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Wineglass mode resonators, their applications and study of their quality factor

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Wineglass mode resonators, their applications and study of their quality factor

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WINEGLASS MODE RESONATORS, THEIR APPLICATIONS AND STUDY
OF THEIR QUALITY FACTOR

by
Arash Vafanejad

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)

August 2015

Copyright 2015 Arash Vafanejad

ii
Acknowledgements

I would like to express my sincere gratitude to my advisor, Dr. Eun Sok Kim,
for his support and guidance in assuring the success of this thesis. Without Dr. Kim’s
kind instruction, this work wouldn’t have been completed. I am very fortunate to have
such a nice advisor, who supported me not only by teaching science and technology,
but also academically and emotionally during tough times in the past years.
I also like to especially thank late Dr. John Choma. He was an encouraging,
helpful and kind soul. Despite the fact that I never had the opportunity to take any of
his classes or work with him on a research project, the pleasure of working as his
teaching assistant is indescribable. I consider him as one of my good friends. May he
rest in peace!
I would also like to thank Professors Povinelli and Shung who have given me
the privilege to be advised through their being a part of my defense committee. I would
also like to thank Professor Hashemi for his attendance and many helpful comments
in my qualifying exam. I also like to show my appreciation to A. Dorian Challoner,
founder of Inertial Wave, for helpful and insightful discussions.
I am indebted to all the members in the USCMEMS group for their camaraderie
and support. My deepest gratitude goes to Dr. Lukas Baumgartel, for his valuable
advice and friendship. I also like to thanks Drs. Anderson Lin, Shih-Jui Chen, Youngki
Choe, Lingtao Wang, and Qian Zhang for their friendship and invaluable help during

iii
the course of PhD studies. Much gratitude goes to my fellow USC MEMSians; Yufeng
Wang, Anton Shkel, Lurui Zhao and Yonkui Tang.
My sincere thanks go to Dr. Donghai Zhu for his valuable and steadfast support
on the state-of-the-art facilities at the USC cleanroom. Thank you to all the friends I
have met in U.S. for their friendship and assistance in research.
Finally, I would thank my parents for their support and encouragement to my
education. The same appreciation is extended to my sister, Donya, my cousins and my
great friends.

iv
Table of Contents
Acknowledgements ...................................................................................................... ii
List of Tables............................................................................................................... vi
List of Figures ............................................................................................................ vii
Abstract ...................................................................................................................... xv
Chapter 1
Introduction .................................................................................................................. 1
1.1 Review of MEMS vibratory gyroscopes .............................................. 1
1.2 Review of hemispherical resonance gyroscopes (HRG) ...................... 3
1.3 Overview of the chapters ..................................................................... 7
1.4 Chapter 1 references ............................................................................. 8

Chapter 2
Design, FEM simulations, and fabrication ................................................................. 10
2.1 Design ................................................................................................ 10
2.1.1 Piezoelectric materials ............................................................... 11
2.1.2 Device size ................................................................................. 12
2.2 FEM simulations ................................................................................ 16
2.2.1 Shallow domes ........................................................................... 18
2.2.2 Eccentricity ................................................................................ 19
2.3 Fabrication.......................................................................................... 22
2.3.1 Fabrication challenges ................................................................ 24
2.4 Device simulations as HRG ............................................................... 26
2.5 Chapter 2 references ........................................................................... 27

Chapter 3
Quality factor in hemispherical resonators and ways to improve it ........................... 29
3.1 Air Damping ...................................................................................... 31
3.2 Anchor loss ........................................................................................ 32
3.3 Material .............................................................................................. 38
3.4 Spurious modes .................................................................................. 42
3.5 Thermoelastic damping (TED) .......................................................... 42
3.6 Perforation effect on f*Q ................................................................... 43
3.6.1 Theory ........................................................................................ 44
3.7 Effect of compressive stress in LPCVD layer on Q ........................... 49
3.8 Effect of size on quality factor and f*Q ............................................. 58
3.9 Chapter 3 references ........................................................................... 61

v

Chapter 4
Utilizing our device as a whole angle measurement unit ........................................... 63
4.1 Coriolis Force ..................................................................................... 63
4.2 Mathematical relations ....................................................................... 64
4.2.1 Simulations for signal envelope without non-idealities ............. 65
4.3 Mode splitting in HRG ....................................................................... 68
4.3.1 Frequency split based on the developed model.......................... 70
4.4 Our device results ............................................................................... 72
4.4.1 Response to forward and backward rotation .............................. 73
4.4.2 Minimum detectable rate of rotation .......................................... 75
4.5 Angular gain in our device ................................................................. 76
4.6 Our device performance compared to state of the art ........................ 77
4.7 Other measurements and metrics for gyroscopic applications ........... 78
4.7.1 Bias stability ............................................................................... 79
4.7.2 Bandwidth .................................................................................. 79
4.7.3 Dynamic range ........................................................................... 79
4.7.4 Angle Random Walk (ARW) ..................................................... 80
4.8 Chapter 4 references ........................................................................... 80
Chapter 5
Measurement setup..................................................................................................... 82
5.1 Ring down time measurement setup .................................................. 82
5.2 Rotation measurement setup .............................................................. 83
5.3 Actuating circuit and detection circuit .............................................. 86
5.5 New ideas ........................................................................................... 88
5.5 Chapter 5 references ........................................................................... 90
Chapter 6
Conclusion and Future Directions .............................................................................. 91
6.1 Larger domes ...................................................................................... 92
6.2 Effect of ZnO layer encapsulation ..................................................... 92
6.3 New packaging for gyroscopic applications ..................................... 93
6.4 New circuit ......................................................................................... 93
6.5 Chapter 6 references ........................................................................... 94
Bibliography ............................................................................................................... 95

vi
List of Tables
Table 3.1 Summary of the simulation and experimental results ........................... 49
Table 4.1 Comparison of our device with state of the art devices ........................ 77
Table 4.2 Different metrics in a gyroscope ........................................................... 79

vii
List of Figures
Figure 1.1 A diagram to summarize the main difference between rate and
rate integrating gyroscopes [1] ................................................................ 3
Figure 1.2 Standing wave pattern superimposed on HRG shell. Odd
numbers show the designed nodal points ................................................ 4
Figure 1.3 Simulated relative displacements on the rim of the device. Red
areas and arrows show the antinodal points and their
displacement direction, respectively. The electrode in between
two antinodal point represents a nodal point which will be used
as the sensing electrode in rotation measurements. ............................... 5
Figure 2.1 Schematic design of the device. Note that the device is
supported by a thin diaphragm .............................................................. 11
Figure 2.2 3D plots of different parameters. These simulations have been
done in order to confirm the trend in our device. More detailed
simulations have been done using COMSOL (Section 2.3) ................. 13
Figure 2.3 A 2D cut of the 3D simulations that shows devices with different
parameters such as: radius, resonant frequency, and quality
factor trend for a given thickness in the n=2 wineglass mode
vibration. ............................................................................................... 14
Figure 2.4 Simulated resonant frequency trend for different thicknesses
and radii ................................................................................................ 15
Figure 2.5 Simulated quality factor trend for different thicknesses and radii
............................................................................................................... 16
Figure 2.6 FEM simulation on the vibration displacement of the dome-
shaped diaphragm that clearly shows a four-node wine-glass
vibration mode ...................................................................................... 17
Figure 2.7 FEM simulations on the effect of the cap size on the resonant
frequency: (a) Definition of h and r used in the simulations, (b)
simulation results and (c) resonant frequency versus h/r ratio for
r = 0.5mm and thickness = 1μm. ........................................................... 18
Figure 2.8 For a perfect circle, eccentricity or e is defined to be zero. As
we move from an ideal circle to an oval shape, the e factor

viii
changes .................................................................................................. 19
Figure 2.9 Different eccentricities change the frequency split but their
effect on absolute resonant frequency and quality factor is
negligible ............................................................................................... 21
Figure 2.10 COMSOL simulations show a leeway in the ideal shape of the
rim of the device.................................................................................... 22
Figure 2.11 Cross-sectional diagram (a) and top-view photo (b) of the dome-
shaped-diaphragm transducer that is suspended by a thin,
flexible membrane, and actuated by four piezoelectric
transducers (and sensed by those plus four additional ones)
placed on the rim of the dome diaphragm ............................................. 22
Figure 2.12 Fabrication steps of dome transducers .................................................. 23
Figure 2.13 Top view of a finished device ............................................................... 24
Figure 2.14 Deposition through shadow mask to overcome the difficulties of
3D photolithography ............................................................................. 24
Figure 2.15 Red mask is the original shadow mask, while the green mask is
for adding more electrodes at the edges. ............................................... 25
Figure 2.16 Comparison between three different insulation layers: (a) device
without any insulation layer, having a ring down time of few μs,
(b) current device with a ring down time up to 70 seconds, (c)
device in study for a ring down time higher than 70 seconds.. ............. 26
Figure 2.17 Nodal point output simulations while device is under 1rad/s
rotation. As total angle increases the envelope of amplitude of
the nodal point also increases linearly. For small angles we can
approximate the total angle of rotation by the envelope of nodal
point signal. Note that the nodal point still is vibrating with the
resonance frequency of the device, only the envelope changes. ........... 27
Figure 3.1 Measured ring down time versus pressure. Below 350mTorr,
lowering the pressure does not improve the quality factor
anymore. ................................................................................................ 32
Figure 3.2 Edge size defined as the distance from the anchored edge of the
device to the rim of the hemispherical resonator .................................. 33

ix
Figure 3.3 For each of the edge sizes, FEM simulation has been done to
confirm the second mode circumferential resonance in the
device .................................................................................................... 34
Figure 3.4 Simulated resonant frequency and ring down time versus the
edge size. For edge sizes larger than 200μm, the edge size does
not have any effect on resonant frequency or quality factor ................. 35
Figure 3.5 Simulated f*Q product in the device versus the edge size .................... 36
Figure 3.6 Simulated two degenerate modes in the n=2 wineglass mode
vibration. The two modes have slight different resonant
frequencies, depending on the dome symmetry, and are aligned
to the side (left figure) and diagonal (right figure) of the square
opening. ................................................................................................. 37
Figure 3.7 Simulated vibration amplitude distribution over the diaphragm
when the square edge size is large enough for the wineglass
mode vibration (that is occurring at the circular edge of the
dome) to be free from the influence of the square edges. ..................... 38
Figure 3.8 AFM measurement on the surface of the dome without any
modification .......................................................................................... 39
Figure 3.9 AFM measurement on the surface of the dome after HF dip ................ 40
Figure 3.10 Surface roughness measurements in a device with higher
compressive stress and after performing HF dip................................... 41
Figure 3.11 Calculated resonant frequency and quality factor vs. the shell
radius with and without a hole on the shell. The thickness is
fixed at 1μm. ......................................................................................... 45
Figure 3.12 (Top) FEM simulation of the wine-glass-mode vibration for the
dome diaphragm. (Bottom) FEM simulation of the wine-glass-
mode vibration for the dome diaphragm with a hole at the dome
apex. The simulation shows that the resonant frequency
increases only by 3% while the quality factor increases by more
than 100%. ............................................................................................ 46
Figure 3.13 Simulated resonant frequency and ring-down time versus the
diameter of a hole at the dome’s apex. The optimum hole size is
about 200µm for the four-node wineglass mode resonance with
a dome transducer whose dimensions are as indicated in Figure

x
3.12. If the hole size becomes larger than 240µm in diameter,
the wineglass mode resonance no longer exists. ................................... 47
Figure 3.14 Photo of the device after a post-process laser cutting that
produced a hole in the middle of the dome diaphragm. ........................ 48
Figure 3.15 Measured decay times of the output signals, when the actuating
signal is turned off, for two dome diaphragms: one without any
hole (5.947 sec) and the other with a hole in the center (8.964
sec). ....................................................................................................... 48
Figure 3.16 Ring down time and resonant frequency simulations for
different compressively-stressed SiN layers for the same size. ............ 50
Figure 3.17 Comparison of measurement results of ring down time for two
different devices, one with higher compressive stress in SiN
layer. . .................................................................................................... 51
Figure 3.18 Measured ring down time vs. the deposition temperature for the
SiN. Deposition in higher temperature leads to higher
compressive stress in the SiN layer and so higher ring down
time. ....................................................................................................... 52
Figure 3.19 Measured resonant frequency vs the dome size for two different
stress levels in the SiN layer. As the radius increases, the
resonant frequency decreases. And the resonant frequencies are
higher for the devices with higher compressive stress in the SiN
layer. ...................................................................................................... 53
Figure 3.20 Measured quality factor vs the dome size. As the radius
increases, the quality factor also increases. And the quality
factor is higher for the devices with higher compressive stress
in the SiN layer...................................................................................... 54
Figure 3.21 Measured ring down times and resonant frequencies of the two
devices with different compressive stresses in the SiN layer,
both having a dome radius of 1,000μm. ................................................ 55
Figure 3.22 Measured ring down times and resonant frequencies of the two
devices with different compressive stresses in the SiN layer,
both having a dome radius of 1,200μm ................................................. 56
Figure 3.23 Measured ring down times and resonant frequencies of the two
devices with different compressive stresses in the SiN layer,

