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PAGE 1 1 DESIGN, FABRICATION AND CHARACTERI ZATION OF A MEMS PIEZORESISTIVE MICROPHONE FOR USE IN AEROACOUSTIC MEASUREMENTS By BRIAN HOMEIJER 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 2008 PAGE 2 2 2008 Brian Homeijer PAGE 3 3 To my family, especially my loving wife, Sara who has been my inspiration and rock through this long process and my parents, who have always encouraged and supported me. PAGE 4 4 ACKNOWLEDGMENTS Financial support for this work has been provided by the Boeing Com pany, monitored by Jim Underbrink. I appreciate the opportunity to work with the people at the Boeing Aero/Noise/Propulsion/Structural Dynamics La boratory. The lessons learned have been invaluable. This project was al so funded in part by the Florida Center for Advanced Aero Propulsion. I would like to thank Dr. Mark Sheplak, my supe rvisory committee chair, for all of his help and guidance, he greatly helped with my research and explori ng several different opportunities, both at UF and at Hewlett Packar d. I would like to also acknowle dge the rest of my graduate committee for their guidance and support: Dr. Louis Cattafesta III, Dr. David Arnold, Dr. Toshi Nishida and Dr. Bhavani Sankar. I have had the great honor of working with many wonderful colleagues during my time here. Much of this work would not have b een possible without the help of several fellow students who started the long trip with me back in 2003: Vijay Chandrasekharan, Ben Griffin and Chris Bahr. I also would like to thank all of the other members of the Interdisciplinary Microsystems Group (IMG). Last, (but certainly not least) I thank my family, especially my wife, Sara; my parents, Leo and Helen Homeijer; and my brothe r, Dan Homeijer. I also tha nk my friends, especially Marie and Kevin Kane, and Heather and Chad Macu szonok. Without poker nights, homebrew and Orlando weekends, I dont know that I could have lasted this long. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.......................................................................................................................10 ABSTRACT...................................................................................................................................15 CHAP TER 1 INTRODUCTION..................................................................................................................17 1.1 Noise Restrictions for C ommercial Airplanes.............................................................. 17 1.2 Aeroacoustic Microphones ...........................................................................................18 1.2.1 Pressure, F ree and Diffuse Field Microphones................................................. 19 1.2.2 Linearity and Total Harmonic Distortion .......................................................... 21 1.2.3 Noise Floor ........................................................................................................21 1.2.4 Requirem ents for an Aeroacoustic Microphone............................................... 22 1.3 Objectives ......................................................................................................................24 1.4 Dissertation Outline ...................................................................................................... 24 2 BACKGROUND.................................................................................................................... 31 2.1 Transduction Schem es..................................................................................................31 2.1.1 Piezoelectric Transduction ................................................................................31 2.1.2 Piezoresistive Transduction ..............................................................................32 2.1.3 Capacitive Transduction.................................................................................... 33 2.1.4 Optical T ransduction......................................................................................... 34 2.2 Chosen Transduction S cheme....................................................................................... 34 2.3 Literature Review .......................................................................................................... 35 2.3.1 Piezoelectric Microphones ................................................................................ 35 2.3.2 Piezoresistive Transducers ................................................................................36 2.3.3 Capacitive Transducers ..................................................................................... 37 2.3.4 Optical T ransducers.......................................................................................... 38 3 TRANSDUCER MODELING AND DESIGN......................................................................47 3.1 Composite Plate Mechanics.......................................................................................... 48 3.1.1 Derivation of Governing Equations ..................................................................48 3.1.2 Equilib rium Equations...................................................................................... 49 3.1.3 Constitu tive Relationship.................................................................................. 50 3.1.4 Nonlinear S olution............................................................................................ 51 3.1.5 Deviation from Linearity.................................................................................. 53 3.1.6 Calculation of Stresses ...................................................................................... 54 PAGE 6 6 3.1.7 Validation Using Finite Elem ent Analysis........................................................ 55 3.2 Electroacoustics ............................................................................................................ 56 3.2.1 Piezoresistors.....................................................................................................56 3.2.2 Wheatstone Bridge ............................................................................................ 62 3.3 Lumped Element Modeling.......................................................................................... 63 3.3.1 LEM of piezoresistive microphone ................................................................... 65 3.3.1.1 Diaphragm ..........................................................................................65 3.3.1.2 Cavity .................................................................................................. 67 3.3.1.3 Vent .....................................................................................................68 3.3.1.4 Equivalent circuit ................................................................................70 3.3.1.5 Cuton frequency and cavity stiffening ..............................................71 3.3.2 FEA verification ................................................................................................71 3.4 Electronic N oise............................................................................................................ 72 3.5 Conclusions ...................................................................................................................73 4 OPTIMIZATION....................................................................................................................87 4.1 Methodology .................................................................................................................87 4.1.1 Objective Function ............................................................................................ 88 4.1.2 Variables ...........................................................................................................88 4.1.3 Constraints ........................................................................................................89 4.2 Optimization Results..................................................................................................... 92 4.2.1 Optim ization with Constant Voltage.................................................................92 4.2.2 Optim ization with a Constant Current Source.................................................. 94 4.2.3 Constraining Devices to a Single Wafer ...........................................................94 4.2.4 Sensitiv ity Analysis........................................................................................... 96 4.2.5 Uncertainty Analysis ......................................................................................... 96 4.3 Conclusion ....................................................................................................................98 5 DEVICE FABRICATION AND PACKAGING.................................................................. 111 5.1 Process Flow Overview ..............................................................................................111 5.2 The MEMS Microphone ............................................................................................. 112 5.3 Microphone Packaging ............................................................................................... 113 5.3.1 Interface Circuitry ...........................................................................................113 5.3.2 Printed Circuit Board ...................................................................................... 114 5.3.3 Assem bled Package......................................................................................... 114 6 RESULTS AND DISCUSSION........................................................................................... 124 6.1 Device Characterization .............................................................................................. 124 6.1.1 Electrical C haracterization.............................................................................. 124 6.1.2 Acoustic Characterization ...............................................................................126 6.2 Experim ental Results.................................................................................................. 128 6.2.1 Electrical C haracterization.............................................................................. 128 6.2.2 Noise Floor ......................................................................................................131 6.2.3 Linearity and Total Harmonic Distortion ........................................................ 131 PAGE 7 7 6.2.4 Frequency Response ....................................................................................... 132 6.3 Model Validation ........................................................................................................132 6.3.1 Variables and Standard Deviations ................................................................. 133 6.3.2 Model Validation Results ................................................................................133 6.4 Conclusion ..................................................................................................................134 7 CONCLUSIONS.................................................................................................................. 154 7.1 Conclusions .................................................................................................................154 7.2 Recommendations for Future Piezoresistive Microphones ........................................155 7.3 Recommendations for Future W ork............................................................................ 156 APPENDIX A COMPOSITE PLATE MECHANICS..................................................................................158 B PROCESS TRAVELER....................................................................................................... 203 C MATLAB FUNCTIONS...................................................................................................... 207 D OPTIMIZED DEVICES.......................................................................................................212 E DETAILED SPECIFICATIONS OF DEVICE PACKAGE................................................ 215 F DETAILS OF EXPERIMENTAL SETUP AND UNCERTAINTY ANALYSIS ...............217 LIST OF REFERENCES.............................................................................................................222 BIOGRAPHICAL SKETCH.......................................................................................................234 PAGE 8 8 LIST OF TABLES Table page 11. Audio and aeroacoustic m icrophone specifications........................................................... 25 12. Commercial microphones used in ae roacoustic testing specifications ..............................25 21. Transduction schemes a nd desired characteristics. ............................................................ 40 22. Piezoelectric microphone specifications in the literature. ................................................. 40 23. Piezoresistive microphone specifi cations in the literature. ................................................ 41 24. Capacitive microphone specifi cations in the literature. ..................................................... 42 25. Optical microphone specifi cations in the literature ........................................................... 44 31. Material parameters and thic knesses used for FEA analysis. ............................................ 74 32. Conjugate power variables for various energy domains.................................................... 74 33. Lumped element modeli ng param eter estimates................................................................ 75 34. Results from FEA analysis com pared to analytical results................................................ 75 41. Upper and lower bounds for all variables.......................................................................... 99 42. Values for devices chosen for fabr ication for constant voltage biasing. ........................... 99 43. Values for devices A, B, and C in constant current m ode (4mA).....................................99 44. Values for devices A, B, and C in constant current m ode (10mA)................................... 99 45. Single wafer constraine d voltage source devices............................................................. 100 46. Single wafer constrained current source (10m A) devices............................................... 100 47. Standard deviation of input parameters........................................................................... 101 48. Mean and 95% confidence intervals for design parameters............................................ 101 49. Statistical properties of desired param eters..................................................................... 101 61. Resistance values of all tested devices............................................................................. 135 62. Resistance values for th e four test resistors. ....................................................................135 63. Values for VDP and line width test structures. ................................................................ 135 PAGE 9 9 64. Values of VDP and line width test structures for the m etal lines.................................... 136 65. Values of Kelvin test structures....................................................................................... 136 66. Thickness of oxide layer using two techniques............................................................... 136 67. Curve fit parameters for test taper resistor for device A.................................................. 136 68. Curve fit parameters for a BUF1A device...................................................................... 136 69. MDP for tested devices.................................................................................................... 136 610. Methods used to determine the fabricated values for all param ete rs of the devices........ 137 611. Results from the radius determination experiment.......................................................... 137 612. Measured values and standard deviation of input param eters......................................... 137 613. Confidence intervals for the realized param eters for a BUF1A device.......................... 137 614. Statistical data for m odel validation PDFs....................................................................... 137 D1. Optimized devices operating on a curren t source with a Gaussian dopant profile. ......... 213 D2. Optimized devices operating on a voltag e source with a Gaussian dopant profile ......... 214 E1. Passive component specifications.................................................................................... 216 F1. Agilent 4155C semiconductor pa ram eter analyzer settings............................................. 217 F2. SRS 560 amplifier settings............................................................................................... 217 F3. SRS 785 spectrum analyzer settings................................................................................ 217 F4. Pulse Multianalyzer setti ngs for linearity testing............................................................. 218 F5. Pulse Multianalyzer settings for frequency response function testing.............................218 PAGE 10 10 LIST OF FIGURES Figure page 11. Number of noise rest rictions at airports. ............................................................................25 12. Perceived noise levels of various aircraft. .........................................................................26 13. Federal Aviation Administration pa rt FAR 36 m easurement locations............................. 26 14. Noise sources of a typical commercial airplane................................................................ 27 15. Schematic crosssection of a general m icrophone structure.............................................. 27 16. Magnitude and phase of a typical ae roacoustic m icrophone frequency response............. 28 17. Example of over damping a free fi eld m icrophone to increase bandwidth.......................28 18. Pressure field microphone flush m ounted in an enclosure................................................29 19. Traveling acoustic wave s in various fields. ....................................................................... 29 110. Deviation of linear and nonlinear solutions....................................................................... 30 111. Noise power spectral density for a typical microphone..................................................... 30 21. Outline of the different trans duction s chemes of MEMS microphones............................ 44 22. Illustration of the piezoelectric effect. ...............................................................................45 23. Example of the piezoresistive effect.................................................................................. 46 24. Variable capacitor schematic............................................................................................. 46 31. Overview of the microphone modeling process................................................................ 76 32. Schematic of composite plate............................................................................................ 76 33. Kirchoff's hypothesis showing the neutral axis and transverse norm al............................. 77 34. Nondimensional center defl ection per unit pressure of devices with varying inplane forces. .................................................................................................................................77 35. Pressure that results in a 5% deviation from linearity for various inplane forces............. 78 36. Analytical deflection of clamped plat e, at the onset of nonlinearity (2000 Pa), com pared to FEA results....................................................................................................78 PAGE 11 11 37. Center deflection per nondimens ional pressure as a function of P* for various values of inplane stresses.............................................................................................................79 38. Description of the Eulers angles....................................................................................... 79 39. Crystallographic dependence of the piezore sistive coefficients for ptype silicon............ 80 310. Piezoresistive factor dependence on dopi ng concentration at room temperature.............. 81 311. Geometry of piezoresistors............................................................................................... .81 312. Differential elements of the arc and taper resistor. ............................................................ 82 313. Sample Gaussian dopant profile........................................................................................82 314. Stressed arc and taper resistors c onfigured in a W heatstone bridge.................................. 83 315. Schematic of MEMS microphone and associated lumped elements................................. 83 316. Example of a the distributed sy stem and the lumped equivlent......................................... 84 317. Equivalent circuit model of the microphone...................................................................... 84 318. Accuracy of first terms of cotangent expansion................................................................ 85 319. Magnitude and phase response of LE M norm alized by the flat band response................. 85 320. Equivalent circuit illustrating th e effect of the cavity com pliance.................................... 86 41. Operational parameter space for a microphone............................................................... 101 42. Multiobjective optimization Pareto fr ont illustrating the tradeoff between minimizing function J1 and J2..........................................................................................102 43. Ideal linear output of a micr ophone or pressure transducer. ............................................ 102 44. Features that are constr ained to be larger than wline.........................................................103 45. MDP vs. Bandwidth for various Pmax constraints............................................................103 46. MDP vs. Pmax for various bandwidth constraints............................................................. 104 47. MDP vs. Bandwidth of various Pmax constraints for a constant current source device... 104 48. MDP vs. Bandwidth of various Pmax constraints for a constant current source device... 105 49. Sensitivity analysis for constant current source v arying 4mA by 3%............................. 105 410. Sensitivity analysis for constant current source v arying 10mA by 10%......................... 106 PAGE 12 12 411. MDP dependence on Hooge parameter for device A......................................................106 412. Dependence of MDP with respect to each variable......................................................... 107 413. Dependence of MDP on silicon thickne ss overlaid with com pression coefficient.......... 107 414. Monte Carlo simulation schematic.................................................................................. 108 415. Uncertainty of MDP of Device A....................................................................................108 416. Uncertainty of Pmax for device A..................................................................................... 109 417. Uncertainty of the bandwidth for device A...................................................................... 110 418. 95% probability yield limit illustration............................................................................ 110 51. Front side process steps.................................................................................................. .115 52. First four masks for microphone fabrication................................................................... 116 53. Last three masks for front side fabrication...................................................................... 117 54. Back side process steps................................................................................................... .118 55. Backside masks for microphone fabrication.................................................................... 118 56. Array of microphone die af ter dicing. Each die is 2mm x 2mm ....................................119 57. Individual type A mi crophone die af ter dicing................................................................ 119 58. Type B device pictured on a dime................................................................................... 120 59. Backside cavity and vent of an individual type A microphone die after dicing.............. 120 510. Packaging for acoustical characterization........................................................................ 121 511. Interface circuitry showing power supply, ac filter and am plifier................................... 121 512. Printed circuit board for mounting the m icrophone and its associated components....... 122 513. Assembled device on PCB. Devi ce is protected under a TO can. .................................. 122 514. Populated PCB board inserted into PWT endplate.......................................................... 123 61. Circuit representation of reversed bias ed p+ doped resistors in an n substrate. .............. 138 62. Van der Pawl test structure schematic............................................................................. 138 63. Line width test structure schem atic.................................................................................. 139 PAGE 13 13 64. Kelvin test structure schematic........................................................................................ 139 65. Experimental setup fo r noise m easurements.................................................................... 140 66. Experimental setup for acoustic characterization. ........................................................... 140 67. Boron concentration in silicon device layer d etermined by SIMS, the accompanying curve fit and the desired model profile............................................................................ 141 68. Input IV curve of 12 BUF1A and 12 BUF1B devices................................................. 141 69. Output IV curve of 12 BUF1A and 12 BUF1B devices.............................................. 142 610. Input IV curve of a BUF1A device with a linear curve fit............................................ 142 611. Output IV curve of a BUF1A device with a linear curve fit ......................................... 143 612. IV curve of diode characteristics of a BUF1A device.................................................. 143 613. IV curve of diode characteristics of a BUF1A device focusing on the reverse region. ..............................................................................................................................144 614. Noise PSD of a test taper resistor.................................................................................... 144 615. Noise PSD from a test taper resistor minus the setup noise and the associated m odel curve fit. The horizonta l line is the thermal noise floor for this device.......................... 145 616. Noise power spectral density of a BUF1A device..........................................................145 617. Noise PSD minus the setup noise and the associated m odel curve fit. The horizontal line is the thermal noise floor for the device.................................................................... 146 618. Sensitivity of BUF1A devices normalized by the bias voltage...................................... 146 619. Total harmonic distor tion of BUF1A m icrophones........................................................ 147 620. Magnitude frequency response for a BUF1A de vice. Vertical dotted lines mark the piecewise FRFs that were stitched together..................................................................... 147 621. Magnitude FRF for each device with 95% CI bounds..................................................... 148 622. Phase response for each device tested............................................................................. 149 623. Phase FRF for each device with 95% CI bounds............................................................. 150 624. Coherence function between device A5 and the reference microphone........................ 151 625. Microphone photograph with and without backlighting. .................................................151 PAGE 14 14 626. Minimum detectable pressure probability de nsity function............................................. 152 627. Sensitivity probability density function........................................................................... 152 628. Voltage noise probability density function......................................................................153 E1. Layout of PCB package...................................................................................................215 PAGE 15 15 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 DESIGN, FABRICATION, AND CHARACTERI ZATION OF A MEMS PIEZORESISTIVE MICROPHONE FOR USE IN AEROACOUSTIC MEASUREMENTS By Brian Homeijer December 2008 Chair: Mark Sheplak Major: Mechanical Engineering With air traffic expected to increase dramatically in the next decade and urban sprawl encroaching on airports, a reduction in the sound radiated from comm ercial airplanes is needed. To lower aircraft noise, manuf acturers perform extensive scale model wind tunnel tests to locate and eliminate sound sources. One of the most important pieces of equipment needed is a robust microphone that is able to withstand large s ound pressure levels on the order of 160 dB SPL, while possessing an operating bandwidth on the or der of 100 kHz and a low noise floor at or below 26 dB SPL. This work attempts to addres s the needs of aircraft manufacturers with the design, fabrication and characte rization of a microelectromech anical systems piezoresistive microphone for use in aeroacoustic measurements. This microphone design addresses many of th e problems associated with previous piezoresistive microphones such as limited dynami c range and bandwidth. This design focuses on improving the minimum detectable pressure over many current technologies without sacrificing bandwidth. To accomplish this, a novel nonlinear circular composite plate mechanics model was employed to determine the stresses in th e diaphragm, which was designed to be in the compressive quasibuckled state. With this model, the effects of residual inplane stresses that result from the microelectronic fabrication process on the sensitivity of the device are predicted. PAGE 16 16 Ion implanted doped silicon was chosen for th e piezoresistors and an integrated circuit compatible fabrication recipe was formulated to minimize the inherent noise characteristics of the material. The piezoresistors are arranged in a Wheatstone bridge configuration with two resistors oriented for tangential current flow and two for radial current flow. A lumped element model was created to predict the dynamic character istics of the microphone diaphragm and the integrated cavity/vent structure. The devi ce geometry was optimized using a sequential quadratic programming scheme. Results pred ict a dynamic range in excess of 120 dB for devices possessing resonant frequencies beyond 120 kHz. Future work includes the completion of the fabrication process and characterization of the microphones. The characterization of the fabricated devi ce revealed two major problems with the piezoresistors. The diffusion of the resistors was too long and resu lted with the resistor thickness being the entire thickness of the di aphragm. The result of this error dropped the sensitivity two orders of magnitude. In addition to the doping profile error, the i nherent noise characteristic of the resistors was also higher then expected. This increased the noise sign ature of the device two orders of magnitude higher then expected. Th ese two factors couple to gether and increase the MDP of the device by 4 orders of magnitude, or 80 dB. The optimized device A had an expected MDP of 24.5 dB The realized device had a MDP of 108dB, or 83.5 dB higher than the desired value. Despite the error in resi stor fabrication, the models devel oped in this dissertation showed that they correctly represent th e realized device and therefor e will be sufficient to design a second generation microphone. PAGE 17 17 CHAPTER 1 INTRODUCTION W ith air traffic expected to increase dramatically in the next decade and urban sprawl encroaching on airports, the Federal Aviation Ad ministration (FAA) has ta ken steps to regulate aircraft noise. For certification, th e US Code of Federal Regulations stipulates that commercial aircraft must pass airworthiness tests, which state the maximum allowable effective perceived noise level (EPNL) that aircraft can emit. The EPNL is the measured noise level normalized to sound duration, atmospheric conditions and jet en gine operating conditions[1]. To lower the noise radiating from aircraft, manufacturers pe rform extensive wind tunn el tests to locate and eliminate sound sources on planes. This industry is in need of a robust, and low cost alternative to instrumentation grade condenser microphones. A microelectromechanical systems (MEMS) microphone has the potential for a substantial cost reduction and does not have the installation drawbacks that the current microphones possess. This work focuses on the development of a robust microelectromechanical systems (MEMS) microphone as a low cost alternative to the industry standard condenser microphones. This chapter begins with an introduction to noise restrictions for comm ercial aircraft flying in US airspace. Next, the differences between audio and aeroacoustic microphones are explained and requirements for aeroacoustic microphones are given. 