJJ Bessel function of first kind o I modified Bessel function of the first kind oN initial in-plane compressive load ( N/m )
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xvi , T rN 1 radial and circumferential thermal force ( N/m ) , T r M 1 radial and circumferential thermal moment ( N ) A areal-density of the composite diaphragm ( kg/m2) ij A elements of the extensional-stiffness matrix ( N/m ) ij B elements of the flexural -extensional coupling matrix ( N ) ijD elements of the flexural-stiffness matrix ( N m ) 2 coefficient of thermal expansion (oC -1) 3 delta function i elements of the resistivity tensor ("'m ) ij % elements of the piezoresistive tensor ( Pa -1) ,r 1 4 radial and circumferential stress components ( N/m2) ,r 1 / radial and circumferential strain components , r 1 5 radial and circumferential curvatures ( m -1) 67,,,, Grztrz 8 (( Greens function ,rz(( location of the source ( m) , rz location of the observer (m ) ou radial displacement of the reference plane ( m ) w transverse displacement of the plate ( m )
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xvii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMOELASTICALLY ACTUATED ACOU STIC PROXIMITY SENSOR WITH INTEGRATED ELECTRICAL THRO UGH-WAFER INTERCONNECTS By Venkataraman Chandrasekaran May 2004 Chair: Mark Sheplak Major Department: Mechanic al and Aerospace Engineering The development of micromachined acoustic proximity sensors for real-time cavity monitoring of underwater highspeed supercavitating vehicles is presented. Lowresistance polysilicon-based electrical thr ough-wafer interconnects have been integrated with the sensor/actuator to enable backside contacts for drive and sense circuitry. The sensor and interconnects were fabricated in a complementary metal-oxide-semiconductor compatible process using deep re active ion etching, producing a 1 mm -diameter, variable thickness (5-10 !m ) composite diaphragm and 20 !m -diameter high-aspect ratio throughwafer vias on a silicon-on-insul ator wafer. The diaphragm in corporates a central resistive heater for thermoelastic actuation and di ffused piezoresistors for sensing acoustic pressure perturbations. The polysilicon through-wafer interc onnects facilitate a rugged bump-bonded sensor package suitable fo r the harsh sea-water application. A coupled thermomechanical model for thermoelastic actuation of circular composite diaphragms has been developed to optimize the sound-radi ation characteristics
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xviii of the acoustic transducer. Specifically, a cl osed-form solution for the 2-D, axisymmetric temperature distribution in the composite di aphragm due to dynamic Joule heating of the diffused central heater was obtained using Greens functi ons and Hankel transforms. Next, a closed-form solution for the thermoel astically-forced vibra tion of the composite diaphragm was obtained usi ng Kirchoffs plate theory. Electrical, mechanical and acoustic char acterization of the device indicates a transmitter source level of 50 dB (ref 20 !Pa ) at an operating frequency of 60 kHz , a receiving sensitivity of 0.98 !V/(V Pa) , a flat frequency response over the measured range of 1-20 kHz , a linear response from 60-140 dB , negligible leakage current for the junction-isolated diffuse d piezoresistors (< 14 pA at -10 V ), low interconnect resistance of 14 "9'and a minimum detectable signal of 36.5 dB for a 1 Hz bin centered at 60 kHz , at a bias of 9 V .
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1 1 CHAPTER 1 INTRODUCTION The goal of this research effort is to develop a microelectromechanical systems (MEMS)-based measurement system to mon itor the state and thickness of the gaseous cavity surrounding an underwater high-speed, supercavitating vehicl e (HSSV). The high velocity of the HSSV is enabled by supe rcavitation, a phenomenon in which a gaseous cavity envelops the majority of the vehicle, thereby considerably reducing the wetted friction drag (Lecoffre 1999). However, to ensu re the stability of the gaseous cavity as the vehicle maneuvers through the water requi res knowledge of the cavity thickness at all times. Thus a real-time measurement of th e overall cavity thickness is required for vehicle guidance and control. The development of a cavity-sensing technology represents a significant challenge due to th e complex nature of supercavitating flows (Senocak 2002) and the associated harsh sea-wa ter environment. This study is part of a larger effort to combine multi-domain de sign methods, novel MEMS structures and advanced digital signal processing (DSP) t echniques to develop a distributed acoustic based cavity-monitoring system to provide feedback information to an active cavity flow control system. This chapter presents an overview of supercavitation and the motivation for developing a cavity monitoring system using MEMS-based acoustic transducers. Cavity Physics As flow is accelerated over the body of a high-speed underwater vehicle the local pressure drop is inversely propor tional to the square of the velocity (Batchelor 1967). At very high velocities, as the pressure continue s to decrease, a point is reached where the
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2 pressure in the flow becomes less than the vapor pressure of the fluid. This causes a phase change in the fluid resulting in th e formation of gas or gas/vapor cavities (Batchelor 1967). The shape of the cavity is determined by several factors including the body creating it, the cavity pressu re and gravitational force (B atchelor 1967). Cavitation causes severe damage to hydrodynamic structures such as pitting of turbine blades, but it also has beneficial applications such as drag reduction in hydrofoil boats and supercavitating vehicles. S upercavitation is an extreme version of the cavitation phenomenon characterized by the formation of a single bubble e nveloping the moving object either partially or completely. Typically, at velocities of over 50 m/s , blunt-nosed cavitators and tip-mounted gasinjection systems produce these low-density gas pockets, encapsulating the vehicle and thus greatly reducing wetted friction drag (Ashley 2001). Cavitating flows in most engineering sy stems are turbulent. The associated dynamics of the cavity interface is governed by the complex interactions between the liquid and gas phases. Senocak (2002) pr ovides a computational model for qualitative and quantitative prediction of the features of turbulent cavitating flows along with a comprehensive review of the previous work in the field. A simplified schematic of the gaseous cavity surrounding the HSSV is shown in Figure 1-1 and serves merely as a guideline for the sensor development. The interface can be roughly classified into three regimes, rangi ng from a smooth/well-defined structure, typically close to the cavitator (Figure 1-1: Region 1) , to an unstable wavy profile resulting from the high-density gradie nts in the gas/water interface (Figure 1-1: Region 2) and ultimately degrading to a complex, multiple-bubbl e, two-phase flow (Figure 1-1: Region 3).
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3 Gas/water interface Reflected waves Incident waves Hull Underwater vehicle Smooth well-defined interface (Region 1) Complex two-phase flows (Region 3) Onset of instability waves (Region 2) U Water Gas Figure 1-1: Schematic of a bump-bonded sensor package placed on the hull of an underwater supercavitating ve hicle for monitoring the state and thickness of the gas/water interface. Cavity Sensing Issues There are several ways of measuring the in terface distance, all of which involve the radiation of energy from the hull towards the gas/water interface, the subsequent reflection of the incident wave at the interface, and the det ection of the reflected wave (Figure 1-1). The unstable nature and poor electromagnetic (EM) reflection coefficient (< 2 %) of the gas/water interface are some of the disadvantages of using laser or EMbased techniques (Antonelli et al. 1999). In contrast, acoustic techniques possess good signal-to-noise characteristics due to the sound hard nature of the interface with a unity reflection coefficient. In addition, as th e cavity structure begins to degrade and the interface possesses waviness or bubble regions , the specular reflection assumption is
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4 strongly dependent on the interface r oughness relative to the wavelength (0) of the radiated energy. If the interface roughness is larger than a quarter wavelength of the radiated energy ( l > 0/4 ), the incident radiation will be scattered at the wavy interface and may not be redirected to th e fixed receiver location (Tol stoy and Clay 1987). In these regions, acoustic-based techniques provide an advantage because it is possible to fabricate compact devices that radiate acous tic energy with wavelengths on the order of several millimeters (Brand et al. 1997). Furthermore, laser-based techniques may not be able to differentiate between a wavy inte rface and multiple-bubble regions. Acoustic techniques may be able to distinguish the two regions due to re sonant scattering of acoustic waves from bubble surfaces (Williams 1999). Since the scattering cross-sections and resonant frequencies are a function of the bubble size, information regarding the presence and size of bubbles may be provided to the flow control system. Thus acoustic techniques present a potentially efficient wa y of measuring the interface proximity as well as discerning the nature of the interface. However, a disadvantage of the technique is that the acoustic velocity is a function of temperature and the medium of propagation (Blackstock 2000), both of which may vary within the gaseous cavity. From a reliability perspective, the measur ement system must be able to withstand the harsh environment associated with supercav itating flows. This necessitates a sensor possessing a hydrophobic barrier and backside el ectrical connections to drive and sense circuitry for increased robustness and pr otection against the corrosive sea-water environment. The sensor system would have to be a distributed network to provide feedback information about the entire cylin drical cavity structur e and would have to provide fast parameter updates (> 50 Hz ) due to the continuously changing interface
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5 conditions. This ideally requires multiple sens ors with matched characteristics for faster and more efficient signal processing. Th ese requirements may be addressed by MEMSbased transducers that offer se veral potential advantages ov er conventional technology in terms of system performance and integration. In particular, the ba tch fabrication process employed for MEMS-based transducers yields devices with matched characteristics due to identical process conditions (Madou 1997). Scope The goal of this research effort is to de sign, fabricate, and ch aracterize a prototype MEMS-based acoustic proximity sensor for monitoring the state and thickness of the cavity surrounding the HSSV. The system should be capab le of measuring cavity thickness ranging from 1 to 10 cm , provide fast parameter updates (>50 Hz ), and be capable of operating in a harsh sea-water enviro nment. The contributions of this effort in the areas of transducer design, fabric ation and packaging are listed below. Research Contributions : Development of a MEMS-based acoustic pr oximity sensor capable of generating and detecting acoustic waves with wavelengths on the order of several millimeters. : Development of an electr ical through-wafer interc onnect (ETWI) technology, and its integration with the se nsor/actuator to enable a rugged bump-bonded transducerpackaging scheme. : Development of a novel analytical mode l for the thermoelastic actuation of composite diaphragms to optimize the sound radiation characteristics of the transducer. Dissertation Organization This dissertation is organized into 7 ch apters. Chapter 1 provides the background, motivation and objectives of th e research effort. Chapter 2 provides a review of the published work on micromachined acoustic reso nators. Chapter 3 describes the firstgeneration acoustic proximity sensor devel oped for the cavity monitoring application
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6 with details of the device fabrication process. Chapter 4 presents the theoretical development of the thermoelastic actuati on and the piezoresistive detection schemes employed for the generation and detection of acoustic waves. Chapter 5 provides a description of the experiments used to char acterize the sensor/actuator, and Chapter 6 presents the results of the characterization. Chapter 7 offers concluding remarks and scope for future work in areas of sensor modeling and fabrication for performance enhancement of the second-generation proximity sensor.
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7 7 CHAPTER 2 LITERATURE REVIEW This chapter provides a review of the published work on micromachined ultrasonic resonators based on both conservative and non-conservative transduction schemes and their potential advantages ove r conventional transducers. Conventional Ultrasonic Transducers Acoustic proximity sensors are used fo r a wide range of non-contact distance measurement applications (Massa 1999). Conventional transducer s use the pulse-echo technique to measure the distan ce to a target. In this met hod, the transducer is excited by a burst of pulses, which are transmitted toward s the target and are subsequently reflected. The time of arrival of the reflected pulse is then estimated via schemes like threshold detection or sliding window techniques (Barshan 2000). Th e time-delay together with the knowledge of the acoustic velocity in the medium provides an estimate of the distance. Several tr ansduction schemes have been used for the generation of acoustic waves at ultrasonic frequencies, but for indus trial applications, ci rcular piezoceramic transducers (e.g., lead zirconium titanate PZT) vibrating in the thickness mode are most commonly used (Manthey et al. 1992). Thes e resonant acoustic tr ansducers exhibit the behavior of an underdamped second-order system , in which the response to a sharp pulse input is characterized by a ring , or in other words, there is finite decay time for the transmitter vibrations. Thus, for a single tr ansmitter/receiver this creates a dead-zone in the vicinity of the transmitting surface where objects cannot be detected, creating a lower bound on the measurable distance (Manth ey et al. 1992). Usually, an additional
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8 quarter-wave matching layer made of a lo w-density polymer is deposited over the transmitting surface for better impedance ma tching with a gas medium and improved radiation efficiency (Mockl et al. 1990). The matching layer results in higher sound pressure level due to larger vibration am plitude and the higher damping of the lowdensity polymer reduces the transmitter d ecay time improving the distance resolution of the transducer over a limited frequency range. MEMS-based proximity sensors have been fabricated by combining standard semiconductor processing techniques with bulk and surface micromachining (Madou 1997) to produce thin, compliant diaphragms on a silicon substrate. MEMS-based transducers offer several potential advantag es over their conventional counterparts, including lower impedance and a wider ba ndwidth of operation. Micromachined transducers utilizing a thin, compliant diaphr agm as the acoustic source provide a better impedance match with air compared to solid piezoceramic elements. Their small size translates to better spatial and temporal resolution and makes them effective for measuring very small distances. The batch fabrication process yi elds sensors with matched characteristics, which would be invalu able for a distributed network of sensors, and significantly lower cost per device (Ar nold et al. 2002). Th e technology also lends itself to a compact flip-chip type sensor p ackaging scheme with backside contacts for drive and sense electronics making it rugged and more suitable for the harsh sea-water environment (Al Sarawi et al. 1998, Heschel et al. 1998). Several different actuation and sensing mechanisms have been reported in the literature for the generation as well as detection of acoustic waves utilizing micromachined diaphragms as the acoustic source and sensor. Reciprocal actuation
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9 mechanisms (piezoelectric, capacitive) are based on conservative power exchange and energy storage and can also be used for sensing. Irreversible schemes such as electroand opto-thermal actuation are based on ener gy dissipation and are usually used in conjunction with irreversible detection schemes, such as piezoresistive detection and others based on optical met hods or tunneling currents (H ornung and Brand 1999). The type of sensor and operating principle used will depend to a large extent on the end application, and is based on factors such as measurement environment, frequency range of interest, achievable actuation force, dete ction sensitivity and si gnal-to-noise ratio. System issues such as power consumption a nd integration with support electronics also play an important role, in addition to manufacturing cost and complexity. A comprehensive review of sili con resonant sensors has been presented by Stemme (1991) and Brand and Baltes (1997). Micromachined Ultrasonic Transducers The following section presents an outline of the various silicon micromachined acoustic transducers reported in the literature. Brand et al. (1993, 1994) and Hornung et al. (1997, 1998) have developed transducers operating around 70-90 kHz . This is a released diaphragm-type transducer created using potassium hydroxide (KOH) etchi ng of the bulk silicon substrate with an electrochemical etch stop to define the diaphragm thickness (Figure 2-1). n Silicon nitride (100) Silicon Figure 2-1: Cross-sectional schematic of a 1 mm x 1 mm thermally actuated membrane resonator (adapted fr om Brand et al. 1997). Thermal oxide p-silicon heating resistor Epi contact p-Silicon piezoresistor n-epitaxial silicon Passivation
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10 The diaphragm is set into vibration via electro -thermal heating of an integrated heater on the diaphragm. The heating establishes a temperature gradient across the layered composite diaphragm causing it to deflect. Th e diaphragm vibration is detected using piezoresistors placed at the edges. The se nsors were operated at their fundamental resonant frequency maintained by a positive feedback circuit. For an average heating power of 100 mW , maximum vibration amplitudes of 300-400 nm and sound pressure levels of 81.9 dB (at a distance of 50 mm ) were obtained. The piezoresistors are arranged in a Wheatstone bridge configuration with a detection se nsitivity of 0.4 !V/V mPa at resonance. Jin et al. (1998, 1999) reported the fabr ication and characterization of surface micromachined capacitive ultrasonic transducer s (cMUT) for use in the megahertz range. The operating principle is based on electrost atic excitation and capacitive detection and the devices were fabricated on a single wafer using sacrificial layer techniques. A typical transducer measures 1.75 mm x 1.75 mm and is formed from an array of 50 x 50 capacitive cells. Each cell consists of a 30 !m -diameter and 0.53 !m -thick diaphragm, forming a parallel plate capacitor with a sealed cavity (Figure 2-2). Silicon Top electrode Oxide/LTO Aluminum Silicon nitride Figure 2-2: Cross-section of a capacitive mi cromachined ultrasonic transducer element (adapted from Jin et al. 1998). Maximum diaphragm displacement of 230 Å/V was reported. Different types of membranes (silicon nitride and polysilicon) as well as vacuum seal ing techniques were Sealed cavity
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11 11 investigated. Vacuum sealing of the cavity reduces the effect s of cavity stiffening (Rossi 1988) associated with operation in air. The di sadvantage of electrostatic actuators is that they require very narrow gaps between the electrodes in order to produce large excitation forces, adding to the fabrication complexity of the device. In addition, during underwater operation, the pressure of the wa ter can cause a static deflec tion of the diaphragm which would be difficult to compensate with a sealed cavity and a narrow gap between electrodes. Figure 2-3: Cross-section of a ferroelectric transducer (ada pted from Bernstein et al. 1997). Bernstein et al. (1996, 1997, 1999, 2000) have designed diaphragm-based ferroelectric ultrasound transducers for highfrequency imaging applications (up to 2 MHz ). Ferroelectric monomorph sonar trans ducers with diaphragm sizes ranging from 0.2 mm to 2 mm were built using sol-gel PZT on micromachined silicon wafers. When the PZT extends due to an applied voltage, th e diaphragm is forced into vibration. And similarly, any deflection of the diaphragm cau sed by acoustic waves creates a voltage in the PZT. A cross-sectional schematic of th e completed transducer is shown in Figure 23. A modified sol-gel process yiel ded crack-free PZT films up to 12 m in thickness, improving the sensitivity of cer tain classes of sensors and producing larger output forces Substrate contact Monomorph contact Top Ti/Pt Polyimide PZT Ti/P t SiO2 p+ silicon Etched cavity 1.4 mm n-type Si
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12 for actuators. The measured sensitivity for a 260 !m x 260 !m device without correcting for the effects of stray capacitance was reported to be 1.4 !V/Pa . Sleva et al. (1994, 1996) have developed se nsors fabricated by spin casting P(VDFTrFE) onto a silicon membrane which is backf illed with epoxy (Figure 2-4). Transducers measuring 2 mm in diameter and operating at 31 MHz were fabricated. The epoxy provides for a soft boundary condition on the back side of the piezo element and yields a wide-band, half-wavelength resonant devi ce suitable for pulse-echo applications. Electrode PVDF-TrFE Dielectric Epoxy pSilicon p+ Silicon Figure 2-4: Cross-sectional schematic of the P(VDF-TrFE)/silicon-based transducer (adapted from Sleva et al. 1996). However, the use of P(VDF-TrFE) material resulted in a transducer with very low sensitivity. A dynamic range of 40 dB and an insertion loss of 48 dB were reported. Lynnworth et al. (1997) desc ribe the use of solid piezoceramic elements for aircoupled transducers in the 50 kHz to 500 kHz range. Solid piezoceramics are good because they are robust and can handle temperatures from -40 °C to +125 °C . A quarterwave impedance matching layer is needed to make these transducers suitable for applications in air. One drawback, however, is that similar to conventional ultrasonic proximity sensors, these transducers are high ly underdamped, which limits the use of the pulse-echo proximity sensing scheme for measuring small distances. For the cavity monitoring application, pi ezoelectric and electrostatic transducers were not chosen because of their high impedance that makes integration with ETWI more
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13 difficult due to the effects of parasitic capaci tance (Scheeper et al. 1994). Thermoelastic actuation and piezoresistive se nsing provide the best compromise among performance, low cost, durability, ease of fabrication a nd simpler interface el ectronics (Hornung and Brand 1999). Since the device relies on a resistive element fo r both sensing and actuation, it inherently possesses lower impedance compared to capacitive or piezoelectric schemes. Thermoelastic act uators also scale favorably with microminiaturization. The amount of thermal mass (inertia) decreases w ith the volume of the structure, which directly translates to a fast er response time. Vibration of micromachined structures (beams, diaphragms) using electrothermal excitation have been demonstrated up to frequencies in the megahertz range (L ammerink et al. 1992, Brand et al. 1997). Secondly, the percentage of thermal energy lost via conduction into the bulk substrate reduces with decreasing thickness of the struct ures, while maintaining a high force per unit area. Thus, the low impedance of the de vices coupled with favorable scaling of the achievable actuation forces and dissipation losses with miniaturization makes them a suitable choice for the cavity monitoring application.