xi
both having a dome radius of 1,400μm. ................................................ 57
Figure 3.24 Simulated resonant frequency and quality factor. Increasing the
dome diameter will decrease the resonant frequency while
increasing the quality factor, since the f*Q is constant. ........................ 58
Figure 3.25 Simulated resonant frequency vs. the size of a dome that is
hemisphere rather than a cap ................................................................. 59
Figure 3.26 Depiction of a device having a partial hemisphere. .............................. 59
Figure 3.27 FEM simulation of the resonant frequency vs. h/r ratio for a
partial hemisphere. At a ratio of h/r (lower than 0.25), the
resonant frequency varies sharply, and at an even lower ratio
(less than 0.1) the wineglass mode resonance will be lost. ................... 60
Figure 4.1 Simulated envelopes of the two travelling waves and their
combination (middle line) that results in a standing wave. For
these simulations, a constant rotational speed of Ω = 1 rad/s is
chosen. The particular direction of the rotation chosen for the
simulations makes the output signals appear to decrease in time.
But if the direction is opposite, the output signals will increase
in time. .................................................................................................. 66
Figure 4.2 Simulated amplitude of the standing wave signal on the nodal
point (blue line) vs time, for a constant rotation rate of 1 rad/s.
This amplitude closely represents the total angle of rotation
(shown in green line) as long as the total angle of rotation is less
than 180. .............................................................................................. 67
Figure 4.3 A closer look at Figure 4.2 with the output signal (blue line)
linearly mapped (1) to start at 0 V at t = 0 sec through an offset
and (2) to match the value at t = 1 sec through choosing one
fitting parameter. Over this relatively small angle of rotation
(i.e., less than 57.29 ), the difference between the actual rotation
angle and the simulated output signal is at most 5  (around
20°~30°). ............................................................................................... 67
Figure 4.4 Simulated envelope of the nodal point signal as the combination
of the two travelling waves, while the ideal dome (with f = 0)
is under a constant rotation of 1 rad/sec. As the dome rotates
360  per 6.28 sec, the output signal is sinusoidal in time with a
period of 10.13 seconds......................................................................... 68

xii
Figure 4.5 Two degenerate modes of the n=2 wineglass mode vibration of
the dome resonator (Top) and schematic of their measurement
setup (Bottom). ...................................................................................... 69
Figure 4.6 Simulated frequency responses of the dome resonator over a
wide frequency range (Top) and over a narrow range near the
resonant frequency of the n=2 wineglass vibration mode
showing the frequency split (Bottom)... ................................................ 69
Figure 4.7 (Top) Simulink™ model setup to simulate the performance of
the dome resonator as a rate-integrating gyroscope, as the two
degenerate frequencies are varied (i.e., as the frequency split Δf
is varied). (Bottom) Two Simulink™-simulated and one
measured output of the dome resonator vs. time, as an angular
rotation at a constant speed of 1 /sec is applied. The measured
output (with small random noises) shows a linearly increasing
output signal up to 4 sec, which matches to a simulated output
for Δf = 0.1Hz. If Δf is a little larger (say 0.3Hz), the beat
frequency is simulated to be much higher than that of the
measured data... ..................................................................................... 71
Figure 4.8 Measured output for an applied rate of 1°/s. Top plot shows the
actual output of the nodal point, and the envelope of the signal
increases, as the total angle of rotation increases under a
constant applied rotation rate. Bottom plot shows just the
envelope of the signal............................................................................ 72
Figure 4.9 Measure plot showing the detected angle vs. noise level in the
system. This noise level will lead to a minimum detectable angle
of 0.15° .................................................................................................. 73
Figure 4.10 Measured effect of changing the rotation direction. The sign of
the slope changes, as the rotation direction is flipped. .......................... 74
Figure 4.11 Measured output to 0.1°/s rotation rate, which turns out to be the
minimum detectable rate for our device with current noise floor
............................................................................................................... 75
Figure 4.12 Inertial Measurement Unit (IMU) composed of three
gyroscopes positioned orthogonally in order to measure the
change in angle in all directions. Our device size makes it a very
suitable candidate for miniaturized IMU [12] ....................................... 78

xiii
Figure 5.1 Schematic to illustrate the relative locations for actuation and
sensing used to measure the decay time. ............................................. 83
Figure 5.2 Measured voltage vs time that shows the device’s ring down,
which is 25 seconds. . ........................................................................... 83
Figure 5.3 Photos of the dome diaphragm transducer on a printed circuit
board with electrical wires connected to the bonding pads
through epoxy: (Left) the dome transducer and the whole PCB,
(Middle) a close-up view, (Right) photo of the transducer and
electronics for actuation and sensing. ................................................... 84
Figure 5.4 Cross-sectional diagram (Left) and top view photo (Right) of the
dome-shaped-transducer that is suspended by a thin flexible
membrane and actuated by four piezoelectric transducers and
sensed by four other ones, placed on the rim of the dome
diaphragm. ............................................................................................. 84
Figure 5.5 Schematic of the measurement setup .................................................... 85
Figure 5.6 Oscilloscope traces of the applied signal at the Actuator 1 (the
curve with very little noise with peak-to-peak value of 1.4V),
measured signal at the Sensor 3 (the curve with noise with peak-
to-peak value of 0.04V). The 180° phase difference between the
two curves shows that the vibration mode is indeed a four-node
wine-glass mode.. .................................................................................. 86
Figure 5.7 Inverting voltage amplifier to generate a signal to drive the
piezoelectrically actuating element. Vout = -Vin(R2/R1), and R3
is recommended to be chosen to be R1R2/(R1+R2) for minimum
error due to input bias current. .............................................................. 87
Figure 5.8 Voltage amplifier for the piezoelectrically sensed signal with
voltage gain of 101. The resistance values can be adjusted for a
different gain, as Vout = Vin(R1+R2)/R1. For example, if the 47
Ω resistor is changed to 522 Ω, while keeping the 4.7 kΩ as it
is, the voltage gain will be 10................................................................ 88
Figure 5.9 Measured noise floor and photo of the packaging scheme used
in the measurements so far. ................................................................... 89
Figure 5.10 Measured noise floor and photo of the new breadboard with
smaller wires and better shielding to decrease the noise level.. ............ 89

xiv
Figure 6.1 Effect of the dome size on the resonant frequency and quality
factor ..................................................................................................... 92
Figure 6.2 A new packaging proposal for our device. The smaller package
will let us do gyroscopic measurement more easily, while the
chamber transparency helps in characterizing the device using
an optical probe. .................................................................................... 93

xv

Abstract
This thesis presents microelectromechanical systems (MEMS) hemispherical
wineglass mode resonator and its utilization as a rate integrating gyroscope. A key
objective for this device was to increase the quality factor. Etensive studies have been
done on different parameters affecting the quality factor in the n=2 mode resonance of
a hemispherical resonator. Mathematical derivation and MATLAB simulation have
been performed to optimize the device dimensions. We could achieve a record
breaking 25 seconds of ring down time (associated with a quality factor of 9.9 million).
A finite element method (FEM) model has been developed, and various
simulations have been performed in order to study the effects of different parameters
on the quality factor. The device size and thickness have been optimized based on
FEM simulations. Also studied were pre- and post-processing methods to improve the
quality factor. Specifically investigated was a post-fabrication laser trimming
technique, with which the quality factor was improved by 50% (experimentally)
through making a hole at the apex of the dome diaphragm. Also, the effect of
compressive stress in the SiN diaphragm on the quality factor was studied.
The feasibility of using these resonators as rate integrating gyroscope was
explored, and for the first time functionality of a MEMS resonator as a rate integrating
gyroscope without the need for complex circuitry. The minimum detectable angle of
rotation was measured to be 0.15º, with EMI noise limiting the detection limit.
1

Chapter 1
Introduction
This thesis presents the design, fabrication, and measurements of a hemispherical resonator
(Chapter 2), study of its quality factor (Chapter 3), and its utilization as a whole angle measurement
unit (Chapter 4). In Chapter 5, the measurements setup is presented. General reviews of these three
areas are covered in this chapter.
In this thesis we are going to discuss design, fabrication, measurement, improvement in
quality factor, and application of a hemispherical resonator. The objective behind the design is to
have the maximum achievable quality factor (Q) in the hemispherical resonator and to use the
device as a rate integrating gyroscope or in whole angle mode. This can be achieved by having the
largest possible dome, at the cost of reduced resonant frequency (f), in a 3” silicon wafer. The
fabrication method for this sensor has been developed and optimized. Different pre- and post-
processing methods have been developed and examined in order to increase the f*Q limit in these
devices. These methods include increasing quality factor of the device by having the optimum size
for domes, studying the effect of compressive stress of the diaphragm on device quality factor, and
post-processing trimming of the device (Chapter 3). Several measurements, including ring down
time measurement and rotation measurements, have been done, and functionality of the device as
a rate integrating gyroscope has been shown (Chapter 4).
1.1 Review of MEMS vibratory gyroscopes
All micromachined vibratory gyroscopes can be classified in two broad types: angle
gyroscope (or rate integrating gyroscopes) and rate gyroscopes (Figure 1.1). Almost all reported
2

micromachined vibratory gyroscopes can be classified as rate gyroscopes [1]. As the name
suggests, in rate gyroscopes it is the angular rate that is measured. Measurement of angle from a
rate gyroscope needs integration of the angular rate with respect to time. Although rate gyroscopes
have extensive applications in automotive, aerospace, and consumer electronics, they have several
drawbacks. Noise and bias in the measured angular rate, when integrated, will cause a diverging
error in the measured angle [2]. On the other hand, rate integrating gyroscopes measure the
orientation angle directly. They typically utilize symmetric shells or continuous isotropic
structures such as a vibrating ring, vibratory cylinder, or the hemispherical shell. In this thesis we
discuss a rate integrating gyroscope that has a hemispherical dome as its sensing core. The macro
version of the hemispherical resonator gyro (HRG) has proven itself to be a reliable technology
for space applications [3]. On the other hand, microscale versions of HRG were not popular due
to lithography issues of a 3D structure and the need of complicated circuitry due to the need of
reinforcing the signal. This drawback has been overcome in recent years due to innovations in
fabrication systems, and HRG has become a viable device as a microscale rate integrating
gyroscope. Also since our device has a very large quality factor we could use it in open loop and
without need for a complex control scheme.
3

Figure 1.1 A diagram to summarize the main difference between rate and rate integrating
gyroscopes [1].

1.2 Review of hemispherical resonance gyroscopes
HRG has its roots back in the late 19th century, when it was mentioned on G.H. Bryan
paper [4]. In his paper he described the different beats as the effect of rotating a vibrating shell
about its symmetry axis. In a vibrating shell such as a wineglass, rotation about its stem will cause
the nodes of vibration on the rim of the shell to move at an angular velocity that is slower than the
shell itself. From this observation, the concept of modern HRG started. As with all the other
Rate Integrating Gyroscope
Sensing errors
Bias Drift
Angle Random Walk
Scale Factor
No
accumulated
error
Rate Gyroscope
Sensing errors
Bias Drift
Angle Random Walk
Scale Factor




Errors are
also
integrated


4

MEMS gyroscopes, HRG operates by exploiting the Coriolis effect. The Coriolis effect is a
deflection of moving objects when they are viewed in a rotating reference frame. In a reference
frame with clockwise rotation, the deflection is to the left of the motion of the object, while with
counter-clockwise rotation, the deflection is to the right. Although concept, theory, and
understanding of HRG were fully available by late 1800, the first commercialized HRG did not
find success until 1990 when a group of researcher at Delco made and tested the first successful
HRG [5].
HRG operation: HRG shape is that of a wineglass or hemispherical shell. If the rim of the
shell gets struck, a standing wave will form (Figure 1.2). The motion of the hemisphere’s rim is in
radial and vertical directions. It is worth mentioning that this ratio is a decisive factor in the quality
factor of the devices. Points with maximum deflection are defined as anti-nodes, and points with
no deflection are defined as nodes. The position of nodes and antinodes are stable with respect to
the shell. If, however, the shell is rotated about its symmetry axis, the standing wave lags the
physical rotation by a precise amount. The lag for hemispherical shells is approximately 0.3 of the
angle rotated with the 0.3 factor termed the geometric scale factor of the gyro. So, for example, if
gyro is rotated by 45°, the standing wave pattern lags by 13.5°. Figure 1.3 shows the relative
displacements on the rim of a hemispherical resonator.

Resonator
nodal point
Standing
wave
Figure 1.2 Standing wave pattern superimposed on HRG shell. Odd
numbers show the designed nodal points.
5

Figure 1.3 Simulated relative displacements on the rim of the device. Red areas and arrows show
the antinodal points and their displacement direction, respectively. The electrode in between two
antinodal point represents a nodal point which will be used as the sensing electrode in rotation
measurements.

MEMS HRG: In spite of the fact that macro-versions of HRG devices have been used in
craft control systems for a long time, HRG MEMS devices have not been around for long due to
difficulty of 3-D structures, lithography and fabrication. At the beginning years of research on
MEMS gyroscopes, some found success in fabricating rate integrating gyroscopes but these
6

devices usually used a ring rather than the whole hemisphere. Although the concept is the same,
the sensitivity of HRG will be much higher, if the whole hemisphere is used. First implementations
of MEMS HRG date back to 1994 [6] when researchers in University of Michigan developed a
vibrating ring gyroscope with 0.5 °/s/√Hz resolution, fabricating it by metal electroforming. This
gyro was electrostatically excited, and the transfer of energy to the secondary flexural mode due
to the Coriolis effect was detected. In the same year (1994) British Aerospace System reported a
single crystal ring gyroscope with a resolution of 0.5°/s/√Hz over a 100Hz bandwidth [7]. In 1997
Delphi reported a vibratory ring gyroscope with electroplated metal ring structure. The ring was
built on top of CMOS circuits and suspended by semicircular rings [8]. In 2002, University of
Michigan reported a 150 µm thick bulk micromachined single crystal silicon vibrating ring
gyroscope with a resolution of 10.4 °/hr/√Hz [9]. Researchers at Georgia Institute of Technology
showed a high frequency xyz-axis single disk silicon gyroscope in 2008 [10]. In recent years and
due to innovation in fabrication technology, complete HRG devices have been designed and
fabricated. Researchers at University of California, Irvine, developed a fabrication method for
microscale glass-blown 3-D spherical shell resonators in 2011 [11]. Later in 2012, the same group
implemented a 3-D spherical shell resonator gyroscope using their developed fabrication method
[12]. In 2012 a fabrication method for 3-D micromachined hemispherical shell resonator was
developed by researchers at Georgia Institute of Technology [13]. They claimed that by further
optimization their device can be used as a micro-hemispherical gyroscope for portable inertial
navigation. In 2012 another method for micromachining 3-D hemispherical features in silicon via
micro electrical discharge machining was introduced [14]. University of Michigan reported a high-
Q birdbath resonator gyroscope in 2013 and showed its functionality as a rate integrating
gyroscope [15]. The same group in 2014 showed a very high quality factor and the highest reported
7

decay time constant of 43 seconds [16]. Most approaches in literature have two key aspects: they
use electrostatic actuation and sensing, and their structure is supported by a pole in the middle of
the dome.
In our device we use piezoelectric actuation and sensing. In this way we will achieve
limitless dynamic range, high sensitivity, and lower power consumption. Also we support the
dome structure by a flexible thin SiN diaphragm which gives the device freedom to vibrate and
bring down the anchor loss considerably (Chapter 3).
1.3 Overview of the Chapters
In Chapter 1, review of the device and a literature review of hemispherical resonant
devices, both in the micro- and macro- world, are described as a brief introduction to the thesis.
Chapter 2 presents the design, simulation, and fabrication of the hemispherical resonator.
In Chapter 3 we describe the device quality factor and what can affect the quality factor.
Approaches on how to increase the quality factor are studied and presented.
Chapter 4 describes how we can use these hemispherical resonators as a rate integrating
gyroscope. Setup and theory are presented in this chapter.
In Chapter 5 measurements for different ideas and approaches are presented.
Finally, Chapter 6 presents conclusions and future research directions.

