design, fabrication and characterization of a mems
TRANSCRIPT
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DESIGN, FABRICATION AND CHARACTERIZATION OF A MEMS PIEZORESISTIVE MICROPHONE FOR USE IN AEROACOUSTIC MEASUREMENTS
By
BRIAN HOMEIJER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
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© 2008 Brian Homeijer
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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.
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ACKNOWLEDGMENTS
Financial support for this work has been provided by the Boeing Company, monitored by
Jim Underbrink. I appreciate the opportunity to work with the people at the Boeing
Aero/Noise/Propulsion/Structural Dynamics Laboratory. The lessons learned have been
invaluable. This project was also funded in part by the Florida Center for Advanced Aero
Propulsion.
I would like to thank Dr. Mark Sheplak, my supervisory committee chair, for all of his help
and guidance, he greatly helped with my research and exploring several different opportunities,
both at UF and at Hewlett Packard. I would like to also acknowledge 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 been 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 brother, Dan Homeijer. I also thank my friends, especially Marie
and Kevin Kane, and Heather and Chad Macuszonok. Without poker nights, homebrew and
Orlando weekends, I don’t know that I could have lasted this long.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS ...............................................................................................................4
LIST OF TABLES...........................................................................................................................8
LIST OF FIGURES .......................................................................................................................10
ABSTRACT...................................................................................................................................15
CHAPTER
1 INTRODUCTION ..................................................................................................................17
1.1 Noise Restrictions for Commercial Airplanes ..............................................................17 1.2 Aeroacoustic Microphones ...........................................................................................18
1.2.1 Pressure, Free and Diffuse Field Microphones .................................................19 1.2.2 Linearity and Total Harmonic Distortion..........................................................21 1.2.3 Noise Floor........................................................................................................21 1.2.4 Requirements for an Aeroacoustic Microphone ...............................................22
1.3 Objectives......................................................................................................................24 1.4 Dissertation Outline ......................................................................................................24
2 BACKGROUND ....................................................................................................................31
2.1 Transduction Schemes ..................................................................................................31 2.1.1 Piezoelectric Transduction ................................................................................31 2.1.2 Piezoresistive Transduction ..............................................................................32 2.1.3 Capacitive Transduction....................................................................................33 2.1.4 Optical Transduction.........................................................................................34
2.2 Chosen Transduction Scheme.......................................................................................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 Transducers ..........................................................................................38
3 TRANSDUCER MODELING AND DESIGN......................................................................47
3.1 Composite Plate Mechanics ..........................................................................................48 3.1.1 Derivation of Governing Equations ..................................................................48 3.1.2 Equilibrium Equations ......................................................................................49 3.1.3 Constitutive Relationship ..................................................................................50 3.1.4 Nonlinear Solution ............................................................................................51 3.1.5 Deviation from Linearity ..................................................................................53 3.1.6 Calculation of Stresses ......................................................................................54
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3.1.7 Validation Using Finite Element 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 Cut-on frequency and cavity stiffening ..............................................71
3.3.2 FEA verification................................................................................................71 3.4 Electronic Noise ............................................................................................................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 Optimization with Constant Voltage.................................................................92 4.2.2 Optimization with a Constant Current Source ..................................................94 4.2.3 Constraining Devices to a Single Wafer ...........................................................94 4.2.4 Sensitivity 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 Assembled Package.........................................................................................114
6 RESULTS AND DISCUSSION...........................................................................................124
6.1 Device Characterization..............................................................................................124 6.1.1 Electrical Characterization ..............................................................................124 6.1.2 Acoustic Characterization ...............................................................................126
6.2 Experimental Results ..................................................................................................128 6.2.1 Electrical Characterization ..............................................................................128 6.2.2 Noise Floor......................................................................................................131 6.2.3 Linearity and Total Harmonic Distortion........................................................131
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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 Work............................................................................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
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LIST OF TABLES
Table page 1-1. Audio and aeroacoustic microphone specifications...........................................................25
1-2. Commercial microphones used in aeroacoustic testing specifications ..............................25
2-1. Transduction schemes and desired characteristics.............................................................40
2-2. Piezoelectric microphone specifications in the literature. .................................................40
2-3. Piezoresistive microphone specifications in the literature.................................................41
2-4. Capacitive microphone specifications in the literature......................................................42
2-5. Optical microphone specifications in the literature ...........................................................44
3-1. Material parameters and thicknesses used for FEA analysis. ............................................74
3-2. Conjugate power variables for various energy domains....................................................74
3-3. Lumped element modeling parameter estimates................................................................75
3-4. Results from FEA analysis compared to analytical results................................................75
4-1. Upper and lower bounds for all variables ..........................................................................99
4-2. Values for devices chosen for fabrication for constant voltage biasing. ...........................99
4-3. Values for devices A, B, and C in constant current mode (4mA). ....................................99
4-4. Values for devices A, B, and C in constant current mode (10mA). ..................................99
4-5. Single wafer constrained voltage source devices.............................................................100
4-6. Single wafer constrained current source (10mA) devices. ..............................................100
4-7. Standard deviation of input parameters. ..........................................................................101
4-8. Mean and 95% confidence intervals for design parameters. ...........................................101
4-9. Statistical properties of desired parameters. ....................................................................101
6-1. Resistance values of all tested devices.............................................................................135
6-2. Resistance values for the four test resistors. ....................................................................135
6-3. Values for VDP and line width test structures.................................................................135
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6-4. Values of VDP and line width test structures for the metal lines. ...................................136
6-5. Values of Kelvin test structures. ......................................................................................136
6-6. Thickness of oxide layer using two techniques. ..............................................................136
6-7. Curve fit parameters for test taper resistor for device A..................................................136
6-8. Curve fit parameters for a BUF1-A device......................................................................136
6-9. MDP for tested devices....................................................................................................136
6-10. Methods used to determine the fabricated values for all parameters of the devices........137
6-11. Results from the radius determination experiment. .........................................................137
6-12. Measured values and standard deviation of input parameters. ........................................137
6-13. Confidence intervals for the realized parameters for a BUF1-A device..........................137
6-14. Statistical data for model validation PDFs.......................................................................137
D-1. Optimized devices operating on a current source with a Gaussian dopant profile. .........213
D-2. Optimized devices operating on a voltage source with a Gaussian dopant profile .........214
E- 1. Passive component specifications....................................................................................216
F-1. Agilent 4155C semiconductor parameter analyzer settings.............................................217
F-2. SRS 560 amplifier settings...............................................................................................217
F-3. SRS 785 spectrum analyzer settings. ...............................................................................217
F-4. Pulse Multianalyzer settings for linearity testing.............................................................218
F-5. Pulse Multianalyzer settings for frequency response function testing.............................218
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LIST OF FIGURES
Figure page 1-1. Number of noise restrictions at airports.............................................................................25
1-2. Perceived noise levels of various aircraft. .........................................................................26
1-3. Federal Aviation Administration part FAR 36 measurement locations.............................26
1-4. Noise sources of a typical commercial airplane. ...............................................................27
1-5. Schematic cross-section of a general microphone structure. .............................................27
1-6. Magnitude and phase of a typical aeroacoustic microphone frequency response. ............28
1-7. Example of over damping a free field microphone to increase bandwidth. ......................28
1-8. Pressure field microphone flush mounted in an enclosure. ...............................................29
1-9. Traveling acoustic waves in various fields. .......................................................................29
1-10. Deviation of linear and nonlinear solutions. ......................................................................30
1-11. Noise power spectral density for a typical microphone.....................................................30
2-1. Outline of the different transduction schemes of MEMS microphones. ...........................44
2-2. Illustration of the piezoelectric effect. ...............................................................................45
2-3. Example of the piezoresistive effect. .................................................................................46
2-4. Variable capacitor schematic. ............................................................................................46
3-1. Overview of the microphone modeling process. ...............................................................76
3-2. Schematic of composite plate. ...........................................................................................76
3-3. Kirchoff's hypothesis showing the neutral axis and transverse normal. ............................77
3-4. Non-dimensional center deflection per unit pressure of devices with varying in-plane forces..................................................................................................................................77
3-5. Pressure that results in a 5% deviation from linearity for various inplane forces. ............78
3-6. Analytical deflection of clamped plate, at the onset of non-linearity (2000 Pa), compared to FEA results....................................................................................................78
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3-7. Center deflection per non-dimensional pressure as a function of P* for various values of in-plane stresses.............................................................................................................79
3-8. Description of the Euler’s angles.......................................................................................79
3-9. Crystallographic dependence of the piezoresistive coefficients for p-type silicon............80
3-10. Piezoresistive factor dependence on doping concentration at room temperature..............81
3-11. Geometry of piezoresistors. ...............................................................................................81
3-12. Differential elements of the arc and taper resistor. ............................................................82
3-13. Sample Gaussian dopant profile. .......................................................................................82
3-14. Stressed arc and taper resistors configured in a Wheatstone bridge. .................................83
3-15. Schematic of MEMS microphone and associated lumped elements. ................................83
3-16. Example of a the distributed system and the lumped equivlent.........................................84
3-17. Equivalent circuit model of the microphone......................................................................84
3-18. Accuracy of first terms of cotangent expansion. ...............................................................85
3-19. Magnitude and phase response of LEM normalized by the flat band response.................85
3-20. Equivalent circuit illustrating the effect of the cavity compliance ....................................86
4-1. Operational parameter space for a microphone. ..............................................................101
4-2. Multiobjective optimization Pareto front illustrating the trade-off between minimizing function J1 and J2. .........................................................................................102
4-3. Ideal linear output of a microphone or pressure transducer.............................................102
4-4. Features that are constrained to be larger than wline. ........................................................103
4-5. MDP vs. Bandwidth for various Pmax constraints. ...........................................................103
4-6. MDP vs. Pmax for various bandwidth constraints.............................................................104
4-7. MDP vs. Bandwidth of various Pmax constraints for a constant current source device. ..104
4-8. MDP vs. Bandwidth of various Pmax constraints for a constant current source device. ..105
4-9. Sensitivity analysis for constant current source varying 4mA by 3%. ............................105
4-10. Sensitivity analysis for constant current source varying 10mA by 10%. ........................106
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4-11. MDP dependence on Hooge parameter for device A. .....................................................106
4-12. Dependence of MDP with respect to each variable. ........................................................107
4-13. Dependence of MDP on silicon thickness overlaid with compression coefficient..........107
4-14. Monte Carlo simulation schematic ..................................................................................108
4-15. Uncertainty of MDP of Device A. ...................................................................................108
4-16. Uncertainty of Pmax for device A. ....................................................................................109
4-17. Uncertainty of the bandwidth for device A......................................................................110
4-18. 95% probability yield limit illustration............................................................................110
5-1. Front side process steps. ..................................................................................................115
5-2. First four masks for microphone fabrication ...................................................................116
5-3. Last three masks for front side fabrication ......................................................................117
5-4. Back side process steps....................................................................................................118
5-5. Backside masks for microphone fabrication....................................................................118
5-6. Array of microphone die after dicing. Each die is 2mm x 2mm. ....................................119
5-7. Individual type A microphone die after dicing. ...............................................................119
5-8. Type B device pictured on a dime. ..................................................................................120
5-9. Backside cavity and vent of an individual type A microphone die after dicing. .............120
5-10. Packaging for acoustical characterization........................................................................121
5-11. Interface circuitry showing power supply, ac filter and amplifier...................................121
5-12. Printed circuit board for mounting the microphone and its associated components. ......122
5-13. Assembled device on PCB. Device is protected under a TO can. ..................................122
5-14. Populated PCB board inserted into PWT endplate. .........................................................123
6-1. Circuit representation of reversed biased p+ doped resistors in an n substrate. ..............138
6-2. Van der Pawl test structure schematic. ............................................................................138
6-3. Line width test structure schematic..................................................................................139
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6-4. Kelvin test structure schematic. .......................................................................................139
6-5. Experimental setup for noise measurements....................................................................140
6-6. Experimental setup for acoustic characterization. ...........................................................140
6-7. Boron concentration in silicon device layer determined by SIMS, the accompanying curve fit and the desired model profile. ...........................................................................141
6-8. Input I-V curve of 12 BUF1-A and 12 BUF1-B devices.................................................141
6-9. Output I-V curve of 12 BUF1-A and 12 BUF1-B devices. .............................................142
6-10. Input I-V curve of a BUF1-A device with a linear curve fit............................................142
6-11. Output I-V curve of a BUF1-A device with a linear curve fit .........................................143
6-12. I-V curve of diode characteristics of a BUF1-A device. .................................................143
6-13. I-V curve of diode characteristics of a BUF1-A device focusing on the reverse region. ..............................................................................................................................144
6-14. Noise PSD of a test taper resistor. ...................................................................................144
6-15. 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..........................145
6-16. Noise power spectral density of a BUF1-A device..........................................................145
6-17. Noise PSD minus the setup noise and the associated model curve fit. The horizontal line is the thermal noise floor for the device....................................................................146
6-18. Sensitivity of BUF1-A devices normalized by the bias voltage. .....................................146
6-19. Total harmonic distortion of BUF1-A microphones........................................................147
6-20. Magnitude frequency response for a BUF1-A device. Vertical dotted lines mark the piecewise FRFs that were stitched together.....................................................................147
6-21. Magnitude FRF for each device with 95% CI bounds.....................................................148
6-22. Phase response for each device tested. ............................................................................149
6-23. Phase FRF for each device with 95% CI bounds.............................................................150
6-24. Coherence function between device A-5 and the reference microphone. .......................151
6-25. Microphone photograph with and without backlighting..................................................151
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6-26. Minimum detectable pressure probability density function.............................................152
6-27. Sensitivity probability density function. ..........................................................................152
6-28. Voltage noise probability density function. .....................................................................153
E-1. Layout of PCB package. ..................................................................................................215
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
DESIGN, FABRICATION, AND CHARACTERIZATION 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 commercial airplanes is needed.
To lower aircraft noise, 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 robust
microphone that is able to withstand large sound pressure levels on the order of 160 dB SPL,
while possessing an operating bandwidth on the order of 100 kHz and a low noise floor at or
below 26 dB SPL. This work attempts to address the needs of aircraft manufacturers with the
design, fabrication and characterization of a microelectromechanical systems piezoresistive
microphone for use in aeroacoustic measurements.
This microphone design addresses many of the problems associated with previous
piezoresistive microphones such as limited dynamic 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 the diaphragm, which was designed to be in the
compressive quasi-buckled state. With this model, the effects of residual in-plane stresses that
result from the microelectronic fabrication process on the sensitivity of the device are predicted.
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Ion implanted doped silicon was chosen for the 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 characteristics of the microphone diaphragm and the
integrated cavity/vent structure. The device geometry was optimized using a sequential
quadratic programming scheme. Results predict 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 device revealed two major problems with the
piezoresistors. The diffusion of the resistors was too long and resulted with the resistor thickness
being the entire thickness of the diaphragm. The result of this error dropped the sensitivity two
orders of magnitude. In addition to the doping profile error, 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. These two factors couple together 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 resistor fabrication, the models developed in this dissertation showed
that they correctly represent the realized device and therefore will be sufficient to design a
second generation microphone.
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CHAPTER 1 INTRODUCTION
With air traffic expected to increase dramatically in the next decade and urban sprawl
encroaching on airports, the Federal Aviation Administration (FAA) has taken steps to regulate
aircraft noise. For certification, the 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 engine operating conditions[1]. To lower the
noise radiating from aircraft, manufacturers perform extensive wind tunnel 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 commercial 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 world 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 and levels has increased 10 fold, (Figure 1-1) [1]. Curfews
designate quiet time around airports during the night. If aircraft are louder then the curfew limit
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then they can only land in the daytime. Currently, 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 requirements, aircraft manufactures have reduced
noise signatures an order of magnitude from the original jet aircraft of the 1970’s (Figure 1-2).
To determine an aircraft’s noise signature, the FAA takes noise samples at every airport in at
least three locations each of which are illustrated in Figure 1-3. The largest improvement to date
has been in the reduction of noise in the lateral and take-off areas [2]. This improvement is
primarily the result of noise reduction in turbofans, 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 requires 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 1-4).
Aircraft manufacturers are focusing on reducing 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 addition researchers are looking
into toboggan fairings to reduce 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 form to another; microphones are transducers that
specifically convert acoustical energy to electrical energy or modulate the electrical energy due
to the acoustic energy. This energy conversion is achieved in different ways, however, one thing
they all have in common is that they first convert acoustical energy to mechanical energy via a
diaphragm. A diaphragm is a thin structure, pictured in Figure 1-5, that vibrates when sound
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waves strike it. A mechanical to electrical transduction scheme then determines how the
mechanical energy is converted into a readable electrical signal.
In addition to the diaphragm and the transduction mechanism, the other 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 1-5). To ensure that microphone output is representative of the acoustic
input, the sensitivity of the microphone must not change with frequency, and, ideally, the
microphone should have no phase shift. Figure 1-6 illustrates a normal frequency response for an
under-damped aeroacoustic microphone. The bandwidth of the microphone is defined as the
range of frequencies where the microphone’s response magnitude is flat. The cut-on frequency
and resonant frequency are also shown in Figure 1-6. The vent channel dominates the low-
frequency response of the microphone, while the impedance of the vent relative to the diaphragm
dictates whether incident pressure will flow through the vent or deflect the diaphragm. The
damping and the resonant frequency of the microphone 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 under-damped system, as shown in Figure 1-6, while no peak is found in an over-damped
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 three types: free, diffuse and pressure field, determined by
their response in an acoustic field. A free field occurs when sound waves can propagate freely
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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 1-9(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 1-9(B). Pressure rises in front of the microphone due to the scattering off of the
microphone, resulting in a higher output level at certain frequencies. The maximum effect is
when the wavelength of a specific frequency is equal to the diameter of the microphone. Figure
1-9(C). Free field microphones take this effect into account by compensating for their own
disturbing presence. The damping of these microphones is increased so that the spectral shape of
the microphone response is opposite to the spectral shape due to scattering, shown in Figure 1-7.
To work correctly a free field microphone must be normally incident to the noise source.
A diffuse field exists if the field is created by sound waves arriving from all locations
simultaneously with equal probability [4]. The diffuse field microphone is designed to respond
uniformly to signals arriving simultaneously from all angles. These devices are also over-
damped like free field microphones.
A pressure field is found in enclosures which are small compared to the wavelengths of
interest [4]. A pressure field microphone should be used in this situation and should be flush
mounted on the enclosure as seen in Figure 1-8. Because the pressure field microphone 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 factor 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
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website. In addition, a nose cone corrector can be mounted onto a free field microphone to
reduce the effect of the angle of incidence.
1.2.2 Linearity and Total Harmonic Distortion
The linearity of the microphone 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 1-10 is a linear (ideal) and non-linear (real) response of a generic microphone to
a single-tone, 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 non-linearity limit the effective maximum pressure. As seen in Figure 1-10,
the microphones actual dynamic response diverges from the linear response above a maximum
pressure, such as mechanical, electro-mechanical, and amplifier non-linearities. The maximum
pressure at which a microphone is considered linear is defined at the point where a 3% difference
between the linear and nonlinear response of the microphone is detected. The non-linearity of the
microphone is given in terms of the total harmonic distortion (THD) with respect to frequency
because the non-linear 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
22
1
nn
pTHD
p
ω
ω
∞
==∑
. (1-1)
1.2.3 Noise Floor
The lower end of the dynamic range, called the minimum detectable signal (MDS) is
limited by the microphone noise as well as noise contributed 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
22
density (PSD) and the total noise power is based on the PSD integrated over the bandwidth of
interest [6]. A typical noise PSD of a piezoresistive microphone is shown in Figure 1-11. When
in thermodynamic equilibrium, the thermal noise of the system is proportional to the dissipation
of the system [6], This is also known as white noise because the PSD is constant over all
frequencies. Under non-equilibrium conditions, flicker noise arises which has an inverse
proportionality to frequency. It occurs in interface electronics and semiconductive materials, and
usually dominates at lower frequencies. The frequency 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 non-flat spectral shape 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 bandwidth, or integrated over a specified bandwidth. A-weighted
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 performance of audio microphones is tuned to the human ear; an
instrument grade aeroacoustic microphone, however, has different requirements, shown in Table
1-1. The human ear has a minimum detectable pressure of 20 Paμ at 2kHz, also known as the
threshold of hearing. The maximum pressure that an ear can be exposed to, known as the
threshold of pain, is 20Pa . The bandwidth of the human ear is from 20Hz to 20kHz [9].
Aeroacoustic microphones must be useable in areas where the sound pressure level (SPL)
to be measured is very high, like near an aircraft jet engine, where sound pressure levels may
reach 170dB . In addition, the FAA requires certification over the frequency range of
23
45 11.2Hz f kHz≤ ≤ for full scale vehicles [1]. Aeroacoustic testing is frequently done using
1 8 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
360 89.6Hz f kHz≤ ≤ [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
20 120Hz f kHz≤ ≤ , specified by the sponsor of the project, The Boeing Company.
Currently, several commercial microphones are used by the aeroacoustic community. The
specifications of some of theses microphones can be seen in Table 1-2. The B&K microphones
are typically used in arrays for sound localization. The Kulite MIC-093 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 dB SPL for a 1 Hz bin
centered at 1 kHz .
Acoustic arrays, which consist of many microphones 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 requires a large number of
microphones, typically in the 100’s [11], which makes MEMS microphones particularly
appealing due to the possible advantages of batch 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 arrays, the microphones must be physically arranged
within one half wavelength of each other to avoid spatial aliasing. However, this is not a
requirement for logarithmically spaced spiral arrays [11]. Phase matching between microphones
24
is another necessity for acoustic arrays because beam forming algorithms use phase information
to localize sound sources. A mismatch between microphone channels can cause error in the
sound localization.
1.3 Objectives
The goal of this project is the design, fabrication, and characterization of a MEMS
piezoresistive microphone for aeroacoustic testing applications. 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 microphone’s 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 was separated into the design, fabrication and
characterization of the microphone and in addition the development of a fabrication 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 summarizes the optimization scheme used to decide
device dimensions. Chapter 5 outlines the process flow and device fabrication, Chapter 6 details
the characterization and model validation and Chapter 7 concludes with a summary of this work
and recommendations for future devices.
25
Table 1-1. Audio and aeroacoustic microphone specifications. Audio Aeroacoustic Max Pressure 120dB 170dB Noise Floor 23-35dBA 26 dBα Bandwidth 20 Hz - 20 kHz 20 Hz - 100 kHzα 1Hz bin centered at 1kHz
Table 1-2. Commercial microphones used in aeroacoustic testing specifications [12], [13].
Kulite MIC-093 B&K 4939 B&K 4138 Desired Boeing Specifications
Diameter 2.4 mm 6.35 mm 3.18mm Max Pressure 194 dBA 164 dB SPL 168 dB SPL 150- 160 dB SPL Noise Floor 100 dBA 5 dB SPLα 18 dB SPLα <26 dB SPLα Dynamic Range 94 dBA 159 dB SPL 150 dB SPL 124 – 134 dB SPLBandwidth ~90 kHz 4 Hz - 100 kHz 6.5 Hz - 140 kHz 100 – 120 kHz α 1Hz bin centered at 1kHz
01965 1970 1975 1980 1985 1990 1995 2000
Year
0
50
100
150
200
250
300
350
400
Figure 1-1. Number of noise restrictions at airports [1].
26
B-52
707-100
DC8-20
CV990ACV880-22
BAC-11
DC9-10DC8-61 737-100
737-200
727-100727-200
747-100
747-200DC10-10
L-1011
A300B2MD-80 747-300
747-400
737-300
A320-100A321
A330A340MD-11
777
A310-300
BAe 146-200
DC10-30
Comet 4
720
Year of initial service
Noiselevel, EPNdB(1,500 ftsideline)
1950 1960 1970 1980 1990 2000 2010 202080
90
100
110
120
Turbojet and early turbofans
First generation turbofan
Second generation turbofan
707-300B
Figure 1-2. Perceived noise levels of various aircraft [2].
Approach
Lateral
Take-off
Cutback~1,000 ft(305 m) 1,476 ft
(450 m)
6,565 ft
(2,000 m)
21,325 ft
(6,500 m)
Approach in noisiest configurationLanding gear extended, full flaps
Takeoff with maximum takeoff thrust rating
Measurement Location
Figure 1-3. FAA part FAR 36 Measurement Locations [1].
27
Figure 1-4. Noise sources of a typical commercial airplane.
Diaphragm
Cavity
Vent
Figure 1-5. Schematic cross-section of a general microphone structure.
28
10-1 100 101 102 103 104 105 106
-20
0
20
40
Mag
nitu
de [d
B]
(ref r
espo
nse
@ 1
kHz)
10-1 100 101 102 103 104 105 106
-150
-100
-50
0
50
Frequency [Hz]
Pha
se [d
eg]
Figure 1-6. Magnitude and phase of a typical aeroacoustic microphone frequency response.
Mag
. Fre
quen
cy
Res
pons
e
Figure 1-7. Example of over damping a free field microphone to increase bandwidth.
29
Figure 1-8. Pressure field microphone flush mounted in an enclosure.
A B
C
20log M
FF
PP
Dλ1
0dB
10dB
Figure 1-9. A) Traveling acoustic waves in a free-field. B) Effect of placing a microphone in the field. C) Increased pressure sensed by the microphone due to its own presence
[4].
30
Linear Solution
Nonlinear Solution
Incident Pressure
Cen
ter D
efle
ctio
n
Figure 1-10. Deviation of linear and nonlinear solutions.
10-1 100 101 102 103 104 105
10-16
10-15
10-14
10-13
10-12
Frequency [Hz]
Noi
se P
ower
Spe
ctra
l Den
sity
Sv [V
2 /Hz]
Figure 1-11. Noise power spectral density for a typical microphone.
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 capacitive. These transduction mechanisms are
described in the following sections. The optical scheme is broken down into intensity
modulating, polarization modulating and phase modulating. The capacitive scheme is separated
into electret and condenser. Figure 2-1 shows the various classifications of MEMS microphones.
2.1.1 Piezoelectric Transduction
Piezoelectricity is defined as the ability of some materials to generate an electric potential
in response to applied mechanical stress [14]. The Curie brothers were the first to discover that
surface charges developed on some crystals, namely crystals with a noncentrosymmetric crystal
structure, when compressed, and that the magnitude of these charges was proportional to the
applied pressure. Hankel later named this phenomenon “piezoelectricity,” and it is historically
referred to as the direct piezoelectric effect.
In addition, strain is also produced 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 relating strain and charge in
a piezoelectric material. Figure 2-2 illustrates how a piezoelectric material expands, or contracts,
when an electrical potential is placed across it.
32
2.1.2 Piezoresistive Transduction
Certain materials go through a fundamental electronic change in resistivity due to applied
stress that exceeds the resistance change that all resistors experience due to stress-induced
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 was first observed by Smith in 1954 for silicon and
germanium [17]. The resistance of any given material is [18]
LRA
ρ= . (2-1)
where , , and L Aρ are the resistivity, length and cross sectional area of the resistor, respectively
(Figure 2-3). When a metal is stressed, the resistance changes due to geometric effects, however,
when a semiconductor material such as silicon is stressed, the resistivity of the material changes
as well [17]. The change in resistance is given by
dR dL dA dR L A
ρρ
= − + . (2-2)
The geometric effects can be arranged in the following manner
( )1 2dR dR
ρν ερ
= + + . (2-3)
For conductive materials such as metals where 0d ρ ρ ≈ , the maximum change in resistance is
limited to 2 times the strain if 0.5ν = . In a piezoresistive material the change in resistance is
given by
ijijkl kl
ij
Tρ
ρΔ
= ∏ , (2-4)
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 compared to a standard metal strain gauge.
2.1.3 Capacitive Transduction
The capacitive transduction scheme relies on the measurement of a change in capacitance
between two electrically charged surfaces. A parallel plate capacitor is discussed here for
simplicity. The capacitance between two parallel plates is given as,
0 ACg
ε= , (2-5)
where A is the surface area, 0ε 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 2-4 for this case is [22]
( )2
2eQF Q
Aε= , (2-6)
where Q is the charge. Condenser microphones are biased with an external voltage source. The
electrical force for this case is [22]
( ) ( )2
2,2e
AVF V xg x
ε= . (2-7)
Electret devices are not susceptible to electrostatic pull-in, however it is difficult to fabricate
devices with a stable embedded charge.
34
2.1.4 Optical Transduction
Optical microphones tend to be more complex because the mechanical energy of the
diaphragm is sensed optically first and then converted into electrical energy. One benefit of
sensing optically is that the transducer is insensitive to electromagnetic interference [23]. One
drawback is that optoelectronics, 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 light-modulating acoustic sensors rather than direct converters
of sound energy to light. They modulate light in three ways: intensity, phase and polarization
modulation [25]. Cook et al. developed the first optical transducer 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 amount of light that is sensed by the receiver
changes as the diaphragm deflects. 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 reflection off nonmetallic surfaces. The
degree of polarization depends on the incident angle and the material that is reflecting the light
[25]. The light is then passed through a polarizing 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 scheme has the best attributes including:
• The ability to withstand harsh environments • The capability of being packaged with a thin profile • Compatibility with integrated circuit fabrication processes
35
Table 2-1 shows the desired traits and which transduction schemes meet each requirement.
2.3 Literature Review
This literature review discusses the important steps that progressed each microphone
transduction scheme. Select non-MEMS 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 Royer et al. developed the
first MEMS piezoelectric microphone using zinc oxide as the piezoelectric material [28].
Between 1989 and 1991 Kim et al. developed a piezoelectric MEMS microphone using a silicon
nitride diaphragm and a zinc oxide piezoelectric material [29], [30]. Theses devices, however,
never achieved a flat frequency response. In 1992, Schellin et al. developed a new type of
microphone [31]. This work used polyurea for the piezoelectric material 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 Kim’s microphone by tweaking the
fabrication process to control stress in the diaphragm, achieving a flat frequency response [32].
In 2003, Ko et al. developed a device with zinc oxide as the piezoelectric material 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 Kim’s design (1989) by using parylene-D for
the diaphragm, which increased the sensitivity [34]. In 2003 Zhao et al. utilized the piezoelectric
material, lead zirconate titanate (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,
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 aeroacoustic applications. Table 2-2 summarizes the realized
performance for each of the devices.
2.3.2 Piezoresistive Transducers
The first piezoresistive microphone was developed by Burns in 1957 for use in the Bell
type 500 telephone [38]. This microphone was deemed too expensive to fabricate compared to
the standard carbon microphones and was never mass produced. Fourteen years later in 1961,
Samaun et al. [39] developed the first MEMS piezoresistive 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 p-
type 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 structure; however, the cavity was only 3 mμ deep and contributed to a
low sensitivity. In 1995, Schellin et al. developed the first microphone to use ion implanted
piezoresistors [42]. Specifically, they were p-type resistors in an n-well silicon diaphragm. In
1998, Sheplak et al. developed a silicon nitride MEMS microphone for aeroacoustic
measurements [43]. This device had a circular diaphragm and was the thinnest diaphragm
(1500A ) reported in the literature. The piezoresistors 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
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 diaphragm however the dynamic range 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. developed an audio MEMS microphone with integrated
electronics and the lowest noise floor reported in the literature of 34dB [47]. The device’s
maximum pressure was not reported. Table 2-3 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 first capacitive microphone [49].
It was an electret microphone and had a rectangular diaphragm with the longer side being 8 mm
long. In 1989 Hohm et al. developed the first 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 rectangular 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 1mm 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 applications. Three years later in
2003, Scheeper et al. developed the first capacitive microphone tailored for use in aeroacoustic
38
measurements [54]. His device used a non-traditional 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 2-4 shows the reported specifications for each of the discussed devices.