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14 14 CHAPTER 3 ACOUSTIC PROXIMITY SENSOR This chapter presents a detailed desc ription of the first-generation acoustic proximity sensor. Details of the integr ated acoustic transducer/ETWI structure, transduction schemes used for the generation and detection of acoustic waves and device fabrication process are described. The transducer consists of a thin single crystal silicon membrane as the acoustic source/sensor for improved impedance matchi ng with the gas medium . The transceiver structure integrates diffused resistors for thermoelastic actuation and piezoresistive detection and boron-doped polys ilicon electrical through-wafe r interconnects (ETWI) for backside contacts. Figure 3-1 shows a top vi ew optical image of th e sensor/actuator, and a cross-sectional schematic is shown in Figur e 3-2. The device structure consists of a 1 mm -diameter, variable thickness (5-10 !m ) circular diaphragm, created using a deep reactive ion etch (DRIE) back end process. This process combined with a silicon-oninsulator (SOI) wafer as the substrate allows for strict geometry control in terms of thickness and diameter of the diaphragm. Two semicircular diffused heaters (60 !m diameter and 0.46 !m -junction depth) are located in the center for thermoelastic actuation of the diaphragm. Four diffused p-type pi ezoresistors are located at the edge of the diaphragm for piezoresis tive detection of the tr ansverse vibrations.
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15 Figure 3-1: Top view micros copic (diffraction interference contrast) image of the acoustic proximity sensor with inte grated through-waf er interconnects. Heater/Actuator Piezoresistors Aluminum Polysilicon through-wafer interconnect Silicon dioxide Silicon nitride passivationSOI Aluminum Figure 3-2: Cross-sectional schemati c of the acoustic proximity sensor. The use of single-crystal silicon provides maximum sensitivity and a lower noise floor compared to polycrystallin e devices (Brysek et al. 1988). In the case of the tapered piezoresistors, two resistors ar e connected in series via a current turn-around as shown in Figure 3-1 (Sheplak et al. 1998). The di mensions of the turnaround are designed such that its contribution to the total resistance is negligible (< 3 %). This arrangement eliminates additional metal lines on the diaphragm. Arc resistor Tapered resistor Heater ETWI Al/Si (1%) Diaphragm (1 mm x 5-10 !m ) Current turn-around
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16 The intended application of the sensor necessitates a rugged package capable of withstanding the harsh seawater environm ent. Therefore, polysilicon-based ETWI measuring 20 !m in diameter have been integr ated with the sensor/actuator (Chandrasekaran et al. 2001, Chow et al. 2002 ) as shown in Figure 3-2. The use of ptype (boron-doped) ETWI enables direct ohmic contact to p-type piezoresistors employed for the acoustic transducer (Pie rret 1996). The ETWI thus enables an integrated bumpbonded sensor packaging scheme with the dr ive and sense circuitry hidden from the harsh cavity environment (Figure 3-3). SOI ASIC Figure 3-3: Traditional wirebonded sensor package versus the more compact and rugged bump-bonded package. The thermoelastic heater and piezoresist ors are connected to the ETWI via 75 !m wide, 1.4 !m -thick aluminum (1% Si) traces. A 7000 Å -thick silicon dioxide film is thermally grown on top of the silicon laye r to provide compressive stress to the diaphragm in addition to serving as a dielectr ic passivation for the piezoresistors. The inplane compressive loading provided by the ther mal oxide layer together with the aspect ratio of the diaphragm may be used to optimize the device sensitivity. A 3500 Å lowstress, plasma enhanced chemical vapor de posited (PECVD) silic on nitride passivation layer provides a protective moisture barrier on the top surface of the device. In the ASIC SOI Wire bond ETWI Bump bond
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17 following sections, the transduction schemes used for the generation and detection of acoustic waves as well as the device fabrication process are discussed. Operating Principles Thermoelastic Actuation Dynamic Joule heating of the diffused centr al resistors creates a time-varying twodimensional temperature distribution acr oss the diaphragm. The non-uniform temperature profile through the thickness of the diaphragm generates integrated thermal forces and moments resulting in the transverse vibration of the diaphragm. The flexuralextensional coupling is furthe r enhanced by the asymmetrical composite structure of the diaphragm. Thus, by applying a time-vary ing instantaneous voltage signal, the diaphragm is forced into vibration. The te mperature distribution in the diaphragm is a function of the driving frequency of the signal and depends on the dept h of penetration of the thermal waves, governed by Fouriers law of heat conduction (Ozisik 1993). The harmonic Joule heating excites the diaphragm at the driving frequency, $ and additionally at twice that frequency, 2$, and at dc. This is due to the non-linear nature of thermoelastic actuation, where the input power has a quadratic dependence on the excitation voltage, causing the power to be re distributed into two other frequency bins (i.e., 0 and 2$). By adding a conditioning dc voltage the ratio of power between the three frequency bins can be controlled as shown by the following equation: 67 67 2 2 2sin1cos2sin. 2 V VVtVtVVt $ $$( (( ;#;<;2 (3.1) However, adding an offset voltage to the harmonic signal adds a static temperature distribution, which consequen tly changes the in-plane st ress field and the resonant frequency of the diaphragm.
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18 The heater is designed with a shallow junction depth of 0.46 !m (5-10 % of the diaphragm thickness) in order to create a temperature gradient across the diaphragm cross-section at the given operating frequencies (< 100 kHz ). Additionally, the circular shape and central placem ent of the heating resistor pair ensure that the fundamental mode of diaphragm is excited and maximum vibration amplitude is achieved (Lammerink et al. 1990). The theoretical development of the thermoelastic actuation scheme using an analytical thermomechanical model will be discussed in Chapter 4. Piezoresistive Detection Piezoresistivity is defined as the change in the resistivity of a material due to a change in the mobility (or number of charge carriers) induced by a mechanical strain to the material (Smith 1954). For an anisotro pic solid, the resistance modulation is a function of the applied stress and the piezo resistive coefficients of the material. Electromechanical transduction of the diaphr agm vibration is thus achieved via four diffused silicon piezoresistors located at th e edge of the diaphr agm in a fully active Wheatstone bridge configurati on. This configuration provi des a linear relation between the bridge voltage output and the input pr essure perturbation pr ovided that the mean resistances in all four legs are equal and that the resistance modulati on in each resistor of a given leg is equal in magnitude, but of opposite signs. For the first-generation transducer the arrangement cons ists of two tapered and two arc-shaped piezoresistors as shown in Figure 3-4. The choice of arc and ta pered resistors is dictated by the circular geometry of the diaphragm (Sheplak et al. 1998). The piezoresistors are designed such that they possess the same nominal resistance ()arctaperRR # and equal but opposite resistance modulation ()arctaperRR)#<). Thus for an undeflected diaphragm, the output
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19 of the Wheatstone bridge is ideally zero, but a deflection caused by an acoustic wave produces a differential voltage output across the bridge. Figure 3-4: Schematic of the arc and tapered piezoresistors arranged in a Wheatstone bridge configuration. The theoretical development of the piezoresistive dete ction scheme for sensing acoustic pressure perturbations is presented in Chapter 4. The remainder of this chapter describes the processing techni ques used for the fabrication of the integrated acoustic transducer/ETWI. Device Fabrication The fabrication process begins with the creation of the elec trical through-wafer interconnects. After the wafer with ETWI is planarized, the ET WI wafer provides the substrate for the subsequent backend co mplementary metal-oxide-semiconductor (CMOS) process to fabricate the acoustic pr oximity sensor. The interconnects were fabricated using a variation of the polysi licon-based ETWI tec hnology demonstrated by Chow et al. (2002) for easier integration with the p-type piezoresistors used for the acoustic transducer. A detailed process traveler is provided in Appendix A. Tapered resistor Arc resistor Aluminum Substrate contact
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20 Electrical Through-Wafer Interconnect In the previous chapter, the need for electrical through-wa fer interconnects to enable a rugged flip-chip type bump-bonde d transducer package was described. Several different technologies have been employed for creating the ETWI. However, most of these techniques are incompatible w ith standard CMOS processing. Anisotropic wet chemical etching is a commonly used t echnique for fabricati ng through-wafer vias, but this technique results in large chip sizes and impedes the subsequent processing with many etching, metallization and lithography e quipment (Goldberg et al. 1994, Linder et al. 1994, Christensen et al. 1996). Copper electroplating methods have also been combined with DRIE to form ETWI for RF a pplications (Chow et al. 1998, Soh et al. 1999, Wu et al. 2000). DRIE has also been combined with chemical vapor deposited (CVD) tungsten for cantilever ar ray applications (Chow et al . 2000). Metal-filled DRIEetched vias permit a smaller ch ip size, relative to the wet-etched contacts, but the metals used in the interconnects are not always compatible with high temperature processing, thus limiting the commercial use of these techniques. The technology to create high density, polys ilicon-based electrical through-wafer interconnects using DRIE was demonstrated by Chow et al. (2002). Using this technique, 20 !m -diameter vias with an aspect ratio of ~25:1 were achieved. Phosphorus-doped polysilicon forms the conducting layer, which is isolated from the substrate with a 2 !m thick thermal oxide layer. This techni que is CMOS compatible and possesses the following advantages: ETWI fabrication prior to device fabrication, the capability to withstand subsequent high temperat ure thin-film deposition, and low resistance/capacitance suitable for most MEMS applications.
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21 The ETWI fabrication process begins with a 450 !m -thick double side polished ntype SOI wafer as illustrated in Figure 3-5 (A). A 2 !m thermally grown silicon dioxide layer acts as the mask for the through-wafer vi a etch (B). In order to maintain a high aspect ratio (25:1) structure, the vias were created by et ching through both sides of the wafer. A front-to-back ali gner was used to create the a ligned two-sided pattern on the wafer and additionally an infra-red camera was used to inspect the alignment. A slight misalignment would result in a reduction of the via diameter, causing it to prematurely plug during the deposition of the polysilicon thin films. DRIE was then performed from the front and the back-side of the wafer fo r roughly equal durations to etch the via through the wafer (Figure 3-5 (B)). n-Si SiO2 (2 !m ) (DRIE mask) (B) SiO2 (0.4 !m ) (A) n-Si n-Si (10 ! m ) Figure 3-5: A schematic of the fabrication process illustrati ng the creation of the throughwafer vias. The etcher uses the Bosch process alternati ng between etching and passivation to create a high-aspect ratio structure. Once the vias were etched completely through the wafer, helium used for cooling flowed through the vias slowing the etch. By monitoring the helium flow, the etch progress was estimated. A small degree of lateral etching resulted in a tapered via profile illustrated in Fi gure 3-6. A timed (30 minutes) overetch was
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22 performed from the frontside of the wafer usi ng an etch recipe with higher passivation to smooth and straighten the via profile. A support wafer was used during the overetch step to prevent helium flow from the wafer chuc k into the vias. After the etching was complete, the oxide mask was stripped using buffered oxide etch (6:1 BOE). 150 ! m Figure 3-6: SEM cross-section after plasma etching vias th rough the wafer (Chow et al. 2002). The etching of the via was followed by confor mal thin film growth/deposition. To electrically isolate the interconnects from the bulk silicon substrate, a 2 !m thermal oxide layer was grown using a dry-we t-dry oxidation process (Figur e 3-7 (C)). This process was used to ensure a high quality silicon-to-silicon dioxide interface. SiO2 (2 !m ) (C) n-Si Figure 3-7: Fabrication sequenc e: Thermal oxide growth for electrical isolation of the interconnects from the silicon substrate. For electrical conduction, 2 !mthick polysilicon was deposited over the oxide using lowpressure chemical vapor deposition (LPC VD). This was followed by boron diffusion
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23 doping of the polysilicon for 2 hours at 1000 oC and then an anneal at 1000 º C to fully drive the dopants th rough the polysilicon (Figure 3-8 (D)). SiO2 (2 !m ) (D) p++ Polysilicon (2 !m ) Figure 3-8: Fabrication seque nce: Deposition and boron di ffusion doping of polysilicon layers for electrical conduction. The boron-doped polysilicon conduc tion layers enable direct ohmic interconnection to ptype piezoresistors employed for the acousti c transducer. However, the boron diffusion doping of polysilicon forms a non-conducting bo rosilicate glass, which is not easily removed using hydrofluoric acid. This is a critical difference between boron (p-type) and phosphorus (n-type) diffusion doping of polys ilicon. The glass resulting from phosphorus doping of polysilicon is easily etched using hydrofl uoric acid. However, in the case of polysilicon doping using boron, the glass must first be oxidized using wet oxidation at 1100 oC for 30 minutes, followed by 60 minutes in (6:1) BOE to etch the oxidized glass. The process of polysi licon deposition and boron diffusion doping was repeated two or three times to achieve a low resistance ETWI. The rest of the via was filled with polysilicon followed by boron di ffusion doping and patterning of the top and bottom surfaces of the wafer (F igure 3-9 (E), Figure 3-10). p++ Polysilicon SiO2 (2 !m ) (E) Figure 3-9: Fabricati on sequence: Patterning the interconnects.
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24 Figure 3-10: SEM illustrating the oxide insula tion and the polysilicon conduction layers of the ETWI. After patterning the ETWI, an isotropic plasma etch was performed on the doped polysilicon to planarize the interconnects and to produce a gradually sloping sidewall. This is critical for the integration of the ETWI with the sensor/actuator since it ensures proper metal coverage over the ETWI (Figure 3-11). Figure 3-11: Through-wafer interconnect integrated with a metal bond pad. Acoustic Sensor/Actuator The SOI wafer with the patterned ETWI se rved as the substrate for the acoustic transducer fabrication process. A thin layer of oxide was grown and patterned to serve as a hard mask for the resistor implanta tions. Boron was implanted (energy: 170 keV , dose: 1E13 cm-2) to achieve p++ regions with a concentration of 1E20 cm-3. After implantation, the wafers were annealed at 1100 oC for 30 minutes to create a junction depth of 0.46 !m . Silicon dioxide Polysilicon Silicon Via Boron-doped polysilicon Aluminum 100 ! m
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25 The wafers were then patterned with the pi ezoresistor mask and again implanted with boron (energy: 85 keV , dose: 8E13 -2cm ) to form p++ regions of similar concentration (Figure 3-12 (F)). After st ripping the oxide mask, a 7000 Å -thick silicon dioxide layer was thermally grown at 950 oC to passivate the resistors a nd also to provide compressive stresses to the diaphragm in order to achiev e the required sensitivity (Figure 3-12 (F)). The diffusion of the resistive heater and piezoresistors was simulated using FLOOPS® (Law and Cea 1998) (Appendix B) to account for the entire thermal budget of the process flow. SiO2 (7000 A ) p++ Silicon (F) Figure 3-12: Fabrication seque nce: Resistor implantation and thermal oxide growth. A 1.4 !mthick layer of aluminum with 1% silicon (to avoid spiking (Pierret 1996)) was sputtered and patterned , once the contact cuts in the oxi de dielectric layer were made (Figure 3-13 (G)). SOISixNy (3400 A ) (H) Aluminum (1.4 !m ) (G) Figure 3-13: Fabrication sequenc e: Sputtering and patterning aluminum (1%-Si), silicon nitride deposition and DRIE to release the diaphragm.
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26 Low-stress nitride (3500 Å ) was then deposited using pl asma-enhanced chemical vapor deposition to form a protective moisture barrie r (Figure 3-13 (H)). The wafers were then patterned on the backside with front-to-back alignment to create the diaphragm. The relative alignment of the pi ezoresistors and the diaphragm is critical to ensure piezoresistor placement at the edge of the diaphragm. DRIE was performed from the backside of the wafer up to the buried oxide layer (Figure 3-13 (H )) and the buried oxide layer was removed using 6:1 BOE. A schema tic of the integrated acoustic sensor/ETWI is illustrated in Figure 3-14. Figure 3-14: 3-D schematic of the integrated acoustic sensor/ETWI. Pol y silicon conduction la y e r Oxide insulation Aluminum Silicon substrate Piezoresistors ETWI Diaphragm
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27 27 CHAPTER 4 DEVICE MODELING The design of an acoustic proximity se nsor in terms of sound radiation characteristics as well as sensitivity to acous tic pressure perturbati on requires a thorough understanding of the overall system behavior . In particular, th e influence of design parameters such as geometry, material prope rties and fabrication-i nduced stresses on the forced vibration characteristic s of the resonating diaphragm needs to be investigated. This chapter presents the theoretical devel opment of the transduction schemes employed for the proximity sensor and the various de sign considerations for optimizing the device performance. The first section describes an analytical model for the thermoelastic actuation of circular composite diaphragms th at was developed to enable the optimization of the sound-radiation characte ristics of the acoustic transducer. The second section focuses on the optimization of the piezoresist ive-sensing scheme for maximum sensitivity to acoustic pressure pertur bation and a low noise floor. Thermoelastic Actuation The operating principle of the thermoelastic actuation scheme and its suitability to micromachined transducers were presented in Chapter 2. Due to a significant number of design parameters including diaphragm radius , thickness of the individual layers, heater geometry, input power, operating frequency a nd in-plane stress, design and optimization based solely on finite element analysis is not practical due to computational time constraints. This necessitates the developmen t of analytical models, since an analytical
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28 28 expression of the diaphragm deflection provi des scaling relations with the different design parameters, enabling quick identification of vibration characteristics over a range of input values. In addition, non-dimensiona l groups of the input parameters can be formed to reduce the number of dependencies, and the analytical model can be better integrated with formal optimiza tion tools (Papil a et al. 2003). In order to develop accurate analytical models applicable to MEMS-based thermal actuators, certain key aspects of the micromachined transducers namely multiple material layers, in-plane heat conduction and fabrication induced stresses have to be considered in the analysis. The composite nature of the vibrating structure is quite important since released cantilevers or diaphragms rarely consis t of a single material. For example, in the case of thermoelastic actuat ors (Brand et al. 1997, Chandr asekaran et al. 2002), in addition to the silicon structural layer, thin films of silicon dioxide and silicon nitride are used for dielectric passivation and to provide a hydrophobic moisture barrier. Piezoelectric actuators (Bernste in et al. 1997) are comprised of PZT and electrode layers deposited over the silicon diaphragm. The electrical, mechanical and thermal properties of the various layers can be significantly di fferent, altering the overall dynamic behavior of the transducer. Another important c onsideration, specifically for micromachined thermal actuators, is that the diffused resi stor typically covers a small area of the diaphragm and the diffusion of heat from the re sistor is both in the in-plane and thickness direction. As will be shown later in the chap ter, neglecting the eff ects of in-plane heat conduction will severely over-predict the vibrat ion amplitude of the diaphragm. Finally, the transducer fabrication pr ocess imparts significant in-pla ne stresses (compressive or tensile) that alter the vibration characteristics of the diaphragm.