8

1.4 Chapter 1 references
[1] A. M. Shkel, "Type I and Type II Micromachined Vibratory Gyroscopes," in Position,
Location, And Navigation Symposium, 2006, pp.586-593.
[2] B. J. Gallacher, "Principles of a micro-rate integrating ring gyroscope," Aerospace and
Electronic Systems, IEEE Transactions on Vol. 48 No. 1, pp. 658-672, 2012.
[3] A. D. Meyer and D. M. Rozelle, “Milli-HRG inertial navigation system,” in Position Location
and Navigation Symposium (PLANS), pp.24-29, 23-26 April 2012.
[4] G. H. Bryan, "On the beats in the vibrations of a revolving cylinder or bell," Proceedings of
the Cambridge Philosophical Society. Vol. 7. No. 1. 1890.
[5] J. Dickinson, C. R. Strandt, "HRG strapdown navigator," in Position Location and Navigation
Symposium, 1990, pp.110-117.
[6] M. W. Putty and K. Najafi, “A micromachined vibrating ring gyroscope,” in Tech. Dig. Solid-
State Sens. Actuator Workshop, June 1994, pp. 213-220.
[7] I. D. Hopkin, “Vibrating gyroscopes automotive sensors.” IEEE Colloquium on Automative
Sensors. Sep. 1994, pp. 1-4.
[8] D. R. Sparks, S. R. Zarabadi, J. D. Johnson, Q. Jiang, M. Chiao, O. Larsen, W. Higdon, and
P. Castillo-Borelley. “A CMOS integrated surface micromachined angular rate sensor: Its
automotive applications”, in Tech. Dig. 9
th
Int. Conf. Solid-state and actuators (Transducers
‘97), Chicago, IL, June 1997, pp. 851-854.
[9] G. He and K. Najafi, “A single crystal silicon vibrating ring gyroscope,” in Proc. The fifteenth
IEEE Int. Conf. on micro Electro Mechanical Systems, Las Vegas, Jan. 2002, pp. 718-721.
[10] H. Johari, J. Shah, F. Ayazi, “High frequency XYZ-axis single-disk silicon gyroscope,” in
Micro Electro Mechanical Systems, 2008. MEMS 2008. IEEE 21st International Conference
on, pp.856-859, 13-17 Jan. 2008.
[11] I. P. Prikhodko, S. A. Zotov, A. A. Trusov and A. M. Shkel, "Microscale Glass-Blown Three-
Dimensional Spherical Shell Resonators," Microelectromechanical Systems, Journal of , Vol.
20, No. 3, pp.691,701, June 2011.
[12] S. A. Zotov, A. A. Trusov, A. M. Shkel, "Three-Dimensional Spherical Shell Resonator
Gyroscope Fabricated Using Wafer-Scale Glassblowing," Microelectromechanical Systems,
Journal of, Vol.21, No.3, pp.509-510, June 2012.
[13] L. D. Sorenson, X. Gao, and F. Ayazi, "3-D micromachined hemispherical shell resonators
with integrated capacitive transducers." in Micro Electro Mechanical Systems (MEMS), 2012
IEEE 25th International Conference on. IEEE, 2012, pp.168-171.
[14] M. L. Chan, P. Fonda, C. Reyes, J. Xie, H. Najar, L. Lin, K. Yamazaki, and D. A. Horsley.
9

"Micromachining 3D hemispherical features in silicon via micro-EDM," in Micro Electro
Mechanical Systems (MEMS), 2012 IEEE 25th International Conference on, pp. 289-292.
[15] J. Cho, J. Yan, J. A. Gregory, H. Eberhart, R. L. Peterson, and K. Najafi. "High-Q fused silica
birdbath and hemispherical 3-D resonators made by blow torch molding," In Micro Electro
Mechanical Systems (MEMS), 2013 IEEE 26th International Conference on, 2013, pp. 177-
180.
[16] J. Cho, T. Nagourney, A. Darvishian, B. Shiari, J.-K. Woo, "Fused Silica Micro Birdbath
Shell Resonators with 1.2 Million Q and 43 Second Decay Time Constant," Hilton Head
workshop, Hilton Head Island, SC, 2014, pp. 103-104.

10

Chapter 2
Design, FEM simulations, and fabrication
This chapter describes the design, simulation, and fabrication steps for a wineglass mode
resonator.
2.1 Design
As it was briefly discussed in Chapter 1, hemispherical resonators have a long history. Even
in the MEMS world they have been around for a while. One of the novel ideas presented in this
thesis is the fact that our device will have no stem as an anchor, and is supported by a thin layer of
SiN, which gives our device the flexibility to vibrate in second circumferential mode on the rim of
the device. On eight selected locations near the edge of the dome, piezoelectric transducers are
formed by depositing metal and piezoelectric transducers (Figure 2.1 shows the device
appearance). A more detailed description of the fabrication steps and challenges is presented in
Section 2.3.

11

Figure 2.1 Schematic design of the device. Note that the device is supported by a thin
diaphragm.

We ultimately want to use this device as a rate integrating gyroscope. Since the accuracy of
this type of gyroscope depends greatly on the dome geometry, we performed different simulations
to study the effect of non-idealities in the device geometry (this point along with other simulations
performed will be presented in Section 2.2).
2.1.1 Piezoelectric materials
Among various MEMS transducers [1], capacitive and piezoelectric transduction methods
are prevailing over others in the industry. Typical capacitive transducers with built-in variable air
gap [2] require a more complex fabrication and residual stress management techniques. Also an
externally applied polarization voltage is required for capacitive sensing. This leads to a tradeoff

12

between the polarization voltage and sensitivity as well as limited dynamic range due to the
electrostatic pull-in effect. In contrast, most piezoelectric transducers can be fabricated by simple
fabrication process and benefit from no polarization voltage (which leads to lower power
consumption), bi-directional sensing/actuation, and high fidelity.
In our approach we used piezoelectric actuation and sensing. The idea behind this choice
was the fact that piezoelectric driving and sensing will give us unlimited dynamic range, high
sensitivity, and lower power consumption.
2.1.2 Device size
In the design of these devices, one of the objectives was to have a large quality factor (Q).
A study of f*Q scaling in wineglass mode resonance of hemispherical resonators showed that if
we increase the size dome size (and so decrease the resonant frequency, f) [3], the quality factor
will increase. Based on this study we designed the largest hemispherical dome that could be
fabricated in a 3” wafer. The designed hemisphere has a radius of 600µm and depth of more than
300µm. The resonant frequency of the designed structure was calculated using the analytical
method in [4]. The calculated resonant frequency of the designed structure is 126kHz which later
was confirmed by a finite element modeling (FEM) simulation using COMSOL.
The equations that we used for resonant frequency and quality factor are the following:
𝜔 2
=
𝑛 2
( 𝑛 2
− 1 )
2
ℎ
2
𝐸 3 ( 1 + 𝜇 ) 𝜌𝑎
∫ s in
− 3
𝜑 t an
2 𝑛 𝜑 2
𝑑𝜑
𝜋 2
0
∫ { ( 𝑛 + cos 𝜑 )
2
+ 2 s in
2
𝜑 }
𝜋 2
0
s in 𝜑 𝑑 𝜑 eq. 2.1
𝑄 − 1
= 2 |
𝑖𝑚 ( 𝜔 )
𝑅𝑒 ( 𝜔 )
| eq. 2.2

13

where different parameters are defined as follows: n is the mode number (in our case 2); h is the
resonator thickness; E is the Young’s modulus; μ is the Poisson’s ratio; 𝜌 is the mass density; a is
the diameter of the device; and φ is the angle of the dome opening (in our case π/2).
In order to determine the best size and thickness to have the maximum quality factor (or
maximum f*Q product) we used MATLAB to plot the aforementioned equations for different
parameters.

Figure 2.2 3D plots of different parameters. These simulations have been done in order to confirm
the trend in our device. More detailed simulations have been done using COMSOL (Section 2.3).

2
2.5
3
3.5
4
0.5
1
1.5
2
2.5
3
x 10
-3
0
2
4
x 10
4
different modes
different radiuses (mm) resonance frequency (Hz)
2
2.5
3
3.5
4
0.5
1
1.5
2
2.5
3
x 10
-3
0
1
2
x 10
8
different modes
different radiuses (mm)
Quality factor
2
2.5
3
3.5
4
0.5
1
1.5
2
2.5
3
x 10
-3
0
5
10
x 10
12
different modes
different radiuses (mm)
f*Q

14

MATLAB simulations showed the trend that in order to increase the quality factor we need
to decrease the size.

Figure 2.3 A 2D cut of the 3D simulations that shows devices with different parameters such as:
radius, resonant frequency, and quality factor trend for a given thickness in the n=2 wineglass
mode vibration.

We also studied the effect of thickness in our devices in order to have a better understanding
of affecting constraints and a better choice of parameters in fabrication of our device.
0 500 1000 1500 2000 2500 3000
10
2
10
4
10
6
Radius ( m)
Resonance frequency (Hz)
0 500 1000 1500 2000 2500 3000
10
6
10
8
10
10
Quality factor
h = 1 m

15

Figure 2.4 Simulated resonant frequency trend for different thicknesses and radii.

200 400 600 800 1000 1200 1400 1600 1800 2000 2200
10
3
10
4
10
5
10
6
10
7
different resonant frequencies
Radius ( m)
resonant frequency (Hz)

h = 1( m)
h = 2( m)
h = 4( m)
h = 8( m)

16

Figure 2.5 Simulated quality factor trend for different thicknesses and radii.

2.2 FEM simulations
Since our structure is not a perfect hemispherical shell, it was necessary to confirm our
analytical results by numerical techniques. For our finite element method (FEM) simulations we
used COMSOL multyphysics. COMSOL was used during design stages of the project for simple
verification of the modes and finding the resonant frequencies. Also, we used COMSOL in later
stages of the project to see the effects of change in different parameters (pre- or post-fabrication),
to achieve the highest quality factor (this will be discussed in more detail in Chapter 3).
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
10
6
10
7
10
8
10
9
different quality factors
Radius ( m)
Quality factor

h = 1( m)
h = 2( m)
h = 4( m)
h = 8( m)

17

FEM simulations of the dome diaphragm resonator having 600µm radius and an average
thickness of 2µm predicted that a 4-node wineglass mode occurs at a frequency of around 126kHz
(Figure 2.6). FEM simulations using COMSOL helped us to have a better understanding of 4 node
wineglass resonance and also confirmed the analytical calculations.

Later use of COMSOL was valuable to confirm our ideas, for example studying the effect
of compressive stress in the SiN layer or perforation in the diaphragm (Chapter 3).
Another helpful use of COMSOL simulations was to study the effect of non-idealities in
the device. We studied two distinct non-idealities: first shallow dome and second eccentricity of
the dome opening.
Figure 2.6 FEM simulation on the vibration displacement of the dome-shaped
diaphragm that clearly shows a four-node wine-glass vibration mode.

18

2.2.1 Shallow domes
We learned from MATLAB simulations that larger domes will result in lower resonant
frequency and higher quality factor. One limitation that we have is the obligation to use 3” standard
wafers. These wafers have the nominal thickness of 400μm and we cannot etch a dome deeper
than that, so the study of shallow domes became of interest.

By performing FEM simulations, the effect of cap size was studied. As seen in Figure 2.7,
by increasing the ratio of radius and shell depth, the resonant frequency becomes lower as h/r goes
from 1 (full hemisphere) to 0.25, by about 3%. Then the resonant frequency increases rapidly, if
the h/r is reduced further. (If the h/r < 0.1, there is no n=2 wineglass mode vibration.) This means
that as long as we keep h/r > 0.25, the resonant frequency is about the same as that for a full
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
148
150
152
154
156
158
160
162
164
166
168
h/r
resonance frequency (kHz)
(a)
(b)
(c)
Figure 2.7 FEM simulations on the effect of the cap size on the resonant frequency:
(a) Definition of h and r used in the simulations, (b) simulation results and (c) resonant
frequency versus h/r ratio for r = 0.5mm and thickness = 1μm.

19

hemisphere. This is an important result, since it will let us use a shallow dome, rather than a full
hemisphere, to achieve a desired resonant. For example, from Figure 2.4 we know that a full
hemisphere with 1,600μm in diameter will give us a resonant frequency lower than 20kHz. But
for a full hemisphere we need to have a thick wafer (more than 800μm in thickness). Now, based
on these results we can fabricate a shallow dome with 1,600μm diameter on a typical 3” wafer that
is 400μm thick, and obtain a resonant frequency less than 20 kHz.
2.2.2 Eccentricity
Another non-ideality that was studied by using COMSOL was the non-ideality in dome
shape, or in better words the eccentricity of dome opening. Based on fabrication technique, etchant
temperature, etching time, and other factors in the fabrication process, we might end up with a
slightly non-ideal circle or an oval (Figure 2.8).

Figure 2.8 For a perfect circle, eccentricity or e is defined to be zero. As we move from an ideal circle to
an oval shape, the e factor changes.

