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Tunable micromachined microwave devices

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UMI Number: 9624862
Copyright 1996 by
Ayon, Arturo Alejo
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TUNABLE MICROMACHINED
MICROWAVE DEVICES
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the requirements for the Degree of
Doctor of Philosophy
by
Arturo Alejo Ay6n
May 1996
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© Arturo Alejo Ayon 1996
ALL RIGHTS RESERVED
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BIO G R A PH IC A L SK ETC H
Arturo Alejo Ay6n Ballesteros was bom in the sunny state of Sonora, Mexico
on July 17th, 1958. Even though he left his native town, Navojoa, in 1969, he is still
looking for an opportunity to return to the Sonoran Desert, and become a high-tech
Hacendado. During his undergraduate years, he started working for Burroughs (now
Unisys) as a technician and learned that even state of the art equipment fails once in a
while in a catastrophic manner. He finished his coursework in Electronic Engineering
in January, 1983, and IBM hired him in June, 1983, only to send him on International
Assignment 5 months later to another sunny and hospitable place: Rochester,
Minnesota. While in the Midwest, life was generous to him bringing several yards of
snow, memorable low temperatures, engines that refused to start in the morning, trips
to Mexico, Canada, Yellowstone Park and Hawaii, and two daughters: Maria
Alejandra and Nancy Yvette. His 1985 homecoming welcome was a management
position at IBM, very warm weather and plenty of sunshine. The arrival of his third
daughter, Dianne, in 1986, was an additional source of joy. Entrepreneurial
opportunities arose and left IBM in 1986 to explore the wonderful world of business
and stress. Trying to stay away from collapsing, he and his family crisscrossed
Mexico twice and visited all archeological sites on the map. When pyramids started to
appear in his dreams he decided to visit the Pacific Ocean every other month.
However, he missed White Christmases so much, that he arrived in Ithaca in 1989. He
received the Masters of Science Degree in 1992, and in the spring of 1996, the degree
of Doctor of Philosophy. As a prelude of what 1995 would bring, his son Arturo
Alessandro was bom in December 8th, 1994.
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To the memory of my father,
with deep admiration and gratitude
To my wife and children,
for the future
iv
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ACK NO W LEDG M ENTS
My wife and children have always provided for an oasis of serenity and
understanding at home. I am particularly lucky for having them, and for enduring
with me the hardships of graduate school.
This thesis could not have been completed without the guidance and financial
support of my advisor, Professor Noel C. MacDonald. I thank him for keeping me
focused when my mind began to wander but also allowing me to think on my own
feet. I will need both skills in the future.
I also thank the chairman of my Special Committee, Professor David A.
Hammer who introduced me to the fascinating world of Plasma Physics. He has
always been a source of inspiration and support, especially in the most difficult
periods while staying at Cornell. I deeply appreciate his sound advice, both personal
and professional and deserves a special thanks for his patience in reading through this
material and giving valuable advice.
I also thank the other members of my committee Drs. David C. Clark and
Clifford Pollock. Dave Clark continues to pique my interest in Nuclear Engineering.
Clifford Pollock awakened my interest in optoelectronics and lasers, his lessons on
how to tackle engineering problems will always be helpful.
Among my colleagues I enjoyed the conversations and discussions with Alex
Atwood, John Mercier, Brian Oliver, the late Gilberto Barreto, David Hong, Trent
Huang, Dan Haronian, Nick Kolias, Taher Saif, John Chong, Robert Mihailovich,
Scott Miller, Sean O’Keefe, Wolfgang Hofmann and Hercules Neves.
Finally, I want to extend my appreciation for the technical support of the staff
of the Cornell Nanofabrication Facility at Cornell University. Without their help, I
would not be able to have accomplished my experimental work.
This research has been partially funded by ARPA and NSF.
v
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TABLE OF CONTENTS
Chapter 1
Introduction
1.1 MEMS Technology.
.
.
.
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.
1
1.2 Thin Film Technology versus High Aspect Ratio Structures
3
1.3 Other Schemes.
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7
1.4 Applications.
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1.5 Synopsis. .
1.6 References.
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Chapter 2
Transmission Line Theory
2.1 Transmission Line Equations.
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17
2.2 The Field-Cells Approach.
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28
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36
2.3 Thin Plane Conductors.
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2.4 Quasi-Static Approximation.
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2.5 Resonance and Q-factor. .
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2.6 Scattering Matrices.
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51
2.7 References.
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Chapter 3
Fabrication Techniques
3.1 SCREAM Process.
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56
3.2 Process Overview..
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58
3.3 Alternative Approaches: Fully Oxidized Structures.
70
3.4 Coplanar Waveguides.
79
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3.5 References.
82
C hapter 4
M easured Performance
4.1 Test Setup and De-Embedding.
4.2 On-Wafer Probing.
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4.3 Measured Performance.
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4.4 Electrically Thin Substrates.
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4.5 Surface Roughness and Other Considerations.
4.6 Impedance Variation.
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4.7 Electromechanical Tuning.
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4.8 Comparison with Conventional Structures.
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4.9 References.
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C hapter 5
Half-Wavelength Micromachined Dipole Antennas
5.1 Introduction.
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5.2 Basic Antenna Parameters.
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5.3 Radiation Caracteristics of Linear Dipoles.
5.4 Microfabrication.
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5.5 Test Setup and Measured Performance
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5.6 References.
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Chapter 6
Conclusions and Future W ork
5.1 Conclusions.
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5.2 Future Work.
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Appendix I
Other Matrix Representations.
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Appendix II
Other Considerations in Deep Trench etches..
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Appendix III
Equations related to CPW and CPWG
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Appendix IV
Surface Roughness.
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151
LIST OF TABLES
Table 4.1: Etch rates for different silicon planes.
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LIST OF FIGURES
Figure 1.1: Cross section o f a slotline, the thickness of the metallic plates is many
times smaller than the thickness of the substrate..
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.
4
Figure 1.2: Microstrip, this geometry is well suited for certain applications such as
filters and periodic structures.
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4
Figure 1.3: Ccplanar Waveguide (CPW), geometry useful for amplifiers. .
4
Figure 1.4: Schematic view of tunable transmission lines.
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Figure 1.5: Cross section o f the geometry presented in this work.
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6
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8
Figure 1.7: Top view of an electromechanically tunable transmission line. .
9
Figure 1.6: Layout of an electrically tunable coplanar transmission line.
Figure 2.1: Equivalent circuit of an infinitesimal length Ax of a transmission line 19
Figure 2.2: All space being considered is divided in unit cells of length /. .
23
Figure 2.3: Conductor carrying a uniform current J. .
24
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Figure 2.4: The conductor is divided in unit cells o f length /.
Figure 2.5: Conducting strips carrying a current/.
Figure 2.6: Field-cell transmission line.
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Figure 2.7: Profile of the sputtered metallic film.
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Figure 2.8: Coordinate system of parallel-plate transmission line.
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25
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3T
Figure 2.9: Z0 versus h/s, 377 s/h is the impedance o f a semi-infinite line includ­
ed for comparison purposes.
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Figure 2.10: Scattering waves in a two-port network.
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39
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Figure 3.1: Overview of the general processing approach presented in this work.
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57
Figure 3.2: Surface Roughness for a PECVD film deposited @ 240°C.
.
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Figure 3.3: Surface Roughness for a PECVD film deposited @ 300°C.
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Figure 3.4: Surface Roughness for a PECVD film deposited @ 360°C.
.
60
Figure 3.5: SEM micrograph showing the effect of leaving residues on top o f the
mesas which are subsequently physically sputtered and produce
micromasking.
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Figure 3.6: SEM micrograph showing the micromasking effect of contaminants
gathered during photolithography.
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Figure 3.7: Auger analysis o f a wafer with abundant presence of “grass”
revealed the presence of aluminum, oxygen and silicon.
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63
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Figure 3.9: Chlorine chemistries provide for an excellent anisotropy.
.
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Figure 3.8: SEM micrograph showing the presence of trenching.
Figure 3.10: SEM micrograph showing the characteristics sought in deep silicon
etches.
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Figure 3.11: SEM micrograph showing the passivation layer that has to be
removed before releasing the structures. .
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Figure 3.12: Removal o f the passivation layer provides for smooth uniformly
released structures.
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Figure 3.13: SF6 chemistries provide for isotropic etches.
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Figure 3.14: SEM micrograph showing a released structure..
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Figure 3.15: Sputtering at a pressure of 9 mT the surface roughness and stress
increase. Noticeable lumps of aluminum form at the rim o f the
structures.
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Figure 3.16: Annealing of an aluminum film to change the stress from
compressive to tensile. Courtesy of J. Drumheller.
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Figure 3.17: SEM micrograph showing a detail of the cantilevered actuators.
70
Figure 3.18: Picture presenting the growth of the structures during oxidation.
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Figure 3.19: After a 4-hour oxidation 0.8 pm beams are fully oxidized.
Figure 3.20: After a 4-hour oxidation there is still a silicon spike in beams 1 pm
wide.
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.
Figure 3.21: For a pitch too small, the oxidation step closes the gap at the top
before the beams are fully oxidized.
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Figure 3.22: The right pitch permits the full oxidation of the beams.
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Figure 3.23: The MIE step damages the structures when the pitch is too wide.
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Figure 3.24: The pitch is crucial to achieve full oxidation of the anchors such that
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they become a single unit and able to survive an MIE step.
Figure 3.25: Large structures can be suspended with the anchor technique. .
76
Figure 3.26: SEM micrograph presenting a detail of the actuators cantilevered
and fully oxidized.
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Figure 3.27: Detail of the cantilevered fully-oxidized parallel-plate lines.
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Figure 3.28: Detail of meandering lines fully oxidized and cantilevered.
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Figure 3.29: SEM micrograph presenting the effect of using PECVD oxide
during oxidation.
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Figure 3.30: Layout for a conventional coplanar waveguide (CPW).
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Figure 3.31: Layout for a grounded coplanar waveguide (CPWG). .
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Figure 4.1: Overview of the test setup.
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Figure 4.2: Modeling the shunt impedance of the launch pads.
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89
Figure 4.3: Comparison of reflection and transmission scattering parameters for
a line and a model including a shunt impedance.
90
Figure 4.4: Theoretical and experimental attenuation.
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Figure 4.5: Extracted and theoretical phase in units of Rad/mm.
93
Figure 4.6: Extracted attenuation for lines fabricated on substrates with different
xii
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thicknesses.
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Figure 4.7: Extracted phase constant for different substrate thicknesses.
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Figure 4.8: Extracted impedance for different substrate thicknesses.
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Figure 4.9: SEM micrograph showing the surface roughness of the sputtered
aluminum film deposited at a pressure of 1.6 mT.
.
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Figure 4.10: Impedance variation with plate height, for a plate separation of
4.7 pm.
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Figure 4.11: Impedance variation with plate separation, for a plate height of
12.7 pm..
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Figure 4.12: Measured and theoretical impedance for a plate height of 12.7 pm.
101
Figure 4.13: Measured and theoretical impedance for a plate height of 10 pm.
101
Figure 4.14: Measured and theoretical impedance for a plate height of 7 pm.
102
Figure 4.15: Measured and theoretical impedance for a plate separation of
8.9 pm. .
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Figure 4.16: Electromechanical
.
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tuning,
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impedance
.
.
variation
102
can
be
accomplished either by opening the plates (Zo increases) or by
closing them (Zo decreases).
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104
Figure 4.17: Electromechanical tuning, when the transmission line array is
perfectly matched permits the optimization of the working point.
105
Figure 4.18: Comparison of transmission parameters s21 for a CPW and one of
the lines presented in this work as measured, before de-embedding
106
Figure 5.1: Coordinate system for an infinitesimal linear dipole of total length 1
115
Figure 5.2: Coordinate system for a linear dipole.
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Figure 5.3: The radiation pattern for a half-wavelength linear dipole.
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Figure 5.4: The variation of radiation resistance with dipole length..
.
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Figure 5.5: The reactance as a function of dipole length for two different dipole
radii.
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Figure 5.6: Schematic cross section of the silicon wafer with the alignment marks
electrochemically etched..
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Figure 5.7: SEM micrograph of an alignment mark. .
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Figure 5.8: Schematic cross section of the masking wafer showing the alignment
marks and the standing pads.
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Figure 5.9: Schematic cross section of the masking wafer showing the alignment
marks, standing pads and antenna well.
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Figure 5.10: Antenna structure microfabricated as described in Chapter 3. .
130
Figure 5.11: After removing the silicon substrate we have the microfabricated
structure on a thin silicon membrane.
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Figure 5.12: SEM micrograph showing the alignment mark and the antenna
after the silicon substrate has been removed
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Figure 5.13: SEM micrograph showing the electrochemical etch underneath the
silicon beams.
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Figure 5.14: After removing the silicon nitride membrane, the structure is freestanding with no substrate underneath.
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Figure 5.15: Test setup for antennas, only one probe is needed to measure the
reflection coefficient. .
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Figure 5.16: Schematic view o f the structure tested in this experiment.
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Figure 5.17: SEM micrograph showing the finished device .
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Figure 5.18: Measured reflection Coeficient..
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Figure AIL 1: SEM micrograph showing the characteristics usually sought in
deep trenches including smooth and steep sidewalls, no trenching
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136
and absence of grass.
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Figure AII.2: SEM micrograph showing the effect on an isolated feature.
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Figure AII.3: SEM micrograph showing the effect on a wall.
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Figure AII.4: SEM micrograph showing the effect only where the walls have a
clearance larger that the “new feature size”.
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Chapter 1
INTRODUCTION
1.1 MEMS Technology
MicroElectroMechanical Systems Technology is the developing field of micro
devices and structures utilized mainly as sensors, actuators and transducers. It is
common to refer to this technology as one of the most promising in both the short and
the long range. In fact, it has been identified by the National Science Foundation as an
area of national importance [1]. The ability to manufacture moving and sensing
mechanisms with feature size in the micrometer regime presents challenges in
1
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2
fabrication techniques, software tools, reliability and repeatability. On the other hand,
it opens the door to new ideas, and the opportunity to test theoretical predictions at a
scale not found in our everyday experience. The number o f applications of MEMS is
growing rapidly and the explosion of novel designs now includes microrobots with
elastic hinges [2], diaphragms [3], micromotors [4], chromatography applications [5],
microvalves [6], piezoelectric structures for optical applications [7], thermally driven
devices [8], microgimbals for disk drives [9], loading devices [10], object imaging
schemes [11] and tunable transmission lines [12], along with actuators, sensors, gears,
transformers, inductors, pumps, switches and accelerometers [13]-[19]. The number
o f fields in science and engineering that benefit from this technology continues to
grow. Commercially available sensors to monitor acceleration, pressure, temperature,
flow, angle, light intensity and magnetic fields, testify to the versatility and enormous
potential of these devices.
Other areas of active research are those of chemical and biological
applications [20]-[24]. In these fields disposable structures are required, especially in
those instances where fluids are directly in contact with the sensing device [20]. This
characteristic, however, presents an additional challenge not only from a cost
competitiveness point of view, but more importantly, because of the problem of
determining and selecting the packaging for these structures. Pressure sensors and
accelerometers are sealed in a package and isolated from the environment. While this
protection increases stability and reliability, it demands new packaging paradigms for
biosensors.
Because of the large number of applications in many different fields the
realm o f MEMS technology has already left its imprint in large scale manufacturing
due to its large array of possibilities. For example in most automobiles manufactured
today, there are close to fifty sensors measuring a variety of parameters from oil
pressure and oxygen content of the exhaust to air intake velocity [25], As the drive
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3
for better, faster and cheaper devices continues, this MEMS will carve an even larger
niche in our everyday life.
1.2 Thin Film Technology versus High Aspect Ratio Structures
Figures 1.1 through 1.3 show some of the conventional thin film microwave
planar geometries: Slotline, Microstrip and Coplanar Waveguide (CPW). The
thickness of the metallic layers is usually of the order of 1 pm while the thickness of
the substrate is on the order of 350 pm. In all three cases the metallic plates are
deposited on the wafer and the horizontal dimension is larger than the vertical
dimension. Thin film technology has proven valuable in many applications, however,
the main disadvantage of using thin films for structural materials is that the internal
stress of the films limits the thickness of those films to a few micrometers. Since the
vertical stiffness o f a structure is proportional to its height, for a thin film structure the
limitation in thickness represents a limitation on the vertical stiffness. This fact places
a stringent constraint on the length that thin film structures can span. On the other
hand, High Aspect Ratio Structures (HARS) (Figure 1.4) are characterized by larger
vertical stiffness compared to thin film technology and, if needed, it can be increased
even more by increasing the aspect ratio of the micromachined devices. Furthermore,
HARS also present large surface areas of the sidewalls and are, therefore, capable of
producing large electrostatic forces. This thesis exploits the aforementioned
advantages of HARS and describes a new micromachined microwave application of
cantilevered structures, namely, a voltage-tunable transmission line coupled to a
dipole antenna. We use comb-like actuators to move the plates of a parallel-plate
waveguide. Since the characteristic impedance of a parallel-plate transmission line is
a function of the plate separation, modifying the separation we are able to vary the
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4
METALLIC PLATES
SUBSTRATE
Figure 1.1: Cross section of a slotline, the thickness of the metallic plates is many
times smaller than the thickness of the substrate.
SIGNAL PLATE
Substrates
GROUND PLATE
Figure 1.2: Microstrip, this geometry is well suited for certain applications such as
filters and periodic structures.
GROUND
SIGNAL
GROUND
SUBSTRATE
Figure 1.3: Coplanar Waveguide (CPW), geometry useful for amplifiers.
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LAUNCH PAD
LAUNCH PAD
Figure 1.4: Cross section of the geometry presented in this work. The height of the
plates is o f the order of 13 pm and the separation is of the order of 5 pm. A
waveguide as presented is compatible with slotlines and microstrips. For a CPW two
lines back-to-back (in parallel) are required. The out-of-plane parallel-plate
transmission lines presented here are, therefore, compatible with all thin-film
geometries but their high aspect ratio permit the electromechanically tunability.
characteristic impedance of the waveguide. This scheme,
active change
of the
therefore,
permits the
electrical characteristics of the waveguide and allows the
introduction o f other applications, such as n-way switching, resonators of variable
length, etc. Variable matching networks could represent an excellent application for
this device. Under this scheme, it is necessary to micromachine transmission lines
only in the vicinity of the intended operating point and use the actuators to find the
optimum performance. In the case of a directly coupled resonator, the resonator
length determines the resonant frequency, but the possibility to optimize the network
presents the advantage of minimizing the reflected signal.
