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TRENCH WAVEGUIDE: A NOVEL THREE-DIMENSIONAL TRANSMISSION LINE FOR MICROWAVE MONOLITHIC INTEGRATED CIRCUITS

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O rd e r N u m b e r 8 8 0 4 5 6 3
Trench waveguide: A novel three-dim ensional transm ission line
for microwave m onolithic integrated circuits
Yarbrough, Allyson Debra, Ph.D.
Cornell University, 1988
C o p y r ig h t © 1 9 8 7 b y Y a rb ro u g h , A lly s o n D e b r a . A ll rig h ts re s e rv e d .
UMI
300 N. Zeeb Rd.
Ann Arbor, M I 48106
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TRENCH W A V E G U ID E :
TR A N S M IS S IO N
A NOVEL TH R E E -D IM E N S IO N A L
LIN E FOR M IC R O W AVE M O N O LITH IC
IN T E G R A T E D
C IR C U IT S
A Thesis
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
Allyson Debra Yarbrough
January 1988
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©
Allyson Debra Yarbrough
1987
ALL RIGHTS RESERVED
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Biographical Sketch
The author was born in Peterborough, England on February 14,
1958.
She attended New Mexico State University in Las Cruces, New
Mexico and was awarded the B.S. Degree in Electrical and Com puter
Engineering in May, 1979.
She has held positions at the National
Astronom y and Ionospheric Center's Arecibo Observatory in Arecibo,
Puerto Rico, at Hewlett-Packard Company's Network Measurements and
Microwave Technology Divisions in Santa Rosa, California, and at IBM
Federal Systems Division in Owego, New York.
She began g rad u a te
studies at Cornell University, Ithaca, New York in August, 1982 and
completed the M.S. Degree in Electrical Engineering in January, 1985.
Her technical interests lie in the areas of microwave integrated circuit
design, fabrication, and characterization.
iii
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D edication
I gratefully dedicate this work to:
My family Fred, Rosalind, Shelly, Brian and Jason
iv
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Acknow ledgem ents
Work was performed in part at the National Nanofabrication Facility
which is supported by the National Science Foundation under Grant ECS8619049.
The project was funded by the Sem iconductor Research
Corporation (SRC) under Project #247-85.
This has been the single most difficult page of the entire thesis to
write. For in this small space I must attempt to express all the gratitude and
admiration I hold for Prof. G. Conrad Dalman, my thesis advisor. Not only
is Prof. Dalman an inspiring teacher and a model microwave engineer, but
he is also an exceptional individual. On innumerable occasions when he
was very busy he always made time for me and showed a genuine interest
in my progress. In instances too numerous to recount here, he gave freely
of his guidance, expertise, encouragement and attention. For these and so
many more courtesies, I say thank you.
As if to make a difficult task impossible, I must sum m arize my
appreciation and esteem for Prof. Charles A. Lee in this same short space.
Let these words constitute a record of his indulgence, am iability and
thorough example.
I also acknowledge the patience and assistance rendered by Prof.
Paul R. Mclsaac.
The participation of Dr. Walt Butler, Dr. Phil Smith, Donald Seielstad
and Harold Keithley of General Electric's Electronics Laboratory proved
invaluable to the success of this work. Their expertise and assistance with
the angle evaporation, plating, and microwave probe m easurements is
greatly appreciated. Bob Byrnes and his simulator CUMQUATS made the
v
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im portant numerical analysis possible.
I wish to thank him for his
indulgence and consideration.
Finally, I express my indebtedness to a host of individuals who have
supported me in so many tangible and intangible ways. Special thanks to
John Scarpulla, Teresa Cheeks, Carl White, John, Mabel and Veronica
W are and Sally Crosby, who have been com rades, confidants and
adopted family. Mark Fiore, Scott Snyder, Paul Chau, Asanga Perera, Jim
Faist, Aiison Schary, Linda Wanamarta, Azeez Al-Omar, and Wayne Lui all
played a role in making my years at Cornell, though strenuous, the most
delightful ones ever.
Thanks to Dan Dinsmore, Bob Soave, and Mark
Skvarla for their expert help in my process development.
I acknowledge
Nellie and Tim W hetten and the rest of the Submicron staff for their
invaluable assistance. I appreciate the help of Bud Addis of the Materials
Science and Engineering Materials Preparation Facility and acknowledge
the special skills of Elaina Jeddry and Teresa Leidenfrost in the thesis
preparation. To all those with whom I have ever shared a crowded office, a
night in the clean room, or a Chariot anchovie pizza, I express warmest
thanks. I enthusiastically look forward to our fulfillment, good fortune, and
continued rapport.
vi
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Table of Contents
Page
Biographical Sketch
iii
Dedication
iv
Acknowledgements
v
List of Tables
viii
List of Figures
ix
1.
Introduction
1
2.
Experimental Procedure and Results
15
2.1
Trench Waveguide Design
16
2.2
Trench Waveguide Fabrication
19
2.3
Trench Waveguide Characterization
62
3.
Discussion
89
3.1
Analysis of Trench Waveguide Transmission Line Parameters
90
3.2
Summary, Conclusions, and Recommendations for
119
Future Work
A ppendix
125
Literature Cited
143
v ii
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List of Tables
Page
2.1
Trench Waveguide Process Schedule
58
2.2
Trench Waveguide Large Scale Model Data
69
Capacitance/Length & Z q
3.1
Trench Waveguide Large Scale Model
118
Characteristic Impedance
A.1
Trench Waveguide Silicon Fabricated Component Data
viii
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126
List of Figures
Page
1.1
MMIC Transmission Lines a) Microstrip;
3
b) Inverted Microstrip;
c) Suspended Stripline; d)Coplanar Waveguide;
e) Slot Line; f) Fin Line; g) Image Line.
1.2.
Cross-sections of a) Coplanar Waveguide and
7
b) Trench Waveguide.
1.3.
Three-dimensional Views of a) Coplanar Waveguide and
8
b)Trench Waveguide.
1.4.
a) Coplanar and Parallel Plate Waveguides;
11
b) Composite of
(a) resembling Trench Waveguide; c) Trench Waveguide.
2.1.
a) Cross-section of shielded microstrip showing
18
magnetic (H) and electric (E) field lines b) Three-dimensional view
of approximate magnetic field surrounding shielded microstrip
line. In (a) and (b) highlighted areas are those of high
charge/current density. (After [Ed 1981] Figure 3.4.)
2.2.
Trench Waveguide Mask.
21
2.3.
Oxidation System Diagram (Tube 1).
22
2.4.
a)<110> Surface Silicon and relevant crystal planes;
24
b) Results of anisotropic ODE KOH etch.
2.5.
Map of Si wafer showing directions of dicing cuts.
25
2.6.
Reactive Ion Etch System Diagram.
27
2.7.
Trench Waveguide alignment mark after Si02 mask etching.
ix
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29
2.8.
Si KOH Etching Results.
32
2.9.
Si KOH etching showing trench depth relative to
33
total wafer thickness (Magnification = 20X).
2.10.
Parallel trenches by Si KOH etch s=50 |im, w=23 pm,
34
b=12 jim.
2.11.
C lose-up view of single trench of Figure 2.10.
35
Note straight, smooth sidewalls.
2.12.
Si KOH etching results s=42 p m ,w = 3 1 pm, b=46 pm.
36
2.13.
Closer view of trenches of Figure 2.12.
37
2.14.
Close-up view of single trench of Figure 2.12.
38
2.15.
Si KOH etching results s=250 pm, w=37 pm, b=95 pm.
39
2.16.
Close-up view of single trench of Figure 2.15.
40
2.17.
Ion Implantation System Diagram.
41
2.18.
Electron Beam Evaporation System Diagram.
43
2.19.
Fixture for Angle Evaporation. (Not to scale)
44
2.20.
Trench structure after two angle evaporations.
46
2.21.
Figure 2.13 structure after angle evaporations.
47
2.22.
Closer view of Figure 2.21 trenches.
48
2.23.
Opposite angle view of Figure 2.22. Distortion on
49
trench left edges not physical; due to tilt.
2.24.
Close-up view of single trench of Figure 2.22. Note
50
smooth metal coverage of sidewall and final tapering
before trench bottom.
2.25.
Opposite angle view of Figure 2.24 trench.
51
2.26.
Additional view of Figure 2.24 trench.
52
2.27.
Close-up view of top trench corner showing smooth
53
x
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metal flow over edge.
2.28. Close-up view of trench sidewall, showing tapering of
54
metal before reaching trench bottom.
2.29.
Close-up view of Figure 2.28.
55
2.30.
Trenches after two angle evaporations.
56
2.31.
Trenches after Au plating.
57
2.32.
Trench Waveguide large scale models and
68
dielectric inserts.
2.33.
C/I vs W (er=2.76, s=0.325 cm, d=1.27 cm).