20

Simulation results showed we still could observe the wineglass mode resonance in the non-
ideal device, and the change that eccentricity (e) made was the alteration in the frequency split
(discussed in Chapter 4 and Chapter 5). Figure 2.9 shows the frequency split in a given mode and
the effect of eccentricity. In Figure 2.10 the frequency split for different ‘e’ values has been
depicted. Based on these FEM simulations, we can tolerate up to 60% of eccentricity without a
frequency split of larger than 10%.

21

Figure 2.9 Different eccentricities change the frequency split but their effect on absolute resonant
frequency and quality factor is negligible.

22

Figure 2.10 COMSOL simulations show a leeway in the ideal shape of the rim of the device.

2.3 Fabrication
Our fabrication approach is based on isotropic etching of silicon. The dome diaphragm is
made of a low stress silicon nitride thin film deposited over isotropically etched silicon cavities.
Figure 2.11a shows the cross sectional view of the dome.

0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
X: 0.2
Y: 0.4261

f/f
larger
%
eccentricity
Figure 2.11 Cross-sectional diagram (a) and top-view photo (b) of the dome-shaped-diaphragm
transducer that is suspended by a thin, flexible membrane, and actuated by four piezoelectric
transducers (and sensed by those plus four additional ones) placed on the rim of the dome
diaphragm.
(a) (b)

23

The spherical cavities are made by isotropic etching of a thick silicon wafer with a
combination of 49% hydrofluoric acid, 70% nitric acid and 99.5% acetic acid (HNA) with a ratio
of 20:35:55 at 50ºC [5-6], using silicon nitride as an etch mask. This combination has given the
best isotropicity, and we have been able to fabricate large dome diaphragms (up to 1 millimeter in
diameter). We cover the cavities by depositing LPCVD SiN on top of them. It is worth noting that
in the later stages of the project we changed the deposition conditions (temperature) to vary the
compressive stress in the SiN layer. Change in the stress helped us increase the f*Q. Transducers
that are used as actuators and sensors for the device are made on eight selected locations near the
circular edge of the dome diaphragm (Figure 2.11b). Metal and piezoelectric thin films are
deposited through a shadow mask. Four of the transducers are used to actuate the dome in 4-node
wine glass mode. The rest of the transducers, which are put on nodal points, are used to sense
rotation in the system. The dome diaphragm transducers fabrication steps are shown in Figure
2.11.

Depositing thick SiN
Patterning Al and SiN Wet isotropic etching of Si
Removing SiN
Depositing SiN
Depositing ZnO,
PECVD nitride and
top electrode
Patterning the
backside
KOH etching of the
backside to release
the dome
transducer
Depositing bottom
electrode via
shadow mask
Depositing Al
Figure 2.12 Fabrication steps of dome transducers.

24

2.3.1 Fabrication challenges
One problem that was stopping MEMS researchers from investigating the HRG in the
silicon world was the difficulty of 3D lithography and, in general, deposition on a 3D structure.
We tackled this issue by using a shadow mask.

Figure 2.14 Deposition through shadow mask to overcome the difficulties of 3D photolithography.
Another fabrication issue that we faced was poor electrical connection near the edges of
the dome rim. In a perfect hemispherical resonator we have a very sharp edge, and depositing
electrodes in that sharp edge can be challenging.
Figure 2.13 Top view of a finished device.

25

In order to overcome this issue, another shadow mask was designed and a second layer of
electrodes was deposited on top of the first layer to substantiate it near the edge.

Figure 2.15 Red mask is the original shadow mask, while the green mask is for adding more
electrodes at the edges.

Another important problem that we faced during the fabrication and later on during
measurement was the low ring down time of our devices. There are different mechanisms
contributing to the ring down time of a typical oscillator like ours. We have mechanical resonance,
which has its own ring down time. On the other hand we basically have a capacitor (C) with a
finite resistance (R) between the two electrode layers sandwiching a piezoelectric ZnO layer,
presenting its own RC time constant. If we do not deposit any insulation layer between the two
electrodes, the electrical ring down time will be τ = RZnOCZnO which is around 10μs. Adding a
thick enough PECVD insulation layer (Figure 2.16) will increase the electrical ring down time to
about 70 seconds [7].

26

A series of tests have been done to study the effect of PECVD layer on the quality factor
of the device. Preliminary experiments to test the difference in the electrical ring down time for
the two cases of Figure 2.16b and Figure 2.16c have not resulted in conclusive data.
2.4 Device simulations as HRG
The most prominent application of these transducers is going to be as a rate integrating
gyroscope. The equations of motion have been reported in literature [8]. Using equations of
motion for our device and the resonant frequency, we can predict the output of the device. As it
will be described in Chapter 4, our device can be used as a rate integrating gyroscope.
If we maintain the n=2 wineglass mode resonance in the system by applying actuation to
the transducers, we will have a nodal point between the two adjacent actuating transducers. When
a rotation is applied to the system the voltage of the nodal point changes, and as rotation continues
the envelope of the nodal point signal will increase proportionally to the total rotation angle. Using
the equations of motion and assuming an applied rotation of 1 rad/s we can predict the output of
our device using MATLAB simulation. Simulation is performed based on MATLAB code from
[9].

ZnO
ZnO
PECVD SiN
ZnO
Figure 2.16 Comparison between three different insulation layers: (a) device without
any insulation layer, having a ring down time of few μs, (b) current device with a ring
down time up to 70 seconds, (c) device in study for a ring down time higher than 70
seconds.
(a) (b) (c)

27

2.5 Chapter 2 references
[1] P. R. Scheeper, A. G. H. van der Donk, W. Olthuis, P. Bergveld, “A review of silicon
microphones,” Sens. Act. A, vol. 44, pp. 1-11, 1997
[2] D. W. Schindel, D. A. Hutchins, L. Zhou and M. Sayre, “The design and characterization of
micromachined air-coupled capacitance transducers,” IEEE Transactions on Ultrasonics,
Ferroelectrics, and Frequency control, Vol. 42, No. 1, pp. 42-50, 1995.
[3] S. Tallur and S. A. Bhave, "Comparison of f-Q scaling in wineglass and radial modes in ring
resonators," in Micro Electro Mechanical Systems (MEMS), 2013 IEEE 26th International
Conference on, 2013, pp.777-780.
[4] S. Y. Choi, Y. H. NA, and J. H. Kim, “Thermoelastic Damping of Inextensional
Hemispherical Shell,” Proc. World Acad. Sci., Eng. Technol. , Vol. 56, No. 32, pp. 198-203,
2009.
[5] H. Zhang; E.S. Kim, "Dome-shaped diaphragm microtransducers," in Proc. Micro Electro
Mechanical Systems, 1995, pp.256.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-3000
-2000
-1000
0
1000
2000
3000
Time (s)
y Amplitude (mv)
Nodal Response: f= 0.2Hz, =54
o
/s, k=0.3, Kt=1.62 V/ m
Figure 2.17 Simulated nodal point output as the device is rotated at 1 rad/s. As the total angle
increases, the envelope of the output amplitude at the nodal point also increases linearly. Note that
the nodal point still vibrates at the resonant frequency, and only the envelope changes.

28

[6] C. H. Han and E. S. Kim, "Simulation of Piezoelectric Dome-Shaped-Diaphragm Acoustic
Transducers." Journal of semiconductor technology and science, Vol. 5, No. 1, pp.17-23,
2005.
[7] P. L. Chen, R. S. Müller, R. M. White, and R. Jolly. "Thin film ZnO-MOS transducer with
virtually DC response,"1980 Ultrasonics Symposium. 1980, pp.945-948.
[8] IEEE Standard Specification Format Guide and Test Procedure for Coriolis Vibratory Gyros,"
IEEE Std 1431-2004, pp.1-78, Dec. 20 2004.
[9] Private communication with A. Dorian Challoner.

29

Chapter 3
Quality factor in hemispherical resonators and ways to improve
it
In this chapter we will study the quality factor in hemispherical resonators. Different
damping mechanisms will be investigated and ways to assuage each mechanism will be discussed
and backed by simulations. For some of the approaches we have done experiments as well. Brief
measurement results and a conclusion will be presented in this chapter.
An important parameter for hemispherical resonators is their quality factor which can be
translated into their ring down time:
Q = πτf eq. 3.1
where Q represents quality factor; f is the resonant frequency; and τ is the ring down time constant.
We define the ring down time as the decay time constant of the mechanical vibration. We actuate
the dome in wineglass mode resonance and measure the exponential decay time constant of the
signal after we cut off the actuation. The actuating signal is a sinusoidal, and after turning off the
input, we will have the same sinusoidal signal with its amplitude decreasing exponentially in time.
We define the time constant in this exponential wave as the ring down time of our device.
A high ring down time guarantees the uninterrupted function of our sensor [1]. Also a larger
quality factor results in better frequency stability and consequently better sensing resolution [2].
30

For a simple resonator, higher Q can be achieved by improving the quality of the resonator
structural materials or by optimizing the resonator shape to decrease the energy loss of the
resonator. In our design we are limited in terms of materials that are available in microfabrication
technology. This means that we need to investigate other damping mechanisms to maximize our
quality factor.
The quality factor is a measure of energy loss in a system. As long as the stored energy is
constant, higher quality factor means lower energy dissipation per cycle. The quality factor of a
resonator, such as ours, that operates in vacuum has two main loss mechanisms: extrinsic and
intrinsic.
The extrinsic loss is due to interactions with the surrounding medium, e.g., viscous losses
and acoustic radiation. The intrinsic loss is due to interactions within the structure or with its
support structure, e.g., support loss, thermoelastic damping (TED), volume loss, and surface loss.
The total energy dissipation in a resonator is calculated by adding the energy dissipation of each
individual loss mechanism. Hence for a given resonant frequency, the quality factor of the system
is obtained from the quality factors attributed to each individual loss mechanism Q i [3]:
1
𝑄 = ∑
1
𝑄 𝑖 =
1
𝑄 𝑎 𝑖𝑟 −
𝑑 𝑎 𝑚 𝑝 𝑖𝑛𝑔 +
1
𝑄 𝑚 𝑎 𝑡 𝑒 𝑟 𝑖𝑎 𝑙 +
1
𝑄 𝑎 𝑛𝑐 ℎ 𝑜𝑟
+
1
𝑄 𝑠 𝑝 𝑢𝑟 𝑖 𝑜𝑢𝑠 −
𝑚 𝑜𝑑𝑒 +
1
𝑄 𝑇 𝐸 𝐷 eq. 3.2
We are going to study each individual damping mechanisms and what we can do to reduce
it. This chapter presents approaches to improve f*Q product.

31

3.1 Air Damping
Air damping in MEMS resonators has been studied, and different analytical approaches
have been reported in literature [4]. All the major approaches to the viscous damping of a solid
body are derived from Navier-Stokes equation and the continuity equation for incompressible
fluids. Viscous damping is caused by energy losses such as those that occur in liquid lubrication
between moving parts or in a fluid forced through a small opening by a piston, as in automobile
shock absorbers. The viscous-damping force is directly proportional to the relative velocity
between the two ends of the damping device. In our device, unlike other hemispherical dome
designs [2-3, 5-7], we do not have air gap that causes squeeze-film viscous damping. We studied
the effect of air pressure on quality factor of our devices. Measurement results showed that at low
enough pressures (lower than 350mTorr), the quality factor reaches its maximum and does not
change even if we keep decreasing the pressure further down (Figure 3.1). Thus, as long as our
device is kept under 350mTorr of pressure, air damping does not have much effect.
32

Figure 3.1 Measured ring down time versus pressure. Below 350mTorr, lowering the pressure
does not improve the quality factor anymore.

3.2 Anchor loss
Because of specific release mechanisms that we have and unlike other hemispherical
resonators [3, 5-7], we do not have a stem or pole. Rather, our device is supported by a thin SiN
layer. This support minimizes the anchor loss and allows the dome to vibrate freely.
We studied the effect of thin-film-support-diaphragm size and shape on the quality factor
by FEM simulations. We increased the edge length in steps to see where it will not affect the Q
anymore. Figure 3.2 shows the support-thin-film-layer length defined as edge size, which we
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
1
2
3
4
5
6
7
ring down time (s)
pressure(Torr)
effect of air pressure
33

changed to see its effect on vibration modes and their quality factors. This study showed the
minimum edge size that does not affect the Q, modes place and direction.

Figure 3.2 Edge size defined as the distance from the anchored edge of the device to the rim of
the hemispherical resonator

The edge size was changed from 25μm to 400μm, and COMSOL simulations assured that
wineglass mode resonance is still in the device.

34

Changing the edge size showed a change in Q and also resonant frequency for shorter
lengths. For edge sizes larger than 200μm, the edge size does not have any effect on resonant
frequency or quality factor. In fabrication we make sure that the edge size is larger than 200μm.

Figure 3.3 For each of the edge sizes, FEM simulation has been done to confirm the second mode
circumferential resonance in the device
35

0 50 100 150 200 250 300 350 400
1.65
1.7
1.75
1.8
1.85
1.9
x 10
5
size at the edges ( m)
frequency (Hz)
0 50 100 150 200 250 300 350 400
20
40
60
80
100
120
ring down time (s)
Figure 3.4 Simulated resonant frequency and ring down time versus the edge size.
For edge sizes larger than 200μm, the edge size does not have any effect on resonant
frequency or quality factor.
36

The hemispherical resonator is supported by a thin film, and therefore has the freedom to
vibrate, but the thin film support itself is anchored on the edges of the hemispherical dome. We
found that when the edge size was short, two resonances were aligned to the side and diagonal
directions of the square edge.
We noticed that even though the edge size larger than 200μm ensures the Q and resonant
frequency free from the influence by the square opening, the two resonances are still aligned to
the edge and diagonal of the square opening, as seen in Figure 3.6. It turns out that the two
degenerate modes are aligned to the electrodes’ locations rather than the orientation of the square
opening (with respect to the electrode locations). We confirmed this fact by further FEM
0 50 100 150 200 250 300 350 400
2
3
4
5
6
7
8
9
10
x 10
12
size at the edges ( m)
f*Q
Figure 3.5 Simulated f*Q product in the device versus the edge size
37

simulations on domes with edge sizes larger than 200 μm by varying the angle between the
electrode position and the edge of the square opening. (For example, Figure 3.6 defines the angle
to be zero, since the red line in the left figure is parallel to the edge.) In all the different angles, the
two degenerate modes are aligned to the electrodes’ locations rather than the orientation of the
square opening with respect to the locations of the electrodes.