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6
Figure 1.5 presents a schematic view of a voltage-tunable waveguide. The
launch pads form a coplanar waveguide (CPW) configuration and have very large
horizontal dimensions in comparison to those needed for the vertically-oriented
Electrostatic
Actuators
Ground
Launch Pads
Parallel-Plate Transmission Lines
Signal
Ground
Electrostatic
Actuators
Figure 1.5: Schematic view of tunable transmission lines.
parallel-plate transmission lines. In this work the dimensions of the ground-pads were
2
2
150 x 200 pm , and those of the middle or signal conductor were 100 x 200 pm . The
width of the original CAD design of the parallel-plates was 1 pm with the separation
normally fixed at ~5 pm. The work presented in this thesis, in order to be compatible
with coplanar waveguide geometry, requires the arrangement of two parallel-plate
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7
transmission lines in parallel. Therefore the total width needed for a parallel-plate
transmission line is < 20 pm, much less than the 400 pm required for the CPW.
Each transmission line consists of two plates, one is the ground plate and the
other, the signal plate. The actuators move the ground plate. It is not necessary for the
motors to be electrically attached to the plates. If needed an additional mask can be
used to remove the metallic film connecting plate and actuator. Another option is to
physically disconnect plate and actuator and allow them to be mechanically in contact
when the device is being operated. The electrical connection of the motors to the
transmission lines causes additional losses at those frequencies where the extra
electrical length permits radiation. Therefore, it is always advisable to have the
actuators electrically disconnected from the transmission lines.
1.3 Other Schemes
The tunability concept has been applied in different schemes, such as,
depositing superconducting layers of YBa2Cu3C>7-x (YBCO) and dielectric layers of
SrTi03 (STO) on LaA103 substrates [26]. Figure 1.6 shows the layout o f the
electrically tunable coplanar transmission line resonator. STO is a material chosen in
the fabrication of tunable transmission lines mainly because of its large electric field
tunability of the dielectric constant and its chemical and structural compatibility with
YBCO. Device operation is based on the modulation of the phase velocity and the
attenuation constant of microwaves propagating along the transmission lines through
the dc electric field induced changes in the dielectric constant of the STO layer,
superconducting YBCO electrodes serving only to limit conductor losses. However,
the temperatures required are of the order of 80° K, and the Q's obtained are in the
vicinity o f 200. This low temperature regime, is still out of the ordinary for most
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
lr
J
1
J
3
<
t
i
>
'-f
Substrate
STO, Dielectric
YBCO, Superconductor
Figure 1.6: Layout for an electrically tunable coplanar transmission line.
applications. With electromechanical tuning, however, the characteristic impedance
can be varied, and we can tune to a prescribed resonant frequency. Additionally, with
a wise selection o f periodic structures the phase velocity can be selected to match
that of an electron beam, for instance, in a traveling wave tube.
Figure 1.7 presents a top view of a micromachined voltage-tunable
transmission line. The structure includes two transmission lines in parallel and sets of
capacitive microactuators to move the ground plates. In this particular design, use of
both banks of microactuators permits a large change in impedance. Smaller variations
are possible by moving only one bank of microactuators.
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9
The design presented in this work has only one requirement: high resistivity
silicon substrates <100>. The process, though, is compatible with standard industry
practices, does not require high temperature steps, and the geometry is a natural
extension of that of coplanar waveguide structures. Therefore, low-loss, compact
devices with superior performance characteristics are achieved without stringent
processing or material requirements. Although surface micromachining [27]-[30] is
the current standard technology, this work presents bulk micromachining as an
alternative
approach and has
an
additional
dimension of freedom. The
specific needs of the user will determine which option is the most appropriate. The
commercial viability in the long run is not a function of approach, but is more a
function o f the compatibility of the structure with the rest of the device into which it
must fit.
Figure 1.7: Top view of an electromechanically voltage-tunable transmission line.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
1.4 Applications
The number of applications of voltage-tunable transmission lines is very large.
A tunable transmission line can be utilized to match a load, or to characterize new
structures and applications such as tunable low-pass filters, and, by cascading several
in series, a variable length stub becomes possible. Also, an array of these devices, in
combination with a splitter, provides for n-way switching. Tunable resonators can be
used as filters, switches and detectors. Variable phase velocity periodic structures find
applications in traveling wave tubes, free electron lasers [31]-[33] and micro­
cyclotrons. With the implementation of active devices it is also feasible to fabricate
video detectors and amplifiers.
It is therefore evident that tunable micromachined transmission lines offer an
excellent alternative to conventional planar circuits [27]-[30].
1.5 Synopsis
Chapter 2 provides the theoretical background for this work, starting with
standard transmission line equations both from the electric circuit theory and from the
electromagnetic field theory viewpoints. The quasi-static approximation is presented,
followed by the theory of
electromagnetic components that exhibit a resonant
behavior. The problem of on-wafer probing is also addressed in this chapter, along
with the theory of scattering matrices.
Chapter 3 describes the processing of voltage-tunable micromachined
microwave devices. The general approach is to follow the SCREAM technology
developed at Cornell [34]. In this chapter the process is reviewed in detail and all
modifications explained within the scope of the research presented in this work.
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11
Chapter 4 introduces the concept of electromechanical tuning and presents the
analysis o f the data gathered with a network analyzer. The performance of the
micromachined transmission lines is compared with the theoretical predictions for
parallel plate transmission lines presented in chapter 2.
Chapter 5 applies the ideas presented in all previous chapter to develop a new
process to microfabricate dipole antennas where a voltage-tunable waveguide is used
as matching network.
Chapter 6 concludes this dissertation by summarizing the work which has
been accomplished, and outlines future applications and research areas.
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12
1.6 References
[1] K. Gabriel, J. Jarvis and W. Trimmer, editors, "A report on the emerging field o f
microdynamics", Workshop on Microelectromechanical Systems Research, Salt Lake
City, Utah, July, 1978
[2] K. Suzuki, I. Shimoyama and H. Miura, "Insect-model based microrobot with
elastic hinges", J-MEMS, Vol. 3, No. 1, March 1994, pp. 4-9.
[3] P. R. Scheeper, W. Olthuis and P. Bergveld," The design, fabrication and testing
of corrugated silicon nitride diaphragms", J-MEMS, Vol. 3, No. 1, March 1994, pp.
36-42.
[4] K. Deng, M. Mehregany and A. S. Dewa, "A simple fabrication process for
polysilicon side-drive micromotors", J-MEMS, Vol. 3, No. 4, Dec. 1994, pp. 126133.
[5] R. R. Reston and E. S. Kolesar, "Silicon-micromachined gas chromatography
system used to separate and detect ammonia and nitrogen dioxide", parts I and II, JMEMS, Vol.3, No. 4, Dec. 1994, pp. 134-154.
[6] P. Barth, "Silicon microvalves for gas flow control", 8th International Conference
on solid-state sensors and actuators, and eurosensors IX, Stockholm, Sweden, June
25-29,1995, pp. 276-279.
[7] H. Toshiyoshi, H. Fujita and T. Ueda, "A piezoelectrically operated optical
chopper by quartz micromachining", J-MEMS, Vol. 4, No. 1, March 1995, pp. 3-9.
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13
[8] P. L. Bergstrom, J. Ji, Y. Liu, M. Kaviany and K. Wise, "Thermally driven phasechange microactuation", J-MEMS, Vol. 4, No. 1, March 1995, pp. 10-17.
[9] V. Temesvary, S. Wu, W. Hsieh, Y. Tai and D. Miu, "Design, fabrication and
testing of silicon microgimbals for super-compact rigid disk drives", J-MEMS, Vol.
4, No. 1, March 1995, pp. 18-27.
[10] M. T. Saif and N. C. MacDonald, "A milli newton micro loading device",
Proceedings of the SPIE's Smart Structures and Materials Conference, 26 Feb-3
March, 1995, San Diego, CA.
[11] E. Kolesar and C. S. Dyson, "Object imaging with a piezoelectric robotic tactile
sensor", J-MEMS, Vol. 4, No. 2, June 1995, pp. 87-96.
[12] A. A. Ay6n, N. Kolias and N. C. MacDonald, "Tunable, micromachined parallelplate transmission lines", 15th Biennial IEEE/Cornell University Conference on
Advanced Concepts in High Speed Semiconductor Devices and Circuits", Ithaca,
New York, August 7-9,1995.
[13] L. S. Fan, Y. C. Tai and R. S. Muller, "IC-process electrostatic micromotors:
design, technology and testing", IEEE Micro Robots Mechanical Systems Workshop,
Salt Lake City, Utah, February, 1988, pp. 1-6.
[14] W. S. N. Trimmer and K. J. Gabriel, "Design considerations for a practical
electrostatic micromotor", Sensors and Actuators, 11,1987, pp. 189-206.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
[15] W. Trimmer and R. Jebens, "Harmonic electrostatic motors", Sensors and
Actuators, 20,1989, pp. 17-24.
[16] H. Fujita and A. Omodaka, "Electrostatic actuators for micromechanics", IEEE
Micro Robots and Tdeoperators Workshop, Hyannis, MA, November, 1987.
[17] L. S. Fan, Y. C. Tai and R. S. Muller, "Pin joints, gears, springs, cranks and other
novel micromechanical structures", IEEE Solid State Sensors and Actuators, Tokyo,
Japan, June, 1987, pp. 849-852.
[18] R. Jebens, W. Trimmer and J. Walker, "Microactuators for aligning optical
fibers", Sensors and Actuators, 20,1989, pp. 65-73.
[19] J. J. Clark, "CMOS magnetic sensor arrays", IEEE Solid State Sensors and
Actuators Workshop, Hilton Head Island, SC, June, 1988, pp.72-75.
[20] J. Bryzek and W. McCulley, “Micromachines on the march”, IEEE Spectrum,
May 1994, pp. 20-31.
[21] S. A. Boppart, B. C. Wheeler and C. S. Wallace, “A flexible perforated
microelectrode array for extended neural recordings”, IEEE Trans, on Biomedical
Eng., Vol. 39, No. 1, January 1992, pp. 37-42.
[22] J. M. Corey, B. C. Wheeler and G. J. Brewer, “Compliance of hippocampal
neurons to patterned substrate networks”, Journal of Neuroscience Research, Vol. 30,
1991, pp. 300-307.
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15
[23] J. L. Novak and B. C. Wheeler, “Multisite hippocampal slice recording and
stimulation using a 32 elemental microelectrode array”, J. of Neuroscience Methods,
Vol 23,1988, pp. 149-159.
[24] B. C. Wheeler and G. J. Brewer, Multineuron Patterning and Recording, in
Enabling Technologies for Cultured Neural Networks, Academic Press, 1994.
[25] P. J. Hesketh and D. J. Harrison, "Micromachining, the fabrication of
microstructures and microsensors", Interface, Vol. 3, No. 4, Winter, 1994, pp. 21-26.
[26] A. T. Findikoglu, Q. X. Jia, H. Campbell, X. D. Wu, D. Reagor, C. B.
Mombourquette and D. McMurry, " Electrically tunable coplanar transmission line
resonators using YBa2Cu307 -x/SrTiC>3 bilayers", Appl. Phys. Lett., Vol. 66, No. 26,
June 1995, pp. 3674-3676.
[27] J. Helszajn, “Microwave Planar Passive Circuits and Filters”, John Wiley and
Sons (1994).
[28] T. Edwards, “Foundations for Microstrip Circuit Design”, Second Edition, John
Wiley and Sons (1992).
[29] E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications,
John Wiley and Sons, 1988.
[30] K. C. Gupta, R. Garg and I. J. Bahl, Microstrip Lines and Slotlines, Artech,
1979.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
[31] A. Gover and P. Sprangle, "A unified approach of magnetic Bremsstrahlung,
electrostatic Bremsstrahlung, Comptron-Raman scattering, and Cerenkov-SmithPurcell free-electron lasers", IEEE J. Quantum Electron., Vol. QE-17, No. 7, July,
1981, pp.l 196-1215.
[32] S. J. Smith and E. M. Purcell, "Visible light from localized surface charges
moving across a grating", Phys. Rev., Second Series, Vol. 92, No. 4, pp. 1069.
[33] T. C. Marshall, Free-Electron Lasers, Macmillan Publishing Co., 1985
[34] Z. Lisa Zhang and N. C. MacDonald, "A RIE process for submicron, silicon
electromechanical structures", J. Micromech. Microeng., 2(1992), pp. 31-38.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 2
TRANSMISSION LINE THEORY
2.1 Transmission Line Equations
A transmission line can be rigorously defined as any structure that
guides a propagating electromagnetic wave between points A and B. It is, therefore, to
a theoretician a set of boundary conditions to Maxwell's equations that allow the
description o f a propagating wave between those points [1]. Since we are transporting
energy from A to B, we desire that the energy lost by the transmission line be kept at a
minimum [2], We also prefer single-mode propagation and small attenuation in the
transmission line. The theory can be developed either from the viewpoint of
17
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18
electromagnetic field theory [3], or from the viewpoint of electric circuit theory [4]. It
is the latter approach we take first in this work. We will also present the theory of
parallel plate transmission lines and make a comparison with quasi-static results. In
order to derive the equations that govern transmission lines, we have to keep in mind
that they differ from ordinary electric networks in the essential feature o f their
dimensions ranging from a substantial fraction of a wavelength to many wavelengths
long. On the other hand, the physical dimensions of electric networks are much
smaller than the operating wavelength [5]. Thus, whereas the ordinary electric circuit
consists o f lumped elements, the transmission line has to be treated as a distributed
parameter network. To analyze a transmission line in terms of ac circuit theory, we
must obtain the equivalent circuit of the line, which, being passive, can only be
composed o f combinations of resistive, capacitive and inductive elements [6]. An
assumption throughout this exercise, is that line is uniform [7], i.e., cross sections of
the line in planes normal to the power flow are the same for all points of the line.
Consider a differential length Ax of a transmission line that is described by
four parameters; the resistance R and the inductance L, which are series elements,
with units ohms per meter and henrys per meter, respectively, and the conductance G
and the capacitance C, which are shunt elements, with units mhos per meter and
farads per meter, respectively. Figure 2.1 shows the equivalent circuit of this line
segment. The quantities V(x,t) and V(x+Ax,t) denote instantaneous voltages at x and
x+Ax, respectively. Similarly, l(x,t) and I(x+Ax,t) denote the instantaneous currents at
x and x+Ax, respectively. Applying Kirchhoff s voltage law to this circuit, we obtain
(x,0 - L h x 4 ~ Kx,t) - RAxI(x,t) - V(x + Ax,0 = 0
o t
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(2.1)
19
Figure 2.1: Equivalent electric circuit o f an infinitesimal length Ax of a transmission
line.
In a similar manner, applying Kirchhoffs current law to the node N in Figure 2.1, we
have
I(x,t) - CAx— V(x + Ax,/) - GAxV(x + Ax,/) - 7(x + Ax,/) = 0
d t
(2.2)
On the limit A x -» 0 , equations (2.1) and (2.2) become,
~ V ( x , t ) = L-^-I(x,t) + RI(x,t)
ox
at
(2.3)
and
- ^ - I ( x A = C^-V(x,f) + GV(xj)
d x
d t
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(2.4)
20
respectively. Equations (2.3) and (2.4) are the general transmission-line equations,
also known as telegrapher's equations. Assuming harmonic time dependence for V
and I, phasors can be used to simplify these equations to ordinary differential
equations. Consider, for example
F (x ,0 = R e[v (x )e^']
(2.5)
I(x,t) = Re[z(x) eJa,\
(2.6)
from which we obtain from equations (2.3) and (2.4),
dv(x)
= (R + jeo L) i(x)
dx
(2.7)
= (G + jo> C) v(x)
(2.8)
We may obtain a single second-order differential equations containing either the
voltage or the current by differentiating either equation with respect to x and
combining with the other equation, namely
d
= (R + jeo L^ G + jeo ^
= y 2 v(x)
= (R + j<oL )(G +jo)Q i(x ) = r 2 i(x).
In equations (2.9) and (2.10), y is the propagation constant, given by
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(2.9)
(2.10)
21
y = tJ(R+jeo L)(G+ ja> C) - a + jfi
(2.11)
where a is the attenuation constant in nepers per meter, and P is the phase constant in
radians per meter. It should be noted at this point, that the field penetration occurring
in nonperfect conductors causes a phase shift that can be represented by an inductance
[8], which is the internal inductance of the transmission line. This inductance is
usually a small fraction of the total inductance. Therefore, for the purpose of this
work, when no reference to external or internal inductance is made, it is to be
assumed that the external inductance is being considered.
The solutions of equations (2.9) and (2.10) are
v(x) = v+(x) + v"(x) = vj e~r x + vj er x
(2.12)
i(x) = i+(x) + i'(x) = %e~r x + i^er x.
(2.13)
In these equations, the plus and minus signs denote waves traveling in the +x and -x
directions, respectively. The constants v j, v^, i£ and
are arbitrary amplitude
constants for those waves. Consider now a matched line, i.e., without standing waves.
This is the equivalent of an infinite line, and, therefore, we only require the solution
traveling in one direction. Under these circumstances, applying equations (2.7) and
(2.8) we obtain
vo _
v~0
*o
io
R + jw L ^
r
y
(2.14)
G + jwC
This is called the characteristic impedance of the line. As was the case with the
propagation constant, Z0 depends only on R, L, G, C and co and not on the length of
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the line. It is important to observe the possibility of obtaining the distributed
parameters o f a transmission line with the knowledge of the propagation constant and
the characteristic impedance [9],
R = Re[y Z0]
(2.15)
(2.16)
L = j - l m l r Z„]
CO
G = Re[-£-]
A)
(2.17)
(2.18)
We will review this topic once more when scattering parameters be introduced.
2.2 The Field-Cells Approach
We can arrive at equation (2.9) directly from Maxwell's equations, using the
concept o f field cells as presented by J. D. Kraus [10]. From this viewpoint, space is
regarded as an array o f field-cell transmission lines in which the upper and lower
surfaces are considered conducting strips of width w and infinite length in the
direction of propagation of the wave, i.e., the x-direction. With this approach
L = p. = inductance per unit length in H/m,
(2.19)
C = e = capacitance per unit length in F/m,
(2.20)
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23
G = <7 = conductance per unit length in mhos/m.
(2 .21)
We can review briefly how to arrive at these results.
In the case o f the equipotential lines in a parallel plate capacitor where A is the
area and s is the separation between the plates the capacitance (neglecting flinging) is
given by
C=
eA
(2.22)
where s is the dielectric constant of the medium. If we now consider a square cross
section equation o f side b and let the separation between the plates be also b (Figure
2 .2)
b
1r
b
Figure 2.2: All space being considered is divided in unit cells o f length /.
then (2.22) simplifies, to read
C=s I
(2.23)
with all the space under consideration divided into field cells, for each cell the
capacitance per unit depth is given by
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24
(2.24)
1 =£
and this is the significance of the value of &: it is the capacitance per unit depth of a
field-cell capacitor.
Let us consider now a uniform current density in a conductor shaped like a bar
whose conductivity, ct, is known (Figure 2.3),
Figure 2.3: Conductor carrying a uniform current J.
Once more, we divide the side of the bar into square areas that represent the end
surface of a conductor cell (Figure 2.4),
1
/ / w
Figure 2.4: The conductor is divided in unit cells of length /.
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25
In this case, the resistance o f a cell is
h
1
R = -------7 = —
a wl a l
(2.25)
from equation (2.25) we can easily obtain
(2.26)
j = a.