70
2.34.
C/i vs W (er=1 .0, s=0.325 cm, d=1.27 cm).
71
2.35.
Zo vs W (er=2.76, s=0.325 cm, d=1.27 cm).
72
2.36.
Z0 vs W (er=1.0, s=0.325 cm, d=1.27 cm).
73
2.37.
Hewlett-Packard 8510A Microwave Network Analyzer
74
system used to test Trench Waveguide.
2.38.
s n Magnitude vs Frequency (er=2.76, s=0.325 cm,
75
d - 1 .27 cm).
2.39.
s n Phase vs Frequency (er=2.76, s=0.325 cm,
76
d=1 .27 cm).
2.40.
S21 Magnitude vs Frequency (er=2.76, s=0.325 cm,
77
d=1.27 cm).
2.41.
S21 Phase vs Frequency (er=2.76, s=0.325 cm,
78
d=1.27 cm).
2.42.
Time Domain Reflection Coefficient Data (8r=2.76,
79
s=0.325 cm, d=1.27 cm).
2.43.
s n Magnitude vs Frequency (er=1.0, s=0.325 cm,
d=1.27 cm).
xi
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80
2.44. s n Phase vs Frequency (£r=1.0, s=0.325 cm,
81
d=1.27 cm).
2.45. S21 Magnitude vs Frequency (er=1.0, s=0.325 cm,
82
d=1.27 cm).
2.46. S21 Phase vs Frequency (er=1.0, s=0.325 cm,
83
d=1.27 cm).
2.47. Time Domain Reflection Coefficient Data (er=1.0,
84
s=0.325 cm, d=1.27 cm).
2.48. Silicon Trench Waveguide s n Magnitude vs Frequency
85
(er=11.8, s=250 (im, w=37 pm, b=95 pm, d=40 pm).
2.49. Silicon Trench Waveguide s n Phase vs Frequency
86
(er=11.8, s=250 pm, w=37 pm, b=95 pm, d=40 pm).
2.50. Silicon Trench Waveguide S21 Magnitude vs Frequency
87
(er=11.8, s=250 |im, w=37 pm ,b=95 pm, d=40 pm).
2.51. Silicon Trench Waveguide S21 Phase vs Frequency
88
(£r=11.8, s=250 pm, w=37 pm, b=95 pm, d=40 |im).
3.1.
General Conformal Mapping Schwarz-Chrisioffel
92
Transformation; a) z-plane (physical plane);
b)w-plane (transformed plane)
3.2.
Schwarz-Cnristoffel Transformation applied to
93
Trench Waveguide; a)Trench Waveguide physical
plane (z-plane); b)Transformed plane (w-plane)
3.3.
a) Finite Element Mesh Triangle; b) Connected Elements
98
3.4.
Simulated and measured C/I vs W (er=2.76, s=0.325 cm,
102
d=1.27 cm).
3.5.
Simulated and measured C/I vs W (£r=1.0, s=0 325 cm,
x ii
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103
d=1.27 cm).
3.6.
Simulated and measured
Z q vs
W (er=2.76, s=0.325 cm,
104
Zq vs
W (er=1.0, s=0.325 cm,
105
d=1.27 cm).
3.7.
Simulated and measured
d=1.27 cm).
3.8.
Typical mesh plot for Trench Waveguide in air.
106
(s=50 (im, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
3.9.
Mesh plot of Figure 3.8 magnified by 2X.
3.10. Typical mesh plot for Trench Waveguide on Silicon.
107
108
(s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
3.11. Mesh plot of Figure 3.10 magnified by 2X.
109
3.12. Plot of equipotential contours for Trench Waveguide in air.
110
(s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
3.13. Plot of Figure 3.12 magnified by 2X.
111
3.14. Plot of equipotential contours and field lines for Trench
112
Waveguide in air. (s=50 pm, w=15 pm, b=50 pm,
d=25 pm, t=5 urn).
3.15. Plot of Figure 3.14 magnified by 2X.
113
3.16. Plot of equipotential contours for Trench Waveguide on
114
Silicon.(s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 um).
3.17. Plot of Figure 3.16 magnified by 2X.
115
3.18. Plot of equipotential contours and field lines for Trench
116
Waveguide on Silicon. (s=50 pm, w=15 pm, b=50 pm,
d=25 pm, t=5 um).
3.19. Plot of Figure 3.18 magnified by 2X.
117
3.20. Cross-section of alternate version of Trench Waveguide.
122
x iii
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A.1
Plot of equipotential contours for large scale
127
Trench Waveguide in air. (w=0.16 cm, s=0.325 cm,
d=1.27 cm).
A.2.
Plot of Figure A.1 magnified by 2X.
128
A.3.
Plot of equipotential contours and field lines for large
129
scale Trench Waveguide in air. (w=0.16 cm, s=0.325 cm,
d=1.27 cm).
A.4.
Plot of Figure A.3 magnified by 2X.
130
A.5.
Plot of equipotential contours for large scale
131
Trench Waveguide on Silicon.(w=0.16 cm, s=0.325 cm,
d=1.27 cm).
A.6.
Plot of Figure A.5 magnified by 2X.
132
A.7.
Plot of equipotential contours and field lines for large
133
scale Trench Waveguide on Silicon. (w=0.16 cm, s=0.325 cm,
d=1.27 cm).
A.8.
Plot of Figure A.7 magnified by 2X.
134
A.9.
Plot of equipotential contours for Trench Waveguide in air.
135
(s=250 jim , w=37 pm, b=95 pm, d=40 pm).
A. 10. Plot of Figure A.9 magnified by 2X.
136
A.11. Plot of equipotential contours and field lines for Trench
137
Waveguide in air. (s=250 pm, w=37 pm, b=95 pm, d=40 pm
A. 12.
Plot of Figure A.11 magnified by 2X.
138
A.13.
Plot of equipotential contours for Trench Waveguide on
139
Silicon.(s=250 pm, w=37 pm, b=95 pm, d=40 pm).
A. 14.
Plot of Figure A. 13 magnified by 2X.
140
A.15.
Plot of equipotential contours and field lines for Trench
141
xiv
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Waveguide on Silicon. (s=250 fim, w=37 pm,
b=95 |im, d=40 p.m)
A.16. Plot of Figure A.15 magnified by 2X.
xv
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142
CHAPTER 1
IN T R O D U C T IO N
1
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2
Monolithic Microwave Integrated Circuits (MMICs) in both Silicon
(Si) and Gallium Arsenide (GaAs) have grown in popularity because they
are reliable, reproducible, small, lightweight, low cost and broadband. But
their growing use creates a need for more varied and flexible designs.
This attitude is borne out by the many uses to which M M IC-like
components are put in Millim eter Wave Integrated Circuit
(MIMIC) and
Very High Speed Integrated Circuit (VHSIC) environments.
Propagating
and processing signals at increasingly higher frequencies and speeds
requires novel interconnect structures to com plem ent conventional
transmission line media.
Depending on the application, these iines may
be of various types and configurations.
Some of the conventional forms
include m icrostrip and inverted m icrostrip, coplanar w aveguide and
suspended stripline, all supporting quasi-TEM mode propagation.
Non-
TEM structures like image line, fine line and slot line are also useful in
MMICs. The type of circuits to be served and the operating frequency are
the primary factors determining which configuration of transmission line is
to be used.
Figure 1.1 shows the basic structure of each.
All the
transmission lines (except image line, where the guiding medium is the
dielectric) consist of a metal pattern deposited on a dielectric for support.
In the figure dark, bold lines indicate metal, while dielectric areas are
shaded. Because they are simply patterns of alternate areas of metal and
bare dielectric on a surface, impedance, for instance, can be determined
by dimensions in a single plane. That is, impedance can be controlled by
the width or ratio of widths of conducting strips and the gaps. This makes
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3
e»
■—
£0
—
,
11—
(a) Microstrip
il lll
(d) Coplanar wavequide
£o
)
£
••••
.
}
_________________
(b) Inverted microstrip
(e) Slot line
e
(c) Suspended stripline
(f) Fin line
O
H
H
(g) Image line
Figure 1.1. MMIC Transmission Lines a) Microstrip; b) Inverted Microstrip;
c) Suspended Stripline; d)Coplanar Waveguide; e) Slot Line; f) Fin Line; g)
Image Line.
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4
the lithographic and m etallization fabrication steps more convenient,
because these are easier to do when patterning is in a single plane.
Briefly, m icrostrip shown in Figure 1.1(a) is a sim ple geometry
consisting of a conductor-backed substrate.
On the upper side a
conducting metal strip is deposited, with the lower conductor serving as a
ground plane. Widely discussed in the literature, microstrip is used over a
frequency range of approximately 1 to 50 GHz.
Sim ilar in structure to microstrip, inverted m icrostrip, shown in
Figure 1.1(b), also consists of a conducting strip placed on top of a
substrate.