Currently the Knowles Acoustics Sisonic microphone is available for purchase. The
specification sheet reports a noise floor of 39 dBA 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 commercially available MEMS microphone [56].
2.3.4 Optical Transducers
In 1991, Garthe et al. developed the first optical microphone [57]. This device used an
intensity modulating scheme and had an integrated waveguide chip fabricated 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 it’s poor performance. In 1994, Chan et al.
developed the first silicon optical pressure sensor [59]. This device had a 10mm square
diaphragm and used a phase modulation scheme. Five 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 circular pressure sensor [61].
It was a phase modulating device and had a 135 mμ 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 applications but was plagued by an inherently high
noise floor of 70dB. In 2005, Bucaro et al. developed a MEMS intensity modulating microphone
with an improved noise floor of 30.6dB [63], significantly lower then Kadirvel’s device;
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 bench [64]. The device was fabricated using
Sandia’s SwIFT-Lite process but it only had a bandwidth 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 Hall et al. reported on a smaller microphone design
with a 24dBa noise floor [66]. Table 2-5 shows the reported specifications for each of the
discussed devices.
40
Table 2-1. Transduction schemes 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 2-2. Piezoelectric microphone specifications in the literature. Author Diaphragm Cavity Sensitivity Dynamic Bandwidth Piezoelectric Microphone/Year Dimensions Depth Range (Predicted) Material Pres. Trans.
J. Medill 1 1/8”α N/R N/R N/R ~1kHz Rochelle Microphone
[27] salt crystals
M. Royer et. al. 1.5mmα x 30μm N/R 250μV/Pa 73dBA-N/R 10 Hz - 10 kHz ZnO Microphone
[28] (0.1 Hz - 10 kHz) 1st fabricatedE. S. Kim et al. 2mmβ x 1.4μm 380μm 80μV/Pa N/R 3 kHz - 30 kHz ZnO Microphone
[29], [67] E. S. Kim et. al. 3.04mmβ x 2.0μm 380μm 1000μV/Pa 50 dBA-N/R 200 Hz - 16 kHz ZnO Microphone
[30] R. Schellin et al. 0.8mmβ x 1.0μm 280μm 4000 μV/Pa N/R 100 Hz - 20 kHz Polyurea Microphone
[40], [68] R. P. Ried et al. 2.5mmβ x 3.5μm ~500μm 920μV/Pa 57 dBA-N/R 100 Hz - 18 kHz ZnO Microphone
[32] S. S. Lee et al. 2mmγ x 4.5μm N/R 3800 μV/Pa N/R 100 Hz - 890 Hz ZnO Microphone
[69], [70] S. C. Ko et al. 3mmβ x 3.0μm N/R 30μV/Pa N/R 1 kHz - 7.3 kHz ZnO Microphone
[33] M. N. Niu et al. 3mmβ x 3.2μm N/R 520μV/Pa N/R 100 Hz - 3 kHz ZnO Microphone
[34] H. J. Zhao et al. 0.6-1mmβ x N/R 370μm 38mV/Pa N/R N/R - 20kHz PZT Microphone
[35] J. Hillenbrand et al. ~0.5cm x 55μm N/R 2-0.5mV/Pa 37-26 dBA - N/R - ~10kHz Cellular Microphone
[36] N/R polypropylene Y. Yang et al. 200-500μmβ x N/R 61-474 N/R N/R - ~30kHz PZT Microphone
[71] ~1μm μV/Pa S. Horowitz et al. 900μmα x 3.0μm 500μm 0.75μV/Pa 47.8 dBδ - 100 Hz - 6.7 kHz PZT Microphone
[37] 169 dB (100 Hz - 50 kHz) α Radius of circular diaphragm β Length of rectangular diaphragm γ Length of cantilever δ 1 Hz bin
41
Table 2-3. Piezoresistive microphone specifications in the literature. Author Diaphragm Cavity Sensitivity Dynamic Bandwidth Microphone/ Year Dimensions Depth (Predicted) Range (Predicted) Pres. Trans.
F. P. Burns type 500 tele N/R 2.0uV/Pa*mA N/R 0 - 2570 Hz Microphone [38] 1.8 cmα (0 - 2590 Hz) 1st appl. of p.r.
S. Samaun et al. 1.2mmα x 5μm N/R 9.5μV/Pa N/R N/R Pres. Trans. [39] 1st SiN H2O bar.
W. H. Ko et al. 2.29mmβ x 20μm 254μm 75nV/Pa*V N/R - 40kPa 0 - 10 kHz Pres. Trans. [72]
R. Schellin et al. 1mmβ x 1μm N/R 4.2μV/Pa*V N/R 100 Hz - 5 kHz Microphone [31], [40]
E. Kalvesten et al. 100μmβ x 0.4μm 3μm 0.09μV/Pa*V 96dBA - N/R 10 Hz - 10 kHz Microphone [41], [73] (0.10μV/Pa*V) (2 mHz - 1 MHz)
E. Kalvesten et al. 300μmβ x 0.4μm 3μm 0.03μV/Pa*V 90dBA - N/R 10 Hz - 10 kHz Microphone [74] (0.02μV/Pa*V) (10 Hz - 0.9 MHz) Cav. Stiff.
R. Schellin et al. 1mmβ x 1.3μm N/R 10μV/Pa*V 61dBA - 128dBA 50 Hz - 20 kHz Microphone [42]
M. Sheplak et al. 105μmα x 0.15μm 10μm 2.24μV/Pa*V 92dBγ - 155dB 300 Hz - 6 kHz Microphone [43], [75] (100 Hz - 300 kHz)
A. Naguib et al. 510μmβ x 0.4μm N/R .18μV/Pa*V- N/R 1 kHz - 5.5 kHz Microphone [44] 1.0μV/Pa*V
A Naguib et al. 710μmβ x 0.4μm N/R 1.0μV/Pa*V N/R 1 kHz - 5.5 kHz Microphone [76]
D. P. Arnold et al. 500μmα x 1.0μm 10μm 0.6μV/Pa*V 52dBγ - 160dB 1 kHz - 20 kHz Microphone [45] (10 Hz - 100 kHz)
C. Huang et al. 710μmβ x 0.38μm ~20μm 1.1μV/Pa*V 54dBγ – 174dB 100 Hz - 10 kHz Microphone [46]
G. Li et al. N/Rβ x 1.0μm ~400μm 10μV/Pa*V 34dBγ - N/R 100 Hz - 8 kHz Microphone [47]
α Radius of circular diaphragm β Length of rectangular diaphragm γ 1 Hz bin
42
Table 2-4. Capacitive microphone specifications in the literature. Author Diaphragm Air Capacitance Bias Sensitivity Dynamic Bandwidth Condenser/ Year Dimensions Gap Voltage Range (Predicted) Electret
W. H Ko et al. 572μmα x 25μm N/R N/R N/R 1.28μV/Pa N/R N/R Condenser [48]
D. Hohm et al. 8.0mmβ x 13μm 20μm 9 pF 350 V 3mV/Pa N/R 100 Hz - Electret [49] 7.5 kHz
A. J. Sprenkels et al. 3.0mmβ x 2.5μm 20μm N/R 300 V 25mV/Pa N/R 100 Hz - Electret [77], [78] 15 kHz
P. Murphy et al. N/R x 1.5μm 25 - 95μm N/R 200 V 4-8mV/Pa N/R 100 Hz - Electret [79] 15 kHz
D. Hohm et al. 0.8mmβ x .25μm 2μm 6 pF 28 V 0.2mV/Pa N/R 200 Hz - Condenser [50] 4.3mV/Pa 20 kHz
J. Bergqvist et al. 2mmβ x 5μm 4μm 3.5 pF N/R 13mV/Pa N/R 500 Hz - Condenser [51] 2 kHz
J. Bergqvist et al. 2mmβ x 6μm 4μm 3.5 pF N/R 6.1mV/Pa N/R 100 Hz - Condenser [51] 5 kHz
J. Bergqvist et al. 2mmβ x 8μm 4μm 3.5 pF N/R 1.4mV/Pa N/R 500 Hz - Condenser [51] 20 kHz
J. Bergqvist et al. 2mmβ x 5.1μm 2μm 5 pF 5 V 1.8mV/Pa 37 dBA - 2 Hz - Condenser [80] 120dB 20 kHz
P. R. Scheeper et al. 2mmβ x 1μm 1μm 20 pF 2 V 1.4mV/Pa N/R 40 Hz - Condenser [81] N/R
P. R. Scheeper et al. 2mmβ x 1μm 3.3μm 5-7 pF 16 V 2mV/Pa 35 dBA - 100 Hz - Condenser [82], [83] N/R 10 kHz
W. Kuhnel et al. 0.8mmβ x .25μm 2μm 1 pF 28 V 1.8mV/Pa N/R 100 Hz - Condenser [84], [85] 20 kHz
T. Bourouina et al. 500μmβ x 1μm 5μm N/R N/R 0.4mV/Pa N/R N/R - Condenser [86] 20 kHz
T. Bourouina et al. 707μmβ x 1μm 5μm N/R N/R 2mV/Pa N/R N/R - Condenser [86] 7 kHz
T. Bourouina et al. 1mmβ x 1μm 5μm N/R N/R 3.5mV/Pa N/R N/R - Condenser [86] 2.5 kHz
T. Bourouina et al. 1mmβ x 1μm 7.5μm N/R N/R 2.4mV/Pa N/R N/R - Condenser [86] 10 kHz
E. Graf et al. N/R 0.46μm N/R 15 V 38mV/Pa N/R N/R - Condenser [87] 10 kHz
J. Bergqvist et al. 1.8mmβ x 8μm 3μm 5.4 pF 28 V 1.4mV/Pa 43 dBA - 300 Hz - Condenser [88] N/R 13 kHz
J. J. Bernstein et al. 1.8mmβ x N/R N/R N/R 5-10 V 16mV/Pa 25 dBA - 300 Hz - Condenser [89] 114 dB 15 kHz
J. J. Bernstein et al. 1mmβ x N/R N/R N/R 5-10 V 16mV/Pa 25 dBA - 70 Hz - Condenser [89] 114 dB 15 kHz
Q. B Zou et al. 1mmβ x 1.2μm 2.6μm 3.6 pF 10 V 14.2mV/Pa 39 dBA - 100 Hz - Condenser [90], [91] N/R 9 kHz
43
Table 2-4. Continued. Author Diaphragm Air Capacitance Bias Sensitivity Dynamic Bandwidth Condenser/ Year Dimensions Gap Voltage Range (Predicted) Electret
Y. B. Ning et al. 2mmβ x 0.5μm 3μm 9.1 pF 6 V 3mV/Pa N/R 100 Hz - Condenser [92] 10 kHz
B. T. Cunningham et al. 1mmα x 0.5μm 2μm 5.1 pF 8 V 2.1mV/Pa N/R 200 Hz - Condenser [52] 10 kHz 1st Circular Mic
M. Pedersen et al. 1.6mmβ x 0.9μm 1.5μm 14.9 pF 15 V 5.1mV/Pa 35 dBA - 100 Hz - Condenser [93] N/R 15 kHz
M. Pedersen et al. 2.1mmβ x 0.9μm 1.5μm 18.5 pF 15 V 8.1mV/Pa 34 dBA - 100 Hz - Condenser [93] N/R 15 kHz
M. Pedersen et al. 2.2mmβ x 1.1μm 3.6μm 10.1 pF N/A 234Hz/Paγ 60 dBA - 100 Hz - Condenser [94] 120 dB 15 kHz
P. C. Hsu et al. 2.6mmβ x 2μm 4μm 16.2 pF 10 V 20mV/Pa N/R 100 Hz - Condenser [95] 10 kHz
M. Pedersen et al. 2.2mmβ x 1.1μm 3.6μm 10.1 pF 14 V 10mV/Pa 27 dBA - 100 Hz - Condenser [96] 120 dB 8 kHz
D. Schafer et al. 0.4mmα x 0.75μm 4μm 0.2 pF 12 V 14mV/Pa 27 dBA - 150 Hz - Condenser [97] N/R 10 kHz
A. Torkkeli et al. 1mmβ x 0.8μm 1.3μm 11 pF 2 V 4mV/Pa 33.5 dBA - 10 Hz - Condenser [98] N/R 12 kHz
P. Rombach et al. 2mmβ x 0.49μm 0.9μm N/R 1.5 V 13mV/Pa 23 dBA - 10 Hz - Condenser [53], [99] 118 dB 20 kHz
X. X. Li et al. 1mmβ x 1.2μm 2.6μm 1.64 pF 5 V 9.4mV/Pa N/R 100 Hz - Condenser [100] 19 kHz
R. Kressmann et al. 1mmβ x 600nm 2μm N/R N/R 2.9mV/Pa 39 dBA - N/R - Electret [101] (Corrugated) 123 dB 20 kHz
P. R. Scheeper et al. 1.95mmα x 0.5μm 20μm 3.5 pF 200 V 22mV/Pa 23 dBA - 251 Hz - Condenser [54] 141 dB 20 kHz
J. J.Neumann et al. 320μmβ 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. (70μm x 190μm) 1μm 3.56 pF N/A 7.3mV/Pa 64 dBA - 0.1 Hz - Condenser [103] x 0.4μm N/R 100 kHz
D. T. Martin et al. 0.23mmα x 2.0μm 2μm 0.74 pF 9 V 0.28mV/Pa 22.5 dBδ - 300 Hz - Condenser [55], [104] 160 dB 20 kHz
Knolwes Electronics 1.5- 7.9mV/Pa 39dBA - 100Hz- Condenser Sisonic Mic [105] 5.5V 10kHz Analog Devices 1.5 - 14.1mV/Pa 32 dBA - 100Hz Condenser
[56] 3.6V 105 dB 12kHz α Radius of circular diaphragm β Length of rectangular diaphragm γ Frequency Modulation δ 1 Hz bin Note: Bias voltage for electrets are the effective voltage
44
Table 2-5. Optical microphone specifications in the literature Author Diaphragm Cavity Sensitivity Dynamic Bandwidth Optical Year Dimensions Depth (Predicted) Range (Predicted) Modulation
D. Garthe N/R 10μm N/R 42dBA-N/R 0-4.3kHz Intensity [57], [106] Non-MEMS
J. A. Dziuban et al. 10mmβ x 0.5mm N/R 2.4/-0.8 μV/Pa (On/Off switch) N/R Intensity [58] Silicon
M. A. Chan et al. 10mmβ x 7μm 50μm 3.75mPa/fringe N/R-164dB N/R Phase [59] First MEMS Psens
A. Kots et al. 1.5mmα x 1.8μm N/R N/R N/R 0 - 15kHz Intensity [60] Non-MEMS
D. C. Abeysinghe et al. 135μmα x 7μm 0.64μm 0.016μV/Pa 0-551kPa N/R Phase [61] Psens
K. Kadirvel et al. 1mmα x 1μm 500μm 0.5mV/Pa 70dBδ-132dB 0 - 6.4kHz Intensity [62] ( 0 -20kHz)
J. A. Bucaro et al. 1.6mm x 1.5μm 200μm N/R 30.6dBδ− N/R 0.1Hz - 10kHz Intensity [63]
N. A. Hall et al. 2.1mmβx 0.80μm N/R N/R (17.5dBA) N/R - 4kHz Intensity [64]
J. H. Song et al. 800μmβ x 5μm N/R N/R N/R 0.1 - 2kHz Intensity [65]
N. A. Hall et al. 1.5mm x 2.25μm N/R N/R 24dBA N/R - 20kHz Diffraction Based [66]
α Radius of circular diaphragm β Length of rectangular diaphragm δ 1 Hz bin
Figure 2-1. Outline of the different transduction schemes of MEMS microphones.
45
A
0V =
V
B
C
V
Figure 2-2. A) A piezoelectric material in equilibrium. B) The inverse piezoelectric effect: material with an applied bias, expanding. C) The inverse piezoelectric effect:
piezoelectric material with a reverse bias, contracting [107].
46
A
B
Figure 2-3. A) An unstressed resistor with a resistance of R. B) Stressed resistor with a resistance of R + ΔR.
Figure 2-4. Variable capacitor schematic.
47
CHAPTER 3 TRANSDUCER MODELING AND DESIGN
This chapter details the design process for a piezoresistive microphone tailored to acoustic
measurements. Figure 3-1 shows the methodology of the design. A piezoresistive microphone
has three main components: diaphragm, cavity and 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 layer 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 pressure 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 piezoresistors due to the stress determined by the
previous model. Both the electroacoustic transduction model and sensor mechanical model are
incorporated utilizing lumped element modeling (LEM) to determine the dynamics of the
multidomain system.
Design optimization, which is discussed in Chapter 4, incorporates the LEM results, design
specifications and manufacturing constraints to yield an ideal device. The mechanical model is
verified using FEA, and a cavity and vent structure is designed to accompany the diaphragm
determined during design optimization.
The mechanical model of the diaphragm and finite 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.
48
3.1 Composite Plate Mechanics
A brief review of previous research into plate mechanics includes the mechanical scaling
of capacitive and piezoresistive pressure transducers presented by Chau and Wise [108].
However they did not discuss the effects of in-plane forces on the stress field of the diaphragm.
Voorthuyzen and Bergveld [109] furthered the field by using a finite-difference model to
investigate the large-deflection characteristics of circular diaphragms in pressure sensors
subjected to in-plane loading over a limited dimensional domain. Sheplak and Dugundji
clarified the non-linear behavior of circular plates under tension by incorporating the structural
mechanics of thin-film diaphragms with large in-plane forces via a fundamental structural model
using von Kármán plate theory. This work shows that the deflection field is a strong function of
both in-plane loading and non-linear restoring forces [110]. The 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 analysis 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. This 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 brief 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 top layer is silicon nitride. The piezoresistors
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 3-2. The transverse
and radial deflections, ( ) ( )0 0 and w r u r , are defined at the reference plane.
The following three assumptions are stated by Kirchoff’s hypothesis and are used in the
following derivation of circular plate deflection under static loading (Figure 3-3) [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 remain perpendicular to the neutral surface after deformation.
In addition, the material is assumed to be transversely isotropic and the circular diaphragm
deflection is assumed to be axisymmetric
3.1.2 Equilibrium Equations
The equilibrium equations are derived in Appendix A and are repeated here for reference
[111]:
0rr N NdNdr r
θ−+ = , (3-1)
rr r
dMrQ M r Mdr θ= + − , (3-2)
and
( )0 0r rdwd dr N rp rQ
dr dr dr⎛ ⎞ + + =⎜ ⎟⎝ ⎠
. (3-3)
where , , , , and r r rN N M M Qθ θ are the in-plane forces, moments and shear in the plate,
respectively. The subscripts r and θ refer to the radial and tangential directions.
50
3.1.3 Constitutive Relationship
Assuming that silicon is transversely isotropic, the constitutive relationships are defined
as
[ ] [ ] [ ]0
0r r rrQ Q Q zθ θ θθ
σ ε κεσ ε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭, (3-4)
where [ ]Q is defined as
[ ] 11 122
21 22
111
Q Q EQQ Q
ννν
⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦
. (3-5)
The terms ε , 0ε and κ are the initial strain due to in-plane forces, elongation strain due to
transverse loading and curvature due to transverse loading, respectively. The forces per unit
length are found by integrating equation (3-4),
T
B
zr r
z
Ndz
Nθ θ
σσ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭∫ . (3-6)
Substituting equation (3-4) into equation (3-6) yields
[ ] [ ] [ ]T T T
B B B
z z zor r rr
oz z z
NQ dz Q dz Q z dz
Nθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭∫ ∫ ∫ . (3-7)
It is convenient to define the extensional stiffness matrix,
[ ] [ ]T
B
z
z
A Q dz= ∫ (3-8)
and the flexural extensional matrix
[ ] [ ]T
B
z
z
B Q zdz= ∫ . (3-9)
Equation (3-7) is compactly written as,
51
[ ] [ ] [ ]o
r r rro
NA A B
Nθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭. (3-10)
The moments per unit length are determined by integrating the stress times its moment arm, z,
over the thickness:
T
B
zr r
z
Mzdz
Mθ θ
σσ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭∫ . (3-11)
Substituting equation (3-4) into equation (3-11) yields
[ ] [ ] [ ] 2T T T
B B B
z z zor r rr
oz z z
MQ zdz Q zdz Q z dz
Mθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭∫ ∫ ∫ . (3-12)
It is now convenient to define the flexural stiffness matrix,
[ ] [ ] 2T
B
z
z
D Q z dz= ∫ . (3-13)
Equation (3-12) is compactly written as,
[ ] [ ] [ ]o
r r rro
MB B D
Mθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭. (3-14)
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 non-linear case, 0w cannot be isolated
from 0u and therefore two coupled nonlinear ODEs are derived, yielding,
2
*2 * * 22 2
1 12r
d d k P Sd d
ξξ ξ ξ ξ ξ
⎛ ⎞Θ Θ Λ+ − + Θ = − + Θ − Θ⎜ ⎟
⎝ ⎠ (3-15)
and
2 * * 2
2 22 23
2r rd S dS d d
d d d dξ ξ ξ
ξ ξ ξ ξ ξ⎛ ⎞Θ Θ Θ Χ
+ = −Λ + − − Θ⎜ ⎟⎝ ⎠
, (3-16)
52
where
*12
*
B hD
Λ = , (3-17)
2 2 2
11 12*
11
A A hA D−
Χ = , (3-18)
and
0 0
2 22 4* * * *0
* * * *
, , , ,
, , , and .2
rr
w ur dW aW Ua h h d h
N a N aN a paS S k PD D D hD
θθ
ξ φξ
= = = Θ = − =
= = = =
(3-19)
The , , and A B D 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 external forces to bending and transverse forces to stretching.
The *D value is a function of the composite matrices defined as
2
* 1111
11
BD DA
= − . (3-20)
Assuming a perfectly clamped plate, the boundary conditions for Θ and rS , , are
( )0 0ξΘ = = , (3-21)
( )1 0ξΘ = = , (3-22)
*
0
0rdSd ξξ
=
= , (3-23)
and
53
*
*121
11 11
1rr
dS A dSd A dξ
ξξξ ξ===
⎛ ⎞ Θ+ − = −Λ⎜ ⎟
⎝ ⎠. (3-24)
The symmetry coefficient ( )Λ is a measure of the symmetry of the composite plate. This
parameter takes into account that the reference plane is not necessarily the same as the neutral
plane. If the composite plate is symmetric about the reference plane then 0Λ = ; the more
asymmetric the plate becomes about the reference plane, the larger Λ becomes. The
approximate range for Λ is 0 0.04≤ Λ ≤ for the composite makeup considered here. The
composite coefficient ( )Χ captures the disparities between the different composite materials.
For a homogenous plate, ( )212 1 νΧ = − . For the composite makeup studied in this dissertation,
Χ deviated approximately 20% from the homogenous value. Equations (3-15) and (3-16) are
then solved using an iterative finite difference 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 ensures a constant sensitivity with respect to pressure, which is
essential to designing a microphone with low distortion. To ensure this, the composite plate
model is used to calculate the percent deviation from linearity of the diaphragm’s center
deflection. The result of a non-linearly deflecting diaphragm is harmonic distortion. A 5%
deviation from linearity is set as the limit for the maximum detectable pressure. Figure 3-4
shows the non-dimensional sensitivity of devices with varying in-plane forces. This figure also
shows the trade-off between having an increase in sensitivity versus an increase in the maximum
pressure to remain linear. Figure 3-5 shows the maximum pressure that can act on a plate with a
given in-plane load to remain linear.
54
As the in-plane 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 initial stress due to fabrication and stress due to
pressure loading,
initial stress stress dueto loading
r r r
θ θ θ
σ σ σσ σ σ
⎧ ⎫ ⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭, (3-25)
where
[ ] [ ]0
0r rrQ Q zθ θθ
σ κεσ κε
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭. (3-26)
The stress due to pressure loading is desired, therefore the non-dimensional radial and tangential
stresses due to loading are defined by
2r
r
SihEa
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
(3-27)
and
2
SihEa
θθ
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
. (3-28)
Solving for and r θΣ Σ yields
2
11r i i
i
dU U dd d
ν η νν ς ξ ξ ξ ξ
⎛ ⎞⎛ ⎞ ⎛ ⎞Π Θ ΘΣ = + + +⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠
(3-29)
and
55
2
11 i i
i
dU U dd dθ ν η ν
ν ς ξ ξ ξ ξ⎛ ⎞⎛ ⎞ ⎛ ⎞Π Θ Θ
Σ = + + +⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠, (3-30)
where
ah
ς = , (3-31)
zh
η = , (3-32)
and iν is the local Poisson’s ratio. Π is defined as
2
2
1 if
if
if
m Si
SiOm SiO
Si
SiNm SiN
Si
E EE
E EEE E EE
⎧⎪ =⎪⎪
Π = =⎨⎪⎪
=⎪⎩
, (3-33)
where mE is the Young’s 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 3-node quadratic, 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 elements are accurate 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 uniform 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.
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 defined in each layer. The geometric and
material values used can be found in Table 3-1. Figure 3-6 shows the deflection of a composite
plate calculated analytically from section 3.1 and compared to results obtained from the FEA
simulation. It is easily seen that there is excellent agreement between the two calculations.
Figure 3-7 illustrates the transition from linear to nonlinear behavior as the plate deflection
becomes large. Once again there is excellent agreement 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 defined 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 applied stress and piezoresistive coefficients ( )ijπ
[113]. For the cubic crystal structure of silicon, the relationship reduces to
1 11 12 12 1
2 12 11 12 2
3 12 12 11 3
23 44 23
13 44 13
12 44 12
0 0 00 0 00 0 01
0 0 0 0 00 0 0 0 00 0 0 0 0
ρ π π π σρ π π π σρ π π π σρ π τρρ π τρ π τ
Δ⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎢ ⎥ ⎪ ⎪Δ⎪ ⎪ ⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎢ ⎥ ⎪ ⎪Δ⎪ ⎪ ⎪ ⎪= ⎢ ⎥⎨ ⎬ ⎨ ⎬Δ ⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪Δ⎢ ⎥⎪ ⎪ ⎪ ⎪
Δ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎣ ⎦ ⎩ ⎭
, (3-34)
where ρΔ is the change in resistivity. The resulting electric field from stressed silicon is [113],
57
( ) ( )( ) ( )
( ) ( )
1 11 1 1 12 2 3 1 44 2 12 3 13
2 11 2 2 12 1 3 2 44 1 12 3 23
3 11 3 3 12 1 2 3 44 1 13 2 23
unstressed stressed
i i i i i
i i i i i
i i i i i
ρ ρπ σ ρπ σ σ ρπ τ τ
ρ ρπ σ ρπ σ σ ρπ τ τ
ρ ρπ σ ρπ σ σ ρπ τ τ
= + + + + +
= + + + + +
= + + + + +
1
2
3
E
E
E
. (3-35)
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 perpendicular to the current and the final term represents the effect
of shear on the electric field [113].
The piezoresistive coefficients are then transformed to an arbitrary axis using the
following transformation matrix [114],
*1 1 1
*2 2 2
*3 3 3
x l m n xy l m n yz l m n z
⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎣ ⎦ ⎩ ⎭⎩ ⎭
, (3-36)
where , , and i i il m n are the direction cosines which are given in terms of Euler’s angles [114],
1 1 1
2 2 2
3 3 3
l m n c c c s s s c c c s s cl m n c c s s c s c s c c s sl m n c s s s c
φ θ ψ φ ψ φ θ ψ φ ψ θ ψφ θ ψ φ ψ φ θ ψ φ ψ θ ψ
φ θ φ θ θ
− + −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − − + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
, (3-37)
where ( )coscθ θ= , etc. These angles are graphically illustrated in Figure 3-8 [19].
For this case, ( )100 silicon is used and the piezoresistors are implanted vertically into the wafer.
Therefore, 0θ = and 0ψ = (refer to Figure 3-8) and the matrix(3-37) reduces to
1 1 1
2 2 2
3 3 3
00
0 0 1
l m n c sl m n s cl m n
φ φφ φ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
, (3-38)
where φ varies from 0 to 2π . Applying a transformation of coordinates to equation (3-35),
using the matrix (3-38), the longitudinal and transverse piezoresistive coefficients are [19],
58
( )( )2 2 2 2 2 211 44 12 11 1 1 1 1 1 12 l m l n m nπ π π π π= + + − + +l (3-39)
and
( )( )2 2 2 2 2 212 44 12 11 1 2 1 2 1 2l l m m n nπ π π π π= − + − + +t . (3-40)
These coefficients are plotted for ( )100 p type silicon in Figure 3-9.
The piezoresistive coefficient also depends on temperature and doping level. . This
relationship is given as a product of the low-doped room temperature value and a piezoresistive
scaling factor ( ),P N T [19] shown as,
( ) ( ), ,N T P N TπΠ = , (3-41)
where N is doping concentration and T is temperature. Many theoretical [19] and experimental
[115-117] studies have shown that the piezoresistive factor ( ),P N T is a function of doping
concentration. Kanda’s model [19] accurately predicts the effect of doping concentration and
temperature for low concentrations. Kanda’s model also shows how at higher doping
concentrations, the effect of temperature on the piezoresistive coefficients is minimized. In fact,
at concentrations above 19 310 # 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]. Kanda’s model however, when compared to
experimental data [115-117], under predicts the decline of ( ),P N T for concentrations above
17 310 # cm . For doping concentration above 17 310 # cm , the experimentally fitted piezoresistive
factor ( ),P N T [118] is used,
( )0.201422 31.53 10,300 log cmP N K
N
−⎛ ⎞×⎜ ⎟⎝ ⎠
. (3-42)
59
The piezoresistive factor is plotted in Figure 3-10 versus concentration at room temperature.
The piezoresistors are designed to take advantage of the crystallographic dependence of p-
type silicon having almost equal and opposite piezoresistive coefficients (Figure 3-9). The
resistors are designed to isolate current either in the tangential direction or in the radial direction.
The geometry of the piezoresistors in the diaphragm can be seen in Figure 3-11. To calculate the
change in resistance, the piezoresistors are divided up into differential elements (Figure 3-12)
and their individual resistances are numerically integrated. For the arc resistor, the resistance of
an unstressed differential element is [18]
arcrddR
dzdrρ θ
= , (3-43)
and the resistance of a stressed element is given by
( )1arc arc l l t trddR d R
dzdrρ θ σ π σ π+ Δ = + + , (3-44)
where ρ is the resistivity of the material.
Summing up the unstressed differential resistors in series (equation (3-43) along the θ
direction) yields,
( ),
as arc
z rR
drdzρ θ
= . (3-45)
Summing up the differential elements in parallel yields,
0
1 jaout
ain
zr
arc ar
drdzR rρ θ
= ∫ ∫ . (3-46)
Integrating in the r direction yields,
0
ln1 j
aoutz
ain
a a
rr dz
R θ ρ
⎛ ⎞⎜ ⎟⎝ ⎠= ∫ . (3-47)
60
The resistivity is [18]
( )
1
n pq N Pρ
μ μ=
+, (3-48)
where q is the charge of an electron ( )1.6 19 q e C= − , nμ and pμ are the mobilities of electrons
and holes respectively, and N and P are the concentration of electrons and holes respectively.
The piezoresistors are doped heavily with boron which is a p-type implant ( )p n . Therefore
equation (3-48) reduces to
1 1
p p pq P q Nρ
μ μ= = , (3-49)
where pN is the doping concentration of boron [18]. The mobility is modeled after an empirical
fit [18],
0min
1p
p
ref
NN
α
μμ μ= +⎛ ⎞
+ ⎜ ⎟⎜ ⎟⎝ ⎠
, (3-50)
where min 0, , , and refNμ μ α are constants depending on the temperature and dopant [18].