The conductivity, therefore, is the conductance per unit depth of a conductor cell.
Finally, consider two flat parallel conducting strips of height h and separation
s, each carrying a current I (Figure 2.5),
I
s
Figure 2.5: Conducting strips carrying a current I.
Again, let us divide the line into a number of field-cell transmission lines arranged in
parallel, o f square cross section (Figure 2.6).
The current, /, in each cell is
i=~I
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(2.27)
26
t i
s
Figure 2.6: Field-cell transmission line,
and the inductance is given by
A juH sl
L = — = —~ — = / / /
i
Hs
(2.28)
where A is the magnetic flux. From equation (2.28) we finally obtain
(2.29)
= ju.
This result allow us to conclude that the permeability can be looked upon as the
inductance per unit length of a transmission line cell filled with the medium of
permeability p.
To apply the previous results, consider an electromagnetic wave with the
electric field polarized in the y-direction, the magnetic field in the z-direction, and
propagation in the x-direction. Then Maxwell's equations become
„ TT T d D
d E
V x H = J + —r - = c E + s
dt
at
aH 2
_
dEy
- = c r E v+ e — -
ax
y
at
and
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(2.30)
27
„ „
<?H
dEy
dH2
V x E = -------- = - u ------ => — - = - / / ---- 2-.
f
d t
d x
d t
(2.31)
We can express equations (2.30) and (2.31) in phasor form to obtain
d x
= - (<r + j a s ) E
(2.32)
and
I?
— £- = -j< O fi H ..
o x
(2.33)
Differentiating equation (2.32) respect to x and using equation (2.31) we obtain
d 1 £„y
d x‘
= j <DH(cr + j o > £ ) E .
(2.34)
At this point, we can introduce definitions (2.19) - (2.21), and equation (2.34)
becomes
32^
—— y = j co L(G + j o ) C ) E y.
ox
y
(2.35)
Upon integration between the upper and lower strips we obtain the potential
difference:
*K* } = j a > L ( G + ] a > C ) \ { x ) .
ox
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(2.36)
28
However, we still have to remove the lossless condition for the conducting strips.
Introducing a finite resistance per unit length, we recover equation (2.9)
ax
= (R + ja> L)(G + jeo Qv(*) = Y 2 v(x).
(2.9)
We, therefore, observe that the same transmission line wave equations are obtained
either applying circuit theory or following a field approach.
2.3 Thin Plane Conductors
The previous two sections presented the general transmission line equations,
both from the circuit theory viewpoint (Section 2.1) and from the standpoint o f field
theory (Section 2.2). However, equations 2.11 and 2.14 do not actually relate y to the
dimensions o f the line. We actually need this relationship in order to compare any
measured response to that predicted by Maxwell’s equations. In order to do this, we
have to apply those equations to our specific geometry and consider the boundaries as
well. Our reference point can be the simple theory of semi-infinite plates found in
elementary textbooks of electromagnetic theory. In this case, the characteristic
impedance is directly proportional to the ratio of the plate separation to the plate
height [5],[7]. In our case, the plates are not semi-infinite, and their thickness is not
many times greater than the skin-depth even at the highest frequency covered in this
work o f 40 GHz. In order to obtain the right equations we start from basic principles
and take into consideration the dimensions of the plates as well as the thickness o f the
deposited metallic film.
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29
The measured average metallic film thickness on the plates is of the order of
0.25 pm for a sputtering step of 60 minutes (Figure 2.7), but this number can be
larger, provided the sputtering time is longer or another technique to deposit the
aluminum, or any other metallic film, is applied. Therefore, the theory for conductors
of any thickness is outlined. Consider the geometry presented in Figure 2.8, two strip
conductors o f thickness t, having a separation b and height h, (h » b), have a current
applied o f the form JeJ0**. It is also assumed that only the vector components Jx, Hy,
Ex and Ez exist, and that they do not have a y-dependence. For this configuration,
Maxwell's equations read
(2.37)
V xH = J + —
dt
(2.38)
(2.39)
V»B = 0.
(2.40)
We also need the constitutive relations [11],
D = sE
(2.41)
B=/*H
(2.42)
and Ohm's law,
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30
r:'5,0t:V
S -Q .S p tt
#081 i
Figure 2.7: Profile of the sputtered metallic film.
J = cr E
and assume s, p and a are constants.
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(2.43)
31
Taking the curl of equation (2.37), applying equations (2.38) and (2.42), and
considering that there is no storage of charge density, we obtain
y
X
------ 1
z
s
Figure 2.8: Coordinate system of parallel-plate transmission line.
V 2 E = m — ( o -E + s — )
dt
dt
(2.44)
or using phasor notation
V 2 E = (y*y//cr - c o 2 /ue) E.
(2.45)
Similarly, taking the curl of equation (2.38), and applying equations (2.37), (2.40) and
(2.41), we get
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32
V 2 H = 0 ' a j u c r - a 2f i s ) U
(2.46)
Having obtained the equations we need, we start the analysis in the dielectric region
between the plates, where a = 0. Then the x-component of equation (2.45) becomes
d 2 Ex
d 2 Ex
2
<2-47)
Since Ex can only be harmonic or exponential, we can observe from Figure 2.8 that it
has to be positive at z = b/2 and negative and of equal amplitude at z = -b/2, this can
only happen if it varies sinusoidally, therefore
Ex = Ate ~yx Sin K,z
(2.48)
Substituting equation (2.48) into equation (2.47) we obtain
There is, however, only a y-component of H, which from equation (2.38),
z - ^ - x - j j = j a s (xEx + z E x ).
(2.50)
From the x-component of equation (2.50) we obtain
3H
a
— JL = j a s Ex => Hy = j a s e ' yx —^-CojK,z.
(/ z
K,
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(2.51)
33
Meanwhile, equating the z-component of equation (2.50) yields
dH > = j ■a s E
——
ox
(2.52)
and from equations (2.51) and (2.52) we find
E; = ~ Y e ~r* A Cos K, z.
(2.53)
This equation completes the analysis in the dielectric. In the conductor we assume
that the conduction current is much larger than the displacement current
as «
(2.54)
<j .
Using equation (2.45) we obtain the x-component of the E-field from
f-J O M vH '.
(2.55)
The general solution for this differential equation is
Ex = e-]'x ( A 2 e~K^ + A3 eK22).
(2.56)
Substituting this value into equation (2.55) we obtain
/ 2 + K 22 = j
co fxa
.
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(2.56a)
34
From equation (2.38) we find
(2.57)
Equating the x-components in equation (2.57) we obtain
(2.58)
Finally, equating the z-components in equation (2.57), the result is
dHy
dx
= <jE2
=>
Ez = - y e - y x ( ^ - e - K' z - ^ - e K' 2 )
K_2
1C
(2.59)
We can now evaluate the constants by applying the appropriate boundary conditions.
Since h » b, the H field external to the line is 0 because the fields from currents in
the +x-direction cancel the fields of the currents in the opposite direction, even though
they add at internal points. Therefore, at z = t + b/2, Hy = 0 and we obtain from
equation (2.58)
(2.60)
On the other hand, at z = b/2 we have a dielectric-conductor boundary. Exploiting the
continuity of the tangential component o f E, Ex at z=b/2, equations (2.48) and (2.56),
together with equation (2.60) gives
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35
ASin( K , £ )
A3 =( e 2Kj' + l ) e
(2.61)
K,tb '
2
Similarly, from the continuity o f the tangential component of H, Hy, equations (2.51)
and (2.58) and employing equation (2.60) we get
4
A xj< a z K2 Cos ( Kj —)
IT
a K , ( e z**f - l ) e
(2-62)
equating equations (2.61) and (2.62), we obtain [12],
Tan( K, - ) =
2
c
— Coth K2 1.
K,
(2.63)
In equation (2.63) the hyperbolic term tends to 1 as the conductor thickness tends to
infinity. In that limit, we recover the better known expression for thick parallel-plate
conductors,
=
2
(2.64)
CF
i\ |
Equation (2.63) can be solved for y numerically and represents the general case of
propagation in a transmission line with thin conductors. We will have need of
equation (2.63) along with equations (2.49) and (2.56a) to compare the predicted
attenuation and the measured attenuation. Exploiting the concept of surface
impedance [13], we can obtain the effective surface resistance and the internal
inductance
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36
Ex ( z = b! 2 )
Z^ “ =
HE,
K,
^
C° ,h ^ '
( 2
' 6
5
)
result based on the assumption that h » b . In reality, the field is not totally confined
within the plates and we need another scheme to account for all the field throughout
space. This is accomplished using conformal mapping.
2.4 Quasi-Static Approximation
Even though section 2.3 addressed the problem of relating the characteristic
impedance as well as the propagation constant to the dimensions of the plates and the
conductivity o f the metallic film involved, conformal mapping can provide for a
quick and rather accurate estimate for the characteristic impedance. The main
drawbacks o f this approach are the conspicuous absence o f frequency dependence,
and the difficulty o f applying this approach to many arbitrary given geometries.
However, the parallel plate capacitor is a well-known problem that has been solved by
several authors and it is outlined here for comparison purposes with the more general
theory developed in section 2.3.
The fundamental problem of conformal mapping is to find an analytic
function w = f(z) that maps a given region of the z-plane [14]-[16] into some
particular region of the w-plane. If the function f(z) = u(x,y) + iv(x,y) is analytic in a
region S, then u(x,y) and v(x,y) satisfy the Cauchy-Riemann conditions [17], [18]
£ “
d x
d y
p. 66)
and
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37
d u (xy)
h
o y
—
dv(xy)
o-------•
ox
(2.67)
If we differentiate equation (2.66) with respect to y and equation (2.67) with respect
to x, based on the continuity of the partial derivatives of second order, we obtain the
Laplace equation for v(x,y)
lj a l-0
oy
ox
(2.68)
similarly, differentiating equation (2.66) with respect to x and equation (2.67) with
respect to y, we obtain the Laplace equation for u(x,y),
*
d X1
J.
d y‘
(2.69)
Harmonic functions, i.e., those satisfying Laplace's equation, are normal to each other
[19], as can be seen if we multiply equation (2.66) by dv/ dx, equation (2.67) by
dv/dy and add the results. We obtain
■+
— = Vu»Vv = 0.
d xd x d y d y
(2.70)
Therefore, the curves u = constant and v = constant are mutually perpendicular.
Finally, the transformation maintains the angle of intersection o f the curves, i.e., it is
conformal. Conformality only fails where the function w(z) or its inverse z(w) is not
analytic. However, these methods are limited to this particular partial differential
equation and to problems that can be reduced to two dimensions [20].
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38
Among the methods employed, the Schwarz-Christoffel transformation maps
the real axis in the z-plane into a general polygon in the w-plane with the upper half
of the z-plane mapping into the region interior to the polygon [21]. In the case of a
parallel-plate arrangement, the capacitance per unit length is given by [22], [23]
c = s K^ farad s
K meter
(2 J))
where K and K1are the complete elliptic integrals of the first kind of modulus k and
k1, respectively. The ratio of height to separation, h/s, is related to k through the
equation
where E(<j»,k') is the incomplete elliptic integral of the second kind of modulus k', and
F(4»,k') is the incomplete elliptic integral of the first kind of amplitude § and modulus
k. The value of <j>can be found by applying
V/ _ EV
Sin2 $ =
— .
v (1 —& )K '
(2.73)
The evaluation of elliptic integrals can be avoided by the use o f the approximate
formula
« h r, s
, 2 tv
C = s - [ 1 + — —(1 + In------- )]
S
TV h
s
and the characteristic impedance can be found by applying
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.74)
Figure 2.4 presents a graph of Z0 versus the ratio of height to separation.
As can be seen in Figure 2.9, the fringe effect increases the capacitance and
lowers the characteristic impedance of the line with respect to the ideal semi-infinite
transmission line. Additional details can be found in the literature [24]-[ 30]. H. A.
Wheeler [31]-[32] has developed relations in terms of ordinary functions, i.e.,
exponential and hyperbolic. R. S. Elliott [33] arrived at a similar expression as
equation (2.74). His approximation, however, provides for a lower bound value.
250
200
150
Zo
7 s/h
100
50
0
1
2
3
4
5
6
h/s Ratio
Figure 2.9: Z0 versus h/s, 377s/h is the impedance of a semi-infinite line included for
comparison purposes.
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40
Inspection of Figure 2.9 reveals not only the relevance of the fringe
capacitance but the expected variation of impedance as the ratio of height/separation
changes. Evidently, the impedance departs more from the ideal semi-infinite plates as
the ratio gets smaller, i.e., as the separation increases or the height decreases.
Finally, Sato and Ikeda [34], have included in their analysis the thickness of
the plates. The capacitance is still given by equation (2.71), but k, the modulus o f the
elliptic integrals, and the dimensions of the plates are now related by
h
/(!,< ? ,)
J
/ ( 0, 1)
T
K 8 x, S 2 )
(2.76)
(2.77)
s “ 2 / ( 0, 1 )
(2.78)
n s 2, k - x)
where
(2.78a)
the 8j are points are found when mapping the z-plane into the auxiliary w-plane, and T
is the thickness of the plates. Their analysis show that the characteristic impedance is
lowered by the thickness of the plates. For the purpose of this work, however, the
ratio of thickness to separation is very small, usually less than 0.1 which implies a
correction of the order of 5% or less. Therefore, the thin plate approximation will
suffice.
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41
2.5 Resonance and Q-factor
Resonant devices find applications in filters, oscillators, tuned amplifiers,
phase equalizers and as frequency meters [35]. Even though with our current
technology it is feasible to fabricate resonant circuit with extremely small lumped
elements, this approach has some drawbacks: dielectric, ohmic and radiative losses
can be beyond tolerance and, lumped parameter circuits usually have severe power
handling restrictions. On the other hand, structures with dimensions comparable to
wavelengths provide a suitable alternative most of the time. We are, therefore, mainly
concerned with short and open circuited sections of transmission lines. Consider an
open-circuited transmission line of length 2n-l quarter-wavelengths at the resonant
frequency and characteristic impedance Z0, in which the input port is located at x = -1
and the open output at x = 0. Then using equations (2.12) and (2.13), the expressions
for the mode voltage and current are
(x) = Ae~rx + B e rx
(2.79)
(2.80)
where A and B are the complex amplitudes of the incident and reflected waves and y
is the propagation constant y = a + j p. To satisfy the boundary condition at x = 0 one
must set A = B. Then equations (2.79 ) and (2.80) become
(x) = 2 A Cosh yx
(2.81)
I (x) = —— Sink y x .
4
(2.82)
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42
From equations (2.81 ) and (2.82 ), we obtain the input impedance
Z,
/(-/)
= Za C o t h ( a + j f i I) = Z0 ?-anh( a l ) J Co t ( P l )
°
\ - j Ta n h ( a l ) C o t ( p I)
(2 .83)
K '
Expression (2.83) can be put in a more tractable form, taking into consideration the
use o f low-loss materials, for which a l « 1, this implies
Z * Z a l J C o t ( P 1') .
°1 - j a l C o t ( p i )
(2.84)
K J
Consider now
(2.85)
co- cor + Aco
where tor is the angular resonant frequency and Aco/cor «
1, then for a propagating
TEM mode we have
yg/ = ( 2n - l ) ^ - ^ + Afi> = ( 2» - l ) - ^ ( l + — )
4 Xr vr
2
cor
(2.86)
on the other hand
C of [ ( 2 » i - l ) ^ ( i + ^ ® ) ] = _ 3 r a » [ ( 2 » - l ) f — ].
2
cor
2 oor
(2.87)
At this point we will assume that the argument in this equation is small enough to
obtain
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43
m
ft A(0 -.71 A<0
Tan[(2n —1)--------- ]« (2k- 1 ) -------- .
2 0)r
2 0)r
r / A
(2.88)
The validity o f the last step comes from the fact that the bandwidth is related to the
quality factor Q through the simple relation
Q=— =1 ^
* BW
<or
(2.89)
where Q indicates the sharpness of the resonance. Usually Q is required as high as
possible but open geometries like the ones presented in this thesis do not achieve Q
larger than a few hundred under optimum conditions. We will go back to the Q factor
in a few lines.
We, therefore, require 2n-l « Q/10 to ensure the argument is much less than
1. Equation (2.84) has now reduced to
, .
.. ^ A6)
a l + j (2n - l ) — —
Z,=Z„ ---------------------- - —C
- r—.
i
^
n n A©
1+ j a I (2n - 1) — —
2 <o.r
(2.90)
The term in the denominator is second order in a and Aco/cor, therefore, to first
approximation, equation (2.90) can be expressed as
^
r
i
^ A
. ( 2 h - 1) n Aco..
Z , = Z 0 [ a l + j (2« —1) — — ] = a IZ0 [ 1 + ; -------— ].
2 CO.
Ct I 2 CD.
,.nn
(2.91)
In order to shed some light on equation (2.91), we recall the theory of a series
resonant circuit RLC, in which the impedance is given by
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44
Z = R + j ( a L — — ).
a t
(2.92)
Very close to resonance, i.e., using condition (2.85), the impedance can be expressed
as
Z = R + j to. L + j AcoL-----— + / - ^ r — + ... = /?+j A<o L + j
(or C
to; C
a; C
(2.93)
where we have employed the equality
(2.94)
=
(Dr C
Applying (2.94) to equation (2.93) we obtain
Z = R + jZ 4 ^ = R(l+A - a; C
(or R C
(2-95)
which resembles equation (2.91). We can further illuminate the meaning of equation
(2.95) by defining the quality factor Q as
-
time - average energy stored in the system
..
Q - a ) . --------------- 2------- —-----------------------------------------------------------------energy loss per unit time in the system
where all factors are computed at the resonance frequency. The Q-factor is understood
to describe the sharpness of the resonance [36] or selectivity of a resonant circuit.
Equation (2.102) will show that circuits with low losses have higher Q's. In the
capacitor the peak energy stored is given by
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45
1 Ja
W- m 2 l ? C
(Z 9 7 )
and the average power lost in the resistor is
Using equations (2.97) and (2.98), the unloaded Q-factor can be expressed as
(2-">
Now we can apply this result to express the impedance as
„ _ . 2 A<»
. 2 AG)/a>r .
_ A©.
Z = R + j - r - = R ( l + j ------- - ~ ^ ) = R ( l + j 2 Q
).
co; C
eor R C
(or
(2 .100)
Utilizing this result in equation (2.91), we can represent the open-circuited
transmission line as a series resonant RLC-citcuit, in which
R = a l Z 0 = ^ j ^ a A r Z0
(2.101)
_ _ { 2 n - \ ) n _ 7C
P
4a l
~ aX~Ya
(2.102)
a>r
4 a>r
c = —7 7 = 77
77------ —
a>‘ L ( 2 w - l ) ncor Z0
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(2.104)
46
where \ is the wavelength of the resonant frequency. Equations (2.101) - (2.104)
show that the resonance frequency has a dependence on the characteristic impedance
and, therefore, changing the impedance implies a shift in the resonant frequency.