In this instance, however, no conductor is placed on the back
side of the substrate. The ground plane conductor is separated by an air
space instead. Most of the field is concentrated in the air space between
the conducting strip and the ground plane. This structure can be operated
at higher frequencies than m icrostrip and utilizes a w ider strip for a
particular characteristic impedance.
The advantages include reduced
conductor losses and less stringent fabrication limits.
Suspended stripline shown in Figure 1.1(c) is achieved by making
the ground plane of the inverted microstrip totally enclose the structure,
with air gaps above and below the substrate. Losses in these transmission
lines are low, but it is easy to excite undesirable TE or TM modes at higher
frequencies.
This configuration does not accommodate shunt-connected
devices.
The last pseudo-TEM transmission line mentioned here is coplanar
waveguide (CPW) shown in Figure 1.1(d). Sharply different from the other
planar lines, all the conductors are situated in the same plane. A center
conducting strip is flanked on either side by ground planes.
Relative to
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microstrip, less field enters the substrate so that it has less influence on the
propagation of the wave. Design equations have been formulated and the
structure has proven convenient in connecting devices in series and in
parallel as they can be placed across the gaps between signal line and
side ground plane.
Slot line, shown in Figure 1.1(e), is com prised of a dielectric
substrate with metallization on one side alone. The slot line is formed by a
separating narrow slot etched into the metal.
Design equations are
available and unique circuits applications have been realized. The main
disadvantage is that it is difficult to achieve characteristic impedances
below 60 Q.
Fin line, shown in Figure 1.1(f), involves a shielding rectangular box
with a substrate centered across two faces. On one side of the substrate a
metal is deposited and a slot formed in the metal produces the fin line
circuit. This configuration is characterized by low losses, high operating
frequencies and sim ple fabrication.
A critical requirem ent is that
m icrowave short circuit to the upper and low er shielding walls is
maintained. This is generally not difficult to achieve in practice.
Image line, shown in Figure 1.1(g), is form ed by depositing a
dielectric strip or slab on a conducting sheet.
Useful at frequencies of
hundreds of GHz, the structure behaves like a dielectric waveguide
propagating TE and TM modes within the dielectric.
Limitations include
incompatibility with active devices, radiation at discontinuities and mutual
coupling.
Trench Waveguide is a novel addition to the list of quasi-TEM planar
transmission lines.
It is a three-dimensional structure that employs a
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
metallization pattern identical to that of CPW, except the intervening gaps
between signal line and ground planes are replaced by trenches etched
into the bulk of the substrate, as shown in Figures 1.2 and 1.3.
The
additional metallization of the sidewalls further distinguishes the Trench
Waveguide.
Note that as the trench depth, b, and metal extension, d,
shrink to zero, Trench Waveguide becomes CPW.
The investigation of
Trench W aveguide on Silicon is described in this work.
The novel
structure has features that can specifically address MMIC requirements.
First, the presence of the m etallized trench sidewalls reduces
current density near the strip edges where it is usually high. This occurs
because in the region where the current crowds, there is now increased
conductor surface area.
With the extension of the top electrodes down
along the trench sidewall, the current redistributes itself over the larger
area, reduces current density and current crowding and so leads to
reduced l2R loss.
The thick, plated electrodes should be several skin
depths thick to keep these ohmic losses to a minimum.
W illiam s and
Schwarz [Wi 1984] have, in fact, shown from numerical analysis on a
sim ilar structure (without metallized sidewalls) that conductor loss is
reduced by a factor of four, compared to conventional CPW. Their results
predict a significant reduction in dielectric loss as well.
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7
w __ 3 — -
w
^
'
Z
.
1
I ’ 1. ' . ' ' '
’. " ' "
I
1.............. ' r . ' . . . . . 7
Al
i
+
I
iv r ^ v X v S :
Al
I
SI Si
SI Si
(b)
Figure 1.2. Cross-sections of a) C oplanar W aveguide and b) Trench
Waveguide.
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8
i
t
T
h
1
hwH^s-H~wH
i
T
T
h
1
T
b
w-
Figure 1.3. Three-dim ensional Views of a) Coplanar W aveguide and
b)Trench Waveguide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9
Second, as with CPW, Trench W aveguide has low radiation loss
because the ground planes that flank the signal line serve to confine more
of the wave near the substrate.
Also like CPW, it has fewer isolation
problems since there is always a ground plane between any two adjacent
transmission lines. This is a useful property since MIC lines are applied in
circuits of increasing density, where components are placed closer and
closer together.
Researchers in the optoelectronics area also agree that
for their low fanout applications, planar transm ission lines are a good
choice.
This is because for short runs between nearby devices, for
example, large fanout of optical interconnects is not required, and planar
lines are sufficiently low-loss and broadband to be used successfully.
Trench Waveguide also makes efficient use of chip surface area.
Unlike other MMIC lines, Trench Waveguide exploits the unused bulk of
the silicon substrate via the etched trenches.
This leaves the limited
surface real estate available for placement of active devices and other
elem ents.
The three-dim ensional nature of the Trench W aveguide is
consistent with the trend in other technologies toward such configurations.
V-Groove silicon solar cells and charge-coupled devices [Ba 1975], [Ko
1980], optical waveguides [Ki 1980], silicon photovoltaic cells [Go 1982],
U-Groove and trench isolation [Ta 1982], [Ch 1983] and V-MOS FETS [Jh
1980] all heavily rely upon and effectively use the bulk of the supporting
silicon substrate.
Those applications requiring low characteristic impedances are also
served by the trench waveguide's three-dimensional nature.
In general,
the Zo for quasi-TEM lines is not defined by absolute strip and gap
dimensions, but by their ratio, w/s, where
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10
s = width of center strip conductor
w = spacing between center conductor and ground plane conductor.
This means that low impedances require wide center conductor strips,
leading to potentially large transmission lines overall, or small gap spacing
which may be difficult to fabricate. Lowering Zo may also be achieved by
shielding the structure or adding a backside ground plane [Gh 1987]. On
the opposite end, to achieve high impedances the ratio of w/s must be
made large. Large w/s implies a small s, or narrow center strip conductor.
Ohmic loss is known to increase with the narrowing of the strip [Wi 1984].
Repeatably fabricating very narrow strips may also be difficult.
In using Trench Waveguide for low Zo applications, it is convenient
to view the set of trenches as two vertically-oriented parallel-plate
waveguides (Figure 1.4(a)).
The characteristic impedance of such a
waveguide is given by
(1 .1 )
where
p = magnetic permeability
6 = dielectric permittivity
w = plate spacing
d = plate width
and
30
it
K'(k)
(1.2)
for CPW where
k
s+2w
(1.3)
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11
C oplanar W aveguide
T
d
w
i
Parallel p la te
w aveguide
Parallel p la te
waveguide
(a)
(b)
w
w
1
¥
lit
(C)
Figure 1.4. a) Coplanar and Parallel Plate Waveguides; b) Composite of
(a) resembling Trench Waveguide; c) Trench Waveguide.
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12
s
=
w =
center conductor strip width
gap width between center conductor and ground plane
conductor
k'(k)~is a rat'° °* cornPlete elliptic integrals of the first kind,
K
1 + Vk’ I
1 - V k 'J
0.707 < k < 1
(1.4)
0< k < 0.707
(1.5)
k' = a/ 1 - k 2 .
(1 .6 )
The additional relevant Trench Waveguide parameters are
b = trench depth
d = trench sidewall metal depth
t = metal thickness
With two such waveguides "in parallel" with the CPW on top, the composite
characteristic impedance is lowered. So for applications requiring Z0 less
than the 35-40
Q.
reliably achieved with CPW, the Trench Waveguide
provides an alternative.
Finally, since ail slectrcdes are accessible from the top surface of
the w afer like CPW, series and shunt connection of com ponents is
feasible.
This elim inates the need for via holes and their associated
parasitics, potentially enhancing wafer yields.
For the same reason,
thicker substrates may be used, making handling processing much easier.
This also contributes to higher wafer yields.
Extensive coverage has been given in the literature to the analysis
and modeling of CPW. Examining this work is relevant since CPW bears
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13
the closest physical resemblance to Trench Waveguide. Schottky contact
and M etal-insulator-sem iconductor (MIS) CPW has been analyzed [Fu
1983], [So 1984], [Ha 1977], as well as CPW on semi-insulating substrates
[Me 1968], [Pe 1976].
Closed form expressions fo r characteristic
im pedance, effective dielectric constant, filling factor and cross-talk
coupling factor have been derived for various configurations of CPW with
shielding [Gh 1987].
Behavior of symmetrical and asym metrical CPW
formats are described in [Ha 1984], [Ko 1984], [Be 1984], and losses in [Pr
1980], [Go 1982], [Kw 1987]. Analysis of microwave planar transmission
lines, including CPW, has been carried out by many researchers using a
wide variety of techniques.