Substituting in for ρ in equation (3-47) yields
( ) ( )( )
min 00 0
ln1
1
j j
aoutz z
painp
a a p
ref
rqN zr
N z dz dzR N z
N
αμ μθ
⎡ ⎤⎛ ⎞ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠= +⎢ ⎥⎛ ⎞⎢ ⎥
+ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
∫ ∫ . (3-51)
Taking the inverse of equation (3-51) results in
61
( ) ( )( )
min 00 0
1
ln
1
j j
aa z z
aout pp
ainp
ref
Rr N zq N z dz dzr N z
N
α
θ
μ μ=
⎛ ⎞+⎜ ⎟
⎝ ⎠ ⎛ ⎞+ ⎜ ⎟⎜ ⎟
⎝ ⎠
∫ ∫. (3-52)
The ion implantation process yields a Gaussian doping profile [119],
( )
2
: 0j
zz
sp s
b
NN z N zN
⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞= ≥⎜ ⎟
⎝ ⎠, (3-53)
where , z , and s j bN N are the surface concentration, junction depth and background
concentration, respectively. A sample dopant profile is shown in Figure 3-13. The fabricated
profile will differ slightly from a Gaussian profile. Performing the same procedure for the
stressed arc resistor (equation (3-44)) yields,
( ) ( ) ( ) ( ) ( )0
11 , ,
r
jaoutl
ain
a a zr
l l t tr
dR Rdzdr
r z r z r z
θ
θ
θ
ρ σ π θ σ π θ
+ Δ =
⎡ ⎤+ +⎣ ⎦
∫∫ ∫
. (3-54)
For the taper resistor, the resistance of an unstressed differential element is
2t
drdRrd dz
ρθ
= , (3-55)
and the resistance of a stressed element is given by
( )2 1t t l l t tdrdR d R
rd dzρ σ π σ πθ
+ Δ = + + . (3-56)
The factor of 2 follows from the fact that there are two legs per taper resistor. Performing the
same procedure for the unstressed taper resistor as for the unstressed arc resistor yields,
62
( ) ( )( )
min 00 0
2 ln1
1
j j
tout
tint z z
wt pp
p
ref
rr
Rq N z
N z dz dzN zN
α
θμ μ
⎛ ⎞⎜ ⎟⎝ ⎠=
+⎛ ⎞
+ ⎜ ⎟⎜ ⎟⎝ ⎠
∫ ∫, (3-57)
Similarly for the stressed taper resistor,
( ) ( ) ( ) ( ) ( )( )0
2
1 , ,
tout
jrtin
l
r
t t zr
l l t t
drR Rdzdr
z r z r z
θ
θ
θρ σ π θ σ π θ
+ Δ =
+ +
∫∫ ∫
. (3-58)
The MATLAB m-files for calculating 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 3-14. For a constant voltage bias ( )bV , the output of the bridge ( )0V yields,
0a a t t
ba a t t
R R R RV VR R R R
⎛ ⎞+ Δ − − Δ= ⎜ ⎟+ Δ + + Δ⎝ ⎠
. (3-59)
The arc and taper resistors are designed to have the same nominal resistance value
( )a tR R R= = . Applying this to equation (3-59) yields,
0 2a t
ba t
R RV VR R R
⎛ ⎞Δ − Δ= ⎜ ⎟+ Δ + Δ⎝ ⎠
. (3-60)
Knowing that the change in resistance is small compared to the mean resistance, the power
consumption for the circuit is
2
bVPR
≈ . (3-61)
If the device is biased with a constant current source ( )bI the response is,
63
0 2a a t t
bR R R RV I+ Δ − − Δ⎛ ⎞= ⎜ ⎟
⎝ ⎠. (3-62)
Again, assuming a balanced bridge, a tR R R= = yields,
0 2a t
bR RV IΔ − Δ⎛ ⎞= ⎜ ⎟
⎝ ⎠. (3-63)
Equation (3-63) reveals that when the device operates with an ideal constant current source
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
2bP I R≈ (3-64)
For a device operating with either a voltage or current supply, a power limitation would be
implemented to keep the overall power consumption below 100mW .
3.3 Lumped Element Modeling
The most accurate and complete way to mathematically describe a physical system is a
physics-based model that is correlated to an analytical 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 mimic 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 understanding of the scaling laws of the system [120-122]. The use
of LEM reduces the complexity of a numeric or analytic expression by dividing a given
distributed system into discrete elements that are based on system interactions with energy [121].
64
Three different types of interactions are accounted for: the storage of kinetic and potential
energy, and the dissipation of energy.
The storage of kinetic and potential energy in a distributed dynamic system requires a
partial differential equation to accurately represent the physics of the problem, because the
spatial and temporal components are 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 method assumes that the static mode shape is similar
to the dynamic mode shape up to the first resonant frequency.
Although nomenclature varies in different energy domains, the mathematics remain
constant. In lumped mechanical systems, kinetic energy is stored via mass while potential
energy is represented as the compliance of a spring and the dissipation of energy is represented
as the losses of a damper. In electrical systems, kinetic energy is represented as the magnetic
field of an inductor, while potential energy is represented as the charge across a capacitor, and
the dissipation of energy is represented as a resistor. 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 analytical, have been developed to solve large
networks of interconnected elements. In these techniques, the interconnected elements are
represented using electrical circuit notation. In lumped element modeling the elements are
denoted using an equivalent circuit form for all of the energy domains, where masses are
represented as inductors, compliances are represented as capacitors, and the dissipative
65
components are represented as resistors. Once the complete equivalent circuit is constructed,
standard circuit analysis techniques can be applied 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 3-2 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 mechanical components: diaphragm, cavity and vent
and can be seen in Figure 3-15. The following sections discuss the impedances 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 plated in order to find the lumped element representation for the
diaphragm. The plate is lumped to a piston of mass, ( )daM , a resistance of ( )daR and a
compliance, ( )daC , shown in Figure 3-16. The area of the piston and the area of the diaphragm
are not equal; rather the piston area is equated to maintain volume 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],
daVolCp
Δ= . (3-65)
where the change in volume is given by
( )2
0 0
a
Vol w r rdrdπ
θΔ = ∫ ∫ . (3-66)
Plugging in the non-dimensional parameters from equation (3-19) yields,
66
( )16
* *0
da
WaC dD P
ξπ ξ ξ= ∫ . (3-67)
The acoustic mass is determined by evaluating the kinetic energy expressed in acoustic conjugate
power variables to the total kinetic energy. This mass is calculated as [125],
( ) 2
0
2a
da A
w rM rdr
Volρ π
⎛ ⎞= ⎜ ⎟Δ⎝ ⎠
∫ , (3-68)
where Aρ is the areal density of the composite plate. Substituting equation (3-66) into (3-68)
and substituting for the non-dimensional variables results with,
( ) 22 15
* *02
Ada
da
WaM dD C P
ξπρ ξ ξ⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
∫ . (3-69)
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],
2
83
airradM
aρπ
= , (3-70)
where, airρ is the density of air. The radiation mass is added to the diaphragm mass to give a
combined diaphragm mass. Assuming the diaphragm is lightly damped, the resonant frequency
of the diaphragm is
1 12dia
da da
fC Mπ
= . (3-71)
The resistance, ( )daR , is caused by damping in the diaphragm. The majority of the damping
contributions are the dissipation to the supports, the dissipation into the surrounding air and
thermomechanical dissipation within the structure. This term is difficult to analytically express
67
accurately and therefore a value of the damping ratio ( )0.03ζ = is taken from previously
fabricated devices with similar size and aspect ratio [37]. The diaphragm resistance is then
calculated[126] as
2 dada
da
MRC
ζ= . (3-72)
The total impedance from Figure 3-16C is
( ) ( ) 1dia da rad da
da
Z j M M Rj C
ω ωω
= + + + . (3-73)
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. The cavity is therefore modeled as a closed cavity
with a sound hard boundary. The specific acoustic impedance in the cavity is [127],
( ) ( )0, cotZ d jZ kdω = − , (3-74)
where d is the distance from the bottom wall. For a short cavity, the cotangent function can be
expanded to yield
( ) 020
1lim , ...3kd
Z kdZ d ja kd
ωπ→
⎛ ⎞= − − +⎜ ⎟⎝ ⎠
. (3-75)
For 0.3kd ≤ all but the first term may be neglected, yielding
( ) ( ) 1,a
Z d Zj C
ω ωω
= = , (3-76)
where
20 0
aVCcρ
= , (3-77)
and V is the volume of the cavity,
68
2V a dπ= , (3-78)
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 (3-75) needs to be accounted for. This term represents an acoustic mass,
023a
dMa
ρπ
= , (3-79)
with the total impedance being,
( ) 1cav a
a
Z j Mj C
ω ωω
= + . (3-80).
A comparison of the full cotangent solution and the leading terms is shown in Figure 3-18.
3.3.1.3 Vent
The vent channel is designed to have a large resistive value with a small mass. To
accomplish this, the length must be as large as possible with a small cross-section. The lumped
acoustic mass for a laminar pipe flow is calculated by integrating equation (3-68) yielding [120],
2
43av
LMa
ρπ
= . (3-81)
Microfabrication processes are not capable of fabricating circular channels and therefore, the
vent channel has a square cross-section. Because of the limited space on a microphone die, the
vent channel is designed as a serpentine. This can be accommodated by calculating an effective
length of the channel by adding all of the major and minor head losses [128]. Equation (3-81) is
estimated using a hydraulic diameter and effective length yielding,
2
163
effav
H
LM
Dρ
π= , (3-82)
69
where HD is the hydraulic diameter and effL is the effective length. The hydraulic diameter is
defined as [128]
4 21H
A hD hP b= =
+, (3-83)
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 account any turns in the channel by equating the
channel with turns to a longer channel 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 HL L nD= + , (3-84)
where L is the length of the vent and n is the number of right angle turns. To determine the
resistance of the vent, volume velocity is expressed as a function of pressure yielding,
4
8aQ V dA pL
πμ
= ⋅ = Δ∫ ∫ . (3-85)
where Q is volume velocity and pΔ is pressure drop. The resistance is then taken from equation
(3-85),
4
8av
LRaμ
π= . (3-86)
Adjusting the resistance for a serpentine square channel yields,
4
128 effav
H
LR
Dμ
π= , (3-87)
The total impedance is
vent av avZ R j Mω= + . (3-88)
70
3.3.1.4 Equivalent circuit
Analyzing Figure 3-15, pressure acts on the diaphragm and vent simultaneously due to the
front side vent. There is a pressure drop across the diaphragm and through the vent and the
resulting flows converge in the compressible fluid back cavity. This results with the diaphragm
and vent impedance being in parallel and the effective impedance is in series with the back
cavity. Using this analysis, the equivalent circuit for the lumped element model is seen in Figure
3-17
The acoustic sensitivity directly relates to the voltage output of the microphone, which is
defined as ( ) ( )cq j pω ω ω , assuming sinusoidal input [75]. Using Kirchhoff circuit theory
(Figure 3-17), a transfer function for the acoustic sensitivity of the multi-domain dynamic system
is
( )( )2
c venta
dia cav vent dia cav
q j ZSj p Z Z Z Z Z
ωω ω
−= =
+ +, (3-89)
The magnitude and phase response for aS is seen in Figure 3-19 and a table outlining the
lumped element parameters is seen in Table 3-3. The piezoresistive transduction scheme is
modeled as a dependant voltage source, DSV and is given by
( ) cDS me
da
qV Sj C
ωω
⎛ ⎞= ⎜ ⎟
⎝ ⎠, (3-90)
where meS is the quasi-static mechanical sensitivity of the device in [ ]V Pa [75]. The
remaining terms represent the pressure drop across the diaphragm [ ]Pa Therefore, the total
dependant source output is in V . For a device connected to an amplifier with a very large input
impedance, the output voltage of the device is equal to the dependant source voltage. In the limit
of zero frequency, the right side of the circuit possesses (Figure 3-17) infinite impedance,
71
therefore, the microphone does not respond to slow changes in pressure. As the frequency of
interest gradually increases, the microphone response will increase linearly with frequency [75].
After a corner frequency defined by,
12c
av da
fR Cπ
= , (3-91)
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 Cut-on frequency and cavity stiffening
To determine the cut-on frequency of the microphone and prevent cavity stiffening, a
cavity and vent structure was designed. Examining Figure 3-17, and assuming the frequency of
interest is above the cut-on frequency, the equivalent circuit reduces to Figure 3-20. The
compliance of the cavity is in series with the compliance 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 3-17, the vent structure needs to have a large resistance to lower the cut-on frequency. If
the resistance of the vent is too low, all of the volume flow will pass through the vent, resulting
in no response of the diaphragm as seen in equation (3-92).
ventc
vent cav
Zq qZ Z
=+
(3-92)
3.3.2 FEA verification
A modal analysis was performed using ABAQUS and compared to the lumped element
model, equation (3-71). This was done for three different size diaphragms designated A, B and
C. The results can be seen in Table 3-4 and all results match to within 1%. An axisymmetric
72
composite plate model meshed with seventy-four 3-node 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 piezoresistive 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 Wheatstone bridge. The two dominant types of noise
in resistors are thermal noise and flicker ( )1 f noise. Thermal noise is given by [130],
4tR b KV k T R f= Δ , (3-93)
where bk is Boltzmann’s constant, kT is the temperature in Kelvin, R is the nominal
resistance, and fΔ is the frequency range over which the noise is calculated. When an external
voltage is applied to imperfect electrical conductors with interfacial or bulk defects, an excess
noise above the thermal equilibrium noise floor is observed. This excess noise exhibits an
inverse frequency 1/ f dependence. The mechanism that generates electrical 1/ f noise is still
debated, however the model used in this work follows the mechanism described by Hooge [131].
Hooge’s 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/ fVS
Nfα
= , (3-94)
where α is the Hooge parameter (determined experimentally), N is the number of carriers in
the resistor, and V is the bias across the resistor. The voltage noise is then given by [131],
2
21/
1lnfR
V fV fNα ⎛ ⎞= ⎜ ⎟
⎝ ⎠, (3-95)
73
where 2f and 1f are the bounds of the frequency range of interest. Using equations (3-93) and
(3-95), the total noise in the Wheatstone bridge at the output is calculated to be
( ) 2 22 1
1
1 1 14 ln8N b K
arc tap
fV k T R f f V N N fα ⎛ ⎞ ⎛ ⎞= − + + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠. (3-96)
The Analog Devices AD624, which has a noise power spectral density of 4 /nV Hz 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
( ) ( ) ( )22 22 1 2 1
1
1 1 14 ln 4 98N b K H
arc tap
fV k T R f f V e f fN N fα ⎛ ⎞ ⎛ ⎞= − + + + − −⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠. (3-97)
For this device, the power spectral density of the noise was calculated for a 1 Hz bin centered at
( )1 kHz dB SPL .
3.5 Conclusions
All portions of the piezoresistive microphone are modeled in this chapter. 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 model determines the overall dynamics of the
microphone including the cavity and vent structure interactions. Given a set of design variables,
the expected performance of a microphone can be calculated. This includes the frequency
response function and onset of nonlinearity. Chapter 4 implements an optimization scheme that
utilizes all of these models to generate a superior device.
74
Table 3-1. Material parameters and thicknesses used for FEA analysis. Material Thickness σ0 E ν
[μm] [MPa] [MPa] [--] Si 2.0 0 150 0.27
SiO2 0.3 -300 70 0.17 SixNy 0.1 100 270 0.24
Table 3-2. Conjugate power variables for various energy domains. Energy Domain Effort Flow Mechanical translation Force, F Velocity, v Fixed-axis rotation Torque, τ Angular velocity, ω Acoustic Pressure, P Volumetric flow, Q Electric circuits Voltage, V Current, I Magnetic circuits MMF, M Flux rate, φ Incompressible fluid flow Pressure, P Volumetric flow, Q Thermal Temperature, T Entropy flow rate, S
75
Table 3-3. Lumped element modeling parameter estimates. Acoustic
impedanceDescription
daM Equivalent acoustic mass lumped as a rigid baffle. ( ) 22 15
* *02
Ada
da
WaM dD C P
ξπρ ξ ξ⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
∫
daC Volume displacement normalized by the pressure. ( )16
* *0
da
WaC dD P
ξπ ξ ξ= ∫
radM Approximating the diaphragm as a piston in an infinite baffle.
2
83
airradM
aρπ
=
Diaphragm [16], [75]
daR 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
0 0aCC
cρ∀
= , where ∀ is volume of the cavity. Cavity [127]
aM 023a
dMa
ρπ
= , where d is the depth of the cavity.
avR Assuming fully developed pressure driven flow.
4
128 effav
H
LR
Dμ
π= , where effL is the effective length and
HD is the hydraulic diameter.
Vent [128], [132]
avM Also assuming fully developed pressure driven flow.
2
163
effav
H
LM
Dρ
π=
Table 3-4. 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%
76
Figure 3-1. Overview of the microphone modeling process.
Figure 3-2. Schematic of composite plate.
77
Figure 3-3. Kirchoff's hypothesis showing the neutral axis and transverse normal.
10-1 100 101 102 103 104 105 106
10-4
10-3
10-2
10-1
100
Non-dimensional pressure (P*)
Cen
ter d
efle
ctio
n pe
r pre
ssur
e (W
0( ξ =
0)/P
)
Figure 3-4. Non-dimensional center deflection per unit pressure of devices with varying in-plane forces.
78
-20 -10 0 10 20 30 40 5010-2
10-1
100
101
102
In-plane force parameter (k*2)
Max
imum
pre
ssur
e to
rem
ain
linea
r (P
* max
)
Figure 3-5. Pressure that results 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) [μm]
Tran
sver
se d
efle
ctio
n (w
0) [ μ
m]
FEAAnalytical
Figure 3-6. Analytical deflection of clamped plate, at the onset of non-linearity (2000 Pa), compared to FEA results.
79
10-1
100
101
102
103
104
105
106
10-4
10-3
10-2
10-1
100
Non-dimensional pressure (P*)
Cen
ter d
efle
ctio
n pe
r pre
ssur
e (W
0(ξ
= 0)
/P)
AnalyticalFEA
Figure 3-7. Center deflection per non-dimensional pressure as a function of *P for various values of in-plane stresses.
φ
φ
θ
θ
ψ
z
x
y
*y
*z
*x
Figure 3-8. Description of the Euler’s angles [19].
80
2e-010
4e-010
6e-010
8e-010
30
210
60
240
90
270
120
300
150
330
180 0
tπ
lπ
110 110
l tπ π= −
Figure 3-9. Crystallographic dependence of the piezoresistive coefficients for p-type silicon [ ]1 Pa [19].
81
Figure 3-10. Piezoresistive factor dependence on doping concentration at room temperature . Blue line corresponds to Kanda’s work [19] and the pink line corresponds to the
work of Harley et al. [118]
a
,t inr
,t outr
,a outr,a inr
aθ
wtθ
tθ
Figure 3-11. Geometry of piezoresistors.
82
Taper Resistor
Arc Resistor
Figure 3-12. Differential elements of the arc and taper resistor.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.161015
1016
1017
1018
1019
1020
Depth into substrate (z) [μm]
Car
rier c
once
ntra
tion
(Np)
[#/c
m3 ]
Figure 3-13. Sample Gaussian dopant profile.
83
t tR R+ Δ a aR R+ Δ
0V
,b bV I
t tR R+ Δa aR R+ Δ
Figure 3-14. Stressed arc and taper resistors configured in a Wheatstone bridge.
( )tpradM
,a aC M
,av avR M
, ,da da daM C R
Figure 3-15. Schematic of MEMS microphone and associated lumped elements.
84
A
p
Area, AeffMda
Cda
B
C
Mda
Cda
Rda
p(t)
Q
Figure 3-16. A) Diaphragm with distributed deflection. B) Lumped diaphragm with equivalent volume flow rate. C) Equivalent circuit for the lumped diaphragm.
aC
DS
BR
OV)(tp
aC
DSV
BR
OV
avM
avR
daCda radM M+
aM
diaZ
ventZ
cavZ qc
daR
Figure 3-17. Equivalent circuit model of the microphone.
85
0 0.5 1 1.5 2 2.5 3-10
-8
-6
-4
-2
0
2
4
6
8
10
kd
Im[Z
cavi
ty/Z
0]
SpringSpring-MassFull Sol.
Figure 3-18. Accuracy of first terms of cotangent expansion.
10-1 100 101 102 103 104 105 106
-20
0
20
40
Mag
nitu
de [d
B]
(ref r
espo
nse
@ 1
kHz)
10-1 100 101 102 103 104 105 106
-150
-100
-50
0
50
Frequency [Hz]
Pha
se [d
eg]
Figure 3-19. Magnitude and phase response of LEM normalized by the flat band response.
86
da radM M+ daCdaR
aC
aM
( )p t dq
Figure 3-20. Equivalent circuit illustrating the effect of the cavity compliance
87
CHAPTER 4 OPTIMIZATION
There are many factors that influence the behavior of the device, including composite lay-
up, aspect ratio, piezoresistor geometry, doping densities and doping profiles. Because of this,
an optimization scheme was employed to determine the variables that yielded optimal
performance specifications. This chapter begins with the methodology used to formulate the
optimization scheme, including the objective function, variables and constraints. Next, the
results are discussed for various cases including 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 operational space of the device over a specified
region shown in Figure 4-1. To increase this area, the optimization scheme must minimize the
minimum detectable pressure, MDP , and simultaneously maximize the maximum detectable
pressure, maxP , and bandwidth, BW , of the device. To accomplish this multiobjective
optimization, the ε -constraint method is implemented to generate a Pareto front [133]. An
example is shown in Figure 4-2. In this figure, the utopia point is defined as the point that
possesses the best obtainable value for each objective function. Points A-D are non-dominated
solutions on the Pareto front and point E is a dominated point within the feasible region of the
design space. Since the maximum detectable pressure and bandwidth have a more specified
desired value, they are chosen to be constraints. The MDP remains the primary objective
function because the goal is to lower this value as much as possible. Mathematically this is
expressed as
88
max 1 2 and i j
MDPP BWε ε− ≤ − ≤
Minimize such that
, (4-1)
where 1ε and 2ε are the iterated constrained values 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,
0me
VSP
Δ= , (4-2)
which can be seen pictorially in Figure 4-3 where 0VΔ is given by equation (3-60) or (3-63) for a
voltage or current source, respectively. The minimum detectable pressure is defined when the
output signal is the same magnitude as the noise level,
min0
N N
me
V V PPS V
= =Δ
, (4-3)
where NV is given by equation (3-97). Expressing minP in dB yields
min20 logref
PMDPP
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠, (4-4)
where,
20refP Paμ= . (4-5)
4.1.2 Variables
As shown in sections 3.1 and 3.2, the design variables are as follows:
1 2 3 , , , ,
, , , , , , , , , , ,j a t wt a in a out t in t out
GeometricH H H a z r r r rθ θ θ−
, (4-6)
1 2 3 1 2 3
, , , , , ,Material
E E E ν ν ν α−
, (4-7)
89
and
, p B B
OtherN V or I−
. (4-8)
To satisfy equation (3-60) and balance the Wheatstone bridge, the mean resistance of the taper
resistor is set equal to the arc resistor. To accomplish this, the following relationship must hold:
, ,
2 log logwta t in a in
a ar r
θθ
⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠. (4-9)
It is important that the resistors are located at the point of highest stress. To ensure this, the outer
radii of the resistors are constrained to be 5 mμ larger then the radius of the diaphragm. This
accounts for any fabrication issues associated 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 fabricated microphone structure is similar to the work
of Chandraschakren [134] whose calculated Hooge parameter was 7 5e − . In addition the work
of Dieme [135] leads to an estimate of the value for the Hooge parameter of 5 6e − . Finally, the
material values are assumed to be constant and are not varied during the optimization. This
reduces the total optimization variables to twelve,
1 2 3 , ,, , , , , , , , , , or j a t a in t in surf B BH H H a z r r N V Iθ θ . (4-10)
4.1.3 Constraints
The constraints are comprised of fabrication and performance constraints. All of the
variables are given upper and lower bounds so that
{ } { } { }1 2 3 , ,, , , , , , , , , ,j a t a in t in surf BLB H H H a z r r N V UBθ θ≤ ≤ . (4-11)
The values for the LB and UB were determined by physical and fabrication limitations and can
be found in Table 4-1. For example, the resistors are implanted into the silicon layer. Therefore
the junction depth cannot be larger then the silicon thickness. Fabrication resolution is also finite
90
and there is a minimum line width that can be achieved. For the fabrication methods used, the
minimum line width ( )linew was estimated to be a very conservative 10 mμ . 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 4-4. The constraints are then
,, , , ,t t gap t a a linew w l w l w≥ , (4-12)
which in terms of the optimization variables are
( )
,
,
,
,
,
,,0,0,
0.
a in line
t in line
a a in line
wt t in line
t in t wt line
a r wa r w
r wr w
r w
θ
θ
θ θ
− + ≤ −
− + ≤ −
− + ≤
− + ≤
− − + ≤
(4-13)
In addition to the geometric constraints, performance constraints are also employed. For the
device response to remain linear, the onset of non-linearity must be higher than that of the
desired operational range. This results with the following constraint,
max20 logref
p PMdBp
⎛ ⎞≥⎜ ⎟⎜ ⎟
⎝ ⎠, (4-14)
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
BW V I= . (4-15)
the current is related to the voltage by the resistance of the Wheatstone bridge, BR , yielding the
following constraint,
2
maxB
B
V WR
≤ , (4-16)
for a constant voltage device or
91
2maxBI R W≤ , (4-17)
for a constant current device. The 3dB+ point for the upper bandwidth limit is calculated from
the LEM in section 3.3 and is constrained to be greater than or equal to a specified BW , that is
( )3
2dB
H j f BWπ+
≥ . (4-18)
The bandwidth of the device was calculated using the lumped element model assuming a
damping ratio of 03.0=ζ . At the 3dB+ point the phase lag is 2 degrees. This value of ζ 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 non-dimensional parameter *ck
is constrained to be below a value of 3.4, before buckling occurs (Recall Figure 3-5). Collecting
all of the variables and constraints with equation (4-1) yields,
{ } { } { }
( )
max 1 2
1 2 3 , ,
, ,
, ,
,
,
,
*
and
, , , , , , , , , ,
1 0
1 0
1 0
1 0
1 0
1 03.4
i j
j a t a in t in surf B
a out a in
line
t out t in
line
a a in
line
wt t in
line
t in t wt
line
c
MDPP BW
LB H H H a z r r N V UB
r rw
r rw
rw
rw
rw
k
V
ε ε
θ θ
θ
θ
θ θ
− ≤ − ≤
≤ ≤
− ++ ≤
− ++ ≤
−+ ≤
−+ ≤
− −+ ≤
− ≤
Minimize such that
2
max
1 0B
RW− ≤
(4-19)
92
To run the optimization, MATLAB’s 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 are discussed in the following sections. The
first mode was for a constant bias voltage applied across the Wheatstone bridge and the second
for a constant current source through the bridge.
4.2.1 Optimization with Constant Voltage
In Figure 4-5, MDP is plotted for a variety of maxP and bandwidth constraints. With
increasing bandwidth constraint, the MDP also increases, as expected. The same trend holds true
for MDP as a function of maxP , shown in Figure 4-6, also as expected. The curves corresponding
to a maximum pressure of 140 and 145 converge at a higher bandwidth because the maxP
constraint is not active. This 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 fabrication are designated devices A, B, and C
and can be seen in Figure 4-5 and Table 4-2. All variables and constraint values can be found for
all devices in the multi-objective 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,
93
( )2 2
bN
s j a ain
VVN z a rθ
∝−
. (4-20)
When the thermal noise is dominant the mean resistance dominates the noise floor[130]. This is
proportional to the following variables,
( )
a s
j ain
NRz a r
θ∝
−. (4-21)
The sensitivity of the device, is proportional to the compliance of the diaphragm and the
compliance of the diaphragm is proportional to the following variables,
2aSens
h∝ . (4-22)
Conversely the bandwidth constraint is proportional to
1BWMC
∝ (4-23)
and the bandwidth is therefore proportional to the following variables
222
1 1BWaaa h
h
∝ = (4-24)
The linearity constraint is inversely proportional to the compliance of the diaphragm. This
results with
max 2
hPa
∝ . (4-25)
From the proportionalities, it can be seen that the sensitivity is inversely proportional to the
bandwidth and maximum detectable pressure. This sets up the trade-off between each value and
results with the Pareto fronts in Figure 4-5 and Figure 4-6.
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: 4 10%mA ± and 10 15%mA ± [137]. Figure 4-7 and Figure 4-8 show MDP
plotted as a function of bandwidth constraint for a variety of maxP constraints for devices running
at 4mA and 10mA , respectively. The MDP of devices A,B and C can be seen in Table 4-3 and
Table 4-4.
Comparing Table 4-2 to Table 4-3 and Table 4-4, 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 counterparts. 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
4-9 and Figure 4-10. Note that a higher input current would violate the power 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, doping concentrations, and junction depths do not
allow the devices A, B and C to be processed on the same silicon wafer. To make fabrication on
the same wafer possible, the devices were optimized 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
95
{ } { } { }
( )
max 1 2
, ,
, ,
, ,
,
,
,
*
2
max
and
, , , , , or
1 0
1 0
1 0
1 0
1 0
1 03.4
1 0
a t a in t in B B
a out a in
line
t out t in
line
a a in
line
wt t in
line
t in t wt
line
c
B
MDPP BW
LB a r r V I UB
r rw
r rw
rw
rw
rw
k
VRW
ε ε
θ θ
θ
θ
θ θ
− ≤ − ≤
≤ ≤
− ++ ≤
− ++ ≤
−+ ≤
−+ ≤
− −+ ≤
− ≤
− ≤
Minimize such that
(4-26)
Table 4-5 and Table 4-6 show the results of an optimization with these added constraints
for voltage and current source (10mA) devices, respectively. The first sub-table is for a set of
devices where the parameters of device A were used as the constraints for devices B and C. The
second sub-table 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 device’s 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 4-6. This was chosen
because the sponsor, Boeing, ultimately would like 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. Additionally, the designs to operate at 10mA
possess similar performance as devices designed with a voltage source.
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 variable in the optimization as well as the noise
figure of merit, the Hooge parameter.
The following results are all for optimized device A from Figure 4-8. As previously stated,
the optimization was performed assuming a Hooge parameter of 5 6e − . Figure 4-11 shows that
errors in the assumed Hooge parameter can have a major impact on MDP. If 1/f noise is
dominant, the voltage noise and MDP scale with the ( )1/ 2α .
The dependence of MDP with respect to all independent variables is shown in Figure 4-12.
The variable with the largest effect on the MDP is the thickness of the silicon. As the thickness
decreases, the stiffness decreases as well and the diaphragm begins to buckle resulting in the
sharp drop in MDP. The linear deflection solution however is invalid in this region. Figure 4-13
shows the effect of MDP on the thickness of silicon 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 performance metrics is derived in this section. The
formulations presented here utilize results obtained in section 4.2.3. The MDP of design A is
analyzed. Furthermore, the predictions for the bandwidth and linearity constraints are explored.
Calculating the MDP of the microphone includes numerical integrals 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
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 manufacturing tolerances and can be found in Table 4-7. These
statistically independent values are fed into the objective function and the distribution of the
objective function is obtained. In this case, uncertainties in the design variables correspond to a
statistical distribution of the noise voltage, sensitivity and their ratio, the MDP. There will also
be a distribution for the resonant frequency and maximum detectable pressure. This process is
illustrated in Figure 4-14 where ix is the optimized design variable, iσ is the standard deviation
for each design variable and Y is a Gaussian distributed random number with mean zero,
variance one and standard deviation one.