However, what is actually measured is [37] the loaded-Q of the circuit,
which is the total Q o f the circuit. In equation (2.105), p is called the coupling factor
(this P should not be confused with that appearing in y = a + jP) and is related to the
reflection coefficient p at the resonance frequency by
B - l + P (<cor )
1-p(a>r )
(2.106)
In order to obtain the unloaded-Q , we first determine the loaded-Q of the circuit [38]
via the equation
(2.107)
where the denominator is the 3dB bandwidth o f the resonator; with the measured
reflection coefficient we obtain p, and applying equation (2.105) we determine the
unloaded-Q. At this point it is obvious that the interest in determining Q0, is that it is
directly related to the attenuation coefficient. Therefore, the variables to which we
have access through measurement, resonant frequency, reflection coefficient and 3dB
bandwidth, permit the calculation of a via the unloaded Q. Reference [36] has a
complete discussion o f different techniques for determining Q. In addition, reference
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47
[38] also presents a least-squares method, that can be applied when there is only a
sampling of the resonance region.
2.6 Scattering Matrices
Scattering parameters are defined in terms of incident and reflected waves,
and they require transmission lines terminated in their characteristic impedance as the
boundary condition [1]. This is particularly important at high frequencies at which
measuring current, for instance, can disturb the circuit under consideration. In the
scattering matrix approach, however, the measurable quantities are the amplitudes and
phase angles o f the waves relative to those of the incident wave. It is this
normalization that provides the symmetry of the scattering matrix [35] (Note: for the
geometry studied in this work Sn = S22 and S2i = S12). Consider a to be the incident
vector and b the reflected vector. Then, in this formulation they are related by [39] [41]
b=Sa
(2.108)
For a two-port network, a and b can be expressed as
a=
b=
(2.109)
bx
(2.110)
h
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48
and the scattering matrix is
5=
5„
5,2
(2.111)
5 2, S 22
where the diagonal elements are reflection coefficients and the off-diagonal elements
are transmission coefficients [10]. According to equation (2.108), the relationship
between the incoming and outgoing waves is described by
(2.112)
6, —a, 5„ + a 2 5,2
b2 = a, 5 2, + a 2 5 22
(2.113)
The arrangement can be seen in Figure 2.10. The S-parameter responses measured
from a lossy unmatched transmission line with parameters y and Z in a Z0 impedance
system [42] are
( Z 2- Z 02 ) S i n h ( r I)
2ZZ0
2ZZo
( Z 2 - Z 2) Sinh ( y I)
(2.114)
where
A = 2 Z Z a Cosh( y l) + ( Z 2 +Z02 ) S i n h ( y I)
(2.115)
Notice that in equations (2.114) and (2.115) Z is the characteristic impedance of the
line we are actually measuring, but Z0 is the known characteristic impedance of the
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49
line used to apply the signal to the device under test. We can convert matrix (2.114) to
the equivalent ABCD matrix [43] to obtain
Cosh(y I)
[ABCD]=
Z S i n h ( y I)
(2.116)
Co s h ( y I)
—Sink ( y /)
S 2i
---------------- >
\f
>i
S 11
„
22
< --------------
Figure 2.10: Scattering waves in a two-port network.
Furthermore, the S-parameters and the ABCD matrix are related [43] via
A = ( l + Sn - S n - A S ) / 2 S n
B = ( \ + SH+S22+ A S ) / 2 S 2i
C = (1-<S'I1- S 22+ A S ) / 2 S 2i Z0
D = - Sxl + S22 - A S ) / 2 S2l
where
A S —Su S22 S2i Si2.
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(2.118)
50
Using equations (2.114) - (2.118) we can express the parameters y and Z of the line
under consideration as
e ^ = 1 — 'l—
(2.119)
IS.21
where
I ( Sjj - S 2, +1) - ( 2 S n ) ‘
i
(2 5 21) 2
(2.120)
and
2=
2 0 ± 5 iIz ^ L
(2.121)
° ( l - S , ? ) 2- ^
furthermore, applying equations (2.15) - (2.18) we can also find the equivalent
network for the line in consideration. From a known ABCD matrix we can find the Sparameters using
g _ A Z0 + B - C Z g - D Za
A Z „ + B + C Z„ + DZ„
S,-, —■
S2l —■
A Z 0 + B + CZ„ + DZ0
- A Z0 + B —C Z 2 + D Z0
S22 —
A Z 0 + B + C Z l + DZ0
2 (AD -BC )Z0
a z 0+ b + c z 2
0 + dz0
Additional relations can be found in Appendix I.
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(2.122)
51
2.7 References
[1] L. N. Dworsky, Modem Transmission Line Theory and Applications, John Wiley
& Sons, 1979.
[2] R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill Book Co.,
1966.
[3] R. E. Collin, Field Theory of Guided Waves, Second Edition, IEEE Press, 1991.
[4] S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields,
Addison-Wesley Publishing Co., 1971.
[5] D. K. Cheng, Field and Wave Electromagnetics, Second Edition, Addison-Wesley
Publishing Company, 1990.
[6] R. E. Matick, Transmission Lines for Digital and Communication Networks,
McGraw-Hill Book Co., 1969.
[7] R. L. Liboff and G. C. Dalman, Transmission lines, Waveguides and Smith
Charts, Macmillan Publishing Co., 1986.
[8] D. J. Griffiths, Introduction to Electrodynamics, Second Edition, Prentice Hall,
Inc., 1989.
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52
[9] R. B. Marks and D. F. Williams, “Characteristic impedance determination using
propagation constant measurement”, IEEE Microwave and Guided Wave Letters,
Vol. l,No. 6, June 1991, pp. 141-143.
[10] J. D. Kraus, Electromagnetics, Fourth Edition, McGraw-Hill, Inc., 1992.
[11] J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons,
1975
I
[12] H. Y. Lee and T. Itoh, "Wideband conductor loss calculation of planar quasiTEM transmission lines with thin conductors using a phenomenological loss
equivalence method", IEEE MTT-S Digest, 1989, pp. 368-370
[13] S. A. Schelkunoff, "The impedance concept and its application to problems of
reflection, refraction, shielding and power absorption", Bell Systems Technical
Journal, Vol 17, 1938, pp. 17-48.
[14] M. D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, Inc., 1978
[15] R. A. Silverman, Introductory Complex Analysis, Dover Publications, Inc., 1972
[16] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill
Publishing Co., Part 1, 1953.
[17] E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., 1968.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
[18] G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable,
Hod Books, 1983.
[19] J. Matthews and R. L. Walker, Mathematical Methods of Physics, Second
Edition, Addison-Wesley Publishing Co., 1970.
[20] G. Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press,
Inc., 1985.
[21] M. A. Evgrafov, Analytic Functions, Dover Publications, Inc., 1966.
[22] A. E. H. Love, "Some electrostatic distributions in two dimensions", Proc.
London Math. Soc., (2), 22 (1924), pp. 337-369.
[23] H. Palmer, "The capacitance of a parallel-plate capacitor by the SchwarzChristoffel transformation", AIEE Trans., Vol. 56, March, 1937, pp. 363.
[24] F. Bowman, "Notes on two-dimensional electric fields problems", Proc. London
Math Soc., (2), 41 (1936), pp. 205-215.
[25] E. Weber, Conformal Mapping Applied to Electromagnetic Field Problems,
Construction & Applications o f conformal maps, National Bureau of Standards,
Applied Mathematics Series, No. 18,1952, pp. 59-69.
[26] W. H. Chang, "Analytical IC metal-line capacitance formulas", IEEE Trans.
MTT, Vol. MTT-24, Sept., 1976, pp. 608-611.
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54
[27] W. Hilberg, " From approximations to exact relations for characteristic
impedances", IEEE Trans. MTT, Vol. MTT-17, No. 5, May, 1969, pp. 259-265.
[28] H. R. Kaupp, "Characteristics of microstrip transmission lines", IEEE Trans.
Electr. Comp., Vol. EC-16, No. 2, April, 1967, pp. 185-193.
[29]
G.
Warner and
R.
Anderson,
"Numerical
conformal
mapping
for
undergraduates", Int. J. Elect. Enging. Educ., Vol. 18,1981, pp. 359-373.
[30] R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and
Applications, Elsevier Science Publishers B. V., 1991.
[31] H. A. Wheeler, " Transmission-line properties o f parallel wide strips by a
conformal- mapping approximation", IEEE Trans. MTT, Vol. MTT-12, May, 1964,
pp. 280-289.
[32] H. A. Wheeler, " Transmission-line properties of parallel-strips separated by a
dielectric sheet", IEEE Trans. MTT, Vol. MTT-13, March, 1965, pp. 172-185.
[33] R. S. Elliott, Electromagnetics, McGraw-Hill Book Co., 1966.
[34] R. Sato and T. Ikeda, Line Constants, Microwave Filters and Circuits, Edited by
A. Matsumoto, Chapter V, Academic Press, 1970, pp. 129-131.
[35] R. S. Elliott, An Introduction to Guided Waves and Microwave Circuits,
Prentice-Hall, Inc., 1993.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
[36] M. Sucher, Measurements of Q, in Handbook of Microwave Measurements,
Third Edition, Edited by M. Sucher and J. Fox, Polytechnic Press, Vol. II, Chapter
VIII.
[37] R. C. Compton, Microwave Integrated Circuits, notes for course EE 433, Fall,
1993.
[38] N. Kolias, Design of Millimeter Wave Integrated Active Antenna Arrays, M. S.
Thesis, Cornell University, August, 1993.
[39] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in
Communications Electronics, John Wiley & Sons, Inc., 1965.
[40] H. J. Carlin and A. B. Giordano, Network Theory, Prentice-Hall, Inc., 1964.
[41] J. Helszajn, Microwave Planar Passive Circuits and Filters, John Wiley & Sons,
1994.
[42] W. R. Eisenstadt and Y. Eo, "S-parameter-based IC interconnect transmission
line characterization", IEEE Trans, on Components, Hybrids and Manufacturing
Technology, Vol. 15, No. 4, August, 1992, pp. 483-490.
[43] K. C. Gupta, R. Garg and R. Chadha, Computer-Aided Design of Microwave
Circuits, Artech House, Inc., 1981.
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Chapter 3
FABRICATION TECHNIQUES
3.1 SCREAM Process
We follow the general guidelines of the SCREAM process [1] developed at
Cornell. SCREAM stands for Single Crystal Reactive Etching and Metallization, and
this methodology has proven to be extremely flexible and readily applicable, with
some appropriate modifications, even to GaAs substrates [2], The basic process is
described in detail elsewhere[3] and a large number of applications has proven the
effectiveness of this method including transmission lines [4], and loading devices [5],
56
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57
among many others. A schematic overview of the general process can be seen in
Figure 3.1.
photorwitf
Single crystal SIBcon
st£sirate(SCS)
■ «
rtamnmniiiw
v iw v r f v v
v w a w
(•)
Pattern transfer
(b)
SP6 Release
(0
Metal
Sktewal oxkte deposition
(PECVO)
(d)
MetaKzation by sputtering
Figure 3.1: Overview of the general processing approach presented in this work.
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58
3.2 Process Overview
1.) SILICON OXIDE DEPOSITION: The process begins with the deposition of a
Plasma Enhanced Chemical Vapor Deposition (PECVD) film of Si02 @ 300°C,
chamber pressure o f 450 mT, 15 seem o f SiH4 and 50 seem of N 2O with the power
set at 50 Watts. We use high resistivity (p > 2000 Q-cm) silicon (<100>) substrates
(nominal thickness 380 ± 20 pm) which have been widely used in other microwave
applications. The deposition time is related to the etch time in a subsequent step,
because this film will be used as the mask to etch the silicon substrate. Even though
the selectivity is normally greater than 20:1 (silicon to oxide), the film thickness must
account for other aspects of dry etching such as faceting. Therefore, there are added
advantages to keeping the film thicker than strictly required; 0.5 pm in addition to the
thickness needed is considered enough.
The temperature of deposition is relevant because the quality of the film, as
well as the compressive stress of the film, increase with temperature; 90 MPa, 110
MPa and 120 MPa are typical for films deposited at 240°C, 300°C and 360°C,
respectively. Figures 3.2 through 3.4 show the surface roughness taken with an
Atomic Force Microscope. Each sample was scanned in three different places, and
each scan was obtained sweeping 10 pm of the surface. The definitions of all
roughness parameters, Ra, Rp, Rt, Rpm and Rtm can be found in appendix IV.
The coefficient of thermal expansion of silicon is » 4.24 times [2] that of
silicon oxide, providing for thermal stresses. Furthermore the flow temperature for the
oxide is 960°C and the deposition is carried at a lower temperature, which, in turn,
provides for intrinsic stresses; therefore, after film deposition the surface of the wafer
is convex [6], representing a possible problem especially for thin membranes. If
required, the stress can be decreased with a 30-minute annealing cycle @ 700°C [7].
S i0 2 is often chosen not only because o f its stability and to provide a base for
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59
resist adhesion and imaging [8], but, more importantly, because it can provide for
electrical passivation as well as isolation and protection o f the silicon substrate [9].
2.) PHOTOLITHOGRAPHY STEP: After the oxide deposition, photosensitive
material (KTI 895i, 34 cs) is spun on the wafer to achieve a thickness of 2.4 pm (2500
rpm), baked on using a 90°C hot-plate for 1 minute, exposed to light of wavelength
365 nm with a 10:1 stepper (0.46 seconds, focus = 249), and developed following the
manufacturers recommendation of a 3 minute immersion in KTI 945.
Figure 3.2: Surface roughness for a PECVD film deposited @ 240°C.
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60
Figure 3.3: Surface Roughness for a PECVD film deposited @ 300°C.
Figure 3.4: Surface roughness for a PECVD film deposited @ 360°C.
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61
3.) SILICON OXIDE ETCH: The topography is now transferred to the oxide layer
using a Magnetic Ion Etching (MIE) machine: 30 seem of CHF3 with 1 kW power
and chamber pressure < 3 mT.
4.) PHOTORESIST STRIPPING: Afterwards, the photoresist is removed with an
oxygen plasma in a barrel etcher (fully isotropic etch). Subsequent steps are very
sensitive to photoresist organic residues left on the wafer (Figures 3.5 and 3.6),
because the chlorine etch can bum the residues which serve as micromasking sites
when sputtered during plasma etching.
Figure 3.5 SEM micrograph showing the effect of leaving residues on top of the
mesas which are subsequently physically sputtered and produce micromasking.
The formation of grass can be catastrophic in the event of an air leak in the
chamber or the use of tap water instead of deionized water during rinsing (wafers are
rinsed after development). In the case of tap water, the large number of impurities
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62
serve as micromasking sites. An
auger analysis was performed on a sample
presenting heavy formation of grass due to a leak (Figure 3.7). The presence of
silicon and oxygen was certainly expected but the results also show a significant
presence of aluminum. The cathode of the Reactive Ion Etcher (RIE) is made of
aluminum and it is believed that the presence of air permitted its physical sputtering.
5.) DEEP SILICON ETCH: The oxide layer is then used as a mask to etch the
silicon substrate with a chemistry rich in chlorine [10] with a small amount of BCI3
to prevent trenching [11] (Figure 3.8). This etch is performed with a RIE machine. It
should be noted that BCI3 also lowers the etch selectivity [12] between silicon and
silicon oxide, therefore, it must be kept at a minimum. Some authors recommend the
removal o f native oxide before etching the actual deep silicon trench [1], [3]. The
[I &
ill ILIA; tisib lltiil!
)■ im ii. i i —
X456
■ 1 c» O \r f a
25.0k V
«" i ■■■<'
NNF
#0 0
Figure 3.6: SEM micrograph showing the micromasking effect of contaminants
deposited on the sample during photolithography.
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63
stoichiometry of native oxide has been reported [13] to be SiO with an even thinner
layer o f Si0.7O0.3 at the interface with the silicon substrate. It is also noteworthy to
point out the possible promotion of formation of native oxide with the use of HF
[13].
ru
a
CD
•rH
CD
AL2
LU
SI2
"O
TD
1
201
401
601
601
1001
1201
E l e c t r o n Energy
1401
1601
1801
(eV)
Figure 3.7: Auger analysis of a wafer with abundant presence of “grass” revealed the
presence o f aluminum, oxygen and silicon.
The flows of Cl2 and BCI3 were set at 60 seem and 1.5 seem respectively. Chlorine
chemistries follow reasonably well relations developed by Zarowin [12], [14]-[17].
According to his approach the anisotropy of the trench is proportional to the ratio of
electric field strength to density (Figure 3.9). Figure 3.10
shows some of the
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64
characteristics sought during deep silicon etches: smooth and steep sidewalls and the
absence of grass or trenching.
Ratios of self-bias to chamber pressure of -11.88 V/mT [3] have been
reported. In an effort to provide for high quality parallel-plates, a ratio of -15 V/mT
was chosen in the same piece of equipment, i.e., self-bias = - 450V,
chamber
pressure = 30 mT. With this selection the measured etch rate is 12 pm/hour.
6 .) REMOVAL OF PASSIVATION LAYER: There is always a residue left on the
walls o f the trench after the chlorine etch which disturbs the subsequent release of the
structures. An RCA (Radio Corporation o f America) clean as proposed in the original
SCREAM paper [1] eliminates the problem.
Figure 3.8: SEM micrograph showing the presence of trenching.
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65
Figure 3.9: Chlorine chemistries provide for excellent anisotropy.
Figure 3.10: SEM micrograph showing the characteristics sought in deep silicon
etches.
A 30 minute immersion in nanostrip [18] followed by a 10 minute clean in bubbling
deionized water can be used, but this step usually does not provide the superior
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66
results o f the RCA step. Figures 3.11 and 3.12 show released structures with and
without a passivation layer, respectively.
Figure 3.11: SEM micrograph showing the passivation layer that has to be removed
before releasing the structures.
Figure 3.12: Removal of the passivation layer provides for smooth uniformly released
structures.
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67
7.) WALL PROTECTION W ITH SILICON OXIDE: Prior to releasing the
structures, an additional 15 minute conformal deposition of PECVD oxide is needed
to protect the walls. This is also done at 300°C, followed by a MIE step to remove the
oxide from the floor of the trenches.
8.) RELEASE STEP: The structure can now be released. We use 120 seem SF6 and
150W power in an RIE machine. Figures 3.13 and 3.14 show the results after a
release step on a wall and on a beam, respectively. With these settings the etch rate is
1.1 |im/minute. It should be noted that the effect of the overhang obtained during
release counters the effect of the top silicon oxide. Thus, it is feasible to obtain
straight beams adjusting the thickness of the oxide film and the height o f the
overhang [19].
9.) ALUMINUM SPUTTERING: After release, the devices are conformally coated
with aluminum in a DC magnetron sputtering system with 2 kW power and by
adjusting the Ar gas flow to obtain a chamber pressure of 1.6 mT. With this low
pressure, the compressive stress of the film is kept a t « 50 MPa [20]. Operating at a
higher pressure, i.e., 9 mT, the thickness of the sidewall film can be increased but the
surface roughness increases considerably (Figure 3.15) and the stress is also much
higher. Once more, this stress can be made tensile with an annealing cycle [20], [21].