(q

Figure 3.6 Simulated two degenerate modes in the n=2 wineglass mode vibration. The two modes have slight
different resonant frequencies, depending on the dome symmetry, and are aligned to the side (left figure) and
diagonal (right figure) of the square opening.
38

Figure 3.7 Simulated vibration amplitude distribution over the diaphragm when the square edge
size is large enough for the wineglass mode vibration (that is occurring at the circular edge of the
dome) to be free from the influence of the square edges.
3.3 Material
Material-related damping can affect the quality factor in two ways. The first is the limit in
the quality factor due to thermoelastic damping (TED) (which will be investigated in Section 3.5).
The second effect is in surface non-idealities. If we have non-idealities in the surface of the device,
the mechanical vibration can be scattered, therefore decreasing the quality factor.
In the second case we measured the surface roughness of the device right after etching
using atomic force microscope. The RMS value of the surface roughness is 125.31nm.
39

Figure 3.8 AFM measurement on the surface of the dome without any modification.
40

We performed an additional HF dip after etching, and this additional etch step improved
our surface roughness from 125nm to 51nm, more than 40%.

Figure 3.9 AFM measurement on the surface of the dome after HF dip.

We also studied the effect of compressive stress on quality factor of our devices (which
will be discussed in more details in Section 3.7). One additional benefit of compressive stress on
the SiN layer was to improve the surface roughness.
41

Figure 3.10 Surface roughness measurements in a device with higher compressive stress and after
performing HF dip

In order to improve the effect of material damping we added two additional steps in our
fabrication. One was an additional HF dip to improve the surface roughness. The second one was
to improve the roughness by introducing compressive stress to SiN layer. The combined effect of
these two approaches improved our surface roughness by a factor of 4.

42

3.4 Spurious modes
Spurious modes happen for different reasons, such as the shape of the device, symmetry in
different directions, slight non-idealities in shape and size, and so on. We performed COMSOL
simulations in order to find the effect of other modes in our device Q and resonant frequency.
Since we used our device in the second circumferential mode, the spurious modes were not close
to the frequency of the resonances of interest, and so their effect can be neglected.
3.5 Thermoelastic Damping (TED)
In absence of air damping and support loss, one of the most influential damping
mechanisms is thermoelastic damping or TED. In any vibrating structure, the strain field causes a
change in the internal energy such that the compressed region becomes hotter (assuming a positive
coefficient of thermal expansion) and the extended region becomes cooler. The mechanism
responsible for thermoelastic damping is the resulting lack of thermal equilibrium between various
parts of the vibrating structure. Energy is dissipated when irreversible heat flow driven by the
temperature gradient occurs.
As described in Chapter 2, one of the main metrics for our design was the maximum quality
factor which, after considering air damping and anchor loss, leaves us with the main damping
mechanisms, TED. So we designed our system for the maximum QTED.
Zener [8] presented an approximate equation, based on thermodynamics, for calculating
the TED-limited Q of a homogeneous and isotropic beam. From that the following equation was
derived in [9]:
𝑄 𝑇 𝐸𝐷 = (
𝑐 𝑝 𝜌 𝐸 𝛼 2
𝑇 0
)
1 + ( 𝜔𝜏 )
2
𝜔𝜏
eq. 3.3
43

where E is the Young’s modulus; 𝜌 is the material density; cp is specific heat at constant
pressure; α is the coefficient of thermal expansion; T0 is the equilibrium temperature; and τ is
the time constant at which the temperature gradient decays in the fundamental Eigen-mode and
is defined as:
𝜏 = (
ℎ
𝜋 )
2
𝑐 𝑝 𝜌 𝑘 eq. 3.4
where k is the thermal conductivity.
Based on Zener’s work on TED in beams [8], Choi et al. studied the TED mechanisms in
a hemispherical shell [10]. In their work they obtained the temperature profile from the heat
conduction equation, and used it to calculate the strain energy, which then was used in calculating
the resonant frequency ( ) through the Rayleigh energy method:
𝜔 2
=
𝑛 2
( 𝑛 2
− 1 )
2
ℎ
2
𝐸 3 ( 1 + 𝜇 ) 𝜌 𝑎 4
∫ 𝑠𝑖 𝑛 − 3
𝜑 𝑡 𝑎 𝑛 2 𝑛 𝜑 2
𝜋 2
0
𝑑𝜑
∫ { ( 𝑛 + 𝑐 𝑜𝑠 𝜑 )
2
+ 2 𝑠𝑖 𝑛 2
𝜑 } 𝑠𝑖 𝑛 𝜑 𝑑𝜑 𝜋 2
0
eq. 3.5
where n is the mode number; h is the diaphragm thickness; a is the radius of the hemisphere; φ is
opening angle of the hemisphere. The Q-factor was then obtained from the complex equation for
the resonant frequency, with 𝑄 − 1
= 2 |
𝐼𝑚 (  )
𝑅𝑒 (  )
|.
3.6 Perforation effect on f*Q
We studied the effect of perforation on the quality factor and resonant frequency of our
devices. A hole (200µm in diameter) at the apex of a hemispherical dome (600µm in diameter)
increased the Q and thereby ring-down time by more than 50% and 100% experimentally and
theoretically, respectively, for a micromachined HRG resonating at about 170 kHz. Adding such
a hole affects the resonant frequency by only a little (a few percent).
44

In case of a microfabricated cantilever, perforation on the cantilever has been studied, and
shown to decrease the stiffness and thereby the resonant frequency (and thus increase the Q, as
f*Q product is constant). The hole in this work has been obtained through using a laser cutting
machine.
Instead of making many holes, we make one hole at the apex of the dome in order to
increase the dome’s Q without changing the resonant frequency much. With a single hole, there
are some unique locations for the hole over a dome diaphragm where the resonant frequency is not
affected much.
3.6.1 Theory
The concept of equivalent stiffness is widely used for analysis of a perforated structure
[11]. Equivalent effective elastic modulus (Eperforated) and Poisson’s ratio of a perforated structure
can be obtained as a function of the elastic modulus (E untouched) and Poisson’s ratio of an
unperforated structure. For a flat diaphragm, O’Donnell et al. [12] calculated the Poisson’s ratio
of 0.5 and
𝐸 𝑝 𝑒 𝑟 𝑓 𝑜 𝑟 𝑎 𝑡 𝑒 𝑑 𝐸 𝑢 𝑛 𝑡 𝑜 𝑢 𝑐 ℎ 𝑒𝑑
= 0 . 15 for a single hole in the particular diaphragm, showing that perforation
causes a reduction of the effective stiffness, which leads to a lower resonant frequency that means
a higher quality factor for a structure that has a constant f*Q.
However, in the case of our HRG (Figure 3.3), a hole on the shell increased the quality
factor, not through a reduction of the resonant frequency, but through reduction in thermoelastic
damping (TED) which is the energy loss due to mechanically-induced temperature gradients in a
vibrating structure.
We used these equations to calculate the resonant frequency and the Q vs. the shell radius
with and without a hole on the shell by using MATLAB. The calculated results (Figure 3.11)
clearly show an improved Q with the hole. But the analytical equation predicts a lower resonant
45

Figure 3.11 Calculated resonant frequency and quality factor vs. the shell radius with
and without a hole on the shell. The thickness is fixed at 1μm.

frequency for a perforated shell, unlike our Finite Element Analysis (FEA) and experimental
results to be described in the next sections.

Simulations show the optimum size and place of a hole for a given dome shape and size.
Figure 3.12 shows two simulation results for a hemispherical shell with and without a hole made
at the dome’s apex.

0 500 1000 1500 2000 2500 3000
0
5000
10000
15000
Resonance frequency changes for h = 1e-06
Radius( m)
Resonance frequency (Hz)

0 500 1000 1500 2000 2500 3000
0
0.5
1
1.5
2
x 10
10
Quality factor changes for h = 1e-06
Radius( m)
Quality factor

Quality factor for solid device
Quality factor for perforated device
Resonant frequency for solid device
Resonant frequency for perforated device
46

Figure 3.12 (Top) FEM simulation of the wine-glass-mode vibration for the dome diaphragm.
(Bottom) FEM simulation of the wine-glass-mode vibration for the dome diaphragm with a hole
at the dome apex. The simulation shows that the resonant frequency increases only by 3% while
the quality factor increases by more than 100%.

47

The hole diameter and location also were varied to find the optimum size and location for
the highest Q. Simulations show (Figure 3.13) that the optimum size (200μm in diameter) and
location (which turned out to be the dome apex) of the hole can result in more than 100%
improvement in the Q with only 3.6% increase in the resonant frequency for a hemispherical dome
whose resonant frequency for the four-node wine glass mode is about 170 kHz. If the holes are
larger than 240µm in diameter, the wineglass mode resonance is lost. If the holes are made closer
to the dome’s edges, rather than the dome apex, the quality factor decreases substantially as the
symmetry in the resonator is decreased.

0 1 2 3
x 10
-4
1.64
1.66
1.68
1.7
1.72
1.74
x 10
5
hole diameter (m)
frequency (Hz)
0 1 2 3
x 10
-4
10
20
30
40
50
60
ring down time (s)
Figure 3.13 Simulated resonant frequency and ring-down time versus the diameter of a hole at the
dome’s apex. The optimum hole size is about 200µm for the four-node wineglass mode resonance
with a dome transducer whose dimensions are as indicated in Figure 3.12. If the hole size becomes
larger than 240µm in diameter, the wineglass mode resonance no longer exists.
48

The Q improvement has been demonstrated through measuring the ring-down times before
and after a laser cutting to create a hole on the dome’s apex. A photo of the shell after forming a
hole (200μm in diameter) in the center is shown in Figure 3.14.

Measurements showed an increase of about 50% in the Q with only a 1.3% increase in the
resonant frequency around 170 kHz, as the ring-down time increased from 5.947 to 8.964 sec
(which corresponds to a Q of 4.5 million). The ring-down-time measurements are presented in
Figure 3.15.

Figure 3.14 Photo of the device after a post-process laser cutting
that produced a hole in the middle of the dome diaphragm.
Figure 3.15 Measured decay times of the output signals, when the actuating signal is turned
off, for two dome diaphragms: one without any hole (5.947 sec) and the other with a hole in
the center (8.964 sec).
49

The measured resonant frequency increased slightly with the hole size according to the
simulations. The experimental Q increase was less than the theoretical Q increase, likely due to
the fact that the hole was not exactly at the apex.

3.7 Effect of compressive stress in LPCVD SiN layer on Q
Another study to investigate the improvement of f*Q was to consider the effect of
compressive stress in our devices. We usually deposit low stress LPCVD SiN, but in this study we
changed the temperature during deposition from 835ºC to 885ºC. COMSOL simulations showed
(Figure 3.16) that higher compressive stress will result in higher f*Q for a given hole size.
Simulated
frequency
Measured
frequency
Simulated ring-
down time
Measured ring-
down time
Without a hole 167.5kHz 158kHz
26.094 sec (Q =
13.73 million)
5.947 sec (Q =
2.95 million)
With a hole 173.5kHz 160kHz
54.020 sec (Q =
29.4 million)
8.964 sec (Q =
4.5 million)
Table 3.1 Summary of the simulation and experimental results.

50

Figure 3.16 Ring down time and resonant frequency simulations for different compressively-
stressed SiN layers for the same size.

We fabricated two sets of resonators with dome diaphragm of about 1mm in diameter (with
small variability). The set with deposition at 835ºC showed a resonant frequency of about 142
kHz, while the set made of thin film SiN that was deposited at 885ºC showed a little higher resonant
frequency of around 149kHz.
Figures 3.17 and 3.18 show the measured time decays of the devices fabricated at different
temperatures.
0 1 2 3 4 5 6
x 10
8
1.6
1.7
1.8
x 10
5
stress (Pa)
resonance frequency (Hz)
0 1 2 3 4 5 6
x 10
8
0
200
400
ring down time (sec)
51

Figure 3.17 Measured ring down times for two different devices having two different levels of
compressive stress in the SiN layer of the dome diaphragm.

0 10 20 30 40 50 60 70 80 90
-5
-4
-3
-2
-1
0
1
2
3
4
5
time(s)
Voltage(V)

prestressed device
low stress device
52

Figure 3.18 Measured ring down time vs. the deposition temperature for the SiN. Deposition in
higher temperature leads to higher compressive stress in the SiN layer and so higher ring down
time.

In this fabrication run we had different dome sizes as well: 1mm, 1.2mm and 1.4mm in
diameter. As expected, the larger devices had lower resonant frequency and higher quality factor
(Figure 3.19).
830 840 850 860 870 880 890
4
4.5
5
5.5
deposition temperature (
o
C)
ring down time (s)
53

Figure 3.19 Measured resonant frequency vs the dome size for two different stress levels in the
SiN layer. As the radius increases, the resonant frequency decreases. And the resonant frequencies
are higher for the devices with higher compressive stress in the SiN layer.

Also the quality factors of the larger devices were higher, and the devices with higher
compressive stress showed larger quality factor (Figure 3.20).
950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
x 10
5
radius ( m)
resonant frequency (Hz)

low stress SiN layer
compressively stressed SiN layer
54

Figure 3.20 Measured quality factor vs the dome size. As the radius increases, the quality factor
also increases. And the quality factor is higher for the devices with higher compressive stress in
the SiN layer.

In all different sizes, we could see that a higher deposition temperature for the SiN, which
means higher compressive stress, will result in higher quality factor.