1.1 Noise Restrictions for Commercial Airplanes In response to the anticipated doubling of wo rld air traffic over the next 20 years [1], noise restrictions continue to grow more stringent. The total number of noise restrictions, comprised of curfews, charges a nd levels has increased 10 fold, ( Figure 11) [1]. Curfews designate quiet tim e around airports during the night. If aircraft are louder then the curfew limit PAGE 18 18 then they can only land in the daytime. Cu rrently, aircraft that do not meet the noise requirements are fined, with increased noise levels corresponding to a larger fine. To meet the increasingly restrictive noise requ irements, aircraft manu factures have reduced noise signatures an order of magnitude from the original jet aircraft of the 1970s ( Figure 12). To determ ine an aircrafts noise signature, the FAA takes noise samples at every airport in at least three locations each of which are illustrated in Figure 13. The largest improvement to date has been in the reduction of noise in the latera l and takeoff areas [2]. This im provement is primarily the result of noise reduction in tur bofans, which are loudest during takeoff. The takeoff noise signature is also less problematic because planes gain altitude quickly, and noise ceases to reach ground level. However, landing re quires a low, slow approach. At this stage, airframe noise is significant because the plane is in a noisy configuration with the landing gear down and flaps fully extended ( Figure 14). Aircraft m anufacturers are focusing on reducin g the noise of commercial aircraft even further. Next generation aircraft engines are being equipped with serrated edges called chevrons for the back of the engine nacelle and exhaust nozzle [2]. In additi on researchers are looking into toboggan fairings to redu ce landing gear noise [3]. To accomplish their goals, aircraft manufactures need a robust aeroacoustic microphone that meets the requirements for aeroacoustic testing. 1.2 Aeroacoustic Microphones Transducers convert energy from one for m to another; mi crophones are transducers that specifically convert acoustical ener gy to electrical energy or m odulate the electrical energy due to the acoustic energy. This energy conversion is achieved in different wa ys, however, one thing they all have in common is that they first c onvert acoustical energy to mechanical energy via a diaphragm. A diaphragm is a thin structure, pictured in Figure 15, that vibrates when sound PAGE 19 19 waves strike it. A mechanical to electrical transduction scheme th en determines how the mechanical energy is converted in to a readable electrical signal. In addition to the diaphragm and the transduc tion mechanism, the othe r important elements of a microphone are the vent and cavity. The vent is used to equilibrate the pressure acting on the device so that the microphone only senses an ac signal, and the cavity connects the vent to the diaphragm ( Figure 15). To ensure that microphone output is representative of the acoustic input, th e sensitivity of the microphone must not change with frequency, and, ideally, the microphone should have no phase shift. Figure 16 illustrates a normal frequency response for an underdam ped aeroacoustic microphone. The bandwid th of the microphone is defined as the range of frequencies where the microphones res ponse magnitude is flat. The cuton frequency and resonant frequency are also shown in Figure 16. The vent channel dominates the lowfrequency response of the m icrophone, while the impe dance of the vent relative to the diaphragm dictates whether incident pressu re will flow through the vent or deflect the diaphragm. The damping and the resonant frequency of the mi crophone dominate the frequency response at high frequencies. The mechanical resonance of the diaphragm is a function of its compliance and mass. The shape of the frequency response close to the resonance frequency is determined by damping in the microphone structure. For example, an distinct resonance peak will be evident for an underdamped system, as shown in Figure 16, while no peak is found in an overdamped system. The frequency response of a microphone can be optimized to allow it to have a larger bandwidth in various acoustic fields. This can be accomplished several ways and is discussed in the following section. 1.2.1 Pressure, Free and Diffuse Field Microphones Microphones are divided into thr ee types: free, diffuse and pressure field, determ ined by their response in an acoustic field. A free fi eld occurs when sound waves can propagate freely PAGE 20 20 without reflections. This type of field can be found outdoors or in an anechoic chamber as long as the sound source is far enough away Figure 19(A) [4]. For this type of application, a free field microphone should be used. When a microphone is placed into the sound field, it modifies that field Figure 19(B). Pressure rises in front of the m icrophone due to the scattering off of the microphone, resulting in a higher ou tput level at certain frequenc ies. The maximum effect is when the wavelength of a specific frequency is equal to the diameter of the microphone. Figure 19(C). Free field m icrophones ta ke this effect into account by compensating for their own disturbing presence. The damping of these microphone s is increased so that the spectral shape of the microphone response is opposite to the spectral shape due to sc attering, shown in Figure 17. To work correctly a free field m icrophone must be normally incident to the noise source. A diffuse field exists if the field is creat ed by sound waves arriving from all locations simultaneously with equal probability [4]. Th e diffuse field microphone is designed to respond uniformly to signals arriving simultaneously fr om all angles. These devices are also overdamped like free field microphones. A pressure field is found in enclosures whic h are small compared to the wavelengths of interest [4]. A pressure field microphone should be used in this situa tion and should be flush mounted on the enclosure as seen in Figure 18. Because the pre ssure field m icrophone has a minimal effect on the field, no corrections are needed to account for the presence of the microphone [4]. All three microphones can be used in any field as long as a correction fa ctor is taken into account. Since the devices were optimized for particular sound fields, the usable bandwidth will decrease. The correction factors for Bruel and Kjaer microphones can be obtained on their PAGE 21 21 website. In addition, a nose c one corrector can be mounted onto a free field microphone to reduce the effect of th e angle of incidence. 1.2.2 Linearity and Total Harmonic Distortion The linearity of the m icrophone explains how, at a fixed, flat band frequency, the microphone output magnitude varies as a function of the amplitude of the incident pressure. Shown in Figure 110 is a linear (ideal) and nonlinear (real) response of a generic m icrophone to a singletone, fluctuating amplitude pressure. In the ideal case, a linear relationship between the output voltage and the amplitude of the incident pressure is shown. In practice, however, a variety of sources of nonlinearity limit the effective maximum pressure. As seen in Figure 110, the m icrophones actual dynamic response diverges from the linear response above a maximum pressure, such as mechanical, electromechan ical, and amplifier nonlinearities. The maximum pressure at which a microphone is considered linear is defined at the point where a 3% difference between the linear and n onlinear response of the microphone is detected. Th e nonlinearity of the microphone is given in terms of the total harmoni c distortion (THD) with respect to frequency because the nonlinear response of the microphone causes distortions in the output. The THD is described as the ratio of the total power in the higher harmonics to the power in the fundamental frequency, and is given as [5], 2 2 2 1 n np THD p (11) 1.2.3 Noise Floor The lower end of the dynam ic range, called th e minimum detectable signal (MDS) is limited by the microphone noise as well as noise cont ributed by the interface circuitry, since it is the output when no input is given [6]. Noise is normally given in terms of a power spectral PAGE 22 22 density (PSD) and the total noise power is base d on the PSD integrated over the bandwidth of interest [6]. A typical noise PSD of a piezoresistive microphone is shown in Figure 111. When in therm odynamic equilibrium, the thermal noise of the system is propor tional to the dissipation of the system [6], This is also known as white noise because the PSD is constant over all frequencies. Under nonequilibrium conditions, flicker noise arises which has an inverse proportionality to frequency. It occurs in inte rface electronics and semiconductive materials, and usually dominates at lower frequencies. The freq uency when the noise PSD of the flicker noise equals the thermal noise is known as the corner frequency [7]. The role of the individual noise sources can be shaped by the dynamic response of the microphone. This can cause a flat thermal noise source to have a nonflat spectral sh ape in the microphone output. The noise floor of a microphone is often stated for a specific bandwidth. As an example, the noise can be specified at a given frequency for a narrow ba ndwidth, or integrated over a sp ecified bandwidth. Aweighted noise, denoted dBA, is another common metric where the noise spectrum is passed through a filter that approximates the response of the human ear, then integrated and converted to dB [8]. 1.2.4 Requirements for an Aeroacoustic Microphone As stated above, the perform ance of audio microphones is tuned to the human ear; an instrument grade aeroacoustic microphone, howev er, has different requirements, shown in Table 11. The hum an ear has a minimum detectable pressure of 20 Pa at 2kHz, also known as the threshold of hearing. The maximum pressure th at an ear can be exposed to, known as the threshold of pain, is 20 Pa. The bandwidth of the human ear is from 20 Hz to 20 kHz [9]. Aeroacoustic microphones must be useable in areas where the sound pressure level (SPL) to be measured is very high, like near an airc raft jet engine, where sound pressure levels may reach 170 dB. In addition, the FAA requires certi fication over the frequency range of PAGE 23 23 4511.2 HzfkHz for full scale vehicles [1]. Aeroac oustic testing is frequently done using 18 scale models. To retain dynamic similarity, the frequency range of interest is enlarged by this scale factor, meaning that acoustic testing is conducted over the range of 36089.6 HzfkHz [10]. However, the microphones should still be useable in full scale flyover applications, which require the microphone bandwidth to extend down to 45Hz. In addition to this requirement, the bandwidth goals for this project extend to a range of 20120 HzfkHz specified by the sponsor of the project, The Boeing Company. Currently, several commercial microphones ar e used by the aeroacoustic community. The specifications of some of theses microphones can be seen in Table 12. The B&K microphones are typ ically used in arrays for sound localization. The Kulite MIC093 is used on turbulence control screen arrays for static engine tests. The goal for the minimum detectable pressure (MDP) of the microphone specified by the sponsor of this project is 26 dBSPL for a 1 Hz bin centered at 1 k Hz. Acoustic arrays, which consist of many micropho nes arranged in a specific geometry, are often used in aeroacoustic measurements to localize the noise source. A selective spatial response can be determined via beam forming signal processing, which enables the acoustic array to listen to a specific area in space [ 10]. This technique re quires a large number of microphones, typically in the 100s [11], wh ich makes MEMS microphones particularly appealing due to the possible advantages of ba tch fabrication for reducing the cost of each microphone. However, additional specifications are needed to enable the use of the microphones in an acoustic array. For regularly spaced arra ys, the microphones must be physically arranged within one half wavelength of each other to avoid spatial alia sing. However, this is not a requirement for logarithmically spaced spiral arrays [11]. Phase matching between microphones PAGE 24 24 is another necessity for acoustic arrays because beam forming algorithms use phase information to localize sound sources. A mismatch between microphone channe ls can cause error in the sound localization. 1.3 Objectives The goal of this project is the design, fa brication, and charac terization of a MEMS piezoresistive microphone for aeroa coustic testing app lications. The microphone should have a sufficient dynamic range and bandwidth for use in scale model aeroacoustic wind tunnel tests and full scale flyovers. The final objective is to have an accurate model to aid in the fabrication of a second generation microphone, therefore the microphones characterization results will be used to adjust the model to account for any errors. The entire project is larger then the scope of this dissertation. The division of labor wa s separated into the design, fabrication and characterization of the microphone and in addition the de velopment of a fabr ication recipe for the piezoresistors to minimize the noise of the device is the remaining portion of the project. 1.4 Dissertation Outline This work is organized into seven chapters. Chapter 1 presents the motivation and goals of the project. Chapter 2 contains a literature review of MEMS microphones. Chapter 3 describes the modeling of the microphone. Chapter 4 summari zes the optimization scheme used to decide device dimensions. Chapter 5 outlines the process flow and device fabrication, Chapter 6 details the characterization and model va lidation and Chapter 7 concludes with a summary of this work and recommendations for future devices. PAGE 25 25 Table 11. Audio and aeroac oustic microphone specifications. Audio Aeroacoustic Max Pressure 120dB 170dB Noise Floor 2335dBA 26 dB Bandwidth 20 Hz 20 kHz 20 Hz 100 kHz 1Hz bin centered at 1kHz Table 12. Commercial microphones used in aeroacoustic te sting specifications [12], [13]. Kulite MIC093B&K 4939 B&K 4138 Desired Boeing Specifications Diameter 2.4 mm 6.35 mm 3.18mm Max Pressure 194 dBA 164 dBSPL 168 dBSPL 150160 dB SPL Noise Floor 100 dBA 5 dB SPL 18 dB SPL <26 dB SPL Dynamic Range 94 dBA 159 dB SPL 150 dB SPL 124 134 dB SPL Bandwidth ~90 kHz 4 Hz 100 kHz6.5 Hz 140 kHz100 120 kHz 1Hz bin centered at 1kHz 0 19651970197519801985199019952000 Year 0 50 100 150 200 250 300 350 400 Figure 11. Number of noise re strictions at airports [1]. PAGE 26 26 B52 707100 DC820 CV990A CV88022 BAC11 DC910 DC861 737100 737200 727100 727200 747100 747200 DC1010 L1011 A300B2 MD80 747300 747400 737300 A320100 A321 A330 A340 MD11 777 A310300 BAe 146200 DC1030 Comet 4 720 Year of initial service Noise level, EPNdB (1,500 ft sideline) 19501960197019801990200020102020 80 90 100 110 120Turbojet and early turbofans First generation turbofan Second generation turbofan707300B Figure 12. Perceived noise leve ls of various aircraft [2]. ApproachLa t e r a lTakeoffCutback ~1,000 ft (305 m) 1,476 ft (450 m) 6 5 6 5 f t ( 2 0 0 0 m )2 1 3 2 5 f t ( 6 5 0 0 m )Approach in noisiest configuration Landing gear extended, full flaps Takeoff with maximum takeoff thrust rating Measurement Location Figure 13. FAA part FAR 36 Measurement Locations [1]. PAGE 27 27 Figure 14. Noise sources of a typical commercial airplane. Diaphragm Cavity Vent Figure 15. Schematic crosssecti on of a general microphone structure. PAGE 28 28 101 100 101 102 103 104 105 106 20 0 20 40 Magnitude [dB] (ref response @ 1kHz) 101 100 101 102 103 104 105 106 150 100 50 0 50 Frequency [Hz]Phase [deg] Figure 16. Magnitude and phase of a typi cal aeroacoustic microphone frequency response. Mag. Frequency Response Figure 17. Example of over damping a free field microphone to increase bandwidth. PAGE 29 29 Figure 18. Pressure field microphon e flush mounted in an enclosure. A B C 20log M FF P P D 10 dB10 dB Figure 19. A) Traveling acousti c waves in a freefield. B) Effect of placing a microphone in the field. C) Increased pressure sensed by the microphone due to its own presence [4]. PAGE 30 30 Linear Solution Nonlinear Solution Incident PressureCenter Deflection Figure 110. Deviation of linear and nonlinear solutions. 101 100 101 102 103 104 105 1016 1015 1014 1013 1012 Frequency [Hz]Noise Power Spectral Density Sv [V2/Hz] Figure 111. Noise power spectral density for a typical microphone. PAGE 31 31 CHAPTER 2 BACKGROUND This chapter discusses the details of the transduction schemes commonly used in MEMS microphones including piezoelectric, piezoresistive, capacitive and optical in section 2.1. A comprehensive literature review of MEMS microphone developments is then given in section 2.2. 2.1 Transduction Schemes This work organizes the various MEMS microphones by transduction mechanism: piezoelectric, piezoresistive, optical, and cap acitive. These transduction mechanisms are described in the following sections. The op tical scheme is broken down into intensity modulating, polarization modulati ng and phase modulating. The cap acitive scheme is separated into electret and condenser. Figure 21 shows the various clas sifications of MEMS m icrophones. 2.1.1 Piezoelectric Transduction Piezoelectricity is defined as th e ability of some materials to generate an electric potential in response to applied mechanical stress [14]. Th e Curie brothers were th e first to discover that surface charges developed on some crystals, namely crystals with a noncentrosymmetric crystal structure, when compressed, and that the magn itude of these charges was proportional to the applied pressure. Hankel later named this pheno menon piezoelectricity, and it is historically referred to as the dire ct piezoelectric effect. In addition, strain is also pr oduced when an electrical field is applied, called the converse piezoelectric effect [15]. This effect is caused by an internal polarization at the atomic level [16]. The piezoelectric charge modulus, d, is the material constant re lating strain and charge in a piezoelectric material. Figure 22 illustrates how a piezoelectr ic m aterial expands, or contracts, when an electrical potential is placed across it. PAGE 32 32 2.1.2 Piezoresistive Transduction Certain materials go through a fundamental electr onic change in resistivity due to applied stress that exceeds the resistance change that all resistors experience due to stressinduced deformations, called the piezoresistive effect. In semiconductor piezoresistive materials, like silicon and germanium, the change in resistivity is caused by a change in the mobility in addition to a change in physical size. This effect wa s first observed by Smith in 1954 for silicon and germanium [17]. The resistance of any given material is [18] L R A (21) where and LA are the resistivity, length and cross se ctional area of the re sistor, respectively ( Figure 23). When a metal is stressed, the resi s tance changes due to geometric effects, however, when a semiconductor material such as silicon is stressed, the resi stivity of the material changes as well [17]. The change in resistance is given by dRdLdAd RLA (22) The geometric effects can be ar ranged in the following manner 12dR d R (23) For conductive materials such as metals where 0 d the maximum change in resistance is limited to 2 times the strain if 0.5 In a piezoresistive material the change in resistance is given by ij ijklkl ijT (24) PAGE 33 33 where is a rank 4 piezoresistance tensor and T is the stress tensor [ 19]. The magnitude of this term can be of the order of 100 times the strain [19]. This allows for a much greater sensitivity in piezoresistive materials comp ared to a standard metal strain gauge. 2.1.3 Capacitive Transduction The capacitive transduction scheme relies on th e measurement of a change in capacitance between two electrically charged surfaces. A parallel plate capacitor is discussed here for simplicity. The capacitance between tw o parallel plates is given as, 0A C g (25) where A is the surface area, 0 is the permittivity of the dielectric material between the plates, and g is the distance between the plates [20]. If a force moves one of the plates, the capacitance changes as the distance between the two plates changes. The two main classes of capacitive sensing are electret and condenser sensing [21]. Electrets are biased with a fixed permanent charge, which is usually implanted into a dielectric layer on the fixed portion of the microphone. The electrical force eF in Figure 24 for this case is [22] 22eQ FQ A (26) where Q is the charge. Condenser microphones are bias ed with an external voltage source. The electrical force for this case is [22] 2 2, 2eAV FVx gx (27) Electret devices are not susceptible to electrostatic pullin, however it is difficult to fabricate devices with a stable embedded charge. PAGE 34 34 2.1.4 Optical Transduction Optical microphones tend to be more comple x because the mechanical energy of the diaphragm is sensed optically first and then conve rted into electrical en ergy. One benefit of sensing optically is that the transducer is inse nsitive to electromagnetic interference [23]. One drawback is that optoelectronic s, typically a photodiode, are needed to convert the optical signal to an electrical signal. This can cause a significant amount of noise [24]. All optical microphones are lightm odulating acoustic sensors rather than direct converters of sound energy to light. They modulate light in three ways: in tensity, phase and polarization modulation [25]. Cook et al. developed the first optical tr ansducer found in the literature in 1979, a lever displacement sensor [26]. Intensity based optical microphones use an emitter, which shines light onto the diaphragm. The am ount of light that is sensed by the receiver changes as the diaphragm defl ects. For microphones operating in the phase modulating scheme, as the diaphragm deflects, the distance the light travels from emitter to receiver changes and is sensed by a shift in phase. Optical microphones that operate using polarization modulation use the fact that unpolarized light can be polarized through reflecti on off nonmetallic surfaces. The degree of polarization depends on the incident angl e and the material that is reflecting the light [25]. The light is then passed through a polar izing filter and the intensity of the light is measured. 2.2 Chosen Transduction Scheme Of the available transduction schemes the piezoresistive scheme was chosen for this project. The piezoresistive transduction sc heme has the best attributes including: The ability to withsta nd harsh environments The capability of being packaged with a thin profile Compatibility with integrated circuit fabrication processes PAGE 35 35 Table 21 shows the desired traits and which tran sduction schemes meet each requirement. 2.3 Literature Review This literature review disc usses the important steps th at progressed each microphone transduction scheme. Select nonMEMS and pressure sensors of historical significance are added to show how the technology developed. 2.3.1 Piezoelectric Microphones In 1953 Medill developed the first miniature piezoelectric microphone for use as a secondary standard in production testing. This microphone measured 1 1/8 in diameter and used Rochelle salt crystals as the piezoelectric material [27]. In 1983 R oyer et al. developed the first MEMS piezoelectric microphone using zinc oxide as the pi ezoelectric material [28]. Between 1989 and 1991 Kim et al. developed a pi ezoelectric MEMS microphone using a silicon nitride diaphragm and a zinc oxide piezoelectric ma terial [29], [30]. Theses devices, however, never achieved a flat frequency response. In 199 2, Schellin et al. developed a new type of microphone [31]. This work used polyurea for th e piezoelectric materi al and achieved a high sensitivity of 4mV/Pa but Schellin et al., like Kim et al., could not achieve a flat frequency response. One year later, in 1993, Ried et al. improved on Kims microphone by tweaking the fabrication process to control stress in the diap hragm, achieving a flat frequency response [32]. In 2003, Ko et al. developed a device with zinc oxide as the piezoelectric ma terial that could be used as a microphone or a microspeaker [ 33]. This device performed poorly in both applications. Also in 2003, Niu et al. improved on Kims design (1989) by using paryleneD for the diaphragm, which increased the sensitivity [34] In 2003 Zhao et al. u tilized the piezoelectric material, lead zirconate titana te (PZT), for the first time in a MEMS microphone. With this material, Zhao et al. achieved a high sensitivity of 38 mV/Pa and a flat frequency response [35]. In 2004, Hillenbrand et al. used a cellular polypropylene material for the piezoelectric crystal, PAGE 36 36 but the device did not reach the sensitivity of Zhao et al. [36]. In 2007, Horowitz et al. developed an aeroacoustic microphone using PZT [37] This device was the first to have a large dynamic range suitable for aer oacoustic applications. Table 22 summarizes the realized perform ance for each of the devices. 2.3.2 Piezoresistive Transducers The first piezoresistive mi crophone was developed by Burns in 1957 for use in the Bell type 500 telephone [38]. This microphone was deem ed too expensive to fabricate compared to the standard carbon microphones and was never ma ss produced. Fourteen years later in 1961, Samaun et al. [39] developed the first MEMS piezor esistive pressure sensor. A pressure sensor is similar to a microphone except it is used to measure an absolute dc pressure instead of a relative ac pressure. This device had a silicon nitride moisture barrier and was intended for use in biomedical applications. It was not until 1992 that Schellin et al. developed the first MEMS piezoresistive microphone [40]. This device was composed of a square diaphragm and four ptype dielectrically isolated polysilicon piezoresistors. In 1994, Kalvesen et al. developed a MEMS microphone for use in turbulent gas flows [ 41]. This device was the first to have an integrated cavity and vent stru cture; however, the cavity was only 3 m deep and contributed to a low sensitivity. In 1995, Schellin et al. de veloped the first microphone to use ion implanted piezoresistors [42]. Specifically, they were ptype resistors in an nwell silicon diaphragm. In 1998, Sheplak et al. developed a silicon nitride MEMS microphone for aeroacoustic measurements [43]. This device had a circul ar diaphragm and was the thinnest diaphragm (1500 A ) reported in the literature. The piezoresist ors were dielectrically isolated and had two types: arc and taper. The arc resistor was designed for current flow in the tangential direction and the taper resistor was designed for current flow in the radial direction. This microphone had PAGE 37 37 a large cavity and winding vent channel integrated into the design. At the same time, Nagiub et al. developed another MEMS microphone for use in aeroacoustic measurements [44]. The device used a rectangular di aphragm however the dynamic ra nge was not reported in the literature. In 2001, Arnold et al. improved on the design of Sheplak et al. (1998) and significantly lowered the noise floor of the device [45]. One year later, Huang et al. improved on the original design of Nagiub ( 1999) with a device that had a similar minimum detectable pressure (MDP) [46]. In 2004 Li et al. develo ped an audio MEMS microphone with integrated electronics and the lowest noise floor reported in the literatu re of 34dB [47]. The devices maximum pressure was not reported. Table 23 outlines the various devices and shows the reported specifications for each. 2.3.3 Capacitive Transducers Ko et al. developed the first MEMS capacitive transducer, a condenser pressure sensor in 1982 [48]. Two years later in 1984 Hohm et al. developed the firs t capacitive microphone [49]. It was an electret microphone and had a rectangu lar diaphragm with the longer side being 8 mm long. In 1989 Hohm et al. developed the firs t condenser microphone [50]. The device was 10 times smaller than the previous effort and operated from 200Hz to 20 kHz. In 1990, Bergqvist et al. developed the first microphone built completely using microfabrication techniques [51]. Their devices were all rectangul ar with a size of 2 mm. In 1997 Cunningham et al. developed the first capacitive microphone with a circular diaphragm [52]. This device was designed for audio applications and had a 1 mm diameter. In 2000, Rombach et al. developed the dual plate capacitive microphone [53]. This microphone used two fixed plates with a diaphragm in the middle to increase the capacitance change resulting with an increase in sensitivity. The noise floor of the device was 23dBA and was designed for audio applica tions. Three years later in 2003, Scheeper et al. developed the first capacitive microphone tail ored for use in aeroacoustic PAGE 38 38 measurements [54]. His device used a nontrad itional octagonal diaphragm and had a dynamic range of 23dB to 141dB. In 2005, Martin et al developed the first dual backplate microphone for aeroacoustics [55]. This device had a dynamic range of 22.5 dB to 160 dB and a bandwidth of about 100 kHz. Table 24 shows the reported specifications for each of the dis cussed devices. Currently the Knowles Acoustics Sisonic mi crophone is available for purchase. The specification sheet reports a noise floor of 39 dB A and a bandwidth of 10 kHz. In September, 2008, Analog Devices Incorporated released the iMEMS condenser microphone. To date it has the best performance of any commerci ally available ME MS microphone [56]. 