Figure 3.16 presents the change in stress from compressive to tensile for an aluminum
film.
10.) PROBE PROTECTION: Finally, a thin film (100 A) of Cr as an adhesion
layer and 450 A o f Au are evaporated to protect the microwave probes. Figure 3.17
presents a detail o f the actuators cantilevered.
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68
•'A
t< . • •
\
• C 2 •?
; 5. 0 I. V . ■
ft y
r< = 0 0 1 4
Figure 3.13: SF6 chemistries provide for isotropic etches.
Figure 3.14: SEM micrograph showing a released structure.
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69
Figure 3.15: Sputtering at a pressure of 9 mT the surface roughness and stress
increase. Noticeable aluminum lumps form at the rim of the structures.
200
lO O
O
So
iS o
200
Figure 3.16: Annealing o f an aluminum film to change the stress from compressive to
tensile. Courtesy of J. Drumheller.
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70
%
Figure 3.17: SEM micrograph showing a detail of the cantilevered actuators.
3.3 Alternative Approaches: Fully Oxidized Structures
Other options have been explored that can provide for additional flexibility in
the manufacture o f these devices and other similar structures. We start depositing a
film o f Low Temperature Oxide (LTO) or Low Pressure Chemical Vapor Deposition
(LPCVD) instead of PECVD oxide, even though we have also used layers of
thermal
oxidation with equally good results. The LPCVD oxide deposition is
accomplished with 104 seem of SiH4 and 75 seem of 0 2 @ 400 °C and chamber
pressure of 275 mT. This deposition is very sensitive to the separation between the
wafers. Therefore, we can obtain samples with various oxide thickness in the same
run, flexibility that other processes do not have. The thermal oxidation or steam
oxidation, is done using 0.4 1/min of C2H3CI3 Trichloroethane (TCA), 4.5 lt/min of
H 2 and 2.5 lt/min of O2 at atmospheric pressure having the temperature as a means to
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71
control the oxidation rate. With the temperature set @ 1150 °C beams 0.8 pm wide
are fully oxidized and mesas and walls develop an oxide layer 1.4 pm thick in 4
hours. Even though the selectivity in chlorine chemistry does not vary significantly
when using PECVD, LTO or Thermal oxide, the films differ in their compressive
stress, which varies from 90 MPa for PECVD deposited at 240 °C to 300 MPa [22]
for thermal oxide grown at 1100 °C. The compressive stress of LTO films vary from
90 MPa to 250 Mpa [23] depending of the position of the wafer in the deposition
furnace. The index o f refraction measured at a light wavelength o f 633 nm, reflecting
the composition of the film, varies from 1.46 for thermal (1.47 for LTO) to 1.51 for
PECVD.
Regardless o f which method of oxidation is used, the photolithography, MIE
transfer, photoresist strip and trench etch steps remain the same.
After the chlorine etch, we proceed with an RCA-cleaning step. This process
involves a basic (5 parts deionized water, 1 part H20 2 and 1 part NH4OH at 75°C)
and an acidic bath (6 parts deionized water, 1 part HC1 and 1 part H20 2 also at 75°C,
followed by 10 parts deionized water and 1 part HF at room temperature) with
bubbling deionized water bath steps after each one. Once the wafers are clean, we
proceed to thermally oxidize the structures, as previously described.
During thermal oxidation silicon atoms combine with oxygen atoms to form
amorphous silica, i.e., a random version of crystalline quartz [24] with a similar index
o f refraction of 1.462 and dielectric constant of 3.9.
Since the molar volume of oxide is 120% larger than the molar volume of Si
[9], the structures grow in all directions and allowances have to be made to have this
growth included (Figure 3.18). In fact, for every thickness x of oxide formed, 0.45x of
silicon is consumed. The exact nature and charge of the diffusion species have not yet
been fully identified but kinetic models based on the steps of transport to the surface,
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72
diffusion through the oxide layer and reaction at the silicon surface, have been
proposed and tested, with reasonably good accuracy [25].
It was found that structures 0.8 pm wide, can be fully oxidized after 4 hours
@ 1150°C (Figure 3.19) while structures 1 pm wide have a spike of silicon left
inside the beams (Figure 3.20). Furthermore, the spacing of the grids expected to
provide the anchors to support the released structures structures is critical in the
process. Openings « 2 pm during the thermal oxidation close first at the top (Figure
3.21) rendering a smooth surface of oxide. (This effect could be used for making
vessels for the transport of liquids for instance). Openings > 2 pm can accommodate
the growth of the beams and permit the full oxidation (Figure 3.22).
Figure 3.18: Picture presenting the growth of the structures during oxidation.
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73
Figure 3.19: After a 4-hour oxidation 0.8 pm grid-beams are fully oxidized.
Figure 3.20: After a 4-hour oxidation there is still a silicon spike in beams 1pm wide.
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74
xg5S8
:
I 1• lO v n ■
2 5.0k V , ' S
flyon
a0058
Figure 3.21: For a grid separation too small, the oxidation step closes the gap at the
top before the grid-beams are fully oxidized.
Figure 3.22: The right grid spacing permits the full oxidation of the grid-beams.
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75
After the thermal oxidation step, the beams are fully oxidized, but the floor also has
oxidized as well. An MIE step is necessary to remove the oxide from the floor of the
trenches. However, it was found that the grid spacing left after oxidation is crucial in
this step as can be seen in Figure 3.23 for a spacing considered too wide. A
combination o f the microloading effect [26], [27] and ions bouncing on the walls
explain the necessity to choose the opening carefully. Therefore, to avoid forming
mesas o f oxide and still be able to remove the oxide on the floor, the opening was
finally fixed at 2 pm. With this selection, it is possible to fully oxidize all beams and
still form anchors to hold the structure after it has been cantilevered, as can be seen in
Figures 3.24 and 3.25. Figures 3.26 and 3.27 present additional details o f these
structures. Because of the growth of the beams, it is not possible to fully oxidize them
and still have straight features. With curved beams, however, this can be
accomplished, and this case is portrayed in Figure 3.28.
Figure 3.23: The MIE step damages the structures when the grid spacing is too wide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.24: The grid spacing is crucial to achieve full oxidation of the anchors such
that they become a single unit and are able to survive an MIE step.
x79O
25.0k V
P ^ o n ' 94
«90 16
Figure 3.25: Large structures can be suspended with the anchor technique.
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Figure 3.26: SEM micrograph presenting a detail of the actuators cantilevered and
fully oxidized.
Figure 3.27: Detail o f cantilevered and fully-oxidized parallel-plate lines.
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78
Figure 3.28: Detail of meandering lines fully oxidized and cantilevered.
At this point it is worth mentioning why it is not possible to use PECVD oxide
as a mask in
thermal oxidation processes, namely, large blisters form on
the surface (Figure 3.29) and many sections peel off. This phenomenon is believed to
be due to the presence of hydrogen in the PECVD oxide film since SiH4 and N 2O are
the compounds utilized in the glow discharge plasma.
After the oxidation and MIE steps, the structures are released and the release
time depends only on the size of the anchors. Once more the structures are
conformally metalized with aluminum, layers of Cr and Au are evaporated and the
samples tested.
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79
i'
x595
i
ii
■tOPpw.
2 5 .9 k V
■
'
”
' I
AVON
*0012
Figure 3.29: SEM micrograph presenting the effect of using PECVD oxide during
oxidation.
3.4 Coplanar Waveguides
In order to compare the performance of these novel structures,
standard
coplanar waveguides (CPW's), as well as CPW's with lower ground planes were
prepared using the same high resistivity silicon wafers as substrates.
CPW's are not
only well understood, but they are production-proven commercially valuable devices.
Equations describing them have been developed using conformal mapping methods
[28]-[29] and can be found in Appendix III. Figures 3.30 and 3.31 present both
layouts. The only difference between them is the metal layer required in the latter case
to avoid the electric field from reaching into the substrate. Therefore, the fabrication
descriptions are the same.
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80
We begin depositing an oxide layer on the silicon wafer. The thickness of this
film is o f relevance because the attenuation decreases as the thickness of this
dielectric layer increases; some results on this will be presented in Chapter 4. We use
a conventional image reversal and lift-off process to form the structures on the wafer,
which we now briefly describe.
Photosensitive material is spun on the wafer, baked before exposure, exposed
and instead of developing, it is image-reversed. This is accomplished by exposing the
wafer to an ambient rich in ammonia which diffuses into the exposed resist. Then the
ammonia binds with the indene carboxilic acid generated by exposure to light and
makes the area insoluble in developer solution. The whole wafer is then floodexposed and the photoresist that was not originally exposed can dissolve during
development. The advantage of this process is that the side wall slope can be tailored
to give an undercut profile for lift-off. In other words, positive resist acts like negative
resist [30], With the photoresist walls presenting an undercut, aluminum is
evaporated, as well as a thin adhesion layer of Cr and finally Au.
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81
4
l
1
1
,v v
v ... .
21
A
■■ '• .’ *■
• r-
'
f
Substrate
Dielectric
Metal Layer
Figure 3.30: Layout for a conventional Coplanar Waveguide (CPW).
Substrate
Metal Layer
Dielectric
Figure 3.31: Layout for a grounded coplanar waveguide (CPWG).
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82
3.5 References
[1] Z. Lisa Zhang and N. C. MacDonald, "A RIE process for submicron, silicon
electromechanical structures", J. Micromech. Microeng., 2 (1992), pp. 31-38.
[2] Z. Lisa Zhang, G. A. Porkolab and N. C. MacDonald, "Submicron, movable
Gallium Arsenide mechanical structures and actuators", Micro Electro Mechanical
Systems '92, Travemunde, Germany, February 4-7,1992, pp. 72-77.
[3] K. Shaw, Scream I: a single crystal silicon, single mask, reactive ion etching
process for microelectromechanical systems, M.S. Thesis, Cornell University, 1993.
[4] A. A. Ayon, N. J. Kolias and N. C. MacDonald, "Tunable micromachined
parallel-plate transmission lines", 15th Biennial IEEE/Cornell University Conference
on Advanced Concepts in High Speed Semiconductor Devices and Circuits, Ithaca,
NY, August 7-9,1995.
[5]
M. T. Saif and N. C. MacDonald, "A milli newton micro loading device",
Proceedings o f the SPIE's Smart Structures and Materials Conference, 26 Feb-3
March, 1995, San Diego, CA.
[6] F. J. Blatt, Principles of Physics, Second Edition, Allyn and Bacon Inc., 1986.
[7] D. Haronian, Private Communication.
[8] D. J. Elliott, Integrated Circuit Fabrication Technology, McGraw-Hill Book
Company, 1982.
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83
[9] E. A. Irene, Silicon oxidation: a process step for the manufacture o f integrated
circuits, in Integrated Circuits: Chemical and Physical Processing, edited by P.
Stroeve, American Chemical Society, 1985.
[10] N. C. MacDonald and A. Jazairy, "Single crystal silicon: application to microoptico-electromechanical devices", SPIE, Vol. 2383, pp. 125-135,1995.
[11] J. Maa, H. Gosenberger and L. Hammer, "Effects on sidewall profile of Si etched
in BCI3/CI2 chemistry", JVST, Vol. 8B, 1990, pp. 581-585.
[12] P. VanDerVoom, Y. Chieh and P. Krusius, “Cl2- and BC13- based two-step ultrafine-line gate polysilicon etch process”, 15th Biennial IEEE/Cornell University
Conference on Advanced Concepts in High Speed Semiconductor Devices and
Circuits, Ithaca, NY, August 7-9, 1995.
[13] C. R. M. Grovenor, A. Cerezo and G. D. W. Smith, Atom probe analysis of
native oxides and the thermal oxide/silicon interface, in Layered Structures, Epitaxy,
and Interfaces, edited by J. M. Gibson and L. R. Dawson, Materials Research Society,
1985.
[14] C. B. Zarowin and R. S. Horwath, “Control of plasma etch profiles with plasma
sheath electric field and RF power density”, J. Electrochemical Soc, Vol. 129, No. 11,
Nov. 1982, pp. 2541-2547.
[15] C. B. Zarowin, "Plasma etch anisotropy- theory and some verifying experiments
relating ion transport, ion energy and etch profiles", J. Electrochemical Soc., Vol.
130, No. 5, May, 1983, pp. 1144-1152.
[16] C. B. Zarowin, "Relation between the RF discharge parameters and plasma etch
rates, selectivity, and anisotropy", JVST, Vol. 2A, pp. 1537-1549,1984
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84
[17] A. Kassam, C. Meadowcroft, C. A. T. Salama and P. Ratnam, “Characterization
of BC13-C12 silicon trench etching”, J. Electrochem. Soc., Vol. 137, No. 5, May 1990,
pp. 1613-1617.
[18] K. Shaw, Private Communication.
[19] M. T. A. Saif and N. C. MacDonald, “Deformation of large MEMS due to
thermal and intrinsic stresses”, SPIE, Vol. 2441, pp. 329-340,1995.
[20] J. Drumheller, Private Communication.
[21] M. Saif, Private Communication.
[22] W. Hofmann, Private Communication.
[23] P. Infante, Private Communication.
[24] B. E. Deal, The thermal oxidation of silicon and other semiconductor materials,
in Semiconductor Materials and Process technology Handbook for VLSI and ULSI,
edited by G. E. McGuire, Noyes Publications, 1988.
[25] W. E. Beadle, J. C. C. Tsai and R. D. Plummer, Editors, Quick Reference
Manual for Silicon Integrated Circuit Technology, John Wiley & Sons, 1985.
[26] M. Sato, S. Kato and Y. Arita, "Effect of gas species on the depth reduction in
silicon deep-submicron trench reactive ion etching", Jap. J. Appl. Phys., Vol. 30, pp.
1549-1555,1991.
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85
[27] D. Chin, S. H. Dhong and G. J. Long, “ Structural effects on a submicron trench
process”, J. Electrochemical Soc., Vol. 132, No. 7, 1985, pp. 1705-1707.
[28] G. Ghione and C. Naldi, "Analytical formulas for coplanar lines in hybrid and
monolithic MIC's", Electronic Letters, Vol. 20, No. 4, pp. 179-181, 16 February,
1984.
[29] G. Ghione and C. Naldi, "Parameters of coplanar waveguides with lower ground
plane", Electronic Letters, Vol. 19, No. 18k, pp. 734-735,1 September, 1983.
[30] Notes for YES vacuum oven, Cornell Nanofabrication Facility.
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Chapter 4
MEASURED PERFORMANCE
4.1 Test Setup and De-Embedding Process
Scattering parameters were measured using a Network Analyzer (HP8510C)
with nominal frequency range from 45 MHz to 50 GHz. A set of coaxial cables
connected the analyzer to the air coplanar probes (Figure 4.1). The cables can operate
up to 50 GHz, but the Cascade Microtech probes can only operate up to 40 GHz.
86
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87
HP8510C
Network Analyzer
Cables (50 GHz)
HP8517A
S-Param eter Test Set
—
Iw K
*
C ascade Microtech
Probe Station
Figure 4.1: Overview of the test setup.
Therefore the frequency limitation in the probe operation determined the highest
frequency to be used. The probes had a characteristic impedance of 50 Q and a pitch
of 150 pm in a coplanar configuration (Ground-Signal-Ground). The Device Under
Test (DUT) was placed on a Cascade Microtech Probe Station where the probes can
be moved in three orthogonal directions. A vacuum pump held the samples on the
platform. The mechanical system includes additional hardware to level the coplanar
probes whenever the three points do not touch the sample at the same time. It is very
important to mention that the measurement (Figure 1.1) includes the launch pads as
well as the out-of-plane parallel-plate transmission lines. We will need to de-embed
the performance of the out-of-plane transmission lines. We will go back to this
problem shortly.
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88
Before a precise measurement is made it is necessary to calibrate the entire
setup. This is necessary because even small temperature variations change the
operating point of the equipment and measurements done on different days with the
same calibrations will not be self-consistent. The calibration is done with a Cascade
Microtech Impedance Standard Substrate (ISS), which contains short circuits,
transmission lines and 50 Q loads all of them on a CPW geometry. This Line-ReflectMeasurement (LRM) calibration defines a reference point at the tip of the coplanar
probes. The ISS consists o f highly accurate microwave reference devices made of
gold on an alumina substrate.
The transmission lines presented in this work, are too small to be measured
directly (< 4 pm). For this reason launch pads with dimensions of 120 pm x 200 pm
are included during the test. As has been reported in the literature, they can be seen as
shunt impedances [l]-[3] that can be subtracted if we measure the launch pads
standing alone. Therefore, all devices always included an additional set of launch
pads standing alone and as close as possible to the devices to be tested. In other words
we measure first a Device Under Test (DUT, which includes the launch pads) and
then a set o f launch pads standing alone.
In order to extract the impedance and propagation constant the data obtained
from the network analyzer is converted to PC format, from here the scattering
parameters S are obtained and converted to Y parameters. We then use [2],
Y EXTRACTED DUT = Y OUT ' Y LAUNCH PADS
(4-1)
We convert once more to S parameters and extract Z 0 and y as was outlined in
Chapter 2.
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89
4.2 On-Wafer Probing
The problem of on-wafer probing has been addressed by several authors and
different techniques have been developed based on their analyses [4]-[16j. Extensive
consideration has also been given to modeling and interpreting interconnections [17][18] and transitions [19]-[21], as well as to the problems of de-embedding and
unterminating [22]-[23]. Algorithms for error correction can also be found in the
literature [24]-[25]. The method based on the subtraction of the effect of the launch
pads [l]-[3] was thought to be appropriate because of the geometry of the
arrangement and also because the transition from quasi-coplanar to parallel-plate can
be modeled as a shunt capacitance [26]-[28]. To prove this assertion we can consider
Figure 4.2, where the pads are being modeled as shunt capacitance in parallel with
a resistance [29]. Between the pads there is a slab of high resistivity silicon, which
like any other dielectric, can be modeled by lumped circuit equivalents. For this
exercise, we considered a resistance value of 1500 Q, and a capacitance value of 0.08
pF. Figure 4.3, presents the measured and the modeled values of transmission and
reflection parameters for a 1000 pm transmission line. The comparison was done with
PUFF ™ considering a microstrip line with silicon as substrate.
LOSSY LINE
C T
Figure 4.2: M odeling the shunt impedance o f the launch pads.
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90
In figure 4.3 we can see that the predicted values of reflection and transmission
scattering parameters follow those of the measured sample.
The successful application of the de-embedding procedure applied in this
work hinges in the presence of these shunt impedances. It is worth mentioning that
some authors [4], in comparing different on-wafer measurements, have reported the
highest accuracy for the arrangement of large pads-small devices.
M odeling o f Performance
0
»
-5 H
-10
-15
t>
‘SSV.
£ -2 0
-25-30-35
-40
0
■at
SI 1
S21
SI 1
S21
5
10
M odel
M odel
Sam ple
Sam ple
15 20 25 30 35
Frequency (GHz)
40
Figure 4.3: Comparison o f reflection and transmission scattering parameters for a line
and a model including a shunt impedance.