The techniques fall into one of several
categories:
1.
Conformal mapping [Co 1987], [Fo 1974], [Gh 1987]
2.
Finite Element Method (FEM) [Pa 1986]
3.
Method of Moments [Na 1986], [Ka 1968]
4.
Green's function techniques [Ve 1985], [Ko 1982], [Da 1984]
5.
Variational methods [Ya 1968], [Ko 1983]
6.
Spectral Domain or Full Wave Method [It 1978a], [It 1978b], [Sy
1979]
7.
Fourier Integral/Fourier Transform Method [Mi 1970], [El 1981],
[Fa 1976]
8.
Discrete Variational Conformal Technique [Di 1986]
9.
Finite Difference Method [Sc 1965]
10.
Boundary Element Method [Ha 1969]
Relative strengths and weaknesses of any of these techniques applied to
arbitrary problem s have already been described in the literature by
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14
researchers in the field and will not be discussed here.
Because of its
flexibility in handling structures of arbitrary geometry and ease of access,
the Finite Element Method was used for analysis in this work.
FEM is a
numerical technique that is virtually unrestricted in the types of problems to
which it may be applied.
Trench Waveguide is markedly different from
other MMIC interconnects in that it has pertinent dimensions not only in the
x- and y-, but in the z-direction as weil. FEM is particularly well-suited to
such a problem.
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CHAPTER 2
EXPERIMENTAL PROCEDURE AND RESULTS
15
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16
2.1
Trench Waveguide Design
Trench waveguide design and the motivation for its geometry were
determ ined by the need for novel
in te rco n n e cts
to
co m plem e n t
conventional types of transmission lines and to provide low impedances in
an MIC environment. As in other MIC transmission lines, the overall loss
consists of conductor loss, dielectric loss and radiative loss. In general, the
largest and most troublesome of these is the conductor loss.
Using semi-insulating or intrinsic material substrates like Silicon or
Gallium Arsenide with dielectric loss tangents of 0.1525 and 1.4 x 10-7,
respectively, dielectric losses are of the order of 0.0632 and 6.1 x 1 0 '8
dB/cm, iespectively, at 10 GHz. These values are obtained by considering
Si with a = 0.01 s/cm, er’ = 11.8 and GaAs with a = 10'8 s/cm, er' = 12.85
and treating these materials as im perfect dielectrics.
In general, th e
dielectric permittivity
e = e '-je "
(2.1)
e' = Er' eo
(2.2)
e" = - ,
co
(2.3)
where
and
Er' is the custom ary relative die lectric constant, a is the
su bstra te
conductivity and co is the operating frequency. The loss tangent,
e"
tan 5 = —
(2.4)
and from (2.2) and (2.3)
e"
a
1
T :------- ;— = ten 5
£
CO E r
Eg
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(2.5)
17
Since the wave number k = co y \ i B '
( 2 .6 )
then a!! quantities requried to calculate attenuation constant (for an
imperfect dielectric) are known:
If the substrates are not insulating, losses can be much higher.
Radiative losses occur because the electric and magnetic fields are
not com pletely confined to the surface and substrate.
Some portion of
each penetrates the air outside, making it possible for energy to radiate
away from the substrate.
Shielding or providing adequate grounding will
reduce the size of this loss component.
Losses associated with the conductors of planar transmission lines
become significant because of the current that crowds along the edges of
the metal strips.
Charge and current density in the highlighted region
shown for microstrip in Figure 2.1 are high, leading to ohmic or resistive
losses [Pu 1981], [Ch 1980], [Sp 1977], [Ka 1981], [Ko 1985]. Also, it is
custom ary and useful to consider the metal as an infinite plane which
supports an electromagnetic wave. This assumption is valid, given that the
metal is at least two skin depths thick. A skin depth, 8, is that distance at
which the value of the fields are 1/e times their value at the surface. Thus,
it is desirable that the conductor be at least 25, so that the wave remains in
a thin layer or skin close to the surface and ohmic losses are again
controlled.
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18
(a)
(b)
Figure 2.1. a) Cross-section of shielded microstrip showing magnetic (H)
and electric (E) field lines b) Three-dimensional view of ap p ro xim a te
magnetic field surrounding shielded microstrip line.
In (a) and (b)
highlighted areas are those of high charge/current density. (After [Ed 1981]
Figure 3.4.)
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19
2 .2
Trench Waveguide Fabrication
This section presents a detailed description of the procedure used
to fabricate the trench waveguide devices.
Steps from mask layout
through device plating are outlined in such a manner so that the process is
self-contained and repeatable. Process flow sheets are provided as well.
Included are brief references to the configuration of the equipment used.
In this way, process param eters may be adjusted for any differences
between the systems used in this work and those used elsewhere.
Most of the processing was carried out at the facilities of the
National Nanofabrication Facility (NFF).
Located in Knight Laboratory at
C ornell U niversity, NFF has capability for full-scale sem iconductor
processing.
Wafer Procurement
The Silicon wafers used to fabricate Trench W aveguide were of a
special crystal orientation.
<110> surface orientation was required
because etching straight-walled trenches depends upon the preferential
etch properties of Silicon in Potassium Hydroxide (KOH) and water.
Suitable wafers with the following characteristics were purchased from
Virginia Semiconductor, Inc. of Fredericsburg, VA: <110> orientation, 10
Q-cm resistivity, 5.08 cm diameter, 0.33 cm thickness, polished on one side
at a cost of $20/wafer (1985 price).
Layout and Production of Mask
The patterns used to define the devices were produced using a
Calm a C om puter Aided Design tool, whose softw are has 0.5 pm
resolution. The masks were made on a David Mann Model 3600 Pattern
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20
Generator using 6.35 cm emulsion plates. Two key factors determined the
features of the masks. The first was actual line widths or the dimensions of
the trench and the center conductor. The other important feature was the
set of alignm ent bars required to make sure the substrate edges were
parallel to the pattern. A logo in each corner of the masks also aided in the
alignment.
The advantage of the process is that there is only a single
lithography level.
Figure 2.2 shows a drawing of mask layout used.
Growth of Silicon Dioxide or Silicon Nitride Mask
Two materials make suitably rugged masks for the trench etching.
Either a thick (> 4000
A) S i0 2 film or a thin (~2400 A) Si3N4 film perform
the function of protecting most of the silicon surface while the trenches are
being etched. A 1 p.m thick SiC>2 film was used in this process.
The system used to oxidize the wafers consists of a quartz tube
surrounded by an electrical resistance heater. A block diagram is shown
in Figure 2.3. The wafers are held vertically in a slotted quartz sled and
pushed into the hot zone of the furnace with a quartz pushrod.
Process
gases are introduced into the system through a quartz injector positioned
at one end of the tube.
The furnace temperature was set to 1050°C.
The wafers were
slowly pushed into the hot zone in nitrogen. The push rate was slow— two
inches per minute— to minimize wafer warpage.
The wafer temperature
was allowed to stabilize in nitrogen for five minutes before the oxidation
step.
For the oxidation, hydrogen and oxygen were injected into the
furnace and reacted to form pyrogenic steam. The H 2 :C>2 flow ratio was
1.8:1 to insure com plete com bustion of the hydrogen.
Steam was
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21
i
i
i
I
i
i
Figure 2.2. Trench Waveguide Mask.
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22
H ’S
So V s
CM
x
z
* 1 0 3H !S
H
CM
41-------- Z
to
©
T“
©
00
A A
CM
A
CD
H
CO
o
CO
A
CO
o
in
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Diagram
CM
o
Figure 2.3. Oxidation System
<1—
(Tube 1).
IOH
23
preferred for the oxidation because the oxidation rate is faster than in dry
oxygen. The wafers were oxidized for 50 minutes, annealed in nitrogen for
10 minutes and finally pulled from the furnace in nitrogen.
Wafer Dicing
The anisotropic preferential etch requires that the features to be
etched lie at an angle of 35.26° on the <110> surface, as shown in Figure
2.4. The safest, easiest, and most accurate way of making certain this is
done is to dice the wafer into appropriate rectangular sample sizes with 2
edges at the 35.26° angle, illustrated in Figure 2.5. The samples are then
already of a size com patible with the fixturing used fo r m icrowave
m easurem ents later.
This can be done using a precision watering
machine with circular saw. In this case the wafers were diced into pieces
1.26 cm x 1.02 cm with the long edge along the 35.26° angle. This greatly
facilitates lithography alignment because now the alignment bars of Figure
2.2 need only be oriented parallel to the sample edges.
This is much
easier to do than attempting to line up the bars at a 35.26° angle to the
wafer edge, with no previous lithography level to reference.
Wafer Clean
Cleaning the samples after dicing is imperative, so that grease and
soil will be removed from the S i0 2 surface. Without adequate cleaning, the
photoresist used for the lithography will not adhere.