2.3.4 Optical Transducers In 1991, Garthe et al. devel oped the first optical microphone [57]. This device used an intensity modulating scheme and had an integrated waveguide chip fabri cated using polymethyl methacrylate (PMMA), though it was not a MEMS device. In 1992 Dziuban et al. were the first to develop a silicon optical device [58]. This transducer had a 10 mm square diaphragm and could only be used as a pressure on/off switch due to its poor performance. In 1994, Chan et al. developed the first silicon optic al pressure sensor [59]. Th is device had a 10mm square diaphragm and used a phase modulation scheme. Fi ve years later in 1999, Kots et al. developed an intensity modulating optical microphone [60]. This was the smallest device to date with a 1.5mm circular diaphragm. In 2001, Abeysinghe et al. developed a ci rcular pressure sensor [61]. It was a phase modulating device and had a 135 m diameter. In 2004, Kadirvel et al. developed a MEMS optical microphone [62]. The microphone used an intensity modulating transduction scheme. It was designed for aeroacoustic applic ations but was plagued by an inherently high noise floor of 70dB. In 2005, Bucaro et al. developed a MEMS intens ity modulating microphone with an improved noise floor of 30.6dB [63], significantly lower then Kadirvels device; PAGE 39 39 however the maximum detectable pressure was not reported. In 2005, Hall et al. developed a device with the lowest detectable pressure in the literature, 17.5 dBA, however this was accomplished in the laboratory on an optical be nch [64]. The device was fabricated using Sandias SwIFTLite process but it only had a ba ndwidth of 4kHz. In 2006, Song et al. reported on an optical microphone based on a reflective micro mirror diaphragm however, the dynamic range was not reported [65]. Finally, in 2007 Ha ll et al. reported on a smaller microphone design with a 24dBa noise floor [66]. Table 25 shows the reported sp ecification s for each of the discussed devices. PAGE 40 40 Table 21. Transduction scheme s and desired characteristics. Characteristic Meet Fail Harsh environment Piezoresistive Capacitive Optical Piezoelectric Thin package profile Piezoresistive Optical Capacitive Piezoelectric IC compatible fabrication Piezoresistive Optical Capacitive Piezoelectric Table 22. Piezoelectric microphone specifications in the literature. Author Diaphragm CavitySensitivity Dynamic Bandwidth Piezoelectric Microphone/ Year Dimensions Depth Range (Predicted) Material Pres. Trans. J. Medill 1 1/8 N/R N/R N/R ~1kHz Rochelle Microphone [27] salt crystals M. Royer et. al. 1.5mm x 30 m N/R 250 V/Pa 73dBAN/R10 Hz 10 kHz ZnO Microphone [28] (0.1 Hz 10 kHz) 1st fabricated E. S. Kim et al. 2mm x 1.4 m 380 m 80 V/Pa N/R 3 kHz 30 kHz ZnO Microphone [29], [67] E. S. Kim et. al. 3.04mm x 2.0 m 380 m1000 V/Pa 50 dBAN/R200 Hz 16 kHz ZnO Microphone [30] R. Schellin et al. 0.8mm x 1.0 m 280 m 4000 V/Pa N/R 100 Hz 20 kHz Polyurea Microphone [40], [68] R. P. Ried et al. 2.5mm x 3.5 m ~500 m920 V/Pa 57 dBAN/R100 Hz 18 kHz ZnO Microphone [32] S. S. Lee et al. 2mm x 4.5 m N/R 3800 V/Pa N/R 100 Hz 890 Hz ZnO Microphone [69], [70] S. C. Ko et al. 3mm x 3.0 m N/R 30 V/Pa N/R 1 kHz 7.3 kHz ZnO Microphone [33] M. N. Niu et al. 3mm x 3.2 m N/R 520 V/Pa N/R 100 Hz 3 kHz ZnO Microphone [34] H. J. Zhao et al. 0.61mm x N/R 370 m 38mV/Pa N/R N/R 20kHz PZT Microphone [35] J. Hillenbrand et al. ~0.5cm x 55 m N/R 20.5mV/Pa 3726 dBA N/R ~10kHz Cellular Microphone [36] N/R polypropylene Y. Yang et al. 200500 m x N/R 61474 N/R N/R ~30kHz PZT Microphone [71] ~1 m V/Pa S. Horowitz et al. 900 m x 3.0 m 500 m 0.75 V/Pa 47.8 dB 100 Hz 6.7 kHz PZT Microphone [37] 169 dB (100 Hz 50 kHz) Radius of circular diaphragm Length of rectangular diaphragm Length of cantilever 1 Hz bin PAGE 41 41 Table 23. Piezoresistive microphone specifications in the literature. Author Diaphragm CavitySensitivity Dynamic Bandwidth Microphone/ Year Dimensions Depth (Predicted) Range (Predicted) Pres. Trans. F. P. Burns type 500 tele N/R 2.0uV/Pa*mAN/R 0 2570 Hz Microphone [38] 1.8 cm (0 2590 Hz) 1st appl. of p.r. S. Samaun et al. 1.2mm x 5 m N/R 9.5 V/Pa N/R N/R Pres. Trans. [39] 1st SiN H2O bar. W. H. Ko et al. 2.29mm x 20 m 254 m75nV/Pa*V N/R 40kPa 0 10 kHz Pres. Trans. [72] R. Schellin et al. 1mm x 1 m N/R 4.2 V/Pa*V N/R 100 Hz 5 kHz Microphone [31], [40] E. Kalvesten et al. 100 m x 0.4 m 3 m 0.09 V/Pa*V 96dBA N/R 10 Hz 10 kHz Microphone [41], [73] (0.10 V/Pa*V) (2 mHz 1 MHz) E. Kalvesten et al. 300 m x 0.4 m 3 m 0.03 V/Pa*V 90dBA N/R 10 Hz 10 kHz Microphone [74] (0.02 V/Pa*V) (10 Hz 0.9 MHz) Cav. Stiff. R. Schellin et al. 1mm x 1.3 m N/R 10 V/Pa*V 61dBA 128dBA50 Hz 20 kHz Microphone [42] M. Sheplak et al. 105 m x 0.15 m 10 m 2.24 V/Pa*V 92dB 155dB 300 Hz 6 kHz Microphone [43], [75] (100 Hz 300 kHz) A. Naguib et al. 510 m x 0.4 m N/R .18 V/Pa*VN/R 1 kHz 5.5 kHz Microphone [44] 1.0 V/Pa*V A Naguib et al. 710 m x 0.4 m N/R 1.0 V/Pa*V N/R 1 kHz 5.5 kHz Microphone [76] D. P. Arnold et al. 500 m x 1.0 m 10 m 0.6 V/Pa*V 52dB 160dB 1 kHz 20 kHz Microphone [45] (10 Hz 100 kHz) C. Huang et al. 710 m x 0.38 m ~20 m 1.1 V/Pa*V 54dB 174dB 100 Hz 10 kHz Microphone [46] G. Li et al. N/R x 1.0 m ~400 m10 V/Pa*V 34dB N/R 100 Hz 8 kHz Microphone [47] Radius of circular diaphragm Length of rectangular diaphragm 1 Hz bin PAGE 42 42 Table 24. Capacitive microphone specifications in the literature. Author Diaphragm Air CapacitanceBias SensitivityDynamic BandwidthCondenser/ Year Dimensions Gap Voltage Range (Predicted)Electret W. H Ko et al. 572 m x 25 m N/R N/R N/R 1.28 V/PaN/R N/R Condenser [48] D. Hohm et al. 8.0mm x 13 m 20 m 9 pF 350 V 3mV/Pa N/R 100 Hz Electret [49] 7.5 kHz A. J. Sprenkels et al. 3.0mm x 2.5 m 20 m N/R 300 V25mV/PaN/R 100 Hz Electret [77], [78] 15 kHz P. Murphy et al. N/R x 1.5 m 25 95 m N/R 200 V 48mV/Pa N/R 100 Hz Electret [79] 15 kHz D. Hohm et al. 0.8mm x .25 m 2 m 6 pF 28 V 0.2mV/PaN/R 200 Hz Condenser [50] 4.3mV/Pa 20 kHz J. Bergqvist et al. 2mm x 5 m 4 m 3.5 pF N/R 13mV/Pa N/R 500 Hz Condenser [51] 2 kHz J. Bergqvist et al. 2mm x 6 m 4 m 3.5 pF N/R 6.1mV/PaN/R 100 Hz Condenser [51] 5 kHz J. Bergqvist et al. 2mm x 8 m 4 m 3.5 pF N/R 1.4mV/Pa N/R 500 Hz Condenser [51] 20 kHz J. Bergqvist et al. 2mm x 5.1 m 2 m 5 pF 5 V 1.8mV/Pa37 dBA 2 Hz Condenser [80] 120dB 20 kHz P. R. Scheeper et al. 2mm x 1 m 1 m 20 pF 2 V 1.4mV/Pa N/R 40 Hz Condenser [81] N/R P. R. Scheeper et al. 2mm x 1 m 3.3 m 57 pF 16 V 2mV/Pa 35 dBA 100 Hz Condenser [82], [83] N/R 10 kHz W. Kuhnel et al. 0.8mm x .25 m 2 m 1 pF 28 V 1.8mV/Pa N/R 100 Hz Condenser [84], [85] 20 kHz T. Bourouina et al. 500 m x 1 m 5 m N/R N/R 0.4mV/PaN/R N/R Condenser [86] 20 kHz T. Bourouina et al. 707 m x 1 m 5 m N/R N/R 2mV/Pa N/R N/R Condenser [86] 7 kHz T. Bourouina et al. 1mm x 1 m 5 m N/R N/R 3.5mV/PaN/R N/R Condenser [86] 2.5 kHz T. Bourouina et al. 1mm x 1 m 7.5 m N/R N/R 2.4mV/Pa N/R N/R Condenser [86] 10 kHz E. Graf et al. N/R 0.46 m N/R 15 V 38mV/PaN/R N/R Condenser [87] 10 kHz J. Bergqvist et al. 1.8mm x 8 m 3 m 5.4 pF 28 V 1.4mV/Pa 43 dBA 300 Hz Condenser [88] N/R 13 kHz J. J. Bernstein et al. 1.8mm x N/R N/R N/R 510 V16mV/Pa25 dBA 300 Hz Condenser [89] 114 dB 15 kHz J. J. Bernstein et al. 1mm x N/R N/R N/R 510 V 16mV/Pa 25 dBA 70 Hz Condenser [89] 114 dB 15 kHz Q. B Zou et al. 1mm x 1.2 m 2.6 m 3.6 pF 10 V 14.2mV/Pa39 dBA 100 Hz Condenser [90], [91] N/R 9 kHz PAGE 43 43 Table 24. Continued. Author Diaphragm Air CapacitanceBias SensitivityDynamic BandwidthCondenser/ Year Dimensions Gap Voltage Range (Predicted)Electret Y. B. Ning et al. 2mm x 0.5 m 3 m 9.1 pF 6 V 3mV/Pa N/R 100 Hz Condenser [92] 10 kHz B. T. Cunningham et al. 1mm x 0.5 m 2 m 5.1 pF 8 V 2.1mV/Pa N/R 200 Hz Condenser [52] 10 kHz 1st Circular Mic M. Pedersen et al. 1.6mm x 0.9 m 1.5 m 14.9 pF 15 V 5.1mV/Pa35 dBA 100 Hz Condenser [93] N/R 15 kHz M. Pedersen et al. 2.1mm x 0.9 m 1.5 m 18.5 pF 15 V 8.1mV/Pa 34 dBA 100 Hz Condenser [93] N/R 15 kHz M. Pedersen et al. 2.2mm x 1.1 m 3.6 m 10.1 pF N/A 234Hz/Pa60 dBA 100 Hz Condenser [94] 120 dB 15 kHz P. C. Hsu et al. 2.6mm x 2 m 4 m 16.2 pF 10 V 20mV/Pa N/R 100 Hz Condenser [95] 10 kHz M. Pedersen et al. 2.2mm x 1.1 m 3.6 m 10.1 pF 14 V 10mV/Pa27 dBA 100 Hz Condenser [96] 120 dB 8 kHz D. Schafer et al. 0.4mm x 0.75 m 4 m 0.2 pF 12 V 14mV/Pa 27 dBA 150 Hz Condenser [97] N/R 10 kHz A. Torkkeli et al. 1mm x 0.8 m 1.3 m 11 pF 2 V 4mV/Pa 33.5 dBA 10 Hz Condenser [98] N/R 12 kHz P. Rombach et al. 2mm x 0.49 m 0.9 m N/R 1.5 V 13mV/Pa 23 dBA 10 Hz Condenser [53], [99] 118 dB 20 kHz X. X. Li et al. 1mm x 1.2 m 2.6 m 1.64 pF 5 V 9.4mV/PaN/R 100 Hz Condenser [100] 19 kHz R. Kressmann et al. 1mmx 600nm 2 m N/R N/R 2.9mV/Pa 39 dBA N/R Electret [101] (Corrugated) 123 dB 20 kHz P. R. Scheeper et al. 1.95mm x 0.5 m 20 m 3.5 pF 200 V22mV/Pa23 dBA 251 Hz Condenser [54] 141 dB 20 kHz J. J.Neumann et al. 320 m x N/R N/R 1 pF N/A 1.4mV/Pa 46 dBA 100 Hz Condenser [102] N/R 6 kHz S. T. Hansen et al. (70 m x 190 m) 1 m 3.56 pF N/A 7.3mV/Pa64 dBA 0.1 Hz Condenser [103] x 0.4 m N/R 100 kHz D. T. Martin et al. 0.23mm x 2.0 m 2 m 0.74 pF 9 V 0.28mV/Pa 22.5 dB 300 Hz Condenser [55], [104] 160 dB 20 kHz Knolwes Electronics 1.57.9mV/Pa39dBA 100HzCondenser Sisonic Mic [105] 5.5V 10kHz Analog Devices 1.5 14.1mV/Pa 32 dBA 100Hz Condenser [56] 3.6V 105 dB 12kHz Radius of circular diaphragm Length of rectangular diaphragm Frequency Modulation 1 Hz bin Note: Bias voltage for electre ts are the effective voltage PAGE 44 44 Table 25. Optical microphone spec ifications in the literature Author Diaphragm Cavity Sensitivity Dynamic Bandwidth Optical Year Dimensions Depth (Predicted) Range (Predicted) Modulation D. Garthe N/R 10 m N/R 42dBAN/R 04.3kHz Intensity [57], [106] NonMEMS J. A. Dziuban et al. 10mm x 0.5mm N/R 2.4/0.8 V/Pa (On/Off switch) N/R Intensity [58] Silicon M. A. Chan et al. 10mm x 7 m 50 m 3.75mPa/fringeN/R164dB N/R Phase [59] First MEMS Psens A. Kots et al. 1.5mm x 1.8 m N/R N/R N/R 0 15kHz Intensity [60] NonMEMS D. C. Abeysin ghe et al. 135 m x 7 m 0.64 m 0.016 V/Pa 0551kPa N/R Phase [61] Psens K. Kadirvel et al. 1mm x 1 m 500 m 0.5mV/Pa 70dB132dB 0 6.4kHz Intensity [62] ( 0 20kHz) J. A. Bucaro et al. 1.6mm x 1.5 m 200 m N/R 30.6dB N/R 0.1Hz 10kHz Intensity [63] N. A. Hall et al. 2.1mmx 0.80 m N/R N/R (17.5dBA) N/R 4kHz Intensity [64] J. H. Song et al. 800 mx m N/R N/R N/R 0.1 2kHz Intensity [65] N. A. Hall et al. 1.5mm x 2.25 m N/R N/R 24dBA N/R 20kHz Diffraction Based [66] Radius of circular diaphragm Length of rectangular diaphragm 1 Hz bin Figure 21. Outline of the different tr ansduction schemes of MEMS microphones. PAGE 45 45 A 0 V V B C V Figure 22. A) A piezoelectric ma terial in equilibrium. B) The inverse piezoelectric effect: material with an applied bi as, expanding. C) The inverse piezoelectric effect: piezoelectric material with a re verse bias, contracting [107]. PAGE 46 46 A B Figure 23. A) An unstressed re sistor with a resistance of R. B) Stressed resistor with a resistance of R + R. Figure 24. Variable capacitor schematic. PAGE 47 47 CHAPTER 3 TRANSDUCER MODELING AND DESIGN This chapter details the design process for a piezoresistive m icrophone ta ilored to acoustic measurements. Figure 31 shows the methodology of the design. A piezoresistive m icrophone has three main components: diaphragm, cavity a nd vent. The diaphragm is a layered composite composed of silicon, silicon dioxide and silicon nitride. The silicon dioxide layer is used to passivate the resistors, and the silicon nitride laye r is used to create a moisture barrier. These materials were chosen because they are standard silicon processing materials. The fabrication of these layers induces stresses in the diaphragm that must be taken into account. A sensor mechanical model is developed to calculate pr essure induced stress in the diaphragm as a function of geometry and fabrication induced stresses. An electroacoustic model then determines the change in resistance of the pi ezoresistors due to the stress determined by the previous model. Both the electroacoustic tran sduction model and sensor mechanical model are incorporated utilizi ng lumped element modeling (LEM) to determine the dynamics of the multidomain system. Design optimization, which is discussed in Chap ter 4, incorporates the LEM results, design specifications and manufacturing co nstraints to yield an ideal devi ce. The mechanical model is verified using FEA, and a cavity and vent st ructure is designed to accompany the diaphragm determined during design optimization. The mechanical model of the diaphragm and fi nite element analysis (FEA) verification is described in section 3.1. The electroacoustic transduction model is derived in section 3.2. A LEM is discussed in section 3.3. Finally, a cavity and vent structure is designed in section 3.4. PAGE 48 48 3.1 Composite Plate Mechanics A brief review of previous re search into plate mechanics includes the mechanical scaling of capacitive and piezoresistive pressure tran sducers presented by Ch au and Wise [108]. However they did not discuss the ef fects of inplane forces on the stress field of the diaphragm. Voorthuyzen and Bergveld [109] furthered the field by using a finitedifference model to investigate the largedeflection characteristics of circular diaphragms in pressure sensors subjected to inplane loading over a limite d dimensional domain. Sheplak and Dugundji clarified the nonlinear behavior of circular plates under tension by incorporating the structural mechanics of thinfilm diaphragms with large inplane forces vi a a fundamental structural model using von Krmn plate theory. This work shows that the deflection field is a strong function of both inplane loading an d nonlinear restoring forces [110]. Th e model presented in this work extends the work of Sheplak and Dugundji [110] by incorporating a composite makeup and compressive stresses in addition to tensile. To determine the stresses in the diaphragm and calculate the resonant frequency, a nonlinear composite plate model was developed. This nonlinear anal ysis of a circular composite diaphragm under a static load is used to determine the behavior of the plate as a function of geometry and fabrication induced stresses. Th is model describes an axisymmetric composite diaphragm made up of transversely isotropic materials. The plate behavior was analyzed using classical laminated theory to derive governing differential equations which were then solved using an iterative finite difference scheme. A br ief description of the derivation is given in the next section and the details are included in Appendix A. 3.1.1 Derivation of Governing Equations The analyzed plate is a composite structure composed of 3 layers. The base layer is silicon, the middle layer is silicon dioxide and the t op layer is silicon nitride. The piezoresistors PAGE 49 49 are implanted into the top of the silicon layer. To simplify the calculation, the reference plane was put at the same layer as the resistors. This plane can be seen in Figure 32. The transverse and radial deflections, 00 and wrur, are defined at the reference plane. The following three assumptions are stated by Kirchoffs hypothesis an d are used in the following derivation of circular plate deflection under static loading ( Figure 33) [111]: straight lines perpendicular to the neutral surface before deformation (i.e. transverse normals) remain straight after deformation, transverse normals do not experience elongation, transverse normals rotate such that they re main perpendicular to the neutral surface after deformation. In addition, the material is assumed to be tr ansversely isotropic and the circular diaphragm deflection is assumed to be axisymmetric 3.1.2 Equilibrium Equations The equilibrium equations are derived in Appe ndix A and are repeated here for reference [111]: 0r rNN dN drr (31) r rrdM rQMrM dr (32) and 00rrdw dd rNrprQ drdrdr (33) where , and rrrNNMMQ are the inplane forces, moments and shear in the plate, respectively. The subscripts r and refer to the radial and tangential directions. PAGE 50 50 3.1.3 Constitutive Relationship Assuming that silicon is transversely isotropi c, the constitutive re lationships are defined as 0 0 rrr rQQQz (34) where Q is defined as 1112 2 21221 1 1QQ E Q QQ (35) The terms 0 and are the initial strain due to inplane forces, elongation strain due to transverse loading and curvature due to transverse loading, respectively. The forces per unit length are found by integrating equation (34), T Bz rr zN dz N (36) Substituting equation (34) into equation (36) yields TTT BBBzzz o rrr r o zzzN QdzQdzQzdz N (37) It is convenient to define the extensional stiffness matrix, T Bz zAQdz (38) and the flexural extensional matrix T Bz z B Qzdz. (39) Equation (37) is compactly written as, PAGE 51 51 o rrr r oN AAB N (310) The moments per unit length are determined by in tegrating the stress times its moment arm, z, over the thickness: T Bz rr zM zdz M (311) Substituting equation (34) into equation (311) yields 2TTT BBBzzz o rrr r o zzzM QzdzQzdzQzdz M (312) It is now convenient to define the flexural stiffness matrix, 2T Bz zDQzdz. (313) Equation (312) is compactly written as, o rrr r oM BBD M (314) 3.1.4 Nonlinear Solution In the linear case it is possible to isolate the transverse deflection 0w and solve the ordinary differential equation (ODE) explicitly. In the nonlinear case, 0w cannot be isolated from 0u and therefore two c oupled nonlinear ODEs are derived, yielding, 2 *2**2 2211 2rdd kPS dd (315) and 2** 2 2 2 223 2rrdSdS dd dddd (316) PAGE 52 52 where 12 B h D (317) 222 1112 11AA h AD (318) and 00 22 24 **** 0 ****, , and 2r rwu rdWa WU ahhdh NaNa Na p a SSkP DDDhD (319) The and ABD matrixes are dependant on the composite makeup and are derived in Appendix A. As the name suggests, the A matrix illustrates how the plate reacts to extensional forces. The D matrix shows how the plate reacts to transverse forces and bending moments and the B matrix shows the reaction of ex ternal forces to bending and transverse forces to stretching. The *D value is a function of the composite matrices defined as 2 11 11 11 B DD A (320) Assuming a perfectly clamped pl ate, the boundary conditions for and rS are 00 (321) 10 (322) 00rdS d (323) and PAGE 53 53 12 1 11 1 11r rdSA d S dAd (324) The symmetry coefficient is a measure of the symmetry of the composite plate. This parameter takes into account that the reference pl ane is not necessarily th e same as the neutral plane. If the composite plate is sy mmetric about the reference plane then 0 ; the more asymmetric the plate becomes about the reference plane, the larger becomes. The approximate range for is 00.04 for the composite makeup considered here. The composite coefficient captures the disparities between th e different composite materials. For a homogenous plate, 2121 For the composite makeup studied in this dissertation, deviated approximately 20% from the homogenous value. Equations (315) and (316) are then solv ed using an iterative finite di fference scheme discussed in Appendix A. 3.1.5 Deviation from Linearity Ideally, a microphone should have a linear response over the entire dynamic range of the device. Having a linear response en sures a constant sensitivity with respect to pressure, which is essential to designing a microphone with low distortion. To ensu re this, the composite plate model is used to calculate the percent devi ation from linearity of the diaphragms center deflection. The result of a nonlinearly defl ecting diaphragm is harmonic distortion. A 5% deviation from linearity is set as the lim it for the maximum detectable pressure. Figure 34 shows the nondim ensional sensitiv ity of devices with varying inp lane forces. This figure also shows the tradeoff between having an increase in sensitivity versus an increase in the maximum pressure to remain linear. Figure 35 shows the maximum pressure that can act on a plate with a given in plane load to remain linear. PAGE 54 54 As the inplane force parameter decreases past 10 the linear range of the plate goes to zero. This is due to the onset of buckling and this model accurately predicts the axisymmetric buckling modes. 3.1.6 Calculation of Stresses The stress in the plate is decomposed into init ial stress due to fabricat ion and stress due to pressure loading, initial stressstress due to loading rrr (325) where 0 0 rr rQQz (326) The stress due to pressure loadi ng is desired, therefore the nondimensional radial and tangential stresses due to loading are defined by 2 r r Sih E a (327) and 2 Sih E a (328) Solving for and r yields 21 1rii idUUd dd (329) and PAGE 55 55 21 1ii idUUd dd (330) where a h (331) z h (332) and i is the local Poissons ratio. is defined as 2 21 if if if mSi SiO mSiO Si SiN mSiN SiEE E EE E E EE E (333) where mE is the Youngs modulus in the given layer. 3.1.7 Validation Using Finite Element Analysis This section details the verification of the composite plate mechanics of the diaphragm using FEA. The FEA was performed using the commercially available ABAQUS software package. The elements used were 3node quadra tic, axisymmetric shell elements. This shell theory allows for finite strains and rotations of the shell [112]. The strain measure used is accurate to second order with regard to strain. These el ements are accura te for scenarios modeled in terms of Kirchhoff stresses with the following assumptions [112]: Only terms up to first order with respect to the thickness direction are included. The thinning of the shell due to stretching is assumed to be unifor m through the thickness. The thinning of the shell is assumed to occur smoothly. All stresses except those acting parallel to the reference surface are neglected. PAGE 56 56 Planar cross sections remain planar. Transverse shears are assumed to be small, and the material response to such deformation is assumed to be linear elastic. The FEA model uses independent stresses de fined in each layer. The geometric and material values used can be found in Table 31. Figure 36 shows the deflection of a composite plate calcu lated analytically from section 3.1 and compared to results obtained from the FEA sim ulation. It is easily seen that there is excellent agreement between the two calculations. Figure 37 illustrates the transition from linear to nonlinear behavior as the plate d eflection becomes large. Once again there is excellent ag reement between the analytical and FEA models well into the nonlinear regime. 3.2 Electroacoustics With the pressure induced stress field known, an electromechanical transduction model is coupled with the mechanical model to obtain the resulting voltage output. 3.2.1 Piezoresistors Piezoresistivity, which is define d as the change of the resistivity of a material due to a change in carrier mobility, as a result of applied mechanical stress. In piezoresistive transduction, resistance modulation is a function of the a pplied stress and piezoresistive coefficients ij [113]. For the cubic crystal structure of silicon, the relationship reduces to 1111212 1 2121112 2 3121211 3 23 44 23 13 44 13 12 4412000 000 000 1 00000 00000 00000 (334) where is the change in resistivity. The resulti ng electric field from st ressed silicon is [113], PAGE 57 57 111111223144212313 211221213244112323 311331212344113223unstressed stressediiiii iiiii iiiii 1 2 3E E E. (335) The first term captures the contribution from unstressed silicon. The second term captures the effect of a stress in the same direction as the current flow i. The third term captures the effect of the normal stresses acting perpen dicular to the current and the final term represents the effect of shear on the electric field [113]. The piezoresistive coefficients are then tr ansformed to an arbitrary axis using the following transformation matrix [114], 111 222 333 x lmnx y lmny zlmnz (336) where ,, and iiilmn are the direction cosines which are given in terms of Eulers angles [114], 111 222 333lmncccssscccssc lmnccsscscsccss lmncsssc (337) where cosc etc. These angles are graphically illustrated in Figure 38 [19]. For this c ase, 100 silicon is used and the piezoresistors are implanted vertically into the wafer. Therefore, 0 and 0 (refer to Figure 38) and the matrix (337) reduces to 111 222 3330 0 001 lmncs lmnsc lmn (338) where varies from 0 to 2 Applying a transformation of coordinates to equation (335), using the m atrix (338), the longitudinal and transverse piezoresistive coefficients are [ 19], PAGE 58 58 222222 114412111111112 lmlnmnl (339) and 222222 12441211121212llmmnnt. (340) These coefficients are plotted for 100 p type silicon in Figure 39. The piezoresistive coefficien t also depends on temperature and doping level. This relationship is given as a product of the lowdoped room temperature va lue and a piezoresistive scaling factor PNT [19] shown as, ,, NTPNT, (341) where N is doping concentration and T is temperature. Many theoretical [19] and experimental [115117] studies have shown th at the piezoresi stive factor PNT is a function of doping concentration. Kandas model [19] accurately predicts the effect of doping concentration and temperature for low concentrations. Kand as model also shows how at higher doping concentrations, the effect of temperature on the pi ezoresistive coefficients is minimized. In fact, at concentrations above 19310#cm, the piezoresistance coefficient is a weak function of temperature [117]. However, the sensitivity declines due to the reduced piezoresistive coefficient at a high doping level [115]. Kandas model however, when compared to experimental data [115117], un der predicts the decline of PNT for concentrations above 17310#cm. For doping concentration above 17310#cm, the experimentally fitted piezoresistive factor PNT [118] is used, 0.2014 2231.5310 ,300logcm PNK N (342) PAGE 59 59 The piezoresistive factor is plotted in Figure 310 versus concentra tion at room temperature. The piezoresistors are designed to take advant age of the crystallogr aphic dependence of ptype silicon having almost equal and opposite piezoresistive coefficients ( Figure 39). The resis tors are designed to isolate cu rrent either in the tangential direct ion or in the radial direction. The geometry of the piezoresistors in the diaphragm can be seen in Figure 311. To calculate the change in resistan ce, the piezoresistors are divided up into differential elements ( Figure 312) and their ind ividual resistances are numerically inte grated. For the arc resistor, the resistance of an unstressed differential element is [18] arcrd dR dzdr (343) and the resistance of a stressed element is given by 1arcarc llttrd dRdR dzdr (344) where is the resistivity of the material. Summing up the unstressed differentia l resistors in series (equation (343) along the direction) yields, ,a sarczr R drdz (345) Summing up the differential elements in parallel yields, 01j aout ainz r arca rdrdz R r (346) Integrating in the r direction yields, 0ln 1jaout z ain aar r dz R (347) PAGE 60 60 The resistivity is [18] 1npqNP (348) where q is the charge of an electron 1.619 qeC n and p are the mobilities of electrons and holes respectively, and N and P are the concentration of elect rons and holes respectively. The piezoresistors are doped heavily w ith boron which is a ptype implant p n. Therefore equation (348) reduces to 11pppqPqN (349) where pN is the doping concentration of boron [18]. The mobility is modeled after an empirical fit [18], 0 min1p p refN N (350) where min0, and refN are constants depending on the temperature and dopant [18]. Substituting in for in equation (347) yields min 0 00ln 1 1jjaout zz p ain p aa p refr q Nz r Nzdzdz R Nz N (351) Taking the inverse of equation (351) results in PAGE 61 61 min 0 001 ln 1jja a zz aout p p ain p refR r Nz q Nzdz dz r Nz N (352) The ion implantation process yields a Gaussian doping profile [119], 2:0jz z s ps bN NzNz N (353) where z, and s jbNN are the surface concentration, junction depth and background concentration, respectively. A sample dopant profile is shown in Figure 313. The fabricated profile will differ slightly fr om a Gaussian profile. Performing the same procedure for the stressed arc resistor (equation (344)) yields, 01 1,,r j aout l ainaa z r lltt rd RR dzdr r zrzrz (354) For the taper resistor, the resistance of an unstressed differential element is 2tdr dR rddz (355) and the resistance of a stressed element is given by 2 1tt llttdr dRdR rddz (356) The factor of 2 follows from the fact that ther e are two legs per taper resistor. Performing the same procedure for the unstressed taper resistor as for the unstressed arc resistor yields, PAGE 62 62 min 0 002ln 1 1jjtout tin t zz wt p p p refr r R q Nz Nzdzdz Nz N (357) Similarly for the stressed taper resistor, 02 1,,tout j r tin lr tt z r llttdr RR dzd r zrzrz (358) The MATLAB mfiles for cal culating the resistance can be found in Appendix C. 3.2.2 Wheatstone Bridge The piezoresistors are arranged in a fully active Wheatstone bridge configuration as seen in Figure 314. For a constant voltage bias bV, the output of the bridge 0V yields, 0 aatt b aattRRRR VV RRRR (359) The arc and taper resistors are designed to have the same nominal resistance value at R RR Applying this to equation (359) yields, 02at b atRR VV RRR (360) Knowing that the change in resistance is small compared to the mean resistance, the power consumption for the circuit is 2bV P R (361) If the device is biased wit h a constant current source b I the response is, PAGE 63 63 02aatt bRRRR VI (362) Again, assuming a balanced bridge, at R RR yields, 02at bRR VI (363) Equation (363) reveals that when the device operate s with an ideal constant current so urce connected to the Wheatstone bridge, the output voltage does not depend on the unstressed resistance value. The power consumption for a constant current source is 2bPIR (364) For a device operating with either a voltage or current supply, a power limitation would be implemented to keep the ove rall power consumption below 100mW. 3.3 Lumped Element Modeling The most accurate and complete way to mathem atically describe a physical system is a physicsbased model that is correlated to an anal ytical expression for the system behavior. FEM does extremely well at predicting system behavior in cases where an analytical approach is impractical. FEM techniques can accurately predict system behavior using a numerical approach, producing results that can accurately mi mic a physical system, but the physical insight obtained is limited. In addition, the FEM results are dependant on the convergence of the iterative calculations as well as the numerical, and it is therefore hard to determine scaling behavior from FEM results. LEM is useful to gain understa nding of the scaling laws of the system [120122]. The use of LEM reduces the complexity of a numeric or analytic expre ssion by dividing a given distributed system into discrete elements that ar e based on system interactions with energy [121]. PAGE 64 64 Three different types of inter actions are accounted for: the storage of kinetic and potential energy, and the dissipation of energy. The storage of kinetic and potential energy in a distribute d dynamic system requires a partial differential equation to accurately represent the physics of the problem, because the spatial and temporal components ar e intrinsically coupled [123], [124 ]. When the wavelength of the signal increases to the point at which it is considerably greater than the length scale of interest, negligible variation in the distribution of energy as a function of space occurs. At this point, the mathematical decoupling of the spatial and temporal components allows for the use of ordinary differential equations [120]. This met hod assumes that the static mode shape is similar to the dynamic mode shape up to th e first resonant frequency. Although nomenclature varies in different energy domains, the mathematics remain constant. In lumped mechanical systems, ki netic energy is stored via mass while potential energy is represented as the compliance of a sp ring and the dissipation of energy is represented as the losses of a damper. In electrical systems, kinetic ener gy is represented as the magnetic field of an inductor, while poten tial energy is represen ted as the charge across a capacitor, and the dissipation of energy is represented as a resist or. In lumped acoustical systems, kinetic and potential energy are represented as acoustical mass and acoustical compliance, while the dissipation of energy is given as an acoustic resistance. Many techniques, both graphical and analytic al, have been developed to solve large networks of interconnected elements. In thes e techniques, the interc onnected elements are represented using electrical circuit notation. In lumped element modeling the elements are denoted using an equivalent ci rcuit form for all of the en ergy domains, where masses are represented as inductors, compliances are repr esented as capacitors, and the dissipative PAGE 65 65 components are represented as resistors. Once th e complete equivalent circuit is constructed, standard circuit analysis techniques can be app lied to find the solution. Power flow between the elements must be considered when managing more than one lumped element. The elements must be represented in terms of conjugate power variables, more specifically referred to as an effort, e, and a flow f where the product ef is power. A table of conjugate power variables is given in Table 32 for several of energy domains. 