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29 Thermally-induced vibration of beams and plates by a time-harmonic heat flux on one surface have been analyzed by severa l authors including Irie and Yamada (1978), Lammerink et al. (1992) and Pa ul and Baltes (1999). Ho wever, there are several limitations to the models proposed by these authors. The analys es by Irie and Yamada (1978) and Lammerink et al. ( 1992) are restricted to homoge nous structures. Paul and Baltes (1999) included varying ma terial properties in their pl ate analysis, however, their thermal analysis was simplified to a hom ogenous beam with a surface heat flux. Furthermore, the in-plane heat conduction e ffects of the actuating heater have not been considered in any of the analyses. In this section, a thermomechanical model that incorporates the composite nature of the diaphragm in both the thermal and mechanic al analysis is presented. The diffused heater used for actuation is more accurately re presented as an internal heat source rather than a surface flux and the anal ysis also includes the effect s of in-plane heat conduction from the edge of the central heater. 2-D Thermomechanical Model The analytical model consists of an ax isymmetric composite diaphragm comprised of three transversely isotropic layers with different thermal and elastic properties as shown in Figure 4-1. The assumption of tr ansversely isotropic elastic properties for silicon is a simplification based on its moderate degree of anisotropy (1.57) + #. For a cubic crystal, such as silicon, the degree of anisotropy is defined as 44 11122 , E EE+# < (4.1) where ijE are the independent elastic moduli. For an isotropic material + has a value of 1 (Brantley 1973). In addition, the angular variation of the el astic constants (< 30%) in
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30 30 the (100) plane has also been neglected in the plate analysis. An axisymmetric diaphragm is a sufficiently accurate representati on of the sensor since it is operated below its fundamental resonant fre quency. The layers are assume d to have isotropic thermal properties and in perfect therma l contact with continuity of temperature and heat flux across the interfaces. The edge of the dia phragm is assumed to be clamped and at ambient temperature. This temperat ure boundary condition is based on the high operating frequencies (60-80 kHz ) at which the diffusion length scale in silicon ( 30 m ) is significantly smaller than th e radius of the diaphragm (500 m ). The clamped boundary condition represents the released diap hragm with its circumference built-in to the silicon substrate. This is an approximation, however, and does not account for the finite compliance of the built-i n edge (Gerlach et al. 1996), wh ich can be incorporated in the model if required. SOI Figure 4-1: Axisymmetric model of the composite diaphragm consisting of three transversely isotropic layers and a diffused resistive heater (H1, H2, H3 and HHeat represent axial distances from the re ference plane. The piezoresistors are not represented in the analytical model). b a r H1 H2 H3 HHeat (2) (3) (1) z Diffused Heater (0,0)
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31 The resistive heater (Figure 4-1) is represente d by a time-harmonic internal source of heat generation 6767676710,,cos,HeatH b o Hgrztgrdrzdzt33$#>> (4.2) where go ( W/m3) is the magnitude of the heat source, $ is the angular frequency and 3 represents the Dirac delta function. The heater creates a dynamic two-dimensional temperature distribution across the diaphrag m. The non-uniform temperature profile through the thickness of the diaphragm genera tes a thermal force and a moment that results in the transverse vibration of the di aphragm. The flexural -extensional coupling is further enhanced by the composite structure of the diaphragm. Assuming an internal heat source that is an arbitrary function of space a llows for a more accurate representation of the exact heater geometry, incl uding the doping profile of the diffused resistor. This was not possible using previous models where th e heater was represented by a surface heat flux (Irie and Yamada 1978, Paul and Baltes 1999). In the following sections, the 2-D temperature distribution in the composite diaphragm is calculated using the Fourier h eat conduction model that is then used to derive the thermoelastic forcing functions . The plate governing equations are then formulated using the equations of moti on and the linear thermoelastic constitutive relations. Heat Conduction Model The governing equation for heat conduction in the ith layer of a multilayered composite diaphragm (Figure 4-1) based on Fouriers law is given by (Ozisik 1993) 67 67 6 7 ,, ,,,,,,,i 2 iii iTrzt 1 Trztgrzti123 kt,? @A .;## BC ? DE (4.3)
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32 where ,i and ki represent the thermal diffusivity and the thermal conductivity of the individual layers, 67,,igrzt represents an internal he at source within a layer and 2 . is the axisymmetric Laplacian operato r in cylindrical coordinates, .2 2 21 r rrrz ? ?? @A .#; BC ? ?? DE (4.4) Each layer of the composite must sati sfy two radial boundary conditions and two transverse boundary conditions. The diaphragm is assumed to be thermally insulated on the top and bottom surfaces, 1 z0T 0 z#? # ? and .33 zHT 0 z#? # ? (4.5) This is a simplification based on the low free-c onvective coefficient of air. Additionally, the heat conduction in the diaphragm is assu med to be symmetric about the center, and the edges are maintained at ambient temperature, i r0T 0 r#? # ? and .i raTT F ## (4.6) The layers are assumed to be in perfect th ermal contact denoted by the continuity of temperature and heat flux, ; at 12 12121TT TTkkzH zz ? ? # ## ?? (4.7) and 3 2 23232; at=. T T TTkkzH zz ? ? ## ?? (4.8) The solution to the two-dimensional, transi ent heat conduction pr oblem described above is obtained using Greens functions and Hankel transforms (details of the derivation are provided in Appendix C) 67 6767 67 67 67 67 111 222 12 ,,ReheatH jt omi iom mn mmnmnm HJrZz be TrztgJbZzdz aJaNj$/ / / /00$@A @A (( # BC BC BC ; DE DEGG > (4.9)
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33 where a and b are the radius of the diaphragm and the heater respectively, H1 and HHeat are axial distances from the reference plan e to the top and the bottom of the heater, Jo and J1 are Bessel functions of the first kind, $ is the frequency of the time-harmonic heat source, 67 22 22sincos,nmnm iimim ii Z zAzBz00 // ,,@A@A #<;< BCBC BCBC DEDE (4.10) nm 0 and m / are eigenvalues, and A , B and 6 7nmN0 are constants obtained from the boundary and interface continuity conditions. Once the unsteady temperature field is known, the integrated thermal forces and mo ments can be computed. In the following sections, the transverse vibration of the composite diaphragm resulting from the timevarying two-dimensional temperat ure distribution is derived. Plate Equations The following analysis is based on Kircho ffs plate theory assuming axisymmetric, small deflection and plane stress normal to the thickness direction. The governing equation describing the thermally-forced vibr ation characteristics of the composite diaphragm is given by (details of the derivation are provid ed in Appendix C) 676711 *4222 11,TT oArrB DwNwwNM A-.;.;#.<. !! (4.11) where (,) wrt is the transverse deflection, 22, wwt # ??!! oN is the initial in-plane compressive load, A is the areal density of the composite, HIHIzAQdz #> is the extensional stiffness matrix, HIHIz B Qzdz #> is the flexural-exten sional coupling matrix,
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34 HIHI2 zDQzdz #> is the flexural stiffness matrix, 2 11 * 11 11 B DD A #< , T rN is the integrated thermal force 67HI,,,T r T zN TrztQdz N12JK # LM NO> (4.12) T r M is the thermal moment 67HI,,,T r T zM TrztQzdz M12JK # LM NO> (4.13) and r 1 2 22## is the coefficient of thermal expansion. For a transversely isotropic material the stiffness matrix is 21 [], 1 1 E Q * * * J K # L M < N O (4.14) where E is the Youngs modulus and * is the Poissons ratio distributions in the composite plate. Equation (4.11) is derive d from the strain-displacement relationship, stress-strain relationship and the equati ons of motion. Although derived for an axisymmetric composite plate, the vector form of Equation (4. 11) is the governing equation for the thermally induced vibration of both rectangular a nd circular plates (Reddy 1996). A flow chart illustrating the sequence of steps in the solution procedure is shown in Figure 4.2. The solution to Equation (4.11) for the case of an axisymmetric circular plate with clamped edges is (details of the derivation are provided in Appendix C) 6767 67 6713 * 4222 *, 1 (,)Re,nmm jt ooom o mn mmwrtcJrcIrJre N D D$P0/ QR/ //RQ@A JK BC LM #;; BC LM BC LM << BC NO DEGG (4.15) where
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35 35 * *,,2 22 o A 2 oN 4D 11 2DN (4.16) 13 * 4222 *, 1 ,onmmom o mn oo mm I aJa cc N JaDJa D (4.17) and 11 3 * 4222 11 *, 1 .nmmommom o mn oo mmJaJaJaJa c N DIaJaJaIa D (4.18) Figure 4-2: A flow chart of the solution procedure.
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36 Results and Discussion In this section, results from the analyt ical model are compared with a coupled thermal-mechanical finite element simulation for verification of the model. The material properties, geometry and input parameters used for comparison of the analytical and finite element model are given in Table 4-1. The initial and boundary conditions were identical in both the analytical and finite element model and th e effect of static heating on the vibration characteristics of the diaphragm wa s neglected. The effect of static heating will be discussed later in the chapter. Table 4-1: Material pr operties and geometry used in the analytical and finite element models (radius of the diaphragm: 500 !m , radius and thickn ess of the heater: 30 !m , 0.5 !m , input power: 20 mW at 50 kHz ). H'(!m )', (10-6 m2/s ) E ( GPa ), * -'( kg/m3)'2 (10-6 / K ) Silicon (1) 8 94.96 150, 0.27 2330 2.8 Silicon dioxide (2) 0.7 0.59 70, 0.17 2200 0.7 Silicon nitride (3) 0.3 0.90 270, 0.27 3000 2.3 The finite element analysis was performed in ABAQUS® using 8-noded axisymmetric thermally-coupled quadrilater al elements (CAX8T). A total of 60,720 elements were used in the model that represents a well refined mesh. Successive refinement varied the value of the plate vi bration amplitude by < 0.1 %. Similarly, a sufficient number of terms were used in the analytical series solutions for temperature (Equation (4.9)) and transverse deflection (Equation (4.15)) to ensure a converged solution. Figure 4-3 shows a plot of the te mperature calculated at a specific location (, rz ) and time using Equation (4.9) versus the number of terms (, mn ) used in the series. The convergence of the analytical solution can be observed for ,35 mn S .
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37 0.0 0.1 0.2 0.3 0.4 0.5 0102030405060Number of Terms (m,n) in the Series Solution) T (K) Figure 4-3: Plot i llustrating the convergence of the analytical series solution for temperature (calculated at 40 rm ! # , 4 zm ! # and 125 t $ %#) The time-harmonic temperature distributi on in the composite diaphragm obtained from the analytical and finite element m odels are plotted in Figure 4-4 and shows good agreement between the two models. The initia l difference in the temperature is due to a transient effect captured by the finite element model that decays exponentially. -2 -1 0 1 2 0102030405060 Time ( ! s)) T (K) Analytical FEM Figure 4-4: Temperature distribution in the composite diaphragm (at 0r# and 0H # ) as a function of time.
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38 This effect can also be shown using the anal ytical model, however, the transient solution is based on the artificially forced ambi ent boundary condition at the edge of the diaphragm, which is an approximation base d on a thermal wavelength argument. The correct transient solution will have to invol ve the surrounding substrate and a continuity (of heat flux) condition at the diaphragm edge. The goal of the analytical model is to study the steady-state response of the dia phragm which agrees well between the two models. In terms of computation time, th e analytical model is significantly faster compared to a refined finite element model. For comparison, the time response (3 cycles) of the composite diaphragm to a sinusoidal heat flux input at a single frequency was calculated using the analytical and finite element models on a Pentium 4® processor running at 2.8 GHz . For a fixed geometry, the average computation time for the coupled thermal-mechanical finite element analysis is around 80 minutes whereas it takes less than 1 minute to execute the analytical code (in Mathcad®). A plot of the temperature profile through the thickness of the composite diaphragm at 0 r # is shown in Figure 4-5, where the ma ximum temperature indicates the location of the diffused heater. The plot illustrates the effect of a composite structure on the thermal gradient across the diaphragm. The temperature gradient is largest across the oxide layer (8 8.7 !m ) due to its poor thermal conduc tivity, which consequently leads to a larger thermal force and moment compar ed to a homogenous structure. The radial temperature distribution along the center of the silicon layer is shown in Figure 4-6. The plot indicates a finite slope for the temper ature distribution across the diaphragm radius and this effect can be captured only by employing a 2-D heat conduction model. Previously reported thermomechanical models assumed a step profile for the temperature
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39 distribution based on a 1-D heat conduction model and neglecte d the effects of in-plane heat conduction from the edge of the heater (Irie and Yamada 1978, Lammerink et al. 1992, Paul and Baltes 1999). The effect of in-p lane heat conduction is discussed later in this section. 0.4 0.6 0.80123456789Depth ( ! m)) T (K) Analytical FEM Figure 4-5: Non-uniform temp erature profile through the thickness of the composite diaphragm (at 0r#). -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0100200300400500 Radius ( ! m)) T (K) Analytical FEM Figure 4-6: Radial te mperature distribution in the co mposite diaphragm along the center of silicon layer.
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40 Figure 4-7 and Figure 4-8 compare the vibra tion amplitude of the composite diaphragm with the finite element model plotted with re spect to time and radius of the diaphragm, respectively. The close agreement between th e models is a validation for the simplifying assumptions (Kirchoffs hypothesis, plane stre ss normal to the z-axis etc) made in the analytical plate model since the finite elem ent model has no such inherent assumptions. -0.20 -0.10 0.00 0.10 0.20 0102030405060 Time ( ! s)Vibration Amplitude (nm) Analytical FEM Figure 4-7: Vibration amplitude of the composite diaphragm (at 0 r # and 0 H #) as a function of time. 0.00 0.04 0.08 0.12 0.16 0100200300400500 Radius ( ! m)Vibration Amplitude (nm) Analytical FEM Figure 4-8: Radial variation of the vibra tion amplitude of the composite diaphragm.
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41 A comparison of the analytical model with an experimental measurement of the diaphragm vibration amplitude is shown in Figure 4-9. The measured vibration amplitude is larger by a factor of 2.7 compar ed to the analytically predicted value. The observed difference may be attributed to seve ral factors including uncertainties in the material properties, diaphragm aspect ratio , fabrication-induced stresses, temperature distribution in the diaphragm and the complia nce of the built-in edge. However, the model still provides a good approximation of th e vibration amplitude and may be further improved with more precise measurements of the device parameters. 0.0 0.2 0.4 0.6 0100200300400500 Radius ( ! m)Vibration Amplitude (nm) Analytical ExperimentalDiaphragm radius: 500 !m Silicon thickness: 10 !m Oxide thickness: 0.7 !m N itride thickness: 0.3 !m Heater radius: 30 !m Heater thickness: 0.46 !m In-plane load: 210 N/m Figure 4-9: Comparison of the an alytical model with an experimental measurement of the diaphragm vibration amplitude for an input power of 39 mW at 40 kHz . Effect of in-plane heat conduction While the non-uniform transverse temper ature profile generates the integrated thermal force and moment, the in-plane heat conduction also has a si gnificant effect on the vibration amplitude of the diaphragm. Equation (4.11) shows that the forcing functions are proportional to the radial sl ope and curvature of the thermal force and moment. Previously reported models (L ammerink et al. 1992, Paul and Baltes 1999)
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42 neglected the radial edge effects of the actua ting heater by assuming a step profile for the radial temperature distribution. This conse quently leads to the us e of a delta function (derivative of the step-function) for the sl ope and its derivative for the curvature. However, as shown by Equation (4.9) and i llustrated in Figure 4-6 the temperature distribution in the diaphragm is not independent of the radius and possesses a finite slope and curvature. -0.2 0.2 0.6 1 00.20.40.60.81 Normalized RadiusNormalized ) T 60000 Hz 6000 Hz 600 Hz Figure 4-10: Normalized radial temperature distribution at the center of a homogenous silicon diaphragm (500 !m -diameter and 10 !m -thick) as a function of driving frequency. Therefore, assuming a step function (with in finite slope) for the radial temperature distribution, is an over-simp lification and severely over pr edicts (x500) the vibration amplitude of the diaphragm for a given h eat flux input. The radial temperature distribution is also a functi on of the input frequency and the effect is more pronounced for lower driving frequencies where the varia tion is more gradual and spread out over a larger radius (Figure 4-10), t hus deviating further from the step-profile assumption. The
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43 diffusion length scale even at an operating frequency of 60 kHz is not negligible for silicon (~ 30 !m ) compared to the heater size (30 !m ). Effect of static heating As explained in the previ ous chapter the inpu t power has three components at dc, $ and 2$ due to its quadratic dependence on the ex citation voltage and th e addition of a dc bias, 67 67 67 22 2 2sin2sincos. 22P PdcVV VVtVVVtt$ $ $$@A (( (( ;#;;< BC BC DE 2#$%$ & #$%$ & (4.19) In the model presented above, the sinusoid al steady-state temp erature distribution resulting from the dynamic input co mponent at the operating frequency 67P $ was derived. The effect of the component at 2$ can be minimized by suitably selecting the ratio of the dc and ac voltages. In order to consider th e effect of static heating 6 7Pdc the model would have to incorporate the subs trate surrounding the diaphragm, which is beyond the scope of the analytical solution. However, the temperature distribution resulting from a combined static and dyna mic input power was modeled using finite element techniques. The model used for the analysis is shown in Figure 4-11 and consists of a homogenous silicon diaphragm (500 !m -radius, 10 !m -thick) that is built into the surrounding silicon substrate (2 mm -wide and 500 !m -thick). Eight-noded axisymmetric thermally-coupled quadrilateral elements (CAX8T) were used similar to the finite element simulation used to verify th e analytical model. The top surfaces of the diaphragm and the surrounding substrate were ke pt insulated. This is an approximation based on the low conductivity thin films (Table 4-1) deposited over the silicon structural layer. The bottom surface of the diaphragm as well as the sides of the substrate were
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44 prescribed a free-convective boundary condition with a heat transfer coefficient of 5 W/m2K and an ambient condition was prescribed at the base of the substrate. Figure 4-11: Axisymmetric finite elemen t model of a releas ed diaphragm (500 !m -radius, 10 !m -thick) with the su rrounding substrate (500 !m -thick), illustrating the temperature distribution due to a combined dc (20 mW ) and ac (20 mW at 50 kHz ) input. The structure was initially mainta ined at ambient temperature (300 K ) and the subsequent temperature distribution due a combined dc (20 mW ) and ac (20 mW at 50 kHz ) input to the diffused heater (30 !m -radius, 0.5 !m -thick) was then calculated. 300 304 308 312 316 050100150200250300 Time ( ! s)Temperature (K) Figure 4-12: Plot of temperatur e as a function of time at the center of the diaphragm due to a combined dc (20 mW ) and ac (20 mW at 50 kHz ) input. Tave
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45 The results of simulation are plotted in Figur e 4-12 that indicates an average temperature rise ( Tave) in addition to a sinusoidal variation with time. The sinusoidal temperature variation is the result of the dynamic power component P ($) and the static component P ( dc ) causes an average temperature rise ( Tave) until thermodynamic equilibrium is reached. Thus at steady state conditions, when the average temperature asymptotes to a fixed value, the net effect of the dc component is a static te mperature elevation that leads to an increase in the overall compressive loading on the diaphragm and consequently affects its resonant frequency a nd vibration amplitude. This e ffect can be incorporated in the analytical model to the first order by appr opriately adjusting the value of the in-plane load (ooNN T) ). The analytical model presented in this ch apter can also be used to predict the vibration characteristic s of beam-type resonators and can be extended to transversely isotropic piezoelectric actuators by substituting the thermal force NT and moment MT with equivalent piezoelectric force and moment. The piezoelectric forcing functions are given by HI31 31 P r f P zd N EQdz d N1UV U V # W XWX YZ YZ> (4.20) and HI31 31,P r f P zd M EQzdz d M1UV UV # WXWX YZ YZ> (4.21) where fE is the transverse electric field and 31d is the piezoelectric coefficient. In the following sections, the sound radia tion from the vibrating diaphragm is investigated using the model of a piston in an infinite baffle.