950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450
4
4.5
5
5.5
6
6.5
7
7.5
radius ( m)
decay time constant (s)

low stress SiN layer
compressively stressed SiN layer
55

830 840 850 860 870 880 890
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
deposition temperature
o
C
ring down time (s)
relation between deposition temperature and decay time constant for r = 1000 m
830 840 850 860 870 880 890
1.4
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
x 10
5
deposition temperature
o
C
resonant frequency (Hz)
relation between deposition temperature and resonant frequency for r = 1000 m
Figure 3.21 Measured ring down times and resonant frequencies of the two devices with
different compressive stresses in the SiN layer, both having a dome radius of 1,000μm.
56

Figure 3.22 Measured ring down times and resonant frequencies of the two devices with
different compressive stresses in the SiN layer, both having a dome radius of 1,200μm
830 840 850 860 870 880 890
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
6.6
deposition temperature
o
C
ring down time (s)
relation between deposition temperature and decay time constant for r = 1200 m
830 840 850 860 870 880 890
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
x 10
5
deposition temperature
o
C
frequency (Hz)
relation between deposition temperature and resonant frequency for r = 1200 m
57

830 840 850 860 870 880 890
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
x 10
5
deposition temperature
o
C
frequency (Hz)
relation between deposition temperature and resonant frequency for r = 1400 m
830 840 850 860 870 880 890
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
deposition temperature
o
C
ring down time (s)
relation between deposition temperature and decay time constant for r = 1400 m
Figure 3.23 Measured ring down times and resonant frequencies of the two devices with
different compressive stresses in the SiN layer, both having a dome radius of 1,400μm
58

3.8 Effect of size on quality factor and f*Q
FEM simulations showed that for wineglass mode resonators, quality factor increases as
resonant frequency decreases. A series of simulations have been done for structures with different
sizes and different resonance frequencies to study the effect of f*Q scaling in wineglass mode in
our structure. The result is presented in Figure 3.23.
Based on the limitation in fabrication and COMSOL simulations, devices with different
dome sizes have been designed and are being fabricated. The next step would be quality factor
measurements in these devices either electronically or by optical means. In this way we can find
the optimum size for our application.

200 400 600 800 1000
0
5
10
x 10
5
Radius ( m)
Resonance frequency (Hz)
200 400 600 800 1000
0
1
2
x 10
6
Quality factor
Figure 3.24 Simulated resonant frequency and quality factor. Increasing the dome
diameter will decrease the resonant frequency while increasing the quality factor,
since the f*Q is constant.
59

One of the challenges in making larger domes is the limitation that we have due to the
thickness of silicon wafers. Since the silicon wafer thickness is limited (400μm) we cannot etch it
more than that, or we will have a hole on the dome diaphragm. On the other hand, large openings
will lead to a partial hemisphere or a cap rather than a full hemisphere (as described in Figure
3.26).

200 300 400 500 600 700 800 900 1000 1100
0
2
4
6
8
10
x 10
5
Resonance frequency changes for h = 1e-06
Radius( m)
Resonance frequency (Hz)
Figure 3.25 Simulated resonant frequency vs. the size of a dome that is hemisphere rather
than a cap
Figure 3.26 Depiction of a device having a partial hemisphere.
60

We discussed the effect of shallow dome in Section 2.2.1, but had some issues during the
fabrication of a large and shallow dome. The main challenge for larger dome fabrication was faced
during release of the dome diaphragm. Due to low (and slightly tensile) stress of LPCVD silicon
nitride, after release the diaphragm will go flat rather than staying in a dome (cap) shape. In order
to overcome this challenge we can do either of the following: (1) use a larger silicon substrate, to
have a hemispherical etched cavity before LPCVD silicon nitride deposition or (2) deposit silicon
nitride with compressive stress.
The first approach has some drawbacks such as higher wafer price and difficulty of
patterning and compatibility with fabrication systems in our cleanroom. On the other hand, the
second approach will give a cheaper alternative in addition to the fact that the compressive stress,
even to the point of wrinkling on the supporting diaphragm, reduces the stiffness of the supporting
diaphragm for the 4-node wineglass mode vibration.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
145
150
155
160
165
170
h/r
Resonance frequency (Hz)
Figure 3.27 FEM simulation of the resonant frequency vs. h/r ratio for a partial
hemisphere. At a ratio of h/r (lower than 0.25), the resonant frequency varies sharply, and
at an even lower ratio (less than 0.1) the wineglass mode resonance will be lost.
61

In addition, simulations showed that the change in the resonant frequency of the dome
resonator as the dome gets shallower is negligible. So, we can find an optimum ratio of the cap
depth and radius which gives us the highest quality factor for a given resonant frequency, and by
having compressive stress in the silicon nitride diaphragm, we can make these larger devices
without any issue after their release.
If we succeed to increase the dome size (to 1,600μm in diameter) and its quality factor, the
ring down time can be in excess of 500 seconds, more than 100 times the highest reported time in
MEMS HRG [12].

3.9 Chapter 3 references
[1] A. D. Challoner, H. H. Ge, and J. Y. Liu. "Boeing Disc Resonator Gyroscope," Position,
Location and Navigation Symposium-PLANS 2014, 2014 IEEE/ION.
[2] L. Sorenson, and F. Ayazi, "Effect of structural anisotropy on anchor loss mismatch and
predicted case drift in future micro-Hemispherical Resonator Gyros," in Position, Location
and Navigation Symposium - PLANS 2014, 2014, pp.493-498.
[3] A. M. Shkel, "Type I and Type II Micromachined Vibratory Gyroscopes," in Position,
Location, And Navigation Symposium, 2006, pp.586-593.
[4] K. Naeli, and O. Brand, “Dimensional considerations in achieving large quality factors for
resonant silicon cantilevers in air,” Journal of Applied Physics, Vol. 105 No.1, 2009.
[5] J. Y. Cho, J. K. Woo, J. Yan, R. L. Peterson, and K. Najafi. "Fused-Silica Micro Birdbath
Resonator Gyroscope (μ-BRG)."Microelectromechanical Systems, Journal of, Vol. 23, No.
1, pp. 66-77, 2014.
[6] J. Cho, J-K. Woo, J. Yan, R. L. Peterson, and K. Najafi. "A high-Q birdbath resonator
gyroscope (BRG)," in Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS &
EUROSENSORS XXVII), 2013 Transducers & Eurosensors XXVII: The 17th International
Conference on, 2013, pp. 1847-1850.
[7] J-K. Woo, J. Y. Cho, C. Boyd, and K. Najafi. "Whole-angle-mode micromachined fused-
silica birdbath resonator gyroscope (WA-BRG)," in Micro Electro Mechanical Systems
(MEMS), 2014 IEEE 27th International Conference on, 2014, pp. 20-23.
62

[8] R. N. Candler, A. Duwel, M. Varghese, S. A. Chandorkar, M. A. Hopcroft, W-T. Park, B.
Kim and T. W. Kenny, "Impact of geometry on thermoelastic dissipation in micromechanical
resonant beams,"Microelectromechanical Systems, Journal of, Vol. 15, No. 4, pp. 927-934,
2006.
[9] S. Y. Choi, Y. H. NA, and J. H. Kim, “Thermoelastic Damping of Inextensional
Hemispherical Shell,” Proc. World Acad. Sci., Eng. Technol. , Vol. 56, No. 32, pp. 198-203,
2009.
[10] J. P. Duncan, and R.W. Upfold, “Equivalent elastic properties of perforated bars and
plates,” Journal of Mechanical Engineering Science, Vol. 5, No. 1, pp. 53-65, 1963.
[11] W. J. O’Donnell, “Analysis of perforated plates,” Doctoral Thesis, University of Pittsburg,
1962.
[12] J. Cho, T. Nagourney, A. Darvishian, B. Shiari, J.-K. Woo, "Fused Silica Micro Birdbath
Shell Resonators with 1.2 Million Q and 43 Second Decay Time Constant," Hilton Head
workshop, Hilton Head Island, SC, 2014, pp. 103-104.

63

Chapter 4
Utilizing our device as a whole angle measurement unit
Previous chapters describe the design, simulation and fabrication of a wineglass mode
resonator. We studied the quality factor and approaches to increase it, but the main benefit of these
devices is their application as a rate integrating gyroscope. Coriolis Vibratory Gyroscopes (CVGs)
can be divided in two classes according to the nature of the physical shape of the resonator and the
vibration modes involved [1]. Class I MEMS CVGs are asymmetric structures, typically
implemented as high Q-factor dual mass tuning forks. Class I CVGs have shown potential for
lower end and medium tactical grade performance. However, the technology appears to have
reached its fundamental limitation with bias uncertainties on the order of 10 ◦/h. In Class II CVGs,
the two modes are identical, allowing for arbitrary positioning of the drive axis or pattern angle.
This unique feature enables whole angle and self-calibration mechanism in addition to a direct rate
measurement [2]. These CVG’s are typically implemented as vibrating bars or axisymmetric
shells. In this chapter we describe the application of our resonator as a Class II CVG.
4.1 Coriolis Force
All micromachined gyroscopes use the Coriolis effect, which is defined as the deflection of
moving objects when the motion is described relative to a rotating reference frame. In a reference
frame with clockwise rotation, the deflection is to the left of the motion of the object; but with
counter-clockwise rotation, the deflection is to the right.
64

The dynamics of the revolving ring or cylinder have been investigated for a long time. In
1890, Bryan discovered that Coriolis forces give rise to splitting of the frequencies of natural
modes of flexural vibrations, leading to precession of standing waves and beats.
Since 1990, the interest in using a vibrating hemispherical shell as the sensing element in
gyroscope applications has increased. The main reasons are lower cost of hemispherical shells over
ring laser gyroscopes or fiber optical gyroscopes for a required whole-angle accuracy [3].
In this chapter we discuss the effect of Coriolis force on equations of motion specifically for
our device, and then we investigate its application as a rate integrating gyroscope.
4.2 Mathematical relations
Analytical relations for vibrations on the rim of the hemispherical shells and their response
to Coriolis force have been studied extensively [4-6]. Dynamics of a hemispherical resonator can
be expressed with Lagrange’s equations of motion for both n=2 wineglass modes.
The equations of motion for an ideal rate integrating shell type gyroscope with perfect
symmetry for the frequency (f) and ring down time (τ) are expressed in terms of the generalized
displacement along the two axes (q1, q2), effective mass (Meff), Coriolis mass (γ = 2MeffAg), yaw-
axis rotation rate (Ωz), and spring constants (𝑘 = 4𝜋 2
𝑓 2
𝑀 ) by [4]:
[
𝑀 0
0 𝑀 ] [
𝑞 ̈ 1
𝑞 ̈ 2
] + [
0 − 2𝛾 𝛺 𝑧 2𝛾 𝛺 𝑧 0
] [
𝑞 ̇ 1
𝑞 ̇ 2
] + [
𝑘 0
0 𝑘 ] [
𝑞 1
𝑞 2
] = [
0
0
] eq. 4.1
And the solution is:
65

 
 
1 0 0
2 0 0
*cos *cos
*sin *cos
z
z
q a dt t
M
q a dt t
M

  

  

   



   





where 𝜔 = √
𝑘 𝑀 ⁄ is the resonant frequency; a is in-axis vibration amplitude; and 𝜑 0
is an offset
phase. The total angle of rotation, Θ, can be found simply from Θ =
𝑡𝑎𝑛 −1
(
𝑞 2
𝑞 1
⁄ )
2𝐴 𝑔 ⁄
, where
𝐴 𝑔 is defined as the angular gain.
In reality we have non-idealities in both the quality factor (and so τ) and resonant frequency.
Although one of the benefits of hemispherical resonators or type II CVGs in general is their close
resonant frequencies, deviation from the ideal case of the same resonant frequency for the two
degenerate modes for the n=2 vibration mode will result in unacceptable beat frequency on the
output signal for whole angle measurement.
4.2.1 Simulations for signal envelope without non-idealities
Using the mathematical relation in the previous section, we can perform MATLAB
simulations for 𝑞 1
and 𝑞 2
and their combination. In the ideal case where Δf is zero, we have the
results shown in Figures 4.1 – 4.4 for a typical vibration of 1 rad/s. Specifically, Figure 4.1 shows
the amplitudes of the traveling waves (namely the 𝑞 1
and 𝑞 2
amplitudes in eq. 4.2) and their sum
which represents the sensor output. Figure 4.2 shows the simulated amplitude of the sum of two
traveling waves versus time, along with the total angle of rotation vs time. Now, we linearly map
the simulated amplitude shown in Figure 4.2 to make it (1) start at 0 V at t = 0 sec through an offset
and (2) match the value at t = 1 sec through choosing a proper fitting parameter. And the linearly
eq. 4.2
66

mapped amplitude is shown in Figure 4.3, after converting radians to degrees. The
mapped/calibrated output signal (Figure 4.3) represents the actual angle of rotation with maximum
error of 5° over a 57.29° total rotation range. The maximum error happens around 20° - 30°, and
is due to the facts (1) that output of the sensor is sinusoidal in time for a constant rotational speed
and (2) that since we use two end points in our linear mapping, the error is more pronounced in
the middle of the range. Figure 4.4 shows the sinusoidal output of the sensor over a long time span
that covers more than 360  total angle of rotation. This sinusoidal nature of the sensor output (even
for an ideal case of f = 0 Hz) means that we can use a linear approximation only for relatively
small angle of rotation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-3
time (s)
output signals amplitude (V)

first wave motion
second wave motion
combination of the two waves or standing wave motion
Figure 4.1 Simulated envelopes of the two travelling waves and their combination (middle line) that
results in a standing wave. For these simulations, a constant rotational speed of Ω = 1 rad/s is chosen.
The particular direction of the rotation chosen for the simulations makes the output signals appear to
decrease in time. But if the direction is opposite, the output signals will increase in time.
67

Figure 4.3: A closer look at Figure 4.2 with the output signal (blue line) linearly mapped (1) to start at
0 V at t = 0 sec through an offset and (2) to match the value at t = 1 sec through choosing one fitting
parameter. Over this relatively small angle of rotation (i.e., less than 57.29 ), the difference between
the actual rotation angle and the simulated output signal is at most 5  (around 20°~30°).
Figure 4.2 Simulated amplitude of the standing wave signal on the nodal
point (blue line) vs time, for a constant rotation rate of 1 rad/s. This
amplitude closely represents the total angle of rotation (shown in green
line) as long as the total angle of rotation is less than 180 .
68