The Monte Carlo simulation was implemented 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 4-15.
The results highlighted 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 4-16 and
Figure 4-17 respectively. The distributions are not assumed to be Gaussian and all statistical
moments can be found in Table 4-9. To determine the 95% probability limits for each
specification, a numerical integration was performed. This integration starts at the mean value
and moves outwards until the value under the PDF is 47.5%. The process is done for values
above and below the mean to yield a total probability of 95%. Figure 4-18 shows this process
98
pictorially. The 95% probability for the performance of the desired specifications can be seen in
Table 4-8. It is important to note that the lower end of the confidence integral is in question
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 multi-objective 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.
99
Table 4-1. Upper and lower bounds for all variables
HSi HSi02 HSiN a Ns zj rain Θa rtin Θt Vb units μm Α A μm #/cm3 μm μm deg μm deg V UB 10 3000 1000 800 1e20 7 800 45 800 25 15 LB 1 300 300 60 1e18 0.1 60 1 60 0.5 1
Table 4-2. 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 4-3. 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 4-4. 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
100
Table 4-5. Single wafer constrained 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 4-6. Single wafer 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
101
Table 4-7. Standard deviation of input parameters.
HSi HSi02 HSiN a Ns zj rain Θa rtin Θt Ib units μm Α A μm #/cm3 μm μm deg μm deg mΑ 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 4-8. Mean and 95% confidence intervals for design parameters.
Mean Lower CI Upper CI units MDP 24.5 11.5 29.1 dB Pmax 150 error 158 dB
Bandwidth 120 31.0 157 kHz
Table 4-9. Statistical properties of desired parameters. Units Mean Variance Skewness Kurtosis
MDP [dB SPL] 24.9 6.9 -0.177 2.37 Pmax [Pa] 633 10200 -0.0272 2.88 BW [kHz] 117.5 777.2 -0.668 3.76
Frequency (Hz)
Bandwidth [Hz]
Dynamic Range[dB SPL]
Operational Space of Device
Figure 4-1. Operational parameter space for a microphone.
102
2J
1J
A
B
C
D
E
Utopia
Λ
Figure 4-2. Multiobjective optimization Pareto front illustrating the trade-off between minimizing function J1 and J2.
prms
V
Sensitivity
pmax
Figure 4-3. Ideal linear output of a microphone or pressure transducer.
103
aw al
,t gapwtw
tl
Figure 4-4. Features that are constrained to be larger than linew .
50 60 70 80 90 100 110 120 130 140 15014
16
18
20
22
24
26
28
30
32
34
Bandwidth (+/- 3dB) [kHz]
MD
P [d
B]
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 4-5. MDP vs. Bandwidth for various Pmax constraints.
104
140 145 150 155 160 165 17014
16
18
20
22
24
26
28
30
32
34
Pmax (5% deviation from linearity) [dB]
MD
P [d
B]
BW = 150kHzBW = 130kHzBW = 110kHzBW = 90kHzBW = 70kHzBW = 50kHz
Figure 4-6. MDP vs. Pmax for various bandwidth constraints.
50 60 70 80 90 100 110 120 130 140 15016
18
20
22
24
26
28
30
32
34
Bandwidth (+/- 3dB) [kHz]
MD
P [d
B]
Pmax = 170dBPmax = 165dBPmax = 160dBPmax = 155dBPmax = 150dBPmax = 145dBPmax = 140dB
Figure 4-7. MDP vs. Bandwidth of various Pmax constraints for a constant current source device (4mA). (Letters correspond to Table 4-3)
105
50 60 70 80 90 100 110 120 130 140 15016
18
20
22
24
26
28
30
32
34
Bandwidth (+/- 3dB) [kHz]
MD
P [d
B]
Pmax = 170dBPmax = 165dBPmax = 160dBPmax = 155dBPmax = 150dBPmax = 145dBPmax = 140dB
Boeing target spec
AC
B
Figure 4-8. MDP vs. Bandwidth of various Pmax constraints for a constant current source device (10mA). (Letters correspond to Table 4-4)
3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.423
24
25
26
27
28
29
30
31
32
Input current [mA]
MD
P [d
B]
Device ADevice BDevice C
Boeing target spec
Figure 4-9. Sensitivity analysis for constant current source varying by 4 10%mA ± .
106
8.5 9 9.5 10 10.5 11 11.522
23
24
25
26
27
28
29
30
Input current [mA]
MD
P [d
B]
Device ADevice BDevice C
Boeing target spec
Figure 4-10. Sensitivity analysis for constant current source varying by 10 15%mA ± .
10-6 10-5 10-4 10-320
25
30
35
40
45
Hooge parameter (α)
MD
P [d
B]
Device Optimizedfor α = 5e-6
Figure 4-11. MDP dependence on Hooge parameter for device A.
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 Variable
MD
P [d
B S
PL]
(1H
z bi
n @
1kH
z)
Sensitivity Analysis
H1H2
H3
aNszj
θarain
θt
rtinIb
Figure 4-12. Dependence of MDP with respect to each variable.
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
20
40
60
HSi (μm)
MD
P [d
B]
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 31
2
3
4
k c*
k*c = 3.4Optimized device
constrained to k*c <= 3.4.
Figure 4-13. Dependence of MDP on silicon thickness overlaid with compression coefficient.
108
1 1x Yσ+
i ix Yσ+
XX Yσ+
Figure 4-14. Monte Carlo simulation schematic [138].
0 100 200 300 400 500 600 700 8000
0.5
1
1.5
2
2.5
3
3.5x 10-3
Pro
babi
lity
Den
sity
Fun
ctio
n of
Pm
in
Minimum Pressure [μPa]
k < 3.6k < 3.8k > 3.8
Figure 4-15. Uncertainty of MDP of Device A.
109
-1000 -500 0 500 1000 1500 2000 25000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-3
Pro
babi
lity
Den
sity
Fun
ctio
n of
Pm
ax
Maximum Pressure [Pa]
k < 3.6k < 3.8k > 3.8
154dB148dB 160dB
Figure 4-16. Uncertainty of maxP for device A.
110
0 20 40 60 80 100 120 140 160 180 2000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Pro
babi
lity
Den
sity
Fun
ctio
n of
Ban
dwid
th
Bandwidth [kHz]
k < 3.6k < 3.8k > 3.8
Figure 4-17. Uncertainty of the bandwidth for device A.
47.5% 47.5%
95% Probability Yield
Figure 4-18. 95% probability yield limit illustration.
111
CHAPTER 5 DEVICE FABRICATION AND PACKAGING
This chapter provides an overview of the fabrication process, describes the realized device,
and details the associated packaging used for acoustic testing. The process flow is designed to
be integrated circuit (IC) compatible. This will allow future fabrication of the device to be
completed at external commercial foundries and will in theory permit the integration of
electronics on the microphone die. The microphone is fabricated using a bulk micromachining
process starting with an SOI wafer and utilizing 9 masks. The process 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 5-1 and the masks can be seen in Figure
5-2 and Figure 5-3 For front side fabrication, 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
3 8 cm− Ω . 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 same 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 n-well dope of the previously etched hole followed by a drive-in
and activation (c). The second is the p-type doping of the piezoresistors followed by a drive-in
and activation(d). After the drive-in 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 backside 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 layer of PECVD nitride is then deposited over the
112
device (i). The two remaining steps 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 5-4 and the masks in Figure 5-5. These steps
begin with a deposit of photoresist (PR) on the front side of the wafer to protect it from damage
(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 attached to the SOI wafer to provide support to the
diaphragms while they are etched (c). The backside 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 nested oxide mask (g). Finally, a Pyrex wafer is
anodically bonded to the wafer to complete the process (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 devices, as well as a variety of
test structures. Before dicing the yield was close to 100%. Problems with the dicing lowered the
yield to 50%. After dicing was completed and against instructions, a jet of water was sprayed
onto the wafers to clean them of particulate. This resulted in the destruction of about half of the
diaphragms. Images of these devices and structures are shown in Figure 5-6 through Figure 5-9.
The final die measures 2mm square and Figure 5-6 shows a close up of a finished wafer after
dicing was completed. Each wafer is divided into two sections; half of the devices are type A
and the remaining are type B. Figure 5-7 shows the front side of a type A die. To get a reference
to the size of the microphone, Figure 5-8 shows a type B device on a dime. The top of this die
contains five bond pads, each 200 μm square, the center pad is the substrate contact and the
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 silicon nitride and silicon
dioxide layers. Figure 5-9 shows the backside cavity and vent structure.
5.3 Microphone Packaging
For acoustical characterization, 4 type A devices were mounted onto a printed circuit board
(PCB). A new endplate for the acoustic characterization setup was fabricated that allows for a
1/4”-condenser reference microphone to be mounted next to the PCB board. A schematic of this
setup can be seen in Figure 5-10. To minimize attenuation of the signal all associated electronics
including the high pass filter, amplifier and power 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 the integrated circuitry shown in Figure 5-11. The
microphone 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 amplifier (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 has a bandwidth of 25 kHz. Due to resistor
mismatch in the bridge, there will be a large 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 are used to supply the
microphone die with the bias voltage for the Wheatstone bridge and to supply the bias for the
substrate to isolate the resistors. To accomplish this, two Linear Technologies LT1963 low noise
114
voltage regulators are used in an adjustable operation mode. For each regulator, a potentiometer
is used to adjust the output voltage supplied to the microphone die.
5.3.2 Printed Circuit Board
The printed circuit board serves a variety of purposes: mounting and positioning the
microphone die into the acoustic test stand, attaching all of the associated circuitry and providing
a backplate for the microphone cavity. Figure 5-12 shows an unpopulated PCB board. These
boards are gold plated and were ordered from Sierra Protoexpress. All components are mounted
on the front side and the microphone 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 tolerance of ±10 %. The resistors are 0805 size
metal film surface mount resistors with a tolerance of ±1 %. For details on all of the
specifications for the interface circuitry see Appendix E.
5.3.3 Assembled Package
The populated PCB board can be seen in Figure 5-13. Two potentiometers are used for the
gain control, one of which has a higher resistance than the other. 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 5-14. For details on all components and
a detailed drawing see Appendix E.
115
(a) Begin with a SOI wafer with a handle wafer thickness of 350 m, BOX thickness of 3000Å, and device wafer thickness of 1.5 m
Silicon350 m
Oxide 3000ÅSilicon 1.5 m
(b) Etch contact pad for substrate bias using RIE
(d) Ion implant the p-type piezoresistors
(e) Grow thermal oxide
(f) PECVD silicon dioxide on backside
(g) Etch contact cuts using BOE
(h) Deposit Al-Si Leads
(i) PECVD silicon nitride
(j) Etch vent hole using RIE
(k) Etch contact cuts using RIE
(c) Ion implant the n-type substrate contact
Figure 5-1. Front side process steps.
116
A B
C
D
Figure 5-2. A) Ground strap mask. B) N-Well Mask. C) Piezoresistor mask. D) Piezoresistor contact mask.
117
A
B
C Figure 5-3. A) Metallization mask. B) Topside vent mask. C) Bond pad mask.
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 5-4. Back side process steps.
A
B Figure 5-5. A) Backside vent path mask. B) Backside cavity mask.
119
Figure 5-6. Array of microphone die after dicing. Each die is 2mm x 2mm.
2 mm
2 mm
Figure 5-7. Individual type A microphone die after dicing.
120
Figure 5-8. Type B device pictured on a dime.
Figure 5-9. Backside cavity and vent of an individual type A microphone die after dicing.
121
Figure 5-10. Packaging for acoustical characterization.
Figure 5-11. Interface circuitry showing power supply, ac filter and amplifier.
122
Figure 5-12. Printed circuit board for mounting 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 5-13. Assembled device on PCB. Device is protected under a TO can.
123
Mount for ¼” condenser
microphone
Mount for BUF1
Figure 5-14. Populated PCB board inserted into PWT endplate.
124
CHAPTER 6 RESULTS AND DISCUSSION
The results for the aeroacoustic microphone designed and fabricated in Chapters 3-5 are
outlined here. The methodology and experimental setup 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 BUF1-A devices and 12 BUF1-B devices are tested for the I-V characteristics
of the bridge as well as the PN diode characteristics. Two four point test resistors are tested for
each type A and B arc and taper resistors. In addition, 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 experiments see Appendix
F.
Electrical characterization consists of several tests described here. Input and output
resistances are measured using an Agilent 4155C semiconductor parameter analyzer (SPA). To
obtain the bridge offset, the resistance of isolated resistors on a test structure are tested with a
four point probe. To check for resistor isolation (Figure 6-1), the I-V characteristics of the p-n
diode from the resistor to the substrate are tested. Also included on the finished wafers are van
der Pawl test structures to determine sheet resistance, contact resistance and line width control.
Schematics of these structures are shown in Figure 6-2, Figure 6-3, and Figure 6-4. The sheet
resistance is determined by the following equation
ln 2 2
abcd dabcs
R RR π += . (6-1)
125
To determine the lateral diffusion via the line width test structure, the resistance is given by
sL LR Rw w
+ Δ=
+ Δ, (6-2)
assuming L dL yields,
1
1s
LR R www
=Δ
+. (6-3)
Taking the first term from a Taylor series expansion yields
1sL wR Rw w
Δ⎛ ⎞= −⎜ ⎟⎝ ⎠
. (6-4)
Finally, the lateral diffusion is given by
1s
R ww wR L
⎛ ⎞Δ = −⎜ ⎟
⎝ ⎠. (6-5)
The Kelvin structure yields the contact resistance defined by
cVRI
= . (6-6)
In addition to the van der Pawl structures, MOS capacitor test structures are included in the set of
test structures. The capacitor test structures verify the dielectric thickness and substrate doping.
Due to the high doping concentration, the MOS capacitor 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
iAC
dε
= , (6-7)
where iC is the capacitance measured in accumulation mode, A is the area of the capacitor and
d is the thickness of the dielectric.
126
The experimental setup for noise experiments is discussed in Dieme et al. [135] and shown
in Figure 6-5. The microphone is placed inside three concentric Faraday cages, which serves the
function of attenuating electromagnetic waves 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 cages 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 curve fit is performed to determine the Hooge parameter discussed
in Chapter 3. Equation (3-95) in terms of PSD is
1f
VSNf β
α Γ
= . (6-8)
Taking the logarithm of equation (6-8) and expanding yields
( ) ( ) ( ) ( ) ( )1
1 21 2
log log log log logf
m mx x by
S V f Nβ α= Γ − + − . (6-9)
From this equation, Γ , β , and α can be solved knowing the total number of active carriers in
the DUT.
6.1.2 Acoustic Characterization
The goal of the acoustic characterization is to determine the frequency response and
linearity of the microphone. For this experiment a PWT (Figure 6-6), with a 1 in x 1 in square
cross section, 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 the DUT and the reference microphone to be exposed to the
127
same incident pressure. This frequency is known as the cut-on frequency of the PWT [127]. For
this PWT, the cut-on frequency in air is 6.7 kHz.
The supporting hardware for the plane wave tube experiments is also shown in Figure 6-6.
The Bruel and Kjaer Pulse multi-analyzer system supplies a function generator to power the
acoustic driver and also receives the input signals from the DUT 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 reference microphone. The microphones are tested
up to 170 dB. The 377A51 microphone has a 3% distortion limit of 192 dB and will therefore be
sufficient for all acoustic measurements [139]. The signal sent to the acoustic driver is amplified
by a Techron 7540 power supply amplifier and the 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 microphone and the coherence
between the signals is recorded to determine the uncertainty in the measurement.
Total harmonic distortion (THD) (equation (1-1)) is measured using a single-tone pressure,
and is used to calculate the linearity of the device. The THD grows as the microphone changes
from linear to non-linear operation. Nonlinearity in the microphone causes output power at
frequencies that are integer multiples of the fundamental frequency [120]. Therefore, the power
measured in the harmonics can be utilized to approximate the THD. However, non- linearities
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 pressure at harmonic frequencies,
128
and acoustic wave propagation turns nonlinear at high sound pressure levels [127].
Consequently, the THD of the microphone is calculated in the presence 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 measures the total acoustic pressure, which
includes the harmonic components, several assumptions are necessary to validate this analysis.
First, the reference microphone must not bring any non-linearities into the system. Additionally,
it is assumed only plane waves are propagating in the PWT and that the total pressure detected
by the reference microphone and the piezoresistive microphone are the same. For the THD
measurements, a 1 kHz tone is used. Because the cut on frequency of the PWT is greater than 6
kHz, the first five harmonic components propagate 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 test structures are presented, including I-V 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 tested via secondary ion mass spectroscopy
(SIMS), which yields the dopant concentration as a function of depth 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 measurement show a gross difference between the desired
129
profile and measured profile, shown in Figure 6-7. This error has an extremely large impact on
the performance of the device. The large error leads to a boron concentration of 35 17e cm at the
backside of the diaphragm. The background concentration of the wafers is 31 15e cm , which
shows that there is no junction isolation on the underside of the resistors. This leads to a plethora
of problems which will be detailed in the later sections. The profile error is due to an error in the
recipe for the drive-in portion of the resistor fabrication. A miscalculation between a computer
simulation and a 1-D 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 6-3. The calculated sheet resistance corresponds with the nominal resistance of the
piezoresistors. The masked line width structures were drawn to be 20 mμ . The calculated value
for the p+ line width structure is 21 mμ which indicates there is a lateral diffusion of 0.5 mμ on
each side of the structure. This value helps determine the state of the fabricated structure to
allow for a more accurate model validation. This lateral diffusion correlates with the SIMS
profile and confirms the error in the dopant profile. In addition, metal line width structures,
summarized in Table 6-4, reveal a 4.1 mμ over etch. This is in agreement with the way the
metal lines were fabricated and gives more confidence in the test structure values. Finally, Table
6-5 shows the contact resistance is 1.6% the value of the resistors and the specific contact
resistivity is in line with the literature for aluminum-silicon 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 6-6. These
two measurements have good agreement between the tests and desired values.
130
Arc and taper four point test resistors for both devices were tested and the results can be
seen in Table 6-2. There is a large discrepancy between 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 Wheatstone
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 out by the interface circuitry, however the mismatch
will couple in power supply noise.
Results of the input and output I-V curves are shown in Figure 6-8 and Figure 6-9,
respectively. These figures show consistency over all devices tested. Table 6-1 shows the input
and output resistances for all devices tested. All device are batch fabricated to be identical. To
get an idea on the yield statistics the mean and standard deviation of the resistance is also shown
in Table 6-1.
Two important aspects of the input and output I-V curves are the linearity and intercept
point, where the intercept point corresponds to the leakage current. Figure 6-10 shows the data
for the input I-V curve as well as the linear fit associated with the data. From this, the linearity
of the I-V curve is apparent. In addition, the I-V curve does not cross at absolute zero. This is
due to the leakage current from the resistor into the substrate. The leakage current is higher then
expected but is 4 orders of magnitude less than current through the resistor. Figure 6-11 shows
the output linearity and leakage current. For this case, the I-V curve has a slight nonlinearity at a
potential of 10V. The leakage current is determined by the reverse diode characteristics of the
device. Figure 6-12 shows an I-V curve measuring current from the substrate contact to the
closest resistor seen in Figure 5-7. For this test, the ground pad was used. Upon closer
inspection the reverse diode characteristics have a strange behavior seen in Figure 6-13. The
leakage current for a reverse bias below 2V is 0.25 Aμ and this increases to 2.25 Aμ above 2V .
131
This behavior is consistent for both released and 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 6-14 shows the noise PSD for a test taper resistor for a type A device
along with the setup noise PSD. Subtracting the setup noise and applying equation (6-9) yields
the Hooge parameter and is shown in Table 6-7. The curve fit is then plotted with the data in
Figure 6-15. Figure 6-16 shows the PSD of the noise for a BUF1-A device. To determine the
Hooge parameter from this data, equation (3-96) is utilized. Table 6-8 shows the results of the
curve fit and Figure 6-17 compares the data to the model fit. The Hooge parameters measured
for both devices is much higher then expected. The measured value is four orders of magnitude
higher than the desired values. This data will be used to validate 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 6-18. The sensitivity is
shown to be constant over the entire pressure range. 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 6-19. Since the maximum
detectable pressure was defined as a 5% deviation from a linear defection of the diaphragm, the
maximum pressure measurement is considered the first point where the THD reaches 5%. This
value for the tested microphones range from 153 160 dB SPL− .
132
6.2.4 Frequency Response
The magnitude and phase frequency response for the tested microphones is found in Figure
6-20 and Figure 6-22, respectively. Each individual frequency response is plotted with it’s
associated 95% confidence interval in Figure 6-21 and Figure 6-23. 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 available. 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 periodic 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 dramatically, resulting in large random errors. The
coherence for one device is shown in Figure 6-24. With the sensitivity and noise floor
calculated, the MDP for each device for a 1 Hz bin centered at 1kHz is shown in Table 6-9.
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 4-14 where ix is the optimized
design variable, iσ 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
133
variables are statistically independent. The major difference between the implementation here
versus that of Chapter 4 is that the values put into 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 predicted with the developed models.
6.3.1 Variables and Standard Deviations
Table 6-10 shows each variable with the associated methods used to determine an accurate
value. Redundancy was used to increase the confidence in the realized value. All of the tests in
Table 6-10 are discussed in the test structure section earlier in this chapter 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 Schott high power snake light and an Olympus microscope with a
Quadracheck 200 measurement system. The snake light was positioned directly under the clear
stage and pointed towards the die. This illuminates the semi-transparent diaphragm from the
backside. The Quadracheck 200 is then used in circle mode to determine the diameter of the
device. Figure 6-25 shows the microphone die with and without illumination. Thirty one type A
devices were tested and the diameters were smaller then expected. Upon further inspection, it
was found that the final 10% of the backside cavity etch, designed to prevent footing, had a
slight taper leading to the smaller diameter. These results can be found in Table 6-11. Finally,
the values and standard deviations used in the Monte Carlo simulation are assembled in Table
6-12.
6.3.2 Model Validation Results
The Monte Carlo simulation was implemented in MATLAB and the random independent
variables were generated using the randn function. The simulation was run for 30,000 iterations
134
and the probability distribution function (PDF) for MDP can be seen in Figure 6-26. The results
yield a PDF with a mean value of 106.4 dB SPL and a standard deviation of 1.51 dB SPL . The
statistical moments can be found in Table 6-14. The uncertainty for the sensitivity, voltage
noise, and maxP can be seen in Figure 6-27, Figure 6-28, and Figure 6-29, respectively. The 95%
probability limits for the design parameters can be seen in Table 6-13. The results from the
model validation agree with the experimental results. The maximum detectable pressure
determined by THD are shown as red lines in Figure 6-29, which fit inside the 95% probability
range for the model. The sensitivities determined in the PWT are also shown in Figure 6-27.
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. Finally, the MDP is experimentally calculated from
the noise floor and sensitivity of the device and is found to be 108 dB SPL 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 resistor thickness being the entire
thickness of the diaphragm. The result of this error dropped the sensitivity two orders of
magnitude. In addition to the doping profile error, 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. These two factors couple together 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 realized device had a MDP of 108dB, or 83.5 dB
higher then the desired value.
135
Table 6-1. Resistance values of all tested devices. Device RinA RoutA RinB RoutB R2(RinA)R2(outA)R2(RinB)R2(RoutB)
# Ω Ω Ω Ω [−−] [−−] [−−] [−−] 1 919 934 1043 1057 0.99996 0.99982 0.99995 0.99982
2 931 931 1031 1030 0.99992 0.9998 0.99982 0.9998
3 946 943 1046 1043 0.99982 0.99982 0.99983 0.99982
4 942 941 1016 1012 0.99982 0.99981 0.9998 0.99981
5 955 929 1017 1015 0.99979 0.99979 0.99984 0.99979
6 936 926 1038 1036 0.99963 0.99975 0.99985 0.99975
7 925 919 1013 1013 0.99975 0.99975 0.99984 0.99975
8 923 914 1008 1011 0.99976 0.99976 0.99984 0.99976
9 945 933 1004 1013 0.99975 0.99978 0.99982 0.99978
10 949 945 1015 1011 0.99981 0.99981 0.99985 0.99981
11 927 926 1005 1007 0.99977 0.99978 0.99981 0.99978
12 916 916 1003 1002 0.99977 0.99978 0.99978 0.99978
R 934 930 1020 1021
95% CI (926-942) (924-936) (1010-1030) (1010-1032) σ 12.9 10.2 15.5 16.6
95% CI (9.14-21.9)(7.23-17.3) (11.0-26.3) (11.8-28.2)
Table 6-2. 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 (674.4-867.6) (1042-1119) (793.2-883.5) (958.4-1429)
*All values in Ω
Table 6-3. Values for VDP and line width test structures.
Rs,design Rs Wp,design Wp Ω/sq Ω/sq μm μm 336 194.6 20.0 21.0
136
Table 6-4. Values of VDP and line width test structures for the metal lines. Rs metal Wmetal Wm,design
Ω/sq μm μm 0.022 15.9 20.0
Table 6-5. Values of Kelvin test structures. Rc ρc ρc,lit Ω Ωcm2 Ωcm2
16.37 2.62E-04 O(E-4-E-5)
Table 6-6. Thickness of oxide layer using two techniques. Design goal MOS capacitor Ellispometer
μm μm μm 0.159 0.160 0.152
Table 6-7. Curve fit parameters for test taper resistor for device A. α β Γ R R2 [--] [--] [--] [Ω] [--]
Value 7.55E-02 1.107 1.933 1050 0.997 CI (6.04 - 9.06)E-02(1.105 - 1.109) (1.927 - 1.939)
Table 6-8. Curve fit parameters for a BUF1-A device. α β Γ R R2 [--] [--] [--] [Ω] [--]
Value 5.13E-02 1.083 2.075 912 0.995 CI (3.79 - 7.01)E-02 (1.081 - 1.086) (2.073 - 2.081)
Table 6-9. 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
137
Table 6-10. Methods used to determine the fabricated values for all parameters of the devices. H1 H2 H3 a Ns zj θa rain θt rtin Ibias
SIMS SIMS SIMS light test SIMS SIMSline
width line
width line
width line
width appliedwafer 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 6-11. Results from the radius determination experiment. BUF1A
adesign ameas 95% CI 113 108 (107.3 - 108.7)
* All measurements in μm.
Table 6-12. Measured values and standard deviation of input parameters. HSi HSi02 HSiN a Ns zj rain Θa rtin Θt Ib μm Α A μm #/cm3 μm μm deg μm deg mΑ
value 1.45 1600 380 109 4e18 Hsi 87 43.3 77 21.5 10 STD 0.05 50 10 1.6 1e17 0.05 0.25 0 0.25 0 0
% 3.5 3.1 2.6 1.5 2.5 3.5 0.29 0 0.32 0 0
Table 6-13. Confidence intervals for the realized parameters for a BUF1-A device. Pmax Sensitivity Vn MDP [Pa] [nV/Pa/V] μV dBSPL (1 Hz bin @ 1kHz)
1000 2200 4.0 5.6 0.201 0.269 104 109
Table 6-14. Statistical data for model validation PDFs. Units Mean Variance Skewness Kurtosis
MDP [dB SPL] 106.4 2.29 0.156 3.15 Sens [μV/Pa] 0.046 2.00E-05 0.279 3.27 Vn [μV] 0.226 3.49E-04 0.561 3.6
Pmax [Pa] 1360 86300 0.542 3.49
138
P+
n
SiO2
R/2 R/2C
Figure 6-1. Circuit representation of reversed biased p+ doped resistors in an n substrate [119].
R R+R R+
Figure 6-2. Van der Pawl test structure schematic.
139
Figure 6-3. Line width test structure schematic.
Figure 6-4. Kelvin test structure schematic.
140
Figure 6-5. Experimental setup for noise measurements.
B&K Pulse Multi-
analyzer
Amplifier
DUT
Reference Microphone
PWT
Driver End Plate
Figure 6-6. Experimental setup for acoustic characterization.
141
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.81014
1015
1016
1017
1018
1019
1020
1021
depth [μm]
Bor
on c
once
ntra
tion
[#/c
m3 ]
SIMS dataCurve fitDesired profileBackground Concentration
Figure 6-7. Boron concentration in silicon device layer determined 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]
Cur
rent
[mA
]
Device ADevice B
RinA = 934Ω
σ = 12.9ΩRinB = 1020Ω
σ = 15.5Ω
Figure 6-8. Input I-V curve of 12 BUF1-A and 12 BUF1-B devices.
142
-10 -8 -6 -4 -2 0 2 4 6 8 10-15
-10
-5
0
5
10
15
Potential [V]
Cur
rent
[mA
]
Device ADevice B
RoutA = 930Ω
σ = 10.2ΩRoutB = 1021Ω
σ = 16.6Ω
Figure 6-9. Output I-V curve of 12 BUF1-A and 12 BUF1-B devices.
-10 -8 -6 -4 -2 0 2 4 6 8 10-20
-10
0
10
20
Potential [V]
Cur
rent
[mA
]
-10 -8 -6 -4 -2 0 2 4 6 8 101.5
1.6
1.7
1.8
Leak
age
Cur
rent
[ μA
]
Curve Fit
data
Figure 6-10. Input I-V curve of a BUF1-A device with a linear curve fit.
143
-10 -8 -6 -4 -2 0 2 4 6 8 10-15
-10
-5
0
5
10
15
Potential [V]
Cur
rent
[mA
]
-10 -8 -6 -4 -2 0 2 4 6 8 101.8
1.9
2
2.1
2.2
2.3
Leak
age
Cur
rent
[ μA
]
Curve Fit
Data
Figure 6-11. Output I-V curve of a BUF1-A 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]
Cur
rent
[mA
]
Figure 6-12. I-V curve of diode characteristics of a BUF1-A device.
144
-12 -10 -8 -6 -4 -2 0-3
-2.5
-2
-1.5
-1
-0.5
0
Potential [V]
Cur
rent
[ μA
]
Figure 6-13. I-V curve of diode characteristics of a BUF1-A device focusing on the reverse region.
10-1
100
101
102
103
104
105
106
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
Frequency [Hz]
Noi
se P
SD
[V2 /
Hz]
setupV = 0.22V = 0.43V = 0.81
Figure 6-14. Noise PSD of a test taper resistor.
145
10-1 100 101 102 103 104 105 10610-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
Frequency [Hz]
Noi
se P
SD
[V2 /
Hz]
V = 0.22V = 0.43V = 0.81
Figure 6-15. 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.
10-1 100 101 102 103 104 105
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-9
Frequency [Hz]
Noi
se P
SD
[V2 /
Hz]
setupV = 0.28V = 0.93V = 1.96V = 3.94V = 9.96
Figure 6-16. Noise power spectral density of a BUF1-A device.
146
10-1
100
101
102
103
104
105
106
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
Frequency [Hz]
Noi
se P
SD
[V2 /
Hz]
V = 0.28V = 0.93V = 1.96V = 3.94V = 9.96
Figure 6-17. 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 1602
3
4
5
6
7
8
9
10
Sen
sitiv
ity [n
V/P
a/V
]
Incident Pressure [dB SPL]
Figure 6-18. Sensitivity of BUF1-A devices normalized by the bias voltage.
147
115 120 125 130 135 140 145 150 155 160 165 1700
5
10
15
20
25
THD
[%]
Incident Pressure [dB SPL]
Figure 6-19. Total harmonic distortion of BUF1-A microphones.
0 1 2 3 4 5 6 7-170
-165
-160
-155
-150
-145
-140
-135
-130
Frequency [kHz]
Mag
nitu
de R
espo
nse
[dB
] (re
1V
/Pa)
Figure 6-20. Magnitude frequency response for a BUF1-A device. Vertical dotted lines mark the piecewise FRFs that were stitched together.