3.3.1 LEM of piezoresistive microphone To create an analytical model for the microphone, the system is broken down into sections. The microphone is composed of three main mechan ical components: diaphragm, cavity and vent and can be seen in Figure 315. The following sections discuss the im pedances of each. 3.3.1.1 Diaphragm The diaphragm is modeled as a mass, spring and a damper. The distributed diaphragm is modeled as a clamped circular pl ated in order to find the lumped element representation for the diaphragm. The plate is lumped to a piston of mass, daM, a resistance of da R and a compliance, daC, shown in Figure 316. The area of the pist on an d the area of the diaphragm are not equal; rather the piston ar ea is equated to maintain volum e velocity continuity between the physical diaphragm and the piston model. The acoustic compliance of the diaphragm is equal to the volume of air displaced by the deflection of the diaphragm [125], daVol C p (365) where the change in volume is given by 2 00aVolwrrdrd (366) Plugging in the nondimensional parameters from equation (319) yields, PAGE 66 66 1 6 ** 0daW a Cd DP (367) The acoustic mass is determined by evaluating the kinetic energy expressed in acoustic conjugate power variables to the total kinetic ener gy. This mass is calculated as [125], 2 02a daAwr M rdr Vol (368) where A is the areal density of the composite plate. Substituting equation (366) into (368) and substituting for the nondim e nsional variables results with, 2 2 1 5 ** 02A da daW a M d DCP (369) There is an additional effective mass that acts on the diaphragm, caused by fluid particles that oscillate with the diaphragm. This radiation mass, radM is given by approximating the diaphragm as a piston in an infinite baffle [120], 28 3air radM a (370) where, air is the density of air. The radiation mass is added to the diaphragm mass to give a combined diaphragm mass. Assuming the diaphr agm is lightly damped, the resonant frequency of the diaphragm is 11 2dia dadaf CM. (371) The resistance, da R is caused by damping in the diaphragm. The majority of the damping contributions are the dissipation to the suppor ts, the dissipation into the surrounding air and thermomechanical dissipation within the structure. This term is difficult to analytically express PAGE 67 67 accurately and therefore a va lue of the damping ratio 0.03 is taken from previously fabricated devices with similar size and aspect ratio [37]. Th e diaphragm resistance is then calculated[126] as 2da da da M R C. (372) The total impedance from Figure 316C is 1dia darad da da Z jMMR jC (373) 3.3.1.2 Cavity The cavity is the open area behind the diaphragm. The cavity is cylindrical in shape and has a backplate that is assumed to be rigid. Th e cavity is therefore m odeled as a closed cavity with a sound hard boundary. The specific acoustic impedance in the cavity is [127], 0,cot Z djZkd, (374) where d is the distance from the bottom wall. For a short cavity, the cotangent function can be expanded to yield 0 2 01 lim, ... 3kdZ kd Zdj akd (375) For 0.3kd all but the first term may be neglected, yielding 1 ,aZdZ j C (376) where 2 00aV C c (377) and V is the volume of the cavity, PAGE 68 68 2Vad (378) a is the radius of the diaphragm, and d is the depth of the cavity. If the cavity is required to be longer, so that 0.3kd does not hold, then the second term in the cotangent expansion of equation (375) needs to be accounted for. This term represents an acoustic mass, 0 23ad M a (379) with the total impedance being, 1cav a a Z jM jC (380). A comparison of the full cotangent solution and the leading terms is shown in Figure 318. 3.3.1.3 Vent The vent channel is designed to have a larg e resistive value with a small mass. To accomplish this, the length must be as large as po ssible with a small crosssection. The lumped acoustic mass for a laminar pipe flow is calculated by integrating equation (368) yielding [120], 24 3avL M a (381) Microfabrication processes are not capable of fabricating circular channels and therefore, the vent channel has a square crosssection. B ecause of the limited space on a microphone die, the vent channel is designed as a serpentine. Th is can be accommodated by ca lculating an effective length of the channel by adding all of the ma jor and minor head losses [128]. Equation (381) is estim ated using a hydraulic diameter and effective length yielding, 216 3eff av HL M D (382) PAGE 69 69 where H D is the hydraulic diameter and effL is the effective length. The hydraulic diameter is defined as [128] 42 1HAh D h P b (383) where A and P are the cross sectional area and perimeter, h is the depth of the vent and b is the width. The effective length takes into ac count any turns in the channel by equating the channel with turns to a longer ch annel without any turns. This device will only have 90 degree turns so the effective length is equal to the physical length plus a correction factor to account for the turns [128] 60eff HLLnD (384) where L is the length of the vent and n is the number of right angl e turns. To determine the resistance of the vent, volume velocity is expressed as a function of pressure yielding, 48a QVdAp L (385) where Q is volume velocity and p is pressure drop. The resist ance is then taken from equation (385), 48avL R a (386) Adjusting the resistance for a serpentine square channel yields, 4128eff av HL R D (387) The total impedance is ventav av Z RjM (388) PAGE 70 70 3.3.1.4 Equivalent circuit Analyzing Figure 315, pressure acts on the diaphr agm and vent simultaneously due to the front side vent. There is a pressure drop across th e diaphragm and through the vent and the resulting flows converge in the compressible fluid back cavity. This resu lts with the diaphragm and vent impedance being in para llel and the effective impedance is in series with the back cavity. Using this analysis, the equivalent circ uit for the lumped element model is seen in Figure 317 The acoustic sensitivity directly relates to the voltage output of the m icrophone, which is defined as cqjp assuming sinusoidal input [75]. Using Kirchhoff circuit theory ( Figure 317), a transfer function for the acoustic sensitivity o f the multidomain dynamic system is 2c vent a diacavventdiacavqj Z S jp ZZZZZ (389) The magnitude and phase response for aS is seen in Figure 319 and a table outlining the lum ped element parameters is seen in Table 33. The piezoresistive transduction scheme is modeled as a dependant voltage source, D SV and is given by c D Sm e daq VS jC (390) where meS is the quasistatic mechanical sensitivity of the device in VPa [75]. The remaining terms represent the pr essure drop across the diaphragm []Pa Therefore, the total dependant source output is in V. For a device connected to an am plifier with a very large input impedance, the output voltage of the device is eq ual to the dependant sour ce voltage. In the limit of zero frequency, the right si de of the circuit possesses ( Figure 317) infinite impedance, PAGE 71 71 therefore, the microphone does not respond to slow changes in pressure. As the frequency of interest gradually increases, the microphone resp onse will increase linearly with frequency [75]. After a corner frequency defined by, 1 2c avdaf R C, (391) the response of the microphone will be in the flat band region and will have zero phase shift until the frequency of the incident pressure is at the resonant frequency of the device. 3.3.1.5 Cuton frequency and cavity stiffening To determine the cuton frequency of the microphone and prevent cavity stiffening, a cavity and vent structure was designed. Examining Figure 317, and assuming the frequency of inte rest is above the cu ton frequency, the equiva lent circuit reduces to Figure 320. The com pliance of the cavity is in series with the co mpliance of the diaphragm. Therefore, if the compliances are of the same order of magnitude the cavity will have a significant restorative force on the back of the diaphragm, resulting in a reduction of sensitivity. Also illustrated in Figure 317, the vent structure need s to have a large resistance to lower the cuton frequency. If the resistance of the vent is too low, all of th e volume flow will pass th rough the vent, resulting in no response of the dia phragm as seen in equation (392). vent c ventcavZ qq ZZ (392) 3.3.2 FEA verification A modal analysis was performed using ABAQ US and compared to the lumped element model, equation (371). This was done for three different size diaphragms designated A, B and C. The results can be seen in Table 34 and all results match to within 1%. An axisymmetric PAGE 72 72 composite plate model meshed with seventyfour 3node quadratic thin (or thick) shell elements (type SAX2) was used. Residual stress was supplied as an initial condition. 3.4 Electronic Noise The dominant source of noise in a piezoresi stive microphone determined by Dieme et al. [129] is the electronic noise of the resistors. Therefore, the lowest detectable signal is determined by the electronic noise of the Wheatst one bridge. The two dominant types of noise in resistors are thermal noise and flicker 1 f noise. Thermal noise is given by [130], 4tRbKVkTRf (393) where bk is Boltzmanns constant, kT is the temperature in Kelvin, R is the nominal resistance, and f is the frequency range over which the noi se is calculated. When an external voltage is applied to imperfect el ectrical conductors with interfacial or bulk defects, an excess noise above the thermal equilibr ium noise floor is observed. This excess noise exhibits an inverse frequency 1/ f dependence. The mechanism that gene rates electrical 1/ f noise is still debated, however the model used in this work follows the mechanism described by Hooge [131]. Hooges mechanism is described as the fluctuation in the bulk mobility of the material. He gave an empirical formula for the noise PSD of 1/ f noise as 2 1/ fV S Nf, (394) where is the Hooge parameter (determined experimentally), N is the number of carriers in the resistor, and V is the bias across the resistor. Th e voltage noise is then given by [131], 2 2 1/ 1lnfRV f V f N (395) PAGE 73 73 where 2 f and 1 f are the bounds of the frequency ra nge of interest. Using equations (393) and (395), the total noise in the Wheatstone bridge at the output is calcu lated to be 2 2 21 11 11 4ln 8NbK arctapf VkTRffV NNf (396) The Analog Devices AD624, which has a noise power spectral density of 4/ nVHz at 1kHz, is used for amplification. The 1 f noise of the amplifier is much lower then that of the resistors and therefore it is neglected. The total noise of the microphone coupled with the amplifier is given by 2 2 2 21 21 11 11 4ln49 8NbK H arctapf VkTRffV eff NNf (397) For this device, the power spectral dens ity of the noise was calculated for a 1 Hz bin centered at 1 kHzdBSPL. 3.5 Conclusions All portions of the piezoresistive microphone ar e modeled in this ch apter. A composite plate model is derived that determines the stress in the diaphragm and the onset of nonlinearity. An electromechanical transduction model determines the resulting change in resistance due to a given stress field, and a lumped element m odel determines the ove rall dynamics of the microphone including the cavity and ve nt structure interactions. Gi ven a set of design variables, the expected performance of a microphone can be calculated. This includes the frequency response function and onset of nonlinearity. Chap ter 4 implements an optimization scheme that utilizes all of these models to generate a superior device. PAGE 74 74 Table 31. Material parameters and thicknesses used for FEA analysis. Material Thickness0 E [ m] [MPa] [MPa] [] Si 2.0 0 150 0.27 SiO2 0.3 300 70 0.17 SixNy 0.1 100 270 0.24 Table 32. Conjugate power vari ables for various energy domains. Energy Domain Effort Flow Mechanical translation Force, F Velocity, v Fixedaxis rotation Torque, Angular velocity, Acoustic Pressure, P Volumetric flow, Q Electric circuits Voltage, V Current, I Magnetic circuits MMF, M Flux rate, Incompressible fluid flow Pressure, P Volumetric flow, Q Thermal Temperature, T Entropy flow rate, S PAGE 75 75 Table 33. Lumped element modeling parameter estimates. Acoustic impedance Description da M Equivalent acoustic mass lumped as a rigid baffle. 2 2 1 5 ** 02A da daW a M d DCP daC Volume displacement normalized by the pressure. 1 6 ** 0 daW a Cd DP radM Approximating the diaphragm as a piston in an infinite baffle. 28 3air radM a Diaphragm [16], [75] da R Do not have an accurate way of modeling the damping of the diaphragm. Damping ratios estimated from experiments of similar previously fabricated devices. aC 2 00 aCC c where is volume of the cavity. Cavity [127] a M 0 23ad M a where d is the depth of the cavity. avR Assuming fully developed pressure driven flow. 4128eff av H L R D where effL is the effective length and H D is the hydraulic diameter. Vent [128], [132] av M Also assuming fully developed pressure driven flow. 216 3eff av HL M D Table 34. Results from FEA analysis compared to analytical results. Resonant Frequency [kHz] Device Analytical FEA % Difference A 253.5 252.1 0.56% B 227.2 225.7 0.66% C 200.1 199.1 0.50% PAGE 76 76 Figure 31. Overview of th e microphone modeling process. Figure 32. Schematic of composite plate. PAGE 77 77 Figure 33. Kirchoff's hypothe sis showing the neutral axis and transverse normal. 101 100 101 102 103 104 105 106 104 103 102 101 100 Nondimensional pressure (P*)Center deflection per pressure (W0( = 0)/P) Figure 34. Nondimensional cent er deflection per unit pressure of devices with varying inplane forces. PAGE 78 78 20 10 0 10 20 30 40 50 102 101 100 101 102 Inplane force parameter (k*2)Maximum pressure to remain linear (P*max) Figure 35. Pressure that resu lts in a 5% deviation from linearity for various inplane forces. 0 20 40 60 80 100 0.12 0.1 0.08 0.06 0.04 0.02 0 Radius (r) [m]Transverse deflection (w0) [m] FEA Analytical Figure 36. Analytical deflec tion of clamped plate, at the onset of nonlinearity (2000 Pa), compared to FEA results. PAGE 79 79 101 100 101 102 103 104 105 106 104 103 102 101 100 Nondimensional pressure (P*)Center deflection per pressure (W0( = 0)/P) Analytical FEA Figure 37. Center deflec tion per nondimensional pr essure as a function of *P for various values of inplane stresses. z x y*y*z* x Figure 38. Description of the Eulers angles [19]. PAGE 80 80 2e010 4e010 6e010 8e010 30 210 60 240 90 270 120 300 150 330 180 0 tl 110 110lt Figure 39. Crystallographic de pendence of the piezoresistive co efficients for ptype silicon 1 Pa [19]. PAGE 81 81 Figure 310. Piezoresistive factor dependence on doping concentration at room temperature Blue line corresponds to Kandas work [ 19] and the pink line corresponds to the work of Harley et al. [118] a, tinr, toutr, aoutr, ainra wt t Figure 311. Geometry of piezoresistors. PAGE 82 82 Taper Resistor Arc Resistor Figure 312. Differential elements of the arc and taper resistor. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1015 1016 1017 1018 1019 1020 Depth into substrate (z) [m]Carrier concentration (Np) [#/cm3] Figure 313. Sample Gaussian dopant profile. PAGE 83 83 tt R R aa R R 0V ,bbVItt R R aa R R Figure 314. Stressed arc and taper resist ors configured in a Wheatstone bridge. tp radM,aaCM,avav R M ,,dadada M CR Figure 315. Schematic of MEMS microphone and associated lumped elements. PAGE 84 84 A p Area, AeffMdaCda B C MdaCdaRda p(t) Q Figure 316. A) Diaphragm with distributed deflection. B) Lump ed diaphragm with equivalent volume flow rate. C) Equivalent circuit for the lumped diaphragm. aCDS BROV)( tp aCDSVBROV avMavRdaC daradMM a M dia Z vent Z cav Z qc da R Figure 317. Equivalent ci rcuit model of the microphone. PAGE 85 85 0 0.5 1 1.5 2 2.5 3 10 8 6 4 2 0 2 4 6 8 10 kdIm[Zcavity/Z0] Spring SpringMass Full Sol. Figure 318. Accuracy of first terms of cotangent expansion. 101 100 101 102 103 104 105 106 20 0 20 40 Magnitude [dB] (ref response @ 1kHz) 101 100 101 102 103 104 105 106 150 100 50 0 50 Frequency [Hz]Phase [deg] Figure 319. Magnitude and phase response of LEM normalized by the flat band response. PAGE 86 86 darad M M daCda R aCa M p tdq Figure 320. Equivalent circuit illustrati ng the effect of the cavity compliance PAGE 87 87 CHAPTER 4 OPTIMIZATION There are many factors that influence the beha vior of the device, including com posite layup, aspect ratio, piezoresistor geometry, doping de nsities and doping profiles. Because of this, an optimization scheme was employed to de termine the variables that yielded optimal performance specifications. This chapter begi ns with the methodology used to formulate the optimization scheme, including the objective func tion, variables and constraints. Next, the results are discussed for various cases includ ing a device operating on a voltage source and current source. In addition, an optimization was then run to determine the feasibility of fabricating multiple devices on a single wafer. Finally, a sensitivity analysis and uncertainty analysis was performed on the final optimized devices. 4.1 Methodology The ultimate goal is to maximize the opera tional space of the device over a specified region shown in Figure 41. To increase this area, th e optim ization scheme must minimize the minimum detectable pressure, M DP, and simultaneously maximize the maximum detectable pressure,maxP, and bandwidth, B W, of the device. To accomplish this multiobjective optimization, the constraint method is implemented to generate a Pareto front [133]. An example is shown in Figure 42. In this figur e, th e utopia point is defined as the point that possesses the best obtainable value for each obje ctive function. Points AD are nondominated solutions on the Pareto front and point E is a domin ated point within the feasible region of the design space. Since the maximum detectable pr essure and bandwidth have a more specified desired value, they are chosen to be constr aints. The MDP remains the primary objective function because the goal is to lower this value as much as possible. Mathematically this is expressed as PAGE 88 88 max1 2 and ijMDP PBW Minimize such that, (41) where 1 and 2 are the iterated constrained va lues varied over a desired range. 4.1.1 Objective Function The sensitivity of the microphone is defined as the ratio of the output voltage to the input pressure, 0meV S P (42) which can be seen pictorially in Figure 43 where 0V is given by equation (360) or (363) for a voltage or current source, respec tively. The m inimum detectable pressure is defined when the output signal is the same ma gnitude as the noise level, min 0NN meVVP P SV (43) where NV is given by equation (397). Expressing minP in dB yields min20logrefP MDP P (44) where, 20refPPa (45) 4.1.2 Variables As shown in sections 3.1 and 3.2, the design variables are as follows: 123 ,,,, ,,,,,,,,,,, j atwtainaouttintoutGeometric HHHazrrrr, (46) 123123 ,,,,,, M aterial EEE (47) PAGE 89 89 and pBBOther NVorI. (48) To satisfy equation (360) and balance the Wheatstone bridge, the mean resistance of the taper resis tor is set equal to the ar c resistor. To accomplish this, th e following relationship must hold: ,,2 loglogwt atinainaa rr (49) It is important that the resistors are located at the point of highest stress. To ensure this, the outer radii of the resistors ar e constrained to be 5m larger then the radius of the diaphragm. This accounts for any fabrication issues as sociated with the back to front side alignment and takes into account compliant boundary conditions. FEA studies have indicated that the compliant region scales with the diaphragm thickness. The fabricat ed microphone structure is similar to the work of Chandraschakren [134] whose calculated Hooge parameter was 75e In addition the work of Dieme [135] leads to an estimate of the value for the Hooge parameter of 56e. Finally, the material values are assumed to be constant a nd are not varied during the optimization. This reduces the total optimization variables to twelve, 123 ,,,,,,,,,,,, or j ataintinsurfBBHHHazrrNVI (410) 4.1.3 Constraints The constraints are comprised of fabrication and performan ce constraints. All of the variables are given upper and lower bounds so that 123 ,,,,,,,,,,,,jataintinsurfBLBHHHazrrNVUB. (411) The values for the LB and UB were determined by physical and fabrication limitations and can be found in Table 41. For example, the resistors are im plan ted into the silicon layer. Therefore the junction depth cannot be larger then the silicon thickness. Fabric ation resolution is also finite PAGE 90 90 and there is a minimum line width that can be ac hieved. For the fabrication methods used, the minimum line width linew was estimated to be a very conservative 10m This also helps prevent problems of punch through in the taper resistor faced by Li [136]. The features for which the line width constraint applies are shown in Figure 44. The constraints are then ,,,,,ttgaptaalinewwlwlw, (412) which in terms of the optimization variables are , ,, 0, 0, 0.ainline tinline aainline wttinline tintwtlinearw arw rw rw rw (413) In addition to the geometric constraints, performance constraints are also employed. For the device response to remain linear, the onset of nonlinearity must be high er than that of the desired operational range. This results with the following constraint, max20logrefp PMdB p (414) where PMdB is the upper limit of the desired range. The power dissipated in the device is constrained to be less then or equal to a specified value maxW. Power is equal to BWVI (415) the current is related to the voltage by the resistance of the Wheatstone bridge, B R yielding the following constraint, 2 max B BV W R, (416) for a constant voltage device or PAGE 91 91 2 max BIRW, (417) for a constant current device. The 3dB point for the upper bandwidt h limit is calculated from the LEM in section 3.3 and is constrained to be great er than o r equal to a specified B W, that is 32dBHjfBW. (418) The bandwidth of the device was calculated using the lumped element model assuming a damping ratio of 03 .0 At the 3dB point the phase lag is 2 degrees. This value of was taken from experimentally determined damping ratios of previous similar devices [37]. Finally, the device needs to remain in the linear regime and therefore the nondimensional parameter ck is constrained to be below a value of 3.4, before buckling occurs (Recall Figure 35). Collecting all of the variables and constraints with equation (41) yields, max1 2 123 ,, ,, ,, and ,,,,,,,,,, 10 10 10 10 10 10 3.4ij jataintinsurfB aoutain line touttin line aain line wttin line tintwt line cMDP PBW LBHHHazrrNVUB rr w rr w r w r w r w k V Minimize such that 2 max10BRW (419) PAGE 92 92 To run the optimization, MATLABs optimization toolbox, specifically the sequential quadratic programming function, fmincon, was utilized. All values are nondimensionalized to range from 0 to 1 which normalizes all search gradients within fmincon and provides better search directions for the software package. 4.2 Optimization Results The optimization was run in two modes that ar e discussed in the following sections. The first mode was for a constant bias voltage ap plied across the Wheatstone bridge and the second for a constant current so urce through the bridge. 4.2.1 Optimization with Constant Voltage In Figure 45, MDP is plotted for a variety of maxP and bandwidth constraints. With increasing bandwidth constraint, the MDP also incr eases, as expected. The same trend holds true for MDP as a function of maxP, shown in Figure 46, also as expect ed. The curves corresponding to a m aximum pressure of 140 and 145 converge at a higher bandwidth because the maxP constraint is not active. Th is occurs because the bandwidth constraint will not allow the diaphragm to become more compliant. For each case the power constraint is also active at 100mW. The devices preliminarily chosen for fabr ication are designated devices A, B, and C and can be seen in Figure 45 and Table 42. All variables and constraint values can be found for all dev ices in the multiobjective optimization in Appendix D. In general, the optimization simultaneously lowers the noise floor and increases the sensitivity. To lower the noise floor when 1/f noise dominates[131] the bias voltage needs to be lowered and the total number of carriers needs to be increased. This is proportional to the following variables, PAGE 93 93 22 b N s jaainV V Nzar (420) When the thermal noise is dominant the mean resi stance dominates the noise floor[130]. This is proportional to the following variables, as jainN R zar (421) The sensitivity of the device, is proportiona l to the compliance of the diaphragm and the compliance of the diaphragm is propor tional to the following variables, 2a Sens h. (422) Conversely the bandwidth constraint is proportional to 1BW M C (423) and the bandwidth is therefore pr oportional to the fo llowing variables 2 2 211BW a a ah h (424) The linearity constraint is inversely proportional to the complia nce of the diaphragm. This results with max 2h P a. (425) From the proportionalities, it can be seen that the sensitivity is inve rsely proportional to the bandwidth and maximum detectable pressure. Th is sets up the tradeoff between each value and results with the Pareto fronts in Figure 45 and Figure 46. PAGE 94 94 4.2.2 Optimization with a Constant Current Source A National Instruments PXI system may be used to power the microphone. This system runs in two modes: 410%mA and 1015%mA [137]. Figure 47 and Figure 48 show MDP plotted as a function of bandwidth constraint for a variety of maxP constraints for devices running at 4mAand 10mA, respectively. The MDP of devices A,B and C can be seen in Table 43 and Table 44. Com paring Table 42 to Table 43 and Table 44, it can be seen that the constant current source devices (running at 4mA) have a higher MDP then their voltage source counterpart. The devices running at 10mA have a lower MDP then those running at 4mA and are almost identical to their voltage driven counterpa rts. A sensitivity analysis was then done on devices A, B, and C varying the current source by the specified 10% and 15% The results can be seen in Figure 49 and Figure 410. Note that a higher input current would violat e the po wer limitation constraint. 4.2.3 Constraining Devices to a Single Wafer To reduce fabrication costs, the production of multiple microphone designs on one wafer is required. However, different thicknesses, dopi ng concentrations, and junction depths do not allow the devices A, B and C to be processed on th e same silicon wafer. To make fabrication on the same wafer possible, the devices were optim ized with an added constraint of each device having the same thickness of silicon, oxide and nitride. The doping concentration and junction depth were also constrained to be the same. The reformulated optimization problem is PAGE 95 95 max1 2 ,, ,, ,, 2 max and ,,,,, or 10 10 10 10 10 10 3.4 10ataintinBB aoutain line touttin line aain line wttin line tintwt line c BMDP PBW LBarrVIUB rr w rr w r w r w r w k V RW Minimize such that (426) Table 45 and Table 46 show the results of an optim ization w ith these added constraints for voltage and current source (10mA) devices, resp ectively. The first subtable is for a set of devices where the parameters of device A were used as the constraints for devices B and C. The second subtable uses parameters from device B, constraining devices A and C, and so forth for the remaining tables. The difference in MDP is the difference between each devices constrained and unconstrained MDP. From this data, the scenario of devices run with a 10mA current source were chosen to fabricate. Specifically, the devices from the first scenario of Table 46. This was chosen because the sponsor, Boeing, ultim ately would lik e to operate the microphones from the current power supply from their National Instruments PXI system. Between the two modes, the devices operating at 10mA had greater performance. Additi onally, the designs to operate at 10mA possess similar performance as devices designed with a voltage source. PAGE 96 96 4.2.4 Sensitivity Analysis To determine the dependence of the results to changes in various parameters and to gain a physical understanding of different aspects of the optimization, a sensitivity analysis was performed. This analysis was performed on each va riable in the optimization as well as the noise figure of merit, th e Hooge parameter. The following results are all for optimized device A from Figure 48. As previously stated, the optim ization was performed assuming a Hooge parameter of 56e Figure 411 shows that errors in the assum ed Hooge parameter can have a major impact on MDP. If 1/f noise is dominant, the voltage noise and MDP scale with the 1/2. The dependence of MDP with respect to all independent vari ables is shown in Figure 412. The variab le with the largest e ffect on the MDP is the thickness of the silicon. As the thickness decreases, the stiffness decreases as well and th e diaphragm begins to buckle resulting in the sharp drop in MDP. The linear deflection solu tion however is invalid in this region. Figure 413 shows the ef fect of MDP on the thickness of silic on as well as the constraint for buckling. The optimized point is at *3.4ck showing that this constraint is active and will not allow the thickness of the silicon layer decrease. 