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46 Generation of Acoustic Waves In order to determine the sound radia tion characteristics of a vibrating microstructure, it is important to calculate the generated acoustic field directly from the forced vibration characteristics of the resonator. To descri be the radiation of spherical waves from a region in space, the radiating re gion can be divided into elements each of which acts as a simple source or a point m onopole (Blackstock 2000). The total radiation received at a point r' is then the sum of radiations from the individual sources. If the radiating region is a bounded surf ace instead of a volume and the radiation is restricted to the hemisphere in front of the source plane such as a piston se t in an infinite rigid baffle the pressure (in the freque ncy domain) is given by 67 6 7 (,) ,, 2jtkR oo sjkcuxye p rdxdy R$$ %<#>>( (4.22) which is known as the Rayleighs integr al (Blackstock 2000). In Equation (4.22) , R rr ( #<'' primed variables represent the lo cation of the source and the unprimed variables represent a field point and kc $ # is the wave number. For a circular piston vibrating with a uniform velocity 6 7,,jt ourue $ 1# the Rayleighs integral is (Figure 413) 67 6 7 2 00,, 2ajtkR ooojkcue p rdd R%$$ 44Q %<#>>' (4.23) where 222sincos Rrr 4 41Q#;< . (4.24) Although Equation (4.23) is an ex act solution, it is difficult to integrate analytically. However, if we assume radiation in the far-field the problem is greatly simplified
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47 47 212sincossincos RrOr rr (4.25) and Rayleighs integral re duces to (Blackstock 2000) 1sin ,,. sinjtkr oooJka jacu pre r (4.26) Figure 4-13: Radiation fro m a circular piston. In the case of the thermoelastic resonator the transverse velocity is not uniform across the radiating surface. In order to study its sound radiation characteristics it can be modeled using a non-uniform piston. Non-uniform pist ons are used to repr esent radiators that cannot vibrate with a uniform velocity e.g., radi ators that are restricted at the edges. For an axisymmetric circular radiat or that is clamped at the edge the normal velocity is given by (Blackstock 2000) 3 2 231,avjt pouue a (4.27) and the corresponding pressure in the far-field is 2 3 348sin ,,, 2 sinav jtkr oooJka kacu prje r ka (4.28) R r z y x a dS 0 d L(x,0,z)
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48 where 4 is defined in Figure 4-13 and av ou is the average velocity amplitude. As shown in Figure 4-14 the clamped radiator is a good approximation for the normal velocity distribution of the thermoel astic resonator operated at its resonant frequency. 0 0.25 0.5 0.75 1 0100200300400500 Radius ( ! m)Normalized Velocity Thermoelastic resonator Clamped radiator Figure 4-14: Comparison of the normal velo city distribution of the thermoelastic resonator operated at resonance and a clamped radiator. The directivity of the radiated acoustic field is characterized by the amplitude directivity factor D , which is defined as the rati o of pressure at any angle 1' to the pressure on the axis of the radiator (1=0 ) for a fixed radial di stance (Blackstock 2000) 67 6 7 67 , , ,0 Pr D Pr 1 1# (4.29) where P is the pressure amplitude. For the case of a clamped radiator this results in 67 6 7 67 3 348sin . sin Jka D ka 1 1 1# (4.30) A graphical representati on of the directional characteristics of the clamped radiator for different values of ka is shown in Figure 4-15. The pl ot indicates that the generated acoustic field is nearly omni-directional, if the acoustic wavelength is large compared to
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49 the physical size of the transducer (0>>a ). With increasing ka , the acoustic field becomes more focused towards a small radial region around the axis of the radiator. 1 0.75 0.5 0.25 0 Figure 4-15: Polar plot of the amplitude direc tivity factor of a clamped circular radiator for different values of ka . For a typical micromachined transduc er with a lateral dimension of 1 mm and operating at frequencies less than a 100 kHz , the generated acoustic field is expected to be omnidirectional. To obtain a more directional acoustic source, th e lateral dimensions of the transducers can be increased. For example, to obtain 3 dB angle of 20o for the main lobe, corresponding to ka = 10 , a circular transducer operating at 100 kHz would need to have a radius of 5 mm . 1 0.75 0.5 0.25 0 Figure 4-16: Comparison of the directiona l characteristics of two transducers ( ka = 5 ): (a) radiator with uniform vibration amplitude , (b) radiator clamped at the edges. ka = 1 ka = 3 ka = 10 (a) (b) 0.5 0.5 0.75 1 0.25 0.75 D 1 ' 0.5 0.5 0.75 1 0.25 0.75 D 1 '
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50 A side effect of highka radiators is the presence of several secondary maxima or minor lobes in the radiation pattern (Bl ackstock 2000). However, non-uniform pistons especially radiators clamped at the edges demonstrate impr oved sidelobe suppression as indicated in Figure 4-16. The rest of the chapter deals with the th eoretical development of the piezoresistive sensing scheme employed for the detecti on of acoustic pressure perturbation. Piezoresistive Sensing Piezoresistivity is defined as the change in the resistivity of a material due to a change in the mobility (or number of charge carriers) induced by a mechanical strain to the material (Smith 1954). In a piezoresistive sensing scheme , the resistance variation is a linear product of the applied stress and the piezoresistive coefficients %, which are functions of the crystal orientation (Ka nda 1982). The large piezoresistivity of monocrystalline silicon, compared to metals strain gauges and its excellent mechanical properties have led to use of piezoresistive silicon sensors to infer parameters such as force, pressure and acceleration. The discovery of the piezoresistance eff ect in silicon and germanium (Smith 1954) marked the beginning of silicon-based transduc ers. Initially, these transducers utilized homogenously-doped silicon strips that were attached to a su pporting structure to make use of the higher gage factor of silicon exclusively. The development of diffusion techniques for the fabrication of piezoresistive se nsors using single cr ystal silicon (Pfann and Thurston 1961, Tufte et al. 1962) led to the use of silicon wafers in bending tests with diffused resistors to measure the maximu m stress at the surface. The integration of strain gauges into the structure has severa l advantages (Sze 1994) including a perfect
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51 transmission of strain and th e process is extremely suitable for miniaturization and batch fabrication. Advances in bulk micromachin ing techniques led to the creation of thin compliant membranes by etching away part of th e silicon resulting in devices with higher sensitivity. Majority of the commercially available pressure sensors today use silicon piezoresistors (Motorola 1998). In this section, an overview of the tr ansducer design based on the piezoresistive effect of monocrystalline si licon is presented. A more fundamental description of piezoresistivity includi ng physical models like the ma ny valley model can be found in several references (Kanda 1982, Sze 1994, Senturia 2001). The normalized change in resistivity, ) for small strains (assuming linear piezoresistive effect) is related to the applie d stress by the piezoresis tive coefficients, .ij % In crystals with cubic symmetry (e.g., silicon and germanium) the resistivity is a scalar and the piezoresistive coefficient matrix can be can be completely defined using three fundamental piezoresistive coefficients ,1112 % % and 44 % (Smith 1954) 1 1 111212 2 2 121112 3 3 121211 4 1 44 5 2 44 3 44 6000 000 000 1 , 00000 00000 000004 %%% 4 %%% 4 %%% 8 % 8 % 8 % -) @A @A JK BC BC LM ) BC BC LM BC BC LM ) # BC BC LM ) BC BC LM BC BC LM ) BC BC LM BC BC ) LM NO DE DE (4.31) where 12, 4 4 and 3 4 represent the normal stresses al ong the cubic crystal <100> axes and 12, 8 8 and 3 8 represent the shear stresses. In most applications, two special cases of uniaxial stress applied to a long, relatively narrow resistor are utilized. The longitudinal piezore sistive coefficient describes the case of a uniaxial stress appl ied in the direction of the current and the
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52 transverse coefficient describe s the case of a stress applied perpendicular to the direction of the current. The longitudinal and transverse coefficients in terms of the fundamental piezoresistive coefficients and direc tion cosines are given by (Kanda 1982) ()()()222222 l111112441111112lmmnnl%1%%%%#<<<;; (4.32) and ()()(),222222 t12111244121212llmmnn%1%%%%#;<<;; (4.33) where (,,)111lmn and 222(,,) lmn are the sets of direct ion cosines between the longitudinal resistor direction (subscript 1) and the crystal axis, and between the transverse resistor directi on (subscript 2) and the crysta l axis. The variation of piezoresistive coefficients as a function of or ientation on the (100) pl ane of p-type silicon is shown in Figure 4-17. Using Equations (4.32) and (4.33) the longitudinal and transverse piezoresistive coefficients can be obtained for any orientation of the piezoresistors in the crystal plane. 20 40 60 80 90 270 1800 Figure 4-17: Room temperature piezoresistive coeffi cients in the (100) plane of p-type silicon (10-11 Pa-1) (Kanda 1982). lt % %#< 110 [ S 110[S l % t %
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53 For our particular sensor, p-type piezoresistors were chosen because of their higher sensitivity along the <110> direction (Figur e 4-17).In addition, the longitudinal and transverse piezoresistive coefficients are equal in magnitude but opposite in sign along the <110> direction (%l = -1.08 %t) making then better suited for full-bridge applications (Senturia 2001). Typically, four piezoresistors are used in a fully active Wheatstone-bridge configuration driven by a consta nt voltage source. In such a configuration, the voltage output of the bridge can be lin early related to the acoustic pr essure fluctuation provided that the mean resistances in all four legs ar e equal and that the resi stance variation in each resistor of a given leg is equal in magnit ude, but opposite in direc tion (Senturia 2001). For a polar geometry, these conditions are achi eved by placing a tapered resistor opposite an arc-shaped resistor (Shepl ak et al. 1998). The equations for resistance modulation of the tapered and arc-shaped resistors are given by ()()()()rltt taperR rr R 4 %14%1 ) #; (4.34) and ()()()()tlrt arcR rr R 4 %14%1 ) #;, (4.35) where R ) is the change in resistance of the piezoresistor with initial resistance R under zero load conditions and the subscripts r and t stand for radial and tangential directions respectively with respect to resistor orienta tion. However, for a resistor of finite dimensions the piezoresistive coefficients a nd the induced stresses wi ll be different at each point depending on its orientation on th e crystal plane and radial location on the diaphragm. In order to account for these vari ations the resistor has to be discretized and the resistance modulation at each point must be calculated using Equations (4.34) and
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54 (4.35). The overall resistance modulation ca n then be obtained by adding the resistance changes of the individual elements either in series or in parallel depending on the geometry of the resistor (Sheplak et al. 1998). An iterative pro cess is used to arrive at the optimum arc and tapered piezoresistor sizes that have equal mean resistance values and equal but opposite sign modulations for maximi zing sensitivity (deta ils of the procedure are provided in Appendix B). In order to design diffused piezoresistors, the non-uniform doping profile of the resistor and the non-uniform stress distribu tion through the diaphragm thickness have to be taken into consideration in addition to surface orientation. According to Kanda (1982), any piezoresistive coefficient can be expressed as a product of the its low-doped room-temperature value %o and a dimensionless piezoresistance factor P(N,T) , which is a function of doping concentr ation and temperature 6 7 6 7,.,.oNTPNT%%# (4.36) In the theoretical model proposed by Kanda (1982), the piezoresistiv e coefficients are weak functions of doping concentration for doping below 1019 cm-3 and then decrease drastically for higher doping levels (Figur e 4-18). According to Harley and Kenny (2000), based on experimental data obtaine d from several authors, the model is reasonably accurate at low c oncentrations but substantiall y underestimates the p-type longitudinal piezoresistive coefficient %l at higher doping concentr ations. A comparison of the theoretical model proposed by Kanda (1982) and the experimental data (at room temperature) by Mason et al . (1962), Tufte and Stelzer (1963) and Kerr and Milnes (1963) is shown in Figure 4-18. For concentration above 1017 cm-3, the data is approximated by a straight line on the semilog plot defined by
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55 log,ab P P@A # BC DE (4.37) where a = 0.2014 and b = 1.53 E22 cm-3. In designing the piezoresistors of the acoustic transducer, the calculations for the relation be tween the piezoresistive coefficients and doping concentration were based on the theoretical model by Kanda (1982). 0 0.2 0.4 0.6 0.8 1 1.21.0E+141.0E+151.0E+161.0E+171.0E+181.0E+191.0E+20Concentration (cm-3)P(N,T) Figure 4-18: Plot of th e piezoresistance factor P(N,T) as a function of impurity concentration for p-type silicon. Th e line is based on a theoretical model by Kanda (1982) and the points are expe rimental data (Harley and Kenny 2000). In the case of diffused resistors the impur ity concentration decreases with depth, ultimately reaching the background concentr ation. The depth at which the doping concentration of the piezores istor reaches the background con centration of the substrate is defined as the junction depth. A plot of the doping profile of the piezoresistor with a surface concentration of 1020 cm-3 and a junction depth of 0.46 !m , obtained using FLOOPS& (Law and Cea 1998), is shown in Figure 4-19. Since the doping concentration decreases with depth, consequently for a diffused resistor the piezoresistance factor increases with depth as shown in Figure 4-20.
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56 1.00E+14 1.00E+16 1.00E+18 1.00E+20 0.00.10.20.30.40.5 De p th ( ! m)Concentration (cm-3) Figure 4-19: Piezoresistor doping profile with a junction depth of 0.46 !m obtained using FLOOPS&. 0 0.2 0.4 0.6 0.8 1 0.00.10.20.30.40.5 Depth ( ! m)P(N,T) Figure 4-20: Plot of the piezo resistance factor as function of depth of the resistor. An effective coefficient % that would yield the same elec tromechanical behavior as the piezoresistance profile %(z) can be defined for the doping profile (Figure 4-19) using P(N,T) (Figure 4-17) and Equation (4.36). Ho wever, a higher contribution to the effective coefficient is from layers where the current flow is higher, which are near the surface as indicated by Figure 4-21. Therefore, the piezoresistance profile is weighted by
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57 the local conductivity () z P and the effective piezoresistive coefficient is defined by (Sze 1994) ()()(),jjxx 00zzdzzdz%%4P#>> (4.38) where xj is the junction depth. 1.0E-02 1.0E+00 1.0E+02 1.0E+04 0.00.10.20.30.40.5 Depth ( ! m)Conductivity Figure 4-21: Conductivity variation thr ough the thickness of the piezoresistor. Table 4-2 compares the effective coefficien ts obtained for the doping profile shown in Figure 4-19 with a surfa ce concentration of 1020 cm-3 and a background concentration of 1014 cm-3 with the piezoresistive coefficients of a uniformly low-doped p-type silicon, at room temperature. Table 4-2: Comparison of the effective piezo resistive coefficients with that of a uniformly low-doped p-type silicon at room temperature (Smith 1954). Units 10-11 1Pa < 10-11 1Pa < 10-11 1Pa< Doping profile (Figure 4-19) 2.9811 % # -0.49712 % # 62.444 % # Uniform doping 6.611 % # -1.112 % # 138.144 % # The reduced sensitivity resulting from the high doping concentration is a tradeoff for decreased temperature sensitivity of the piezor esistors (Smith 1954). Since in addition to the dependence on doping levels, the piezoresistive coefficien ts also vary non-linearly
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58 with temperature (Smith 1954, Tufte and Long 1963). However, at high doping concentrations, the temperature dependence of the piezoresistive coefficients reduces considerably. Tufte and Stelzer (1963) have shown that as doping concentration rises, particularly above 1020 cm-3, the piezoresistive coefficients become almost independent of temperature variations between 80 oC and 100 oC . To calculate the resistance modulation of the arc and tapered piezoresistors (using Equations (4.34) & (4.35)) th e stress distribution in the si licon diaphragm needs to be investigated. The mechanical behavior of the composite diaphr agm is modeled using Kirchoffs plate theory described previously in the chapter. For axisymmetric, small deflections of a composite plat e subjected to a combination of in-plane compressive load No and a uniform pressure load p , the governing equations in terms of in-plane ( uo) and transverse ( w ) displacement are (Gururaj 2003) 2 11 22* 1111 2oooouuuN B p w r rrrrADrr?? ? @A ;<#<; BC ??? DE (4.39) and 32 322**11 . 2oN wwwwpr rrrrrDrD???? ;<#<< ???? (4.40) Equations (4.39) and (4.40) ar e solved analytically to obtain an expression for the inplane and transverse plate displacements 67 67 21 11 *2 1111 2okr J B pa a urra ADkJk@A @A BC BC DE BC #<< BC BC DE (4.41) and 67 67 67 422 *2211 , 42ookr paar a r DkakJJk w Jk J K @A @A L M BC BC @A < DE L M BC BC L M BC DE L M BC DE N O< #< (4.42) where
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59 2 2 * oNa k D# (4.43) is the compression parameter. The radial and circumferential stress components are 67 67 67 67 67 2 1 11 *2 111 1 121 1 1 2 , 1 1 21o o orrakrkr JJk B pa raa ADkJk rz akrkr JkJ p raa z NJkE* * * *4 *@A @A @A@A << BC BCBC BC DEDE BC BC <;;; BC BC BC BC DE BC BC @A @A@A << BC BCBC BC DEDE BC BC ;; BC BC BC BC BC DE DE# < and (4.44) 67 67 67 67 67 *21 2 11 111 1 121 ,.1 1 2 1 1 21oo okrkr aa D rz krkr aaa JJk B pa r AkJk a JkJ p r z NJkE11** * ** *4 *UV @A @A@A \\ BC BCBC DEDE \\ BC \\ BC \\ BC \DE\ WX @A @A@A \\ BC BCBC \\ DEDE BC \\ BC \\ BC \\ DE YZ<< <;;; << ;;# < (4.45) The stress distribution in a cl amped circular plate subjected to an in-plane compressive stress of 300 MPa , induced by the thermal oxide layer, and a uniform pressure load of 200 Pa is plotted in Figure 4-22. In order to achieve maximum sensitivity to acoustic pressure perturbations, the effective placem ent of the piezoresistors in regions of maximum strain is crucial. The plot indica tes that the maximum radial stress is at the clamped edge and decreases towards the center of the diaphragm with the inflection point at = 0.3 mm . The corresponding circumfere ntial stress also has to be considered for equal and opposite resistance m odulation of the arc and tapered resistors.