The samples were placed in successive ultrasonic baths of acetone,
methanol and de-ionized water for 1.5 minutes each.
They were blown
with dry nitrogen and baked dry for 30 minutes at 150°C.
Then the
samples were placed in a reactive ion etch (RIE) system to slightly etch
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24
(110) Face
{111} Faces
109.47
70.53
= 35.26
(a)
(b)
Figure 2.4. a)<110> Surface Silicon and relevant crystal planes; b)
Results of anisotropic ODE KOH etch.
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25
35.26
Figure 2.5. Map of Si wafer showing directions of dicing cuts.
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26
back the S i0 2, which would serve as the mask.
Any organics remaining
were removed by this step.
The RIE took place in an Applied Materials system, a parallel plate
configuration with the upper electrode plate grounded and the lower
electrode (where the samples sit) at a negative potential. The system is
diagrammed in Figure 2.6.
Pertinent Si02 etching parameters are listed
below:
CHF3 Gas
30 SCCM
30 mTorr
0.25 mw/cm2
-547 v dc
13.56 MHz
Lithography
The lithography required to pattern the S i0 2 mask will be outlined
next.
To enhance adhesion of the photoresist, Shipley Company's C30
Primer was puddled on the wafer, allowed to sit for at least 10 sec, and
then spun for 30 seconds at 5000 rpm.
Shipley 1450J photoresist was
spun for 30 seconds at 5000 rpm, followed by a 30 minute soft bake at
90°C. This yields a film of about 1.5 pm thickness.
The exposure tool used in this fabrication process was a Nikon
Mask Aligner with UV source of ~436 nm wavelength. The alignment bars
served to align the pattern itself to the wafer edges. A 30-second exposure
at 12 m w/cm 2 was used, followed by a 60 second developm ent in
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* • # .i,
■
r i ,skl
Plasma
si 0_ Plate
Samples
Bell Jar
2 k *.
^
♦ it;
mmm
Etch Gas
In
rf power
in
Matching
Network
Water In
Baffle
6" Diffusion
Pump
Water Out
Figure 2.6. Reactive Ion Etch System Diagram.
28
Shipley's MF 312 CD 27 photoresist developer.
The sam ples were
agitated during the development, with a thorough rinse following.
The sam ples were descum m ed to remove any undeveloped
photoresist in the RIE system already described for 30 seconds in 0 2, 30
seem, 80 mtorr, -484v dc. The resist was hard-baked for 30 minutes at
90°C, after which time more 1450 J was applied to the backs of the wafer
to protect them during the S i0 2 etch. This resist was also spun at 5000
rpm for 30 seconds, then baked for one hour at 90°C.
S i 0 2 Mask Patterning
Once the resist has been hard-baked, the samples are ready for the
S i0 2 mask to be patterned. This is done using buffered hydrofluoric acid
(BHF) 6:1 mixture H20:H F. This solution is highly toxic, so extreme caution
must be exercised while using it. Mandatory, diligent use of gloves, apron
and face shield cannot be overlooked. No glassware must be used; only
nalgene or teflon beakers and tweezers.
Running water during etching is
also an appropriate safety habit. The samples were etched for 4.5 minutes
at room temperature followed by a rigorous 5 minute rinse in de-ionized
water.
The samples were blown with dry nitrogen.
A photograph of the
S i02 mask etched in this manner is shown in Figure 2.7.
KOH Silicon Etch
Etchants which have different etch rates on different crystal planes
are called anisotropic [Pe 1982].
The etch mask used was Si02 with
orientation dependent etch (ODE) KOH. KOH in water is a caustic solution
that rapidly attacks exposed <110> Silicon. Due to the crystallography of
the material, <111> crystal planes are at 90° from the <110> planes and
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29
Figure 2.7. Trench Waveguide alignment mark after Si02 mask etching.
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30
form the sidewalls of the trenches [St 1970].
< 1 1 1> surfaces etch
extremely slowly, thus little lateral etching of the trenches will take place
provided that the features to be etched lie at 35.26° from the <110> flat.
The anisotropy of the etch accounts for the very straight sidewalls of the
trenches.
Wet chem ical etching of Silicon consists of three reactions:
oxidation, dissolution and diffusion.
As with any other reaction the etch
rate is detrem ined by the rate of the slowest stage of the reaction.
Changing the reaction conditions changes the rates of the subreactions.
Reaction conditions are changed by varying oxidizing agent, dilutent,
substrate
dopant, te m p era ture,
concentration
of o xid izing
concentration of dilutent, stirring the mixture or adding catalysts.
agent,
For this
application the critical parameters are the temperature and concentration
of KOH. Reducing the concentration of the oxident causes the etchant to
become more anisotropic and selective.
The effect is to cause etching
toward the damage on the surface. Thus, as the trench bottom is defined,
the etch prefers to proceed in the direction of the dam aged surface,
increasing the trench depth. Increasing the temperature also increases the
etch rate of the Silicon. However, the rate of attack on the SIO 2 or Si'3 N 4
etch mask increases also.
Thus a moderate tem perature of 80°C was
used in this work. The etchant concentration used consisted of 44 mg of
KOH dissolved in 100 ml deionized H2 O for each device of approximately
1.3 cm 2 area.
KOH is very caustic and safety procedures like those
described above must be used. The samples were placed in the solution
on a hot plate for ~ 1.5 hr. This procedure yielded silicon etch rates of up
to 1.1 jim/min. The Si02 mask was etched at approximately 100 A/min.
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31
When the etch was complete, the samples were thoroughly rinsed, blown
dry and the remaining SiC>2 mask stripped using BHF for about two
minutes,
Figures 2.8 through 2.16 show scanning electron microscope
(SEM) photographs of typical etching results.
trench sidewalls achieved by this ODE.
Note the straight, smooth
No undercut, typical with wet
etches, is observable.
Ion Implantation
The w afers from V irginia Sem iconductor were p-type material
(Boron doped) of 10- Q-cm resistivity. These substrates had the required
crystal orientation to facilitate the trench etching, but were effectively
conducting substrates, unsuitable to place transm ission lines on.
High
resistivity substrates of <110> orientation were not commercially available.
In order to produce high resistivity substrates, counter-doping or
com pensation using a Lithium implant technique was performed.
This
involved subjecting the samples to an implant dose of 1014/cm 3 of Lithium
with no mask, as it was desired to bombard the entire surface. This was
done using an Al/Veeco 300 R ion implanter with 250 kev energy particles
with a 0.2 |ia/cm 2 beam density.
The configuration of the implanter is
shown in Figure 2.17.
The technique used to counter-dope the material relies on the fact
that Lithium is an interstitial impurity for Silicon. There are interstitial sites
in each unit cell of the Silicon lattice- each of which is a void available to
receive a diffusing Lithium atom.
With interstitial diffusion the Lithium
moves through the crystal lattice from one interstitial site to the next
adjacent one.
So given that the implanted Lithium (n-type) concentration
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32
Figure 2.8. Si KOH Etching Results.
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33
Figure 2.9. Si KOH etching showing trench depth relative to total water
thickness (Magnification = 20X).
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34
Figure 2.10. Parallel trenches by Si KOH etch s=50 p.m, w=23 (im, b=12
jim.
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35
Figure 2.11. Close -up view of single trench of Figure 2.10. Note straight,
smooth sidewalls.
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36
Figure 2.12. Si KOH etching results s=42 nm , w=31 jim, b=46 |im.
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37
Figure 2.13. Closer view of trenches of Figure 2.12.
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38
HU-WD
3 3 ’-j f‘l
<&
Figure 2.14. Close-up view of single trench of Figure 2.12.
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Figure 2.15. Si KOH etching results s=250 pm, w=37 |im, b=95 |im.
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40
Figure 2.16. Close-up view of single trench of Figure 2.15.
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RESOLVING
APERTURE ACCELERATION
^ANALYZER
MAGNET
/
Y SCANNER
WAFER
X SCANNER
ION
BEAM
! S'
SOURCE
DIFF PUMP
GAS SOURCE
TRIPLET
QUADRUPOLE
LENS
llllli
M IS
ION SOURCE
L POWER SUPPLY
ION SOURCE
HIGH VOLTAGE
TERMINAL
END STATION
DIFF PUMP
BEAM LINE
DIFF PUMP
J
Figure 2.17. Ion Implantation System Diagram.
FARADAY CAGE
42
is close to the original Boron (p-type) doping, it is possible to change the
resistivity of the material. That is, each Boron atom is compensated by a
Lithium atom, thus reducing the conductivity.
The samples in this work
were placed on a hot plate at 300°C under a beaker for
the Lithium .
~6
hours to diffuse
In this manner resistivity, determ ined by 4-point probe
measurements, was changed from the original
Q-cm to
10
Q -cm
>100
resistivity.