4.2.5 Uncertainty Analysis The uncertainty for the theoretical performan ce metrics is derived in this section. The formulations presented here utilize results obta ined in section 4.2.3. The MDP of design A is analyzed. Furthermore, the predictions for the bandwidth and linearity co nstraints are explored. Calculating the MDP of the microphone includ es numerical integral s and therefore an explicit analytical formula cannot be obtained. To calculate the uncertainty of the design metrics a Monte Carlo simulation is employed. The Monte Carlo method involves assuming PAGE 97 97 distributions for all of the input uncertainties and then randomly perturbing each input variable with a perturbation drawn from its uncertainty distribution [138]. The standard deviation for each variable is estimated from manuf acturing tolerances and can be found in Table 47. These statistically independent values a re fed into the objective func tion and the distribution of the objective function is obtained. In this case, unc ertainties in the design variables correspond to a statistical distribution of the noise voltage, sensit ivity and their ratio, th e MDP. There will also be a distribution for the resonant frequency and maximum detectable pressure. This process is illustrated in Figure 414 where i x is the optimized design variable, i is the standard deviation for each design variable and Y is a Gaussian distributed random number with mean zero, variance one and sta ndard deviation one. The Monte Carlo simulation was implemente d in MATLAB and the random independent variables were generated using the randn function. The simulation was run for 100,000 iterations and the probability distribution function (PDF) for MDP can be seen in Figure 415. The results highligh ted in yellow and red are cases where the composite diaphragm is buckled and close to buckling, respectively. Since the device is designed to be in the linear regime, the optimization uses a linear solver to calculate the deflections and stresses within the diaphragm. These solutions are invalid because the linear solution starts to deviate from the nonlinear solution at about 3.6k The PDFs for maxP and bandwidth can be seen in Figure 416 and Figure 417 respectively. The dist rib utions are not assumed to be Gaussian and all statistical moments can be found in Table 49. To determine the 95% probability limits for each specification a numerical integra tion was performed. This integr ation starts at the mean value and moves outwards until the value under the PD F is 47.5%. The process is done for values above and below the mean to yield a total probability of 95%. Figure 418 shows this process PAGE 98 98 pictorially. The 95% probability for the performan ce of the desired specifi cations can be seen in Table 48. It is important to note that the lo wer end of the confidence integral is in q uestion because of the error in the linear solution. The confidence integral was calculated by numerically integrating under the PDF. 4.3 Conclusion This chapter implemented a multiobjective optimization scheme to help design a superior device. Devices were optimized for voltage sources as well as current sources. A secondary optimization was also completed constraining the devices to have multiple on each silicon wafer. A sensitivity analysis was also conducted to determine the effect of each variable on the primary objective function, MDP. Finally an uncertainty analysis was completed using a Monte Carlo simulation to help determine the uncertainty of the various objective functions. PAGE 99 99 Table 41. Upper and lower bounds for all variables HSi HSi02 HSiN a Ns zj rain artin tVb units m A m cm m m deg m deg V UB 10 3000 1000 8001e20 7 80045 800 25 15 LB 1 300 300 60 1e18 0.1 60 1 60 0.5 1 Table 42. Values for devices chosen for fabrication for constant voltage biasing. Device BW [kHz]Pmax [dB]MDP [dB]Dyn Range [dB] A 120 150 24.5 125.5 B 120 160 28.1 131.9 C 100 150 23.3 126.7 Table 43. Values for devices A, B, and C in constant current mode (4mA). Device BW [kHz]Pmax [dB]MDP [dB]Dyn Rng [dB] A 120 150 26.6 123.4 B 120 160 29.7 130.3 C 100 150 25.2 124.8 Table 44. Values for devices A, B, and C in constant current mode (10mA). Device BW [kHz]Pmax [dB]MDP [dB]Dyn Rng [dB] A 120 150 24.8 125.3 B 120 160 28.6 131.5 C 100 150 23.7 126.3 PAGE 100 100 Table 45. Single wafer constr ained voltage source devices. Optimize for device A and constrain thickness for devices B and C Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 24.7 N/A 125.3 B 120 160 31.2 2.9 128.8 C 100 150 24.7 1.2 125.3 Optimize for device B and constrain thickness for devices A and C Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 28.3 3.7 121.7 B 120 160 28.3 N/A 131.7 C 100 150 25.8 2.3 124.2 Optimize for device C and constrain thickness for devices A and B Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 26.1 1.5 123.9 B 120 160 30.0 1.7 130.0 C 100 150 23.5 N/A 126.5 Table 46. Single wafe r constrained current source (10mA) devices. Optimize for device A and constrain thickness for devices B and C Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 24.8 N/A 125.3 B 120 160 31.2 2.7 128.8 C 100 150 24.8 1.1 125.3 Optimize for device B and constrain thickness for devices A and C Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 28.6 3.8 121.4 B 120 160 28.6 N/A 131.5 C 100 150 26.1 2.4 123.9 Optimize for device C and constrain thickness for devices A and B Device BW [kHz] Pmax [dB] MDP [dB] Difference in MDP Dyn Range [dB] A 120 150 26.3 1.5 123.7 B 120 160 30.1 1.5 129.9 C 100 150 23.7 N/A 126.3 PAGE 101 101 Table 47. Standard devi ation of input parameters. HSi HSi02 HSiN a Ns zj rain artin tIb units m A mcmmmdeg mdeg m STD 0.05 50 10 1 1.00E+18 0.01 1 0.5 1 0.5 1 % 3.5 3.1 3.3 0.9 3.5 2.8 0.6 1.1 1.2 1.3 10 Table 48. Mean and 95% confiden ce intervals for design parameters. Mean Lower CI Upper CI units MDP 29.1 dB Pmax 150 error 158 dB Bandwidth 120 31.0 157 kHz Table 49. Statistical prope rties of desired parameters. Units Mean VarianceSkewnessKurtosis MDP [dB SPL]24.9 6.9 0.177 2.37 Pmax [Pa] 633 10200 0.02722.88 BW [kHz] 117.5 777.2 0.668 3.76 Frequency (Hz) Bandwidth [Hz] Dynamic Range [dB SPL] Operational Space of Device Figure 41. Operational parameter space for a microphone. PAGE 102 102 2 J 1 J A B C D E Utopia Figure 42. Multiobjective optimization Pare to front illustrating the tradeoff between minimizing function J1 and J2. prmsV Sensitivity pmax Figure 43. Ideal linear output of a microphone or pressure transducer. PAGE 103 103 aw al tgapwtwtl Figure 44. Features that are constrained to be larger than linew. 50 60 70 80 90 100 110 120 130 140 150 14 16 18 20 22 24 26 28 30 32 34 Bandwidth (+/3dB) [kHz]MDP [dB] Pmax = 170 dB Pmax = 165 dB Pmax = 160 dB Pmax = 155 dB Pmax = 150 dB Pmax = 145 dB Pmax = 140 dB A B C Figure 45. MDP vs. Bandwidth for various Pmax constraints. PAGE 104 104 140 145 150 155 160 165 170 14 16 18 20 22 24 26 28 30 32 34 Pmax (5% deviation from linearity) [dB]MDP [dB] BW = 150kHz BW = 130kHz BW = 110kHz BW = 90kHz BW = 70kHz BW = 50kHz Figure 46. MDP vs. Pmax for various bandwidth constraints. 50 60 70 80 90 100 110 120 130 140 150 16 18 20 22 24 26 28 30 32 34 Bandwidth (+/3dB) [kHz]MDP [dB] Pmax = 170dB Pmax = 165dB Pmax = 160dB Pmax = 155dB Pmax = 150dB Pmax = 145dB Pmax = 140dB Figure 47. MDP vs. Bandwidth of various Pmax constraints for a constant current source device (4mA). (Letters correspond to Table 43) PAGE 105 105 50 60 70 80 90 100 110 120 130 140 150 16 18 20 22 24 26 28 30 32 34 Bandwidth (+/3dB) [kHz]MDP [dB] Pmax = 170dB Pmax = 165dB Pmax = 160dB Pmax = 155dB Pmax = 150dB Pmax = 145dB Pmax = 140dB Boeing target spec A C B Figure 48. MDP vs. Bandwidth of various Pm ax constraints for a constant current source device (10mA). (Letters correspond to Table 44) 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 23 24 25 26 27 28 29 30 31 32 Input current [mA]MDP [dB] Device A Device B Device C Boeing target spec Figure 49. Sensitivity analysis for constant current source varying by 410% mA PAGE 106 106 8.5 9 9.5 10 10.5 11 11.5 22 23 24 25 26 27 28 29 30 Input current [mA]MDP [dB] Device A Device B Device C Boeing target spec Figure 410. Sensitivity analysis for constant current source varying by 1015% mA 106 105 104 103 20 25 30 35 40 45 Hooge parameter ()MDP [dB] Device Optimized for = 5e6 Figure 411. MDP dependence on Hooge parameter for device A. PAGE 107 107 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 5 10 15 20 25 30 Independant VariableMDP [dB SPL] (1Hz bin @ 1kHz)Sensitivity Analysis H1 H2 H3 a Ns zj a rain t rtin Ib Figure 412. Dependence of MDP with respect to each variable. 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0 20 40 60 HSi (m)MDP [dB] 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 2 3 4 kc k*c = 3.4 Optimized device constrained to k*c <= 3.4. Figure 413. Dependence of MDP on silicon th ickness overlaid with compression coefficient. PAGE 108 108 11 x Y ii x Y X X Y Figure 414. Monte Carlo simulation schematic [138]. 0 100 200 300 400 500 600 700 800 0 0.5 1 1.5 2 2.5 3 3.5 x 103 Probability Density Function of PminMinimum Pressure [Pa] k < 3.6 k < 3.8 k > 3.8 Figure 415. Uncertainty of MDP of Device A. PAGE 109 109 1000 500 0 500 1000 1500 2000 2500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 103 Probability Density Function of PmaxMaximum Pressure [Pa] k < 3.6 k < 3.8 k > 3.8 154dB 148dB 160dB Figure 416. Uncertainty of maxP for device A. PAGE 110 110 0 20 40 60 80 100 120 140 160 180 200 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Probability Density Function of BandwidthBandwidth [kHz] k < 3.6 k < 3.8 k > 3.8 Figure 417. Uncertainty of the bandwidth for device A. 47.5%47.5% 95% Probability Yield Figure 418. 95% probability yield limit illustration. PAGE 111 111 CHAPTER 5 DEVICE FABRICATION AND PACKAGING This chapter provides an overview of the fabric ation process, describ es the realized device, and details the associated packaging used for acous tic testing. The proces s flow is designed to be integrated circuit (IC) compatible. This w ill allow future fabrication of the device to be completed at external commercial foundries a nd will in theory perm it the integration of electronics on the microphone die. The microphone is fabricated using a bulk micromachining process starting with an SOI wafe r and utilizing 9 masks. The pr ocess flow was divided into two parts: front and backside processes. The front side process has ten steps and the backside process has eight. 5.1 Process Flow Overview The microphone process flow is shown in Figure 51 and the masks can be seen in Figure 52 and Figure 53 For front side fabrica tion, the starting material is a bare SOI wafer from the foundry (a). This wafer is a (100) N type phosphorous doped wafer with a resistivity of 38cm. The first step is to etch down to the handle wafer (b). This step is needed to attach both the handle wafer and device wafer to the sa me potential to avoid a floating ground. The next three steps are outsourced to the MEMS and Nanotechnology Exchange company based in Reston, Virginia. The first is an nwell dope of the previously etched ho le followed by a drivein and activation (c). The second is the ptype dop ing of the piezoresistors followed by a drivein and activation(d). After the drivein is completed, the wafers undergo the oxidation step (e) and then return to UF for the remaining processes. The next step is to deposit plasma enhanced chemical vapor deposition (PECVD) oxide on the back side of the wafer for use in a nested mask at a later step (f). Then the contact cuts in the oxide (g) are etched followed by a metallization step (h). To passivate against moisture, a laye r of PECVD nitride is then deposited over the PAGE 112 112 device (i). The two remaining step s are to etch the front side of the vent hole (j) and finally the contact cuts for the bond pads (k). The backside process steps are seen in Figure 54 and the masks in Figure 55. These steps begin with a deposit of photoresist (PR) on the front side of the wa fer to protect it from da mage (a). Next, the backside PECVD oxide is etched to create a nested mask for the cavity and vent structure (b). A carrier wafer is then attach ed to the SOI wafer to provide support to the diaphragms while they are etched (c). The b ackside is etched down to the buried oxide layer (BOX), creating the cavity and backside vent hole (d,e). The nested oxide mask is employed to etch the vent channel into the backside of the wafer (f). A BOE is then used to release the diaphragm, open the vent hole and remove the nest ed oxide mask (g). Finally, a Pyrex wafer is anodically bonded to the wafer to complete the pr ocess (h). A detailed process traveler is provided in Appendix B. 5.2 The MEMS Microphone After the final fabrication step each wafer holds over 1200 devi ces, as well as a variety of test structures. Before dicing the yield was close to 100%. Probl ems with the dicing lowered the yield to 50%. After dicing was completed and ag ainst instructions, a je t of water was sprayed onto the wafers to clean them of particulate. This resulted in the destruc tion of about half of the diaphragms. Images of these devices and structures are shown in Figure 56 through Figure 59. The final die m easures 2mm square and Figure 56 shows a close up of a finished wafer after dicing was com pleted. Each wafe r is divided into two sections; half of the devices are type A and the remaining are type B. Figure 57 shows the front side of a type A die. To get a reference to the size of the m icrophone, Figure 58 shows a type B device on a dime. The top of this die contain s five bond pads, each 200 m square, the center pad is th e substrate contact and the PAGE 113 113 remaining four pads are the connections to the Wheatstone bridge. On the bottom of the die the front side vent is visible. The yellow color is due to the thin films of s ilicon nitride and silicon dioxide layers. Figure 59 shows the backside cavity and ven t structure. 5.3 Microphone Packaging For acoustical characterization, 4 type A devi ces were mounted onto a printed circuit board (PCB). A new endplate for the acoustic characteri zation setup was fabricated that allows for a 1/4condenser reference microphone to be mounted next to the PCB board. A schematic of this setup can be seen in Figure 510. To minimize attenuation of the signal all associated electronics including the high pass filter, amplifier and pow er supplies can be integrated onto the PCB, reducing the distance from the microphone to the amplifier. 5.3.1 Interface Circuitry The microphone is characterized with th e integrated circuitry shown in Figure 511. The m icrophone is represented by a Wheatstone bridge. The incident pressure will modulate the value of the resistors creating a differential output. To amplify this signal, the piezoresistive microphone is connected to an Analog Devices instrumentation amp lifier (AD625). Even though the device is designed to have a bandwidth of 100 kHz, the test setup is only operable to 6.7kHz and therefore the circuitry does not need to have a functional bandwidth of 100 kHz. The gain is set at 1000V/V and the amplifier ha s a bandwidth of 25 kHz. Due to resistor mismatch in the bridge, there will be a larg e DC component to the microphone output voltage. Therefore, the output of the microphone is high pass filtered before amplification by a single pole filter with a corner frequency of 1.6 Hz. Two voltage regulators ar e used to supply the microphone die with the bias voltage for the Whea tstone bridge and to supply the bias for the substrate to isolate the resistor s. To accomplish this, two Lin ear Technologies LT1963 low noise PAGE 114 114 voltage regulators are used in an adjustable operation mode. For each regulator, a potentiometer is used to adjust the output volta ge supplied to the microphone die. 5.3.2 Printed Circuit Board The printed circuit board serves a variet y of purposes: mounting and positioning the microphone die into the acoustic test stand, attaching all of the a ssociated circuitry and providing a backplate for the microphone cavity. Figure 512 shows an unpopulated PCB board. These boards are gold plated and were ordered from Si erra Protoexpress. All components are mounted on the front side and the micr ophone is mounted on the reverse. This minimizes the roughness of the PCB inside the acoustic setup to limit any acoustical scattering. The capacitors are 0805 size ceramic surface mount capacitors with a tole rance of 0 %. The resistors are 0805 size metal film surface mount resistors with a tolera nce of %. For details on all of the specifications for the interf ace circuitry see Appendix E. 5.3.3 Assembled Package The populated PCB board can be seen in Figure 513. Two potentiom eters are used for the gain control, one of which has a higher resistance than the o ther. This configuration was chosen to allow for a course and fine tune of the gain value. The microphone end of the PCB fits into the plane wave tube (PWT) endplate as shown in Figure 514. For details on all components and a detailed drawing see Appendix E. PAGE 115 115 (a) Begin with a SOI wafer with a handle wafer thickness of 350 m, BOX thickness of 3000, and devic e wafer thickness of 1.5 m Silicon 350 m Oxide 3000 Silicon 1.5 m (b) Etch contact pad for s ubstrate bias using RIE (d) Ion implant the ptype piezoresistors (e) Grow thermal oxide (f) PECVD silicon dioxide on backside (g) Etch contact cuts using BOE (h) Deposit AlSi Leads (i) PECVD silicon nitride (j) Etch vent hole using RIE (k) Etch contact cuts using RIE (c) Ion implant the ntype substrate contact Figure 51. Front side process steps. PAGE 116 116 A B C D Figure 52. A) Ground strap mask. B) NWell Mask C) Piezoresistor ma sk. D) Piezoresistor contact mask. PAGE 117 117 A B C Figure 53. A) Metallizati on mask. B) Topside vent mask. C) Bond pad mask. PAGE 118 118 Pyrex (b) Etch backside silicon dioxide for nested mask using BOE (d) Deposit photoresist for cavity etch (carrier wafer not shown) (e) Etch cavity and backside vent hole using DRIE (carrier wafer not shown) (f) Etch vent hole path using RIE (carrier wafer not shown) (g) Etch to release diaphragm and open vent hole using BOE. Carrier wafer to be removed after this step (h) Anodic bond wafer to Pyrex backplate (a) Spin protective layer of PR on top. Front to back align for oxide mask etch (c) Attach carrier wafer using PR and cool grease Carrier Wafer Figure 54. Back side process steps. A B Figure 55. A) Backside vent pa th mask. B) Backside cavity mask. PAGE 119 119 Figure 56. Array of microphone die after dicing. Each die is 2mm x 2mm. 2 mm 2 mm Figure 57. Individual type A microphone die after dicing. PAGE 120 120 Figure 58. Type B de vice pictured on a dime. Figure 59. Backside cavity and vent of an individual type A micr ophone die after dicing. PAGE 121 121 Figure 510. Packaging for acoustical characterization. Figure 511. Interface circuitry showi ng power supply, ac filter and amplifier. PAGE 122 122 Figure 512. Printed circuit board for mounti ng the microphone and its associated components. s BNC Connectors Potentiometers to adjust voltage to microphone AD 625 amplifier High pass filter Microphone under TO can Potentiometers to adjust amp gain LT1963 voltage regulators Figure 513. Assembled device on PCB. Device is protected under a TO can. PAGE 123 123 Mount for condenser microphone Mount for BUF1 Figure 514. Populated PCB board inserted into PWT endplate. PAGE 124 124 CHAPTER 6 RESULTS AND DISCUSSION The results for the aeroacoustic m icrophone desi gned and fabricated in Chapters 35 are outlined here. The methodology and experimental se tup for the electrical and acoustic testing is discussed. Next, the experimental results are shown and finally the corresponding model validation is presented. 6.1 Device Characterization 6.1.1 Electrical Characterization In total, 12 BUF1A devices and 12 BUF1B devices are tested for the IV characteristics of the bridge as well as the PN di ode characteristics. Two four point test resistors are tested for each type A and B arc and taper resistors. In ad dition, two Van der Pawl, line width and Kelvin structures, 2 MOS capacitors, 2 test PN diodes, and 2 metal Van der Pawl and line width structures are tested. Finally a large p+ doped region was created to test the dopant profile via secondary ion mass spectroscopy (SIMS) analysis. For details on all e xperiments see Appendix F. Electrical characterization consists of seve ral tests described here. Input and output resistances are measured using an Agilent 4155C semiconductor parameter analyzer (SPA). To obtain the bridge offset, the resist ance of isolated resi stors on a test structur e are tested with a four point probe. To check for resistor isolation ( Figure 61), the IV characteristics of the pn diode from the resistor to the s ubstrate are tested. Also included on the finished wafers are van der Pawl test structures to determine sheet resist ance, contact resistance and line width control. Schematics of these structures are shown in Figure 62, Figure 63, and Figure 64. The sheet resistance is determ ined by the following equation ln22abcddabc sRR R (61) PAGE 125 125 To determine the lateral diffusion via the line wi dth test structure, th e resistance is given by sLL RR ww (62) assuming LdL yields, 1 1sL RR w w w (63) Taking the first term from a Ta ylor series e xpansion yields 1sLw RR ww (64) Finally, the lateral diffusion is given by 1sRw ww RL (65) The Kelvin structure yields th e contact resistance defined by cV R I (66) In addition to the van der Pawl structures, MOS cap acitor test structures are included in the set of test structures. The cap acitor test structures ve rify the dielectric thickness and substrate doping. Due to the high doping concentration, the MOS cap acitor is unable to determine the substrate doping. The thickness of the dielectric layer in the capacitor is the layer of silicon dioxide on the diaphragm. This thickness is determined by the following equation iA C d (67) where iC is the capacitance measured in accumulation mode, A is the area of the capacitor and d is the thickness of the dielectric. PAGE 126 126 The experimental setup for noise experiments is discussed in Dieme et al. [135] and shown in Figure 65. The microphone is placed inside thr ee concentric Faraday cages, which serves the function of attenuating electrom agnetic wave s and reducing external electromagnetic interference [129]. The output of the microphone is amplified using a Stanford Research Systems SR560 inside two of the three Faraday cag es and then sent to a Stanford Research Systems SR785 spectrum analyzer that measures the output noise power spectral density. The voltage noise of the experimental setup is then subtracted from the total noise including the DUT. A multivariable linear curv e fit is performed to determin e the Hooge parameter discussed in Chapter 3. Equation (395) in terms of PSD is 1 fV S Nf (68) Taking the logarithm of equation (68) and expanding yields 1 1 2 12logloglogloglogfm m xxb ySVfN (69) From this equation, and can be solved knowing the tota l number of active carriers in the DUT. 6.1.2 Acoustic Characterization The goal of the acoustic char acterization is to determine the frequency response and linearity of the microphone. For this experiment a PWT ( Figure 66), with a 1 in x 1 in square cross sectio n, was used. An acoustic driver and the DUT and a reference microphone are placed at opposite ends of the tube. The microphones are oriented such that they are at normal incidence to the planer incident pressure wave. Due to tube geometry, below a certain frequency, only plane waves propagate, causing th e DUT and the reference microphone to be exposed to the PAGE 127 127 same incident pressure. This frequency is known as the cuton frequenc y of the PWT [127]. For this PWT, the cuton frequency in air is 6.7 kHz The supporting hardware for the plane wave tube experiments is also shown in Figure 66. The Bruel and Kjaer Pulse m ultianalyzer system supplies a function generator to power the acoustic driver and also receives the input signals from the DU T and reference microphone. This system also executes the data analysis functions and records the data. A PCB Piezoelectronics 377A51 condenser microphone is used as the re ference microphone. The microphones are tested up to 170 dB. The 377A51 microphone has a 3% distor tion limit of 192 dB a nd will therefore be sufficient for all acoustic measurem ents [139]. The signal sent to the acoustic driver is amplified by a Techron 7540 power supply amplifier and th e driver is a BMS 4590P compression driver [140]. To determine the frequency response of the DUT, the generator is set to a periodic random signal and the FFT analyzer is configured to match this signal. The Pulse system is set to take 300 averages, use 1 Hz bins, no overlap and no windowing during the measurement. The frequency response is measured with respect to the reference micr ophone and the coherence between the signals is recorded to determ ine the uncertainty in the measurement. Total harmonic distortion (THD) (equation (11)) is measured using a singletone pressure, and is used to calculate the lin earity of the device. The TH D grows as the m icrophone changes from linear to nonlinear operation. Nonlinear ity in the microphone causes output power at frequencies that are integer multip les of the fundamental frequenc y [120]. Therefore, the power measured in the harmonics can be utilized to approximate the THD. However, nonlinearities are produced in the test setup as the incident pressure increases. For example, when powered by a single tone, the acoustic driver produces significant sound pre ssure at harmonic frequencies, PAGE 128 128 and acoustic wave propagation turns nonlinea r at high sound pressure levels [127]. Consequently, the THD of the microphone is calculated in the pr esence of external nonlinearities. The harmonic components caused by the experimental setup are subtracted from the DUT output signal during the estimation of the total harmonic distortion of the piezoresistive microphone. Because the reference microphone m easures the total acoustic pressure, which includes the harmonic components, several assumptions are necessary to validate this analysis. First, the reference microphone must not bring a ny nonlinearities into the system. Additionally, it is assumed only plane waves ar e propagating in the PWT and that the total pressure detected by the reference microphone and the piezoresis tive microphone are the same. For the THD measurements, a 1 kHz tone is used. Because the cut on fr equency of the PWT is greater than 6 kHz, the first five harmonic components propaga te as plane waves, and are used in the THD calculations. It is assumed that power in higher harmonics is negligible. For this experiment, the incident pressure is incrementally increased until the THD reaches 5 % This pressure is considered the maximum input pressure of the microphone. 6.2 Experimental Results The characterization results for the microphone are shown in this section. First, electrical characterization of devices and te st structures are presented, in cluding IV curves and noise floor results. This is followed by acoustical characterization which includes linearity, THD and frequency response results. 6.2.1 Electrical Characterization The profile of the p+ doped region was te sted via secondary ion mass spectroscopy (SIMS), which yields the dopant c oncentration as a function of de pth into the substrate [141]. The values it measures are the actual atom count which is not necessarily the same as the number of activated ions. The results of this measur ement show a gross difference between the desired PAGE 129 129 profile and measured profile, shown in Figure 67. This error has an extremely large impact on the perform ance of the device. The large error leads to a bor on concentration of 3517 ecm at the backside of the diaphragm. The bac kground concentration of the wafers is 3115 ecm which shows that there is no junction isolation on the underside of the resist ors. This leads to a plethora of problems which will be detailed in the later secti ons. The profile error is due to an error in the recipe for the drivein portion of the resistor fabrication. A miscalculation between a computer simulation and a 1D diffusion equation calculation was not discovered until after the microphone was fully fabricated The results for the van der Pawl and line width structures for the p+ resistors are shown in Table 63. The calculated sheet resistance corresponds with th e nom inal resistance of the piezoresistors. The masked line widt h structures were drawn to be 20 m The calculated value for the p+ line width structure is 21 m which indicates there is a lateral diffusion of 0.5 m on each side of the structure. This value helps de termine the state of the fabricated structure to allow for a more accurate model validation. Th is lateral diffusion correlates with the SIMS profile and confirms the error in the dopant profile. In additi on, metal line width structures, summarized in Table 64, reveal a 4.1 m over etch. This is in agreement with the way the metal lines were fabricated and gives more conf idence in the test structure values. Finally, Table 65 shows the contact re s istance is 1.6% the value of the resistors and the specific contact resistivity is in line with the literature for aluminumsilicon in contact with doped silicon [119]. To determine the thickness of the oxide layer, a MOS capacitor structure was tested and the thickness was also measured with an ellispometer. The results can be found in Table 66. These two m easurements have good agreement be tween the tests and desired values. PAGE 130 130 Arc and taper four point test resistors for both devices were tested and the results can be seen in Table 62. There is a large discrepancy betw een the resistance values of the arc and taper resistor due to the dopant profile error. This error results in a 17% dc offset in the W heatstone bridge for both device A and B. While this is not desired, the device is only designed to measure ac signals and the dc offset will be filtered ou t by the interface circuitr y, however the mismatch will couple in power supply noise. Results of the input and output IV curves are shown in Figure 68 and Figure 69, respectively. These figures show c onsistency over all devices tested. Table 61 shows the input and output resistances for all device s tested. All d evice are batch fabricated to be identical. To get an idea on the yield statistics the mean and st andard deviation of the re sistance is also shown in Table 61. Two i mportant aspects of the input and output IV curves are the linearity and intercept point, where the intercept point co rresponds to the leakage current. Figure 610 shows the data for the input IV curve as well as the linear fit associated with the data. From this, the linearity of the IV curve is apparent. In addition, the IV curve does not cross at absolute zero. This is due to the leakage current from the resistor into the substrate. The leakag e current is higher then expected but is 4 orders of magnitude less than current through the resistor. Figure 611 shows the outpu t linearity and leakage current. For this case, the IV curve has a slight nonlinearity at a potential of 10V. The leakage current is determ ined by the reverse diode characteristics of the device. Figure 612 shows an IV curve measuring current from the substrate contact to the closes t resistor seen in Figure 57. For this test, the ground pad was used. Upon closer inspection the reverse diode characterist ics have a strange behavior seen in Figure 613. The leakage current for a rev erse bias below 2V is 0.25 A and this increases to 2.25 A above 2V. PAGE 131 131 This behavior is consistent for both released an d unreleased resistors. The cause of this is believed to be due to the lack of junction isolation due to the dopant profile error. 