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60 -500000 -250000 0 250000 500000 00.10.20.30.40.5Radius (mm)Stress (N/m2) Radial Circumferential Figure 4-22: Radial and circumferential stre ss distribution in the composite diaphragm subjected to in-plane (300 MPa -compressive) and transverse (200 Pa ) loading. An iterative process is used to arrive at the final geometry and placement of the piezoresistors.
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61 61 CHAPTER 5 EXPERIMENTAL CHARACTERIZATION This chapter presents a detailed descrip tion of the experimental set-up used to characterize the acoustic proxim ity sensor and the electrical through-wafer interconnects. Device characterization included measurements of the current vs. voltage characteristics of the diffused piezoresistors and the ETWI, electrical isolation characteristics of the silicon dioxide layer surrounding the ETWI, overall noise floor, vibration and sound radiation characteristics a nd the response of the diaphragm to acoustic pressure perturbation. These measurements have been broadly classified as electrical, mechanical and acoustic characterization and are de tailed in the following sections. Electrical Characterization Electrical characterization in cluded current vs. voltage measurements to investigate the ohmic behavior of the diffused resist ors and the ETWI, capacitance vs. voltage measurements to test the isolation character istics of the silicon dioxide insulating the ETWI, and measurement of the overall device noise floor. To enable the electrical testing of the ETWI and to extract average prop erties, several test structures consisting of chains of interconnects connected by su rface polysilicon lines were fabricated. Current vs. Voltage Characteristics Current vs. voltage (I-V) measurements were performed on the diffused resistors and on different ETWI chains to determine th e resistance of the diffused piezoresistors and the average interconnect resistance. An interconnect chain consists of several ETWI (specifically 6, 22, 62 and 100) c onnected in series by surface polysilicon lines patterned
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62 on the top and bottom surfaces of the wafer (Fig ure 5-1). Separate test structures were also fabricated to estimate the resistance of the surface polysilicon lines, which is later subtracted from the overall resistance of th e chain to compute the average interconnect resistance. Figure 5-1: Chain of 6, 22, 62 and 100 ETWI connected in series via surface polysilicon lines (dark and bright lines indicate the top and the bottom surface of the wafer respectively). N 6 N 22 N 62 N 100 Bond pads Top-surface p ol y silicon lines Bottom-surface p ol y silicon lines ETWI
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63 In addition, I-V measurements were al so obtained across the pn junction formed between the diffused resistors and the silicon substrate to obtain its forward and reverse bias characteristics. The reverse bias char acteristics of the pn junction determine the leakage current from the junction-isolated resist ors into the substrate. Minimal leakage is essential for effective Joule he ating of the diaphragm and low piezoresistor noise floor. All measurements were made using a Hewlett Packard 4155B semiconductor parameter analyzer and a wafer level probe stat ion using bias voltages ranging from 10 V to 10 V with 0.1 V increments, while monitoring the current at each voltage step. Capacitance vs. Voltage Characteristics Interconnect capacitance affects the propaga tion delay of signals as well as the capacitive loading of the sensor output. The p-type polysilicon interconnect dielectrically isolated from the n-type substrate form s a metal-oxide-semiconductor capacitor (MOSC) where the polysilicon acts as a metal gate a nd the silicon layer is the substrate. The voltage-dependent MOSC capacitance was char acterized using a Hewlett Packard 4294A vector impedance meter with a 1 MHz small signal frequency. Similar to the I-V characterization, the bias voltage (in this case, across the ETWI and the substrate) was swept from 20 V to 20 V with 0.1 V increments, while monitoring the capacitance at each voltage step. An open and short circuit calibration, prior to the measurements, was performed with the vector impedance meter to eliminate the capacitive contribution of the connecting leads and the test setup. Noise Floor Spectra The study of the electrical noise floor of a device is required to determine the minimum detectable signal (MDS). Measur ements of the noise power spectral density (PSD) were made in a Faraday cage (Figure 5-2) using low-noise test equipment. The
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64 64 aim of the experimental setup was to isol ate the random physical noise of the device under test from deterministic interference (Bha radwaj 2001). Deterministic sources arise from capacitive coupling of electromagnetic in terference (EMI) to the device and cabling, with the ac power line being the major contributor (60 Hz and its harmonics). The purpose of shielding is to isolate the intern al signal path by intercepting the capacitive current and shunting it to ground (Bhardwa j 2001). The Faraday cage considerably reduced the interference, permitting analysis of the noise PSD. The sensor was configured similar to the operating conditions with a 9 V bias across the Wheatstone bridge. As illustrate d in Figure 5-2, a battery powered Stanford Research Systems SR560 differential amplif ier with specified noise voltage of 4 nV/Hz was used to amplify the differential voltage from the bridge with a gain of 10,000. This in conjunction with a battery (9 V ) as the voltage source minimizes the 60 Hz line interference. The differential measurement technique rejects the common contamination signal in the two inputs. Figure 5-2: Schematic repr esentation of the device noise measurement setup.
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65 After proper grounding of all measurement equipment, the noise PSD was measured using a SR785 dynamic spectrum analyzer using 500 averages. In order to prevent any offset voltage the different grounds namely, the signal ground, the ground of the power supply and the ground of the instru mentation were maintained at the same potential. In order to maintain sufficient resolution at low frequencies, a low frequency range of 0-1.6 kHz with a 2 Hz bin was used. This frequency resolution effectively confines the 60 Hz power line interference and its harmonics to the measurement bins and prevents it from being spilled into adjacent fr equency bins. Once isolated these samples can be filtered out, this techni que is referred to as selective filtering. Larger frequency ranges of 25.6 kHz with 8 Hz bin were used for the higher fre quencies. In order to extract the device noise floor, the se t-up noise PSD was measured by shorting the differential outputs of the amplifier and was subt racted from the total noise PSD 22. D UTEMIPSDPSDPSD #< (5.1) Mechanical Characterization Mechanical characterizati on involved investigating th e effects of diaphragm geometry and fabrication induced stresses on the resonant frequency, vibration amplitude and buckling behavior of the diaphragm. M easurements of the mechanical characteristics of the diaphragm were obtained using optical interferometric techniques. The sensors were excited using a combination of ac (7 Vpk) and dc (7 V ) voltages. The vibration amplitude of th e diaphragm was then measured using a Polytec PI scanning laser vibrometer (MSV 300) in combination with a Stanford Research Systems (SR785) spectrum analyzer. The laser vibrometer uses an optical interferometric technique to enable non-contact measurement of surface vibr ations. The operation of the vibrometer
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66 is based on a helium neon laser that is fo cused on the vibrating diaphragm and is subsequently scattered back and coupled into the interferometer. The interferometer compares the phase and frequency of the object beam (reflected from the vibrating surface) with that of the inte rnal reference beam. The frequency difference corresponds to the instantaneous velocity and the phase di fference is proportional to the instantaneous position of the vibrating surface. Measurements of the transverse vibration of the acoustic resonator were made by fitting the vibrometer onto an Olympus microscope with a 10x objective producing a laser spot size of 20 !m on the diaphragm surface and by scanning the laser across the entire diaphr agm exact vibration mode shapes were obtained. The compressive stresses generated in the thermal oxide layer (Madou 1997), due to a difference in the thermal expansion coe fficients of silicon a nd silicon dioxide, can cause buckling of the thin diaphragms if the net in-plane load exceeds the critical buckling load (Leissa 1993, Soderkvist and Lindberg 1994). Buckling is characterized by a static deflection of the diaphragm and wa s observed for diaphragms below a thickness of 8 !m . The dependence of buckling height on th e aspect ratio of the diaphragm, for a fixed value of the in-plane compressive st ress, was characterized using a Wyko optical profilometer (NT1000). In addition, the static heating of the diaphragm induces additional compressive stresses that can alter its buc kled height, resonant freque ncy and vibration amplitude. The effect of varying static power on the static and dynamic behavior of the diaphragm was also characterized using the vibrometer and the optical profilometer. The NT1000 is courtesy of University of Florida s Major Analytical Instrume ntation Center (MAIC). URL: http://www.mse.ufl.edu/~maic/Wyko.htm.
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67 Acoustic Characterization-Receiver The dynamic response of the sensor to acoustic pressure perturbation was characterized in a plane wave tube (PWT ) (Chandrasekaran et al. 2000). The PWT consists of a rigid-walled duct that s upports planar (0,0 mode) acoustic waves propagating along the length of the duct. For linear lossless acoustic motion in a rigidwalled square duct, the fundamental mode (0 ,0) or the plane wave propagates at all frequencies. The higher order modes can pr opagate only when the width of the duct is greater than half th e acoustic wavelength 6 72 D0[ (Rossi 1988). Therefore, below the first cut-on frequency ( f < co/ 2D ), the duct will propagate only plane waves and the higher order modes are evanescent. Thus, se nsors placed at the same axial location from the acoustic driver sense the same acoustic pres sure field. This permits the calibration of an acoustic sensor by comparing the output to a reference microphone with a known response. The sensors were calibrated in two different PWTs, a 25.4 mm x 25.4 mm normal incidence PWT and an 8.5 mm x 8.5 mm grazing incidence PWT. The sensor and a reference microphone (Brüel and Kjær Type 4138) were flush mounted at the same axial distance from the acoustic driver. The sensor was biased at 9 V , and the differential output of the Wheatstone bridge was connected to a SR560 pr eamplifier. The amplified signal was then fed into a SR785 dynamic sp ectrum analyzer for data processing. The low-frequency cut-on for the first non-planar (1,0), (0,1) mode is 20 kHz for the grazing incidence PWT and 6.7 kHz for the normal incidence tube. Thus, the usable bandwidth is limited to 20 kHz and 6.7 kHz respectively.
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68 The normal incidence PWT was also used to measure the linear response of the sensor to varying sound pressure level. M easurement of the device linearity is required to determine the dynamic range of operati on of the sensor. In a proximity-sensing scheme, the transmitter, typically an under-dam ped second-order system, can have a gain of greater than 40 dB at resonance and a linear response of the receiver over the entire range would be required for spectral anal ysis of the measured signal. A 1 kHz tone at varying amplitudes, monitored by the refe rence microphone, was used to excite the sensor, which was biased at 9 V . The rms output voltage of the sensor at each sound pressure level was recorded. Acoustic Characterization-Transmitter The end application requires an array of sensors to be us ed in a network to monitor the state of the entire cavity surrounding the HSSV. When used as an array, minimal cross-talk between the sensors is required fo r efficient real-time monitoring. Cross-talk refers to the portion of the acoustic radiation that reaches the receive r directly from the transmitter. The directional behavior of the generated acoustic field and the spacing between the sensors will determine the crosstalk. The acoustic field generated by the vibration of the thermoelastically actuated diaphragm was characterized in a free-field environment. In order to characterize the transmitted acoustic field, two sets of measurements were obtained. In both cases, the transmitting sensor was fixed and oriented such that the diaphragm surface is vertical. The transmitter was then excited with a combination of harmonic ac (9 Vpk) and dc (9 V ) voltages. The acoustic field was measured using a Brüel and Kjær Type 4138 microphone. To measure the directionality of the generated acoustic field, the microphone was positioned at a fixed radial distance of 25 mm from the transmitter and a jig was cons tructed that allowed the microphone to
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69 be revolved around the transmitter at the fixe d radius. Sound pressure measurements were made at 2o intervals, from 0o to 90o, averaged, and recorded. To obtain the variation of sound pressure level with distance from the transmitter, the microphone was positioned directly opposite th e transmitter on a single axis traverse with a precision of 1 !m . The initial position was set at 3 mm from the transmitting diaphragm and then varied to 50 mm along the axis of the diaphragm. At each position, sound pressure measurements were obtained using the microphone and a spectrum analyzer. Proximity Sensing A continuous-wave phase-shift technique (Li et al. 2002) was used to demonstrate proof-of-concept proximity sensing using the micromachined transducer as the acoustic transmitter. The transducer and a microphone (Brüel and Kjær Type 4138) were positioned on a 1-D traverse with a precision of 1 !m facing a sound hard boundary, which was initially set at a distance of 20 mm . The amplitude and phase of the acoustic waves generated by the transducer (operating at 69 kHz ) and subsequently reflected from the boundary was recorded using the microphone , while altering the distance between the transducer/microphone and the boundary. Th e distance moved by the boundary is then calculated from the measured phase-shift afte r subtracting the cross-talk between the transducer and the microphone. In order to estimate the cross-ta lk the sound hard boundary was removed and the magnitude and phase of the acoustic radiation was measured using the microphone. The phase of the vector difference between the two measurements (with and without the boundary) corresponds to the phase of the reflected wave.
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70 70 CHAPTER 6 RESULTS AND DISCUSSION This chapter presents the results from the electrical, mechanical and acoustic characterization of the acoustic transducer/ETWI. These include current vs. voltage characteristics of the diffused resistors, re sistor/substrate (pn) junction and the ETWI, capacitance vs. voltage characte ristics of the oxide insulati ng the ETWI, overall device noise floor, diaphragm vibra tion characteristics as well as acoustic transmitting and receiving characteristics. Devices with va rying diaphragm thicknesse s and with front and backside contacts were tested (Table 6-1). Table 6-1: List of devices used for the ch aracterization (The thickness of the silicon dioxide and silicon ni tride layers are 0.7 !m and 0.3 !m respectively). Device label Thickness of the silicon layer (!m ) C1 (front-side contacts)10 C3 (with ETWI) 8 C9 (front-side contacts)5 Electrical Characterization Current vs. Voltage Characteristics The reverse bias characteri stics of the pn junction formed between the boron-doped piezoresistors and the n-type silicon substrat e determines the leakage current from the resistors into the substrate. Minimal leakage is essential for effectiv e Joule heating of the resistors as well as low piezo resistor noise floor. Results of the I-V characterization (Figure 6-1) indicate neglig ible leakage current (< 14 pA ) up to a reverse bias voltage of 10 V . The small current value in the forward bias mode (deviating from the ideal diode
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71 10 V . The small current value in the forward bias mode (deviating from the ideal diode behavior) is due to the large se ries resistance of the lightly doped silicon substrate. The reverse bias breakdown voltage for the pn junction is greater than 30 V as shown in Figure 6-2. BiasVoltage(V) Current(A) -10 -5 0 5 10 0 1E-07 2E-07 3E-07 -1.2E-10 -9E-11 -6E-11 -3E-11 0ReverseBias ForwardBias Figure 6-1: Forward and reverse bias charac teristics of the pn junction, indicating negligible leakage current (< 14 pA at -10 V ) (Device C3). -100 -80 -60 -40 -20 0 -40-30-20-100 Voltage (V)Current (nA) C1 C9 Figure 6-2: Reverse bias breakdown voltage of the pn junction for devices C1 and C9. I-V measurements across the diffused resi stors indicate an average resistance of around 2350 " for the actuating heater, 6300 " for the arc resistor and 7300 " for the
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72 tapered resistor. The resistan ce values obtained for the different devices are tabulated in Table 6-2. Table 6-2: Resistance of the diffused resistors (in k"). Device Arc 1 Taper 1 Arc 2 Taper 1 Heater C 1 6.26 7.46 6.13 7.34 2.34 C 3 6.55 7.44 6.64 7.14 2.62 C 9 6.29 7.53 6.07 7.76 2.10 Average 6.33 7.35 6.28 7.27 2.35 Theoretical 4.62 4.62 4.62 4.62 1.50 The interconnects display ohmic behavior as indicated by the linear variation of current and voltage in Figure 6-3. Av erage resistance values ranging from 10 " to 14 "'were obtained for each interconnect de pending on the thic kness of the doped polysilicon layers (Figure 6-4). -10 -6 -2 2 6 10 -0.1-0.0500.050.1 Current (A)Voltage (V) N6 N22 N62 Figure 6-3: I-V characteristics fo r chains of 6, 22 and 62 ETWI. This is comparable to the n-type ETWI (14 " (Chow et al. 2002)7'and is satisfactory for use with our piezoresistive sensors, since th e contribution to the ove rall resistance is less than 0.6 %. The theoretical minimum resistivity of boron-doped polysilicon is 2000 !"cm (Kamins 1990), which corresponds to 25 "'for a 20 !m diameter, 400 !m long
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73 conductor assuming uniform doping. However, an increase of 3 !m in the diameter can reduce the resistance to 12 ", which is consistent with our results. y = 13.81x + 6.5505 y = 8.2902x + 2.41020 200 400 600 800 1000 020406080 Number of ViasResistance (Ohms) Figure 6-4: Plot illustrating th e linear variation in the resist ance of the ETWI with (a) 4 !m -thick and (b) 6 !m -thick doped polysilicon layers (slope indicates average ETWI resistance). Although, the vias were designed to be 20 !m in diameter, several factors including mask erosion, overetch steps, simultaneous lateral etching can cause an increase in the diameter. In addition, doping crowding effects caused by oxidation can result in increased conduction. Capacitance vs. Voltage Characteristics The high frequency capacitance vs. voltage curve for varying bias voltage (-20 V to 20 V ) between the substrate and the ETWI is show n in Figure 6-5. This curve is typical for an n-type substrate MOSC where an asymptotic maximum capacitance is observed at large positive voltages when the n-type substrate is in accumulation and a minimum capacitance is seen at negativ e voltages when the substrate is inverted. The measured capacitance is lower than the theoretical ETWI capacitance (C how et al. 2002) due to the floating SOI substrate used in the ETWI process. In the SOI wafer used for the (a) (b)
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74 fabrication of the acoustic transducer, the ac tive silicon layer is separated from the bulk substrate by a thin (4000 Å ) oxide layer. Thus requiring separate n+ contacts for the active silicon layer and the bulk substrate. In the first generation de sign, a contact to the bulk substrate was not provided. 1.35 1.37 1.39 1.41 1.43 -20-1001020 Voltage (V)Capacitance (pF) Figure 6-5: High-frequency (1 MHz ) capacitance vs. voltage characteristics for a chain of 22 interconnects. Noise Floor Spectra The study of the electrical noise floor of a device is required to determine the minimum detectable signal (MDS). The volta ge noise PSD of the se nsor is plotted in Figure 6-6. As indicated, the voltage noise PSD is dominated by 1/f noise at low frequencies. The MDS was calculated by taki ng the square root of the noise PSD and dividing by the sensitivity of the device to obt ain pressure. Additional plots of the set-up noise from the amplifier and EMI and a sensor with front-side contacts are also shown for comparison. Since one application of the ETW I is to connect the sensors small-signal output to nearby signal-conditioni ng electronics, the noise contributed by the interconnect itself must be minimal.