Angle Evaporation
In order to coat the sidewalls of the trenches without coating the
trench bottom, a multiple angle evaporation technique was used.
The
electron beam deposition system used fo r the angle evaporation is a
Varian 3140 cryogenic pumped unit with multiple hearth electron guns
diagrammed in Figure 2.18.
Each gun is directed by separate deposition
controllers that operate on a feedback produced by a resonating quartz
crystal sensor determining the rate of evaporation.
Since the evaporator
deposits line of sight, the samples were placed on a specially angled
stage, shown in Figure 2.19 on a stationary platform 61 cm directly above
the source material with the trenches horizontal to the source. This places
the sidew alls at a 45° angle to the source and prevents metal from
depositing on the trench bottoms, if they are sufficiently deep. Evaporation
was performed at 4 x 10 -7 Torr background pressure at a rate of
6
A/sec,
0.054 A emission c u rre n t. Because the unit was not set up with the ability
to operate a stage with an X, Y, or Z axis, the samples were allowed to cool
to room temperature after the first evaporation of 500
A titanium and
1000
A gold and the system backfilled with dry nitrogen. Then the samples were
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43
Sample
Shutter
Vacum
Chamber
Shutter
/ -/“ Shutter
//
Source 1
Source 2
Cryogenic
Pump
Figure 2.18. Electron Beam Evaporation System Diagram.
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44
Figure 2.19. Fixture for Angle Evaporation. (Not to scale)
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45
rotated 180° to deposit on the opposing sidewall with the same parameters
stated above.
Figures 2.20 through 2.30 are scanning electron m icro­
scope (SEM) photographs of the trench waveguide after com pletion of
both evaporations.
Figure 2.25 shows dramatically that the metal indeed
deposits on the trench sidewalls, tapers to nothing -2 /3 of the way down,
and in fact does not deposit on the trench bottom.
Electroplating
Particularly in microwave applications, thick metal lines (> 2 pm) are
required to avoid high ohmic losses. Since e-beam evaporation of such a
thickness is prohibitive, gold electroplating was used to build up the
electrodes.
Using the film deposited by angle evaporation as a plating
base, a Selrex BDT gold plating bath with arsenic brightener was set up.
Using a current to maintain a density of J = 2.95 mA/cm2, the electrodes
were plated up to a thickness of -
1 .8
pm at a rate of
2000
A/min The bath
tem perature was maintained at 25°C, pH level of 8.7-9.0.
shows the Trench W aveguide after plating.
Figure 2.31
This step com pletes the
fabrication process. Table 2.1 is a Trench Waveguide Process Schedule
summarizing the steps described above.
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46
Figure 2.20. Trench structure after two angle evaporations
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47
Figure 2.21. Figure 2.13 structure after angle evaporations.
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48
Figure 2.22. Closer view of Figure 2.21 trenches.
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49
Figure 2.23. Opposite angle view of Figure 2.22. Distortion on trench left
edges not physical; due to tilt.
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50
Figure 2.24. Close-up view of single trench of Figure 2.22. Note smooth
metal coverage of sidewall and final tapering before trench bottom.
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Figure 2.25. Opposite angle view of Figure 2.24 trench.
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52
Figure 2.26. Additional view of Figure 2.24 trench.
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53
Figure 2.27. Close-up view of top trench corner showing smooth metal
flow over edge.
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54
Figure 2.28. Close-up view of trench sidewall, showing tapering of metal
before reaching trench bottom.
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55
Figure 2.29. Close-up view of Figure 2.28.
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Figure 2.30. Trenches after two angle evaporations.
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57
Figure 2.31. Trenches after Au plating.
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58
Table 2.1
Trench Waveguide Process Schedule
Process Parameters
Process Step_________
Wafer 5 cm diameter silicon
into process samples
1.26 cm x 1 . 0 2 cm with long edges
along < 1 1 0 > crystal plane
Clean wafer samples
Organic
Ultrasonic 1.5 min
ACE
MET
Dl H20
Soak 1.5 min in Dl H20
Rinse in Dl H20
Blow dry with N2
RIE
S i0 2 mask film
Clean RIE chamber
0 2
30 seem
30 mTorr
0.25 mw/cm 2
-548 vdc
tuning = 285
loading = 946
15 min
RIE clean samples
chf3
30 seem
30 mTorr
0.25 mw/cm 2
-547 vdc
tuning = 285
loading = 946
1 min
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59
Table 2.1 (continued)
Lithography
Apply photoresist (PR)
Sample Backs
Puddle Shipley C30 Primer 10 sec
Spin C30 Primer 30 sec 5 K rpm
Spin Shipley 1450J PR 30 sec 5K rpm
Pre-Bake 45 min 90°C
Sample Fronts
Puddle Shipley C30 Primer 10 sec
Spin C30 Primer 30 sec 5 K rpm
Spin Shipley 1450J PR 30 sec 5K rpm
Pre-Bake 30 min 90°C
Expose
Nikon 436 nm wavelength Mask
Aligner, 30 sec (12 mw/cm2)
Develop
Shipley MF 312 CD 27 developer
Undiluted 60 sec
Rinse well in Dl H 2 O
Blow dry with N2
Examine
Optical Microscope
Assure that developed areas are
fully clear up to both edges of sample
PR De-sum
Clean RIE chamber
Same as in RIE clean step
PR De-sum
02
30 seem
80 mTorr
0.25 mw/cm 2
-484 vdc
tuning = 285
loading = 946
1 min
Post-Bake
90°C 65 min
Etch SiC>2 mask
BHF 6:1 10 min
Rinse > 5 min
Examine
Leitz Film Thickness Microscope
< 0.1 nm in clear (trench) areas
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60
Table 2.1 (continued)
T re n c h E tc h in g
44 gm KOH
100 ml Dl H20
Heat in beaker on hot plate
Deposit 1 sample at 80°C
> 80 min
Rinse 1 min
Examine
Leitz Microscope
< 0.1 nm
Strip Remaining
Etch Mask
10 min RIE (CHF 3 ) with parameters
given in RIE clean step
5 min RIE (O 2 ) with parameters
given in RIE clean step
C o u n te r- D o p in g
Ion Implantation
Mount on aluminum holders with PR
dry baked 90°C 15 min
Sample front side up
Li
250 kev
0.2 pA
7° tilt
Remove with ACE soak
Perform organic clean
BHF dip 1 min
Lithium Diffusion
Place glass slides on hot plate
Heat to 300°C
Deposit 1 sample under overturned
cover of small petri dish
6 hours
Test Resistivity
4-Point Probe Measurement
>5 current levels
Assure > 100 Q-cm
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61
Table 2.1 (continued)
A n g le
E v a p o ra tio n
Mount samples
P la tin g
Cover ends with aluminum foil
Affix samples to angled fixture
with beryllium-copper clips with
trenches parallel to 90° corner of fixture
Mount fixture to evaporation sample
holder with trenches parallel to source
Deposit :
Ti, 500A, 6 A/sec, 4x10' 7Torr, 0.054A
Au, 1000A, 6 A/sec, 4x10-7 Torr, 0.054A
Rotate samples 180°, repeat depositions
Selrex BDT Gold Plating Bath
As brightener
pH 8.7-9.0
Thief: 6.45 cm 2 anode
Sample: cathode
Adjust current to maintain 2.95 mA/cm 2
at thief
2000 A/min
Assure > 2 jj.m thickness via Alpha Step
Profile measurement.
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62
2 .3
Trench Waveguide Characterization
Several parameters were of interest in characterizing the trench
w aveguide com ponents fabricated.
These included ch aracteristic
im pedance, capacitance/unit length, and scattering param eters (sparameters).
From TEM transmission line theory, characteristic impedance
(Zq)
may be expressed equivalently as
Zo = VpL
or
(2 . 8 )
or
(2.9)
(2 . 1 0 )
where L is the inductance/unit length of the line, and C is the capacitance/
unit length.
vp is the phase velocity of the wave along the line atop a
substrate with dielectric permittivity er. If the dielectric is absent, or the
idealized case of an air-filled transmission line, then the wave has phase
velocity equal to that of light, c. The Zq of this air-filled line becomes
Z 01 = cL
or
(2 . 11 )
or
(2 . 12 )
(2.13)
L is again the inductance/unit length, unchanged from the case above
since the m agnetic perm eability, po. remains unity.
C i is now the
capacitance/unit length for the air-filled structure. From equation (2.8)
(2 . 8 )
Solving (2.12) for L leads to
L = Z 0 i/c.
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(2.14)
63
Thus (2.8) becomes
<2 -15>
Next substituting (2.13) for Z 0 1 , (2.8) yields
rrn — r
Z°
- v c cCi C
T
c2 C l C
-V
= 37^
<216)
This useful result means that if capacitance/length of a transmission line
with dielectric er =1 (C-i) and dielectric er * 1 (C) can be measured, then Zo
can be calculated.