6.2.2 Noise Floor The noise floor of the device is characterized to determine the minimum detectable signal. To show consistency over different structures, a test arc resistor is evaluated in addition to a full device bridge. Figure 614 shows the noise PSD for a test ta per resistor for a type A device along with the setup noise PSD. Subtracting the setup noise a nd applying equation (69) yields the Hooge param eter and is shown in Table 67. The curve fit is then plotted with the data in Figure 615. Figure 616 shows the PSD of the noise fo r a BUF1A device. To determ ine the Hooge parameter from this data, equation (396) is utilized. Table 68 shows the results of the curve fit and Figure 617 compares the data to the mode l fit. The Hooge param eters measured for both devices is much higher th en expected. The measured value is four orders of magnitude higher than the desired values. Th is data will be used to valid ate the models from Chapter 3; however, a different fabrication technique should be used for any future generation devices. 6.2.3 Linearity and Total Harmonic Distortion The results of the linearity measurements are shown in Figure 618. The sensitivity is shown to be constant over the entire pressure rang e. As the SPL increases the error in the measurement drops significantly. Details on the uncertainty analysis can be found in Appendix F. The estimated total harmonic distortion (THD) is shown in Figure 619. Since the maximum detectable pressure was defined as a 5% deviatio n from a linear defection of the diaphragm, the maximum pressure measurement is considered th e first point where the THD reaches 5%. This value for the tested microphones range from 153160 dBSPL PAGE 132 132 6.2.4 Frequency Response The magnitude and phase frequency respons e for the tested microphones is found in Figure 620 and Figure 622, respectively. Each individual frequency response is plotted with its associated 9 5% confidence interval in Figure 621 and Figure 623. During the test a 10 V bias was applied over the Wheatstone bridge and the amplifier gain was set to 1000. The response is plotted over the range of 300 Hz to 6.7 kHz. The magnitude is shown to be constant to within 2 dB and the phase is matched to within 2 over most of the frequency range. These results proved difficult to obtain with the equipment ava ilable. Due to the high noise floor of the device, the compression driver was unable to output sufficiently high sound pressure levels over any bandwidth larger then 800 Hz. The FRF was obtained piecewise and the total FRF was concatenated to obtain the entire range for the PWT. The Pulse mulitanalyzer settings were set to 1 Hz bins, no overlap and no windowing. The driver was set to peri odic random starting at 300 Hz to 1.1 kHz with 1 Hz bin widths and run for 300 averages. This method was still insufficient for lower frequencies 1000Hz In this region of the data, the signal to noise ratio is very low and the coherence dropped dramatic ally, resulting in large random errors. The coherence for one device is shown in Figure 624. With the sensitivity and noise floor calcu lated, the MDP for each device for a 1 Hz bin centered at 1kHz is shown in Table 69. 6.3 Model Validation A Monte Carlo simulation was implemented to determine the validity of the model described in chapter 3. This method is similar to the simulation executed for the uncertainty of the optimization in chapter 4. This process is illustrated in Figure 414 where i x is the optimized design variable, i is the standard deviation for each design variable and Y is a Gaussian distributed random number with mean zero and unit variance. The final assumption is that the PAGE 133 133 variables are statistically inde pendent. The major difference between the implementation here versus that of Chapter 4 is that the values put in to this simulation are the actual realized values from the fabricated device. The values for the device were determined from the test structures used earlier in this chapter. In addition, a test was run to determine that actual diameter of the diaphragm. The ultimate goal for the validation is to show that the fabricated device, with poor performance, can be predicte d with the developed models. 6.3.1 Variables and Standard Deviations Table 610 shows each variable with the associated methods used to determine an accurate value. Redundancy was used to increase the confid ence in the realized value. All of the tests in Table 610 are discussed in the te st structure section earlier in this ch apter except for the light test to determine the radius of the device. To determine the actual size of the diaphragm a technique is derived using a Sc hott high power snake light and an Olympus microscope with a Quadracheck 200 measurement system. The snak e light was positioned directly under the clear stage and pointed towards the die. This illumi nates the semitransparent diaphragm from the backside. The Quadracheck 200 is then used in circle mode to determine the diameter of the device. Figure 625 shows the microphone die with and wi thout illum ination. Thirty one type A devices were tested and the diameters were sma ller then expected. U pon further inspection, it was found that the final 10% of the backside cavity etch, desi gned to prevent footing, had a slight taper leading to th e smaller diameter. These results can be found in Table 611. Finally, the values and standard deviations used in the Monte Carlo simulation are assem bled in Table 612. 6.3.2 Model Validation Results The Monte Carlo simulation was implemente d in MATLAB and the random independent variables were generated using the randn function. The simulation was run for 30,000 iterations PAGE 134 134 and the probability distribution function (PDF) for MDP can be seen in Figure 626. The results yield a PDF with a m ean value of 106.4 dBSPL and a standard deviation of 1.51 dBSPL. The statistical moments can be found in Table 614. The uncertainty for the sensitivity, voltage noise, and maxP can be seen in Figure 627, Figure 628, and Figure 629, respectively. The 95% probability lim its for the design parameters can be seen in Table 613. The results from the m odel validation agree with the experimental results. The maximum detectable pressure determined by THD are shown as red lines in Figure 629, which fit inside the 95% probability range f or the model. The sensitivities determined in the PWT are also shown in Figure 627. These values also fit within the 95 % probability bounds from the model validation. The voltage noise fits inside the probability bounds but this is because the Hooge parameter is experimentally calculated and fed into the model validation. Fi nally, the MDP is experimentally calculated from the noise floor and sensitivity of the device and is found to be 108 dBSPL which also fits within the 95% probability bounds of the model validation. 6.4 Conclusion The characterization of the device revealed a major problem with the fabrication. The diffusion of the resistors was too long and resulted with the resi stor thickness being the entire thickness of the diaphragm. The result of this error dropped the sensi tivity two orders of magnitude. In addition to the doping profile erro r, the inherent noise characteristic of the resistors was also higher then expected. This increased the noise signature of the device two orders of magnitude higher then expected. Th ese two factors couple to gether and increase the MDP of the device by 4 orders of magnitude, or 80 dB. The optimized device A had an expected MDP of 24.5 dB (1 Hz bin @ 1kHz). The re alized device had a MD P of 108dB, or 83.5 dB higher then the desired value. PAGE 135 135 Table 61. Resistance values of all tested devices. Device RinA RoutA RinB RoutB R2(RinA)R2(outA)R2(RinB)R2(RoutB) # 1 919 934 1043 1057 0.999960.999820.99995 0.99982 2 931 931 1031 1030 0.999920.99980.99982 0.9998 3 946 943 1046 1043 0.999820.999820.99983 0.99982 4 942 941 1016 1012 0.999820.999810.9998 0.99981 5 955 929 1017 1015 0.999790.999790.99984 0.99979 6 936 926 1038 1036 0.999630.999750.99985 0.99975 7 925 919 1013 1013 0.999750.999750.99984 0.99975 8 923 914 1008 1011 0.999760.999760.99984 0.99976 9 945 933 1004 1013 0.999750.999780.99982 0.99978 10 949 945 1015 1011 0.999810.999810.99985 0.99981 11 927 926 1005 1007 0.999770.999780.99981 0.99978 12 916 916 1003 1002 0.999770.999780.99978 0.99978 R 934 930 1020 1021 95% CI (926942) (924936) (10101030)(10101032) 12.9 10.2 15.5 16.6 95% CI (9.1421.9)(7.2317.3)(11.026.3)(11.828.2) Table 62. Resistance values for the four test resistors. Arc A Taper A Arc B Taper B Value 778.6 1084 834.8 1175 Value 763.4 1078 841.9 1212 R 771.0 1081 838.4 1194 95% CI (674.4867.6)(10421119) (793.2883.5)(958.41429) *All values in Table 63. Values for VDP and line width test structures. Rs,design Rs Wp,design Wp /sq /sq m m 336 194.6 20.0 21.0 PAGE 136 136 Table 64. Values of VDP and line widt h test structures for the metal lines. Rs metal Wmetal Wm,design /sq m m 0.022 15.9 20.0 Table 65. Values of Kelvin test structures. Rc c c,lit cm2 cm2 16.37 2.62E04O(E4E5) Table 66. Thickness of oxide layer using two techniques. Design goalMOS capacitor Ellispometer m m m 0.159 0.160 0.152 Table 67. Curve fit parameters fo r test taper resistor for device A. R R2 [] [] [] [] Value 7.55E02 1.107 1.933 1050 0.997 CI (6.04 9.06)E02(1.105 1.109)(1.927 1.939) Table 68. Curve fit parameters for a BUF1A device. R R2 [] [] [] [] Value 5.13E02 1.083 2.075 912 0.995 CI (3.79 7.01)E02(1.081 1.086)(2.073 2.081) Table 69. MDP for tested devices. Minimum Detectable Pressure 1 2 3 4 [dB SPL] [dB SPL] [dB SPL] [dB SPL] 108.0 108.0 109.1 108.4 for a 1 Hz bin centered at 1 kHz PAGE 137 137 Table 610. Methods used to determine the fabric ated values for all para meters of the devices. H1 H2 H3 a Ns zj a rain t rtin Ibias SIMS SIMS SIMS light testSIMSSIMS line width line width line width line width applied wafer specs ellipsometry ellipsometry MOS capacitor In addition, two contact resistances are added to each resistor to account for the current flowing to and from each resistor. The calculated Hooge parameter is also used. Table 611. Results from the radius determination experiment. BUF1A adesign ameas 95% CI 113 108 (107.3 108.7) All measurements in m. Table 612. Measured values and sta ndard deviation of input parameters. HSi HSi02 HSiN a Ns zj rain a rtin t Ib m A m cm m m deg m deg m value 1.45 1600 380 109 4e18 Hsi 87 43.377 21.5 10 STD 0.05 50 10 1.6 1e17 0.05 0.250 0.25 0 0 % 3.5 3.1 2.6 1.5 2.5 3.5 0.290 0.32 0 0 Table 613. Confidence intervals for the realized parame ters for a BUF1A device. Pmax Sensitivity Vn MDP [Pa] [nV/Pa/V] V dBSPL (1 Hz bin @ 1kHz)1000 2200 4.0 5.6 0.201 0.269 104 109 Table 614. Statistical data for model validation PDFs. Units Mean VarianceSkewnessKurtosis MDP [dB SPL]106.4 2.29 0.156 3.15 Sens [ V/Pa] 0.046 2.00E050.279 3.27 Vn [ V] 0.226 3.49E040.561 3.6 Pmax [Pa] 1360 86300 0.542 3.49 PAGE 138 138 P+ n SiO2 R/2 R/2 C Figure 61. Circuit representati on of reversed biased p+ doped re sistors in an n substrate [119]. RR RR Figure 62. Van der Pawl test structure schematic. PAGE 139 139 Figure 63. Line width test structure schematic. Figure 64. Kelvin te st structure schematic. PAGE 140 140 Figure 65. Experimental setup for noise measurements. B&K Pulse Multianalyzer Amplifier DUT Reference Microphone PWT Driver End Plate Figure 66. Experimental set up for acoustic characterization. PAGE 141 141 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1014 1015 1016 1017 1018 1019 1020 1021 de p th [ m ] Boron concentration [#/cm3] SIMS data Curve fit Desired profile Background Concentration Figure 67. Boron concentra tion in silicon device layer dete rmined by SIMS, the accompanying curve fit and the desired model profile. 10 8 6 4 2 0 2 4 6 8 10 15 10 5 0 5 10 15 Potential [V]Current [mA] Device A Device B RinA = 934 = 12.9RinB = 1020 = 15.5 Figure 68. Input IV curve of 12 BUF1A and 12 BUF1B devices. PAGE 142 142 10 8 6 4 2 0 2 4 6 8 10 15 10 5 0 5 10 15 Potential [V]Current [mA] Device A Device B RoutA = 930 = 10.2RoutB = 1021 = 16.6 Figure 69. Output IV curve of 12 BUF1A and 12 BUF1B devices. 10 8 6 4 2 0 2 4 6 8 10 20 10 0 10 20 Potential [V]Current [mA] 10 8 6 4 2 0 2 4 6 8 10 1.5 1.6 1.7 1.8 Leakage Current [A] Curve Fit data Figure 610. Input IV curve of a BU F1A device with a linear curve fit. PAGE 143 143 10 8 6 4 2 0 2 4 6 8 10 15 10 5 0 5 10 15 Potential [V]Current [mA] 10 8 6 4 2 0 2 4 6 8 10 1.8 1.9 2 2.1 2.2 2.3 Leakage Current [A] Curve Fit Data Figure 611. Output IV curve of a BU F1A device with a linear curve fit 20 15 10 5 0 5 10 0.2 0 0.2 0.4 0.6 0.8 1 1.2 Potential [V]Current [mA] Figure 612. IV curve of diode ch aracteristics of a BUF1A device. PAGE 144 144 12 10 8 6 4 2 0 3 2.5 2 1.5 1 0.5 0 Potential [V]Current [A] Figure 613. IV curve of diode characteristics of a BUF1A device focusing on the reverse region. 101 100 101 102 103 104 105 106 1018 1017 1016 1015 1014 1013 1012 1011 1010 Frequency [Hz]Noise PSD [V2/Hz] setup V = 0.22 V = 0.43 V = 0.81 Figure 614. Noise PSD of a test taper resistor. PAGE 145 145 101 100 101 102 103 104 105 106 1018 1017 1016 1015 1014 1013 1012 1011 1010 Frequency [Hz]Noise PSD [V2/Hz] V = 0.22 V = 0.43 V = 0.81 Figure 615. Noise PSD from a test taper resistor minus the setup noise and the associated model curve fit. The horizontal line is the thermal noise floor for this device. 101 100 101 102 103 104 105 1016 1015 1014 1013 1012 1011 1010 109 Frequency [Hz]Noise PSD [V2/Hz] setup V = 0.28 V = 0.93 V = 1.96 V = 3.94 V = 9.96 Figure 616. Noise power spectral density of a BUF1A device. PAGE 146 146 101 100 101 102 103 104 105 106 1018 1017 1016 1015 1014 1013 1012 1011 1010 Frequency [Hz]Noise PSD [V2/Hz] V = 0.28 V = 0.93 V = 1.96 V = 3.94 V = 9.96 Figure 617. Noise PSD minus the setup noise and the associated model curve fit. The horizontal line is the thermal noise floor for the device. 115 120 125 130 135 140 145 150 155 160 2 3 4 5 6 7 8 9 10 Sensitivity [nV/Pa/V]Incident Pressure [dB SPL] Figure 618. Sensitivity of BUF1A devices normalized by the bias voltage. PAGE 147 147 115 120 125 130 135 140 145 150 155 160 165 170 0 5 10 15 20 25 THD [%]Incident Pressure [dB SPL] Figure 619. Total harmonic di stortion of BUF1A microphones. 0 1 2 3 4 5 6 7 170 165 160 155 150 145 140 135 130 Frequency [kHz]Magnitude Response [dB] (re 1V/Pa) Figure 620. Magnitude frequency response for a BUF1A device. Vertical dotted lines mark the piecewise FRFs that were stitched together. PAGE 148 148 0 1 2 3 4 5 6 7 155 150 145 Mag FRF [dB]Frequency [kHz] 0 1 2 3 4 5 6 7 155 150 145 Mag FRF [dB]Frequency [kHz] 0 1 2 3 4 5 6 7 155 150 145 Mag FRF [dB]Frequency [kHz] 0 1 2 3 4 5 6 7 155 150 145 Mag FRF [dB]Frequency [kHz] Figure 621. Magnitude FRF for each device with 95% CI bounds. PAGE 149 149 0 1 2 3 4 5 6 7 20 15 10 5 0 5 10 15 20 Frequency [kHz]Phase Response [deg] Figure 622. Phase respons e for each device tested. PAGE 150 150 0 1 2 3 4 5 6 7 20 10 0 10 20 Phase [deg]Frequency [kHz] 0 1 2 3 4 5 6 7 20 10 0 10 20 Phase [deg]Frequency [kHz] 0 1 2 3 4 5 6 7 20 10 0 10 20 Phase [deg]Frequency [kHz] 0 1 2 3 4 5 6 7 20 10 0 10 20 Phase [deg]Frequency [kHz] Figure 623. Phase FRF for each device with 95% CI bounds. PAGE 151 151 0 1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency [kHz]Coherence Figure 624. Coherence function between device A5 and the reference microphone. A B Figure 625. A) Picture of a microphone die with topside lighti ng. B) Picture of a microphone die with backside lighting. PAGE 152 152 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Probability Density Function of MDPMDP [Pa] Figure 626. Minimum detectable pre ssure probability density function. 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0 10 20 30 40 50 60 70 80 90 100 Probability Density Function of SensitivitySensitivity [V/Pa] Figure 627. Sensitivity probability density function. PAGE 153 153 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0 5 10 15 20 25 Probability Density Function of Voltage NoiseVoltage Noise [ V] Figure 628. Voltage noise probability density function. 500 1000 1500 2000 2500 3000 0 0.5 1 1.5 x 103 Probability Density Function of PmaxMaximum Pressure [Pa] Figure 629. Maximum detectable pre ssure Probability density function. PAGE 154 154 CHAPTER 7 CONCLUSIONS The m ain contributions of this research as we ll as a summary of the research objectives are given in this chapter. In addition, ideas for extending this work and suggestions for improvements are proposed. 7.1 Conclusions As urban sprawl encroaches on airports and a dr amatic increase in air traffic is expected, there is a great need for a reduction in commercia l aircraft noise. Aircra ft manufacturers perform extensive scale model wind tunnel tests to locate and eliminate sound sources. One of the most important pieces of equipment needed is a robus t microphone that is able to withstand large sound pressure levels 160dBSPL, has an operating bandwidth on the order of 100kHz, and has a low noise floor 26dBSPL. Many microphones have been developed and published in the literature. Most of these are either a proof of concept or tailored to the audi o range (refer to chapte r 2). In the past decade MEMS microphone development has focused on devi ces specific to the conditions inside a wind tunnel and within the proximity of a jet engine [37], [45], [5 5], [62], [75]. Existing MEMS microphones meet one of several requirements, and either have a sufficient noise floor, maximum pressure or bandwidth. However, with the exception of Martin et. al. [55] no MEMS microphone developed to date ap proaches both the lower and uppe r end of the Bruel and Kjaer 4138 dynamic range while maintaining a sufficiently high bandwidth. However, Martins device [55] is susceptible to moisture, making it unsui table for many aeroacoustic applications. This work focused on designing and fabricating a piezoresistive device designed to be impervious to moisture. PAGE 155 155 The major contribution of this work has been to develop a model and optimization scheme for a piezoresistive microphone that allows a de signer to maximize the operational space of the device to meet their needs. This model has been verified with a fabricated device. This device unfortunately had major problems with the resistor doping profile, leading to a high noise floor exceeding 100 dBSPL. The models from Chapters 3 and 4 show that with proper resistors a device can obtain a noise floor below 30 dBSPL while maintaining a maximum pressure of 160 dBSPL and a bandwidth greater than 100 kHz. 7.2 Recommendations for Future Piezoresistive Microphones Based on the design, fabrication and hand ling of MEMS microphones in this project, several suggestions are provided for future device s. Due to the problems with the noise levels associated with ionimplanted resistors, a sw itch to solid source diffusion resistors is recommended. The advantage of this method is th at there is no damage to the silicon substrate associated with the solid source diffusion process, resulting in a drop in the overa ll noise of the device. Now that the process flow has been designed and tested, the next generation microphone should be transitioned to a commercial foundry. This will increase the yield and decrease undesirable characteristics such as contact resistance due to more reliable equipment and a more controlled environment. With this switch it is essential to incorporate short loops into project timelines to identify potential problems early in the fabrication process, which is extremely important for the piezoresistors. A short l oop to determine the dopant profile and Hooge parameter would be invaluable in determining project directions. Finally, the development of electronic through wafer interconnects ( ETWI) is suggested. Aeroacoustic devices by design are subject to high flow rates acr oss the diaphragm. With the current microphone, wire bonds are needed to electrica lly connect the die to the device circuitry. These wires limit the smoothness of the face of the microphone p ackage and can contribute to PAGE 156 156 errors in measurements. Incorporating ETWI would allow the microphone face to be smoother and limit the possibility of th e electrical connections bei ng exposed to the elements. 7.3 Recommendations for Future Work This section details the recommendations for increasing the characterization capabilities for testing microphones that are designed for aeroacoustic purposes. The microphones designed in this dissertation have a pr edicted resonant frequency over 200 kHz. The LV with spark source setup is only capable of measuring the resonant frequency of devices up to ~180 kHz. Ideally the spark source contai ns frequency content well beyond the resonant frequency of the device. Further development of a source should be done. In addition to the spark source, a setup should be constructed to rigidly attach the pressure coupler designed by K. Kadirvel [142] to the optical table. This will reduce vibrations in th e setup and help isolate the vibrations of the diaphragm. Currently, frequency response meas urements are only possible up to 20 kHz with He injected into the PWT. Minimally, a technique should be prepared to characterize a microphone to at least 100 kHz. Ideally, the frequency response shoul d be measured over the full bandwidth of the microphone. To accomplish this, an acoustic source is needed that can produce sound over the entire frequency range. Audio drivers operate up to20 kHz and specialized tweeters can reach frequencies up to 30 kHz. An ionophone similar to Franssons design [143] is recommended for the audio driver. Also, a new e xperimental setup is needed in order to produce a controlled sound field. The wavelength of sound in air is 3.4 mm at 100 kHz. Therefore, a plane wave tube would be impractical, because a MEMS and reference microphone would not be able to fit in the required small cross section. Consequently a free field measurement is probably the best option. To accomplish this, the DUT a nd reference microphone are installed into a large PAGE 157 157 baffle and placed in an anechoic chamber. A noi se source is set far enough away to ensure a spherically spreading wave hits the baffle and that the refe rence and DUT microphone see the same sound field. This method is described by King et. al.[144]. The methods for measuring the total harmonic distortion is an addi tional area that needs improvement. For future experiments, the micr ophone should have incident pressure that is a pure sine wave, and the harmonics generated would therefore only be caused by nonlinearities in the microphone. Limitations in the experimental setup in this study produced nonlinearities for sound pressure le vels approaching 160 dB, but if an ideal amplifie r, signal generator, and acoustic driver there would sti ll be nonlinearities due to the propagation of the high amplitude sound wave. Preferably, an experi mental setup should be developed that predistorts the signal sent to the acoustic driver, resu lting in a pure sine wave is received by the microphone, should be developed. R. Holman demonstrated the feasibi lity of using a feedforwa rd loop for generating a predistorted signal to make a pur e sine wave for a synthetic jet actuator [145]. This technique could be used in the acoustic test setup for th e THD characterization. In addition, it has proved difficult to obtain sound pressure levels above 165 dB for anything except a pure sine wave. To increase the capabilitie s of testing at high s ound pressure levels, the PWT should be equipped with an array of 2 or 4 drivers. This will increase the SPL by 6 and 12 dB respectively. PAGE 158 158 APPENDIX A COMPOSITE PLATE MECHANICS Linear Theory The plate being analyzed is a composite struct ure made up of 3 layers. The base layer is silicon Si, the middle layer is silicon dioxide 2SiO and the top layer is silicon nitride SiN. All of the following equations will be derived from a reference axis shown in Figure 32. The GreenLagrange Strain Tensor for the rr and direction are as follows [111], 2221 2rr z rruuuu E rrrr (A1) and 222 22 2111111 22 2rrzr rruuuuuuu Euuuu rrrrrr (A2) Applying von Karman plate theory and symmetry, equations (A1) and (A2) can be simplified. Fi rst equation (A1) will be reduced. Applying symmetry sets ,0 u If the transverse normals rotate a moderate amount (1015deg), then the term 2ru r is second order with respect to ru r and can be neglected [111]. (Note: sin within 1% up to 15deg). The term 2zu r however, can only be neglected if the deflection of the membrane is small. Therefore, equation (A1) simplifies as follows 21 2rr rruu E rr 2 neglectedu r 2 symmetry zu r small deflection or PAGE 159 159 r rrdu dr (A3) When the GreenLagrange equations are simplifie d, the notation of strain is switched from E to Equation (A2) can be greatly simplified due to symmetry: 1ru u E r r 2 22 sym.1111 2rzu uu rrr 2 2 sym.1 22r ru u uuu r 2 sym. 2.r rru uu rr (A4) If the radial displacement normalized by the radi us is much, much smaller than one, then the second term can be neglected because it is second order with respect to ru r Therefore, equation (A4) becomes ru r (A5) The radial and axial displacements can be relate d to displacements at a reference axis by the following equations taken from classical plate theory [146] 0 0,rdwr urzurz dr (A6) and 0,zurzwr (A7) where 00 and wu are the axial and radial displacements at the reference axis respectively. The reference axis was chosen to be at the locati on of the piezoresistors. Therefore, equations (A3) and (A5) in terms of reference ax is displacem ents and neglecting higher order terms are written as PAGE 160 160 initialstrain or resulting residualfrom strainloading, ,rr rrrrrz rz (A8) where 0 0 curvature axial term term due to due to bending stretching rr rr rrz (A9) with 0 0 rrdur dr (A10) 2 0 2,rrdwr dr (A11) 0 0, ur r (A12) and 01 dwr rdr (A13) Constitutive Relationship It is assumed that silicon is transversely isotropic. The validity of this assumption is based on silicons small degree of anis otropy. Anisotropy is defined as 44 11122 E EE (A14) where ijE are the independent elastic moduli. A purely isotropi c material has 1 where silicon has 1.57 [147]. Silicon dioxide and silicon nitride are amorphous and therefore PAGE 161 161 isotropic [148]. The constitutive relationships are then defined as (note that the redundant subscript will be dropped from and rr for the rest of the document) 0 0 rrr rQQQz (A15) where Q is defined as 1112 2 21221 1 1 QQ E Q QQ (A16) The forces per unit length are found by integrating equation (A15) as follows T Bz rr zN dz N (A17) Substituting equation (A15) into equation (A17) yields TTT BBBzzz o rrr r o zzzN QdzQdzQzdz N (A18) It is convenient to de fine two new matrices: T Bz zAQdz (A19) and T Bz z B Qzdz (A20) where A and B are the extensional stiffness matrix and the flexuralextensional matrix due to coupling respectively. Equation (A18) can now be written as o rrr r oN AAB N (A21) PAGE 162 162 The moments per unit length can be solved by in tegrating the stress times its moment arm, z over the thickness, T Bz rr zM zdz M (A22) Substituting equation (A15) into equation (A22) yields 2TTT BBBzzz o rrr r o zzzM QzdzQzdzQzdz M (A23) It is now convenient to define the flexural stiffness matrix as follows 2T Bz zDQzdz (A24) Equation (A23) can now be written as o rrr r oM BBD M (A25) The transverse loads and moments are also grouped in the following manner 0 0 force due initial force to loadingrr rNN N NN N (A26) and 0 0 moment due initial moment to loadingrr rMM M MM M (A27) where 0 0,rrN A N (A28) PAGE 163 163 ,o r rr oN AB N (A29) 0 0,rrM B M (A30) and .o r rr oM BD M (A31) The initial compression of the plate is applied such that r and therefore000rNNN and 000r M MM Material Parameters There are three material parameters that are required in order to solve the governing differential equations. In order to solve for them in a composite plate, they must be solved for in a piecewise manner. Using the convention from Figure 32 the elements of the extensional stiffness m atrix are given by and 0 123 11 222 123 0 0 112233 12 222 123 0, 111 111MT BM MT BMzz zz zz zzEEE Adzdzdz EEE Adzdzdz (A32) which given the location of the reference plane reduces to and 123 11123 222 123 112233 12123 222 123, 111 111 EEE AHHH EEE AHHH (A33) Note that these matrices are symmetric, therefore 11221221 and QQQQ The elements of the flexural extensional matrix due to coupling are PAGE 164 164 and 0 123 11 222 123 0 0 123 12 222 123 0, 111 111MT BM MT BMzz zz zz zzEEE B zdzzdzzdz EEE B zdzzdzzdz (A34) which given the location of the reference plane reduces to and 222 123 1112332 222 123 222 112233 1212332 222 1232, 212121 2. 212121 EEE BHHHH H EEE BHHHH H (A35) The elements of the flexural stiffness matrix are and 0 222 123 11 222 123 0 0 222 123 12 222 123 0, 111 111MT BM MT BMzz zz zz zzEEE Dzdzzdzzdz EEE Dzdzzdzzdz (A36) which given the location of the reference plane reduces to and 333 123 111233232 222 123 223 112233 121233232 222 1233, 313131 3. 313131 EEE DHHHH HHH EEE DHHHH HHH (A37) Derivation of Governing Equations Tension Case The equilibrium equations were derived in the linear homogenous plate document and are repeated here for convenience: 0rrdNNN drr (A38) r rrdM QMrM dr (A39) PAGE 165 165 and 00rrdw dd rNrprQ drdrdr (A40) A pure displacement differential equation governing the composite plate is desired. To achieve this, all three of the governing equations, as well as equations (A21) and (A25) and their derivates need to be incorporated. The derivation starts by integrating the governing equation (A40) at the reference axis to yield 2 00 2rrdw pr rNrQ dr (A41) Substituting equation (A39) into (A41) results in 20 2r r rMM dM dwpr rN r dr rdr (A42) Equations (A21) and (A25), as well as rdM dr are substituted into equation (A42) to yield 0 022 2 00000000 1101211 22 2 32 00000 12111112 32 11121212 1rr N rr Mdwdudw duduudwdw pr rAuArrB drdrdr drdrrdrdr dwdwdwdwdw Br DrAA dr drdrrdrdr BBBB 010.M (A43) Next, equation (A21) is substituted in to the governing equation (A38) resulting in 0 02 000 111112 22 32 000 121111 32211 111 0.r N r Nduduu AAA drrdrrr dwdwdw AA B rd r r d r r d r (A44) Reducing equation (A44) and solving for the u term s yields PAGE 166 166 23 2 000000 11 2232 2 1111 1 duduudwdwdw B drrdrrAdrrdrrdr (A45) Equation (A45) is substituted into equation (A43) and the higher order terms are neglected to yield, 00 11dwdu rA drdr 00 12 ... HOTdwu rA drr 2 ... 322 00000 11 322 112 1HOTpr dwdwdwdwdw B rr Adrdrrdrdrdr 2 0 11 12 ... HOTdw B B dr ... 