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75 1.00E-18 1.00E-16 1.00E-14 1.00E-12 1.00E-10 110100100010000100000 Frequency (Hz)Noise PSD (Vrms2/Hz) Sensor 1 Backside Contacts Sensor2 Frontside Contacts Set up Figure 6-6: Noise power spectral density of the sensor at a bridge bias of 9 V (devices tested C1 and C3). The results indicate a negligible noise contribution from the ETWI. The 1/f noise intersects the thermal noise (= 1.34 mPaHz ) at approximately 60 kHz , making the device only Johnson noise limited at the ope rating frequencies, with a MDS of 36.5 dB for a 1 Hz bin centered at 60 kHz . The spikes in the data are due to the deterministic interference at 60 Hz and 20 kHz and their harmonics. Mechanical Characterization Surface vibration measurements were pe rformed on multiple devices using the scanning laser vibrometer. A visualization of the first three vibration modes of the thermoelastic resonator in response to a periodic chirp signal with a 3 V dc offset is shown in Figure 6-7. Figure 6-8 shows a plot of the vibration amplitude measured at the center of the diaphragm as a function of excitati on frequency. In th is case, the diaphragm was excited using a harmonic ac voltage (7 V ) at varying frequencies superimposed on a dc voltage (7 V ) to reduce frequency doubling of the output signal. Johnson Noise
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76 Figure 6-7: Visualization of the first three vi bration modes of the thermoelastic resonator using the scanning laser vibrometer. 0.1 1 10 100 1000 255075100Frequency (kHz)Vibration Amplitude (nm) Figure 6-8: Plot of the vibration amplitude m easured at the center of the diaphragm as a function of excitation frequency for vary ing thickness of the silicon layer. As seen in Figure 6-8, the diaphrag ms exhibit a gain of more than 20 dB in the vibration amplitude at resonance. The measured vibra tion amplitude and quality (Q) factor of the resonators are tabulated in Tabl e 6-3. The Q-factor of the re sonator is defined as the ratio of the vibration energy to the energy dissi pated per cycle (Stemme 1990). It can be calculated from the amplitude-f requency spectrum (Figure 6-8) by dividing the resonant frequency of the diaphragm by th e frequency bandwidth at the 3 dB attenuation point (Stemme 1990) C9 C3 C1 (0, 0) mode Frequency: 58 kHz Maximum amplitude: 50 nm (1, 0) mode Frequency: 108 kHz Maximum amplitude: 100 pm (2, 0) mode Frequency: 186 kHz Maximum am p litude: 7 p m
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77 3dB.resf Q f # ) (6.1) Table 6-3: Mechanical characteristic s of the thermoelastic resonators. Device label Silicon thickness (!m ) Buckling height (!m ) Resonant frequency ( kHz ) Amplitude at resonance ( nm ) Quality factor C1 10 0 90 11.3 60 C3 8 2 55 167 37 C9 5 8 69 108 34.5 Measurements made on transducers with vary ing silicon thickness indicate a relationship between the diaphragm thickness and the vibr ation characteristics (amplitude, resonant frequency and buckling height) as shown in Ta ble 6-3. As the thickness is reduced from 10 !m to 8 !m , an increase in the vibration amplitude (15x) is observed. This is due to a net increase in the in-plane compressive load on the diaphragm. Since the compressive stress induced by the thermal oxide layer is fixed, a decrease in the silicon thickness proportionately increases the overall compressi ve load on the diaphragm and due to the slope wr ?? of the deformed plate, the in-plane compressive load ( No) produces an added bending effect. Consequently, an incr ease in the in-plane load increases its contribution to the overall bending effect. This is true for in-plane loads less than the critical buckling load ( No < Ncr). According to linear buckling theory (Timoshenko and Krieger 1959 and Soderkvist and Lindberg 1994) the (static) transverse deflection due to in-plane compressive loads is ze ro in the pre-buckling state ( No < Ncr) and unlimited with an unknown direction in the post-buc kled state. It can be seen from Equation (4.42) that a value of the stiffness parameter, k corresponding to the first root of the Bessel function 676710Jk # produces a singularity in the transver se deflection of the plate. The compressive load at which the singularity occu rs is known as the cr itical buckling load
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78 78 * 214.68.crD N a (6.2) The critical buckling loads for various diaphragm thicknesses assuming clamped and simply supported boundary conditions are tabulated in Table 6-4. The boundary conditions represent limiting cases and in pr actice, the built-in e dge of the diaphragm would have a finite rotational compliance. Table 6-4: Critical buckling loads. Device label Silicon thickness (m ) Critical buckling load ( N/m ) (simply supported edge) Critical buckling load ( N/m ) (clamped edge) C1 10 295.2 1031.8 C3 8 161.7 565.3 C9 5 48.3 168.9 In micromachined resonators, the fabricati on-induced stress is usually constant, the value being fixed by the process parameters. In this case th e compressive stress in the oxide layer is assumed to be 300 MPa which corresponds to a compressive load of 210 N/m . Therefore, the critical buckling load corresponds to a minimum thickness of the diaphragm below which it would buckle under th e compressive load as shown in Figure 6-9. However, this is based on the assumption of full symmetry and the omission of deflection-induced axial strain. In reality, factors such as the asymmetrical vibrating structure and transverse loading can cause si gnificant transverse deflection before the critical buckling load as seen from the expe rimental data (Device C3). The buckling like transverse deflection, defined as quasibuckling, of compressive ly and transversely loaded structures has been investigated by Soderkvist and Lindberg (1994).
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79 79 0.1 1.0 10.0 100.0 68101214 Diaphragm Thickness ( m)Vibration Amplitude (nm) Critical Buckling Load Figure 6-9: Vibrati on amplitude of a 1 mm -diameter diaphragm calculated at the center as function of the diaphragm thickness. According to Soderkvist and Lindberg (1994), the vibration amplitude is finite at the buckling load with a smooth tran sition to one of the stable po st-buckling states. In the post-buckled state, the vibration amplitude is no longer a function of the diaphragm thickness but instead depends on the buckled height of the diaphragm (Hornung and Brand 1999). The vibration amplitude decrease s with increasing buckling height (or inplane load) of the strain-hardened diaphragm. This effect is illustrated in Figure 6-8 by the comparatively smaller vibration amplitude of the thinnest diaphragm (Device C9) with a buckled height of 8 m . It has been found that the resonant fre quency reaches a minimum (Soderkvist and Lindberg 1994) and the vibration amplitude is at its maximum value for diaphragms with compressive loads close to the critical buckl ing load as shown in Figure 6-10 and Figure 6-11. In addition to the significant increase in the vibration amplitude, a lower resonant frequency translates to longer wavelengths (3.8 mm at 90 kHz , 6.23 mm at 55 kHz ), which would be beneficial for operating the sensors in the wavy interface.
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80 80 20 40 60 80 100 02468Buckling Height ( m)Resonant Frequency (kHz) Figure 6-10: Resonant frequency of the thermo elastic resonators as a function of their buckled height measured at the center (data represents 7 devices). 0 40 80 120 160 200 0246810Buckling Height ( m)Vibration Amplitude (nm) Figure 6-11: Vibration amplitude of the ther moelastic resonator as a function of the buckled height both measured at the cen ter of the diaphrag m (data represents 7 devices). In addition to the fabrication-induced st ress, the static component of the input power produces a static temperat ure distribution that introduce s additional stresses in the diaphragm.
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81 81 0 1 2 3 4 00.250.50.751 Diameter (mm)Height ( m) 0 mW 10 mW 32.5 mW Figure 6-12: Deflection mode shape as a function of static power. The additional stress induced by the stat ic heating, observed from the increasing buckled height (Figure 6-12), affects the resonant frequenc y and vibration amplitude of the diaphragm. A plot of the change in the buckling height a nd resonant frequency caused by the change in the static powe r are shown in Figure 6-13 and Figure 6-14. y = 0.0281x R2 = 0.94460 0.2 0.4 0.6 0.8 051015202530 Static Power (mW)Buckled Height ( m) Figure 6-13: Change in the buckled height of the diaphragm as a func tion of static power.
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82 y = 134.19x + 53571 R2 = 0.974154000 56500 59000 61500 64000 020406080 Static Power (mW)Resonant Frequency (Hz) Figure 6-14: Resonant frequency of the di aphragm as a function of static power. Acoustic Characterization-Receiver The results of the acoustic characterization of the transducer used in a receiving mode are presented in the following section. Linearity The dynamic response of the sensor to varying sound pressure levels up to 140 dB (ref 20 !Pa ) obtained in the normal incidence plane wave tube is shown in Figure 6-15. 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 6080100120140 Sound Pressure Level (dB ref 20 ! Pa)Output Voltage (mVrms) Figure 6-15: Plot illustrating the device linearity in sensing acoustic pressure perturbations (up to 140 dB ) at a frequency of 1 kHz (Device C3).
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83 The results indicate a linear response to acoustic pressure perturbations over four orders of magnitude (60-140 dB ). The measured sensitivity at 1 kHz is 1.415 T 0.0002 !V/V.Pa , or equivalently, -118.4 dB T 1.4 dB ref 1 V/V.Pa . Frequency Response Figure 6-16 shows the magnitude frequency re sponse of the sensor calibrated in the grazing incidence PWT and the normal incidence tube using a constant amplitude tone of 110 dB . The plot indicates a flat frequency response with an average sensitivity of 0.98 !V/(V.Pa) or equivalently 120.2 dB re 1 V/(V.Pa ) with a standard deviation of 1 dB over measured frequency range (compared to a theoretical sensitivity of 1.28 !V/V.Pa ). A flat frequency response is required for correlation an d spectral analysis of the measured data. -120 -110 -100 -90 -80 05101520 Frequency (kHz)FRF (dB ref 1V/Pa) Grazing Incidence Tube Normal Incidence Tube Figure 6-16: Magnitude of the sensor freque ncy response function to a constant sound pressure level of 110 dB (Device C3). The results obtained from the grazing inci dence PWT indicate data scatter at low frequencies. These variations may be attributed to the non-ideal propagation characteristics of the tube, including a co mpliant boundary condition and an area change at the tube termination. The same frequency range (1-6 kHz ) tested in the normal
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84 incidence tube indicates a flat sensor response. Th e scatter observed near 20 kHz is due to the propagation of higher order modes. Acoustic Characterization-Transmitter The directional characteristic of the gene rated acoustic field is shown in Figure 617. The plot indicates a drop in sound pre ssure level with increasing angle from the diaphragm axis. At an angle of 90o the sound pressure level is reduced to 40% of the maximum on-axis value. Theoretical ly, a sensor with Helmholtz number ka<1 should exhibit omni-directional radiati on pattern as explained in Chap ter 4. However, this holds for a radiator mounted on an infinite baffle. The observe d focusing of the sound field may be attributed to the packaging of the transducer, which is recessed in a square depression of 0.5 mm . A detailed schematic of the tr ansducer package is provided in Appendix D. NormalisedSoundPressure 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 110 20 30 40 50 60 70 80 Figure 6-17: Directivity of the generate d acoustic field at a frequency of 60 kHz (Device C3) (0.55ka #). Sound pressure measurements versus distance from the diaphragm surface are plotted in Figure 6-18. A transmitter source level of 50 dB (ref 20 !Pa ) at an operating
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85 frequency of 60 kHz was obtained by extrapolating the graph to 1 m . The plot also shows the characteristic inverse relation between sound pressure level and distance. 60 70 80 90 100 110100 Distance (mm)Sound Pressure Level (dB) Sensor 1 Backside Contacts Sensor 2 Frontside Contacts "1/distance" fit Figure 6-18: Sound pressure level of generate d acoustic field as a function of radial distance, at a frequency of 60 kHz (Device C3 and C9). The far-field for the transducer operating at 60 kHz is established at a distance of 0.15 mm , known as the Rayleigh di stance (Blackstock 2000) 2. 2oka R# (6.3) The oscillations in the data are due to s cattering from the microphone surface that results in the formation of a standing wave betw een the transmitter and the microphone. The scattering effect is reduced with increasing distance from the transmitter. Table 6-5 compares the physical dimensions and operational char acteristics of the micromachined acoustic resonator with that of a conventional ultrason ic transducer. The performance of the commercial transducer is superior in terms of transmitting and receiving sensitivities, however, it should be noted that the design of the micromachined transducer is not optimized. The main goa l of the first-generation acoustic proximity
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86 sensor was prove the manufacturing feasibility of a transducer integrat ed with electrical through-wafer interconnects. Table 6-5: Comparison of the MEMS-based acoustic transducer with a conventional ultrasonic transducer. MEMS-Based transducer (Device C3) Conventional transducer (Massa E-152) Dimensions 1 mm -diameter, 9 !m -thick (resonator) 5 mm x 5 mm (chip size) 11.1 mm -diameter, 10.1 mm -thick (packaged transducer) Resonant frequency 55 kHz 73 kHz Bandwidth (untuned) 1 kHz 1 kHz Transmitting sensitivity (untuned) (ref: 20 !Pa at 0.305 m ) 55 dB (at 60 kHz ) 88 dB (at 73 kHz ) Receiving sensitivity (untuned) (ref: 1 V/Pa ) -101 dB (off resonance)* -42 dB (at resonance) Total beam angle 110 deg 60 deg Power 40 mW 10 mW * Predicted sensitivity at resonance = -81 dB . Proximity Sensing Figure 6-19 compares the true distance of a sound-hard boundary with that obtained using the continuous-wave (CW) phase-shift technique. 0 10 20 30 40 50 01020304050 True Distance (mm)Measured Distance (mm) Initial boundary distance Figure 6-19: Plot of the true distance versus the measured distance obtained using a CW phase-shift technique.
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87 The measured phase of the acoustic wa ve can be related to the time-delay 8 by 2, f ^ %8 # (6.4) where f is the operating frequency. The time de lay together with the knowledge of the acoustic velocity in the medium is used to calculate the distance to the boundary. This experiment, however, is only a proof-of-con cept demonstration of proximity sensing using the acoustic transducers and the actual performance of the system would depend to a large extent on the algorithm used (Barshan 2000, Li et al. 2002).
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88 88 CHAPTER 7 CONCLUSIONS AND FUTURE WORK This chapter summarizes the work presented in this thesis. A first-generation acoustic proximity sensor utilizing thermoel astic actuation and piezoresistive detection with integrated electrical through-wafer el ectrical interconnects has been developed. Concluding remarks are presented and potenti al paths for future work are discussed. Conclusions A thermoelastically actuated acoustic proximity sensor with integrated polysilicon-based electrical through-wafer interconnects has been developed. The rigorous theoretical modeling, fabrication and preliminary characteri zation of the device were presented. The analytical model for th e thermoelastic actuation of the composite diaphragm incorporates for the first time se veral key aspects relevant to micromachined thermal actuators that have not been cons idered in previously reported models. Specifically the composite structure of the actuator has been considered in both the thermal and the mechanical analysis, the diffused heater has been more accurately represented with an internal heat source and finally the effects of in-plane heat conduction and fabrication-indu ced stresses on the vibrati on characteristics of the diaphragm have been included in the analysis . Comparison of the model with a coupled thermal-mechanical finite element simu lation shows excellent agreement with significantly faster computation time for the analytical model. The analytical model developed here can be easily incorporated in to optimization tools in Matlab thus enabling
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89 faster design optimization. The structure of the analytical model is modular and can be easily adapted to include second order eff ects such as convective boundary conditions, compliance of the plate boundary and the eff ects of radiation without significantly altering the analysis procedure. The main goal of the theoretical analysis, however, was to provide a tool that quickly identifies to the first-order the optim um operating point of the resonator given the large number of design parameters. It has been observed from experimental data that maximum sound pressu re level of the radiated acoustic field is obtained for resonators with in-plane compre ssive load very close to the buckling load and the analytical modeling of the vibr ation of a buckled diaphragm may be mathematically very intensive. If the an alytical model provides an estimate of the optimum design parameters, it can be furthe r refined using numerical techniques and incorporating more realistic sensor operati ng conditions. For instance, the analytical model can be used to identify the buckling load for a given set of design parameter. Next using finite element technique s the vibration characteristics of the diaphragm at the buckling load can be calculated while also relaxing the simplifying assumptions of the analytical model. The combination of analy tical and numerical techniques may be more optimal than pursuing only one technique entirely. Preliminary electrical, mechanical and acoustic characterization of the device indicate a transmitter source level of 50 dB (ref 20 !Pa ) at an operating frequency of 60 kHz , a receiving sensitivity of 0.98 !V/(V Pa) , a flat frequency response over the measured range of 1-20 kHz , a linear response from 60-140 dB , negligible leakage current for the junction-isolated diffused piezoresistors (< 14 pA at -10 V ), low
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90 interconnect resistance of 14 "9'and a minimum detectable signal of 36.5 dB for a 1 Hz bin centered at 60 kHz , at a bias of 9 V . While the first-generation acoustic transducer is not optimized in terms of performance and does not yet reach the specifications of a commercial ultrasound transducer, the manufacturing feasibility of the MEMS-based transducer with integrated electrical through-wafer interc onnects has been demonstrated. The advantages of the MEMS-based transducer in te rms of performance and device packaging can be compared to commercial transducers after the devel opment of an optimized second-generation transducer. Future Work Future work in the area of device modeli ng could involve impr oving the analytical model by incorporating more realistic comp liant boundary conditions, effects of damping and geometric non-linearities in the strain-displacement relationship. The physical validity of the model can then be accessed by comparing with direct measurements of the diaphragm vibration obtained us ing a laser vibrometer. Howe ver, a direct comparison of the model with experimental data will requir e more detailed and precise measurements of the transducer geometry and in-plane stresses . Finally, the model could be coupled with formal optimization tools to maximize the sound radiation efficiency of the resonator. The fabrication of the ETWI presented here is a preliminary i nvestigation into ptype ETWI for MEMS sensors/actuators. Accurate C-V characteristics and the ETWI noise floor could not be extracted due to the lack of a proper contact to the bulk silicon substrate in the SOI wafer process. Futu re work should focus on fabrication process modifications to better suit a SOI wafer. In-situ doped polysilicon deposition techniques
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91 could be explored for both the n-type (usi ng a silane:phosphine:nit rogen mixture) and ptype (using a mixture of silane and borontrichloride) ETWI and compared to the existing method in terms of ETWI characteristic s as well as deposition ra tes. Planarity is a critical issue for integra tion of the pre-process ETWI with an accompanying MEMS sensor/actuator. Techniques such as chemical mechanical polishing c ould be investigated towards this end. For performance enhancement in a pr oximity-sensing scheme, directional acoustic sources could be investigated by incr easing the size of the diaphragm relative to the generated acoustic wavelength. This can potentially reduce cross-talk between the transducers enabling faster algorithms. Sepa rate transmitters and receivers should be considered instead of an in tegrated transducer since the optimum diaphragm design for the generation of acoustic waves and its detec tion are significantly different. There is a significant level of capacitive coupling (on th e order of the acousti c signal) between the transducers (transmitter and receiver) that needs to the addressed through better packaging and by modifying the transducer fa brication process to include a top-side metal layer to shield the electromagnetic signal.
