DESIGN, FABRICATION AND CHARACTERIZATION OF A MEMS ...

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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

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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



= + + + + +

= + + + + +

= + + + + +

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


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


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)


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)


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


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)


*

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)


( )( )

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

qq

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


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

204

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));

211

Resarc = (thetaa)/log(a/rain)*(1./trapz(z,1./rho)); DRtea = RplusDRtea - Restea; DRarc = RplusDRarc - Resarc; global DynRange Resistance Sensitivity Vnoise MDP if mode == 0 DVo=((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')

212

APPENDIX D OPTIMIZED DEVICES

213

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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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.

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