148
0 1 2 3 4 5 6 7-155
-150
-145
Mag
FR
F [d
B]
Frequency [kHz]
0 1 2 3 4 5 6 7-155
-150
-145
Mag
FR
F [d
B]
Frequency [kHz]
0 1 2 3 4 5 6 7-155
-150
-145
Mag
FR
F [d
B]
Frequency [kHz]
0 1 2 3 4 5 6 7-155
-150
-145
Mag
FR
F [d
B]
Frequency [kHz]
Figure 6-21. Magnitude FRF for each device with 95% CI bounds.
149
0 1 2 3 4 5 6 7-20
-15
-10
-5
0
5
10
15
20
Frequency [kHz]
Pha
se R
espo
nse
[deg
]
Figure 6-22. Phase response for each device tested.
150
0 1 2 3 4 5 6 7-20
-10
0
10
20
Pha
se [d
eg]
Frequency [kHz]
0 1 2 3 4 5 6 7-20
-10
0
10
20
Pha
se [d
eg]
Frequency [kHz]
0 1 2 3 4 5 6 7-20
-10
0
10
20
Pha
se [d
eg]
Frequency [kHz]
0 1 2 3 4 5 6 7-20
-10
0
10
20
Pha
se [d
eg]
Frequency [kHz]
Figure 6-23. Phase FRF for each device with 95% CI bounds.
151
0 1 2 3 4 5 6 70.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency [kHz]
Coh
eren
ce
Figure 6-24. Coherence function between device A-5 and the reference microphone.
A B
Figure 6-25. A) Picture of a microphone die with topside lighting. B) Picture of a microphone die with backside lighting.
152
2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Pro
babi
lity
Den
sity
Fun
ctio
n of
MD
P
MDP [Pa]
Figure 6-26. Minimum detectable pressure probability density function.
0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.0750
10
20
30
40
50
60
70
80
90
100
Pro
babi
lity
Den
sity
Fun
ctio
n of
Sen
sitiv
ity
Sensitivity [μV/Pa]
Figure 6-27. Sensitivity probability density function.
153
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.340
5
10
15
20
25
Pro
babi
lity
Den
sity
Fun
ctio
n of
Vol
tage
Noi
se
Voltage Noise [μV]
Figure 6-28. Voltage noise probability density function.
500 1000 1500 2000 2500 30000
0.5
1
1.5x 10-3
Pro
babi
lity
Den
sity
Fun
ctio
n of
Pm
ax
Maximum Pressure [Pa]
Figure 6-29. Maximum detectable pressure Probability density function.
154
CHAPTER 7 CONCLUSIONS
The main contributions of this research as well 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 dramatic increase in air traffic is expected,
there is a great need for a reduction in commercial aircraft noise. Aircraft 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 robust 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 audio range (refer to chapter 2). In the past decade
MEMS microphone development has focused on devices specific to the conditions inside a wind
tunnel and within the proximity of a jet engine [37], [45], [55], [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 approaches both the lower and upper end of the Bruel and Kjaer
4138 dynamic range while maintaining a sufficiently high bandwidth. However, Martin’s device
[55] is susceptible to moisture, making it unsuitable for many aeroacoustic applications. This
work focused on designing and fabricating a piezoresistive device designed to be impervious to
moisture.
155
The major contribution of this work has been to develop a model and optimization scheme
for a piezoresistive microphone that allows a designer 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 handling of MEMS microphones in this project,
several suggestions are provided for future devices. Due to the problems with the noise levels
associated with ion-implanted resistors, a switch to solid source diffusion resistors is
recommended. The advantage of this method is that there is no damage to the silicon substrate
associated with the solid source diffusion process, resulting in a drop in the overall 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 loop 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 across the diaphragm. With the
current microphone, wire bonds are needed to electrically connect the die to the device circuitry.
These wires limit the smoothness of the face of the microphone package and can contribute to
156
errors in measurements. Incorporating ETWI would allow the microphone face to be smoother
and limit the possibility of the electrical connections being 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 predicted 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 contains 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 the setup and help isolate the vibrations of the
diaphragm.
Currently, frequency response measurements 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 should 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 to 20 kHz and specialized tweeters can
reach frequencies up to 30 kHz . An ionophone similar to Fransson’s design [143] is
recommended for the audio driver. Also, a new experimental 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 and reference microphone are installed into a large
157
baffle and placed in an anechoic chamber. A noise source is set far enough away to ensure a
spherically spreading wave hits the baffle and that the reference 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 additional area that needs
improvement. For future experiments, the microphone should have incident pressure that is a
pure sine wave, and the harmonics generated would therefore only be caused by non-linearities
in the microphone. Limitations in the experimental set-up in this study produced non-linearities
for sound pressure levels approaching 160 dB , but if an ideal amplifier, signal generator, and
acoustic driver there would still be non-linearities due to the propagation of the high amplitude
sound wave. Preferably, an experimental setup should be developed that pre-distorts the signal
sent to the acoustic driver, resulting in a pure sine wave is received by the microphone, should be
developed. R. Holman demonstrated the feasibility of using a feed-forward loop for generating a
pre-distorted signal to make a pure sine wave for a synthetic jet actuator [145]. This technique
could be used in the acoustic test setup for the 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 capabilities of testing at high sound 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.
158
APPENDIX A COMPOSITE PLATE MECHANICS
Linear Theory
The plate being analyzed is a composite structure 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 3-2. The
Green-Lagrange Strain Tensor for the rr and θθ direction are as follows [111],
2 2 21
2r r z
rru u u uEr r r r
θ⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
(A-1)
and
2 2 22 2
2
1 1 1 1 1 1 2 22
r r z rr r
u u u u u u uE u u u ur r r r r r
θ θ θθθ θ θθ θ θ θ θ θ
⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦. (A-2)
Applying von Karman plate theory and symmetry, equations (A-1) and (A-2) can be simplified.
First equation (A-1) will be reduced. Applying symmetry sets , 0uθ θ∂
=∂
. If the transverse
normals rotate a moderate amount (10-15deg), then the term 2
rur
∂⎛ ⎞⎜ ⎟∂⎝ ⎠
is second order with respect
to rur
∂∂
and can be neglected [111]. (Note: sinθ θ≈ within 1% up to 15deg). The term 2
zur
∂⎛ ⎞⎜ ⎟∂⎝ ⎠
,
however, can only be neglected if the deflection of the membrane is small. Therefore, equation
(A-1) simplifies as follows
21
2r r
rru uEr r
∂ ∂⎛ ⎞= + ⎜ ⎟∂ ∂⎝ ⎠
2
neglected
urθ∂⎛ ⎞+ ⎜ ⎟∂⎝ ⎠
2
symmetry
zur
∂⎛ ⎞+ ⎜ ⎟∂⎝ ⎠small deflection
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
or
159
rrr
dudr
ε = . (A-3)
When the Green-Lagrange equations are simplified, the notation of strain is switched from E to
ε . Equation (A-2) can be greatly simplified due to symmetry:
1r uuErr
θθθ θ
∂= +
∂
22 2
sym.
1 1 1 12
r zuu ur r r
θ
θ θ θ
⎡ ⎤∂∂ ∂⎛ ⎞⎛ ⎞ ⎛ ⎞+ + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦
22
sym.
1 2 2 rr
u uu u ur
θθ θθ θ
∂ ∂+ − +
∂ ∂2
sym.
2
.
r
r r
u
u ur r
⎛ ⎞⎜ ⎟
+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞= + ⎜ ⎟⎝ ⎠
(A-4)
If the radial displacement normalized by the radius is much, much smaller than one, then the
second term can be neglected because it is second order with respect to rur . Therefore,
equation (A-4) becomes
rurθθε = . (A-5)
The radial and axial displacements can be related to displacements at a reference axis by the
following equations taken from classical plate theory [146]
( ) ( ) ( )00,r
dw ru r z u r z
dr= − , (A-6)
and
( ) ( )0,zu r z w r= , (A-7)
where 0 0 and w u are the axial and radial displacements at the reference axis respectively. The
reference axis was chosen to be at the location of the piezoresistors. Therefore, equations (A-3)
and (A-5) in terms of reference axis displacements and neglecting higher order terms are written
as
160
( )( )
initial strainor resultingresidual fromstrain loading
,,
rr rr rrr zr zθθ θθ θθ
ε ε εε ε ε
⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎪ ⎪ = +⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎩ ⎭ ⎩ ⎭⎩ ⎭
, (A-8)
where
0
0
curvatureaxial termterm due todue tobendingstretching
rr rrrr zθθ θθθθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭, (A-9)
with
( )00
, rr
du rdr
ε = (A-10)
( )20
2 ,rr
d w rdr
κ = − (A-11)
( )00 ,u r
rθθε = (A-12)
and
( )01 .dw r
r drθθκ = − (A-13)
Constitutive Relationship
It is assumed that silicon is transversely isotropic. The validity of this assumption is
based on silicon’s small degree of anisotropy. Anisotropy is defined as
44
11 12
2EE E
ϕ =−
, (A-14)
where ijE are the independent elastic moduli. A purely isotropic material has 1ϕ = where
silicon has 1.57ϕ = [147]. Silicon dioxide and silicon nitride are amorphous and therefore
161
isotropic [148]. The constitutive relationships are then defined as (note that the redundant
subscript will be dropped from θθε and rrε for the rest of the document)
[ ] [ ] [ ]0
0r r rrQ Q Q zθ θ θθ
σ ε κεσ ε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭, (A-15)
where [ ]Q is defined as
[ ] 11 122
21 22
111
Q Q EQQ Q
ννν
⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦
. (A-16)
The forces per unit length are found by integrating equation (A-15) as follows
T
B
zr r
z
Ndz
Nθ θ
σσ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭∫ . (A-17)
Substituting equation (A-15) into equation (A-17) yields
[ ] [ ] [ ]T T T
B B B
z z zor r rr
oz z z
NQ dz Q dz Q z dz
Nθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭∫ ∫ ∫ . (A-18)
It is convenient to define two new matrices:
[ ] [ ]T
B
z
z
A Q dz= ∫ (A-19)
and
[ ] [ ]T
B
z
z
B Q zdz= ∫ , (A-20)
where [ ]A and [ ]B are the extensional stiffness matrix and the flexural-extensional matrix due to
coupling respectively. Equation (A-18) can now be written as
[ ] [ ] [ ]o
r r rro
NA A B
Nθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭. (A-21)
162
The moments per unit length can be solved by integrating the stress times its moment arm, z,
over the thickness,
T
B
zr r
z
Mzdz
Mθ θ
σσ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭∫ . (A-22)
Substituting equation (A-15) into equation (A-22) yields
[ ] [ ] [ ] 2T T T
B B B
z z zor r rr
oz z z
MQ zdz Q zdz Q z dz
Mθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭∫ ∫ ∫ . (A-23)
It is now convenient to define the flexural stiffness matrix as follows
[ ] [ ] 2T
B
z
z
D Q z dz= ∫ . (A-24)
Equation (A-23) can now be written as
[ ] [ ] [ ]o
r r rro
MB B D
Mθ θ θθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧ ⎫= + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎩ ⎭. (A-25)
The transverse loads and moments are also grouped in the following manner
0
0
force dueinitial forceto loading
r r rN N NN N Nθ θ θ
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭, (A-26)
and
0
0
moment dueinitial momentto loading
r r rM M MM M Mθ θ θ
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭, (A-27)
where
[ ]0
0
,r rNA
N θ θ
εε
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ (A-28)
163
[ ] [ ] ,o
rr ro
NA B
N θθ θ
κεκε
⎧ ⎫ ⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭⎩ ⎭ ⎩ ⎭ (A-29)
[ ]0
0
,r rMB
M θ θ
εε
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ (A-30)
and
[ ] [ ] .o
rr ro
MB D
M θθ θ
κεκε
⎧ ⎫ ⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭⎩ ⎭ ⎩ ⎭ (A-31)
The initial compression of the plate is applied such that r θε ε= and therefore 0 0 0rN N Nθ= = and
0 0 0rM M Mθ= = .
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 3-2 the elements of the extensional
stiffness matrix are given by
and
01 2 3
11 2 2 21 2 30
01 1 2 2 3 3
12 2 2 21 2 30
,1 1 1
,1 1 1
M T
B M
M T
B M
z z
z z
z z
z z
E E EA dz dz dz
E E EA dz dz dz
ν ν ν
ν ν νν ν ν
= + +− − −
= + +− − −
∫ ∫ ∫
∫ ∫ ∫ (A-32)
which given the location of the reference plane reduces to
and
1 2 311 1 2 32 2 2
1 2 3
1 1 2 2 3 312 1 2 32 2 2
1 2 3
,1 1 1
.1 1 1
E E EA H H H
E E EA H H H
ν ν νν ν νν ν ν
= + +− − −
= + +− − −
(A-33)
Note that these matrices are symmetric, therefore 11 22 12 21 and Q Q Q Q= = . The elements of the
flexural extensional matrix due to coupling are
164
and
01 2 3
11 2 2 21 2 30
01 2 3
12 2 2 21 2 30
,1 1 1
,1 1 1
M T
B M
M T
B M
z z
z z
z z
z z
E E EB zdz zdz zdz
E E EB zdz zdz zdz
ν ν ν
ν ν ν
= + +− − −
= + +− − −
∫ ∫ ∫
∫ ∫ ∫ (A-34)
which given the location of the reference plane reduces to
and ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 21 2 311 1 2 3 3 22 2 2
1 2 3
2 2 21 1 2 2 3 312 1 2 3 3 22 2 2
1 2 3
2 ,2 1 2 1 2 1
2 .2 1 2 1 2 1
E E EB H H H H H
E E EB H H H H H
ν ν ν
ν ν νν ν ν
−= + + +
− − −
−= + + +
− − −
(A-35)
The elements of the flexural stiffness matrix are
and
02 2 21 2 3
11 2 2 21 2 30
02 2 21 2 3
12 2 2 21 2 30
,1 1 1
.1 1 1
M T
B M
M T
B M
z z
z z
z z
z z
E E ED z dz z dz z dz
E E ED z dz z dz z dz
ν ν ν
ν ν ν
= + +− − −
= + +− − −
∫ ∫ ∫
∫ ∫ ∫ (A-36)
which given the location of the reference plane reduces to
and ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
3 3 31 2 311 1 2 3 3 2 3 22 2 2
1 2 3
2 2 31 1 2 2 3 312 1 2 3 3 2 3 22 2 2
1 2 3
3 ,3 1 3 1 3 1
3 .3 1 3 1 3 1
E E ED H H H H H H H
E E ED H H H H H H H
ν ν ν
ν ν νν ν ν
= + + + +− − −
= + + + +− − −
(A-37)
Derivation of Governing Equations
Tension Case
The equilibrium equations were derived in the linear homogenous plate document and are
repeated here for convenience:
0r rdN N Ndr r
θ−+ = , (A-38)
rr r
dMQ M r Mdr θ= + − , (A-39)
165
and
( )0 0r rdwd dr N rp rQ
dr dr dr⎛ ⎞ + + =⎜ ⎟⎝ ⎠
. (A-40)
A pure displacement differential equation governing the composite plate is desired. To achieve
this, all three of the governing equations, as well as equations (A-21) and (A-25) and their
derivates need to be incorporated. The derivation starts by integrating the governing equation
(A-40) at the reference axis to yield
2
0 02r r
dw prr N rQdr
+ + = . (A-41)
Substituting equation (A-39) into (A-41) results in
( )2
02
r rr
M M dMdw prrN rdr r dr
θ−⎛ ⎞+ + + =⎜ ⎟
⎝ ⎠. (A-42)
Equations (A-21) and (A-25), as well as rdMdr
, are substituted into equation (A-42) to yield
( )
( )0
0
2 220 0 0 0 0 0 0 0
11 0 12 112 2
2 3 20 0 0 0 0
12 11 11 123 2
11 12 12 1
2
1
r
r
N
r r
M
dw du dw d u du u dw d wprr A u A r r Bdr dr dr dr dr r dr dr
dw d w d w dw dwB r D r A Adr dr dr r dr dr
B B B B
θ
θ θ
ε ε
ε ε ε ε
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ + + + − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
⎛ ⎞⎛ ⎞− − + − + +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
+ + − +( )0
1 0.M θ
=
(A-43)
Next, equation (A-21) is substituted into the governing equation (A-38) resulting in
( )
( )
0
0
20 0 0
11 11 122 2
3 20 0 0
12 11 113 2 2
1 1
1 1 1 0.
r
N
r
N
d u du u A A Adr r dr r r
d w d w dwA A Br dr r dr r dr
θ
θ
ε ε
ε ε
⎛ ⎞+ − + +⎜ ⎟
⎝ ⎠
⎛ ⎞− + − + − =⎜ ⎟
⎝ ⎠
(A-44)
Reducing equation (A-44) and solving for the u terms yields
166
2 3 2
0 0 0 0 0 0112 2 3 2 2
11
1 1 1d u du u d w d w dwBdr r dr r A dr r dr r dr
⎛ ⎞+ − = + −⎜ ⎟
⎝ ⎠. (A-45)
Equation (A-45) is substituted into equation (A-43) and the higher order terms are neglected to
yield,
0 011
dw dur Adr dr
⎛ ⎞⎜ ⎟⎝ ⎠
0 012
. . .H O T
dw ur Adr r
⎛ ⎞+ ⎜ ⎟⎝ ⎠
2
. . .
3 2 20 0 0 0 011
3 2 211
2
1
H O T
pr
d w d w dw dw d wB r rA dr dr r dr dr dr
+
⎛ ⎞ ⎛ ⎞+ + − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
20
11 12
. . .H O T
dwB Bdr
⎡ ⎤⎢ ⎥ ⎛ ⎞⎢ ⎥ − ⎜ ⎟⎢ ⎥ ⎝ ⎠⎢ ⎥⎣ ⎦ . . .
3 20 0 0 0
11 0 0 03 2
1 0.
H O T
r
cancel
d w d w dw dwr D r N M Mdr dr r dr dr θ
⎛ ⎞− + − + + − =⎜ ⎟
⎝ ⎠
(A-46)
Rearranging equation (A-46) by grouping in terms of 0w produces
3 2 2
* 0 0 0 03 2 * 2
12
d w d w N dw prD r rdr dr D r dr
⎛ ⎞⎛ ⎞+ − + =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
, (A-47)
where
2
* 1111
11
BD DA
= − . (A-48)
Equation (A-47) is differentiated and divided by r to yield
4 3 2
* 0 0 0 0 0 04 3 * 2 2 * 2
2 1 1 1d w d w N d w N dwD pdr r dr D r dr r D r dr
⎡ ⎤⎛ ⎞ ⎛ ⎞+ − + − − =⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦
. (A-49)
Equation (A-49) is now solely a function of 0w .The Laplacian and Biharmonic operators are
defined in polar coordinates as
[ ] [ ] [ ]22
2
1d ddr r dr
∇ = + , (A-50)
and
167
[ ] [ ] [ ] [ ] [ ]4 3 24
4 3 2 2 3
2 1 1d d d ddr r dr r dr r dr
∇ = + − + . (A-51)
Substituting the Laplacian and Biharmonic operators into equation (A-49) yields
* 4 20 0 0D w N w p∇ − ∇ = . (A-52)
The composite plate is axis-symmetric and clamped around the perimeter and therefore is subject
to the following boundary conditions:
0
0
0
0
0
( ) 0,
0
0 ,
( 0) .
r a
r
w r aclampeddw
dr
dw symmetrydr
w r
=
=
= = ⎫⎪
⎛ ⎞ ⎬=⎜ ⎟ ⎪⎝ ⎠ ⎭⎫⎛ ⎞ = ⎬⎜ ⎟
⎝ ⎠ ⎭= < ∞
(A-53)
Compression Case
For the case of in-plane compression 0CN− is substituted into 0N in equation (A-49) to yield
4 3 2
* 0 0 0 0 0 04 3 * 2 2 * 2
2 1 1 1C Cd w d w N d w N dwD pdr r dr D r dr r D r dr
⎡ ⎤⎛ ⎞ ⎛ ⎞+ + − + + =⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦
. (A-54)
Recognizing the Laplacian and Biharmonic operators from equations (A-50) and (A-51)
produces
* 4 20 0 0CD w N w p∇ + ∇ = . (A-55)
Note that the boundary conditions for both cases remains the same.
Solution of Governing Equation
Tension Case
It is possible to write equation (A-52) in the following manner due to the linearity of the
Laplacian
168
2 2 00 0* *
N pw wD D
⎛ ⎞∇ ∇ − =⎜ ⎟⎝ ⎠
. (A-56)
First, the homogeneous solution is solved
200
1 1 0dwd d dr r wr dr dr r dr dr
χ⎡ ⎤⎛ ⎞⎛ ⎞ − =⎢ ⎥⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠⎣ ⎦. (A-57)
Upon integrating equation (A-57), where 0*
ND
χ = , the following differential equation is
obtained
200 1 2
1 lndwd r w C r Cr dr dr
χ⎛ ⎞ − = +⎜ ⎟⎝ ⎠
. (A-58)
Next the homogeneous solution of equation (A-58) is found as
( ) ( )0 3 0 4 0Hw C I r C K rχ χ= + . (A-59)
Variation of parameters is used to find the particular solution of equation (A-58) [149]:
( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )1 2 0 1 2 0
0 0 00 0 0 0
ln ln, ,P
C r C K r C r C I rw I r dr K r dr
W I r K r W I r K rχ χ
χ χχ χ χ χ+ +
= − +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
∫ ∫ , (A-60)
where W[ ] in equation (A-60) is the Wronskian. The Wronskian of the Bessel functions is [150]
( ) ( )0 01,W I r K rr
χ χχ
= −⎡ ⎤⎣ ⎦ . (A-61)
Implementing the identity from (A-61) and rearranging in terms of the constants, the following
equation is obtained:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )0 1 0 0 0 0
2 0 0 0 0
ln ln
.
Pw C I r r r K r dr K r r r I r dr
C I r rK r dr K r rI r dr
χ χ χ χ χ χ
χ χ χ χ χ χ
⎡ ⎤= −⎣ ⎦⎡ ⎤+ −⎣ ⎦
∫ ∫∫ ∫
(A-62)
The integrals multiplying the constant C2 are evaluated as follows
( ) ( ) ( ) ( )0 1 0 1, rI r dr rI r rK r dr rK rχ χ χ χ χ χ= = −∫ ∫ . (A-63)
169
The integrals multiplying the constant C1 are evaluated using integration by parts as follows
and ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1 0
0 1 0
1ln ln
1ln ln .
r r I r dr rI r r I r
r r K r dr rK r r K r
χ χ χ χχ
χ χ χ χχ
= −
= − −
∫
∫ (A-64)
The evaluations are then plugged into equation (A-62)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 0 1 0 0 1 0
2 0 1 0 1
1 1ln ln
.
Pw C I r rK r r K r K r rI r r I r
C I r rK r K r rI r
χ χ χ χ χ χχ χ
χ χ χ χ
⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦+ − −⎡ ⎤⎣ ⎦
(A-65)
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, the particular solution is
( )0 1 22
1 lnPw C r Cχ
= − + . (A-66)
Incorporating the constant outside of equation (A-66) into 1 2and C C , the total solution to the
homogeneous equation (A-57) is now written as
( ) ( )0 1 2 3 0 4 0( ) lnw r C r C C I r C K rχ χ= + + + . (A-67)
Note that solution (A-67) could have been found from the superposition of the solutions to the
following two equations obtained from the factored version of (A-56)
21 1 0w wχ∇ − = , (A-68)
and
22 0w∇ = . (A-69)
Therefore, the solution is
0 1 2w w w= + . (A-70)
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 (A-56) will illustrate. Using
definition (A-70), the homogeneous part of equation (A-56) is written
( )2 2 21 1 2 2 0w w w wχ χ∇ ∇ − + ∇ − = . (A-71)
Restricting w1 and w2 to the following
21 1 0w wχ∇ − = (A-72)
and
22 0w∇ = , (A-73)
equation (A-71) becomes
( )22 0wχ∇ = . (A-74)
With N0 and D* as constants, equation (A-74) simplifies to the restriction (A-73). Thus the
separation of equation (A-56) into (A-68) and (A-69) proves correct. The particular solution to
equation (A-56) is found by assuming a polynomial of the 4th order:
4 3 20 1 2 3 4 5( )pw r c r c r c r c r c= + + + + . (A-75)
Substituting this solution into equation (A-56) yields a particular solution of
2
00
( )4pprw rN
= − . (A-76)
The total solution is then
( ) ( )2
0 1 2 3 0 4 00
( ) ln4prw r C r C C I r C K rN
χ χ= + + + − . (A-77)
Subjecting equation (A-77) to the boundary conditions found in equation (A-53) yields
2
0 00
0 1 0
( ) ( )( ) 1 12 2 ( ) ( )
I a I rpa a rw rN a I a I a
χ χχ χ χ
⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎛ ⎞= − + −⎢ ⎥⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥ ⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭
. (A-78)
171
Equation (A-45) is now used to solve for the radial displacement u . Substituting 0w in equation
(A-45) yields
( )( )
2120 0 0 11
2 20 11 1
12
I rd u du u Bpadr r dr r N A I a
χχ
χ+ − = . (A-79)
The homogenous part of the solution is Cauchy’s equation which has the solution of
( ) 20 1H
cu r c rr
= + . (A-80)
To solve for the particular solution the method of variation of parameters is used. This method
states
( ) ( ) ( )2 10 1 2p
u F r u F ru r u u
W W= − +∫ ∫ , (A-81)
where
( )( )
1
2
1211
0 11 1
,1 ,
( ) ,2
u r
u rI rBpaF r
N A I aχ
χχ
=
=
=
(A-82)
and
1 2
1 2
2u u
W du du rdr dr
= = − . (A-83)
Using the recurrence relationship
( ) ( ) ( )1 12I z I z I zzν ν νν
− +− = , (A-84)
yields the particular solution
( )( )
1110
0 11 12p
I rBpauN A I a
χχ
= . (A-85)
172
The boundary conditions for the radial displacement are as follows
( )( )
0 ,
0 0 ;
u a clamped
u symmetry
= →
= → (A-86)
which yields
( ) ( )( )
1110
0 11 12I rBpa ru r
N A I a aχχ
⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠. (A-87)
Equations (A-78) and (A-87) are non-dimensionalized using the following definitions:
0 0
2 4* *0 11
* *11
, , , ,
1, , , and 2 na
w ur dWW Ua h h d
N a Bpa hk aD hD a h A
ξξ
χ ς η
= = = Θ = −
= = Ρ = = =
(A-88)
such that
( )2 * **
0 0*2 * * *
1 0
1 ( ) ( )1( ) 12 ( ) ( )
I k I kWk k I k I k
ξ ξξ⎧ ⎫− ⎡ ⎤Ρ ⎪ ⎪= + −⎨ ⎬⎢ ⎥
⎣ ⎦⎪ ⎪⎩ ⎭, (A-89)
( )( )
**1
*2 *1
naI kPU
k I k
ξη ς ξ⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠
, (A-90)
and
( )( )
**1
*2 *1
I kPk I k
ξξ
⎛ ⎞⎜ ⎟Θ = −⎜ ⎟⎝ ⎠
. (A-91)
Note that naη is the normalized distance 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 (A-55) can be written as
173
2 2 00 0* *
CN pw wD D
⎛ ⎞∇ ∇ + =⎜ ⎟⎝ ⎠
. (A-92)
First, the homogeneous solution is solved.
200
1 1 0d d d dwr r wr dr dr r dr dr
χ⎡ ⎤⎛ ⎞⎛ ⎞ + =⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦
. (A-93)
Upon integrating equation (A-93), where 0*CN
Dχ = , the following differential equation is
obtained,
200 1 2
1 lnd dwr w C r Cr dr dr
χ⎛ ⎞ + = +⎜ ⎟⎝ ⎠
. (A-94)
Next the homogeneous solution of equation (A-94) is found as
( ) ( )0 3 0 4 0Hw C J r C Y rχ χ= + . (A-95)
Variation of parameters is used to find the particular solution of equation (A-94) [149]:
( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )1 2 0 1 2 0
0 0 00 0 0 0
ln ln, ,P
C r C Y r C r C J rw J r dr Y r dr
W J r Y r W J r Y rχ χ
χ χχ χ χ χ+ +
= − +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
∫ ∫ . (A-96)
The Wronskian of these particular Bessel functions is [150]
( ) ( )0 02,W J r Y r
rχ χ
πχ=⎡ ⎤⎣ ⎦ . (A-97)
Implementing the identity from (A-97) and rearranging in terms of the constants, the following
equation is obtained:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 0 0 0 0
2 0 0 0 0
ln ln2
.2
Pw C J r r r Y r dr Y r r r J r dr
C J r rY r dr Y r rJ r dr
π χ χ χ χ χ χ
π χ χ χ χ χ χ
⎡ ⎤= − +⎣ ⎦
⎡ ⎤+ − +⎣ ⎦
∫ ∫
∫ ∫ (A-98)
The integrals multiplying the constant C2 are evaluated as follows
174
( ) ( ) ( ) ( )0 1 0 1, rJ r dr rJ r rY r dr rY rχ χ χ χ χ χ= =∫ ∫ . (A-99)
The integrals multiplying the constant C1 are evaluated using integration by parts as follows
and ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1 0
0 1 0
1ln ln
1ln ln .
r r J r dr rJ r r J r
r r Y r dr rY r r Y r
χ χ χ χχ
χ χ χ χχ
= +
= +
∫
∫ (A-100)
The evaluations are then plugged into equation (A-98)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 0 1 0 0 1 0
2 0 1 0 1
1 1ln ln2
.2
Pw C J r rY r r Y r Y r rJ r r J r
C J r rY r Y r rJ r
π χ χ χ χ χ χχ χ
π χ χ χ χ
⎡ ⎤⎛ ⎞ ⎛ ⎞= − + + +⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
+ − +⎡ ⎤⎣ ⎦
(A-101)
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, the particular solution is
( )0 1 22
1 lnPw C r Cχ
= − + . (A-102)
Incorporating the constant outside of equation (A-102) into 1 2and C C , the total solution to the
homogeneous equation (A-93) is now written as
( ) ( )0 1 2 3 0 4 0( ) lnw r C r C C J r C Y rχ χ= + + + . (A-103)
The particular solution to equation (A-92) is found by assuming a solution of the form:
4 3 20 1 2 3 4 5( )pw r c r c r c r c r c= + + + + . (A-104)
Substituting this solution into equation (A-92) yields a particular solution of
2
00
( )4pprw rN
= . (A-105)
The total solution is then
175
( ) ( )2
0 1 2 3 0 4 00
( ) ln4prw r C r C C J r C Y rN
χ χ= + + + + . (A-106)
Subjecting equation (A-106) to the boundary conditions found in equation (A-53) yields
2
0 00
0 1 0
( ) ( )( ) 1 12 2 ( ) ( )
J a J rpa a rw rN a J a J a
χ χχ χ χ
⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎛ ⎞= − + −⎢ ⎥⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥ ⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭
. (A-107)
Equation (A-45) is now used to solve for the radial displacement u . Substituting 0w in equation
(A-45) yields
( )( )
2120 0 0 11
2 20 11 1
12
J rd u du u Bpadr r dr r N A J a
χχ
χ+ − = . (A-108)
The homogenous part of the solution is Cauchy’s equation which has the solution of
( ) 20 1H
cu r c rr
= + . (A-109)
To solve for the particular solution, the method of variation of parameters (equation (A-81)) is
used where
( )( )
1
2
1211
0 11 1
,1 ,
( ) ,2
u r
u rJ rBpaF r
N A J aχ
χχ
=
=
=
(A-110)
and
1 2
1 2
2u u
W du du rdr dr
= = − . (A-111)
Using the recurrence relationship
( ) ( ) ( )1 12J z J z J zzν ν νν
− ++ = , (A-112)
yields the particular solution
176
( )( )
1110
0 11 12p
J rBpauN A J a
χχ
= − . (A-113)
The boundary conditions for the radial displacement are the same as in the tension case found in
equation (A-86) which yields
( ) ( )( )
1110
0 11 12J rBpa ru r
N A a J aχχ
⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠. (A-114)
Equations (A-107) and (A-114) are non-dimensionalized using the definitions found in equation
(A-88):
( )2 * **
0 0*2 * * *
1 0
1 ( ) ( )1( ) 12 ( ) ( )
J k J kWk k J k J k
ξ ξξ⎧ ⎫− ⎡ ⎤−Ρ ⎪ ⎪= + −⎨ ⎬⎢ ⎥
⎣ ⎦⎪ ⎪⎩ ⎭, (A-115)
( )( )
**1
*2 *1
naJ kPU
k J k
ξη ς ξ⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠
, (A-116)
and
( )( )
**1
*2 *1
J kPk J k
ξξ
⎛ ⎞⎜ ⎟Θ = −⎜ ⎟⎝ ⎠
. (A-117)
Non-dimensional Stresses
The stress in the composite plate is defined by equation (A-15). The stress is then
decomposed into initial stress and stress due to loading
initial stress stress dueto loading
r r r
θ θ θ
σ σ σσ σ σ
⎧ ⎫ ⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭, (A-118)
where
[ ] [ ]0
0r rrQ Q zθ θθ
σ κεσ κε
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭. (A-119)
177
The non-dimensional radial and tangential stresses due to loading are defined by
2r
r
SihEa
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
, (A-120)
and
2
SihEa
θθ
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
. (A-121)
Note that SiE is Young’s modulus of the bulk silicon. To solve for the non-dimensional stress,
0 and ε κ are non-dimensionalized by
0 0
0 0
2 2
2
2
,
,
,
1 .
r r
r r
dU aEd hU aE
hd W aKd h
dW aKd h
θ θ
θ θ
εξ
εξ
κξ
κξ ξ
= =
= =
= − =
= − =
(A-122)
Tension Case
Solving for the non-dimensionalized strains and curvatures yields
( )( )
( )( )
* * **0 10
*2 * *1 1
1nar
k I k I kPEk I k I k
ξ ξη ςξ
⎛ ⎞− ⎜ ⎟= − +⎜ ⎟⎝ ⎠
, (A-123)
( )
( )**
10*2 *
1
1naI kPE
k I kθ
ξη ςξ
⎛ ⎞− ⎜ ⎟= −⎜ ⎟⎝ ⎠
, (A-124)
( )( )
( )( )
* * **0 1
*2 * *1 1
1r
k I k I kPKk I k I k
ξ ξ
ξ
⎛ ⎞⎜ ⎟= − +⎜ ⎟⎝ ⎠
, (A-125)
and
178
( )
( )**
1*2 *
1
1I kPK
k I kθ
ξ
ξ
⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠
. (A-126)
The radial non-dimensionalized stress due to loading is
( )( ) ( ) ( ) ( )
( )( )( )
* **1 0*
*2 2 * *1 1
11 11
nar i i
i
I k I kP kk I k I k
ξ ξη ην ν
ξν
⎧ ⎫−Π ⎪ ⎪Σ = + + − −⎨ ⎬− ⎪ ⎪⎩ ⎭
, (A-127)
and the tangential stress due to loading
( )( ) ( ) ( ) ( )
( )( )( )
* **1 0*
*2 2 * *1 1
11 11
nai i i
i
I k I kP kk I k I kθ
ξ ξη ην ν ν
ξν
⎧ ⎫−Π ⎪ ⎪Σ = + − − −⎨ ⎬− ⎪ ⎪⎩ ⎭
, (A-128)
where iν is the local Poisson’s ratio. Π is defined as
22
1 if
if
if
m Si
SiOm SiO
Si
SiNm SiN
Si
E EE
E EEE E EE
⎧=⎪
⎪⎪Π = =⎨⎪⎪ =⎪⎩
, (A-129)
where mE is the Young’s modulus in the local region of calculation.