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92 92 APPENDIX A PROCESS TRAVELER The process flow for the fabrication of th e integrated acoustic proximity sensor and ETWI is presented. Wafer : 4 n-type (100) SOI, 4000 Å -thick buried oxide layer, 12 !m -thick silicon overlayer, 3-5 ohm-cm . Masks: " Labels LM " Bond pads BPM " Holes HM " Piezoresistors PRM " Heater HTM " Contact openings through oxide front surface C1M " Contact openings through oxide back surface C2M " Metal front surface MIM " Metal back surface M2M " Contact through nitridePM " DRIE DRM " Scribe lines SM " Signal mask SGM " Substrate contact SCM Process Steps Start with SOI wafer (n-Si CZ <100> 3-5 #-cm ) with 13 !m silicon on 4000 Å BOX. Scribe and etch a. Scribe wafers. b. DI rinse. c. Measure wafer thickness. 1. Pattern and etch holes for bond pads and through wafer vias
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93 93 a. Pre-oxidation clean. b. Grow 1000 Å of dry SiO2 at 1100 oC on both sides of the wafer. c. Coat and pattern resist/oxide on scri bed side (BPM). This step puts alignment marks in the oxide. Singe 150 oC for 15 min Coat HMDS 2000 rpm for 30 sec Coat resist (1m , SPR3612) spin at 5700 rpm for 45 sec Align (EV) using BPM for 2 sec , hard contact Develop Bake 110 oC for 60 min Descum (Drytek) 30 sec to clear holes BHF (6:1) 90 sec (etch rate 900 Å/min ) Piranha 20 min , dump rinse, spin dry d. Coat and pattern resist/oxide on non-sc ribed side (BPM). This step puts alignment marks in the oxide. e. Coat and pattern resist using the th rough hole mask on both sides (HM). Align with alignment ma rks of step 2c-2d. Singe 150 oC for 5 min Coat 10 m on scribed side HMDS on both sides Coat resist (SPR220) at 1700 rpm for 70 sec and 5000 rpm for 1 sec for edge bead removal Bake on hotplate 120 sec Coat 10 m on non-scribed side Softbake 110 oC oven for 90 min Let wafers sit in air for 10+ hrs Expose with through hole mask on scribed side (HM) Align (EV) in hard contact mode and expose for 30 sec Expose with through hole mask on non-scribed side (HM) Develop both sides LDD26W by hand ~ 3-5 min Note: Dont do post bake, as it tends to crack the resist (either immediately or in the STS etcher) 2. Etch through-wafer vias (TWV) usin g deep reactive ion etch (DRIE) a. Etch alignment marks. STS etch 2 min on scribed side and 2 min on non-scribed side b. Etch from scribed side up to the BOX tape over major resist blemishes.
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94 94 c. Etch oxide using buffered HF (or Plasma). d. Etch from non-scribed side halfwa y tape over resist blemishes. STS etch for ~210 min e. Continue with step (b) finish TWV. STS etch for ~210 min and 30 min timed overetch with support wafer f. Ash strip resist. 3. Clean etch polymer Acetone, isopropanol and blow dry 5 min oxygen plasma 20 min piranha 5 min oxygen plasma 4. Deposit thin films a. Deposit LPCVD-oxide isolation layer. Oxidation diffusion clean su lfuric/peroxide, HF, HCL/peroxide Deposit 2 m oxide (wet) at 1150 oC b. Deposit LPCVD-polysilicon ground conduction layer. LPCVD diffusion clean Deposit 2 m polysilicon 2 hrs and 45 min Inspect make sure light passes through all holes c. Dope polysilicon with boron. LPCVD diffusion clean Boron dope polysilicon d. Clean borosilicate glass formed ove r the polysilicon and diffusion clean. e. Repeat polysilicon depos ition and doping (4b 4d). 5. Fill TWV holes a. Deposit 3 m of polysilicon. b. Dope surface polysilicon with boron. LPCVD diffusion clean Boron dope polysilicon Clean borosilicate glass formed over the polysilicon and diffusion clean c. Drive in boron.
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95 95 LPCVD diffusion clean Anneal at 1000 oC for 1 hr 6. Pattern polysilicon bond pads a. Coat both sides with 10 m resist. b. Expose scribed side with SGM mask (Clear field). c. Expose non-scribed side with SGM mask (Clear field). d. Develop. e. Plasma etch polysilicon from the top surface, stopping on the oxide. f. Etch polysilicon from the backside, stopping on the oxide. Note: Repeat (e) and (f) in steps of 1-2 m , to prevent wafer bow. g. Strip resist. h. Etch pads to a height of 1-2 m or if possible make it planar with the oxide (use profile-meter to measure height). 7. Pattern and diffuse heaters Spin photoresist on front surface and pattern heater mask. Alignment marks make sure that the heater resistor is aligned to the center of the diaphragm. This mask also creates alignment marks for the next layer. a. Coat and pattern heater mask on scribed side (HTM) using SPR220. b. Etch oxide. c. Heater implantation. Ion implant dopant: boron, energy: 170 keV , dose: 1e13 cm-2 7 degree tilt, blanket front su rface using photoresist as a screen (This forms the p++ heater of the resonator) d. Ash photoresist. e. Anneal at 1100 oC for 130 min in inert ambient. 8. Pattern and diffuse piezoresistors Spin photoresist on front surface and pattern piezoresistors mask. Alignment marks make sure that the piezoresistors are aligned to the edge of the diaphragm. This mask also creates alignm ent marks for the next layer. a. Coat and pattern piezoresistor ma sk on scribed side using SPR 220. b. Etch oxide.
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96 96 c. Piezoresistor implantation. Ion implant dopant: boron, energy: 85 keV , dose: 8E13 cm-2 d. Ash photoresist. 9. Oxide growth Pre-oxidation clean top and bottom surfaces. Grow 7000 Å of high-quality passiva tion oxide (dry-wet-dry) Temp = 950 oC , time=100 min Temp = 950 oC , time=245 min Temp = 950 oC , time=100 min 10. Substrate contact a. Spin photoresist on front surface and pa ttern substrate contact (SCM) mask. b. Etch Oxide. c. Ion implantdopant: phosphorus, energy: 85 keV , dose: 8E13 cm-2. d. Ash strip resist. e. Anneal at 900 oC for 15 min in inert ambient. 11. Pattern front and back contact cuts a. Spin photoresist on front and back su rfaces and pattern contact cut masks (CIM & C2M). b. De-scum using oxygen plasma Note: contact holes PR, 1 min to insure clearing. c. Etch oxide. d. Ash photoresist. 12. Metallization a. Pre-metal piranha clean. Very short HF dip, do not remove more than 50 Å . b. Sputter 1.4 m Al (1% Si) to avoid "spiking". c. Coat both sides with 10 m resist. d. Softbake 110 oC oven for 90 min . e. Let wafers sit in air for 10+ hrs . f. Expose scribed side with M1M.
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97 97 g. Expose non-scribed side with M2M. h. Develop both sides. i. Etch aluminum both sides. j. Clean up Al/Si etch freckle etch fo r a few minutes (to remove silicon precipitates). k. Strip resist. Acetone, isopropanol and blow dry O2 asher l. Forming gas anneal. 13. Nitride passivation Deposit PECVD-nitride 3300 Å on the top surface. 14. Pattern nitride contacts a. Spin photoresist on front surface and pattern contact cut mask (PM). b. Etch nitride dry etch. c. Ash photoresist. 15. DRIE for membrane release (will include cleave lines) Spin photoresist on both surfaces and pattern back surface for DRIE (DRM). This produces the thin silicon diaphr agm of the proximity sensor. a. Coat and pattern diaphrag m mask on non-scribed si de (DRM) using SPR 220. b. Plasma etch silicon (~400 µm ) up to the BOX. The burie d oxide serves as the etch stop for the DRIE. c. Etch oxide (6:1 BOE). d. Ash photoresist.
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98 98 APPENDIX B PIEZORESISTOR DESIGN FLOOPS® Input Files The FLOOPS® input files used for simulating the process of ion implantation and anneal are included here. (A) Heater This program simulates the final (boron) dopi ng profile in the sili con layer after ion implantation, activation anneal and th e growth of a thermal oxide layer ****************************************************** line x loc=-0.005 tag=oxi spacing=0.01 line x loc=0 tag=top spacing=0.01 line x loc=3 spacing=0.01 tag=bot1 line x loc=3.1 spacing=0.01 tag=bot2 region oxide xlo=oxi xhi=top region silicon xlo=top xhi=bot1 region silicon xlo=bot1 xhi=bot2 init strip oxide implant boron dose=1e13 energy=170 sel z=log10(Boron+1) plot.1d diffuse temp=1100 time=130 diffuse temp = 950 dry time = 100.0 diffuse temp = 950 wet time = 245.0 diffuse temp = 950 dry time = 100.0 sel z=log10(Boron+1) plot.1d bound !cle set cout [open resdata2 w] puts $cout [print.1d] close $cout sel z=log10(8.79e14) plot.1d !cle ******************************************************
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99 (B) Piezoresistors This program simulates the final (boron) dopi ng profile in the sili con layer after ion implantation, activation anneal and th e growth of a thermal oxide layer ****************************************************** line x loc=-0.005 tag=oxi spacing=0.01 line x loc=0 tag=top spacing=0.01 line x loc=3 spacing=0.01 tag=bot1 line x loc=3.1 spacing=0.01 tag=bot2 region oxide xlo=oxi xhi=top region silicon xlo=top xhi=bot1 region silicon xlo=bot1 xhi=bot2 init strip oxide implant boron dose=8e13 energy=85 sel z=log10(Boron+1) plot.1d bound !cle diffuse temp=900 time=15 diffuse temp = 950 dry time = 100.0 diffuse temp = 950 wet time = 245.0 diffuse temp = 950 dry time = 100.0 sel z=log10(Boron+1) plot.1d bound !cle set cout [open prdata w] puts $cout [print.1d] close $cout sel z=log10(8.79e14) plot.1d !cle ******************************************************
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100 100 Resistance Calculation This section provides details of the discre tization technique used to calculate the resistance of an arc-shaped piezoresistor under zero load condition. Figure B-1: A schematic of the discretized ar c piezoresistor and the equivalent resistor model illustrating the series connection of the elements in the circumferential direction and the parallel connection of the segments in the radial direction. The resistor is discretized into infini tesimal elements (Figure B-1) of length, dlrd and area, , dAtdr where r is the radius of the cente r point of the element and t is the thickness of the resistor. The resistance of an individual element is given by ,elementrd R tdr (B.1) where is the resistivity. For a given arc angle, d the resistors are in parallel in the radial direction and in series in the circumferential direction as shown in Figure B-1. The Element of the arc resistor Segment element
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101 total resistance of the elements in series can be obtained by adding the resistances of the individual elements, which in the lim iting case is replaced by an integral 0.segmentrdr R tdrtdr1 1 1 --##> (B.2) The total resistance of the piezoresistor is then obtained by adding (integrating) the segments in parallel 1 ,b ar arc rtdr R r 1# > (B.3) that results in 67 , lnarc baR trr 1# (B.4) where ar and br represent the inner and outer radius of the resistor and 1 is the arc angle. A similar procedure is adopted for the tapered resistor 6 7 ln .ba taperrr R t1# (B.5) The resistance modulation of the individual elements is obtained by substituting the values of % , 4 and R in the following equations ()()()()rltt taperR rr R 4 %14%1 ) #; (B.6) and ()()()().tlrt arcR rr R 4 %14%1 ) #; (B.7) After calculating the resistance modulation of an individual element the procedure described above is used to calculate the tota l resistance modulation of the piezoresistor. An iterative process is used to arrive at th e optimum arc and tapered piezoresistor sizes (Figure B-2) that have equal mean re sistance values and equal but opposite sign modulations.
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102 102 Figure B-2: Piezore sistor geometry and layout (all dimensions are in mm ).
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103 103 APPENDIX C THERMAL ACTUATION OF A COMPOSITE DIAPHRAGM In the following sections, the 2-D temperature distribution in the composite diaphragm is calculated using the Fourier heat conduction mode l, which is then used to derive the thermoelastic forcing functions . The plate governing equations are then formulated using the equations of moti on and the linear thermoelastic constitutive relations. Heat Conduction Model Figure C-1: Axisymmetric model of the composite diaphragm consisting of three transversely isotropic layers and a diffused resistive heater (H1, H2, H3 and HHeat represent distances from the reference plane). The governing equation for heat conduction in the individual layers of a 3-layered composite diaphragm (Figure C-1) based on Fouriers law is gi ven by (Ozisik 1993) 2 1 1111 11 (,,);0, T TgrztzH kt (C.1) 2 2 2212;, T THzH t (C.2) b a r H2 H3 HHeat (2) (3) (1) Diffused Heater (0,0) z H1
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104 and 2 3 3323;, T THzH t, ? .#[[ ? (C.3) where ,' and k represent the thermal diffusivity and the thermal conductivity of the individual layers, 1(,,) grzt represents an internal heat source within layer 1 6767676711 0,,cos,HeatH b o Hgrztgrdrzdzt33$#>> (C.4) and 2. is the axisymmetric Laplacian operator in cylindrical coordinates, 2 2 21 .r rrrz ? ?? @A .#; BC ? ?? DE (C.5) The diaphragm is assumed to be thermally insulated on the top and bottom surfaces, 1 00zT z#? # ? and 330.zHT z#? # ? (C.6) Additionally, the heat conduction in the diaphragm is assumed to be symmetric about the center, and the edge is maintained at ambient temperature, 00i rT r#? # ? and .i raTT F ## (C.7) The layers are assumed to be in perfect th ermal contact denoted by the continuity of temperature and heat flux, 12 12121 ; at TT TTkkzH zz ? ? # ## ?? (C.8) and 3 2 23232; at=. T T TTkkzH zz ? ? ## ?? (C.9) The solution to the two-dimensional, transi ent heat conduction pr oblem described above is obtained using Greens functions. In or der to simplify the solution procedure, a variable transformati on is first performed . TTT F #<) (C.10) This transformation shifts the temperature at the diaphragm edge to zero, thus making all the boundary conditions homogenous.