4.3 Mode splitting in HRG
For the n=2 wineglass mode vibration of the dome diaphragm, there are two degenerate
modes, as indicated in Fig. 4.5. If there is non-symmetry in the diaphragm, the two modes will
vibrate at slightly different frequencies. With the fabricated dome resonator of which the n=2
wineglass mode vibration was at 126 kHz, we only observed essentially one resonant frequency
for both modes (as the precision of our frequency measurement was limited to about 1 Hz). We
even intentionally generated the two different modes through the two different actuation schemes
shown in Fig. 4.5. This observation of only one resonant frequency at 126 kHz can be construed
as direct evidence that the frequency split between the two degenerate modes was indeed less than
1 Hz (a ppm level), the minimum detectable frequency split that we could obtain with our
measurement system. This ppm level is an amazingly low frequency split for an as-fabricated
device, which is proven to be extremely symmetrical, likely due to HNA’s isotropic etching.
Figure 4.4 Simulated envelope of the nodal point signal as the combination of the two travelling
waves, while the ideal dome (with f = 0) is under a constant rotation of 1 rad/sec. As the dome
rotates 360  per 6.28 sec, the output signal is sinusoidal in time with a period of 10.13 seconds.
69

10
6
20
40
60
80
100
120
140
160
, rad/s
|H|, dB
10
5
10
6
10
7
-100
-50
0
50
100
150
, rad/s
|H|, dB
Figure 4.5 Two degenerate modes of the n=2 wineglass mode vibration of
the dome resonator (Top) and schematic of their measurement setup
(Bottom).
Figure 4.6 Simulated frequency responses of the dome resonator over a wide frequency range
(Top) and over a narrow range near the resonant frequency of the n=2 wineglass vibration
mode showing the frequency split (Bottom).
70

4.3.1 Frequency split based on the developed model
Based on the equations mentioned in Section 4.2, we developed a Simulink™ model. The
idea behind this model is to monitor the effect of non-idealities on the output of the system. The
benefit of using a Simulink™ model is that we do not need to rely on approximations and we can
numerically solve the governing differential equations of motion, namely:
[
𝑀 0
0 𝑀 ] [
𝑞 ̈ 1
𝑞 ̈ 2
] + [
0 − 2𝛾 𝛺 𝑧 2𝛾 𝛺 𝑧 0
] [
𝑞 ̇ 1
𝑞 ̇ 2
] + [
𝑘 1
0
0 𝑘 2
] [
𝑞 1
𝑞 2
] = [
0
0
]
which in open form is:
𝑀 𝑞 ̈ 2
− 2𝛾 𝛺 𝑧 𝑞 ̇ 2
+ 𝑘 𝑞 1
= 0
𝑀 𝑞 ̈ 1
+ 2𝛾 𝛺 𝑧 𝑞 ̇ 1
+ 𝑘 𝑞 2
= 0
In these equations we define 𝜔 1
=
√
𝑘 1
𝑀 ⁄ and 𝜔 2
=
√
𝑘 2
𝑀 ⁄ , and Δ𝜔 = 𝜔 1
− 𝜔 2
is
defined as the frequency split (Δ𝜔 = Δ𝑓 ∗ 2𝜋 ). The quality factor (and thus the ring down time) is
about the same for both modes, since we could not measure any difference in the ring down times
when we activated the two modes as illustrated in Figure 4.5.
By applying different k1 and k2 (i.e., different Δf’s) in the Simulink™ model shown in Fig.
4.7, we simulated how the dome resonator performs as a rate-integrating gyroscope for an applied
rotation at a constant rotational speed of 1 °/sec. We compared the simulated outputs of the dome
resonator for Δf = 0.1Hz and 0.3Hz with a measured output in Fig. 4.7. The measured output (with
small random noises) shows a linearly increasing output signal up to 4 sec (for an applied rotation
at a constant rotational speed of 1 °/sec). This result matches with a simulated output for Δf =
0.1Hz. If Δf is a little larger (say 0.3Hz), the simulated beat frequency is much higher than that of
eq. 4.3
eq. 4.4
71

the measured data. Thus, we estimate the dome-resonator’s frequency split to be 0.1Hz, a mere 0.8
ppm of the resonant frequency.

Figure 4.7 (Top) Simulink™ model setup to simulate the performance of the dome resonator as a
rate-integrating gyroscope, as the two degenerate frequencies are varied (i.e., as the frequency split Δf
is varied). (Bottom) Two Simulink™-simulated and one measured output of the dome resonator vs.
time, as an angular rotation at a constant speed of 1 /sec is applied. The measured output (with small
random noises) shows a linearly increasing output signal up to 4 sec, which matches to a simulated
output for Δf = 0.1Hz. If Δf is a little larger (say 0.3Hz), the beat frequency is simulated to be much
higher than that of the measured data.
Measured data
Simulation for Δf = 0.1 Hz
Simulation for Δf = 0.3 Hz
72

4.4 Our device results
With our fabricated resonator in a vacuum chamber, supported on a rotation table, we
measured the resonator’s response to rotation by monitoring the output signal at one of the nodal
points. As the resonator was rotated at a rate of 1°/s, the piezoelectrically sensed signal was
recorded in a computer after amplification. As plotted in Figure 4.8, the output voltage of the nodal
point linearly increases as the net angle of rotation is increased (with the angular rate kept constant
at about 1°/s). This result shows the functionality of our resonator as a “rate integrating gyroscope”.

A close-up view on the measured data in Figure 4.8 near zero net rotation angle is shown
in Figure 4.9 from which we see that the minimum detectable angle (or the point where the sensed
signal surpasses the noise) is about 0.15°. The sub-degree detectability is highly encouraging for
a microfabricated device.
0 0.2 0.4 0.6 0.8 1
-20
-10
0
10
20
angle (Degrees)
output signal (mV)
0 0.2 0.4 0.6 0.8 1
-5
0
5
10
angle (Degrees)
envelope of the output signal (mV)
Figure 4.8 Measured output for an applied rate of 1°/s. Top
plot shows the actual output of the nodal point, and the
envelope of the signal increases, as the total angle of
rotation increases under a constant applied rotation rate.
Bottom plot shows just the envelope of the signal.
73

4.4.1 Response to forward and backward rotation
We have measured our resonator’s response when we rotate the device in one direction for
a time and then reverse the direction of rotation. As depicted in Figure 4.9, when the device is
rotated for 3 seconds in one direction and then the direction of rotation is reversed (with a faster
rate of change), the output voltage of the nodal point is also decreased as the rotation angle is
reversed.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
2
4
6
8
10
angle (Degrees)
envelope of the output signal (mV)

output signal when rotation is applied
noise floor
Figure 4.9 Measure plot showing the detected angle vs.
noise level in the system. This noise level will lead to a
minimum detectable angle of 0.15°

74

By linear fitting of the envelope of the signal and comparing it with the angular rate and
absolute angle, we find the unamplified sensitivity of 4.84 mv/º/s. This sensitivity can be compared
with those of published rate-integrating gyroscopes [8, 9]. However, comparison of sensitivities
for piezoelectric and capacitive sensing is not straightforward, as the sensitivity of capacitive
sensing depends on the polarization voltage (or how the capacitance change is detected), while
with piezoelectric sensing unamplified sensitivity can easily be defined due to its capability of
producing a voltage difference without any help from a polarization voltage or circuitry.

Figure 4.10 Measured effect of changing the rotation direction. The sign
of the slope changes, as the rotation direction is flipped.
75

4.4.2 Minimum detectable rate of rotation
We also characterized our device to see what its minimum detectable angular rate in °/s is.
As can be seen in Figure 4.11, our device can detect down to 0.1°/s. At this point, the minimum
detectable angular rate and minimum detectable angle are limited by the relatively large noise level
that is mostly due to electromagnetic interferences (EMI) that comes from the fact that our device
has not yet been packaged with an EMI shielding box. It will be relatively straightforward to reduce
the EMI by an order of magnitude.

0 0.02 0.04 0.06 0.08 0.1
-10
0
10
angle (Degrees)
output signal (mV)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
2
4
6
8
angle (Degrees)
envelope of the output signal (mV)
Figure 4.11 Measured output to 0.1°/s rotation rate, which turns out to be the
minimum detectable rate for our device with current noise floor.
76

4.5 Angular gain in our device
The physical parameters of the device are calculated using COMSOL software. The shell
is modeled using the multiphysics module of COMSOL. Modal simulation is performed to obtain
the modal amplitude of all elements for the n = 2 wine-glass modes. Then we can calculate the
Meff as:
𝑚 𝑖 = ∫ 𝜌 (𝑢 𝑖 2
+ 𝑣 𝑖 2
+ 𝑤 𝑖 2
)
𝑣𝑜𝑙𝑢𝑚𝑒 𝑑𝑉
where u, v and w represent modal amplitudes in x, y and z directions respectively. For our fabricated
wineglass resonator, Meff is calculated to be 28.8 μg. The angular gain, Ag, is calculated as:
𝐴 𝑔 = 𝑘 𝑖𝑗
=
1
√
𝑚 𝑖 𝑚 𝑗 ∫ 𝜌 (𝑢 𝑖 𝑣 𝑗 + 𝑢 𝑗 𝑣 𝑖 )
𝑣𝑜𝑙𝑢𝑚𝑒 𝑑𝑉
The Ag of a low-aspect-ratio shell is small, because the shell flexes significantly in the
vertical direction (Z-axis) and less in the lateral directions (X-Y axis). When the shell is rotated
around the Z-axis, only small Coriolis force is generated since the motion of the shell is mostly
parallel to the axis of rotation. But the Ag of a high-aspect-ratio shell increases because the shell
flexes more in the lateral directions and less in the vertical direction. When the shell is rotated
around the Z-axis, a large Coriolis force is generated since the motion of the shell is mostly
perpendicular to the axis of rotation. For the fabricated wineglass geometry, Ag is calculated to be
0.21 [10].

eq. 4.5
eq. 4.6
77

4.6 Our device performance compared to state of the art
One of the most prominent aspects of our device is the ability of detecting the angle of
rotation. Most state-of-the-art rate integrating gyroscopes cannot measure the total angle directly
without backing electronics since they have low quality factor and large frequency split. Due to
low quality factor of these devices, the resonance should be reinforced through force rebalance
[5]. Our device, on the other hand, has a very high quality factor, and is capable of detecting total
rotation angle without any force rebalance. A comparison of our device and state-of-the-art devices
is presented in Table 4.1.
Table 4.1: Comparison of our devices with state of the art devices in MEMS world
Dome
diameter
(mm)
Resonant
frequency
(kHz)
Quality
factor
Ring down
time
Sensitivity
to angular
rate
(mv/º/s)

Minimum
detectable
angle
Our device
[7]
1.5 126 9.9 million 25s 4.84 0.15º
[8] 5 NA 0.26 million 8s 27.8 NA
[9] 1.2 412 8000 6ms NA NA
[11] 1 945 470 158μs 2.1 NA
[12] 2.5 8.7 1.2 million 43s NA NA

78

4.7 Other measurements and metrics for gyroscopic applications
The application domain of miniaturized gyroscopes is one of the fastest growing segments
in the micro-sensor industry. For example, a multitude of applications exist in the automotive
sector including navigation, rollover detection, and antiskid and safety systems.
MEMS gyroscopes also can be used for navigation purposes and be a part of inertial
measurement units (IMUs). IMUs typically use three accelerometers and three gyroscopes placed
orthogonally along the coordinate’s axes to gather information about an object’s direction and
heading.

The application space of microgyroscopes can be divided based on performance
requirements. Table 4.2 summarizes the performance requirements for rate-grade and navigation-
grade gyroscopes [13]. The majority of automotive applications fall under the rate requirements
whereas higher-precision navigation-grade devices are suitable for IMUs and high-end
applications.
IMU concept, Northrop Grumman
Figure 4.12 Inertial Measurement Unit (IMU) composed of three gyroscopes positioned
orthogonally in order to measure the change in angle in all directions. Our device size makes
it a very suitable candidate for miniaturized IMU [12].
79

Table 4.2: Different metrics in a gyroscope
Rate grade Navigation grade
Bandwidth (Hz) >70 ~10-100
Bias drift (deg.h
-1
) 10-1000 <0.1
Scale factor accuracy 0.1-1% <10 ppm
Full scale range (deg.s
-1
) 50-1000 >500
Shock level (g) 1000 10
3
-10
4

Angle random walk (deg.h
-1/2
) >0.5 <0.005

4.7.1 Bias stability
An important measure of long-term stability of a gyroscope is its bias drift. Very similar to
offset in analog circuits, the bias drift of a gyroscope is composed of systematic and random
components. The systematic components arise from temperature variations, linear accelerations,
vibrations, and other environmental factors.
4.7.2 Bandwidth
The bandwidth of the gyroscope determines the response time for a step input, i.e., the time
for the output to settle within a certain range of the expected value for an input step function.
4.7.3 Dynamic range
Dynamic range refers to the range of input values over which the sensor can detect.
Typically it can be calculated as the ratio between the maximum input (full scale range) and noise
floor.
80

4.7.4 Angle Random Walk (ARW)
Another important parameter in a rate gyroscope is angle random walk (ARW). The ARW
from a rate gyroscope is equivalent to white noise in the angular rate output. Specifications are
sometimes given for random-walk noise in gyro sensors, but mostly for the integral of the noise,
not in the noise itself. In our device that operates in a rate integrating mode, ARW is not important.