Knowing C i and C also allows effective permittivity,
eeff, to be calculated.
In transmission lines consisting of more than one
dielectric, eeff relates various velocities, wavelengths and impedances. For
a transmission line
Vp=vro
(2.17)
and when the substrate is air vp = c and
C = VLCT'
(2.18)
Dividing (2.18) by (2.17) yields
i
=
V
^
-
°
r
^
=
^
2
<2
- i
9
>
The capacitance ratio C/Ci is known as eeff, so that
C
/ c \0
eel,=c 7 = ( ^ ) ■
(2 .20 )
This parameter also specifies the guided wavelength in the transmission
line. In air
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64
( 2 . 21 )
c = f^o
where f is frequency and A.o >s the free space wavelength. Otherwise
v p = fA.g
(2 . 22 )
with
being the wavelength in the transmission line.
Using (2.19) and (2.20) in (2.18) yields the result
(2.23)
eeff = ( £ ) 2 = ( r -)2
which leads to
(2.24)
So it can be seen that the wavelength in the m ultiple-dielectric line is
shorter than the free space w avelength.
In addition, eeft relates
characteristic impedances. From (2.10) and (2.13)
2 0 = vjlio yields c =
zqv ^
(2.25)
and
Z01 = 5 ^ - yields
Taking a ratio of the two expressions,
C
Z 01c
C i Z0vp
(2.26)
(2.27)
or
(2.28)
This result shows that
(2.29)
or
(2.30)
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65
Thus, knowledge of Z 01 in air and the effective perm ittivity allows
characteristic impedance,
In order to
Z q,
to be computed.
m easure capacitance/length
Waveguide, large scale models were used.
(C//) of the Trench
Shown in Figure 2.32 is a
diagram of the models, consisting of aluminum stock sitting atop a low
perm ittivity platform or "substrate".
The aluminum pieces simulate the
Trench Waveguide metallization pattern and the spaces between them are
analogous to the "trenches", which can be filled with plexiglass dielectric of
er = 2.76.
Using a standard com parison technique, the H ew lett-Packard
4342A Q-meter was used to determine C, capacitance, for the dielectricfilled Trench W aveguide model.
C i, capactiance for the "air" dielectric
model, was also measured. Data was taken for five different lengths of the
transmission line.
Note that each line has fringing fields at both ends.
These non-uniform fields w ill contribute to the capacitance values
measured, while it is C //fo r a uniform section of line that is desired. Figure
2.32 outlines the concept used to eliminate the effect of this fringing field
capacitance.
Model A can be viewed as a transmission line section of
length /i situated between some fixed lumped stray capacitances at planes
a-a and b-b.
Model B is a line section of length
h.
situated between the
same end capacitances. Though the line lengths vary, the end effects are
the same.
Using this approach with at least two line lengths, the
capacitance/length can be determined free from end effects.
given change in length
capacitance C/2 -
h
-
1\
In short, a
= A/ produces a corresponding change in
= AC. Observing the ratios AC/A/ yields the desired
capacitance/length for both the air-filled and dielectric-filled cases. These
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
values are listed in Table 2.2 for five lengths of transmission lines and are
the values known as C and C i
in equation (2.16).
C haracteristic
impedance was calculated using this expression, with the results also
included in Table 2.2. Plots of C// vs trench width and Zo vs trench width
are shown in Figures 2.33-2.36.for a 7.6 cm-long model with s=0.325 cm
and d=1.27cm.
As verification of the measurement technique, independent, direct
m easurem ents of Zq were made using the H ewlett-Packard 851OA
Microwave Network Analyzer (851 OA) including its' Time Domain feature.
A full 2-port calibration was performed at the ends of the APC-7 ™ test port
cables and the large scale Trench Waveguide model then connected to
the network analyzer as shown in Figure 2.37. Measurements were made
over a frequency range optimum for time domain resolution and including
several resonances of the device (45 MHz-9.045 GHz). The time domain
Low Pass -Step mode was used, which displays the magnitude of and
round-trip-tim e to an im pedance mismatch.
The network analyzer
im pedance environm ent is 50Q , while the Trench W aveguide was
designed to yield a characteristic impedance lower than 5 0 ft.
So the
reflection measured by the network analyzer is that due to the mismatch of
the Trench Waveguide input and indicates its reflection coefficient, p. The
corresponding impedance is easily calculated from p, yielding the Zo of the
Trench Waveguide under test. Plots of s-n, S21 and the time domain data
for the 7.6 cm-long model appear in Figures 2.38-2.47.
(The sloping s n
phase data includes a phase offset due to the line's finite length.) Applying
this method to the 7.6 cm-long model with w=0.16 cm, s=0.325 cm for
instance, yielded
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
Zo = 13.01Q
dielectric-filled
Zo = 17.98Q
air-filled
These results agree reasonably with the values of
Zo = 11 -94£2
dielectric-filled
Zo = 16.66£2
air-filled
gotten by the Q-m eter method. The clear agreem ent of these results
confirmed that the method of characterization was valid. As will be shown
later, computer simulations further verified the results.
Because of their small size (s=250 pm, w=37 pm, b=95 pm, d=40
pm), the silicon fabricated Trench W aveguide was measured using an
851 OA with a Cascade Microtech Probe™ over the 100 MHz-18 GHz
frequency range. The CPW probe with 9-mil pitch footprint was placed
directly on the surface of the Trench Waveguide under test to obtain the
de-im bedded s-parameters [GI1983], [St1980]. The probes were placed
approxim ately 2450 pm apart. The data for the forward s-parameters is
shown in Figures 2.48-2.51. Tabular data may be found in the Appendix.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s
Xt
T3
O)
CM
Figure 2.32. Trench Waveguide large scale models and dielectric inserts.
w
‘co
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE 2.2
TRENCH WAVEGUIDE LARGE SCALE MODEL DATA
CAPACITA ^CE7LENGTH & Z0
1
cm
w
cm
capacitance capacitance AI
c(diel) pf c1(air) pf cm
AC
Pf
...
_
Q-Meter C/L
C1 =
Ac1
C=
pf
Ac/Al
ZC
(diel)
(air)
12.63497
22.37439
32.18697
34.88895
20.8333
34.5423
46.62
52.4934
AC1/AI
3.7
3.7
3.7
3.7
0.16
0.3
0.445
0.559
14.2
8
5.9
5.025
6.5
3.97
2.92
2.5
5.7
5.7
5.7
5.7
0.16
0.3
0.445
0.559
22.9
2
8.9
7.9
9.7
5.9
4.35
3.77
2
8.7
4.6
3
2.875
7.6
7.6
7.6
7.6
0.16
0.3
0.445
0.559
30.3
17.3
12
10.2
13.5
7.7
5.9
4.9
1.9
1.9
1.9
1.9
7.4
4.7
3.1
2.3
3.8
1.8
1.55
1.13
3.8947
2.4737
1.6316
1.2105
2 11.94331
0.9474 21.77445
0.8158 28.89254
0.5947 39.28522
16.6667
35.1852
40.8602
56.0472
9.6
9.6
9.6
9.6
0.16
0.3
0.445
0.559
38.7
21.8
15.2
12.7
17.6
9.9
7.3
2
2
2
4.1
2.2
1.4
1.3
4.2
2.25
6.2
8.4
4.5
3.2
2.5
2.05 11.35997
1.1 21.18806
0.7 31.49704
0.65 36.98001
16.2602
30.303
47.619
51.2821
11.7
11.7
11.7
11.7
0.16
0.3
0.445
0.559
49.4
21.2
12.6
18.8
15.8
2
2
2
2.1
2.1
8.9 2.1
7.6 2.1
3.2
4.35
1.93
2.3
1.43
1.5
1.27 1.4375
1.6
1.25
1.6
0.965
0.715
0.635
10.7
3.6 5.0952
1.7143
11.27859
19.4444
3.6
3.1
1.6 1.7143
1.4 1.4762
0.7619
0.6667
29.16667
33.60108
43.75
50
o>
u>
70
C/L(pF/cm)
4
3
2
1
0.1
0.2
0.4
0.3
0.5
W(cm)
Figure 2.33. C/I vs W (er=2.76, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.6
71
3
C/L(pF/cm)
£=
1.0
2
1
0
0.1
0.2
0.3
0.4
0.5
W(cm)
Figure 2.34. C/I vs W (er=1.0, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.6
72
40 n
£=2.76
ZO(Q)
30 -
20
-
0.1
0.2
0.4
0.3
0.5
0. 6
W(cm)
Figure 2.35. Zq vs W (er=2.76, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
60 -i
e=1.0
50 -
g
o
N
30 -
20
-
0.1
0. 2
0 .4
0.3
0.5
W(cm)
Figure 2.36.