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91
4.3 Measured Performance
Figure 4.4 presents the measured and the theoretical attenuation of a parallelplate transmission line with thin conductor as derived in Chapter 2. As can be seen,
the measured values follow the expected ones very closely. The excess attenuation is
believed to be due to surface waves trapped in the silicon substrate with nominal
thickness of 380 ±20 pm.
The predicted values are obtained using equation (2.63)
h
*
Tan ( K, - ) =
2
cr
K
— Coth K2 1
K,
(2.63)
According to this expression, the attenuation has a very strong dependence on film
thickness and only a mild dependence on the conductivity of the film. Therefore, it
would be advantageous to be able to grow films much thicker than the average
0.25 pm on each wall presented in this experiment. Aluminum sputtering, however,
has the significant drawback that it is thicker at the top (1.3 pm after sputtering 60
minutes) of the structure and thinner on the walls (0.25 pm average thickness) as can
be seen in Figure 2.2. Furthermore,
lumps of aluminum tend to form at the top of
the structure and preclude the possibility of just extending the sputtering time until
the film thickness on the walls is considered enough. The electroless plating
technique could be successfully applied in this case, even though this particular
experiment is not reported in this work. Electroless plating can provide for film
thicknesses o f 1 pm and beyond. Transmission lines fully cantilevered but not
expected to move, could even accommodate films several microns thick, bringing the
attenuation to a fraction o f the values reported in this work, for example 0.17 dB/mm
at 10 GHz for a 1 pm film, compared to 0.6 dB/mm found in this experiment.
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92
The aforementioned equation (2.63) is also used to obtain the predicted phase
for the microwave signal, and this is presented in Figure 4.5, along with the measured
values. The lines show little dispersion as can be desired for an interconnect or simple
transmission line.
Extracted Attenuation
Theory
Measured
-
0.2
-0.4
-
0.8
0
5
10
15 20 25 30 35
Frequency (G H z)
40
Figure 4.4: Theoretical and experimental attenuation.
4.4 Electrically Thin Substrates
All testing was done with transmission lines on silicon substrates that are
380 ± 20 pm thick. It is feasible to think that surface waves are being trapped by this
high dielectric substrate and contributing to the attenuation. In order to determine
whether or not electrically thin substrates are the best alternative [30]-[31], an
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93
experiment was setup involving wafers that were etched back 80, 190 and 290 pm
respectively. An additional sample was etched back 330 pm but it was found that the
electrical
testing
could not be performed because the
thin membrane
Extracted Phase
Theory
M easured
0 .8-
0 .2 -
0
5
10
15 20 25 30
Frequency (GHz)
35
40
Figure 4.5: Extracted and theoretical phase in units of Rad/mm.
bends when the microwave probes touch it. Another technique must be applied for
these extremely thin membranes. The etch-back procedure is done depositing 3000 A
of a low tensile stress LPCVD (Low Pressure Chemical Vapor Deposition) silicon
nitride film («150 MPa ) [32] to be used as a mask in a subsequent wet etch. This
deposition is performed with 10 seem of NH3 and 47 seem of SiH2Cl2 @ 850°C and
a chamber pressure of 150 mT. The stress of the film can be altered by changing the
flows and the temperature of the deposition; the standard, high stress nitride
deposition («800 MPa) [33], is performed with flows of 90 seem o f NH 3 and 30 seem
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94
of SiH2Cl2 @ 800°C. Besides their different stress, these nitride films also differ in
their index o f refraction which is 2.0 (measured @ 633 nm) in the low stress case and
2.15 (measured @ 633 nm) for the standard film. It should be noted that PECVD
(Plasma Enhanced Chemical Vapor Deposition) nitride films of comparable
thickness, do not have an acceptable masking performance in a KOH solution. The
PECVD films have pinhole defects and etching does occur even on the side of the
wafer that the film is supposed to protect.
After the nitride film has been deposited, with a photolithography step, we
open windows o f 10 mm^ on the back of the wafer and immerse the sample in a
solution of 440 g KOH + 1200 ml deionized water @ 80°C. This solution provides
for an etch rate o f « 2pm /minute (Table 3.1). The final opening on the other side of
the wafer is not 10 mm^ but smaller due to the fact that the KOH etch is not fully
anisotropic and proceeds at an angle of 54.7° respect to the surface of the wafer.
Empirical equations have been worked out [34] to determine the final opening as a
function of wafer thickness:
A f ~ A 0- 2 2 T
(4.2)
where A f is the final aperture, A0 is the original opening and T is the thickness of the
wafer.
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95
Table 4.1: Etch rates for different silicon planes.
Plane
Etch Rate (pm/hour)
100
122
111
0.85
331
182
After this step, the nitride film can be removed using a MIE machine with 30
seem CHF3 and 1 kW power and chamber pressure < 3 mT. The nitride film when
not removed from the wafer can be used to counterbalance the compressive stress of
the oxide film required as a mask to etch silicon. In general, there is an added benefit
in not removing this film.
With substrates made electrically thin, we proceed with the rest of the process
as discussed in Chapter 3. Figures 4.6 through 4.8 present the measured impedance,
attenuation and phase constant for devices fabricated on the aforementioned thin
substrates. According to Figures 4.6 and 4.7, the extracted values of attenuation and
phase constant follow very closely the predicted values. Furthermore, Figure 4.7 also
suggests that the extra attenuation located around 20 GHz is not related to the
thickness of the substrate. All three figures lead to the conclusion that for the purpose
o f cantilever interconnects and tunable microwave devices, the thickness of the
substrate plays only a minor role in the final performance of these structures.
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96
Extracted Attenuation
Etch Back
—
0.2
•••■
80 urn
190 um
290 um
0.4
0.8
0
5
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.6: Extracted attenuation for lines fabricated on substrates with different
thicknesses.
Extracted Phase
Etch Back jim
600-
200
80
-
0
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.7: Extracted phase constant for different substrate thicknesses.
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97
Extracted Impedance
140
Etch Back fim
120
---------
_100
80
190
290
S 80
N 60
40
20
0
0
5
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.8: Extracted impedance for different substrate thicknesses.
4.5 Surface Roughness and other Considerations
The effect of surface roughness on TEM modes has been covered in detail by
several authors [35]-[38] and the results have been analyzed using perturbation
methods that give approximate solutions for surfaces of arbitrary waveshape, as well
as using Bessel-series methods which provide for exact solutions for sinusoidal
surfaces.
In general, the effect of surface roughness can be understood as a surface
displacement that modifies all transmission line parameters. Roughness, however, has
been shown to have a second order effect on surface impedance [35] and equation
(2.65) is therefore accurate enough, except for very demanding applications.
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98
The modification o f the characteristic impedance cannot be so easily
neglected, and it can be proved that taking surface roughness into consideration, to
the first order in surface displacement, the characteristic impedance is given by [35],
(4.3)
where Z00 is the characteristic impedance for a smooth surface, ti 0 is the impedance of
free space; Ad is the total surface displacement, and p is the perimetral length of the
rough conductor. The ratio o f (y\J2p) for our plates range from 3.67 Q/pm for plates
12.7 pm tall to 6.28 Q/pm for plates 7 pm tall and this ratio times the equivalent
surface displacement is the change in the characteristic impedance for a smooth
surface. The surface roughness measured is < 0.15 pm. Therefore, we can expect a
correspondingly small increase in the value of the characteristic impedance and the
maximum correction < 2 Q (Figure 4.9).
X 14 1 e.e •' • ■2 S.flk’v
# $ 6 1 .1
Figure 4.9: SEM micrograph showing the surface roughness of the sputtered
aluminum film. This is a top view of one of the plates in a transmission line.
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99
The main role on impedance modification is played by fringing. In equation
(2.74) it was assumed a large ratio of height to separation, but as the ratio becomes
smaller, theoretical values should depart more and more from the measured values
and this will be seen in the following section.
4.6 Impedance Variation
Conventional transmission lines present a definite impedance, and this value is
selected beforehand and set during processing. In this sense, our devices can also be
fabricated to meet impedance specifications as can be seen in Figures 4.10 and 4.11.
In Figure 4.10 the separation is chosen and the etch time is adjusted to obtain a
working point for the device. Alternatively, the height can be fixed and the separation
adjusted to achieve a pre-determined value for the characteristic impedance as can be
seen in Figure 4.11. Very high impedance values can be obtained by evaporating a
metal film instead o f sputtering once all processing has been done (in this case a thin
layer is deposited only on top of the structures, i.e., the separation to height ratio is
very large). Values exceeding 100Q for a single transmission line have been
measured using this approach. All values obtained are compared to those predicted by
h r ,
s
2
C = e - [ 1 + — - (1 + In
i
S
TV h
tv
/i.,
)]
» _
(2.74)
s
equation (2.74) which takes fringing into consideration, and this is seen in Figures
4.12 through 4.15. It is evident that the measured values drift from those predicted as
the ratio of height to separation decreases. This tendency is expected because of the
growing fringing importance.
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100
Separation = 4.7 pm
Height in |tm
150
120
-
12.7
90
6030
0
5
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.10: Impedance variation with plate height, for a plate separation of 4.7 pm.
Plate H eight = 12.7 pm
Sep. in pm
150-
4.7
120
90
60
30
0
5
10
15 20 25 30 35
Frequency (G H z)
40
Figure 4.11: Impedance variation with plate separation, for a plate height of 12.7 pm.
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101
Plate H e i g h t = 12.7 pm
150
;
Sep = 4.7 pm
1
:
1
1
t
120
j
Theory
1
1
1
1
k
S 90
T
O
11
!
i
j
V
N 60
I
30
;
..... ; ..... r .....
i
0
i
1
0
5
10
15 2 0 25 30
F re q u en cy (G H z )
35
40
Figure 4.12: Measured and theoretical impedance for a plate height of 12.7 pm.
Plate H eigh t = 1 0 pm
Sep = 4.7 pm
180
Measured
Theory
150
G120
60
30
0
0
5
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.13: Measured and theoretical impedance for a plate height of 10 pm.
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102
Plate H eig h t = 7 pm
Sep = 4.7 nm
150-
120
Theory
Measured
-
906030-
15 20 25 30 35
Frequency (GHz)
5
40
Figure 4.14: Measured and theoretical impedance for a plate height of 7 pm
Plate H eig h t = 12.7 pm
Sep = 8.9 pm
180-
-------------------
1 5 0 -i
Measured
Theory
G120
!» ■
60H
30
0
0
10
15 20 25 30 35
Frequency (GHz)
40
Figure 4.15: Measured and theoretical impedance for a plate separation of 8.9 pm.
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103
4.7 Electromechanical Tuning
The most remarkable aspect of this experiment is the electromechanical
tunability o f the structures. To this effect, banks of finger capacitors are added to the
structure to move the plates, thus, changing the impedance o f the transmission line
arrangement, as can be seen in Figure 4.16. The results reported in Figure 4.16,
correspond to two different transmission line arrays: in one o f them the plates are
opened by application o f the voltage (Z0 increases), in the other the plates are brought
closer by applying the voltage (Z0 decreases). For this reason, the 0 V line and the 70
V line of decreasing impedance match perfectly because they correspond to the same
device, whereas the line of 70 V of increasing impedance has a slightly different
performance. Evidently the lines were not perfectly identical. It is worth mentioning
that the number o f finger capacitors determines the potential required to achieve a
determined variation for the characteristic impedance [39]-[40], and it is feasible to
operate with single digit voltages provided that the number o f finger capacitors is
large enough. This, in turn, implies additional space for the structure. If necessary a
compromise has to be worked out between the area that can be allotted to the device,
and the maximum impedance variation expected from the device.
Finally, it is necessary to underline once more that the tested structure actually
comprises two lines in parallel. Therefore, the total impedance change-per-line is
twice that reported in the previous figures. This large impedance variation is thought
to suffice for most applications. There is room, however, for larger variations. Since
the spacing between finger capacitors also plays a role in the voltage required for a
specific change, in general, it is necessary to maintain this gap as small as practically
possible [39]. With the metal-sputtering approach, the gap has to be maintained larger
than strictly required due to the lumps of aluminum forming on the rims of the beams.
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104
In the case o f a directly coupled resonator, the length determines the resonant
frequency but we can still optimize the performance of the circuit by moving the
plates until the point of perfect matching is encountered as can be seen in Figure 4.17.
It should be noted that this figure also permits the observation that even extremely
small variations can be achieved with the techniques covered in this work.
4.8 Comparison with Conventional Structures
In order to compare the performance of the micromachined transmission lines
presented in this work with conventional structures, we also fabricated Coplanar
Waveguides on high resistivity silicon wafers. We started depositing 2.3 pm of
E lectro m ech a n ica l T uning
100
— Open (70 V )
— R ef. (0 V)
80-
—
C lo se (70 V )
a 60N 4020
-
0
5
10
15 2 0 25 30 35
F requ en cy (G H z)
40
Figure 4.16: Electromechanical Tuning, impedance variation can be accomplished
either by opening the plates (Z0 increases) or by closing the plates (Z0 decreases).
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105
R eso n an t F req u en cy
-
10
-
10 v
15 v
-5 0 -6 0
2 5 2 5 .5 2 6 2 6 .5 2 7 2 7 .5 2 8 2 8 .5 2 9
F re q u e n c y (G H z)
Figure 4.17: Electromechanical Tuning, when the transmission line array is perfectly
matched permits the optimization of the working point.
PECVD Si0 2 @ 240°C. The CPW lines were then made using the lift-off process
described in Chapter 3. The thickness of the thermally evaporated aluminum was
fixed at 0.457 pm. We also evaporated 100 A of Cr and 480 A of Au to protect the
probes from collecting unwanted aluminum residues. The equations describing CPW
and CPWG can be found in appendix III. As can be seen in Figure 4.18, the
transmission parameters S2i are comparable even though the film thickness in our
work is only 0.25 pm. According to equation (2.63) with only twice the film
thickness, the performance our devices would be better almost over the entire range of
frequencies.
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106
T R A N S M I S S I O N IN dB
— CPW
— LINE
-0 .5 «
rT3
-1 .5 -
2
-
-2.5
0
5
10
15 20 25 30 35
Frequency (G H z)
40
Figure 4.18: Comparison o f transmission parameter S21 for a CPW and one of the
lines presented in this work as measured, before de-embedding.
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107
4.9 References
[1] W. R. Eisenstadt and Y. Eo, “S-parameter-based IC interconnect transmission line
characterization”, IEEE Trans, on Components, Hybrids and Manufacturing
Technology, Vol. 15, No. 4, August 1992, pp. 483-490.
[2] P. J. van Wijnen, H. R. Claessen and E. A. Wolsheimer, “A new straightforward
calibration and correction procedure for “on-wafer” high frequency S-parameter
measurements (45 MHz -1 8 GHz)”, IEEE 1987 BCTM, pp. 70-73.
[3] H. Cho and D. E. Burk, “A three-step method for the de-embedding o f highfrequency S-parameter measurements”, IEEE Trans, on Electron Devices, Vol. 38,
No. 6, June 1991, pp. 1371-1375.
[4] A. Fraser, R. Gleason and E. W. Strid, “GHz on-silicon-wafer probing calibration
methods”, IEEE 1988 Bipolar Circuits and Technology Meeting, pp. 154-157.
[5] M. Roos, “A measurement and calibration technique for accurate measurement of
amplifier S parameters”, 1987 IEEE MTT-S Digest, pp. 449-451.
[6] C. A. Hoer and G. F. Engen, “Calibrating a dual six-port or four-port for
measuring two-ports with any connectors”, 1986 IEEE MTT-S Digest, pp. 665-668.
[7] G. F. Engen and C. A. Hoer, “”Thru-Reflect-Line”: an improved technique for
calibrating the dual six-port automatic network analyzer”, IEEE Trans. MTT, Vol.
MTT-27, No. 12, December 1979, pp.987-993.
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108
[8] J. Pla, W. Struble and F. Colomb, “On-wafer calibration techniques for
measurement o f microwave circuits and devices on thin substrates”, 1995 IEEE MTTS Digest, pp. 1045-1048.
[9] J. S. Kasten, M. B. Steer and R. Pomerleau, “Enhanced Through-Reflect-Line
characterization of two-port measuring systems using free-space capacitance
calculation”, IEEE Trans. MTT, Vol. 38, No. 2, February 1990, pp. 215-217.
[10] G. Dawe and L. Raffaelli, “Characterization of active and passive millimeterwave monolithic elements by on-wafer probing”, 1989 IEEE MTT-S Digest, pp. 413415.
[11] E. W. Strid, “26 GHz wafer probing for MMIC development and manufacture”,
Microwave Journal, August 1986, pp. 71-82.
[12] K. E. Jones, E. W. Strid and K. Reed Gleason, “mm-wave wafer probes span 0 to
50 GHz”, Microwave Journal, April 1987, pp. 177-183.
[13] M. Fossion, I. Huynen, D. Vanhoenacker and A. Vander Vorst, “A new simple
calibration method for measuring planar lines parameters up to 40 GHz”, 22nd
European Microwave Conference, Espoo, Finland, 24-27 August, 1992, pp. 180-185.
[14] D. Williams and R. B. Marks, “Accurate Transmission Line characterization”,
IEEE Microwave and Guided Wave Letters, Vol. 3, No. 8, August 1993, pp. 247-249.
[15] Y. Shih, “Broadband characterization of conductor-backed coplanar waveguide
using accurate on-wafer measurement techniques”, Microwave Journal, April 1991,
pp. 95-105.
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109
[16] D. Williams and R. B. Marks, “On-wafer impedance measurement on lossy
substrates”, IEEE Microwave and Guided wave Letters, Vol. 4, No. 6, June 1994, pp.
175-176.
[17] J. R. Brews, “Transmission line models for lossy waveguide interconnections in
VLSI”, IEEE Trans, on Electron Devices, Vol. ED-33, No. 9, September 1986, pp.
1356-1365.
[18] S. B. Goldberg, M. B. Steer, P. D. Franzon and J. S. Kasten, “Experimental
electrical characterization of interconnects and discontinuities in high-speed digital
systems”, IEEE Trans, on Components, Hybrids and Manufacturing Technology, Vol.
14, No. 4, December 1991, pp. 761-765.
[19] D. F. Williams and T. H. Miers, “A coplanar probe to microstrip transition”,
IEEE Trans, on MTT, Vol. 36, No. 7, July 1988, pp. 1219-1223.
[20] S. R. Pennock, C. M. D. Rycroft, P. R. Shepherd and T. Rozzi, “Transition
Characterization for de-embedding purposes”, 17th European Microwave Conference,
Rome, Italy, 7-11 September, 1987, pp. 355-360.
[21] K. C. Gupta, R. Garg and R. Chadha, Computer-Aided design of Microwave
Circuits, Artech, 1981.
[22] D. F. Williams and T. H. Miers, “De-embedding coplanar probes with planar
distributed standards”, IEEE Trans. MTT, Vol. 36, No. 12, December 1988, pp. 18761880.