32 0000 11000 321 0.HOT r canceldwdwdwdw rDrNMM drdrrdrdr (A46) Rearranging equation (A46) by grouping in terms of 0w produces 32 2 0000 32*21 2 dwdwNdw p r Dr r drdrDrdr (A47) where 2 11 11 11 B DD A (A48) Equation (A47) is differentiated and divided by r to yield 43 2 000000 43*22*22111 dwdwNdwNdw Dp drrdrDrdrrDrdr (A49) Equation (A49) is now solely a function of0w .The Laplacian and Biharmonic operators are defined in polar coordinates as 2 2 21 dd drrdr (A50) and PAGE 167 167 432 4 43223211 dddd drrdrrdrrdr (A51) Substituting the Laplacian and Biharmonic operators into equation (A49) yields *4 2 000DwNwp (A52) The composite plate is axissymmetric and clam ped around the perimeter a nd therefore is subject to the following boundary conditions: 0 0 0 0 0()0 0 0, (0).ra rwra clamped dw dr dw symmetry dr wr (A53) Compression Case For the case of inplane compression0 CN is substituted into 0N in equation (A49) to yield 43 2 000000 43*22*22111CCdwdwNdwNdw Dp drrdrDrdrrDrdr (A54) Recognizing the Laplacian and Biha rmonic operators from equations (A50) and (A51) produces *4 2 000 CDwNwp (A55) Note that the boundary conditions for both cases remains the same. Solution of Governing Equation Tension Case It is possible to write equation (A52) in the following manner due to the linear ity of the Laplacian PAGE 168 168 22 0 00 **N p ww DD (A56) First, the homogeneous solution is solved 2 0 011 0dw ddd rrw rdrdrrdrdr (A57) Upon integrating equation (A57), where 0 *N D, the following differential equation is obtained 2 0 0121 ln dw d rwCrC rdrdr (A58) Next the homogeneous solution of equation (A58) is found as 03040HwCIrCKr (A59) Variation of parameters is used to fi nd the particular solution of equation (A58) [149]: 120 120 00 0 00 00ln ln ,,PCrCKr CrCIr wIr drKr dr WIrKr WIrKr (A60) where W [ ] in equation (A60) is the Wronskian. The Wronskian of the Bessel functions is [150] 001 WIrKr r (A61) Implementing the identity from (A61) and rearranging in terms of the constants, the following equation is o btained: 010 00 0 20 0 0 0ln ln .PwCIrrrKrdrKrrrIrdr CIrrKrdrKrrIrdr (A62) The integrals multiplying the constant C2 are evaluated as follows 0101, rIrdrrIrrKrdrrKr (A63) PAGE 169 169 The integrals multiplying the constant C1 are evaluated using integration by parts as follows and 010 0101 ln ln 1 ln ln rrIrdrrIrrIr rrKrdrrKrrKr (A64) The evaluations are then plugged into equation (A62) 0101 001 0 2010111 lnln .PwCIrrKrrKrKrrIrrIr CIrrKrKrrIr (A65) The term multiplying C2 is recognized as the Wronskian divided by itself. The natural log term also multiplies the Wronskian, in the fi rst term. The other terms multiplying C1 cancel each other out. Therefore, th e particular solution is 012 21 lnPwCrC (A66) Incorporating the consta nt outside of equation (A66) into 12and CC the total solution to the homogeneous equation (A57) is now written as 0123040()ln wrCrCCIrCKr (A67) Note that solution (A67) could have been found from the s uperposition of the solutions to the following two equations obtained fr om the factored version of (A56) 2 110 ww (A68) and 2 20 w (A69) Therefore, the solution is 012www (A70) PAGE 170 170 A solution method of this sort is possible because of the linearity of the Laplacian operator. The following decompositions of homogeneous part of equation (A56) will illustrate. Using definition (A70), the homogeneous part of equation (A56) is written 22 2 11220 wwww. (A71) Restricting w1 and w2 to the following 2 110 ww (A72) and 2 20 w (A73) equation (A71) becomes 2 20 w (A74) With N0 and D* as constants, equation (A74) simplifies to the restriction (A73). Thus the separation of equation (A56) into (A68) and (A69) proves correct. Th e particular solution to equation (A56) is found by assuming a polynomial of the 4th order: 432 012345()pwrcrcrcrcrc (A75) Substituting this solution into equation (A56) yields a part icular solution of 2 0 0() 4p p r wr N (A76) The total solution is then 2 0123040 0()ln 4 p r wrCrCCIrCKr N (A77) Subjecting equation (A77) to the boundary conditions found in equation (A53) yields 2 00 0 010()() ()11 22()() IaIr paar wr NaIaIa (A78) PAGE 171 171 Equation (A45) is now used to solve for the radial displacement u. Substituting 0w in equation (A45) yields 2 1 2 000 11 22 01111 2 I r duduu B pa drrdrrNAIa (A79) The homogenous part of the solution is Ca uchys equation which has the solution of 2 01Hc urcr r (A80) To solve for the particular solu tion the method of variation of pa rameters is used. This method states 21 012puFruFr uruu WW (A81) where 1 2 1 2 11 0111, 1 (), 2 ur u r I r B pa Fr NAIa (A82) and 12 122uu W dudu r drdr (A83) Using the recurrence relationship 112 I zIzIz z (A84) yields the particular solution 1 11 0 01112p I r B pa u NAIa (A85) PAGE 172 172 The boundary conditions for the radial displacement are as follows 0, 00;uaclamped usymmetry (A86) which yields 1 11 0 01112Ir B p ar ur NAIaa (A87) Equations (A78) and (A87) are nondimensionalized using the following definitions: 00 2 4 ** 0 11 ** 11, 1 and 2nawu rdW WU ahhd Na B pah ka DhDahA (A88) such that 2 ** 00 *2*** 101 ()() 1 () 1 2()() IkIk W kkIkIk (A89) 1 *2 1 naIk P U k Ik (A90) and 1 *2 1Ik P k Ik (A91) Note that na is the normalized distan ce of the neutral axis from the reference frame. Compression Case The solution to the compression case is very similar to the tension case but the details were left in for ease of following. Equation (A55) can be written as PAGE 173 173 22 0 00 **CN p ww DD (A92) First, the homogeneous solution is solved. 2 0 011 0 ddddw rrw rdrdrrdrdr (A93) Upon integrating equation (A93), where 0 *CN D, the following differential equation is obtained, 2 0 0121 lnddw rwCrC rdrdr (A94) Next the homogeneous solution of equation (A94) is found as 03040HwCJrCYr (A95) Variation of parameters is used to fi nd the particular solution of equation (A94) [149]: 120 120 00 0 00 00ln ln ,,PCrCYr CrCJr wJr drYr dr WJrYr WJrYr (A96) The Wronskian of these particul ar Bessel functions is [150] 002 ,WJrYr r (A97) Implementing the identity from (A97) and rearranging in terms of the constants, the following equation is o btained: 010 00 0 20000ln ln 2 2PwCJrrrYrdrYrrrJrdr CJrrYrdrYrrJrdr (A98) The integrals multiplying the constant C2 are evaluated as follows PAGE 174 174 0101, rJrdrrJrrYrdrrYr (A99) The integrals multiplying the constant C1 are evaluated using integration by parts as follows and 010 0101 ln ln 1 ln ln rrJrdrrJrrJr rrYrdrrYrrYr (A100) The evaluations are then plugged into equation (A98) 0101 001 0 2010111 lnln 2 2PwCJrrYrrYrYrrJrrJr CJrrYrYrrJr (A101) The term multiplying C2 is recognized as the Wronskian divided by itself. The natural log term also multiplies the Wronskian, in the first term. The other terms multiplying C1 cancel each other out. Therefore, th e particular solution is 012 21 lnPwCrC. (A102) Incorporating the consta nt outside of equation (A102) into 12and CC, the total solution to the homogeneous equation (A93) is now written as 0123040()ln wrCrCCJrCYr (A103) The particular solution to equation (A92) is found by assuming a solution of the form: 432 012345()pwrcrcrcrcrc (A104) Substituting this solution into equation (A92) yields a part icular solution of 2 0 0() 4p p r wr N (A105) The total solution is then PAGE 175 175 2 0123040 0()ln 4 p r wrCrCCJrCYr N. (A106) Subjecting equation (A106) to the boundary conditions found in equation (A53) yields 2 00 0 010()() ()11 22 ()()JaJr paar wr NaJaJa (A107) Equation (A45) is now used to solve for the radial displacement u. Substituting 0w in equation (A45) yields 2 1 2 000 11 22 01111 2Jr duduu B pa drrdrrNAJa (A108) The homogenous part of the solution is Ca uchys equation which has the solution of 2 01Hc urcr r (A109) To solve for the particular solution, the method of variation of parameters (equation (A81)) is used where 1 2 1 2 11 0111, 1 (), 2 ur u r Jr B pa Fr NAJa (A110) and 12 122uu W dudu r drdr (A111) Using the recurrence relationship 112JzJzJz z (A112) yields the particular solution PAGE 176 176 1 11 0 01112pJr B pa u NAJa (A113) The boundary conditions for the radial displacement are the same as in the tension case found in equation (A86) which yields 1 11 0 01112Jr B par ur NAaJa (A114) Equations (A107) and (A114) are nondimensionalized usi ng the definitions found in equation (A88): 2 ** 00 *2*** 101 ()() 1 () 1 2()()JkJk W kkJkJk (A115) 1 *2 1naJk P U k Jk (A116) and 1 *2 1Jk P k Jk (A117) Nondimensional Stresses The stress in the composite plate is defined by equation (A15). The stress is then decom posed into initial stre ss and stress due to loading initial stressstress due to loadingrrr (A118) where 0 0rr rQQz (A119) PAGE 177 177 The nondimensional radial and tangential stresses due to loading are defined by 2r r Sih E a (A120) and 2Sih E a (A121) Note that SiE is Youngs modulus of the bulk silicon. To solve for the nondimensional stress, 0 and are nondimensionalized by 00 00 22 2 2, 1 .rr rrdUa E dh Ua E h dWa K dh dWa K dh (A122) Tension Case Solving for the nondimensionalized strains and curvatures yields *** 01 0 *2 ** 111na rkIkIk P E k I kIk (A123) 1 0 *2 11naIk P E k I k (A124) *** 01 *2 ** 111rkIkIk P K k I kIk (A125) and PAGE 178 178 1 *2 11 Ik P K k Ik (A126) The radial nondimensionalized stress due to loading is ** 10 *2 2** 111 11 1na rii iIkIk P k k IkIk (A127) and the tangential stress due to loading ** 10 *2 2** 111 11 1na iii iIkIk P k k IkIk (A128) where i is the local Poissons ratio. is defined as 2 21 if if if mSi SiO mSiO Si SiN mSiN SiEE E EE E E EE E (A129) where mE is the Youngs modulus in the local region of calculation. Compression Case Solving for the nondimensionalized strains and curvatures yields *** 01 0 *2 ** 111na rkJkJk P E k JkJk (A130) 1 0 *2 11naJk P E k Jk (A131) *** 01 *2 ** 111rkJkJk P K k JkJk (A132) and PAGE 179 179 1 *2 11 Jk P K k Jk (A133) Using equations (A130) through (A133) yields the radial nondim ensionalized stress due to loading ** 10 *2 2** 111 11 1na rii iJkJk P k k JkJk (A134) and the tangential stress due to loading ** 10 *2 2** 111 11 1na iii iJkJk P k k JkJk (A135) Nonlinear Theory The GreenLagrange Strain Tensor for the rr and direction are as follows [111], 2221 2rr z rruuuu E rrrr (A136) and 222 22 2111111 22 2rrzr rruuuuuuu Eu u u u rrrrrr (A137) Applying von Karman plate theory and symmetry, equations (A136) and (A137) can be sim plified. First equation (A136) will be reduced. Applying symmetry sets 0 u If the transverse normals rotate a moderate amount 1015 then the term 2ru r is second order with respect to ru r and can be neglected [111]. (Note: sin within 1% up to 15 ). The PAGE 180 180 term 2zu r however, cannot be neglected because of the large deflection of the plate. Therefore, equation (A136) simplifies as follows 21 2rr rruu E rr 2 neglectedu r 2 symmetryzu r or 21 2rz rruu rr (A138) When the GreenLagrange equations are simplifie d, the notation of strain is switched from to E Equation (A137) can be greatly si mplified due to symm etry: 1ru u E r r 2 22 sym.1111 2rzu uu rrr 2 2 sym.1 22r ru u uuu r 2 sym. 2.r rru uu rr If the radial displacement normalized by the radius is much smaller than one, then the second term can be neglected because it is second order with respect to ru r Therefore, equation (A137) becomes ru r (A139) The radial and axial displacements can be relate d to displacements at a reference axis by the following equations taken from classical plate theory [146]: 0 0,rdwr urzurz dr (A140) PAGE 181 181 and 0,zurzwr (A141) where 00 and wu are the axial and radial displacements at the reference axis respectively. Therefore, equations (A138) and (A139) in terms of reference axis displacements and neglecting higher order term s are written as initialstrain or resulting residualfrom strainloading, ,rr rrrrrz rz (A142) where 0 0 curvature axial term term due to due to bending stretching rr rr rrz (A143) and 2 2 000 0 2 00 01 2 1 .rr rrdurdwr dwr drdr dr urdwr rrdr (A144) Governing Differential Equations The equilibrium equations repe ated here for convenience: 0rrdNNN drr (A145) r rrdM rQMrM dr (A146) and 00rrdw dd rNrprQ drdrdr (A147) PAGE 182 182 A pure displacement differential equation govern ing the composite plate is desired. The derivation starts by integrat ing the governing equation (A147) to yield 20 2rrpr rNrQ (A148) Note that at 0r the integration constant is equa l to zero. Substituting equation (A146) into (A148) and noting the convention dw dr results in 20 2r r rMM dM dwpr rNr drrdr (A149) Equations (A21) and (A25), as well as rdM dr are substituted into equation (A149) to yield 0 03 2 000 0 110 12 22 2 0000 0 11 12 2 32 0000 11 1112 32 111222 13 22 1rr N r Mdwdudw dw rp r rA u A drdrdr dr duduudwdw rB B drdrrdr dr dwdwdwdw rD r A A drdrrdrdr BB 012110.r MBB (A150) Next, equation (A21) is substituted in to the governing equation (A145) resulting in 0022 22 0000000 1112 222 32 000 11 322 11121211111 22 11 11 0.rr NNduduudwdwdw dw AA drrdrrdrdrrdrrdr dwdwdw B drrdrrdr AAAA rr (A151) Reducing equation (A151) and solving for the u terms yields PAGE 183 183 23 2 000000 11 2232 2 11 2 2 000 12 2 1111 1 1 1 2 duduudwdwdw B drrdrrAdrrdrrdr dwdwdw A Ardrdrdr (A152) Equation (A152) is substituted into equation (A150) to yield, 32 2 000 00 0 11 12 11 2 32 2 000 000 11 12 11 322 2 11 11 2 3 00 12 31 222 111 1 2 3 2 dwdudw dwu dw rp r rA r AB drdrdr drr dr dwdwdw dwdwdw BA rB AdrrdrrdrArdrdrdr dwdw Br dr dr 2 000 110 21 0. dwdwdw DrN drrdrdr (A153) Rearranging equation (A153) by grouping of linear and nonlinear terms produces 32 2 0000 32*2 2 2 00000 12 11121101211 2 111 2 1 30, 22 dwdwNdw pr Drr drdrDrdr dwdwdudwdw A r BBrAuArB drdrAdrdrdr (A154) where 2 11 11 11 B DD A (A155) Equation (A154) is then differe ntiated and divided by r to yield 43 2 000000 43*22*2 2 23 2 00 000 1112 121111 11 23 2 11 2 0000 12 2 2 00 11 22111 1 3 1 1 dwdwNdwNdw Dp drrdrDrdrrDrdr dwdw dwdwdw BA BBBrB rdrdrA drdrdr udwdwdu A rdrrdrdr dwdudw A drdrr 23 22 0000000 2231 0. 22 dudwdudwdwdw drdrdrdrdrdrrdr (A156) PAGE 184 184 The linear terms of equation (A156) resembles the Laplacian and Biharmonic operators which are defined in polar coordinates as 2 2 21 dd drrdr (A157) and 432 4 43223211 dddd drrdrrdrrdr (A158) Substituting the Laplacian and Biharmonic operators into equation (A156) yields 2 *42 00 1112 0001211 2 11 2 322 0000000 111112 322 23 222 000000000 11 2221 3 1 131 22 dwdw BA DwNwBB rdrdrA dwdwdwudwdwdu BrBA drdrdrrdrrdrdr dwdudwdudwdudwdwdw A drdrrdrdrdrdrdrdrrdr p (A159) Equation (A159) is a fourth or der nonlinear O DE with 00and wu the dependant variables. It is not possible to isolate 0w as in the linear case and therefor e this equation can not be solved analytically. It is important to note that 0N is defined as an inplane tension. To apply an inplane compression, it is convenient to define a compression parameter0CN where 00CNN (A160) Equation (A159) is then written as PAGE 185 185 2 *42 00 1112 0001211 2 11 2 322 0000000 111112 322 2 222 000000000 11 2221 3 1 131 22Cdwdw BA DwNwBB rdrdrA dwdwdwudwdwdu BrBA drdrdrrdrrdrdr dwdudwdudwdudwdwdw A drdrrdrdrdrdrdrdrrdr 3. p (A161) Mixed Form Tension Case To facilitate the calculations, a nondimensiona l, mixed form of the equations is derived here. Equation (A154) is rearranged to yield 32 2 0000 32*2 2 000 1112 1211 2 11 0 2 000 11121 2 131 22 0. 1 2 dwdwNdw pr Drr drdrDrdr dwdwdw BA BB Ardrrdrdr dw r dr dudwu AA drdrr (A162) Recalling that 2 2 00000 11121112 211 2rdudwudwdw NA ABB drdrrdrrdr (A163) and 0dw dr (A164) yields 22 0 2*2 2 12 1112 111 2 1 0. 2rN ddpr Drr drdrDr A rNBB A (A165) PAGE 186 186 Dividing by and Dr arranges equation (A165) in the same fash ion as in the nonlinear hom ogenous plate for a comparison 2* 2 0 12 2*2***11 22rN NB ddpr drrdrDrDDrD (A166) where 12 121112 11A B BB A (A167) Equation (A166) is checked against the 1st mixed ODE for a homogenous plate for validity. It can be seen the only difference is th e last term; however, the constant 12 B when calculated for a homogenous plate is equal to zero. To solve for the 2nd mixed PDE we need to recall equation (A144): 2 000 00 1 2rdurdwrur drdr r (A168) The derivative of 0 is taken and then substituted into 0 r to yield the compatibility equation for extensional strain due to loading: 2 0 00 01 0 2rdd w r dr dr (A169) Equation (A29) is then solved for 00and r to yield 1112 12121111 11121211 0 22 1112 r rd ANANBABABABA drr AA (A170) and 1112 11121211 12121111 0 22 1112 rd ANANBABABABA drr AA (A171) PAGE 187 187 Equations (A170) and (A171) are then substituted in to the com patibility equation (A169) to yield 22 2 22 2* 1112 12 22 11 13 2rrdNdN AA dd rrBr drdrdrdrrA (A172) Equation (A172) is then checked for va lidity by com paring it to the 2nd mixed PDE for a homogenous plate. It is known from the equation that 12 B is equal to zero for a homogenous plate. When the homogenous values for 1112and AA are entered into the constant1 from equation (A172), it reduces to E h, and therefore equation (A172) completely reduces to the hom ogenous equation. Equations (A166) and (A172) are nondimensionalized by the f ollowing parameters: 00 22 24 **** 0 ****, , and 2r rwu rdWa WU ahhdh NaNa Na p a SSkP DDDhD (A173) such that 2 *2 ** 2 2211 2rdd kPS dd (A174) and 2** 2 2 2 223 2rrdSdSdd dddd (A175) where 12 B h D (A176) and PAGE 188 188 222 1112 11AA h AD (A177) and are important parameters in determining the behavior of the plate. deals with the symmetry of the composite plate and therefore will be referred to as the symmetry coefficient. This parameter takes into account th at the reference axis is not necessarily the neutral axis. If the composite plate is symmetric about the defined reference axis then 0 and the more asymmetric the plate becomes the larger becomes. The approximate range for is from zero to about .04. represents the disparities of the properties of the different materials of the plate. If the plate is homogenous then 2121 For a composite plate, changes accordingly, with approximately 20% deviation from the homogenous value of the substrate. The following are the nondimensional boundary conditions for equations (A174) and (A175): 00 (A178) 10 (A179) 00rdS d (A180) Equation (A175) is second order with respect to rS and therefore needs one more boundary condition. The final boundary condition come s from the radial displacement boundary condition: 0 ura (A181) Recall equation (A168), if 0u, the resulting tangential extensional strain due to loading is 00 ra (A182) The subsequent equation is found by substituting equation (A171) into (A182) PAGE 189 189 11 12 11121211 12121111 0 22 11121 0r ra ra ra rad ANANBABABABA dra AA .(A183) Recall equation (A145) and substitute 00 and rrNNNNNN 0 000r rdNN rNNNN dr (A184) noting that 00 dN dr and solving for N yields r rdN NrN dr (A185) Substitute equation (A185) into equation (A183) to yield 12121111 12 12 11111r rBABA dNAd aNB drA drAa 0 ra (A186) Equation (A186) is then nondime nsionalized to yield 12 1 11 1 11r rdSA d S dAd (A187) The solution for U is found by first substituting 0dw dr into equation (A152) 2 22 000 1112 2222 111111 1 2 duduu BA dd d drrdrrAdrrdrrArdr (A188) Nondimensionalizing equation (A188) yields 22 12 2222 1111 1, 2naA dUdUUddd ddddAd (A189) where PAGE 190 190 h a (A190) and 11 111naB Ah (A191) Note that na is the nondimensionalized distance from the reference frame to the neutral axis. U is not constant through the thickn ess of the plate. To solve for U at the neutral axis, equation (A140) is nondimensionalized to yield rUU (A192) where z h (A193) Compression Case For the compression case equation (A166) is 2* 2 0 12 2*2***11 22C rN NB ddpr drrdrDrDDrD (A194) This equation is then nondimensionalized to yield 2 *2**2 2211 2crdd kPS dd (A195) where 2 0 C cNa k D (A196) All of the remaining equations are the identical to the tension case. PAGE 191 191 Finite Difference Solution Equations (A174) and (A175) are solved in a strai ghtforward m anner using a finite difference scheme. The ensuing 2nd order accurate central diffe rencing schemes for uniform discretization are used as approximations for the first and s econd derivatives: 2 11() 2nn ndfff x dxx (A197) and 2 2 11 222nnn ndffff x dx x (A198) Before and rS can be solved using the finite differen ce method, a coordinate transformation is made to increase the number of gr id points in the edge zone. As was seen in the linear case, a large number of points are needed to capture the behavior of the plate in the edge zone. The coordinate shift used in th is case is as follows [151] ln 1 ln 1 (A199) where 1 is the stretching parameter. As 1 more points are bunched near 1 Solving equation (A199) for yields 1 1 1 1 1 1 (A200) In addition to equation (A200), the first and second order de rivatives are calculated using the chain ru le: PAGE 192 192 dd d ddd (A201) and 2 22 2 222dddd dddd dddddddd (A202) where 222 1 ln 1 d d (A203) and 2 2 2 224 1 ln 1 d d (A204) Tension Case Equation (A174) is transformed using equations (A201) through (A204) and by writing as a function of is formulated as 2 22 *2**2 2 22111 2rddddd kS ddddd (A205) with boundary conditions 00 (A206) and 10 (A207) Note that is as a function of not multiplied by Likewise, equation (A175) is manipulated to reveal PAGE 193 193 2 2** 2 22 22 2 22 2 223 2rrdS dS ddd ddddd ddddd ddddd (A208) with boundary conditions 00rdS d dd (A209) and 12 1 11 1 11r rdSA ddd S ddA dd (A210) Multiplying by 2 and applying the central difference equations (A197) and (A198) (where and fx ) to equation (A205) yields 11nnnnnnncabd, (A211) where 2 2 **1 21, 2nnn r n nn nakS (A212) ,nnnb (A213) ,nnnc (A214) and 3 n nd (A215) Note that because of the nonlinearity in the la st term of equation (A205) the n in equation (A212) will be imputing the answer from the previ ous calculation. The error caused by this will be m inimal. The commonality parameters in equations (A212) through (A214) are defined as PAGE 194 194 2 n n nd d (A216) and 2 2 21 2n nn n ndd dd (A217) The discretized boundary conditions are written as 00 (A218) and 0N (A219) where n=N corresponds to 1 Discretizing equation (A208) yields, 11***nnnnrnrnrngSeSfSh (A220) where 2nne (A221) nnnf (A222) nnng (A223) and 2 1121 2nnnnnnnnnnh (A224) The new commonality parameter is defined as 2 2 21 3 2n nn n ndd dd (A225) PAGE 195 195 The boundary conditions (BC) at 0 and 1 for equation (A208) contain first order derivates and therefore are obt ained by using third order accu rate forward and backward differencing schemes respectively [152]: 3 1231 111892 6n nnnndf f fffOx dxx (A226) and 3 1231 111892 6n nnnndf f fffOx dxx (A227) Implemented equations (A226) and (A227), the discretized boundary conditions are written as 0123****1118920rrrrSSSS (A228) and 12312 11 **** 12361 11 1892111892NNNNrrrrNNNN NA A SSSS (A229) Now that the finite difference equations are established, they can be collected into matrix form 11 1 222 2 333 3 333 3 2222 11100......00 00......0 00......... 00..................... ..................00... .........00 0......00 00......00NNN N NNNN NNNab cab cab cab cab ca 110 2 3 3 2 11... ...N N NNNdc d d d d db (A230) and PAGE 196 196 0 1 2 2 1* 111 222 222 1111118920...00 00......0 00......... 00.................. ... ..................00 ... .........00 0......00 00...02918N Nr r r NNN r NNN rS S gef S gef gef S gef S S 1 2 2 1 *0 ... ...NN N rh h h h (A231) where 12 1161 11,NA A dd (A232) and 123111892.NNNNH (A233) The and rS equations are coupled. They are solved using an iteration sche me where an initial guess of from the linear case is used to solve for rS in equation (A231). The rS obtained is then used to solve equation (A230) for the nonlinear The resulting is compared to the initial guess. If the error is less that 0.05% then and *rS are taken as the solutions, otherwise, the scheme repeats where the calculated becomes the initial guess. After the iteration is complete the solution for is used to solve for U. Transforming equation (A189) to coordinates yields PAGE 197 197 2 22 2 22 2 22 2 22 12 111 1 1 2nadUdU U dd dd dd A d Ad (A234) with boundary conditions 10, 00.Uc l a m p e d U symmetry (A235) Applying the finite difference equations (A197) and (A198) yields 11nnnnnnnmUkUlUq (A236) where 21nnk (A237) nnnl (A238) nnnm (A239) and 1 12 11 11 2 1 211 2 1 2nn annnn nnn ann nn annnnA q A (A240) The discretized boundary conditions for U are PAGE 198 198 00, 0.NU U (A241) The U equations are then collected into matrix form 11 1 222 2 333 3 333 3 2222 11100......00 00......0 00......... 00..................... ..................00... .........00 0......00 00......00NNN N NNNN NNNkl U mkl U mkl U mklU mklU mkU 110 2 3 3 2 11... ...N N NNNqmU q q q q qlU (A242) The derivative of U is needed for the stresses and therefore is calculated below. Numerical differentiation for points 1 to 1 nnN yields 111 2nn n ndU UU d (A243) Solving for dU d at the center of the plate 0n results with 210 0 01 43 2dU UUU d (A244) and at the edge of the plate nN yields 121 34. 2NNN N NdU UUU d (A245) The solutions for the deflection, W, curvature, and tangential load, *S are found from the numerical solutions of the deflection slope, and radial load, rS. They are calculated using numerical differentiating and inte grating schemes of error at least second order. The following review the exact equations for these parameters in terms of and rS and their numerical PAGE 199 199 expressions. Note that in these equations, N is the total number of points. The solutions for *S and W use second order error central diffe rencing schemes for points two through N1 and second order error forward and backward diffe rence schemes for the boundary conditions at 10 and 1N respectively. Nondimensionalizing equation (A185) yields **r rdS SS d (A246) Implementing numerical di fferentiation for points 1 to 1 nnN yields, 11** *1 2nn n nrrr n nSS S S (A247) Solving for S at the center of the plate 0n results with, 0* 0S 100*** 0 01rrrSSS (A248) S at the edge of the plate nN utilizes the boundary condition (A181) to yield, 12 123 111 111892 6N NNNNNr NA SS A (A249) Nondimensional curvature is defined as d d (A250) Numerical differentiation for points 1 to 1 nnN yields 111 2nn n n (A251) Solving for at the center of the plate 0n results with PAGE 200 200 02 1 0 01 43 2 (A252) and at the edge of the plate nN yields 121 34 2NN N N N (A253) The solution for deflection is found using a thir d order error trapezoidal integrating scheme [153]. The integrating scheme starts at the edge of the plate such where the deflection is zero. The resulting deflection vector is then flipped so that the center deflection is designated as 0W. 1 112ii iiiiWd WW (A254) where 1:iNniN Remember that 0 and 0W at 1 Compression Case Equation (A195) is transformed using equations (A201) through (A204) and by writing as a function of is formulated as 2 22 *2**2 2 22111 2crddddd kS ddddd (A255) Multiplying by 2 and applying the central difference equations (A197) and (A198) (where and fx ) to equation (A255) yields 11nnnnnnncabd, (A256) where 2 2 **1 21, 2nnnc r n nn nakS (A257) PAGE 201 201 ,nnnb (A258) ,nnnc (A259) and 3 *n nd (A260) All of the remaining equations are identical to the tension case. Nondimensional Stresses The stress in the composite plate is defined by equation (A15). The stress is then decom posed into initial stre ss and stress due to loading initial stressstress due to loadingrrr (A261) where 0 0rr rQQz (A262) The nondimensional radial and tangential stresses due to loading are defined by 2r r Sih E a (A263) and 2Sih E a (A264) Note that SiE is Youngs modulus of the bulk silicon. To solve for the nondimensional stress, 0 and are nondimensionalized by PAGE 202 202 00 00 22 2 2, 11 .rr rrdUa E dh Ua E h dWda K ddh dWa K dh (A265) Solving for and r yields 21 1rii idUU d (A266) and 21 1ii idUU d (A267) where i is the local Poissons ratio. is defined as 2 21 if if if mSi SiO mSiO Si SiN mSiN SiEE E EE E E EE E (A268) where mE is the Youngs modulus in the local region. PAGE 203 203 APPENDIX B PROCESS TRAVELER Wafer: 4 ntype (100) SOI, 3000 thick buried oxide layer, 1.5m thick silicon device layer and a 350m thick handle layer, 35cm. Masks: Ground strap mask (GSM) NWell mask (NWM) Piezoresistor mask (PRM) Piezoresistor contact mask (PCM) Metallization mask (MTM) Topside vent mask (TVM) Bond pad mask (BPM) Backside vent path mask (BVP) Backside cavity mask (BCM) Process Steps: 1) Etch down to handle wafer for ground strap a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec d) Preexposure bake (Hot plate) 95C for 60 sec e) Align (MA6) using GSM for 9.7 sec, hard contact f) Develop (AZ 300MIF) for 60 sec g) Post exposure bake (Oven) 95C for 60 min h) Etch Si to BOX layer (DRIE) recipe BUF1 Ground strap mask i) Acetone/Methanol/DI wash to remove PR j) BOE (6:1) for 7 min k) Acetone/Methanol/DI wash 2) NWell implant a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec d) Preexposure bake (Hot plate) 95C for 60 sec e) Align (MA6) using NWM w.r.t. GSM for 9.7 sec, hard contact f) Develop (AZ 300MIF) for 60 sec g) Post exposure bake (Oven) 95C for 60 min h) Light O2 ash (RF600W, O2400sccms) for 60 sec i) Implant (Phosphorus, 20 E keV, 2414Qecm ) at MEMS exchange j) Acetone bath to remove PR for 3 hours PAGE 204 204 3) Piezoresistors and Oxidation a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec d) Preexposure bake (Hot plate) 95C for 60 sec e) Align (MA6) using PRM w.r.t. GSM for 9.7 sec, hard contact f) Develop (AZ 300MIF) for 60 sec g) Post exposure bake (Oven) 95C for 60 min h) Light O2 ash (RF600W, O2400sccms) for 60 sec i) Implant (Boron, 5 E keV, 2614Qecm ) at MEMS exchange j) Acetone/Methanol/DI wash to remove PR k) RCA clean at MEMS exchange l) Furnace anneal 2, 1050NTC for 300min at MEMS exchange m) Dry/Wet/Dry oxidation 950TC for 65 min, 21 min, and 65 min respectively n) Chemical/Mechanical Polish to re move backside oxide at ICEMOS 4) Piezoresistor contact cut a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec d) Preexposure bake (Hot plate) 95C for 60 sec e) Align (MA6) using PCM w.r.t. GSM for 9.7 sec, hard contact f) Develop (AZ 300MIF) for 60 sec g) Post exposure bake (Oven) 95C for 60 min h) BOE (6:1) for 8 min i) Acetone/Methanol/DI wash 5) Metallization a) Sputter 1m Al (1%Si) to avoid spikin g using gun 3 for best results b) Acetone/Methanol/DI wash c) Coat HMDS for 5 min d) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec e) Preexposure bake (Hot plate) 95C for 60 sec f) Align (MA6) using MTM w.r.t. PCM for 9.7 sec, hard contact g) Develop (AZ 300MIF) for 60 sec h) Post exposure bake (Oven) 95C for 60 min i) Aluminum etch (Ashland 16:1:1:2) 40C for 2 min j) Acetone/Methanol/DI wash 6) Nitride passivation and top vent etch a) Acetone/Methanol/DI wash PAGE 205 205 b) Plasma enhanced chemical vapor depo sition silicon nitrid e recipe MN300A c) Acetone/Methanol/DI wash d) Coat HMDS for 5 min e) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec f) Preexposure bake (Hot plate) 95C for 60 sec g) Align (MA6) using TVM w.r.t. PCM for 9.7 sec, hard contact h) Develop (AZ 300MIF) for 60 sec i) Post exposure bake (Oven) 95C for 60 min j) Etch nitride layer (Uni xaxis RIE/ICP) SF6 and O2 k) Etch oxide layer (Unixa xis RIE/ICP) CHF3 and O2 l) Etch Si to BOX layer (DRIE) recipe BUF1 Ground strap mask m) Acetone/Methanol/DI wash 7) Bond pad contact cut a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1m AZ1512) spin at 4000 rpm for 50 sec d) Preexposure bake (Hot plate) 95C for 60 sec e) Align (MA6) using BPM w.r.t. PCM for 9.7 sec, hard contact f) Develop (AZ 300MIF) for 60 sec g) Post exposure bake (Oven) 95C for 60 min h) Etch nitride layer (Uni xaxis RIE/ICP) SF6 and O2 i) Acetone/Methanol/DI wash 8) Backside vent path a) Acetone/Methanol/DI wash b) Coat HMDS on front side for 5 min c) Coat resist on front side for protection (10m AZ9260) spin at 2000 rpm for 50 sec d) Preexposure bake (Oven) 95C for 15 min e) Coat HMDS on back side for 5 min f) Coat resist on back side (1m AZ1512) spin at 4000 rpm for 50 sec g) Preexposure bake (Oven) 95C for 30 min h) Align front to back (EVG) using BVP w.r.t. GSM at Georgia Tech i) Develop (AZ 400MIF) 3:1 dilution with DI j) Post exposure bake (Oven) 95C for 60 min k) Etch Si (DRIE) 10m recipe BUF1VTPH l) Acetone/Methanol/DI wash to remove PR from both sides m) Coat HMDS on carrier wafer for 5 min n) Coat resist on carrier wafer top surface (10m AZ9260) spin at 2000 rpm for 50 sec o) Attach top side of wafer to carrier wafer p) Bake (Oven) 1 95C for 5 min 9) Backside cavity and vent through hole PAGE 206 206 a) Coat HMDS for 5 min b) Coat resist on back side (10m AZ9260) spin at 2000 rpm for 50 sec c) Preexposure bake (Oven) 95C for 30 min d) Align (MA6) using BCM w.r.t. BVP for 150 sec, hard contact e) Develop (AZ 300MIF) for 6 min f) Post exposure bake (Oven) 95C for 60 min g) Etch Si (DRIE, SOI kit) 350m recipe BUF1BAO2 h) Acetone/Methanol/DI wash i) Methanol spray, DI dip and BOE (6:1) etch for 8 min j) Release carrier wafer (AZ400K PR stripper) at 40C 10) Forming gas anneal and back plate bond a) Forming gas anneal (N2/H2, 450TC ) for 60 min at MEMS exchange b) Acetone/Methanol/DI wash c) Coat resist on front side to protect Al(10m AZ9260) spin at 2000 rpm for 50 sec d) RCA clean SOI and Pyrex e) Acetone/Methanol/DI wash to remove PR f) Anodic bond Pyrex wafer to backside of SOI wafer PAGE 207 207 APPENDIX C MATLAB FUNCTIONS The f ollowing code is the objective function used in the optimization scheme to determine MDP. function [Obj] = sensitivitybox(X) %%note this code is assu ming only p type doping!!! % note this code assumes T=300K only!!!! % %% constants Nb = 1e15; kb = 1.3806503e23; %[Nm/K] T=300; %[K] f2=1000.5; % [Hz] f1 = 999.5; % [Hz] % f2=1050; % [Hz] % f1 = 950; % [Hz] global hooge COUNT mode COUNT = COUNT + 1; global E nu wline zincr tincr N sigma0 DUB DLB PM bb H(1) = X(1)*(DUB(1)DLB(1))+DLB(1); H(2) = X(2)*(DUB(2)DLB(2))+DLB(2); H(3) = X(3)*(DUB(3)DLB(3))+DLB(3); H(4) = sum(H); a = X(4)*(DUB(4)DLB(4))+DLB(4); ah = a/H(4); Nsurf = X(5)*(DUB(5)DLB(5))+DLB(5); zj = X(6)*(DUB(6)DLB(6))+DLB(6); thetaa = X(7)*(DUB(7)DLB(7))+DLB(7); raout = a; rain = X(8)*(DUB(4)DLB(4))+DLB(4); thetat = X(9)*(DUB(9)DLB(9))+DLB(9); rtout = a; rtin = X(10)*(DUB(4)DLB(4))+DLB(4); if mode == 0 Vb = X(11)*(DUB(11)DLB(11))+DLB(11); else if mode == 1 cur = X(11)*(DUB(11)DLB(11))+DLB(11); end end dlmwrite('nondimX.txt', [COUNT X], 'append') PAGE 208 208 if mode == 0 dlmwrite('dimX.txt', [COUNT H(1) H(2) H(3) a Nsurf zj thetaa rain thetat rtin Vb], 'append') else if mode == 1 dlmwrite('dimX.txt', [COUNT H(1) H(2) H(3) a Nsurf zj thetaa rain thetat rtin cur], 'append') end end if rain >= a rain = a wline; % COUNT errormessage = 1; dlmwrite('errors.txt', [COUNT errormessage], 'append') end if rtin >= a rtin = a wline; % COUNT errormessage = 2; dlmwrite('errors.txt', [COUNT errormessage], 'append') end B11=E(1)*H(1)^2/(2*(1nu(1)^2))+E(2)*H(2)^2/(2*( 1nu(2)^2))+E(3)/(2*(1nu(3)^2))*(H(3)^2+2*H(3)*H(2)); A11=E(1)*H(1)/(1nu(1)^2)+E(2)*H(2)/(1 nu(2)^2)+E(3)*H(3)/(1nu(3)^2); D11=E(1)*H(1)^3/(3*(1nu(1)^2))+E(2 )*H(2)^3/(3*(1nu(2)^2))+E(3)/(3*(1nu(3)^2))*(H(3)^3+3*H(3)*H(2)*(H(3)+H(2))); Dstar=D11B11^2/A11; k=sqrt(sigma0*H(2)/Dstar)*a; p = PM(bb); % good check. set this to say 1000 and the results should be the same P = p*a^4/(2*H(4)*Dstar); z=[0:zj/(zincr1):zj*(1)]; % from the surface (z=0) to the junction depth % Nr = Nsurf.