Compression Case
Solving for the non-dimensionalized strains and curvatures yields
( )( )
( )( )
* * **0 10
*2 * *1 1
1nar
k J k J kPEk J k J k
ξ ξη ςξ
⎛ ⎞⎜ ⎟= − +⎜ ⎟⎝ ⎠
, (A-130)
( )
( )**
10*2 *
1
1naJ kPE
k J kθ
ξη ςξ
⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠
, (A-131)
( )( )
( )( )
* * **0 1
*2 * *1 1
1r
k J k J kPKk J k J k
ξ ξ
ξ
⎛ ⎞− ⎜ ⎟= − +⎜ ⎟⎝ ⎠
, (A-132)
and
179
( )
( )**
1*2 *
1
1J kPK
k J kθ
ξ
ξ
⎛ ⎞− ⎜ ⎟= −⎜ ⎟⎝ ⎠
. (A-133)
Using equations (A-130) through (A-133) yields the radial non-dimensionalized stress due to
loading
( )( ) ( ) ( ) ( )
( )( )( )
* **1 0*
*2 2 * *1 1
11 11
nar i i
i
J k J kP kk J k J k
ξ ξη ην ν
ξν
⎧ ⎫−− Π ⎪ ⎪Σ = + + − −⎨ ⎬− ⎪ ⎪⎩ ⎭
, (A-134)
and the tangential stress due to loading
( )( ) ( ) ( ) ( )
( )( )( )
* **1 0*
*2 2 * *1 1
11 11
nai i i
i
J k J kP kk J k J kθ
ξ ξη ην ν ν
ξν
⎧ ⎫−− Π ⎪ ⎪Σ = + − − −⎨ ⎬− ⎪ ⎪⎩ ⎭
. (A-135)
Nonlinear Theory
The Green-Lagrange Strain Tensor for the rr and θθ direction are as follows [111],
2 2 21
2r r z
rru u u uEr r r r
θ⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
(A-136)
and
2 2 22 2
2
1 1 1 1 1 1 2 22
r r z rr r
u u u u u u uE u u u ur r r r r r
θ θ θθθ θ θθ θ θ θ θ θ
⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦(A-137)
Applying von Karman plate theory and symmetry, equations (A-136) and (A-137) can be
simplified. First equation (A-136) will be reduced. Applying symmetry sets , 0uθ θ∂
=∂
. If the
transverse normals rotate a moderate amount ( )10 15° − ° , then the term 2
rur
∂⎛ ⎞⎜ ⎟∂⎝ ⎠
is second order
with respect to rur
∂∂
and can be neglected [111]. (Note: sinθ θ≈ within 1% up to 15° ). The
180
term 2
zur
∂⎛ ⎞⎜ ⎟∂⎝ ⎠
, however, cannot be neglected because of the large deflection of the plate.
Therefore, equation (A-136) simplifies as follows
21
2r r
rru uEr r
∂ ∂⎛ ⎞= + ⎜ ⎟∂ ∂⎝ ⎠
2
neglected
urθ∂⎛ ⎞+ ⎜ ⎟∂⎝ ⎠
2
symmetry
zur
⎡ ⎤⎢ ⎥∂⎛ ⎞⎢ ⎥+ ⎜ ⎟∂⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
or
21
2r z
rru ur r
ε ∂ ∂⎛ ⎞= + ⎜ ⎟∂ ∂⎝ ⎠. (A-138)
When the Green-Lagrange equations are simplified, the notation of strain is switched from
to E ε . Equation (A-137) can be greatly simplified due to symmetry:
1r uuErr
θθθ θ
∂= +
∂
22 2
sym.
1 1 1 12
r zuu ur r r
θ
θ θ θ
⎡ ⎤∂∂ ∂⎛ ⎞⎛ ⎞ ⎛ ⎞+ + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦
22
sym.
1 2 2 rr
u uu u ur
θθ θθ θ
∂ ∂+ − +
∂ ∂2
sym.
2
.
r
r r
u
u ur r
⎛ ⎞⎜ ⎟
+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞= + ⎜ ⎟⎝ ⎠
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 rur . Therefore, equation
(A-137) becomes
rurθθε = . (A-139)
The radial and axial displacements can be related to displacements at a reference axis by the
following equations taken from classical plate theory [146]:
( ) ( ) ( )00,r
dw ru r z u r z
dr= − , (A-140)
181
and
( ) ( )0,zu r z w r= , (A-141)
where 0 0 and w u are the axial and radial displacements at the reference axis respectively.
Therefore, equations (A-138) and (A-139) in terms of reference axis displacements and
neglecting higher order terms are written as
( )( )
initial strainor resultingresidual fromstrain loading
,,
rr rr rrr zr zθθ θθ θθ
ε ε εε ε ε
⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎪ ⎪ = +⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎩ ⎭ ⎩ ⎭⎩ ⎭
, (A-142)
where
0
0
curvatureaxial termterm due todue tobendingstretching
rr rrrr zθθ θθθθ
ε κεε κε
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭, (A-143)
and
( ) ( ) ( )
( ) ( )
2 20 0 00
2
0 00
1 , ,2
1, .
rr rr
du r dw r d w rdr dr dr
u r dw rr r drθθ θθ
ε κ
ε κ
⎛ ⎞= + = −⎜ ⎟
⎝ ⎠
= = −
(A-144)
Governing Differential Equations
The equilibrium equations repeated here for convenience:
0r rdN N Ndr r
θ−+ = , (A-145)
rr r
dMrQ M r Mdr θ= + − , (A-146)
and ( )0 0r rdwd dr N rp rQ
dr dr dr⎛ ⎞ + + =⎜ ⎟⎝ ⎠
. (A-147)
182
A pure displacement differential equation governing the composite plate is desired. The
derivation starts by integrating the governing equation (A-147) to yield
2
02r r
prrN rQφ− + + = . (A-148)
Note that at 0r = the integration constant is equal to zero. Substituting equation (A-146) into
(A-148) and noting the convention dwdr
φ = − results in
( )2
02
r rr
M M dMdw prrN rdr r dr
θ−⎛ ⎞+ + + =⎜ ⎟
⎝ ⎠. (A-149)
Equations (A-21) and (A-25), as well as rdMdr
, are substituted into equation (A-149) to yield
( )
( )0
0
3 20 0 0 0
11 0 12
2 220 0 0 0 0
11 122
3 20 0 0 0
11 11 123 2
11 12
2 2
1 32 2
1
r
r
N
r
M
dw du dw dwr prr A u Adr dr dr dr
d u du u dw dwr B Bdr dr r dr dr
d w d w dw dwr D r A Adr dr r dr dr
B B
θ
θ
ε ε
ε ε
⎛ ⎞⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎡ ⎤⎛ ⎞ ⎛ ⎞+ + − + −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎛ ⎞
− + − + +⎜ ⎟⎝ ⎠
+ + ( )0
12 11 0.r
M
B Bθ
θε ε− + =
. (A-150)
Next, equation (A-21) is substituted into the governing equation (A-145) resulting in
( ) ( )0 0
2 22 20 0 0 0 0 0 0
11 122 2 2
3 20 0 0
113 2 2
11 12 12 11
1 1 12 2
1 1
1 1 0.r r
N N
d u du u dw d w dw dwA Adr r dr r dr dr r dr r dr
d w d w dw Bdr r dr r dr
A A A Ar rθ θε ε ε ε
⎛ ⎞⎛ ⎞ ⎛ ⎞+ − + + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞
− + −⎜ ⎟⎝ ⎠
+ + − + =
(A-151)
Reducing equation (A-151) and solving for the u terms yields
183
2 3 20 0 0 0 0 011
2 2 3 2 211
2 20 0 012
211
1 1 1
112
d u du u d w d w dwBdr r dr r A dr r dr r dr
dw dw d wAA r dr dr dr
⎛ ⎞+ − = + −⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
. (A-152)
Equation (A-152) is substituted into equation (A-150) to yield,
3 220 0 0 0 0 0
11 12 11
23 2 20 0 0 0 0 011 12
113 2 2 211 11
2 30 0
12 3
12 2 2
1 1 112
32
dw du dw dw u dwr prr A r A Bdr dr dr dr r dr
d w d w dw dw dw d wB A rBA dr r dr r dr A r dr dr dr
dw d wB rdr dr
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − + − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎛ ⎞− −⎜ ⎟⎝ ⎠
20 0 0
11 02
1 0.d w dw dwD r Ndr r dr dr
⎛ ⎞+ − + =⎜ ⎟
⎝ ⎠
(A-153)
Rearranging equation (A-153) by grouping of linear and non-linear terms produces
3 22* 0 0 0 0
3 2 * 2
2 20 0 0 0 012
11 12 11 0 12 11211
12
1 3 0,2 2
d w d w N dwpr D r rdr dr D r dr
dw dw du dw d wA rB B r A u A r Bdr dr A dr dr dr
⎛ ⎞⎛ ⎞− + − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞+ ⎜ − + + + − ⎟ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
(A-154)
where
2
* 1111
11
BD DA
= − . (A-155)
Equation (A-154) is then differentiated and divided by r to yield
4 3 2* 0 0 0 0 0 0
4 3 * 2 2 * 2
22 3 20 0 0 0 011 12
12 11 11 112 3 211
20 0 0 0
12 2
20 0
11 2
2 1 1 1
1 3
1
1
d w d w N d w N dwD pdr r dr D r dr r D r dr
dw d w dw d w d wB A B B B r Br dr dr A dr dr dr
u d w dw duAr dr r dr dr
dw d u dwAdr dr r
⎡ ⎤⎛ ⎞ ⎛ ⎞− + − + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤ ⎛ ⎞+ − − − − ⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦⎛ ⎞
+ +⎜ ⎟⎝ ⎠
+ +2 32 2
0 0 0 0 0 0 02 2
3 1 0.2 2
du d w du dw d w dwdr dr dr dr dr dr r dr
⎛ ⎞⎛ ⎞ ⎛ ⎞+ + + =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
(A-156)
184
The linear terms of equation (A-156) resembles the Laplacian and Biharmonic operators which
are defined in polar coordinates as
[ ] [ ] [ ]22
2
1d ddr r dr
∇ = + , (A-157)
and
[ ] [ ] [ ] [ ] [ ]4 3 24
4 3 2 2 3
2 1 1d d d ddr r dr r dr r dr
∇ = + − + . (A-158)
Substituting the Laplacian and Biharmonic operators into equation (A-156) yields
2* 4 2 0 0 11 12
0 0 0 12 11211
23 2 20 0 0 0 0 0 0
11 11 123 2 2
2 32 2 20 0 0 0 0 0 0 0 0
11 2 2 2
1 3
1
1 3 12 2
dw d w B AD w N w B Br dr dr A
dw d w d w u d w dw duB r B Adr dr dr r dr r dr dr
dw d u dw du d w du dw d w dwAdr dr r dr dr dr dr dr dr r dr
⎡ ⎤∇ − ∇ − − −⎢ ⎥
⎣ ⎦
⎛ ⎞ ⎛ ⎞+ + − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− + + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
.p⎛ ⎞
=⎜ ⎟⎜ ⎟⎝ ⎠
(A-159)
Equation (A-159) is a fourth order non-linear ODE with 0 0and w u the dependant variables. It is
not possible to isolate 0w as in the linear case and therefore this equation cannot be solved
analytically. It is important to note that 0N is defined as an in-plane tension. To apply an in-
plane compression, it is convenient to define a compression parameter 0CN where
0 0CN N= − . (A-160)
Equation (A-159) is then written as
185
2* 4 2 0 0 11 12
0 0 0 12 11211
23 2 20 0 0 0 0 0 0
11 11 123 2 2
22 2 20 0 0 0 0 0 0 0 0
11 2 2 2
1 3
1
1 3 12 2
Cdw d w B AD w N w B B
r dr dr A
dw d w d w u d w dw duB r B Adr dr dr r dr r dr dr
dw d u dw du d w du dw d w dwAdr dr r dr dr dr dr dr dr r dr
⎡ ⎤∇ + ∇ − − −⎢ ⎥
⎣ ⎦
⎛ ⎞ ⎛ ⎞+ + − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− + + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
3
.p⎛ ⎞
=⎜ ⎟⎜ ⎟⎝ ⎠
(A-161)
Mixed Form
Tension Case
To facilitate the calculations, a non-dimensional, mixed form of the equations is derived
here. Equation (A-154) is rearranged to yield
3 22* 0 0 0 0
3 2 * 2
20 0 011 12
12 11 2110
20 0 0
11 12
12
1 3 12 2
0.12
d w d w N dwpr D r rdr dr D r dr
dw dw d wB A B BA r dr r dr drdwr
dr du dw uA Adr dr r
⎛ ⎞⎛ ⎞− + − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞− −⎜ ⎟
⎜ ⎟+ =⎜ ⎟⎛ ⎞⎛ ⎞⎜ ⎟+ + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
(A-162)
Recalling that
2 2
0 0 0 0 011 12 11 122
1 12r
du dw u d w dwN A A B Bdr dr r dr r dr
⎛ ⎞⎛ ⎞= + + − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (A-163)
and
0dwdr
φ = − , (A-164)
yields
2 2* 0
2 * 2
2 1211 12
11
12
1 0.2r
Nd d prD r rdr dr D r
Ar N B BA
φ φ φ
φ φ
⎛ ⎞⎛ ⎞+ − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞− + − =⎜ ⎟
⎝ ⎠
(A-165)
186
Dividing by * and D r arranges equation (A-165) in the same fashion as in the non-linear
homogenous plate for a comparison
2 *2
0 122 * 2 * * *
1 12 2
rN N Bd d prdr r dr D r D D rD
φ φφ φ φ⎛ ⎞+ − + = − + −⎜ ⎟⎝ ⎠
, (A-166)
where
* 1212 11 12
11
AB B BA
= − . (A-167)
Equation (A-166) is checked against the 1st mixed ODE for a homogenous plate for validity. It
can be seen the only difference is the last term; however, the constant *12B when calculated for a
homogenous plate is equal to zero. To solve for the 2nd mixed PDE we need to recall equation
(A-144):
( ) ( ) ( )20 0 00 0
1 , 2r
du r dw r u rdr dr rθε ε
⎛ ⎞= + =⎜ ⎟
⎝ ⎠. (A-168)
The derivative of 0θε is taken and then substituted into 0
rε to yield the compatibility equation for
extensional strain due to loading:
20
0 0 01 02r
d dwrdr dr
θθ
ε ε ε ⎛ ⎞+ − + =⎜ ⎟⎝ ⎠
(A-169)
Equation (A-29) is then solved for 0 0and r θε ε to yield
( ) ( )11 12 12 12 11 11 11 12 12 11
02 2
11 12
r
r
dA N A N B A B A B A B Adr r
A A
θφ φ
ε− + − + −
=−
, (A-170)
and
( ) ( )11 12 11 12 12 11 12 12 11 11
02 2
11 12
rdA N A N B A B A B A B Adr r
A A
θ
θ
φ φ
ε− + − + −
=−
. (A-171)
187
Equations (A-170) and (A-171) are then substituted into the compatibility equation (A-169) to
yield
( )
2 2 22 22 * 11 12
122 211
1
32
r rd N dN A Ad dr r B rdr dr dr dr r A
φ φ φ φ⎛ ⎞ −+ = − + − −⎜ ⎟
⎝ ⎠ (A-172)
Equation (A-172) is then checked for validity by comparing it to the 2nd mixed PDE for a
homogenous plate. It is known from the Θ equation that *12B is equal to zero for a homogenous
plate. When the homogenous values for 11 12and A A are entered into the constant ( )1 from
equation (A-172), it reduces to Eh , and therefore equation (A-172) completely reduces to the
homogenous equation. Equations (A-166) and (A-172) are non-dimensionalized by the
following parameters:
0 0
2 22 4* * * *0
* * * *
, , , ,
, , , and ,2
rr
w ur dW aW Ua h h d h
N a N aN a paS S k PD D D hD
θθ
ξ φξ
= = = Θ = − =
= = = =
(A-173)
such that
2
*2 * * 22 2
1 12r
d d k P Sd d
ξξ ξ ξ ξ ξ
⎛ ⎞Θ Θ Λ+ − + Θ = − + Θ − Θ⎜ ⎟
⎝ ⎠, (A-174)
and
2 * * 2
2 22 23
2r rd S dS d d
d d d dξ ξ ξ
ξ ξ ξ ξ ξ⎛ ⎞Θ Θ Θ Χ
+ = −Λ + − − Θ⎜ ⎟⎝ ⎠
, (A-175)
where
*12
*
B hD
Λ = , (A-176)
and
188
2 2 2
11 12*
11
A A hA D−
Χ = . (A-177)
and Λ Χ are important parameters in determining the behavior of the plate. Λ deals with the
symmetry of the composite plate and therefore will be referred to as the symmetry coefficient.
This parameter takes into account that the reference axis is not necessarily the neutral axis. If the
composite plate is symmetric about the defined reference axis then 0Λ = and the more
asymmetric the plate becomes the larger Λ becomes. The approximate range for Λ is from zero
to about .04. Χ represents the disparities of the properties of the different materials of the plate.
If the plate is homogenous then ( )212 1 νΧ = − . For a composite plate, Χ changes accordingly,
with approximately 20% deviation from the homogenous value of the substrate. The following
are the non-dimensional boundary conditions for equations (A-174) and (A-175):
( )0 0ξΘ = = (A-178)
( )1 0ξΘ = = (A-179)
*
0
0rdSd ξξ
=
= (A-180)
Equation (A-175) is second order with respect to rS , and therefore needs one more boundary
condition. The final boundary condition comes from the radial displacement boundary
condition:
( ) 0u r a= = . (A-181)
Recall equation (A-168), if 0u = , the resulting tangential extensional strain due to loading is
( )0 0r aθε = = (A-182)
The subsequent equation is found by substituting equation (A-171) into (A-182)
189
( ) ( ) ( ) ( )( )11 12 11 12 12 11 12 12 11 110
2 211 12
1
0r r ar a r a
r a
dA N A N B A B A B A B Adr a
A A
θ
θ
φ φε
== ==
− + − + −= =
−.(A-183)
Recall equation (A-145) and substitute 0 0 and r rN N N N N Nθ θ= + = +
( ) ( ) ( )0
0 0 0rr
d N Nr N N N N
dr θ
++ + − + = , (A-184)
noting that 0 0dNdr = and solving for Nθ yields
rr
dNN r Ndrθ = + . (A-185)
Substitute equation (A-185) into equation (A-183) to yield
( )12 12 11 11*1212
11 11
1rr
B A B AdN A da N Bdr A dr A a
φ φ−⎛ ⎞+ − = − −⎜ ⎟
⎝ ⎠( ) 0r aφ = =
. (A-186)
Equation (A-186) is then non-dimensionalized to yield
*
*121
11 11
1rr
dS A dSd A dξ
ξξξ ξ===
⎛ ⎞ Θ+ − = −Λ⎜ ⎟
⎝ ⎠. (A-187)
The solution for U is found by first substituting 0dwdrφ = − into equation (A-152)
2 2 2
0 0 0 11 122 2 2 2
11 11
1 1 12
d u du u B Ad d ddr r dr r A dr r dr r A r dr
φ φ φ φ φφ⎛ ⎞⎛ ⎞
+ − = − + − − − −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
. (A-188)
Non-dimensionalizing equation (A-188) yields
2 2
122 2 2 2
11
1 1 1 ,2na
Ad U dU U d d dd d d d A d
ς ηξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
⎧ ⎫⎡ ⎤⎛ ⎞⎛ ⎞Θ Θ Θ Θ Θ⎪ ⎪+ − = − + − + Θ − +⎨ ⎬⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
(A-189)
where
190
ha
ς = , (A-190)
and
11
11
1na
BA h
η = . (A-191)
Note that naη is the non-dimensionalized distance from the reference frame to the neutral axis.
U is not constant through the thickness of the plate. To solve for U at the neutral axis, equation
(A-140) is non-dimensionalized to yield
rU U ης= + Θ , (A-192)
where
zh
η = . (A-193)
Compression Case
For the compression case equation (A-166) is
2 *2
0 122 * 2 * * *
1 12 2
C rN N Bd d prdr r dr D r D D rD
φ φφ φ φ⎛ ⎞+ + − = − + −⎜ ⎟⎝ ⎠
. (A-194)
This equation is then non-dimensionalized to yield
2
*2 * * 22 2
1 12c r
d d k P Sd d
ξξ ξ ξ ξ ξ
⎛ ⎞Θ Θ Λ+ + − Θ = − + Θ − Θ⎜ ⎟
⎝ ⎠ (A-195)
where
2
* 0*
Cc
N akD
= (A-196)
All of the remaining equations are the identical to the tension case.
191
Finite Difference Solution
Equations (A-174) and (A-175) are solved in a straightforward manner using a finite
difference scheme. The ensuing 2nd order accurate central differencing schemes for uniform
discretization are used as approximations for the first and second derivatives:
21 1 ( )2
n n
n
df f f xdx x
ϑ+ −−⎡ ⎤ = + Δ⎢ ⎥ Δ⎣ ⎦, (A-197)
and
( )2
21 12 2
2n n n
n
d f f f f xdx x
ϑ+ −⎡ ⎤ − += + Δ⎢ ⎥ Δ⎣ ⎦
. (A-198)
Before and rSΘ can be solved using the finite difference method, a coordinate transformation is
made to increase the number of grid 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 this case is as follows [151]
ln
1ln1
β ξβ ξ
ξββ
⎧ ⎫+⎨ ⎬−⎩ ⎭=⎧ ⎫+⎨ ⎬−⎩ ⎭
, (A-199)
where ( )1β > is the stretching parameter. As 1β → , more points are bunched near 1ξ = .
Solving equation (A-199) for ξ yields
1 11
1 11
ξ
ξ
ββξ βββ
⎡ ⎤⎡ ⎤+ −⎢ ⎥⎢ ⎥−⎣ ⎦⎢ ⎥= ⎢ ⎥⎡ ⎤+⎢ ⎥+⎢ ⎥−⎢ ⎥⎣ ⎦⎣ ⎦
. (A-200)
In addition to equation (A-200), the first and second order derivatives are calculated using the
chain rule:
192
( ) ( )d d dd d d
ξξ ξ ξ
= (A-201)
and
( ) ( ) ( ) ( ) 22 22
2 2 2
d d d dd d d dd d d d d d d d
ξ ξ ξξ ξ ξ ξ ξ ξ ξ ξ
⎡ ⎤ ⎛ ⎞ ⎛ ⎞= = +⎜ ⎟ ⎜ ⎟⎢ ⎥
⎝ ⎠⎝ ⎠⎣ ⎦, (A-202)
where
2 2
21ln1
ddξ βξ ββ ξ
β
=⎧ ⎫+⎡ ⎤− ⎨ ⎬⎣ ⎦ −⎩ ⎭
(A-203)
and
2
2 22 2
41ln1
dd
ξ βξξ ββ ξ
β
=⎧ ⎫+⎡ ⎤− ⎨ ⎬⎣ ⎦ −⎩ ⎭
. (A-204)
Tension Case
Equation (A-174) is transformed using equations (A-201) through (A-204) and by writing
ξ as a function of ξ is formulated as
( ) ( )( ) ( )
2 2 2*2 * * 2
22 2
1 1 12r
d d d d d k Sd d d d dξ ξ ξ ξ ξξ ξ ξ ξ ξξ ξ ξ ξξ ξ
⎛ ⎞⎛ ⎞⎛ ⎞ Θ Θ Λ⎜ ⎟⎜ ⎟+ + − + Θ = −Ρ + Θ − Θ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠, (A-205)
with boundary conditions
( )0 0ξΘ = = (A-206)
and
( )1 0ξΘ = = . (A-207)
Note that ( )ξ ξ is ξ as a function of ξ , not ξ multiplied by ξ .
Likewise, equation (A-175) is manipulated to reveal
193
( ) ( ) ( )
( ) ( ) ( )
2 2 * *22 2
2 2
2 2 22
2 2
3
,2
r rd S dSd d dd d d d d
d d d d dd d d d d
ξ ξ ξξ ξ ξ ξ ξ ξξ ξ ξ ξ ξ
ξ ξ ξξ ξ ξ ξξ ξ ξ ξ ξ ξ ξ
⎛ ⎞⎛ ⎞+ + =⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞⎛ ⎞ Θ Θ Θ Χ⎜ ⎟−Λ + + − − Θ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
(A-208)
with boundary conditions
*
0
0rdSdd d ξ
ξξ ξ
=
⎛ ⎞=⎜ ⎟
⎝ ⎠, (A-209)
and
*
*121
11 11
1rr
dS Ad d dSd d A d dξ
ξξ
ξ ξξ ξ ξ ξ=
==
⎛ ⎞⎛ ⎞ ⎛ ⎞Θ+ − = −Λ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠. (A-210)
Multiplying by ( )2ξ ξ and applying the central difference equations (A-197) and (A-198) (where
and f x ξ→ Θ → ) to equation (A-205) yields
( ) ( ) ( )1 1n n n n n n nc a b d− +Θ + Θ + Θ = , (A-211)
where
( )( ) ( ) ( )2 2* * 12 1 ,
2nn n r nn n na k Sξ ξ ξ ξ ξ ξ⎛ ⎞= − Ω + + + − Λ Θ⎜ ⎟
⎝ ⎠ (A-212)
,n n nb = Ω + Γ (A-213)
,n n nc = Ω − Γ (A-214)
and
( )3*n n
d ξ ξ= −Ρ . (A-215)
Note that because of the non-linearity in the last term of equation (A-205) the nΘ in equation
(A-212) will be imputing the answer from the previous calculation. The error caused by this will
be minimal. The commonality parameters in equations (A-212) through (A-214) are defined as
194
( ) 2
nn
n
dd
ξ ξ ξξ ξ
⎛ ⎞⎛ ⎞⎜ ⎟Ω = ⎜ ⎟⎜ ⎟Δ ⎝ ⎠⎝ ⎠
(A-216)
and ( ) ( )22
2
12n n n
n n
d dd dξ ξξ ξ ξ ξ
ξ ξ ξ⎛ ⎞⎛ ⎞⎛ ⎞
Γ = +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟Δ ⎝ ⎠ ⎝ ⎠⎝ ⎠. (A-217)
The discretized boundary conditions are written as
0 0Θ = , (A-218)
and
0NΘ = , (A-219)
where n=N corresponds to 1ξ = . Discretizing equation (A-208) yields,
( ) ( ) ( )1 1
* * *n n nn r n r n r ng S e S f S h
− ++ + = , (A-220)
where
2n ne = − Ω , (A-221)
n n nf = Ω + Π , (A-222)
n n ng = Ω − Π , (A-223)
and
( ) ( ) ( ) ( ){ }2
1 1 2 12n n n n n n n n n nh
ξ ξ + −Χ Λ
= − Θ − Θ Ω + Γ + Θ Ω − Γ − Θ Ω + . (A-224)
The new commonality parameter is defined as
( ) ( )22
2
1 32n n n
n n
d dd dξ ξξ ξ ξ ξ
ξ ξ ξ⎛ ⎞⎛ ⎞⎛ ⎞
Π = +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟Δ ⎝ ⎠ ⎝ ⎠⎝ ⎠. (A-225)
195
The boundary conditions (BC) at 0 and 1ξ ξ= = for equation (A-208) contain first order
derivates and therefore are obtained by using third order accurate forward and backward
differencing schemes respectively [152]:
[ ] ( )31 2 3
1 11 18 9 26
nn n n n
df f f f f O xdx x + + += − + − + + Δ
Δ (A-226)
and
[ ] ( )31 2 3
1 11 18 9 26
nn n n n
df f f f f O xdx x − − −= − + − + Δ
Δ. (A-227)
Implemented equations (A-226) and (A-227), the discretized boundary conditions are written as
0 1 2 3
* * * *11 18 9 2 0r r r rS S S S− + − + = (A-228)
and
( ) ( )1 2 3
12
11 * * * *1 2 3
6 111 18 9 2 11 18 9 2
N N N Nr r r r N N N N
N
AA
S S S Sξ
ξ ξ − − − − − −
⎛ ⎞⎛ ⎞Δ −⎜ ⎟⎜ ⎟
⎝ ⎠⎜ ⎟+ − + − = −Λ Θ − Θ + Θ − Θ⎜ ⎟∂ ∂⎜ ⎟⎜ ⎟⎝ ⎠
. (A-229)
Now that the finite difference equations are established, they can be collected into matrix form
1 1 1
2 2 2 2
3 3 3 3
3 3 3 3
2 2 2 2
1 1 1
0 0 ... ... 0 00 0 ... ... 0
0 0 ... ... ...0 0 ... ... ... ... ... ... ...... ... ... ... ... ... 0 0 ...... ... ... 0 00 ... ... 0 00 0 ... ... 0 0
N N N N
N N N N
N N N
a bc a b
c a b
c a bc a b
c a
− − − −
− − − −
− − −
Θ⎡ ⎤ ⎡⎢ ⎥ Θ⎢ ⎥⎢ ⎥ Θ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
Θ⎢ ⎥⎢ ⎥ Θ⎢ ⎥
Θ⎢ ⎥⎣ ⎦ ⎣
1 1 0
2
3
3
2
1 1
...