[23] R. F. Bauer and P. Penfield, “De-embedding and unterminating”, IEEE Trans.
MTT, Vol. MTT-22, No. 3, March 1974, pp. 282-288.
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110
[24] A. A. M. Saleh, “Explicit formulas for error correction in microwave measuring
sets with switching-dependent port mismatches”, IEEE Trans. Instrumentation and
Measurement, Vol. IM-28, No. 1, March 1979, pp. 67-71.
[25] O. J. Davies, R. B. Doshi and B. Nagenthiram, “Correction of microwavenetwork analyzer measurements of 2-port devices”, Electronic Letters, Vol. 9, No. 23,
November 1973, pp. 543-544.
[26] J. R. Whinnery and H. W. Jamieson, “Equivalent circuits for discontinuities in
transmission lines”, Proc. IRE, February 1944, pp. 98-114.
[27] R. N. Simons and G. E. Ponchak, “Modeling of some coplanar waveguide
discontinuities”, 1988 IEEE MTT-S Digest, pp. 297-300.
[28] R. Sato and T. Ikeda, Line Constants, in Microwave Filters and Circuits, Edited
A. Matsumoto, Chapter V, Academic Press, 1970, pp. 129-131.
[29] A. R. V. Hippel, Dielectrics and Waves, John Wiley and Sons, 1954, pp. 86-91.
[30] N. J. Kolias, Design of millimeter wave integrated active antenna arrays, M.S.
Thesis, Cornell University, 1993.
[31] J. F. Luy and P. Russer, Editors, Silicon-Based Millimeter-Wave Devices,
Springer-Verlag, 1994.
[32] Phil Infante, Private Communication.
[33] Phil Infante, Private Communication.
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Ill
[34] A. Amar, R. L. Lozes, Y. Sasaki, J. C. Davis and R. E. Packard, "Fabrication of
submicron apertures in thin membranes of silicon nitride", JVST, Vol. B 11(2),
Mar/Apr 1993, pp. 259-262.
[35] A. E. Sanderson, “Effect o f surface roughness on propagation of the TEM mode”
in Advances in Microwaves, edited by L. Young, Academic Press, 1971, pp. 1-57.
[36] S. P. Morgan, Jr., “Effect of surface roughness on eddy current losses at
microwave frequencies”, Journal of Applied Physics, Vol. 20, April 1949, pp. 352362.
[37] A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall
impedance”, Proc. Inst. Elec. Eng., Part B, Vol. 102,1955, pp. 698-708.
[38] H. M. Barlow, “High-frequency impedance of a practical metal surface and the
effect of a thin coating of dielectric over it”, Electronic Letters, Vol. 6, No. 13, June
1970, pp. 413-415.
[39] W. Hofmann, L. Cheng and N. C. MacDonald, “Design and fabrication of
micromachined electron guns (MEGS) using multiple-level planar tungsten process”,
SPIE 1995 symposium on Micromachining and Microfabrication, 23-24 October,
1995.
[40] A. A. Ay6n, N. Kolias andN. C. MacDonald, "Tunable, micromachined parallelplate transmission lines", 15th Biennial IEEE/Cornell University Conference on
Advanced Concepts in High Speed Semiconductor Devices and Circuits", Ithaca,
New York, August 7-9, 1995.
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Chapter 5
Half-Wavelength Micromachined
Dipole Antennas
5.1 Introduction
We now apply the microfabrication tools described in previous chapters to the
design and microfabrication of half- wavelength dipole antennas.
An antenna is a structure designed to radiate and receive energy effectively
[1]. It is a transducer between a free-space wave and a guided wave and vice-versa
[2]. In order to optimize the performance of an antenna we have to understand its
electrical characteristics because antennas always are part of a complete electronic
112
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113
circuit. This fact places constraints on the antenna design because of limitations on
the size and shape o f the surface area available for the radiating aperture, the size and
shape of the volume available for the antenna [3], the presence of obstacles and other
conducting elements, interference from other sources of electromagnetic energy in the
microcircuit, etc. These constraints are reflected in the physical design of the antenna,
but the physical geometry determines the performance of the antenna. Therefore,
some of the desired performance specifications have to be compromised.
5.2 Basic Antenna Parameters
All antenna types involve the basic principle that radiation is produced by
accelerated charges [4], The outgoing electromagnetic wave has a radiation pattern
that depends on the geometry of the antenna. This radiation pattern has a dependence
in both 6 and <|>and the purpose of the design is to obtain a pattern as close as possible
as that prescribed by the specific application. In general it is not feasible to tailor
radiation patterns precisely, but excellent approximations can be made.
Directivity is also associated with the radiation pattern and is defined as the
ratio o f the maximum power density to its average value over a sphere [5]. Directivity
is also called maximum directive gain by some authors. For an antenna radiating the
same power in all directions (isotropic) the directivity has a value of 1. Gain is also
widely used to account for ohmic losses (which heat the antenna) in the efficiency of
an antenna. For a lossless antenna the gain equals the directivity, in general [6] gain
and directivity are related according to
G = C0D
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(5.1)
114
where 0<Co<l is the efficiency factor o f the antenna. The radiation patterns usually
have several lobes, and special techniques have to be applied to maintain the main
lobe much higher than all the others.
Before proceeding any further it is necessary to point out the basic
assumptions made in this work. First, the dipole antenna in this experiment of
rectangular cross section with cross section (cross section = width x height = 2.5 pm
x 12 pm and X is
10 000 pm) dimensions « X can be approximated to a cylindrical
dipole antenna o f radius equal to one-fourth of the height [7], [8]. Second, the current
distribution along this symmetrical dipole antenna of small radius (length/radius >
1000) can be assumed to be sinusoidal [4], [9], [10]. We now proceed to derive the
necessary expressions for a dipole antenna.
5.3 Radiation Characteristics o f Linear Dipoles
Consider an infinitesimal linear wire (total length « X) oriented along the zaxis, positioned symmetrically at the origin (Figure 5.1) and carrying a constant
current represented by
7(z ')= /0 z'
(5.2)
Then the vector potential at the point of observation (x, y, z) is given by [11]
(5.3)
where
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115
(5.4)
r = J x 2+ y 2+ z 2
\i0 is the permittivity o f free space and k is the wavenumber.
z
X
Figure 5.1: Coordinate system for an infinitesimal linear dipole of total length /.
We can express equation (5.3) in spherical coordinates using [11]
=
Sin 9 Cos<f> Sin 9 Sin p Cos 9
Cos 9 Cos<f> Cos9 Sinp - S i n 9
1---1
w
Ao
-Sin ip
Cos<j>
0
4
Ay
A .
to obtain
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116
u I le~Jkr
Ar = ^ - 2
Cos e
4 it r
(5.6)
u I le~Jkr
Ag= - Mo Sin 6
4n r
(5.7)
A^O.
(5.8)
We can now apply [12]
B = //0 H = V x A
(5.9)
to obtain
Hr = H g= 0
h
k l a l Sin G
1
..
*r = J — An
a r
( I +j kt trt ) 6
(5.10)
(5-n )
Finally, applying Ampere's law in phasor form [12], and when J - 0,
E = — ^— V x H
ja>e0
we obtain
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(5.12)
117
(5, 4)
] _ )e- , r
E v= 0
(5.15)
\Mo
k
where rja = I— is the impedance of free space, and co~
4 m0^0
The complex Poynting vector associated with these fields is given by
S = i(E x H -)= i(£ „ « ;r-£ rH ;e)
(5.16)
integrating equation (5.16) over a sphere of radius r we obtain the total complex
power, given by
p = # S«ds=770^
h i
(1 -7
__ 1_
( kr)
(5.17)
The imaginary part in equation (5.17) determines the reactive power of the antenna
while the real part is the time-average power radiated [13]. We can conclude,
therefore, that the loss o f power by radiation is due to the resistive part and this
explains the name radiation resistance,
R rad
[14], Therefore,
P = Eradiated + ha clive
and we can equate [15]
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(5.18)
118
(5.19)
P.radiated
and using equations (5.17) and (5.19) we obtain
P-md = 8 0 ;r 2 (-jj- ) 2 Q
(5.20)
According to equation (5.20) the radiation resistance of an infinitesimal dipole is too
low. For instance, for I = A/60 the predicted radiation resistance is only 0.22 Q.
This is the reason the use of electrically small antennas is limited to applications
where there are space limitations [16]. It is also worth mentioning that C. Balanis [13]
calculated the radiation resistance for small dipoles of lengths A/50 < I < A/10
assuming a triangular current distribution and found that the radiation resistance is
given by
= 20 * 2 ( j ) 2 n
(5.21)
Even in this regime the predicted radiation resistance is too small, for / = A/10 the
radiation resistance is only 1.97 Q.
Apparently as the length of the antenna increases the radiation resistance
increases quadratically. However in the previous derivation we assumed / «
X. We
can go back and remove this constraint to extend the results to antennas with sizes
comparable to the wavelength. In this case the current is given by
/ = / „ S i n [ k { - * z ’) ] e J,0(,-slc)
2
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(5.22)
119
the retarded values are needed because the antenna can be seen as composed of a
large number o f infinitesimal dipoles and the phase difference of all their
contributions must be taken into account (Figure 5.2). Then in the far field region
where kr »
1 the distance to the observation point is given by
s = tJx 2 + y 2 +
(z-z')2
(5.23)
and the fields are given by
,„
.60 7 tl Sin 0 _iks , ,
d E0= j
e J dz’
sX
1/2
Figure 5.2: Coordinate systems for a linear dipole.
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(5.24)
Evidently it is only necessary to calculate either Eg or H^. Therefore, we now have
j o»»io nJ 0>t
Sink(--z')
H i —y °
{ f-------* J
OX
1J
(5.27)
o Sink(—+z')
f --------2-------e~ja ,s ,c dz'}
+
-
S
1,2
and from Figure 5.2 we can use
(5.28)
s=r-z'C os0
to obtain
/ Smff p j w
r1^
j
{ J e ^ ^ ' ‘Sink(L-z’)dz’
0
+ |
- in
j
en*>cose)z'/c
(5.29)
Sink(-+z')dz'}
we can apply to equation (5.29) the relations [17]
QU
f e au Sin(b+cu) d u = —t — =-[a5/n(&+CM)-cCoj(h+cn)]
J
a +c
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(5.30)
121
to obtain from the contribution of the integrals
2
kl
kl
Integrals = —
[ Cos (— CosQ ) - C o s — ]
k Sin 0
2
2
(5.31)
and finally
J
e J a( l - r l c )
C0S{
kl
kl
COSQ ) - C o S ----
H d= j —------------- [------- 2------------------ 2 i
*
2nr
L
Sin 0
i
(532)
K
Applying equation (5.25) we obtain
kl
kl
6 0 / e J<0(-, ~rlc) ^ 0,s'(— C o s Q ) - C o s —
E . = j - ^ - r-----1— 2—
------------- * - ]
(5-33)
The first factor gives the instantaneous magnitude of the fields as functions of
distance and antenna impedance while the term in brackets gives the far-field pattern.
For instance, for / = ?J2 the pattern factor, shown in Figure 5.3, is simply [18]
cosy-cos e ]
F<' 0 ) =
v ~ a9
Sin
(5-34)
As was previously done with the infinitesimal dipole we can determine the radiation
resistance by applying equations (5.16) and (5.17) to obtain
?=<£[ S»ds=<$ —(E xH *)»ds
s
s 2
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(5.35)
122
Figure 5.3: The radiation pattern for a half-wavelength linear dipole.
we can use equations (5.32) and (5.33) to obtain
.kl„
^ ,kL,2
n [C oj(— CosQ ) - C o s ( — )]'
/> = 3 0 /2 J ---------2----2----- dQ
J
Sin 9
(5.36)
Using equation (5.19), we obtain
n
* - = 60
/
■k l „
[Cos(— CosQ ) - C o s ( — )]"
—
'—
a re
—
~
d
e
(5 3 7 )
finally with the substitutions x = Cos Qand dx = - Szw QdQ [19] it can be proved that
the radiation resistance is given by [20]-[22]
And = 60{y + L o g k l - C i (kl)
+^ Sin (k l) [S, ( 2 k l ) - 2 S , (kl)]
1
kl
+ ± C o s(k l)[y+ L o g (-j)+ C l (2kl)-2C i (kl)]}
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(5.38)
123
where y = 0.577 is Euler's constant,
(5.39)
and
(5.40)
00
Figure 5.4 presents the computed values for the radiation resistance as a function of
dipole total length. For the specific case of a half-wavelength dipole the radiation
resistance is 73.13 Q. It can be readily seen that the radiation resistance decreases
rapidly as the length of the dipole becomes smaller. Finally to obtain the directivity
of a small dipole we can use equation (5.16) to obtain the maximum power density
and equation (5.17) to obtain the average power radiated over a sphere o f radius r.
From these, we obtain
The directivity of a half-wavelength dipole can be obtained in a similar way. Using
equations (5.32) and (5.33) we determine S, then along with equation (5.36) we
obtain [13]
D half - wavelength
4 7?
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(5.42)
124
Dipole Radiation Resistance
250-
0
1
2
3
4
Normalized Dipole Length (length/A)
5
Figure 5.4: The variation of radiation resistance with dipole length.
We can now conclude that both the higher radiation resistance and improved
directivity of a half-wavelength dipole make it more suitable for our purposes,
relative to a small dipole. There is also a reactance associated with a half-wavelength
dipole which can be extracted using the methodology previously outlined and is
found to be [19], [23] j 42.5 Q. The general expression for the reactance is
X = — [2S, ( k l ) + C o s ( k l ) { 2 S , ( k l ) - S t ( 2 kl)}
4n
- S i n ( k l ) { 2 C , ( k l ) - C , (2 k l ) - C ,
(5.43)
2 k R2
and this equation is presented in Figure 5.5 for different values of the equivalent
radius R. Evidently the reactance for a half-wavelength linear dipole is not very
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125
sensitive to variations in the equivalent radius of the structure. This particular effect
can be seen as providing for flexibility during processing.
This inductive reactance of a half-wavelength linear dipole is eliminated by
shortening the dipole to about 0.475 X [18]. At this length the radiation resistance is
only 67 Q. At this point it is already evident what the advantages of a half-wavelength
dipole are:
1.) the radiation resistance is higher than that of any smaller dipole,
2 .) the directivity is better compared to that of any smaller dipole and
3.) the reactance is insensitive to the radius of the dipole.
Dipole Reactance
300
200
100-
«
o -
100
-
-200
0
2
3
4
Normalized Dipole length (length/A)
1
5
Figure 5.5: The reactance as a function of dipole length for two different dipole radii.
The larger fluctuating curve corresponds to a radius of .03 X while the other curve
corresponds to a radius of .0003 X.
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126
The advantages in radiation resistance and directivity are evident, but the advantage
as far as the reactance is concerned is extremely important because we can
concentrate the fabrication effort in controlling the impedance of the lines without
having to worry about additional electrical implications that could have a deleterious
effect on the performance o f the antenna. Finally, we proceed to describe the
processing in detail.
5.4 Microfabrication
The processing is based in the work presented in previous chapters. There is,
however, one additional consideration: silicon has a high dielectric constant and in
order to improve radiation efficiency we need to remove all silicon underneath the
antenna structure. This implies that the processing being done on both sides o f the
silicon wafer has to be perfectly aligned. In order to accomplish this objective we
require alignment marks that are visible from either side o f the wafer and this is the
starting point for this new scheme.
(a.) Alignment marks. We begin with an RCA cleaning procedure prior to
depositing on a blank wafer a 3000 A layer of LPCVD low-stress silicon nitride
which will serve as a mask for electrochemical etching. This step is followed by a
photolithography step to transfer the topography to the photoresist. This is done using
KTI 895i, 50 cs spun at 4000 rpm (2.4 pm), baked on a hot-plate @ 90°C for 60
seconds, exposed (6 seconds, focus = 251) and developed 3 minutes in OCG 945. The
alignment marks are then transferred to the nitride layer using an RIE machine with
30 seem CHF3, 1 seem 0 2 at 30 mT and 90W power. The photoresist is stripped and
the sample is etched electrochemically using 660 g of KOH in 1400 ml of deionized
water @ 85°C (Figures 5.6 and 5.7). The alignment marks start out as squares of 1000
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127
pm per side. When the etch is done the squares measure 460 pm per side. The
measured etch rate is 2 pm/minute.
H H Silicon
Hi Silicon Nitride
Figure 5.6: Schematic cross section of the silicon wafer with the alignment marks
electrochemically etched. The thickness of the silicon nitride membrane is 3000 A
while the wafer is 360 pm thick.
(b.) Masking Wafer: Standing Pads. In order to obtain the radiation resistance
predicted by equation (5.39) it is necessary to avoid the formation of conducting
planes during aluminum sputtering. This implies that the metal being sputtered has to
be deposited only on the antenna structure and not on tire rest of the wafer. In order to
accomplish this, a second wafer can be used as a mask. This shadowing wafer is
going to be placed on top of the wafer with the antenna, and in order to eliminate the
possibility of the surface of the shadowing wafer damaging the antenna, we
electrochemically etched standing pads. The height of the standing pads determines
the separation between the two wafers (Figure 5.8). The standing pads are squares that
start out measuring 9000 pm per side.
(c.) M asking Wafer: Antenna Wells. The final step in the preparation of the
shadowing wafer is to fabricate the openings that will allow the sputtering of the
antenna structure. This is done electrochemically. Therefore we proceed with a
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128
Figure 5.7: SEM micrograph of an alignment mark. The electrochemical etch
proceeds until there is only a silicon nitride membrane left.
HZ) Silicon
■
Silicon Nitride
Figure 5.8: Schematic cross section of the masking wafer showing the alignment
marks and the standing pads. The height of the pads was fixed at 120 pm
second RCA cleaning procedure and deposit an additional layer of LPCVD lowstress
silicon
nitride
to
be used
as
a
mask during the KOH etch. The
photolithography step follows the description in (a), the only difference being the
utilization o f the alignment marks to place the
antenna wells
in
the proper
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129
locations. We then transfer the topography to the silicon nitride layer, strip the resist
and electrochemically etch the antenna wells (Figure 5.9).
Silicon
Figure 5.9: Schematic cross section of the masking wafer showing the alignment
marks, the standing pads and the antenna well. The height of the pads was fixed at
120 pm.
(d.) Antenna Structure. Having placed alignment marks and having prepared a
shadowing wafer we can now proceed to microfabricate the antenna structure. The
process for this was presented in Chapter 3: deposit a layer of PECVD silicon oxide
followed by a photolithography step, transfer the topography to the oxide, strip the
resist, make the deep trench in silicon with chlorine chemistry, remove the
passivating layer using nanostrip followed by an RCA cleaning procedure, deposit a
thin layer of PECVD oxide and remove the silicon oxide on the floor (Figure 5.10).