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105 Greens Function Solution Technique The use of Greens function for solvi ng non-homogenous, transient heat conduction problems with energy generation, non-homoge nous boundary and initial conditions can be found in several references (Morse and Feshback 1953, Carslaw and Jaeger 1986, Beck et al. 1992, Ozisik 1993). In this se ction the technique is used to solve a twodimensional, transient heat conduction probl em with internal energy generation in a three-layer composite medium. In order to define a Greens function the following auxiliary problem is considered in the same region 676767 21i iiG Grrzztt kt , 33388? @A (( .;<<<#S BC ? DE (C.11) with the requirement that 0iG # for t 8 [ known as the causality condition (Morse and Feshback 1953). This is similar to Equa tion (C.1) except the s ource term in Equation (C.11) is represented by a delta function th at defines a line source located at (,) rz (( releasing its energy instantaneously at t=8. The boundary conditions are the homogenous versions of the original problem. In this case, they are similar to Equations (C.6) (C.9), 1 00zG z#? # ? , 330,zHG z#? # ? 00i rG r#? # ? , 0,i raG# # 12 12121; at , GG GGkkzH zz ? ? ### ?? and 3 2 23232; at=. G G GGkkzH zz ? ? ## ?? The function 67,,,,iGrztrz 8 (( satisfying the auxiliary problem (Equation (C.11)) is referred to as the Greens function for th e two-dimensional tran sient heat conduction
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106 problem described by Equations (C.1) (C.3). The physical significance of the Greens function is that it represents the temperature at any location ( r, z ) and at time t due to an instantaneous source at (,) rz (( releasing its energy at t=8, in a region that is initially ( t<8) at zero temperature and with homogenous boundary conditions. The elegance of this technique is that the solution to the orig inal problem can be represented entirely in terms of the Greens function and the same function can be used to handle additional non-homogeneities arising from the boundary or initial conditions provided the geometry of the problem remains unaltered. Theref ore, once the Greens function is known, the temperature distribution in the re gion can be readily computed. The derivation of the temperature distribu tion or in other words the derivation of the solution to the non -homogenous transient heat conduction problem in terms of the Greens function can be found in several refe rences (Morse and Feshback 1953, Carslaw and Jaeger 1986, Beck et al. 1992, Ozisik 1968 ) and is presented here for the case of a three-dimensional problem with non-homogenou s boundary conditions and internal heat generation (the derivation has been adapte d from Ozisik (1968)). The governing equations satisfied by T (Equations (C.1) (C.3)) and G (Equation (C.11)) are expressed as functions of (r and 8'9 211 (,),, T Tgt k88 ,8 ? (( .;#[ ?r (C.12) and 6767 211 , G Gtt3388 ,,8 ? (( .;<<#~~>>>> >>rrr rrrr (C.15) The volume integral in Equation (C.15) is replaced by a surface integral via Greens theorem 67 22,ll RTG GTTGdGTds nn @A ?? ((( .<.#< BC ?? DE>>r (C.16) where ln ?? represents differentiation along a direction normal to the boundary surface l . The term involving delta functions can be simplified to 67676767,,,RtdTdTt838838((( <<#>>rrrrr (C.17) and evaluating the term on the right -hand side at the limits gives H I 67000 0 tGTGTGF888### ( #<#>>>>r rrrr(C.19) where the first term on the right-hand side is the contribution of th e initial condition and the second and third terms represent contri butions from the energy generation and nonhomogenous boundary conditions. For the ca se of a composite region the Greens function G is replaced by Gij, where i represents a layer in the composite (Figure C-1) and j represents the contribution of the non-homoge nous terms in the layers. For example, G12 represents the Greens function in layer 1 du e to an impulsive source in layer 2 or in other words the temperature distribution in laye r 1 due to an impulsive source in layer 2. Thus, for a two-dimensional transient heat c onduction problem in a three-layer composite slab with energy generation in layer 1, zero initial c ondition and homogenous boundary conditions the solution is of the form (Ozisik 1993) 67 67 671 1 1 0,,,,,,,,.t ii rzTrztGrztrzgrzrdrdz k8, 888(( # ( (((((( #?>>> (C.20) The next step in the solution procedure is to determine the analytical form of the Greens function. Several techniques have been reported in the literature for determining Greens functions including Laplace transf orms, method of images and separation of variables (Morse and Feshback 1953, Ozisik 1968, Carslaw and Jaeger 1986, Beck et al. 1992). This analysis is based on the separa tion of variables technique adapted from Ozisik (1968). Separation of variables is a commonly used technique for solving homogenous transient heat conduction problems subject to a prescrib ed initial condition and it has been shown by Ozisik (1993) that the solution to the homogenous problem can be rearranged to obtain the Greens function at 0 8 # , 6 7,,,,0 Grztrz ( ( . Additionally, by
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109 replacing t with (t-8) in the homogenous solution, the Greens function for the nonhomogenous transient heat conduction problem can be obtained. Thus the Greens function solution technique requires the solution to the homogenous heat conduction problem in the composite medium described by 67 6 7 2,, ,,,1,2,3,i iirzt rzti t1 ,1? .## ? (C.21) with homogenous versions of the boundary conditions described by Equations (C.6) (C.9), 1 00zz1#? # ? , 330,zHz1#? # ? (C.22) 00i rr1#? # ? , 6 70,ia1 # (C.23) 12 12121; at , kkzH zz 1 1 11 ? ? ### ?? (C.24) and 3 2 23232; at=, kkzH zz 1 1 11 ? ? ## ?? (C.25) and a constant initial condition of 6 70o 1 1 # throughout the region. To solve Equation (C.21) analytically, it is first simplified by reducing the number of independent variables to two, such that 67, zt11#. The r variable is eliminated via the application of finite Hankel transform (Sneddon 1951) on the governing equation and the boundary conditions. The transformed equations are th en solved using separation of variables. Finite Hankel Transform The technique of finite Hankel transform is used to eliminate the r variable from Equation (C.21) thus reducing the number of independent variables. The Hankel transform and its inverse pair are mathematically defined as
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110 6767670,,,,a omztrrztJrdr1/1/#> (C.26) and 67 6 7 67 2 2 12 ,,,om ii m mJr rzt a Ja/ 11 /# J K N OG (C.27) where m / are roots of 670omJa/ # and is based on the radial boundary conditions for the problem. The transforme d Laplacian operator in r , 2 r . is reduced to 67 2 2 2 01a ommrJrdr rrr11 / /1@A ?? ;#< BC ?? DE> (C.28) for a boundary condition of ()0 ra 1 # #. Rewriting Equation (C.21) in terms of the transformed variable 67,, zt1/ gives 2 2 2.ii miiizt 1 1 /,1, ? ? <;# ? ? (C.29) Equation (C.29) can now be solved usi ng the separation of variables technique. Assuming 67 6 7 6 7,,iiztZzt1/#_, the separated equation is of the form 2 22 211 .i im iZ Zzt , /0@A ? ?_ < ##< BC ?_? DE (C.30) where 0 is the separation constant. The solution for 6 7t _ is of the form 6721 ttce 0 <_# (C.31) and 67,,i Z z 0 / satisfies the following eigenvalue problem 2 2 2 20,i mi iZ Z z0 / ,@A ? ; <# BC ? DE (C.32) subject to 10 at 0, Z z z ? # # ? (C.33) 12 12121; at , ZZ Z ZkkzH zz ? ? ### ?? (C.34) 3 2 23232; at , Z Z Z ZkkzH zz ? ? ### ?? (C.35)
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111 and 3 30 at . Z zH z ? ## ? (C.36) The general solution to th e eigenvalue problem is 67 11 22 22 22sincos,1,2,3,nmnm inminmminmm iiZzAzBzi00 // ,,@A@A #<;<# BCBC DEDE (C.37) where i is an individual la yer of the composite, 6 7,inminm Z Zz0` and the subscript nm implies an infinite number of discrete eigenvalues 123 mmmnm 0 000 [ [[* for every value of 6712 mm / ///[[* and corresponding eigenfunctions .inmZ The six coefficients ( A1,2,3 and B1,2,3) in Equation (C.37) are determined from the six boundary and interface conditions. However, the coefficients can be determined only as a multiple of an arbitrary constant, since the system of equations (Equations (C .33) (C.36)) is homogenous. The transcendental equation fo r the determination of the eigenvalues 0nm results from the requirement of a non-triv ial solution for the system of homogenous equations. The eigenvalues are therefore obtained by setting the determinant of the coefficients A1,2,3 and B1,2,3 equal to zero. Substituting the eigenfunction into Equation (C.33) results in 1111 2222 2222 2222 11 1111cos0sin00,nmmmnmmmAB0000 //// ,,,,@A@A @A@A@A@A BCBC <<<<<# BCBCBCBC BCBC DEDEDEDE BCBC DEDE (C.38) which implies 10nmA#. We set 11nmB # and solve for the other coefficients in terms of 1. B Substituting the eigenfunction into th e interface continu ity conditions at 1zH# (Equation (C.34)) for temperat ure and heat flux gives
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112 67 111 222 222 222 12121 1221cossincos,mnmmnmmHAHBH000 /// ,,,@A@A@A @A@A@A BCBCBC <#<;< BCBCBC BCBCBC DEDEDE BCBCBC DEDEDE (C.39) and 111 222 222 222 1212121 122sincossinmnmmnmmkHAHBH000 /// ,,,@A@A@A @A@A@A BCBCBC <<#<<< BCBCBC BCBCBC DEDEDE BCBCBC DEDEDE (C.40) respectively, where 11 22 22 22 1212 12.mmkkk00 // ,,@A@A #<< BCBC DEDE Similarly, substituting the eigenfunction into the interface continuity conditions at 2zH # (Equation (C.35)) gives 11 22 22 22 2222 22 11 22 22 22 3232 33sincos sincosnmmnmm nmmnmmAHBH AHBH00 // ,, 00 // ,,@A@A @A@A BCBC <;<# BCBC BCBC DEDE BCBC DEDE @A@A @A@A BCBC <;< BCBC BCBC DEDE BCBC DEDE (C.41) and 11 22 22 22 232222 22 11 22 22 22 3232 33cossin cossin,nmmnmm nmmnmmkAHBH AHBH00 // ,, 00 // ,,JK @A@A @A@A LM BCBC <<<# BCBC LM BCBC DEDE BCBC LM DEDE NO @A@A @A@A BCBC <<< BCBC BCBC DEDE BCBC DEDE (C.42) where 1 1 22 2 2 22 2323 23.mmkkk00 // ,,@A @A #<< BC BC DE DE The boundary condition at 3zH# gives
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113 11 22 22 22 3333 33cossin0.nmmnmmAHBH00 // ,,@A@A @A@A BCBC < <<# BCBC BCBC DEDE BCBC DEDE (C.43) Equations (C.38) (C.43) can be represented in matrix form HI1 2 2 3 31 0 0 0 0 0 B A B M A B # JK J K LM L M LM L M LM L M # LM L M LM L M LM L M N O NO (C.44) where HI 111 222 222 222 111 122 111 222 222 222 12111 122cossincos00 sincossin00 0mmm mmmHHH kHHH M000 /// ,,, 000 /// ,,,@A@A@A @A@A@A BCBCBC <<<<< BCBCBC BCBCBC DEDEDE BCBCBC DEDEDE @A@A@A @A@A@A BCBCBC <<<<< BCBCBC BCBCBC DEDEDE BCBCBC DEDEDE # 11 11 2222 22 22 2222 2222 2233 11 22 22 22 232232 22sincossincos 0cossincosmmmm mmHHHH kHkH0000 //// ,,,, 00 // ,,@A@A @A@A @A@A @A@A BCBC BCBC <<<<<< BCBC BCBC BCBC BCBC DEDE BCBCDEDE BCBC DEDE DEDE @A@A @A@A BCBC <<<< BCBC BCBC DEDE BCBC DEDE 11 22 22 22 22 33 11 22 22 22 33 33. sin 000cossinmm mmHH HH00 // ,, 00 // ,,J K L M L M L M L M L M L M L M L M L M L M L M L M L M L M @A@A @A@A L M BCBC << BCBC L M BCBC DEDE BCBC L M DEDE L M @A@A L M @A@A BCBC L M <<< BCBC BCBC L M DEDE BCBC L M L M DEDE N O (C.45) The complete solution for the transformed variable 1 in any layer i is then constructed as 67672,,,nmt imnmi nztCeZz01/<#G (C.46) which satisfies the differential Equation (C .29) and the boundary conditions and it is further constrained to sa tisfy the initial condition 6767,.omnmi nzCZz1/#G (C.47) The constant nmC is obtained by using the orthogonalit y relation of the eigenfunctions and is of the form 67 11 ,i iz i nmio i ni zk CZzdz N1 ,;#G > (C.48)
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114 where 67 67 673 12 12222 3 12 123 123 0.H HH n HHk kk NZzdzZzdzZzdz,,, ( ((((( #;;>>> (C.49) Taking the inverse Hankel transform, defined by 67 6 7 67 22 12 ,,om ii m mJr rzt aJa/ 11 /#G of Equation (C.46) gives the hom ogenous temperature distribution 67,, rzt1 as 67 67 67 67 67 67 67 672 3 12 12. 22 1 3 12 123 123 0021 ,,nmt om ii mn mn H HH a oom HHJr rzteZz aJaN k kk rJrdrZzdzZzdzZzdz0/ 1 / 1/ ,,,<# K J ((((((((( ;; M L L M N OGG >>>>. (C.50) The solution of the homogenous problem define by Equation (C.21) can now be expressed in terms of Greens functions as 67 6767 6767 67671 3 2 1 211 0 00 2233 00 00,,,,,,, ,,,,,,,,,,.H a ii zr H H aa ii zHrzHrrztGrztrzFrzdrdz GrztrzFrzdrdzGrztrzFrzdrdz8 8818 88# (( ## ## (((( ####(((((( #; (((((((((((( ;>> >>>> (C.51) where 67,iFrz is the initial condition in a given la yer. Comparing Equation (C.51) with Equation (C.50) we can extract the Green s function for the non-homogenous problem 67 6 7 67 6 7 67 67672() 1 1 1 22 112 ,,,,.nmt om i omi mn mnmrJrZz k GrztrzJrZze aJaN 0 8/ 8/ ,/0<< ( (( (( #GG Using the Greens function the steady-state te mperature distribution in the individual layers of the composite diaphragm resulting fr om the time-harmonic joule heating of the diffused heater is evaluate d using Equation (C.20) as
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115 67 6767 67 67 67 67 111 222 12 ,,ReheatH jt omi iom mn mmnmnm HJrZz be TrztgJbZzdz aJaNj$/ / / /00$@A @A (( # BC BC BC ; DE DEGG > (C.52) where a and b are the radius of the diaphragm and the heater, respectively, go ( W/m3) is the magnitude of the heat source £, nm 0 and m / are eigenvalues, Jo and J1 are Bessel functions of the first kind, $ is the frequency of the time-harmonic heat source. Once the unsteady temperature field is known, the integrated thermal forces and moments can be computed. In the following sections, the transverse vibration of the composite diaphragm resulting from the time-varying two-dimensi onal temperature distri bution is derived. Plate Analysis The equilibrium equations for any radial sect ion of the axisymmetric plate (Figure C-2) are given by (Timoshenko and Woinows ky-Krieger 1959, Leissa 1993, Reddy 1996) 67 671 0rrNNN rr1 ? ; <# ? (C.53) 67 671 0rrrMQMM rr1 ? < ;<# ? (C.54) and 67 11 ,roAw rQNrw rrrrr-??? @A <# BC ??? DE!! (C.55) where 22wwt#??!!, rN and N 1 are force resultants (Figure C-2) in the radial and circumferential directions ;.rr zzNdzNdz1144##>> (C.56) Similarly, r M and M 1 are the moment resultants (Figure C-2) £ A joule heat fraction of 1 was assumed in the ther momechanical model, i.e., the thermal power was set equal to the electrical power I2R .
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116 ;,rr zz M zdzMzdz1144##>> (C.57) rQ is the transverse shear force resultant (Figure C-2) 1,rrz zQdz8#> (C.58) and A is the areal density (kg/m2) of the composite plate .A zdz--# > (C.59) In the equations of motion only the inertial term associated with the transverse acceleration of the plate has been retained and the initial in-plane compressive force,roNNN1##, is assumed to be much larger than the incremental forces induced by the deflection of the plate. r rN Ndr r ? ; ? r rM M dr r ? ; ?rN N 1 r M M 1 rQ r rQ Qdr r ? ; ? N Nd1 1 1 1? ; ? M M d1 1 1 1? ; ? d 1 h 0 z # Figure C-2: Force and moment resultants on an element of the circular plate. The radial and transverse displacements based on Kirchoffs plate theory are 6767 ,,,ow urzturtz r ? #< ? (C.60) and 6 7 6 7,,,,owrztwrt # (C.61) where the radial displacement u is a linear function of the axial coordinate z and the transverse displacement w is independent of thickness. The subscript o denotes
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117 reference plane values. The radial and ci rcumferential strain-displacement relationship can then be expressed as o rrrrrz / /5#; (C.62) and ,oz 1 1111 / /5#; (C.63) where r5 and 1 5 are the radial and circumferential curvatures 2 2rw r5 ? #< ? and 1 . w rr15 ? #< ? The strains in the reference plane (0) z # are o o rru r/ ? # ? and .o ou r11/# The thermoelastic stress-strain relationships for a transversely isotropic, linear elastic material are 671112 1222,,,o rrrr rr oQQ zTrzt QQ1111 11452 / 452 /@A UV UVUVUV JK #;< BC WXWXWXWX LM BC NO YZYZYZ YZ DE (C.64) where [ Q ] is the material stiffness matrix, r 1 2 22 # # is the coefficient of thermal expansion and 67,, Trzt is the non-uniform temperature di stribution. For a transversely isotropic material the stiffness matrix is 21 [], 1 1 E Q * * * J K # L M < N O where E is the Youngs modulus and * is the Poissons ratio distributions in the composite plate. The force and moment resultants are obtained by integrating the constitutive relations (Equati on (C.64)) through the thickness of the composite plate and are given by
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118 HIHIoT rr rrr oTN N AB N N11 1115 / 5 / U VUV UVUV #;< WXWXWXWX YZYZ YZYZ (C.65) and HIHI,ToT r rrrr ToTMM BD MM1 11115 / 5 /UVUVUV UV #;< WXWXWXWX YZ YZYZYZ (C.66) where HIHIzAQdz#> is the extensional stiffness matrix, HIHIz B Qzdz#> is the flexuralextensional coupling matrix and HIHI2zDQzdz#> is the flexural stiffness matrix. The thermoelastic coupling generates both a thermal force 67HI,,T r T zN TrztQdz N12JK # LM NO> (C.67) and a thermal moment 67HI,,.T r T zM TrztQzdz M12JK # LM NO> (C.68) The governing displacement equations are deri ved by substituting the strain-displacement relations (Equations (C.62) and (C.63)) and the constitutive relations (Equations (C.65) and (C.66)) into the Equations of motion (Equations (C.53) (C.55)). The resulting equations in terms of the radial and tran sverse displacements are of the form: 672 32 1111 22322111T ooo ruuu www ABN rrrrrrrrrr UV UV ?? ???? ;<<;<# WXWX ?????? YZ YZ (C.69) and 67 6732 432 1111 322343223 22 2221211 11 .oooo TT orrAuuuu wwww BD rrrrrrrrrrrrr ww NMMw rrrrrr-UV UV ??? ???? ;<;<;<; WXWX ??????? YZ YZ @A ???? <;#;; BC ???? DE!! (C.70) Replacing ou in Equation (C.70) with w from Equation (C.69) a nd simplifying results in a governing equation exclusively in te rms of the transverse displacement w
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119 676711 *4222 11,TT oArrB DwNwwNM A-.;.;#.<.!! (C.71) where 2 11 * 11 11 B DD A#<. Equation (C.71) represents the thermally forced vibration of the composite plate where the forcing functions arise from the non-uniform temperature and material property distributions in the plat e. The general soluti on to Equation (C.71) consists of two parts, a complementary solu tion to the homogenous part of the equation and a particular solution sa tisfying the forcing functions. The forcing functions are derived from the temperature field 67676767,,Re,jt ionmmomi mnTrztgfeJrZz$0//@A # BC DEGG (C.72) where 67 67 67 67 6711 1 222 12 11 ,..heaterH m nmm nmmmnm HbJb f Zzdz jaJaN/ 0/ 0$//0@A @A @A (( # BC BC BC BC BC ; DE DE DE> (C.73) The generated thermal force and moment accord ing to Equations (C.67) and (C.68) are 67676767671()() 1112,,i iH Tjtii ronmmomii mni HNrtgfeJrQQZzdz$0//2<@A #; BC BC DEGGG > (C.74) and 67676767671()() 1112,,.i iH Tjtii ronmmomii mni H M rtgfeJrQQZzzdz$0//2<@A #; BC BC DEGGG > (C.75) Therefore the governing e quation is of the form, 676742 ***1 ,jt o A nmmom mnN wwweJr DDD$ P 0//.;.;#GG!! (C.76) where
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120 67676767 67671 1()() 1112 2 11 ()() 1112 11,,.i i i iH ii ii i H nmmmnmmo H ii ii i HQQZzzdz fg B QQZzdz A2 P0//0/ 2< <@A @A BC ; < BC BC BC DE # BC @A BC ; BC BC BC BC DE DEG > G > (C.77) Assuming a harmonic time-dependence for the plate deflection, of the form 6 7 6 7,Re(),jtwrtwre$# (C.78) results in a differential equation in terms of the spatial coordinates 67672 42 ***1 ()()(),.o A nmmom mnN wrwrwrJr DDD-$ P0//.;.<#GG (C.79) The homogenous part of the governing E quation (C.79) can be expressed as 6 7 6 722220, wRQ . <.;# (C.80) where 2* 2 *24 11 2o A oN D DN-$ R J K # ;< L M L M N O (C.81) and 2* 2 *24 11. 2o A oN D DN-$ Q J K # ;; L M L M N O (C.82) The complete solution to the homogenous equation is obtained by superimposing the solutions to the equations 6 722 10 wQ . ;# (C.83) and 6 722 20. wR . <# (C.84) Equations (C.83) and (C.84) represent the 0th order Bessels equation and Modified Bessels equation respectively and ha ve general solutions of the form 6 7 6 7 6 7112 oowrcJrcYr Q Q#; (C.85) and 6 7 6 7 6 7234.oowrcIrcKr R R#; (C.86) Thus the complementary solution to Equation (C.79) is of the form
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121 67 6 7 6 7 6 7 6 71234.oooowrcJrcYrcIrcKr Q QRR#;;; (C.87) In order to satisfy the forcing functio ns, a particular solution, of the form 6 7 6 75 pomwrcJr / # (C.88) is assumed, where 5c is an arbitrary constant. The part icular solution is then substituted into Equation (C.79) and simplified to solve for the constant 5c 6 7 5 * 4222 *, 1 .nmm o mn mmc N D DP0/ / /RQ# <~~