...
N
N
N N N
d cdd
dd
d b
−
−
− −
− Θ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
− Θ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
, (A-230)
and
196
0
1
2
2
1
*
*
1 1 1*
2 2 2
*2 2 2
*1 1 1
11 18 9 2 0 ... 0 00 0 ... ... 0
0 0 ... ... ...0 0 ... ... ... ... ... ... ...... ... ... ... ... ... 0 0 ...... ... ... 0 00 ... ... 0 00 0 ... 0 2 9 18
N
N
r
r
r
N N N r
N N N r
S
Sg e fSg e f
g e f Sg e f S
S
−
−
− − −
− − −
− −⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
− − ϒ⎢ ⎥⎣ ⎦
1
2
2
1
*
0
...
...
N
N
N
r
hh
hh
−
−
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ Η⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
, (A-231)
where
( )
12
11
6 111 ,
N
AA
d d
ξ
ξ ξ
⎛ ⎞Δ −⎜ ⎟
⎝ ⎠ϒ = + (A-232)
and
( )1 2 311 18 9 2 .N N N NH − − −= −Λ Θ − Θ + Θ − Θ (A-233)
The * and rSΘ equations are coupled. They are solved using an iteration scheme where an initial
guess of Θ from the linear case is used to solve for *rS in equation (A-231). The *
rS obtained is
then used to solve equation (A-230) for the non-linear Θ . The resulting Θ is compared to the
initial guess. If the error is less that 0.05% then Θ and *rS are taken as the solutions, otherwise,
the scheme repeats where the Θ calculated becomes the initial guess. After the iteration is
complete the solution for Θ is used to solve for U . Transforming equation (A-189) to ξ
coordinates yields
197
( ) ( )
( ) ( )
( )
22 2
22 2
22 2
22 2
12
11
1
1
,
12
na
d U dU Ud d
d dd d
A dA d
ξ ξ ξξ ξ ξ ξ ξξ ξ ξ ξ
ξ ξ ξηξ ξ ξ ξ ξξ ξ ξ ξ
ςξ
ξ ξξ ξ
⎛ ⎞⎛ ⎞∂ ∂ ∂⎜ ⎟+ + − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠
⎧ ⎫⎛ ⎞⎛ ⎞⎛ ⎞Θ ∂ Θ ∂ ∂ Θ⎪ ⎪⎜ ⎟⎜ ⎟+ + −⎜ ⎟ ⎜ ⎟⎪ ⎪⎜ ⎟∂ ∂ ∂⎝ ⎠⎪ ⎝ ⎠ ⎪⎝ ⎠− ⎨ ⎬⎛ ⎞⎪ ⎪⎛ ⎞ Θ Θ ∂⎜ ⎟+Θ − +⎪ ⎪⎜ ⎟⎜ ⎟∂⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭
(A-234)
with boundary conditions
( )( )
1 0 ,
0 0 .
U clamped
U symmetry
ξ
ξ
= = →
= = → (A-235)
Applying the finite difference equations (A-197) and (A-198) yields
( ) ( ) ( )1 1n n n n n n nm U k U l U q− ++ + = , (A-236)
where
( )2 1n nk = − Ω + (A-237)
n n nl = Ω + Γ (A-238)
n n nm = Ω − Γ (A-239)
and
( ) ( )
( ) ( )
( ) ( )
1
12
11
1
12
12 1 12
12
n na n n n n
n n na n n
n na n n n n
AqA
ς η ξ ξ
ς η ξ ξ
ς η ξ ξ
+
−
⎧ ⎡ ⎤⎛ ⎞Θ − Ω + Γ − Θ Ω⎪ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎪⎪ ⎡ ⎤⎛ ⎞⎛ ⎞⎪= +Θ Ω + − Θ −⎢ ⎥⎜ ⎟⎨ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎪ ⎣ ⎦⎪ ⎡ ⎤⎛ ⎞⎪ +Θ − Ω − Γ + Θ Ω⎜ ⎟⎢ ⎥⎪ ⎝ ⎠⎣ ⎦⎩
(A-240)
The discretized boundary conditions for U are
198
0 0,0.N
UU
==
(A-241)
The U equations are then collected into matrix form
1 1 1
2 2 2 2
3 3 3 3
3 3 3 3
2 2 2 2
1 1 1
0 0 ... ... 0 00 0 ... ... 0
0 0 ... ... ...0 0 ... ... ... ... ... ... ...... ... ... ... ... ... 0 0 ...... ... ... 0 00 ... ... 0 00 0 ... ... 0 0
N N N N
N N N N
N N N
k l Um k l U
m k l U
m k l Um k l U
m k U
− − − −
− − − −
− − −
⎡ ⎤ ⎡⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣
1 1 0
2
3
3
2
1 1
...
...
N
N
N N N
q mUqq
q l U
−
−
− −
−⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
−⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
. (A-242)
The derivative of U is needed for the stresses and therefore is calculated below. Numerical
differentiation for points 1 to 1 n n N= = − yields
( )1 11
2 n nn n
dU U Ud
ξξ ξ ξ + −
∂= −
∂ Δ (A-243)
Solving for dUdξ at the center of the plate ( )0n = results with
( )2 1 00 0
1 4 32
dU U U Ud
ξξ ξ ξ
∂= − + −
∂ Δ, (A-244)
and at the edge of the plate ( )n N= yields
( )1 21 3 4 .
2 N N NN N
dU U U Ud
ξξ ξ ξ − −
∂= − +
∂ Δ (A-245)
The solutions for the deflection, W , curvature, Ψ , and tangential load, *Sθ , are found from the
numerical solutions of the deflection slope, Θ , and radial load, *rS . They are calculated using
numerical differentiating and integrating schemes of error at least second order. The following
review the exact equations for these parameters in terms of Θ and *rS and their numerical
199
expressions. Note that in these equations, N is the total number of points. The solutions for *Sθ
and W use second order error central differencing schemes for points two through N-1 and
second order error forward and backward difference schemes for the boundary conditions at
1 0 and 1Nξ ξ= = respectively. Nondimensionalizing equation (A-185) yields
*
* *rr
dSS Sdθ ξ
ξ= + . (A-246)
Implementing numerical differentiation for points 1 to 1 n n N= = − yields,
( ) ( )1 1
* * * *12n n n nr r rn
n
S S S Sθξξ ξξ ξ + −
⎛ ⎞∂= − +⎜ ⎟∂ Δ⎝ ⎠
. (A-247)
Solving for Sθ at the center of the plate ( )0n = results with,
( )0
*0
Sθ ξ ξ= ( )1 0 0
* * *
00
1r r rS S Sξ
ξ ξ⎛ ⎞∂
− +⎜ ⎟∂ Δ⎝ ⎠. (A-248)
Sθ at the edge of the plate ( )n N= utilizes the boundary condition (A-181) to yield,
( )* *121 2 3
11
1 11 18 9 2 6N NN N N N r
N
AS SAθ
ξξ ξ − − −
⎛ ⎞∂= −Λ Θ − Θ + Θ − Θ +⎜ ⎟∂ Δ⎝ ⎠
. (A-249)
Nondimensional curvature is defined as
dd
ξξ ξ
⎛ ⎞∂ ΘΨ = ⎜ ⎟∂⎝ ⎠
. (A-250)
Numerical differentiation for points 1 to 1 n n N= = − yields
( )1 11
2n n nn
ξξ ξ + −
∂Ψ = Θ − Θ
∂ Δ. (A-251)
Solving for Ψ at the center of the plate ( )0n = results with
200
( )0 2 1 00
1 4 32
ξξ ξ
∂Ψ = −Θ + Θ − Θ
∂ Δ, (A-252)
and at the edge of the plate ( )n N= yields
( )1 21 3 4
2N N N NN
ξξ ξ − −
∂Ψ = Θ − Θ + Θ
∂ Δ. (A-253)
The solution for deflection is found using a third 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
1 12i i
i i i i
W d
W W
ξ
ξ ξ−− −
= − Θ
Θ − Θ= + −
∫ (A-254)
where [ ], 1:i N n i N= − = . Remember that 0Θ = and 0W = at 1ξ = .
Compression Case
Equation (A-195) is transformed using equations (A-201) through (A-204) and by writing
ξ as a function of ξ is formulated as
( ) ( )( ) ( )
2 2 2*2 * * 2
22 2
1 1 12c r
d d d d d k Sd d d d dξ ξ ξ ξ ξξ ξ ξ ξ ξξ ξ ξ ξξ ξ
⎛ ⎞⎛ ⎞⎛ ⎞ Θ Θ Λ⎜ ⎟⎜ ⎟+ + + − Θ = −Ρ + Θ − Θ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (A-255)
Multiplying by ( )2ξ ξ and applying the central difference equations (A-197) and (A-198) (where
and f x ξ→ Θ → ) to equation (A-255) yields
( ) ( ) ( )1 1n n n n n n nc a b d− +Θ + Θ + Θ = , (A-256)
where
( )( ) ( ) ( )2 2* * 12 1 ,
2nn n c r nn n na k Sξ ξ ξ ξ ξ ξ⎛ ⎞= − Ω − + + − Λ Θ⎜ ⎟
⎝ ⎠ (A-257)
201
,n n nb = Ω + Γ (A-258)
,n n nc = Ω − Γ (A-259)
and ( )3*n n
d ξ ξ= −Ρ . (A-260)
All of the remaining equations are identical to the tension case.
Non-dimensional Stresses
The stress in the composite plate is defined by equation (A-15). The stress is then
decomposed into initial stress and stress due to loading
initial stress stress dueto loading
r r r
θ θ θ
σ σ σσ σ σ
⎧ ⎫ ⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭ ⎩ ⎭, (A-261)
where
[ ] [ ]0
0r rrQ Q zθ θθ
σ κεσ κε
⎧ ⎫⎧ ⎫ ⎧ ⎫= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭. (A-262)
The non-dimensional radial and tangential stresses due to loading are defined by
2r
r
SihEa
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
, (A-263)
and
2
SihEa
θθ
σΣ =
⎛ ⎞⎜ ⎟⎝ ⎠
. (A-264)
Note that SiE is Young’s modulus of the bulk silicon. To solve for the non-dimensional stress,
0 and ε κ are non-dimensionalized by
202
0 0
0 0
2 2
2
2
,
,
,
1 1 .
r r
r r
dU aEd hU aE
hd W d aKd d h
dW aKd h
θ θ
θ θ
εξ
εξ
κξ ξ
κξ ξ ξ
= =
= =
Θ= − = = Ψ =
= − = Θ =
(A-265)
Solving for and r θΣ Σ yields
2
11r i i
i
dU Ud
ν η νν ς ξ ξ ξ
⎛ ⎞⎛ ⎞ ⎛ ⎞Π ΘΣ = + + Ψ +⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠
(A-266)
and
2
11 i i
i
dU Udθ ν η ν
ν ς ξ ξ ξ⎛ ⎞⎛ ⎞ ⎛ ⎞Π Θ
Σ = + + Ψ +⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠ (A-267)
where iν is the local Poisson’s ratio. Π is defined as
22
1 if
if
if
m Si
SiOm SiO
Si
SiNm SiN
Si
E EE
E EEE E EE
⎧=⎪
⎪⎪Π = =⎨⎪⎪ =⎪⎩
, (A-268)
where mE is the Young’s modulus in the local region.
203
APPENDIX B PROCESS TRAVELER
Wafer: 4” n-type (100) SOI, 3000 Å thick buried oxide layer, 1.5 mμ thick silicon device layer and a 350 mμ thick handle layer, 3 5 cm− Ω . Masks: Ground strap mask (GSM) N-Well 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 (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec d) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C 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) N-Well implant
a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec d) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min h) Light O2 ash (RF-600W, O2-400sccms) for 60 sec i) Implant (Phosphorus, 20E keV= , 24 14Q e cm= ) at MEMS exchange j) Acetone bath to remove PR for 3 hours
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3) Piezoresistors and Oxidation a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec d) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min h) Light O2 ash (RF-600W, O2-400sccms) for 60 sec i) Implant (Boron, 5E keV= , 26 14Q e cm= ) at MEMS exchange j) Acetone/Methanol/DI wash to remove PR k) RCA clean at MEMS exchange l) Furnace anneal ( )2 , 1050N T C= ° for 300min at MEMS exchange
m) Dry/Wet/Dry oxidation ( )950T C= ° for 65 min, 21 min, and 65 min respectively n) Chemical/Mechanical Polish to remove backside oxide at ICEMOS
4) Piezoresistor contact cut
a) Acetone/Methanol/DI wash b) Coat HMDS for 5 min c) Coat resist (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec d) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min h) BOE (6:1) for 8 min i) Acetone/Methanol/DI wash
5) Metallization
a) Sputter 1 mμ Al (1%-Si) to avoid spiking using gun 3 for best results b) Acetone/Methanol/DI wash c) Coat HMDS for 5 min d) Coat resist (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec e) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min i) Aluminum etch (Ashland 16:1:1:2) - 40°C for 2 min j) Acetone/Methanol/DI wash
6) Nitride passivation and top vent etch
a) Acetone/Methanol/DI wash
205
b) Plasma enhanced chemical vapor deposition silicon nitride – recipe – MN300A c) Acetone/Methanol/DI wash d) Coat HMDS for 5 min e) Coat resist (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec f) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min j) Etch nitride layer (Unixaxis RIE/ICP) SF6 and O2 k) Etch oxide layer (Unixaxis 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 (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec d) Pre-exposure bake (Hot plate) – 95°C 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) - 95°C for 60 min h) Etch nitride layer (Unixaxis 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 (10 mμ , AZ9260) – spin at 2000 rpm for 50 sec d) Pre-exposure bake (Oven) – 95°C for 15 min e) Coat HMDS on back side for 5 min f) Coat resist on back side (1 mμ , AZ1512) – spin at 4000 rpm for 50 sec g) Pre-exposure bake (Oven) – 95°C 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) - 95°C for 60 min k) Etch Si (DRIE) 10 mμ - 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 (10 mμ , AZ9260) – spin at 2000 rpm for 50 sec o) Attach top side of wafer to carrier wafer p) Bake (Oven) 1– 95°C for 5 min
9) Backside cavity and vent through hole
206
a) Coat HMDS for 5 min b) Coat resist on back side (10 mμ , AZ9260) – spin at 2000 rpm for 50 sec c) Pre-exposure bake (Oven) – 95°C 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) - 95°C for 60 min g) Etch Si (DRIE, SOI kit) 350 mμ - 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 40°C
10) Forming gas anneal and back plate bond
a) Forming gas anneal (N2/H2, 450T C= ° ) for 60 min at MEMS exchange b) Acetone/Methanol/DI wash c) Coat resist on front side to protect Al(10 mμ , 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
207
APPENDIX C MATLAB FUNCTIONS
The following code is the objective function used in the optimization scheme to determine MDP.
function [Obj] = sensitivitybox(X) %%note this code is assuming only p type doping!!! % note this code assumes T=300K only!!!! % %-------------------------------------------------------------------------- % constants Nb = 1e15; kb = 1.3806503e-23; %[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')
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*(1-nu(1)^2))+E(2)*H(2)^2/(2*(1-nu(2)^2))+E(3)/(2*(1-nu(3)^2))*(H(3)^2+2*H(3)*H(2)); A11=E(1)*H(1)/(1-nu(1)^2)+E(2)*H(2)/(1-nu(2)^2)+E(3)*H(3)/(1-nu(3)^2); D11=E(1)*H(1)^3/(3*(1-nu(1)^2))+E(2)*H(2)^3/(3*(1-nu(2)^2))+E(3)/(3*(1-nu(3)^2))*(H(3)^3+3*H(3)*H(2)*(H(3)+H(2))); Dstar=D11-B11^2/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/(zincr-1):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=1.6e-19; etaNref=2.4; etamumin=-0.57; etaalpha=-.146; Nref=2.35e17;
209
mumin=54.3; mu0=406.9; etamu0=-2.23; alpha=.88; pi11=6.6e-11; pi12=-1.1e-11; pi44=138.1e-11; 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 with 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=[(gamma-thetaa/2):(thetaa/(tincr-1)):(gamma+thetaa/2)]; phit=[(gamma+(thetat-thetawt)/2):(thetawt/(tincr-1)):(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 both arc and tapered resistors pila=pi11-2*(pi11-pi12-pi44)*(la(:,1).^2.*ma(:,1).^2+ma(:,1).^2.*na(:,1).^2+na(:,1).^2.*la(:,1).^2); pita=pi12+(pi11-pi12-pi44)*(la(:,1).^2.*la(:,2).^2+ma(:,1).^2.*ma(:,2).^2+na(:,1).^2.*na(:,2).^2); pilt=pi11-2*(pi11-pi12-pi44)*(lt(:,1).^2.*mt(:,1).^2+mt(:,1).^2.*nt(:,1).^2+nt(:,1).^2.*lt(:,1).^2); pitt=pi12+(pi11-pi12-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);
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:(a-rain)/(N-1):a]; rt=[rtin:(a-rtin)/(N-1):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)./(1-nu(1)^2).*((1+nu(1))+(1-nu(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)./(1-nu(1)^2).*((1+nu(1))-(1-nu(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)./(1-nu(1)^2).*((1+nu(1))+(1-nu(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)./(1-nu(1)^2).*((1+nu(1))-(1-nu(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));
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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=((DRarc-DRtea)/(2*Resarc+DRarc+DRtea)).*(Vb*1e6); else if mode ==1 DVo= (cur*1e6)/2*(DRarc-DRtea); end end Sens=DVo/p; Sensitivity = Sens; Resistance = Resarc; Na = trapz(z,Nr)*1e6*(.5*thetaa*(raout^2-rain^2)); %carrier concentration times the volume % Na = Nr*1e6*(.5*thetaa*zj*(raout^2-rain^2)); %carrier concentration times the volume Nt = trapz(z,Nr)*1e6*(thetawt*(rtout^2-rtin^2)); %carrier concentration times the volume % Nt = Nr*1e6*(thetawt*zj*(rtout^2-rtin^2)); %carrier concentration times the volume % Vn = sqrt(4*kb*T*Resarc*(f2-f1)+1/8*hooge*Vb^2*(1/Na+1/Nt)*log(f2/f1))*1e6; if mode == 0 Vn = sqrt(4*kb*T*Resarc*(f2-f1)+1/8*hooge*Vb^2*(1/Na+1/Nt)*log(f2/f1)+(4e-9*(f2-f1))^2)*1e6; else if mode == 1 Vn = sqrt(4*kb*T*Resarc*(f2-f1)+1/8*hooge*(cur*Resarc)^2*(1/Na+1/Nt)*log(f2/f1)+(4e-9*(f2-f1))^2)*1e6; end end Vnoise=Vn; MDP = 20*log10((Vn/Sens)/(20e-6)); Obj = MDP; DynRange = 20*log10(p/(Vn/Sens)); data = [COUNT DynRange Resistance Sensitivity Vnoise MDP k thetawt]; dlmwrite('objectives.txt', data, '-append')
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APPENDIX D OPTIMIZED DEVICES
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Table D-1. Optimized devices operating on a current source (10mA) with a Gaussian dopant profile and Nb = 1e15 [#/cm3]. Bandwidth Pmax MDP Dyn Range H1 H2 H3 a a/h D* Nsurf zj θa rain θt θwt rtin warc larc wtap ltap wgap Ra Isupply Sens Noise FL k Power Pmaxact D. Rg. Act Act. BW
[kHz] [dB SPL] [dB SPL] [dB SPL] [μm] [A] [A] [μm] [--] [mNmm] [1/cm3] [μm] [deg] [μm] [deg] [deg] [μm] [μm] [μm] [μm] [μm] [μm] [Ω] [mA] [μV/Pa] [nV] [--] [W] [dB SPL] [dB SPL] [kHz]150 170 32.7 137.3 2.85 300 300 200 69 0.335 1.4E+19 0.64 29.4 169 12.8 9.3 156 31 87 25 44 9 1000 10.0 8.2 7.09 1.04 0.1 170 137.5 150150 165 31.1 133.9 2.15 300 300 172 78 0.147 1.9E+19 0.51 31.5 144 14.0 9.8 132 28 79 23 41 9 1000 10.0 10.1 7.24 1.35 0.1 165 134.1 150150 160 29.5 130.5 1.84 2052 300 130 62 0.109 2.3E+19 0.43 39.9 103 19.2 13.0 92 27 72 21 38 10 1000 10.0 12.4 7.42 3.08 0.1 160 130.5 150150 155 27.7 127.3 1.48 2203 300 104 60 0.062 3.0E+19 0.36 44.8 81 22.1 14.0 71 23 63 17 33 10 1000 10.0 15.9 7.73 3.40 0.1 155 127.2 150150 150 26.0 124.0 1.14 1280 300 91 70 0.027 3.9E+19 0.30 45.0 71 23.4 14.1 61 20 56 15 30 10 1000 10.0 20.2 8.11 3.40 0.1 150 123.8 150150 145 25.4 119.6 1.00 977 300 85 76 0.018 4.2E+19 0.26 45.0 66 22.9 13.4 60 20 52 14 25 10 1000 10.0 23.3 8.67 3.40 0.1 147 122.1 150150 140 25.4 114.6 1.00 977 300 85 76 0.018 4.2E+19 0.26 45.0 66 22.8 13.3 60 20 52 14 25 10 1000 10.0 23.3 8.67 3.40 0.1 147 122.0 150140 170 32.4 137.6 3.05 300 300 215 69 0.410 1.3E+19 0.67 28.5 181 12.4 9.3 169 33 90 27 46 9 1000 10.0 8.4 7.02 1.01 0.1 170 137.7 140140 165 30.8 134.2 2.30 300 300 185 78 0.180 1.7E+19 0.54 30.5 155 13.6 9.8 143 30 82 24 42 9 1000 10.0 10.4 7.16 1.31 0.1 165 134.4 140140 160 29.1 130.9 1.98 2310 300 138 62 0.136 2.1E+19 0.45 39.1 109 18.8 13.1 99 29 74 23 39 10 1000 10.0 12.8 7.32 3.11 0.1 160 130.8 140140 155 27.3 127.7 1.59 2376 300 111 60 0.076 2.7E+19 0.38 43.8 87 21.5 14.1 76 25 66 19 35 10 1000 10.0 16.5 7.60 3.40 0.1 155 127.6 140140 150 25.5 124.5 1.23 1368 300 97 70 0.034 3.6E+19 0.32 45.0 76 22.8 14.1 66 21 60 16 32 10 1000 10.0 20.9 7.90 3.40 0.1 150 124.3 140140 145 24.4 120.6 1.00 903 300 88 79 0.018 4.3E+19 0.27 45.0 69 23.5 13.9 60 19 54 15 28 10 1000 10.0 24.9 8.30 3.40 0.1 146 121.7 140140 140 24.5 115.5 1.00 903 300 88 79 0.018 4.3E+19 0.27 45.0 69 23.4 13.9 60 19 54 15 28 10 1000 10.0 24.9 8.30 3.40 0.1 146 121.7 140130 170 32.1 137.9 3.29 300 300 231 69 0.510 1.1E+19 0.70 27.5 195 12.0 9.2 183 36 94 29 48 9 1000 10.0 8.7 6.96 0.97 0.1 170 138.1 130130 165 30.4 134.6 2.48 300 300 199 78 0.223 1.5E+19 0.57 29.4 167 13.1 9.7 155 32 86 26 44 9 1000 10.0 10.7 7.09 1.26 0.1 165 134.7 130130 160 28.7 131.3 2.14 2604 300 147 61 0.172 1.8E+19 0.48 38.2 116 18.5 13.2 106 31 78 25 41 10 1000 10.0 13.2 7.22 3.14 0.1 160 131.2 130130 155 26.8 128.2 1.72 2554 300 120 60 0.095 2.4E+19 0.40 42.7 93 21.0 14.2 83 27 69 21 37 10 1000 10.0 17.1 7.47 3.40 0.1 155 128.1 130130 150 25.0 125.0 1.32 1471 300 105 70 0.042 3.2E+19 0.34 44.5 81 22.1 14.2 72 23 63 18 33 10 1000 10.0 21.6 7.72 3.40 0.1 150 124.8 130130 145 23.5 121.5 1.01 845 300 92 82 0.018 4.2E+19 0.27 45.0 72 23.5 14.2 62 20 56 15 30 10 1000 10.0 26.9 8.08 3.40 0.1 145 121.4 130130 140 23.5 116.5 1.00 831 300 91 82 0.018 4.3E+19 0.27 45.0 71 23.5 14.2 62 20 56 15 30 10 1000 10.0 27.1 8.10 3.40 0.1 145 121.2 130120 170 31.7 138.3 3.56 300 300 251 69 0.646 1.0E+19 0.74 26.5 212 11.7 9.1 201 39 98 32 50 9 1000 10.0 8.9 6.89 0.94 0.1 170 138.4 120120 165 30.0 135.0 2.69 300 300 216 79 0.282 1.4E+19 0.60 28.3 181 12.7 9.6 170 35 89 28 46 9 1000 10.0 11.0 7.01 1.22 0.1 165 135.1 120120 160 28.4 131.6 2.04 300 300 185 88 0.126 1.9E+19 0.49 30.5 154 14.0 10.2 143 31 82 25 42 9 1000 10.0 13.6 7.14 1.57 0.1 160 131.8 120120 155 26.3 128.7 1.86 2761 300 130 60 0.120 2.2E+19 0.43 41.5 100 20.4 14.2 91 29 73 23 39 10 1000 10.0 17.7 7.35 3.40 0.1 155 128.5 120120 150 24.5 125.5 1.44 1590 300 113 70 0.053 2.9E+19 0.36 43.3 88 21.5 14.3 78 25 66 19 35 10 1000 10.0 22.5 7.57 3.40 0.1 150 125.4 120120 145 23.0 122.0 1.10 913 300 99 81 0.023 3.8E+19 0.29 45.0 77 22.8 14.3 67 22 60 17 32 10 1000 10.0 27.9 7.85 3.40 0.1 145 121.9 120120 140 22.5 117.5 1.00 760 300 95 86 0.018 4.2E+19 0.27 45.0 74 23.2 14.3 64 21 58 16 31 10 1000 10.0 29.9 7.98 3.40 0.1 143 120.7 120110 170 31.4 138.6 3.89 300 300 274 69 0.835 8.7E+18 0.79 25.4 232 11.3 9.0 221 42 103 35 53 9 1000 10.0 9.2 6.83 0.90 0.1 170 138.8 110110 165 29.7 135.4 2.93 300 300 236 79 0.364 1.2E+19 0.64 27.1 198 12.3 9.5 188 38 94 31 48 9 1000 10.0 11.4 6.93 1.17 0.1 165 135.5 110110 160 28.0 132.0 2.22 300 300 202 89 0.162 1.6E+19 0.52 29.2 168 13.5 10.1 158 34 86 28 44 9 1000 10.0 14.1 7.06 1.51 0.1 160 132.2 110110 155 25.9 129.1 2.04 3000 300 141 60 0.155 1.9E+19 0.45 40.2 109 19.9 14.3 100 32 77 25 41 10 1000 10.0 18.4 7.22 3.40 0.1 155 129.0 110110 150 24.0 126.0 1.57 1731 300 124 70 0.069 2.5E+19 0.38 42.0 96 20.9 14.3 86 28 70 22 37 10 1000 10.0 23.4 7.42 3.40 0.1 150 125.8 110110 145 22.4 122.6 1.20 994 300 108 81 0.030 3.3E+19 0.31 43.8 84 22.0 14.4 74 24 64 19 34 10 1000 10.0 29.2 7.66 3.40 0.1 145 122.5 110110 140 21.4 118.6 1.00 691 300 99 90 0.018 4.1E+19 0.27 45.0 77 22.8 14.4 67 22 60 17 32 10 1000 10.0 33.3 7.85 3.40 0.1 142 120.2 110100 170 31.0 139.0 4.28 300 300 301 69 1.107 7.5E+18 0.84 24.3 255 11.0 8.9 246 46 108 38 55 9 1000 10.0 9.5 6.76 0.86 0.1 170 139.2 100100 165 29.2 135.8 3.22 300 300 260 79 0.481 1.0E+19 0.68 25.9 218 11.9 9.4 209 41 99 34 51 9 1000 10.0 11.8 6.86 1.12 0.1 165 135.9 100100 160 27.6 132.4 2.44 300 300 223 89 0.213 1.4E+19 0.56 27.8 186 13.0 9.9 177 37 90 31 47 9 1000 10.0 14.6 6.97 1.45 0.1 160 132.6 100100 155 25.4 129.6 2.23 3000 300 157 62 0.198 1.6E+19 0.49 38.3 122 19.1 14.2 114 36 82 28 44 10 1000 10.0 19.1 7.09 3.36 0.1 155 129.4 100100 150 23.4 126.6 1.73 1899 300 136 70 0.091 2.2E+19 0.41 40.6 105 20.3 14.4 96 31 74 24 40 10 1000 10.0 24.5 7.28 3.40 0.1 150 126.4 100100 145 21.8 123.2 1.32 1090 300 119 81 0.040 2.9E+19 0.34 42.4 92 21.3 14.5 83 27 68 21 36 10 1000 10.0 30.6 7.49 3.40 0.1 145 123.0 100100 140 20.3 119.7 1.00 624 300 104 95 0.017 3.9E+19 0.27 44.3 80 22.5 14.5 71 24 62 18 32 10 1000 10.0 37.5 7.75 3.40 0.1 140 119.6 10090 170 30.6 139.4 4.76 300 300 335 70 1.513 6.3E+18 0.91 23.1 284 10.6 8.8 276 51 115 42 59 9 