At this point we depart from the process presented in Chapter 3. Instead of releasing
the structure, we deposit an additional layer of LPCVD silicon nitride (after the
corresponding RCA cleaning procedure) and then electrochemically remove the
silicon substrate underneath the antenna structure (Figure 5.11). At the end of this
step we have fabricated a structure that is 12 pm high sitting on a silicon nitride
membrane that is 0.3 pm thick (Figure 5.12). Figure 5.13 shows how the
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130
electochemical etch proceeds into the beams of the structure but stops when the etch
fronts meet.
Figure 5.10: Antenna structure microfabricated as described in Chapter 3.
x l9 2 ,
. i O'C1>.»'<*•M
25.8k V a f l v o n ' 9 6
»0886.
Figure 5.11: SEM micrograph showing the antenna sitting on the nitride membrane.
The total thickness of the wafer can be readily seen as a clear band across the picture.
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Figure S.]2: SEM micrograph showing the alignment mark and the antenna after the
silicon substrate has been removed.
Figure 5.13:
SEM micrograph showing the electrochemical etch underneath the
silicon beams.
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Figure 5.14: After removing the silicon nitride membrane, the structure is free­
standing with no substrate underneath.
(d.) Free-Standing Structures (no Substrate). We now proceed to etch the silicon
nitride membrane using an RIE step with 30 seem CHF3, 1 seem Oz at 30 mT and
applying 90 W power (Figure 5.14). An additional RIE isotropic step (SFg) is required
in order to electrically disconnect the launch pads. The antenna structure is now
substrate-free. We place the masking wafer on top of the wafer with the structures and
sputter aluminum with the settings described in Chapter 3. After this step the masking
wafer is fully covered with aluminum but the wafer with the corresponding antenna
structures receives the aluminum only on the structure of interest. We finish the
process depositing 100 A of Cr and 450 A of Au to protect the microwave probes.
Testing is done after removing the masking wafer.
5.5 Test Setup and Measured Performance
The test setup is similar to the one used for transmission lines. However, in
this case we need only one probe (Figure 5.15) to measure the reflection coefficient
which is related to the load impedance [24]-[26] according to
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133
(5.44)
where Z 0 is the characteristic impedance of the transmission line. Prior to performing
the test we proceed with a conventional Through-Reflect-Line (TRL) calibration. The
geometry of the transmission lines selected for this experiment consists of a single
parallel-plate waveguide. This is, therefore, comparable to microstrip or slotline
geometries. Figure 5.16 shows the general layout for the structure in this experiment
and Figure 5.17 an SEM micrograph of the finished device.
HP8510C
Network Analyzer
HP8517A
S-Parameter Test Set
DUT
C ascade Microtech
Probe Station
Figure 5.15: Test setup for antennas, we only need one probe to measure the
reflection coefficient.
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134
Electrostatic
Actuators
Launch
Pads
Dipole
Electrostatic
Actuators
Figure 5.16: Schematic view (not at scale) of the structure used in this experiment: the
launch pads measure 120 x 230 pm, the length of the waveguide used as matching
network (BC) is 1000 pm, the length between the launch pads and the antenna (AE) is
3500 pm and the total dipole length (d) is 5000 pm. The pads at the end of the dipole
antenna are squares of 50 pm per side.
Upon calibration of the equipment we measured the response. Figure 5.18
shows the performance in the frequency range of interest, as well as the values
predicted by equations (5.38) and (5.43) which take into consideration the equivalent
radius of the antenna [27]-[33]. As can be seen, the measured results follow closely
the predicted values.
It should be noted that special care has to be given to the length p of
the waveguide that feeds the antenna. The original antenna design had a p length of
3.5 mm but in the final experiment the p-length was fixed at 2 mm and a larger
number of antenna structures withstood all processing. This is due to the fact that the
vertical stiffness of a suspended structure is inversely proportional to the cube of the
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135
length it spans, therefore, a shorter p produces a larger vertical stiffness. There is,
however, an added advantage in keeping p > 2 mm: standard dipole antenna theory
predicts a lower radiation resistance as the dipole antenna gets closer to the area
occupied by the actuators where the presence of the sputtered metal modifies the
boundary conditions in the solution of Maxwell’s equations. Specific applications
will determine the length p that can be tolerated.
Figure 5.17: SEM micrograph showing the finished device. In the bottom o f the
picture we can see the voltage-tunable waveguide and the arrays of interdigitated
capacitors similar to those presented in Figure 1.7. After the matching network a
section of parallel-plate transmission line feeds the dipole antenna. The ends of the
antenna are being held in place by pads measuring 50 pm x 50 pm. After removing
the silicon nitride membrane a (see page 129) short RIE step with SF6 is required to
permit the antenna end pads to be electrically disconnected from the silicon substrate.
The bending o f section p can be controlled by using higher aspect ratio structures and
by minimizing the thickness of the silicon oxide film.
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136
Reflection Coefficient
0
Measured
Theoiy
-1
-7
-8
15
20
25
30
Frequency (GHz)
35
Figure 5.18: Measured performance of the antenna, the reflection coefficient Sn is
given in dB. With an applied voltage of 15 V, Su at resonance decreases 1.5 dB.
This work can now be readily extended to include antenna arrays for omnidirectional
reception. In order to accomplish this, we can microfabricate 7 (or more) dipole
antennas positioned around a circle. Another alternative is to microfabricate an array
of dipoles in parallel within a fully oxidized frame to avoid the high dielectric
constant of silicon. This work can also find radar applications. In this case the
structure has to be stiff enough to withstand the electrostatic pressure, therefore, the
waveguides would not be voltage-tunable, but we can still use the mechanical
resonance frequency to operate the device.
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137
5.6 References
[1]
D. K. Cheng, "Field and Wave Electromagnetics", Second Edition, AddisonWesley Publishing Company, 1990.
[2]
J. D. Kraus, "Electromagnetics", Fourth Edition, McGraw-Hill, Inc., 1992.
[3]
D. R. Rhodes, "Synthesis of Planar Antenna Sources", Clarendon Press,
Oxford, 1974.
[4]
J. D. Kraus, "Antennas", Second Edition, McGraw-Hill, Inc., 1988.
[5]
S. Ramo, J. R. Whinnery and T. Van Duzer, "Fields and Waves in
Communication Electronics", John Wiley and Sons, Inc., 1965.
[6]
R. W. P. King, H. R. Mimno and A. H. Wing, "Transmission Lines Antennas
and Waveguides", McGraw-Hill Book Company, Inc., 1945.
[7]
Y. T. Lo, "A note on the cylindrical antenna of noncircular cross section", J.
Appl. Phys., Vol. 24, pp. 1338-1339,1953.
[8]
C. W. Harrison and R. King, “The radiation field of a symmetrical centerdriven antenna of finite cross section”, Proc. IRE, Vol. 31, Dec. 1941, pp.
693-697.
[9]
R. King and C. W. Harrison, "The distribution of current along a symmetrical
center-driven antenna", Proc. IRE, Vol. 31, pp. 548-566, October 1943.
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138
[10]
"The Handbook of Antenna Design", A. W. Rudge, K. Miine, A. D. Olver and
P. Knight, Editors, Peter Peregrinus, Ltd., 1983.
[11]
D. J. Griffiths, "Introduction to Electrodynamics", Second Edition, Prentice
Hall, 1989.
[12]
J. D. Jackson, "Classical Electrodynamics", Second Edition, John Wiley &
Sons, 1975.
[13]
C. A. Balanis, "Antenna Theory Analysis and Design", Harper & Row
Publishers, 1982.
[14]
S. A. Schelkunoff and H. T. Friis, "Antennas Theory and Practice", John
Wiley & Sons, 1952.
[15]
J. R. Wait, "Introduction to Antennas and Propagation", Peter Peregrinus,
Ltd., 1986.
[16]
J. T. Bolljahn and R. F. Reese, "Electrically small antennas and the lowfrequency aircraft antenna problem", IRE Trans. Antennas and Propagation,
October 1953.
[17]
G. B. Thomas and R. L. Finney, "Calculus and Analytic Geometry", AddisonWesley Publishing Co., 1979.
[18]
L. V. Blake, "Antennas", Artech House, Inc., 1984.
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139
[19]
R. W. P. King, "The theory o f Linear Antennas", Harvard University Press,
1956.
[20]
R. Bechmann, "Calculation of electric and magnetic field strengths of any
oscillating straight conductors", Proc. IRE, Vol. 19, No. 3, pp. 461-466,
March, 1931.
[21]
A. A. Pistolkors, "The radiation resistance of beam antennas", Proc. IRE, Vol.
17, No. 3, pp. 562-579, March, 1929.
[22]
P. S. Carter, "Circuit relations in radiating systems and applications to antenna
problems", Proc. IRE, Vol. 20, No. 6, pp. 1004-1041, June, 1932.
[23]
H. Jasik, Editor, "Antenna Engineering Handbook", McGraw-Hill Book Co.,
1961.
[24]
W. L. Weeks, Antenna Engineering, McGraw-Hill Book Co., 1968.
[25]
Microwave Antenna Measurement, Edited by J. S. Hollis, T. J. Lyon and L.
Clayton, Scientific Atlanta, 1969.
[26]
R. E. Collin and F. J. Zucker, Antenna theory, McGraw-Hill Book Co., 1969.
[27]
R. King, “Coupled antennas and transmission lines”, Proc. IRE, Vol. 31, Nov.
1943, pp. 626-640.
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140
[28]
R. C. Hansen, “Fundamental limitations in antennas”, Proc. IEEE, Vol. 69,
No. 2, Feb. 1981, pp. 170-182.
[29]
R. King and C. W. harrison, “The impedance of short, long, and capacitively
loaded antennas with a critical discussion of the antenna problem”, J. Applied
Physics, Vol. 15, Feb. 1944, pp. 170-185.
[30]
R. King and F. G. Blake, “The self-impedance of a symmetrical antenna”,
Proc. IRE, Vol. 30, July 1942, pp. 335-349
[31]
R. W. P. King, R. B. Mack and S. S. Sandler, Arrays of cylindrical dipoles,
Cambridge University Press, 1968.
[32]
S. A. Schelkunoff, “Theory of antennas o f arbitrary size and shape”, Proc.
IRE, Vol. 29, Sep. 1941, pp. 493-521
[33]
S. A. Schelkunoff, “Antenna theory and experiment”, J. Applied Physics, Vol.
15, Jan. 1944, pp. 54-60.
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Chapter 6
CONCLUSIONS AND FUTURE
WORK
6.1 Conclusions
We have reviewed in the previous chapters, the performance of high-aspectratio voltage-tunable microwave devices as well as the corresponding processing
approaches. Even though thin film schemes, i.e. coplanar waveguides, microstrips and
slotlines, have already carved out their own niche in current technology, they are only
well suited for certain applications, while large structures fully cantilevered require
MEMS-compatible devices, and this is the arena where our work fits in. The work
141
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142
presented provides an alternative with good performance, ease o f manufacture, and
compatibility with presently used microwave devices. Fully oxidized structures are an
additional option when small attenuation is required. The results presented are a direct
consequence o f high-aspect-ratio silicon structures being aggressively exploited
because HARS produce large forces, they can span lengths on the order of millimeters
and help to decouple vibrational modes. The list of achievements is already very large
as was mentioned in Chapter 1.
6.2 Future Work
The devices explored in this thesis are only the working principle for a family
o f microwave structures that may now be fabricated, such as: filters, detectors,
amplifiers and antenna arrays with omnidirectional reception. They will find direct
application in advanced military and commercial devices due to their performance,
functionality and low. cost. The work presented here fills the gap for devices with
electrical characteristics that can be electromechanically changed. They are also
sturdy and reliable because silicon has an excellent array of mechanical properties
making it suitable for microelectromechanical solutions. The growing number of
applications testifies to the enormous versatility of this material..............and we shall
see more in the near future.
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APPENDIX I
Other Matrix Representations
Even though scattering parameters are used in this work, there are other
equivalent matrix arrangements, among them impedance and admittance. ABCD
parameters have been mentioned. They have the advantage of permitting the analysis
of cascaded elements simply by multiplying their individual ABCD matrices. This
direct matrix multiplication is not feasible with scattering matrices because part of the
variables at a port are independent and the others are dependent. Instead of using
ABCD matrices we can arrange all parameters at port 1 to be dependent on the
variables at port 2. In this case we obtain similar properties to ABCD. We accomplish
this by defining a new set of parameters called transfer scattering or T-parameters,
which are related to S-parameters through
7Ji = ( —Su S22 + Sn S21) / S2]
(2.123)
and
*^11 = ^12^ ^22
= (^11 T22 —Ti2 T2]) / T22
521 = 1 / T22
(2.124)
522 = —T2i / r 22
143
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144
T-parameters are usually chosen instead of ABCD because o f the simplicity o f the
equations involved.
Z and Y matrices on the other hand, provide for a standard visualization o f the
network in terms o f conventional impedance and admittance devices. The necessary
equations for both cases are included here:
_ (*11 1 ) ( *22 + 1 ) ”•*12 *21
s„ =
(*11 + 1 ) (*22 + 1 ) ~ *12 *21
2 z2l
(*11 + 1 ) (z22 + 1 ) “ *12 *21
621
( 1+ s , . ) ( i - s 22) + 5 12 S*
( 1- Sn ) ( l - S 22) ~ s a S21
2 5 2,
( l - 5 „ ) ( l - 5 22 ) - 5 ,2 52,
2z“
(* 11+ 1 ) ( z 22 + 1 )
Z I2 Z2l
J
O
_ ( *11 + ^ ) ( * 2 2 — 1 ) — *12 *21
On
—---------------------------------(*11 + 0 (*22 + 1 )
*12 *21
12 —
25”
( l - S u ) ( l - S 22) - S l2S2l
_ ( 1+ S22 ) (1 —l^n ) + Sy2 S2i
222 (1 - 5 „ ) ( I - S 22 ) - S ]2 S2t
(A.I-4)
and
S _ (1-^11 ) U + >22 )+>12>21
(1 + ^11 ) ( l + y22 ) - y , 2 y21
S = _______ ~2y,2_______
12 (1 + ^u )(1+^22)-^12 >21
S
s
= ________________ - 2
21
yn
v
y*
0
y 21
+ > » ) ( 1 + ^ 2 2 ) ~ > 1 2 >21
( l + S22) ( l - S u) + S l2S2l
(1 + Sl{) (1 + S22) - 5 12 52,
... U
22
1 -^ 2 2
) + > 1 2 >21
+ > ll ) 0 + > 2 2
) “ >12 >21
-2 5n
y '2 (1 + 5„) (1 + 5 22) - 5 ,2 52,
—2 5 2,_________
(1 + 5„) (1 + S22) - 5,2 Sn
0
+ > |1 ) (
—( 1 ~
y22
^ +
+ ^ 12 ^ 21
(1 + 5„) (1 + S22) - 5,2 52,
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(AI 5)
APPENDIX II
Other considerations in Deep Trench Etches
Deep trenches are conventionally done without varying the bias-to-pressure
ratio. This approach (Figure AII.l) permits the micromachining of very steep
sidewalls with smooth transitions at the floor where they join the substrate.
Figure AII.l: SEM micrograph showing the characteristics usually sought in deep
trenches including smooth and steep sidewalls, no trenching and absence of grass.
These transitions usually have a small radius of curvature. When drastic changes
occur during the etch process, the formation of tapered steps becomes feasible, as is
shown in Figures AII.2 and AII.3. The effect presented was achieved by changing the
working point from -450 V/30 mT to -140 V/140 mT. According to the Zarowin
model, when going from the first working point with a bias/pressure ratio of
-15 V/mT to the second one with a ratio of only -1 V/mT, we are increasing the
isotropy of the etch. Whether or not this particular effect can be controlled to achieve
145
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146
a prescribed radius of curvature at the bottom remains an open question and
additional experiments would be needed to settle this matter. With the new working
point, we continued the etch for 10 minutes, evidently the ion bombardment
2 5 .0 k V
HHF-
«080
Figure AII.2: SEM micrograph showing the effect o f drastically changing the etching
operating point on an isolated feature.
HHH j
HH ■
HH HI
^H H H H H
Figure AII.3: SEM micrograph showing the effect on a wall.
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147
adjusted to a “new feature size” before continuing the etch. This effect can be seen on
isolated features as well as at the end of mesas. Finally it is shown in Figure AIL4 that
open areas larger than the “new feature size” or larger than the new radius of
curvature of the smooth transition are needed to observe this effect. In Figure AII.4,
the outermost walls show the effect but the interior walls do not.
Figure AII.4: SEM micrograph showing the effect only where the walls have a
separation larger than the “new feature size”.
As can be seen in Figure AII.4, the distance in the topographic features can be a
limitation in the application of this effect.
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APPENDIX III
EQ U A TIO N S R ELA TED TO CPW A N D CPW G
The impedance of a conventional CPW can be determined using
Zc, „ =
(AIII.l)
where,
, er - l K(k' )K(k, )
eEFF= 1+ - Lz
V '
EFF
2 K(k)K(k{)
(AIII.2)
v
also
(AIII.3)
k=-
and
S ,n « * 1 )
*> =
( ^ I L 4)
T T
S in h (^ )
In equations (AIII.l) through (AIII.4) s is the width of the signal conductor, p is the
opening between the ground and the signal conductors, and er and t are the dielectric
constant and the height of the substrate respectively. K is the complete elliptic integral
148
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149
constant and the height o f the substrate respectively. K is the complete elliptic integral
of the first kind and k and k ’ are related via
k' =V( l - k 2 )
(AIII.5)
Additional equations applicable to Coplanar Striplines can be found in the original
paper by Ghione and Naldi (Electronic Letters, Vol. 20, No. 4,1984, pp.179-181).
For a Coplanar Waveguide with lower ground plane (CPWG) the impedance
is given by
7
60 7t
■‘CPWG -
I
1
v tu \
K(k) { K
W
K ( t ) AT(Jfef)
tr \
t/U U .O J
where
u K(k’) K ( k l ) i
_
W ) W j )
-
u K ( t ) W , )
m l1 )
K ( k ) K( k {)
also
T anh< f >
T a n h (^
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(AHI.8)
150
k is still given by equation (AIII.3). The reader is referred to the original paper by
Ghione and Naldi for a complete discussion o f the derivation of these equations:
Electronic Letters, Vol. 19, No. 18,1983, pp. 734-735.
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APPENDIX IV
Surface R oughness
We can start defining the arithmetic mean H as the sum of all height values
Hi divided by the number of data points n in the profile
(AIV.l)
1=1 ±H ,
n 1=0
Ra
is then the arithmetic mean of the deviations in height from the profile mean
value which can be expressed as
(AIV.2)
Rt is the maximum peak to valley height in the profile:
D
17 _ JJ
^max ^min
(AIV.3)
Rp is the maximum height o f the profile above the mean line:
(AIV.4)
RP = Hmax - H
Rtm and Rpm are the mean values of R{ and Rp respectively:
151
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152
*» = - £ ( * , ) ,
m -r
i =l
(AIV.5)
1 m
(AIV.6)
« /.l
and
are usually considered more representative o f the complete profile.
Additional information can be found in the User’s Manual for TMX 2000 Discoverer
Scanning Probe Microscopy, TopoMetrix, Version 3.05.
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