4.8 Chapter 4 references
[1] IEEE Standard Specification Format Guide and Test Procedure for Coriolis Vibratory Gyros,"
IEEE Std 1431-2004, pp.1-78, Dec. 20 2004.
[2] A. A. Trusov, G. Atikyan, D. M. Rozelle, A. D. Meyer, S. A. Zotov, B. R. Simon, A. M.
Shkel, "Force rebalance, whole angle, and self-calibration mechanization of silicon MEMS
quad mass gyro," in Inertial Sensors and Systems (ISISS), 2014 International Symposium on,
2014, pp. 1-2, 25-26.
[3] J. Dickinson, C. R. Strandt, "HRG strapdown navigator," in Position Location and Navigation
Symposium, 1990, pp.110-117.
[4] J. Y. Cho, J. K. Woo, J. Yan, R. L. Peterson, and K. Najafi. "Fused-Silica Micro Birdbath
Resonator Gyroscope (μ-BRG)."Microelectromechanical Systems, Journal of, Vol. 23, No.
1, pp. 66-77, 2014.
[5] J. Cho, J-K. Woo, J. Yan, R. L. Peterson, and K. Najafi. "A high-Q birdbath resonator
gyroscope (BRG)," in Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS &
EUROSENSORS XXVII), 2013 Transducers & Eurosensors XXVII: The 17th International
Conference on, 2013, pp. 1847-1850.
[6] C. O. Chang, J. J. Hwang, and C. S. Chou. "Modal precession of a rotating hemispherical
shell," International journal of solids and structures. Vol. 33 No.19 pp. 2739-2757, 1996.
[7] A. Vafanejad and E. S. Kim, “Sub-degree angle detection using dome –shaped diaphragm
resonator with wine-glass mode vibration,” in Proc. Hilton Head, 2014, pp. 391-394.
[8] M. L. Chan, P. Fonda, C. Reyes, J. Xie, H. Najar, L. Lin, K. Yamazaki, and D. A. Horsley.
"Micromachining 3D hemispherical features in silicon via micro-EDM," in Micro Electro
Mechanical Systems (MEMS), 2012 IEEE 25th International Conference on, pp. 289-292.
81

[9] L. D. Sorenson, X. Gao, and F. Ayazi, "3-D micromachined hemispherical shell resonators
with integrated capacitive transducers." in Micro Electro Mechanical Systems (MEMS), 2012
IEEE 25th International Conference on. IEEE, 2012, pp.168-171.
[10] "Private communication with A. Dorian Challoner."
[11] S. A. Zotov, A. A. Trusov, A. M. Shkel, "Three-Dimensional Spherical Shell Resonator
Gyroscope Fabricated Using Wafer-Scale Glassblowing," Microelectromechanical Systems,
Journal of, Vol.21, No.3, pp.509-510, June 2012.
[12] J. Y. Cho, and K. Najafi. "A high-q all-fused silica solid-stem wineglass hemispherical
resonator formed using micro blow torching and welding." In Micro Electro Mechanical
Systems (MEMS), 2015 28th IEEE International Conference on, pp. 821-824.
[13] D. M. Rozelle, "The hemispherical resonator gyro: From wineglass to the planets." in Proc.
19th AAS/AIAA Space Flight Mechanics Meeting. 2009, pp. 1157-1178.
[14] F. Ayazi, M.F. Zaman and A. Sharma, "Vibrating Gyroscopes," in Comprehensive
Microsystems: Fundamentals, Technology and Applications, Vol. 2, pp. 181-208, Edited by
Y.B. Gianchandani, O. Tabata, and H. Zappe, Oxford, Elsevier Ltd.

82

Chapter 5
Measurement Setups
This chapter describes the measurement setups and circuits used for this project. We had
two main setups for the measurements. The first one was quality factor (ring down time)
measurement setup and the second one was rotation measurement setup.
5.1 Ring down time measurements
One important requirement for rate-integrating gyroscopes in an aircraft’s inertial
navigation system is a large decay time constant, which means a high Q [1]. High Q means that
the actuating and sensing elements do not have to work simultaneously and also that the bias drift
is low, since high Q means low difference between two decay times of the two degenerate modes.
We measured the decay time constant of our fabricated dome-shaped-diaphragm resonators by
actuating the resonators at their resonant frequencies, stopping the actuation, recording the
decaying amplitudes, and fitting the recorded amplitudes with an exponential function, Aoexp(-
t/τ). As can be seen in Figure 5.2, it took 25 seconds for the amplitude of the sensed signal to drop
by 63% after the signal to the actuators was turned off. Such a large decay time corresponds to a
Q of 9.9x10
6
.

83

5.2 Rotation measurement setup
For this measurement, we apply a certain input voltage to the four electrodes (e.g., #1, 3,
5, and 7 electrodes indicated in Figure 5.4) and measure the voltage on the nodal point (e.g., #4).
Figure 5.2 Measured voltage vs time that shows the device’s ring
down, which is 25 seconds.
Figure 5.1 Schematic to illustrate the relative locations for actuation
and sensing used to measure the decay time.
84

The change in the envelope of the nodal point signal will be representative of the total angle of
rotation.
The dome resonator’s characteristics were evaluated at a vacuum pressure of <50mTorr at
room temperature. The resonator was mounted on a printed circuit board (PCB), through which
electrical connections were made (Figure 5.3).

A standing wave in a wine-glass mode was generated by energizing four piezoelectric
actuators located on the anti-nodal lines of the wine-glass mode. Two of the four actuators (#1 and
5 in Figure 5.4) were energized with an identical signal, while the other two (#3 and 7 in Figure
5.2) were actuated by a signal with 180° phase lag.

Figure 5.3 Photos of the dome diaphragm transducer on a printed circuit board with electrical
wires connected to the bonding pads through epoxy: (Left) the dome transducer and the whole
PCB, (Middle) a close-up view, (Right) photo of the transducer and electronics for actuation and
sensing.

Figure 5.4 Cross-sectional diagram (Left) and top view photo (Right) of the dome-shaped-
transducer that is suspended by a thin flexible membrane and actuated by four piezoelectric
transducers and sensed by four other ones, placed on the rim of the dome diaphragm.

85

The vibration of the dome diaphragm was measured by monitoring the electrical signal
output of the piezoelectric sensors (i.e., any of the numbered electrodes in Figure 5.4) and
embedded on the dome diaphragm rim. A schematic of the measurement setup is shown in Figure
5.5.

To confirm the wine-glass mode vibration, we compared the phase difference between the
sensor signals and the signal applied to the actuators 1 and 5 in Figure 5.2. We applied an electrical
signal to a pair of the actuators and picked up the signals from either the Sensor 3 or Sensor 7 (as
illustrated in Figure 5.5). We observe that the measured signal from the Sensor 3 has a 180° phase
difference from the signal applied to the Actuator 1, as can be seen in Figure 5.6, as expected.

Figure 5.5 Schematic of the setup for measuring rotation angle.
Sensor electrodes
Forcer electrodes
Function
generator
86

5.3 Actuating circuit and detection circuit
For the actuation part of the system we used a regular opamp with high voltage output. We
used a Texas Instruments LM7171 as the op-amp. One of the reasons behind choosing this op-
amp is its large output swing (±15V). The opamp was configured in a feedback scheme with a gain
of -10. In the case of the rotation measurement setup (Figure 5.3), we first used the opamp output
as the input of the second op-amp in order to have the 180º difference in the applied voltage
electrodes that are placed 90º from each other on the rim of the device. In this case the diode at the
input would not be needed since the LM7171 does not have a high input resistance.
Figure 5.6 Oscilloscope traces of the applied signal at the Actuator 1 (the curve with
very little noise with peak-to-peak value of 1.4V), measured signal at the Sensor 3 (the
curve with noise with peak-to-peak value of 0.04V). The 180° phase difference
between the two curves shows that the vibration mode is indeed a four-node wine-glass
mode.
-1.5 -1 -0.5 0 0.5 1 1.5
x 10
-5
-0.04
-0.02
0
0.02
0.04
time(s)
sensor output voltage (V)
-1.5 -1 -0.5 0 0.5 1 1.5
x 10
-5
-1
-0.5
0
0.5
1
input applied voltage (V)

applied signal at actuator 1
measured signal at sensor 3
Applied Signal at Actuator 1
Measured Signal at Sensor 3
87

As a preamp for the signal from the piezoelectric sensing element, we used a LT1793
opamp in the configuration depicted in Figure 5.8. This opamp is chosen because of its high input
resistance, and a diode is needed to provide a DC leak path so that the opamp may be biased
properly. Without the diode, the opamp output quickly drifts to the power supply voltage.

Figure 5.7 Inverting voltage amplifier to generate a signal to drive the piezoelectrically
actuating element. Vout = -Vin(R2/R1), and R3 is recommended to be chosen to be
R1R2/(R1+R2) for minimum error due to input bias current.
88

5.4 New ideas
One of the bottlenecks in achieving the minimum detectable angle is the noise floor in our
system. With proper shielding and a better designed circuitry we can achieve lower minimum
detectable angle. The prelimianry results shown in Figure 5.10 indicates that the noise level is
reduced by a factor of two(??) from 6.6mVpeak-to-peak (Figure 5.9) to 3.2mVpeak-to-peak (Figure 5.10).

Figure 5.8 Voltage amplifier for the piezoelectrically sensed signal with voltage gain of 101.
The resistance values can be adjusted for a different gain, as Vout = Vin(R1+R2)/R1. For example,
if the 47 Ω resistor is changed to 522 Ω, while keeping the 4.7 kΩ as it is, the voltage gain will
be 10.
89

Figure 5.10 Measured noise floor and photo of the new breadboard
with smaller wires and better shielding to decrease the noise level.
Figure 5.9 Measured noise floor and photo of the packaging
scheme used in the measurements so far.
90

5.5 Chapter 5 references
[1] D. M. Rozelle, "The hemispherical resonator gyro: From wineglass to the planets." in Proc.
19th AAS/AIAA Space Flight Mechanics Meeting. 2009, pp. 1157-1178.

91

Chapter 6
Conclusion and Future Directions
Hemispherical resonators have been designed, fabricated, and characterized. Quality
factors (Q) of these devices have been studied extensively, and various methods to improve the Q
have been explored both theoretically and experimentally. Measurements have confirmed
simulations, and viability of these approaches has been proven. However, there are still several
approaches that need to be explored to increase the Q even further.
The devices have been demonstrated to work as a rate-integrating hemispherical resonant
gyroscope (HRG). This demonstration is the first, to our knowledge, for microfabricated HRG, as
there has been no report of using hemispherical resonators as a rate integrating gyroscope without
complex electronics and extensive feedback scheme. One of the devices had a ring down time of
25 seconds, and held the record of the longest ring down time for microscale HRG up to 2014. The
ring down time corresponds to a Q of 9.9 million (at a remarkably high frequency of 126 kHz for
an n=2 wineglass mode vibration), and makes the device suitable for gyroscope applications.
Non-idealities for the hemispherical resonators have been studied in the course of this
research, quantifying the effects of various non-idealities on the frequency split and quality factor
for the n=2 wineglass mode vibration. Particularly, the frequency split was confirmed to be about
0.8 ppm for the high-Q 126-kHz device. Also, the device properties and functionalities have been
analyzed theoretically, modeled, simulated, and compared with measurements.
The following sections describe potential future directions to increase the Q and the
device’s performance as a rate-integrating gyroscope.
92

6.1 Larger domes
In Chapter 2, the effect of larger domes is discussed extensively, and the quality factor is
shown to increase by simply scaling the dome size, as repeated in Figure 6.1. This approach is
currently being pursued in our group.

Figure 6.1 Effect of the dome size on the resonant frequency and quality factor
6.2 Effect of ZnO layer encapsulation
Encapsulating ZnO film completely with electrically insulating layers was noted to
increase electrical charge leak from hours to days [1], and was discussed in Chapter 2. Though the
first attempt to encapsulate ZnO film did not show a large improvement on the Q, this approach
also is being actively pursued with different thicknesses on the top and bottom insulating layers.

0 500 1000 1500 2000 2500 3000
10
2
10
4
10
6
Radius ( m)
Resonance frequency (Hz)
0 500 1000 1500 2000 2500 3000
10
6
10
8
10
10
Quality factor
h = 1 m
93

6.3 New packaging for gyroscopic applications
At this point we do not own a rate table, nor has capacity to vacuum-package the devices,
and testing the devices for gyroscope applications has been challenging. Chapter 3 presents the
experimental data showing that the Q of our device is not sensitive to ambient pressure as long as
it is kept under 350mTorr. This offers us flexibility in the packaging, and a simple small bell jar
can be used for packaging of the device. With a smaller transparent jar shown in Fig. 6.2, we can
rotate the device (contained in the jar) more easily. Also, the transparency of the chamber will help
us to characterize the device better with a laser Doppler displacement meter or other optical means.

Figure 6.2 A new packaging proposal for our device. The smaller package will let us do gyroscopic
measurement more easily, while the chamber transparency helps in characterizing the device using
an optical probe.

6.4 New circuit
Section 5.4 describes the potential benefits of a compact circuit for improving the noise
floor. A new PCB design with minimum wire length as well as small and compact packaging can
94

help us to improve the minimum detectable angle that is currently limited by electromagnetic
interference (EMI) in the sensing circuitry.
6.5 Chapter 6 references
[1] P. L. Chen, R. S. Müller, R. M. White, and R. Jolly. "Thin film ZnO-MOS transducer with
virtually DC response,"1980 Ultrasonics Symposium. 1980, pp.945-948.

95
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Abstract (if available)

Abstract This thesis presents microelectromechanical systems (MEMS) hemispherical wineglass mode resonator and its utilization as a rate integrating gyroscope. A key objective for this device was to increase the quality factor. Etensive studies have been done on different parameters affecting the quality factor in the n=2 mode resonance of a hemispherical resonator. Mathematical derivation and MATLAB simulation have been performed to optimize the device dimensions. We could achieve a record breaking 25 seconds of ring down time (associated with a quality factor of 9.9 million). ❧ A finite element method (FEM) model has been developed, and various simulations have been performed in order to study the effects of different parameters on the quality factor. The device size and thickness have been optimized based on FEM simulations. Also studied were pre‐ and post‐processing methods to improve the quality factor. Specifically investigated was a post‐fabrication laser trimming technique, with which the quality factor was improved by 50% (experimentally) through making a hole at the apex of the dome diaphragm. Also, the effect of compressive stress in the SiN diaphragm on the quality factor was studied. ❧ The feasibility of using these resonators as rate integrating gyroscope was explored, and for the first time functionality of a MEMS resonator as a rate integrating gyroscope without the need for complex circuitry. The minimum detectable angle of rotation was measured to be 0.15º, with EMI noise limiting the detection limit.

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Asset Metadata

Creator Vafanejad, Arash (author)

Core Title Wineglass mode resonators, their applications and study of their quality factor

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/17/2015

Defense Date 03/23/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag gyroscope,MEMS,OAI-PMH Harvest,resonator,wineglass mode

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kim, Eun Sok (committee chair), Povinelli, Michelle L. (committee member), Shung, Koping (committee member)

Creator Email vafaneja@usc.edu,vafanejad.arash@gmail.com

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