1000 10.0 9.9 6.70 0.82 0.1 170 139.5 9090 165 28.8 136.2 3.58 300 300 289 79 0.655 8.8E+18 0.74 24.6 243 11.4 9.3 235 46 104 38 54 9 1000 10.0 12.3 6.79 1.07 0.1 165 136.3 9090 160 27.1 132.9 2.71 300 300 248 90 0.289 1.2E+19 0.60 26.4 207 12.4 9.8 199 41 95 34 49 9 1000 10.0 15.2 6.88 1.39 0.1 160 133.0 9090 155 24.9 130.1 2.46 3000 300 177 64 0.258 1.4E+19 0.52 36.4 137 18.3 14.0 131 40 87 32 47 10 1000 10.0 19.8 6.97 3.31 0.1 155 129.9 9090 150 22.9 127.1 1.93 2106 300 151 70 0.124 1.9E+19 0.44 39.0 116 19.6 14.4 108 35 79 27 42 10 1000 10.0 25.6 7.14 3.40 0.1 150 126.9 9090 145 21.1 123.9 1.47 1208 300 132 81 0.054 2.5E+19 0.36 40.8 101 20.6 14.5 93 30 72 24 38 10 1000 10.0 32.2 7.33 3.40 0.1 145 123.7 9090 140 19.6 120.4 1.12 692 300 115 95 0.024 3.4E+19 0.30 42.6 88 21.7 14.6 80 27 66 20 35 10 1000 10.0 39.7 7.55 3.40 0.1 140 120.3 9080 170 30.2 139.8 5.35 300 300 377 70 2.145 5.1E+18 0.99 21.9 320 10.2 8.7 314 56 122 48 63 8 1000 10.0 10.3 6.64 0.77 0.1 170 139.9 8080 165 28.4 136.6 4.03 300 300 325 80 0.926 7.2E+18 0.80 23.2 274 11.0 9.1 268 51 111 43 57 9 1000 10.0 12.8 6.71 1.01 0.1 165 136.7 8080 160 26.6 133.4 3.05 300 300 280 90 0.407 1.0E+19 0.65 24.9 234 11.9 9.7 227 46 101 38 52 9 1000 10.0 15.9 6.80 1.32 0.1 160 133.6 8080 155 24.4 130.6 2.74 3000 300 203 66 0.349 1.1E+19 0.57 34.2 157 17.4 13.8 152 45 94 37 50 10 1000 10.0 20.6 6.85 3.26 0.1 155 130.4 8080 150 22.3 127.7 2.17 2364 300 169 69 0.176 1.5E+19 0.48 37.3 130 19.0 14.5 124 39 85 31 46 10 1000 10.0 26.9 7.00 3.40 0.1 150 127.5 8080 145 20.5 124.5 1.66 1356 300 148 81 0.077 2.1E+19 0.40 39.0 114 19.8 14.6 107 34 77 27 41 10 1000 10.0 33.9 7.16 3.40 0.1 145 124.3 8080 140 18.8 121.2 1.26 776 300 129 94 0.034 2.8E+19 0.32 40.8 99 20.8 14.7 92 30 71 23 38 10 1000 10.0 42.0 7.36 3.40 0.1 140 121.0 8070 170 29.7 140.3 6.12 300 300 431 70 3.190 4.0E+18 1.09 20.6 367 9.8 8.5 363 64 132 54 68 8 1000 10.0 10.7 6.57 0.72 0.1 170 140.4 7070 165 27.9 137.1 4.60 300 300 372 80 1.373 5.8E+18 0.88 21.8 314 10.5 9.0 310 58 119 49 62 8 1000 10.0 13.4 6.64 0.95 0.1 165 137.3 7070 160 26.1 133.9 3.48 300 300 320 91 0.600 8.1E+18 0.71 23.2 268 11.4 9.5 264 52 109 44 56 9 1000 10.0 16.7 6.71 1.24 0.1 160 134.0 7070 155 23.9 131.1 3.11 3000 300 236 69 0.492 9.2E+18 0.63 31.9 183 16.5 13.5 181 52 102 43 55 9 1000 10.0 21.5 6.74 3.19 0.1 155 131.0 7070 150 21.7 128.3 2.49 2695 300 193 69 0.262 1.2E+19 0.53 35.5 148 18.4 14.5 144 45 92 36 49 10 1000 10.0 28.3 6.87 3.40 0.1 150 128.1 7070 145 19.8 125.2 1.90 1545 300 169 81 0.115 1.7E+19 0.44 37.1 129 19.1 14.6 124 40 84 32 45 10 1000 10.0 35.9 7.01 3.40 0.1 145 125.0 7070 140 18.1 121.9 1.45 884 300 148 94 0.050 2.3E+19 0.36 38.8 113 20.0 14.7 107 35 76 27 41 10 1000 10.0 44.7 7.17 3.40 0.1 140 121.7 7060 170 29.2 140.8 7.14 300 300 503 70 5.045 3.0E+18 1.23 19.2 430 9.4 8.4 429 73 144 63 74 8 1000 10.0 11.2 6.51 0.67 0.1 170 140.9 6060 165 27.4 137.6 5.37 300 300 434 80 2.165 4.4E+18 0.99 20.2 368 10.1 8.8 367 66 130 56 67 8 1000 10.0 14.1 6.57 0.89 0.1 165 137.8 6060 160 25.5 134.5 4.05 300 300 374 91 0.942 6.3E+18 0.80 21.5 314 10.8 9.3 313 60 118 51 61 8 1000 10.0 17.6 6.63 1.16 0.1 160 134.6 6060 155 23.4 131.6 3.58 3000 300 280 72 0.733 7.1E+18 0.70 29.3 219 15.6 13.2 220 61 112 51 60 9 1000 10.0 22.5 6.64 3.10 0.1 155 131.5 6060 150 21.0 129.0 2.90 3000 300 227 70 0.405 9.6E+18 0.59 33.3 173 17.6 14.4 173 54 101 43 54 10 1000 10.0 29.9 6.74 3.38 0.1 150 128.8 6060 145 19.1 125.9 2.23 1798 300 197 81 0.181 1.3E+19 0.49 35.0 150 18.4 14.6 147 47 92 38 49 10 1000 10.0 38.2 6.85 3.40 0.1 145 125.7 6060 140 17.3 122.7 1.69 1028 300 172 94 0.079 1.8E+19 0.40 36.6 131 19.2 14.8 127 41 84 33 45 10 1000 10.0 47.8 6.99 3.40 0.1 140 122.5 6050 170 28.7 141.3 8.57 300 300 604 70 8.683 2.3E+18 1.39 17.7 519 8.9 8.1 521 85 160 73 83 7 1000 10.0 11.8 6.43 0.61 0.1 170 141.4 5050 165 26.8 138.2 6.44 300 300 522 80 3.716 3.2E+18 1.13 18.6 444 9.6 8.6 447 78 144 67 75 8 1000 10.0 14.9 6.49 0.81 0.1 165 138.3 5050 160 24.9 135.1 4.85 300 300 450 92 1.608 4.6E+18 0.91 19.7 379 10.3 9.1 382 71 130 61 68 8 1000 10.0 18.6 6.54 1.06 0.1 160 135.2 5050 155 22.8 132.2 4.23 3000 300 343 75 1.179 5.2E+18 0.79 26.5 269 14.6 12.7 276 74 125 61 66 9 1000 10.0 23.7 6.54 2.99 0.1 155 132.2 5050 150 20.5 129.5 3.42 3000 300 279 74 0.643 7.1E+18 0.67 30.2 214 16.5 14.1 218 65 113 54 61 9 1000 10.0 31.3 6.61 3.30 0.1 150 129.3 5050 145 18.3 126.7 2.68 2151 300 236 81 0.311 9.8E+18 0.56 32.6 179 17.7 14.6 181 57 102 46 55 10 1000 10.0 40.9 6.71 3.40 0.1 145 126.4 5050 140 16.4 123.6 2.04 1229 300 206 94 0.135 1.4E+19 0.46 34.1 156 18.3 14.8 156 50 93 40 50 10 1000 10.0 51.4 6.81 3.40 0.1 140 123.4 50
p p g ( ) p b [ ]
214
Table D-2. Optimized devices operating on a voltage source with a Gaussian dopant profile and Nb = 1e15 [#/cm3]. Bandwidth Pmax MDP Dyn Range H1 H2 H3 a a/h D* Na zj θa rain θt θwt rtin warc larc wtap ltap wgap Ra Vb Sens Noise FL k Power Pmaxact D. Rg. Act Act. BW
[kHz] [dB SPL] [dB SPL] [dB SPL] [μm] [A] [A] [μm] [--] [mNmm] [1/cm3] [μm] [deg] [μm] [deg] [deg] [μm] [μm] [μm] [μm] [μm] [μm] [Ω] [V] [μV/Pa] [nV] [--] [W] [dB SPL] [dB SPL] [kHz]150 170 32.5 137.5 2.85 300 300 200 69 0.335 1.3E+19 0.57 36.2 173 12.2 8.5 145 109 27 22 55 9 1664 12.9 10.4 8.71 1.04 0.1 170 137.7 150150 165 30.9 134.1 2.15 300 300 172 78 0.147 1.7E+19 0.47 37.4 147 13.6 9.2 123 96 25 20 49 10 1544 12.4 12.3 8.66 1.35 0.1 165 134.2 150150 160 29.4 130.6 1.82 1754 300 132 65 0.103 2.1E+19 0.41 43.8 107 18.4 12.1 89 82 25 19 43 10 1382 11.8 14.4 8.52 2.99 0.1 160 130.6 150150 155 27.7 127.3 1.48 2111 300 105 61 0.061 2.9E+19 0.35 45.0 83 21.8 13.7 70 65 21 17 35 10 1161 10.8 17.2 8.34 3.38 0.1 155 127.2 150150 150 26.0 124.0 1.14 1280 300 91 70 0.027 3.9E+19 0.30 45.0 72 23.4 14.0 61 56 19 15 30 10 1041 10.2 20.7 8.29 3.40 0.1 150 123.8 150150 145 25.4 119.6 1.00 977 300 85 76 0.018 4.4E+19 0.27 45.0 65 23.2 13.7 60 51 20 14 25 10 911 9.5 22.0 8.14 3.40 0.1 147 122.1 150150 140 25.4 114.6 1.00 977 300 85 76 0.018 4.4E+19 0.27 45.0 65 23.1 13.6 60 51 20 14 25 10 912 9.6 21.9 8.15 3.40 0.1 147 122.1 150140 170 32.1 137.9 3.05 300 300 215 69 0.410 1.1E+19 0.60 35.6 186 11.9 8.5 156 115 29 23 59 9 1716 13.1 10.8 8.72 1.01 0.1 170 138.0 140140 165 30.6 134.4 2.30 300 300 185 78 0.180 1.6E+19 0.49 36.9 158 13.1 9.0 133 102 27 21 52 9 1617 12.7 12.9 8.72 1.31 0.1 165 134.6 140140 160 29.0 131.0 1.97 2097 300 139 63 0.131 1.9E+19 0.42 43.9 113 18.2 12.2 94 87 26 20 45 10 1444 12.0 15.2 8.56 3.06 0.1 160 131.0 140140 155 27.2 127.8 1.59 2315 300 112 60 0.075 2.6E+19 0.36 45.0 89 21.3 13.7 74 70 23 18 37 10 1224 11.1 18.3 8.38 3.39 0.1 155 127.6 140140 150 25.5 124.5 1.23 1368 300 97 70 0.034 3.5E+19 0.31 45.0 77 22.7 13.9 65 60 20 16 33 10 1103 10.5 22.0 8.33 3.40 0.1 150 124.4 140140 145 24.4 120.6 1.00 903 300 88 79 0.018 4.3E+19 0.27 45.0 69 23.5 14.0 60 54 19 15 28 10 991 10.0 24.8 8.25 3.40 0.1 146 121.7 140140 140 24.5 115.6 1.00 903 300 88 79 0.018 4.3E+19 0.27 45.0 69 23.4 13.9 60 54 19 15 28 10 992 10.0 24.7 8.26 3.40 0.1 146 121.7 140130 170 31.8 138.2 3.29 300 300 231 69 0.510 1.1E+19 0.62 35.3 201 11.3 8.2 169 124 30 24 62 9 1822 13.5 11.4 8.83 0.97 0.1 170 138.4 130130 165 30.2 134.8 2.48 300 300 199 78 0.223 1.4E+19 0.51 36.5 171 12.6 8.9 144 109 28 22 55 9 1697 13.0 13.6 8.78 1.26 0.1 165 134.9 130130 160 28.6 131.4 2.13 2365 300 149 62 0.166 1.7E+19 0.44 43.7 121 17.8 12.2 101 93 28 21 48 10 1518 12.3 16.1 8.62 3.08 0.1 160 131.4 130130 155 26.7 128.3 1.72 2552 300 120 60 0.095 2.3E+19 0.38 45.0 95 20.7 13.6 79 75 24 19 40 10 1296 11.4 19.5 8.44 3.40 0.1 155 128.1 130130 150 25.0 125.0 1.32 1471 300 105 70 0.042 3.1E+19 0.32 45.0 83 22.0 13.9 70 65 22 17 35 10 1173 10.8 23.5 8.38 3.40 0.1 150 124.9 130130 145 23.5 121.5 1.01 845 300 92 82 0.018 4.2E+19 0.27 45.0 72 23.5 14.1 61 57 20 15 30 10 1059 10.3 27.8 8.34 3.40 0.1 145 121.4 130130 140 23.5 116.5 1.00 831 300 91 82 0.018 4.2E+19 0.27 45.0 72 23.5 14.1 61 56 19 15 30 10 1054 10.3 27.9 8.34 3.40 0.1 145 121.2 130120 170 31.4 138.6 3.56 300 300 251 69 0.646 9.3E+18 0.64 34.8 218 10.9 8.0 184 133 32 26 67 9 1911 13.8 12.0 8.90 0.94 0.1 170 138.8 120120 165 29.7 135.3 2.69 300 300 216 79 0.282 1.2E+19 0.53 35.9 186 12.1 8.7 157 117 30 24 59 9 1786 13.4 14.4 8.85 1.22 0.1 165 135.4 120120 160 28.1 131.9 2.32 2677 300 161 61 0.214 1.5E+19 0.46 43.5 131 17.3 12.2 109 99 30 23 52 10 1600 12.7 17.0 8.68 3.11 0.1 160 131.8 120120 155 26.2 128.8 1.86 2761 300 130 60 0.120 2.0E+19 0.40 45.0 103 20.1 13.5 86 81 26 20 43 10 1381 11.8 20.8 8.50 3.40 0.1 155 128.7 120120 150 24.5 125.5 1.44 1590 300 113 70 0.053 2.7E+19 0.34 45.0 90 21.3 13.8 75 71 23 18 38 10 1254 11.2 25.3 8.44 3.40 0.1 150 125.4 120120 145 22.9 122.1 1.10 913 300 99 81 0.023 3.6E+19 0.29 45.0 78 22.7 14.1 66 61 21 16 33 10 1134 10.7 30.0 8.40 3.40 0.1 145 121.9 120120 140 22.5 117.5 1.00 760 300 95 86 0.018 4.0E+19 0.27 45.0 75 23.1 14.1 63 59 20 16 31 10 1095 10.5 31.5 8.38 3.40 0.1 143 120.7 120110 170 31.0 139.0 3.89 300 300 274 69 0.835 8.1E+18 0.67 34.3 239 10.4 7.9 202 143 35 28 72 9 2011 14.2 12.7 8.97 0.90 0.1 170 139.2 110110 165 29.3 135.7 2.93 300 300 236 79 0.364 1.1E+19 0.56 35.4 204 11.5 8.5 172 126 32 25 64 9 1885 13.7 15.3 8.93 1.17 0.1 165 135.8 110110 160 27.6 132.4 2.53 3000 300 175 61 0.280 1.3E+19 0.48 43.1 142 16.8 12.1 118 107 32 25 56 10 1695 13.0 18.1 8.75 3.13 0.1 160 132.3 110110 155 25.8 129.2 2.00 2346 300 146 65 0.140 2.0E+19 0.39 45.0 119 18.1 12.4 98 93 28 21 49 10 1536 12.4 22.0 8.58 3.29 0.1 155 129.1 110110 150 23.9 126.1 1.57 1731 300 124 70 0.069 2.3E+19 0.36 45.0 98 20.6 13.7 82 77 25 20 41 10 1346 11.6 27.2 8.51 3.40 0.1 150 125.9 110110 145 22.3 122.7 1.20 994 300 108 81 0.030 3.2E+19 0.30 45.0 85 21.9 14.0 72 67 23 18 36 10 1220 11.0 32.4 8.46 3.40 0.1 145 122.6 110110 140 21.4 118.6 1.00 691 300 99 90 0.018 3.9E+19 0.27 45.0 78 22.8 14.2 66 61 21 16 33 10 1140 10.7 35.9 8.43 3.40 0.1 142 120.2 110100 170 30.5 139.5 4.28 300 300 301 69 1.107 7.0E+18 0.71 33.7 264 9.9 7.6 224 155 37 30 78 9 2125 14.6 13.5 9.07 0.86 0.1 170 139.6 100100 165 28.8 136.2 3.22 300 300 260 79 0.481 9.5E+18 0.59 34.7 225 11.0 8.3 191 136 35 28 69 9 1998 14.1 16.3 9.01 1.12 0.1 165 136.2 100100 160 27.2 132.8 2.44 300 300 223 89 0.213 1.3E+19 0.49 36.0 191 12.3 9.0 162 120 32 25 61 9 1868 13.7 19.5 8.96 1.45 0.1 160 132.9 100100 155 25.2 129.8 2.23 3000 300 157 62 0.198 1.5E+19 0.44 45.0 126 18.4 13.1 105 99 31 24 53 10 1615 12.7 24.0 8.70 3.36 0.1 155 129.6 100100 150 23.3 126.7 1.73 1899 300 136 70 0.091 2.0E+19 0.38 45.0 108 19.8 13.6 90 85 28 21 45 10 1453 12.1 29.4 8.59 3.40 0.1 150 126.5 100100 145 21.7 123.3 1.32 1090 300 119 81 0.040 2.7E+19 0.32 45.0 94 21.1 13.9 79 74 25 19 40 10 1323 11.5 35.3 8.55 3.40 0.1 145 123.1 100100 140 20.2 119.8 1.00 624 300 104 95 0.017 3.7E+19 0.26 45.0 81 22.4 14.2 69 64 22 17 34 10 1193 10.9 41.3 8.48 3.40 0.1 140 119.6 10090 170 30.1 139.9 4.76 300 300 335 70 1.513 6.0E+18 0.75 33.0 294 9.4 7.4 250 170 41 32 85 9 2250 15.0 14.3 9.16 0.82 0.1 170 140.0 9090 165 28.4 136.6 3.58 300 300 289 79 0.655 8.1E+18 0.62 34.1 251 10.4 8.0 214 149 37 30 75 9 2126 14.6 17.4 9.11 1.07 0.1 165 136.8 9090 160 26.7 133.3 2.71 300 300 248 90 0.289 1.1E+19 0.52 35.3 214 11.6 8.7 181 132 35 28 67 9 1995 14.1 21.0 9.06 1.39 0.1 160 133.4 9090 155 24.6 130.4 2.46 3000 300 177 64 0.258 1.3E+19 0.46 44.3 143 17.5 12.8 119 110 34 27 58 10 1756 13.3 25.9 8.81 3.31 0.1 155 130.2 9090 150 22.7 127.3 1.93 2106 300 151 70 0.124 1.7E+19 0.40 45.0 120 19.0 13.5 100 94 31 24 50 10 1579 12.6 32.0 8.70 3.40 0.1 150 127.1 9090 145 21.0 124.0 1.47 1208 300 132 81 0.054 2.3E+19 0.34 45.0 104 20.2 13.8 88 82 27 21 44 10 1438 12.0 38.5 8.63 3.40 0.1 145 123.8 9090 140 19.5 120.5 1.12 692 300 115 95 0.024 3.1E+19 0.28 45.0 91 21.5 14.1 77 71 24 19 38 10 1304 11.4 45.4 8.57 3.40 0.1 140 120.4 9080 170 29.6 140.4 5.35 300 300 377 70 2.145 5.0E+18 0.81 31.6 332 8.9 7.3 286 183 45 36 91 8 2250 15.0 14.9 9.02 0.77 0.1 170 140.5 8080 165 27.8 137.2 4.03 300 300 325 80 0.926 6.7E+18 0.66 33.2 284 9.8 7.8 243 164 41 33 83 9 2250 15.0 18.6 9.18 1.01 0.1 165 137.3 8080 160 26.1 133.9 3.05 300 300 280 90 0.407 9.0E+18 0.55 34.5 242 10.9 8.4 206 146 38 30 74 9 2141 14.6 22.6 9.17 1.32 0.1 160 134.0 8080 155 24.5 130.5 2.38 529 300 235 95 0.202 1.3E+19 0.44 36.3 201 12.2 9.1 170 128 34 27 65 9 2068 14.4 27.8 9.29 2.09 0.1 155 130.7 8080 150 22.0 128.0 2.17 2364 300 169 69 0.176 1.4E+19 0.42 45.0 135 18.2 13.3 113 106 34 26 56 10 1730 13.2 35.0 8.82 3.40 0.1 150 127.8 8080 145 20.3 124.7 1.66 1356 300 148 81 0.077 1.9E+19 0.36 45.0 118 19.3 13.6 99 92 30 23 49 10 1580 12.6 42.4 8.74 3.40 0.1 145 124.5 8080 140 18.7 121.3 1.26 776 300 129 94 0.034 2.6E+19 0.30 45.0 102 20.5 14.0 86 80 27 21 43 10 1437 12.0 50.4 8.67 3.40 0.1 140 121.1 8070 170 29.1 140.9 6.12 300 300 431 70 3.190 4.0E+18 0.89 30.0 380 8.5 7.1 332 199 51 41 99 8 2250 15.0 15.6 8.87 0.72 0.1 170 141.1 7070 165 27.3 137.7 4.60 300 300 372 80 1.373 5.5E+18 0.73 31.5 325 9.3 7.6 282 179 46 37 90 8 2250 15.0 19.5 9.02 0.95 0.1 165 137.9 7070 160 25.5 134.5 3.48 300 300 320 91 0.600 7.4E+18 0.59 33.2 278 10.3 8.2 239 161 42 34 81 9 2250 15.0 24.3 9.19 1.24 0.1 160 134.6 7070 155 23.4 131.6 3.11 3000 300 236 69 0.492 8.3E+18 0.52 41.8 193 15.4 12.0 161 140 43 34 75 9 2097 14.5 30.6 9.09 3.19 0.1 155 131.4 7070 150 21.3 128.7 2.49 2695 300 193 69 0.262 1.1E+19 0.45 44.4 155 17.4 13.1 129 120 38 30 64 10 1895 13.8 38.6 8.95 3.40 0.1 150 128.5 7070 145 19.5 125.5 1.90 1545 300 169 81 0.115 1.5E+19 0.39 45.0 135 18.4 13.4 113 106 34 26 56 10 1753 13.2 47.1 8.88 3.40 0.1 145 125.3 7070 140 17.9 122.1 1.45 884 300 148 94 0.050 2.1E+19 0.33 45.0 117 19.5 13.8 99 92 31 24 49 10 1600 12.6 56.3 8.80 3.40 0.1 140 121.9 7060 170 28.5 141.5 7.14 300 300 503 70 5.045 3.0E+18 1.00 28.3 444 8.1 7.0 394 219 59 48 109 8 2250 15.0 16.4 8.73 0.67 0.1 170 141.6 6060 165 26.7 138.3 5.37 300 300 434 80 2.165 4.3E+18 0.81 29.6 381 8.8 7.4 336 197 54 43 98 8 2250 15.0 20.5 8.85 0.89 0.1 165 138.5 6060 160 24.9 135.1 4.05 300 300 374 91 0.942 5.8E+18 0.66 31.2 325 9.6 7.9 285 177 49 39 89 8 2250 15.0 25.7 9.00 1.16 0.1 160 135.2 6060 155 22.8 132.2 3.58 3000 300 280 72 0.733 6.5E+18 0.56 39.8 231 14.3 11.5 195 160 49 39 85 9 2250 15.0 33.2 9.14 3.10 0.1 155 132.1 6060 150 20.5 129.5 2.90 3000 300 227 70 0.405 8.7E+18 0.49 43.3 183 16.4 12.8 153 138 44 34 74 10 2100 14.5 42.8 9.11 3.38 0.1 150 129.2 6060 145 18.7 126.3 2.23 1798 300 197 81 0.181 1.2E+19 0.42 44.2 157 17.4 13.2 132 122 39 31 65 10 1949 14.0 52.7 9.03 3.40 0.1 145 126.1 6060 140 17.0 123.0 1.69 1028 300 172 94 0.079 1.6E+19 0.35 45.0 136 18.4 13.6 115 107 35 27 57 10 1807 13.4 63.7 8.97 3.40 0.1 140 122.8 6050 170 27.9 142.1 8.57 300 300 604 70 8.683 2.3E+18 1.12 26.4 535 7.5 6.6 483 247 69 56 122 7 2250 15.0 17.2 8.58 0.61 0.1 170 142.2 5050 165 26.0 139.0 6.44 300 300 522 80 3.716 3.1E+18 0.93 27.6 458 8.2 7.2 412 221 63 52 110 8 2250 15.0 21.8 8.69 0.81 0.1 165 139.1 5050 160 24.2 135.8 4.85 300 300 450 92 1.608 4.4E+18 0.75 29.0 392 9.0 7.7 350 199 58 47 100 8 2250 15.0 27.3 8.81 1.06 0.1 160 136.0 5050 155 22.1 132.9 4.23 3000 300 343 75 1.179 4.9E+18 0.64 36.7 284 13.2 11.1 246 181 59 48 96 9 2250 15.0 35.0 8.90 2.99 0.1 155 132.9 5050 150 19.8 130.2 3.42 3000 300 279 74 0.643 6.4E+18 0.54 40.8 227 15.2 12.4 193 161 52 42 86 9 2250 15.0 46.4 9.10 3.30 0.1 150 129.9 5050 145 17.7 127.3 2.68 2151 300 236 81 0.311 8.7E+18 0.46 43.2 190 16.4 13.0 159 143 46 36 77 10 2192 14.8 59.8 9.22 3.40 0.1 145 127.0 5050 140 16.0 124.0 2.04 1229 300 206 94 0.135 1.2E+19 0.39 43.9 164 17.3 13.3 139 126 42 32 67 10 2036 14.3 72.7 9.13 3.40 0.1 140 123.8 50
g g
215
APPENDIX E DETAILED SPECIFICATIONS OF DEVICE PACKAGE
The following is the layout of the PCB package for testing in the PWT.
Figure E-1. Layout of PCB package.
216
Table E- 1. Passive component specifications. Value Size Type
LT1963 R 1 kΩ 0805 Thin metal film C 10 μF 0805 Ceramic
Pot 10 kΩ 4 mm sq Cermet
AD625 R 20 kΩ 0805 Thin metal film R 20 kΩ 0805 Thin metal film
Pot 100 Ω 4 mm sq Cermet Pot 10 Ω 4 mm sq Cermet C 10 μF 0805 Ceramic C 10 μF 0805 Ceramic
AC filter R 147 kΩ 0805 Thin metal film R 147 kΩ 0805 Thin metal film C 0.68 μF 0805 Ceramic C 0.68 μF 0805 Ceramic
Mic C 10 μF 0805 Ceramic C 10 μF 0805 Ceramic
Power C 10 μF Radial Electrolitic
217
APPENDIX F DETAILS OF EXPERIMENTAL SETUP AND UNCERTAINTY ANALYSIS
Experimental Setup
I-V Measurements Setup
The I-V measurements are taken with an Agilent 4155C semiconductor parameter
analyzer. The settings used are shown in Table F-1.
Table F-1. Agilent 4155C semiconductor parameter analyzer settings. Rin and Rout Diode
Voltage sweep -10V to 10V -20V to 10V points 1001 501
ΔV 20mV 60mV Speed Medium Long
Noise Setup
The noise signal from the microphone under test is amplified by a SRS 560 low noise
amplifier. The settings are shown in Table F-2. The signal is analyzed with a SRS 785 spectrum
analyzer and the settings are shown in Table F-3.
Table F-2. SRS 560 amplifier settings. BUF1 Test arc
Gain 1000 1000 Filter 0.03 Hz - 1MHz 0.03 Hz - 1MHz
Coupling AC AC Setting Low Noise Low Noise
Table F-3. 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
218
Acoustic Setup
There are two setups for the acoustic setup. The first is for the linearity testing. The
settings for the Pulse Multianalyzer are found in Table F-4. Do to limitations in the experimental
setup, the frequency response function was found piecewise and the settings are in Table F-5.
Table F-4. Pulse Multianalyzer settings for linearity testing. FFT Settings
Freq Range 300 Hz - 6.7 kHz # Bins 6400
Bin Width 1 Hz Overlap None
# Averages 300 Window None
Generator Settings
Signal Type Sine wave Frequency 1kHz Amplitude Increasing
Table F-5. 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
219
Uncertainty Analysis
Resistance Values
Since the devices are batch fabricated to yield identical devices, the devices tested for input
and output resistance values which are used to determine 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 ; / 2n nx
st stx x
N Nα αμ
⎡ ⎤− ≤ < +⎢ ⎥
⎣ ⎦, (F-1)
and 2 2
22 2; / 2 ;1 / 2
xn n
ns ns
α α
σχ χ −
⎡ ⎤≤ <⎢ ⎥
⎢ ⎥⎣ ⎦, (F-2)
where N is the number of samples, 1n N= − , t is from the t distribution table, χ is from the
χ distribution table, x is the sample mean and s is the sample standard deviation.
Frequency Response Function
The uncertainty in the frequency response function is comprised of random error in the
measurement as well as bias error due to the analog to digital conversion. For this analysis, it is
assumed that there is no error in the reference microphone used in the measurements. The
normalized random error for the magnitude frequency response function is calculated from the
following equation [126]
( )( )1/ 22ˆ. . 1ˆ
ˆ 2xy xy
r xyxy dxy
s d H fH
nH
γε
γ
⎡ ⎤ −⎣ ⎦⎡ ⎤ = ≈⎣ ⎦ . (F-3)
The standard deviation for the phase in radians is calculated using [126]
( )( )1/ 221ˆ. .2
xyxy
xy d
fs d
n
γφ
γ
−⎡ ⎤ ≈⎣ ⎦ . (F-4)
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 error associated with this is defined as ½ of the bin size. For each
experiment it is important to note the scale for the signal input. For example with the pulse
analyzer, if the signal input was scaled to 707.1mV, the range of discretation is from -707.1mV
to 707.1mV for a range of 1.414V. This system has 16 bits so the bin with is 21.57μV.
Therefore the bias error is ½ of the bin width or 10.79μV. 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 6-18 includes the bias error and it is shown be small 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 (6-9) and the uncertainty
in the total 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. The total number of carriers for an arc resistor is found by assuming a
Gaussian profile with the following equation,
( )2
2 2
0
12
jj
xz z
sa a aout ain s
b
NN r r N dxN
θ
⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞= − ⎜ ⎟
⎝ ⎠∫ . (F-5)
Therefore the uncertainty of aN is related to the uncertainty of j, , , z , ,a aout ain sr r Nθ and
bN . To avoid calculating all of the partial differentials a Monte Carlo simulation is done to
221
calculate the standard deviation of aN . The standard deviations of the independent variables are
taken from Table 6-12. The Monte Carlo simulation was run with 1,000,000 iterations to yield
1.56 9N eσ = . (F-6)
The standard deviation of the Hooge parameter is calculated from the following equation
2 2
N bN bαα ασ σ σ∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
(F-7)
where
10b
Nα∂
=∂
(F-8)
and
110bbNbα −∂
=∂
(F-9)
and bσ is obtained from the least squares fit.
222
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[19] Y. Kanda, "A graphical representation of the piezoresistance coefficients in silicon,"
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BIOGRAPHICAL SKETCH
Brian grew up in Stanhope, NJ, a small town an hour west of New York City, with his
parents, Leo and Helen Homeijer, and brother Dan. After graduating from Lenape Valley
Regional High School in 1999, he attended Lehigh University 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.