*ones(1,length(z)); %creates a uniform profile Nr = Nsurf*(Nsurf/Nb).^((z/zj).^2); % creates a gaussian profile % constant parameters global q mumin mu0 alpha q=1.6e19; etaNref=2.4; etamumin=0.57; etaalpha=.146; Nref=2.35e17; PAGE 209 209 mumin=54.3; mu0=406.9; etamu0=2.23; alpha=.88; pi11=6.6e11; pi12=1.1e11; pi44=138.1e11; gamma=45*pi/180; % mean diaphram orienation relative to the crystal axis % calculates T variation (Pierret) mu=mumin+mu0./(1+(Nr./Nref).^alpha ); %mobility [cm^2/(V*s)] rho=1./((q.*mu.*Nr).*100); %resistivity [ohms*m] % tangents of the arc and tapered resistor wi th reference to the crystal structure thetawt=2*log(a/rtin)*log(a/rain)/(thetaa); % to make the resistors have the same resistance thetawt is determined by this equation phia=[(gammathetaa/2):(thetaa/( tincr1)):(gamma+thetaa/2)]; phit=[(gamma+(thetatthetawt)/2):(thetawt/ (tincr1)):(gamma+(thetat+thetawt)/2)]; % direction cosines la(:,1)=cos(phia); la(:,2)=sin(phia); ma(:,1)=sin(phia); ma(:,2)=cos(phia); na=zeros(length(phia),2); lt(:,1)=cos(phit); lt(:,2)=sin(phit); mt(:,1)=sin(phit); mt(:,2)=cos(phit); nt=zeros(length(phit),2); % piezoresistive coeff for bot h arc and tapered resistors pila=pi112*(pi11pi12pi44)*(la(:,1).^2.*ma(:,1).^2+ma(:,1).^2.*na(:,1).^2+na(:,1).^2.*la(:,1).^2); pita=pi12+(pi11pi12pi44)*(la(:,1).^2.*la(:,2).^2+ma(:,1).^2.*ma(:,2).^2+na(:,1).^2.*na(:,2).^2); pilt=pi112*(pi11pi12pi44)*(lt(:,1).^2.*mt(:,1).^2+mt(:,1).^2.*nt(:,1).^2+nt(:,1).^2.*lt(:,1).^2); pitt=pi12+(pi11pi12 pi44)*(lt(:,1).^2.*lt(:,2).^2+mt(:,1).^2.*mt(:,2).^2+nt(:,1).^2.*nt(:,2).^2); Pnt=.2014*log10(1.5e22./Nr); %calculates doping dependance R = find(Pnt > 1); PAGE 210 210 Pnt(R) = 1; for m=1:zincr PILa(:,m) = pila.*Pnt(m); PITa(:,m) = pita.*Pnt(m); PILt(:,m) = pilt.*Pnt(m); PITt(:,m) = pitt.*Pnt(m); end eta=z/H(4); sig=1/ah; etana=B11/(H(4)*A11); ra=[rain:(arain)/(N1):a]; rt=[rtin:(art in)/(N1):a]; rand=ra/a; rtnd=rt/a; for m=1:zincr for M=1:N sigmara(M,m) = E(1)*sig^2*P/k ^2*(eta(m)etana)./(1nu( 1)^2).*((1+nu(1))+(1nu(1))/rand(M).*besselj(1,k*rand(M))./besselj (1,k)k*besselj(0,k*rand(M))./besselj(1,k)); sigmata(M,m) = E(1)*sig^2*P/k^2*(eta(m)etana)./(1nu(1)^2).*((1+nu(1))(1nu(1))/rand(M).*besselj(1,k*rand( M))./besselj(1,k)nu(1)*k*besselj(0,k*rand(M))./besselj(1,k)); sigmart(M,m) = E(1)*sig^2*P/k^2*(eta(m)etana)./(1nu(1)^2).*((1+nu(1))+(1nu(1))/rtnd(M).*besselj(1,k*rtnd(M))./besselj(1,k)k*besselj( 0,k*rtnd(M))./besselj(1,k)); sigmatt(M,m) = E(1)*sig^2*P/k^2*(eta(m)etana)./(1nu(1)^2).*((1+nu(1))(1nu(1))/rtnd(M).*besselj(1,k*rtnd( M))./besselj(1,k)nu(1)*k*besselj(0,k*rtnd(M))./besselj(1,k)); % sigmara(M,m) = 0; % sigmata(M,m) = 0; % sigmart(M,m) = 0; % sigmatt(M,m) = 0; for B=1:tincr integrandt(M,B,m)=1/(rho(m)*(1+sigmart(M, m).*PILt(B,m)+sigmatt(M,m).*PITt(B,m))); integranda(M,B,m)=1/(rho(m)*(1+sigmara(M, m).*PITa(B,m)+sigmata(M,m).*PILa(B,m))); Ra(:,B)=ra(:); end end end RplusDRtea=2*trapz(rt',1./((trapz(phit, trapz(z,integrandt,3),2)).*rt')); RplusDRarc=trapz(phia,1./(trapz(ra,(trapz(z,integranda,3)./Ra),1))); Restea = 2*log(a/rtin)/(thetawt)*(1./trapz(z,1./rho)); PAGE 211 211 Resarc = (thetaa)/log(a/rain)*(1./trapz(z,1./rho)); DRtea = RplusDRtea Restea; DRarc = RplusDRarc Resarc; global DynRange Resistance Sensitivity Vnoise MDP if mode == 0 DVo=((DRarcDRtea)/(2*Resarc+DRarc+DRtea)).*(Vb*1e6); else if mode ==1 DVo= (cur*1e6)/2*(DRarcDRtea); end end Sens=DVo/p; Sensitivity = Sens; Resistance = Resarc; Na = trapz(z,Nr)*1e6*(.5*thetaa* (raout^2rain^2)); %carrier concentration times the volume % Na = Nr*1e6*(.5*thetaa*zj*(ra out^2rain^2)); %carrier co ncentration times the volume Nt = trapz(z,Nr)*1e6*(thetawt*(rtout^2rtin^2)); %carrier concentration times the volume % Nt = Nr*1e6*(thetawt*zj*(rtout^2rtin^2)); %carrier concentration times the volume % Vn = sqrt(4*kb*T*Resarc*(f2f1)+1/8*hooge *Vb^2*(1/Na+1/Nt)*log(f2/f1))*1e6; if mode == 0 Vn = sqrt(4*kb*T*Resarc*(f2f1)+1/8*hooge*Vb^2*(1/Na+1/Nt)*log(f2/f1)+(4e9*(f2f1))^2)*1e6; else if mode == 1 Vn = sqrt(4*kb*T*Resarc*(f2f1)+1/8*hooge*(cur*Resarc)^2*(1/Na+1/Nt)* log(f2/f1)+(4e9*(f2f1))^2)*1e6; end end Vnoise=Vn; MDP = 20*log10((Vn/Sens)/(20e6)); Obj = MDP; DynRange = 20*log10(p/(Vn/Sens)); data = [COUNT DynRange Resistance Se nsitivity Vnoise MDP k thetawt]; dlmwrite('objectives.txt', data, 'append') PAGE 212 212 APPENDIX D OPTIMIZED DEVICES PAGE 213 213Table D1. Optimized devices oper ating on a current source (10mA) with a Gaussian dopant profile and Nb = 1e15 [#/cm3]. BandwidthPmaxMDPDyn RangeH1H2H3aa/hD*Nsurfz j arain t wtrtinwarclarcwtapltapw g apRaIsuppl y SensNoise FLkPowerPmaxactD. Rg. ActAct. BW[kHz][dB SPL][dB SPL][dB SPL] [ m] [A][A] [ m] [][mNmm] [1/cm 3 ][ m] [deg] [ m] [deg][deg] [ m][ m][ m][ m][ m][ m][ ] [mA] [ V /Pa] [nV][][W][dB SPL][dB SPL][kHz]15017032.7137.32.85 300300 200690.3351.4E+190.6429.416912.89.315631872544 9 100010.08.27.091.04 0.1170 137.5 150 15016531.1133.92.15 300300 172780.1471.9E+190.5131.514414.09.813228792341 9 100010.010.17.241.35 0.1165 134.1 150 15016029.5130.51.842052 300 130620.1092.3E+190.4339.910319.213.09227722138 10 100010.012.47.423.08 0.1160 130.5 150 15015527.7127.31.482203 300 104600.0623.0E+190.3644.88122.114.07123631733 10 100010.015.97.73 3.400.1155 127.2 150 15015026.0124.01.141280 300 91700.0273.9E+190.30 45.0 7123.414.16120561530 10 100010.020.28.11 3.400.1150 123.8 150 15014525.4119.61.00977 300 85760.0184.2E+190.26 45.0 6622.913.46020521425 10 100010.023.38.67 3.400.1 147122.1 150 15014025.4114.61.00977 300 85760.0184.2E+190.26 45.0 6622.813.36020521425 10 100010.023.38.67 3.400.1 147122.0 150 14017032.4137.63.05 300300 215690.4101.3E+190.6728.518112.49.316933902746 9 100010.08.47.021.01 0.1170 137.7 140 14016530.8134.22.30 300300 185780.1801.7E+190.5430.515513.69.814330822442 9 100010.010.47.161.31 0.1165 134.4 140 14016029.1130.91.982310 300 138620.1362.1E+190.4539.110918.813.19929742339 10 100010.012.87.323.11 0.1160 130.8 140 14015527.3127.71.592376 300 111600.0762.7E+190.3843.88721.514.17625661935 10 100010.016.57.60 3.400.1155 127.6 140 14015025.5124.51.231368 300 97700.0343.6E+190.32 45.0 7622.814.16621601632 10 100010.020.97.90 3.400.1150 124. 3 140 14014524.4120.61.00903 300 88790.0184.3E+190.27 45.0 6923.513.96019541528 10 100010.024.98.30 3.400.1 146121.7 140 14014024.5115.51.00903 300 88790.0184.3E+190.27 45.0 6923.413.96019541528 10 100010.024.98.30 3.400.1 146121.7 140 13017032.1137.93.29 300300 231690.5101.1E+190.7027.519512.09.218336942948 9 100010.08.76.960.97 0.1170 138.1 130 13016530.4134.62.48 300300 199780.2231.5E+190.5729.416713.19.715532862644 9 100010.010.77.091.26 0.1165 134.7 130 13016028.7131.32.142604 300 147610.1721.8E+190.4838.211618.513.210631782541 10 100010.013.27.223.14 0.1160 131.2 130 13015526.8128.21.722554 300 120600.0952.4E+190.4042.79321.014.28327692137 10 100010.017.17.47 3.400.1155 128.1 130 13015025.0125.01.321471 300 105700.0423.2E+190.3444.58122.114.27223631833 10 100010.021.67.72 3.400.1150 124.8 130 13014523.5121.51.01845 300 92820.0184.2E+190.27 45.0 7223.514.26220561530 10 100010.026.98.08 3.400.1145 121.4 130 13014023.5116.51.00831 300 91820.0184.3E+190.27 45.0 7123.514.26220561530 10 100010.027.18.10 3.400.1 145121.2 130 12017031.7138.33.56 300300 251690.6461.0E+190.7426.521211.79.120139983250 9 100010.08.96.890.94 0.1170 138.4 120 12016530.0135.02.69 300300 216790.2821.4E+190.6028.318112.79.617035892846 9 100010.011.07.011.22 0.1165 135.1 120 12016028.4131.62.04 300300 185880.1261.9E+190.4930.515414.010.214331822542 9 10 0010.013.67.141.57 0.1160 131.8 120 12015526.3128.71.862761 300 130600.1202.2E+190.4341.510020.414.29129732339 10 100010.017.77.35 3.400.1155 128.5 120 12015024.5125.51.441590 300 113700.0532.9E+190.3643.38821.514.37825661935 10 100010.022.57.57 3.400.1150 125.4 120 12014523.0122.01.10913 300 99810.0233.8E+190.29 45.0 7722.814.36722601732 10 100010.027.97.85 3.400.1145 121.9 120 12014022.5117.51.00760 300 95860.0184.2E+190.27 45.0 7423.214.36421581631 10 100010.029.97.98 3.400.1 143120.7 120 11017031.4138.63.89 300300 274690.8358.7E+180.7925.423211.39.0221421033553 9 100010.09.26.830.90 0.1170 138.8 110 11016529.7135.42.93 300300 236790.3641.2E+190.6427.119812.39.518838943148 9 100010.011.46.931.17 0.1165 135.5 110 11016028.0132.02.22 300300 202890.1621.6E+190.5229.216813.510.115834862844 9 100010.014.17.061.51 0.1160 132.2 110 11015525.9129.12.04 3000300 141600.1551.9E+190.4540.210919.914.310032772541 10 100010.018.47.22 3.400.1155 129.0 110 11015024.0126.01.571731 300 124700.0692.5E+190.3842.09620.914.38628702237 10 100010.023.47.42 3.400.1150 125.8 110 11014522.4122.61.20994 300 108810.0303.3E+190.3143.88422.014.47424641934 10 100010.029.27.66 3.400.1145 122.5 110 11014021.4118.61.00691 300 99900.0184.1E+190.27 45.0 7722.814.46722601732 10 100010.033.37.85 3.400.1 142120.2 110 10017031.0139.04.28 300300 301 691.1077.5E+180.8424.325511.08.9246461083855 9 100010.09.56.760.86 0.1170 139.2 100 10016529.2135.83.22 300300 260790.4811.0E+190.6825.921811.99.420941993451 9 100010.011.86.861.12 0.1165 135.9 100 10016027.6132.42.44 300300 223890.2131.4E+190.5627.818613.09.917737903147 9 100010.014.66.971.45 0.1160 132.6 100 10015525.4129.62.23 3000300 157620.1981.6E+190.4938.312219.114.211436822844 10 100010.019.17.093.36 0.1155 129.4 100 10015023.4126.61.731899 300 136700.0912.2E+190.4140.610520.314.49631742440 10 100010.024.57.28 3.400.1150 126.4 100 10014521.8123.21.321090 300 119810.0402.9E+190.3442.49221.314.58327682136 10 100010.030.67.49 3.400.1145 123.0 100 10014020.3119.71.00624 300 104950.0173.9E+190.2744.38022.514.57124621832 10 100010.037.57.75 3.400.1140 119.6 100 9017030.6139.44.76 300300 335701.5136.3E+180.9123.128410.68.8276511154259 9 100010.09.96.700.82 0.1170 139.5 90 9016528.8136.23.58 300300 289790.6558.8E+180.7424.624311.49.3235461043854 9 100010.012.36.791.07 0.1165 136.3 90 9016027.1132.92.71 300300 248900.2891.2E+190.6026.420712.49.819941953449 9 100010.015.26.881.39 0.1160 133.0 90 9015524.9130.12.46 3000300 177640.2581.4E+190.5236.413718.314.013140873247 10 100010.019.86.973.31 0.1155 129.9 90 9015022.9127.11.932106 300 151700.1241.9E+190.4439.011619.614.410835792742 10 100010.025.67.14 3.400.1150 126.9 90 9014521.1123.91.471208 300 132810.0542.5E+190.3640.810120.614.59330722438 10 100010.032.27.33 3.400.1145 123.7 90 90 14019.6120.41.12692 300 115950.0243.4E+190.3042.68821.714.68027662035 10 100010.039.77.55 3.400.1140 120.3 90 8017030.2139.85.35 300300 377702.1455.1E+180.9921.932010.28.7314561224863 8 100010.010.36.640.77 0.1170 139.9 80 8016528.4136.64.03 300300 325800.9267.2E+180.8023.227411.09.1268511114357 9 100010.012.86.711.01 0.1165 136.7 80 8016026.6133.43.05 300300 280900.4071.0E+190.6524.923411.99.7227461013852 9 100010.015.96.801.32 0.1160 133.6 80 8015524.4130.62.74 3000300 203660.3491.1E+190.5734.215717.413.815245943750 10 100010.020.66.853.26 0.1155 130.4 80 8015022.3127.72.172364 300 169690.1761.5E+190.4837.313019.014.512439853146 10 100010.026.97.00 3.400.1150 127.5 80 8014520.5124.51.661356 300 148810.0772.1E+190.4039.011419.814.610734772741 10 100010.033.97.16 3.400.1145 124.3 80 8014018.8121.21.26776 300 129940.0342.8E+190.3240.89920.814.79230712338 10 100010.042.07.36 3.400.1140 121.0 80 7017029.7140.36.12 300300 431703.1904.0E+181.0920.63679.88.5363641325468 8 100010.010.76.570.72 0.1170 140.4 70 7016527.9137.14.60 300300 372801.3735.8E+180.8821.831410.59.0310581194962 8 100010.013.46.640.95 0.1165 137.3 70 7016026.1133.93.48 300300 320910.6008.1E+180.7123.226811.49.5264521094456 9 100010.016.76.711.24 0.1160 134.0 70 7015523.9131.13.11 3000300 236690.4929.2E+180.6331.918316.513.5181521024355 9 100010.021.56.743.19 0.1155 131.0 70 7015021.7128.32.492695 300 193690.2621.2E+190.5335.514818.414.514445923649 10 10 0010.028.36.87 3.400.1150 128.1 70 7014519.8125.21.901545 300 169810.1151.7E+190.4437.112919.114.612440843245 10 100010.035.97.01 3.400.1145 125.0 70 7014018.1121.91.45884 300 148940.0502.3E+190.3638.811320.014.710735762741 10 100010.044.77.17 3.400.1140 121.7 70 6017029.2140.87.14 300300 503705.0453.0E+181.2319.24309.48.4429731446374 8 100010.011.26.510.67 0.1170 140.9 60 6016527.4137.65.37 300300 434802.1654.4E+180.9920.236810.18.8367661305667 8 100010.014.16.570.89 0.1165 137.8 60 6016025.5134.54.05 300300 374910.9426.3E+180.8021.531410.89.3313601185161 8 100010.017.66.631.16 0.1160 134.6 60 6015523.4131.63.58 3000300 280720.7337.1E+180.7029.321915.613.2220611125160 9 100010.022.56.643.10 0.1155 131.5 60 6015021.0129.02.90 3000300 227700.4059.6E+180.5933.317317.614.4173541014354 10 100010.029.96.743.38 0.1150 128.8 60 6014519.1125.92.231798 300 197810.1811.3E+190.4935.015018.414.614747923849 10 100010.038.26.85 3.400.1145 125.7 60 6014017.3122.71.691028 300 172940.0791.8E+190.4036.613119.214.812741843345 10 100010.047.86.99 3.400.1140 122.5 60 5017028.7141.38.57 300300 604708.6832.3E+181.3917.75198.98.1521851607383 7 100010.011.86.430.61 0.1170 141.4 50 5016526.8138.26.44 300300 522803.7163.2E+181.1318.64449.68.6447781446775 8 100010.014.96.490.81 0.1165 138.3 50 5016024.9135.14.85 300300 450921.6084.6E+180.9119.737910.39.1382711306168 8 100010.018.66.541.06 0.1160 135.2 50 5015522.8132.24.23 3000 300 343751.1795.2E+180.7926.526914.612.7276741256166 9 100010.023.76.542.99 0.1155 132.2 50 5015020.5129.53.42 3000300 279740.6437.1E+180.6730.221416.514.1218651135461 9 100010.031.36.613.30 0.1150 129.3 50 5014518.3126.72.682151 300 236810.3119.8E+180.5632.617917.714.6181571024655 10 100010.040.96.71 3.400.1145 126.4 50 5014016.4123.62.041229 300 206940.1351.4E+190.4634.115618.314.815650934050 10 100010.051.46.81 3.400.1140 123.4 50 pp g( )p b [ ] PAGE 214 214Table D2. Optimized devices operating on a volt age source with a Gaussi an dopant profile and Nb = 1e15 [#/cm3]. BandwidthPmaxMDPDyn RangeH1H2H3aa/hD*Naz j arain t wtrtinwarclarcwtapltapw g apRaVbSensNoise FLkPowerPmaxactD. Rg. ActAct. BW[kHz][dB SPL][dB SPL][dB SPL] [ m] [A][A] [ m] [][mNmm] [1/cm 3 ][ m] [deg] [ m] [deg][deg] [ m][ m][ m][ m][ m][ m][ ] [V] [ V /Pa] [nV][][W][dB SPL][dB SPL][kHz]15017032.5137.52.85 300300 200690.3351.3E+190.5736.217312.28.5145109272255 9 166412.910.48.711.04 0.1170 137.7 150 15016530.9134.12.15 300300 172780.1471.7E+190.4737.414713.69.212396252049 10 154412.412.38.661.35 0.1165 134.2 150 15016029.4130.61.821754 300 132650.1032.1E+190.4143.810718.412.18982251943 10 138211.814.48.522.99 0.1160 130.6150 15015527.7127.31.482111 300 105610.0612.9E+190.35 45.0 8321.813.77065211735 10 116110.817.28.343.38 0.1155 127.2 150 15015026.0124.01.141280 300 91700.0273.9E+190.30 45.0 7223.414.06156191530 10 104110.220.78.29 3.400.1150 123.8 150 15014525.4119.61.00977 300 85760.0184.4E+190.27 45.0 6523.213.76051201425 10 9119.522.08.14 3.400.1 147122.1 150 15014025.4114.61.00977 300 85760.0184.4E+190.27 45.0 6523.113.66051201425 10 9129.621.98.15 3.400.1 147122.1 150 14017032.1137.93.05 300300 215690.4101.1E+190.6035.618611.98.5156115292359 9 171613.110.88.721.01 0.1170 138.0 140 14016530.6134.42.30 300300 185780.1801.6E+190.4936.915813.19.0133102272152 9 161712.712.98.721.31 0.1165 134.6 140 14016029.0131.01.972097 300 139630.1311.9E+190.4243.911318.212.29487262045 10 144412.015.28.563.06 0.1160 131.0 140 14015527.2127.81.592315 300 112600.0752.6E+190.36 45.0 8921.313.77470231837 10 122411.118.38.383.39 0.1155 127.6 140 14015025.5124.51.231368 300 97700.0343.5E+190.31 45.0 7722.713.96560201633 10 11 0310.522.08.33 3.400.1150 124.4 140 14014524.4120.61.00903 300 88790.0184.3E+190.27 45.0 6923.514.06054191528 10 99110.024.88.25 3.400.1 146121.7 140 14014024.5115.61.00903 300 88790.0184.3E+190.27 45.0 6923.413.96054191528 10 99210.024.78.26 3.400.1 146121.7 140 13017031.8138.23.29 300300 231690.5101.1E+190.6235.320111.38.2169124302462 9 182213.511.48.830.97 0.1170 138.4 130 13016530.2134.82.48 300300 199780.2231.4E+190.5136.517112.68.9144109282255 9 169713.013.68.781.26 0.1165 134.9 130 13016028.6131.42.132365 300 149620.1661.7E+190.4443.712117.812.210193282148 10 151812.316.18.623.08 0.1160 131.4 130 13015526.7128.31.722552 300 120600.0952.3E+190.38 45.0 9520.713.67975241940 10 129611.419.58.44 3.400.1155 128.1 130 13015025.0125.01.321471 300 105700.0423.1E+190.32 45.0 8322.013.97065221735 10 117310.823.58.38 3.400.1150 124.9 130 13014523.5121.51.01845 300 92820.0184.2E+190.27 45.0 7223.514.16157201530 10 105910.327.88.34 3.400.1145 121.4 130 13014023.5116.51.00831 300 91820.0184.2E+190.27 45.0 7223.514.16156191530 10 105410.327.98.34 3.400.1 145121.2 130 12017031.4138.63.56 300300 251690.6469.3E+180.6434.821810.98.0184133322667 9 191113.812.08.900.94 0.1170 138.8 120 12016529.7135.32.69 300300 216790.2821.2E+190.5335.918612.18.7157117302459 9 178613.414.48.851.22 0.1165 135.4 120 12016028.1131.92.322677 300 161610.2141.5E+190.4643.513117.312.210999302352 10 160012.717.08.683.11 0.1160 131.8 120 12015526.2128.81.862761 300 130600.1202.0E+190.40 45.0 10320.113.58681262043 10 138111.820.88.50 3.400.1155 128.7 120 12015024.5125.51.441590 300 113700.0532.7E+190.34 45.0 9021.313.87571231838 10 125411.225.38.44 3.400.1150 125.4 120 12014522.9122.11.10913 300 99810.0233.6E+190.29 45.0 7822.714.16661211633 10 113410.730.08.40 3.400.1145 121.9 120 12014022.5117.51.00760 300 95860.0184.0E+190.27 45.0 7523.114.16359201631 10 109510.531.58.38 3.400.1 143120.7 120 11017031.0139.03.89 300300 274690.8358.1E+180.6734.323910.47.9202143352872 9 201114.212.78.970.90 0.1170 139.2 110 11016529.3135.72.93 300300 236790.3641.1E+190.5635.420411.58.5172126322564 9 188513.715.38.931.17 0.1165 135.8 110 11016027.6132.42.53 3000300 175610.2801.3E+190.4843.114216.812.1118107322556 10 169513.018.18.753.13 0.1160 132.3 110 11015525.8129.22.002346 300 146650.1402.0E+190.39 45.0 11918.112.49893282149 10 153612.422.08.583.29 0.1155 129.1110 11015023.9126.11.571731 300 124700.0692.3E+190.36 45.0 9820.613.78277252041 10 134611.627.28.51 3.400.1150 125.9 110 11014522.3122.71.20994 300 108810.0303.2E+190.30 45.0 8521.914.07267231836 10 12 2011.032.48.46 3.400.1145 122.6 110 11014021.4118.61.00691 300 99900.0183.9E+190.27 45.0 7822.814.26661211633 10 114010.735.98.43 3.400.1 142120.2 110 10017030.5139.54.28 300300 301691.1077.0E+180.7133.72649.97.6224155373078 9 212514.613.59.070.86 0.1170 139.6 100 10016528.8136.23.22 300300 260790.4819.5E+180.5934.722511.08.3191136352869 9 199814.116.39.011.12 0.1165 136.2 100 10016027.2132.82.44 300300 223890.2131.3E+190.4936.019112.39.0162120322561 9 186813.719.58.961.45 0.1160 132.9 100 10015525.2129.82.23 3000300 157620.1981.5E+190.44 45.0 12618.413.110599312453 10 161512.724.08.703.36 0.1155 129.6 100 10015023.3126.71.731899 300 136700.0912.0E+190.38 45.0 10819.813.69085282145 10 145312.129.48.59 3.400.1150 126.5 100 10014521.7123.31.321090 300 119810.0402.7E+190.32 45.0 9421.113.97974251940 10 132311.535.38.55 3.400.1145 123.1 100 10014020.2119.81.00624 300 104950.0173.7E+190.26 45.0 8122.414.26964221734 10 119310.941.38.48 3.400.1140 119.6 100 9017030.1139.94.76 300300 335701.5136.0E+180.7533.02949.47.4250170413285 9 2250 15.0 14.39.160.82 0.1170 140.0 90 9016528.4136.63.58 300300 289790.6558.1E+180.6234.125110.48.0214149373075 9 212614.617.49.111.07 0.1165 136.8 90 9016026.7133.32.71 300300 248900.2891.1E+190.5235.321411.68.7181132352867 9 199514.121.09.061.39 0. 1160 133.4 90 9015524.6130.42.46 3000300 177640.2581.3E+190.4644.314317.512.8119110342758 10 175613.325.98.813.31 0.1155 130.2 90 9015022.7127.31.932106 300 151700.1241.7E+190.40 45.0 12019.013.510094312450 10 157912.632.08.70 3.400.1150 127.1 90 9014521.0124.01.471208 300 132810.0542.3E+190.34 45.0 10420.213.88882272144 10 143812.038.58.63 3.400.1145 123.8 90 9014019.5120.51.12692 300 115950.0243.1E+190.28 45.0 9121.514.17771241938 10 130411.445.48.57 3.400.1140 120.4 90 8017029.6140.45.35 300300 377702.1455.0E+180.8131.63328.97.3286183453691 8 2250 15.0 14.99.020.77 0.1170 140.5 80 8016527.8137.24.03 300300 325800.9266.7E+180.6633.22849.87.8243164413383 9 2250 15.0 18.69.181.01 0.1165 137.3 80 8016026.1133.93.05 300300 280900.4079.0E+180.5534.524210.98.4206146383074 9 214114.622.69.171.32 0.1160 134.0 80 8015524.5130.52.38529 300 235950.2021.3E+190.4436.320112.29.1170128342765 9 206814.427.89.292.09 0.1155 130.780 8015022.0128.02.172364 300 169690.1761.4E+190.42 45.0 13518.213.3113106342656 10 173013.235.08.82 3.400.1150 127.8 80 8014520.3124.71.661356 300 148810.0771.9E+190.36 45.0 11819.313.69992302349 10 158012.642.48.74 3.400.1145 124.5 80 8014018.7121.31.26776 300 129940.0342.6E+190.30 45.0 10220.514.08680272143 10 14 3712.050.48.67 3.400.1140 121.1 80 7017029.1140.96.12 300300 431703.1904.0E+180.8930.03808.57.1332199514199 8 2250 15.0 15.68.870.72 0.1170 141.1 70 7016527.3137.74.60 300300 372801.3735.5E+180.7331.53259.37.6282179463790 8 2250 15.0 19.59.020.95 0.1165 137.9 70 7016025.5134.53.48 300300 320910.6007.4E+180.5933.227810.38.2239161423481 9 2250 15.0 24.39.191.24 0.1160 134.6 70 7015523.4131.63.11 3000300 236690.4928.3E+180.5241.819315.412.0161140433475 9 209714.530.69.093.19 0.1155 131.4 70 7015021.3128.72.492695 300 193690.2621.1E+190.4544.415517.413.1129120383064 10 189513.838.68.95 3.400.1150 128.5 70 7014519.5125.51.901545 300 169810.1151.5E+190.39 45.0 13518.413.4113106342656 10 175313.247.18.88 3.400.1145 125.3 70 7014017.9122.11.45884 300 148940.0502.1E+190.33 45.0 11719.513.89992312449 10 160012.656.38.80 3.400.1140 121.9 70 6017028.5141.57.14 300300 503705.0453.0E+181.0028.34448.17.03942195948109 8 2250 15.0 16.48.730.67 0.1170 141.6 60 6016526.7138.35.37 300300 434802.1654.3E+180.8129.63818.87.4336197544398 8 2250 15.0 20.58.850.89 0.1165 138.5 60 6016024.9135.14.05 300300 374910.9425.8E+180.6631.23259.67.9285177493989 8 2250 15.0 25.79.001.16 0.1160 135.2 60 60 15522.8132.23.58 3000300 280720.7336.5E+180.5639.823114.311.5195160493985 9 2250 15.0 33.29.143.10 0.1155 132.1 60 6015020.5129.52.90 3000300 227700.4058.7E+180.4943.318316.412.8153138443474 10 210014.542.89.113.38 0.1150 129.2 60 6014518.7126.32.231798 300 197810.1811.2E+190.4244.215717.413.2132122393165 10 194914.052.79.03 3.400.1145 126.1 60 6014017.0123.01.691028 300 172940.0791.6E+190.35 45.0 13618.413.6115107352757 10 180713.463.78.97 3.400.1140 122.8 60 5017027.9142.18.57 300300 604708.6832.3E+181.1226.45357.56.64832476956122 7 2250 15.0 17.28.580.61 0.1170 142.2 50 5016526.0139.06.44 300300 522803.7163.1E+180.9327.64588.27.24122216352110 8 2250 15.0 21.88.690.81 0.1165 139.1 50 5016024.2135.84.85 300300 450921.6084.4E+180.7529.03929.07.73501995847100 8 2250 15.0 27.38.811.06 0.1160 136.0 50 5015522.1132.94.23 3000300 343751.1794.9E+180.6436.728413.211.1246181594896 9 2250 15.0 35.08.902.99 0.1155 132.9 50 5015019.8130.23.42 3000300 279740.6436.4E+180.5440.822715.212.4193161524286 9 2250 15.0 46.49.103.30 0.1150 129.9 50 5014517.7127.32.682151 300 236810.3118.7E+180.4643.219016.413.0159143463677 10 219214.859.89.22 3.400.1145 127.0 50 5014016.0124.02.041229 300 206940.1351.2E+190.3943.916417.313.3139126423267 10 203614.372.79.13 3.400.1140 123.8 50 gg PAGE 215 215 APPENDIX E DETAILED SPECIFICATIONS OF DEVICE PACKAGE The f ollowing is the layout of the PCB package for testing in the PWT. Figure E1. Layout of PCB package. PAGE 216 216 Table E1. Passive co mponent specifications. Value Size Type LT1963 R 1 k 0805 Thin metal film C 10 F 0805 Ceramic Pot 10 k 4 mm sq Cermet AD625 R 20 k 0805 Thin metal film R 20 k 0805 Thin metal film Pot 100 4 mm sq Cermet Pot 10 4 mm sq Cermet C 10 F 0805 Ceramic C 10 F 0805 Ceramic AC filter R 147 k 0805 Thin metal film R 147 k 0805 Thin metal film C 0.68 F 0805 Ceramic C 0.68 F 0805 Ceramic Mic C 10 F 0805 Ceramic C 10 F 0805 Ceramic Power C 10 F Radial Electrolitic PAGE 217 217 APPENDIX F DETAILS OF EXPERIMENTAL SETU P AND UNCERTAINTY ANALYSIS Experimental Setup IV Measurements Setup The IV measurements are taken with an Agilent 4155C semiconductor parameter analyzer. The settings used are shown in Table F1. Table F1. Agilent 4155C se miconduc tor parameter analyzer settings. Rin and Rout Diode Voltage sweep10V to 10V 20V to 10V points 1001 501 V 20mV 60mV Speed Medium Long Noise Setup The noise signal from the microphone under te st is amplified by a SRS 560 low noise amplifier. The settings are shown in Table F2. The signal is an alyzed with a SRS 785 spectrum analyzer and the settings are shown in Table F3. Table F2. SRS 560 a mplifier settings. BUF1 Test arc Gain 1000 1000 Filter 0.03 Hz 1MHz0.03 Hz 1MHz Coupling AC AC Setting Low Noise Low Noise Table F3. SRS 785 spectrum analyzer settings. Parameter Value Value Value Value Freq Range 12.5 Hz 200 1600 12800 Bin Width 15.625 mHz 250 mHz 2 Hz 16 Hz # Averages 50 120 2400 10000 Window Hanning Hanning Hanning Hanning PAGE 218 218 Acoustic Setup There are two setups for the acoustic setup. The first is for the linearity testing. The settings for the Pulse Mu ltianalyzer are found in Table F4. Do to limitations in the experimental setup, th e frequency response function was found piecewise and th e settings are in Table F5. Table F4. Pulse Multia nalyzer settings for linearity testing. FFT Settings Freq Range300 Hz 6.7 kHz # Bins 6400 Bin Width 1 Hz Overlap None # Averages 300 Window None Generator Settings Signal TypeSine wave Frequency 1kHz Amplitude Increasing Table F5. Pulse Multianalyzer settings for frequency response function testing. FFT Settings Start Freq [kHz] 0.3 1.1 1.9 2.7 3.5 4.3 5.1 5.9 End Freq [kHz] 1.1 1.9 2.7 3.5 4.3 5.1 5.9 6.7 # Bins 800 800 800 800 800 800 800 800 Bin Width [Hz] 1 1 1 1 1 1 1 1 Overlap None # Averages 300 300 300 300 300 300 300 300 Window None Generator Settings Signal Type Periodic Random Start Freq 0.3 1.1 1.9 2.7 3.5 4.3 5.1 5.9 End Freq 1.1 1.9 2.7 3.5 4.3 5.1 5.9 6.7 # Bins 800 800 800 800 800 800 800 800 Bin Width [Hz] 1 1 1 1 1 1 1 1 Amplitude Max PAGE 219 219 Uncertainty Analysis Resistance Values Since the devices are batch fabricated to yield identical devices, the de vices tested for input and output resistance values which are used to de termine a mean and standard deviation. In addition the confidence intervals for the true mean and standard deviations are calculated from the following equations [126] ;/2 ;/2nn xst st xx NN (F1) and 22 2 22 ;/2;1/2x nnnsns (F2) where N is the number of samples, 1nN t is from the t distribution table, is from the distribution table, x is the sample mean and s is the sample standard deviation. Frequency Response Function The uncertainty in the frequency response f unction is comprised of random error in the measurement as well as bias error due to the anal og to digital conversion. For this analysis, it is assumed that there is no error in the refere nce microphone used in the measurements. The normalized random error for the magnitude frequency response function is calculated from the following equation [126] 1/2 2 .. 1 2xy xy rxy xyd xysdH f H n H (F3) The standard deviation for the phase in radians is calculated using [126] 1/2 21 .. 2xy xy xydf sd n (F4) PAGE 220 220 The 95% CI are then calculated by multiplying the standard deviation by 2 since the number of samples is greater then 31. In analog to digital conversion, the magnitude of each data value must be put into a discrete digital bin. The bias er ror associated with this is defined as of the bin size. For each experiment it is important to note the scale fo r the signal input. For example with the pulse analyzer, if the signal input was scaled to 707.1m V, the range of discretation is from 707.1mV to 707.1mV for a range of 1.414V. This syst em has 16 bits so the bin with is 21.57V. Therefore the bias error is of the bin width or 10.79V. To determine the bias error in the frequency response function this voltage error is divided by the known incident pressure from the reference microphone. Figure 618 includes the bias error and it is shown be sm all relative to the random error at lower sound pressure levels. Hooge Parameter The uncertainty of the Hooge parameter is propagated from the uncertainty in the measurement, the uncertainty in the least squares fit b from equation (69) and the uncertainty in the tota l number of carriers N. The error in the noise measurement is assumed negligible due to the large number of averages in the data (2400). The uncertainty is then calculated from the other two sources. 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Ozisik, Heat conduction New York, NY: Wiley, 1993. [153] K. Atkinson, 1993, Second ed. New York: John Wiley & Sons, Inc., 1985. PAGE 234 234 BIOGRAPHICAL SKETCH Brian grew up in Stanhope, NJ, a small town an hour west of New York City, with his parents, L eo and Helen Homeijer, and brother Dan. After graduating from Lenape Valley Regional High School in 1999, he attended Lehigh Un iversity in Bethehem, Pennsylvania. After obtaining a degree in mechanical engineering in 2003, he was accepted into the mechanical engineering graduate program at the University of Florida. During his tenure at UF, he has worked with Dr. Mark Sheplak on the design of MEMS transducers. Upon graduation, Brian will begin work as a research and development engineer in the Imaging and Printing Division of Hewlett Packard in Corvallis, Oregon. 