Z q vs
W (er=1.0, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0. 6
74
Source
o
Sweep Out
Qo o
»
CQ
E
a>
<n
Stop Sweep
HP 851 OA Network Analyzer
3
N
t/
HP-IB
co Cables
d>
o
T“
in
i ®
=3
nfl
TO EXTERNAL
HP-IB, COMPUTER
^ D is p la y • IF Bus
Sweep In
Test Set
Test Set
Interconnect
£
Figure 2.37. Hewlett-Packard 851 OA Microwave Network Analyzer system
used to test Trench Waveguide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
S i ]_
lo g
REF 0 . 0
dB
1
5 .0
dB/
V - 1 5 . 8 5 2 dB
u <
MARKER
j.35
M AG
GHz
START
STOP
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.38. s n Magnitude vs Frequency (er=2.76, s=0.325 cm, d=1.27
cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
S n
REF
1
V
u <
M AR
^
0 .0
®
100.0
8 7 .3 9 1
«/
°
KER
..35
D
START
STOP
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.39. s n Phase vs Frequency (er=2.76, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
S2 1
i og
REF 0 . 0
dB
1
2 .0
dB/
V - 0 . 5 4 7 S dB
MAG
MAR
GH
START
STOP
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.40. S21 Magnitude vs Frequency (er=2.76, s=0.325 cm, d=1.27
cm).
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78
S21
REF
1
V
o <
MARKER
1. 35
4
0 .0 9
100.0
• /
i n . 12 °
GHzl
D
START
STOP
0 .0 4 5 0 0 0 0 0 0
3 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.41. S21 Phase vs Frequency (er=2.76, s=0.325 cm, d = 1 .2 7 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
S n
R.
REF 2 0 . 0 m U ni t s
2
1 0 0 .0 m U n its /
V - 5 8 S . 9 6 mU.
^vEeugPE-MBLecimic
u <
'
START
STOP
-1 5 0 .0
p*
1 .5
n*
Figure 2.42. Time Domain Reflection Coefficient Data (er=2.76, s=0.325
cm, d *1 .27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
S n
lo g
REF 0 . 0
dB
1
1 9 .9
dB/
V - 2 2 . S 8 S dB
M AG
u <
GHz
1-*
START
STOP
Figure 2.43.
cm).
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
s n Magnitude vs Frequency (er=1.0, s=0.325 cm, d=1.27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
l
REF
1
V
^
0 .0
°
100.0
97 .5
• /
°
u <
GHz
O
START
STOP
0 . 0 4 -5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.44. s n Phase vs Frequency (er=1.0, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
S2 1
REF 0 . 0
log
-0 .S 1 7 9
>ȣ)
MAG
dB
ARKER
dB
Hz
D
START
STOP
Figure 2.45.
cm).
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
S21 Magnitude vs Frequency (er=1.0, s=0.325 cm, d=1.27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
S21
REF
X
> n
V
4
0 .0
•
100.0 • /
S 3 .9 4 9
°
MARKER
1 . 98
D
START
STOP
0 .0 4 5 0 0 0 0 0 0
9 .0 4 5 0 0 0 0 0 0
GHz
GHz
Figure 2.46. S21 Phase vs Frequency (er=i .0, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
s n
REF 2 0 .0 m U ni t s
2
1 0 0.0 m U n i t * /
- 4 7 1 . 2 6 nvU .
V
> n
wHvEouirde-A
START
STOP
-1 5 0 .0
ps
1 .0
n*
Figure 2.47. Time Domain Reflection Coefficient Data (er=1.0, s=0.325 cm,
d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
-
10-
-1 4 -
-1 6 -
-18
0
10000
2 00 0 0
FREQ(MHZ)
Figure 2.48. Silicon Trench Waveguide s n Magnitude vs Frequency
(er=11.8 , s*250 urn, w»37 jxm, b*95 |j.m, d=*40 |im).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
200 -j
0
10000
20000
FREQ (MHz)
Figure 2.49. Silicon Trench Waveguide s n Phase vs Frequency (er=11.8,
s=250 nm, w *3 7
b=95 nm, d=40 |im).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
|S21| (dB)
-8 i
-
10-
-
12-
-1 4 -
-1 6 -
-1 8 -
-
20-
-22
0
10000
200 00
FREQ (MHz)
Figure 2.50. Silicon Trench Waveguide S21 Magnitude vs Frequency
(er* 11 . 8 , s=250 |im ,w =37 jim, b=95 nm, d=40 nm).
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
88
40 -i
30 o
W
<
20-
0.
10
-
0
10000
2 00 00
FREQ (MHz)
Figure 2.51. Silicon Trench Waveguide S21 Phase vs Frequency (er=11.8,
s=250 n.m, w *37 nm, b=95 p.m, d»40 |i.m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
D IS C U S S IO N
89
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90
3.1
Analysis
of
Trench
Waveguide
T ransm issio n
Line
Parameters
To analyze the Trench Waveguides in this work, the commonly used
quasi-TEM approximation was made. With no component of E or H field in
the direction of propagation, the TEM fields satisfy Laplace's equation in
two dimensions:
(3.1)
2
—
2
+
—
-0
(3.2)
This facilitates finding expressions for important parameters of the Trench
W ave g u id e
c o m p o n e n ts:
c a p a cita n ce /le n g th
and
c h a ra c te ris tic
impedance (Zo).
One popular analytical technique is that of conform al mapping,
whereby a configuration in the physical plane (z-plane) is mapped onto the
transformed plane (w-plane).
There the configuration is usually sim pler
than the original and has a readily determ ined, known solution.
This
transformation is accomplished via a mapping or transformation function, w
= f(z).
Knowing the solution in the transformed (w) plane leads to the
desired solution in the physical (z) plane by using the inverse of the
mapping function. Special properties of conformal maps are:
(1 )
adjacent points in the z-plane transform to adjacent points in the wplane;
(2)
the conjugate functions (those that satisfy Laplace's equation, i.e.
potential and flux) are normal to one another in both planes;
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91
(3)
distances are merely scaled when mapped from one plane to
another and angles are preserved along with their sense.
Now when a problem may be posed as bounded by a polygon in the zplane, a special transform ation may be used.
The Schwarz-Cristoffel
transformation maps'the boundary of a polygon onto the real line in the wplane, the interior of the polygon onto the upper half of the w-plane and the
exterior onto the lower half of the w-plane.
This transformation and its
properties are described elsewhere [Gi 1958], [Ma 1977], [Si 1984]. It has
the general form
dZ
— = A(w-a)
aw
(o/ ti)-1
(w-b)
(P/7t)-1
(w-c)
(y/7i )-1
(3.3)
where A is a scaling constant and there is a factor appearing for each side
of the polygon, as shown in Figure 3.1.
a, p, y, ... are the polygon's
interior angles and a, b, c ... are the w-plane coordinates of the vertices in
the original z-plane. Figure 3.2 shows how the Schwarz-Christoffel (SC)
transformation has been applied to the Trench Waveguide configuration.
The general Schwarz-Cristoffel Transform ation (for straightening eight
right angles) is
A(w-a ) a/7t_1 ( w - b f * - 1 ( w - c p
(w-e ) ^
" 1
1
(w-d ) 8/7t_1 •
(w-f) ^ 71-1 ( w - g ) ^ " 1 (w-h )T1/7t-1 .
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(3.4 )
92
y
w =c
w =b
w =d
W = +00
:w = w =a
x
0
(a)
V
1
1
A
B
0
»
1
C
D
u
(b)
Figure 3.1. General Conformal Mapping Schwarz-Christoff el
Transformation; a) z-plane (physical plane); b)w-plane (transformed plane)
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93
z • plane
k
)
-e
e
x
_oo
. OO
-a
«
a
„
_________
-ci
-di
di ^
c>
Tpl
w - plane
-OO -A
-C -D
Figure 3.2. Schwarz-Christoffel Transformation applied to Trench
Waveguide; a)Trench Waveguide physical plane (z-plane); b)Transformed
plane (w-plane)
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94
In particular,
+m
m, - 1/2 rw + R i /2 .
•A f w + m/ i /2 f\ w' ~
f - 1/2 ^x-w- .+—
QW
( W - E ) 1/2 ( w - D ) - 1/2 ( w - C ) - 1/2 ( w - B ) 1/2
(3.5)
Sim plifying,
^ •=
A(w2 - B2)1/2 (W2 - C 2)-1/2 (W2 . D2)-1/2 (w 2 . £2)1/2
(3.6)
Integrating both sides yields,
Z =
A
f w (t2 -
JQ
B2)1/2 (t2 . C2)-1/2 (t2 . D2)-1/2 (t2 . £2)1/2 dt
(3.7)
subject to these conditions:
z(w = 0 ) =
(3.8)
0
z(w = ± E) = a
(3.9)
z(w = ± D = ± 1/2) = a - j
(3.10)
z(w = ± C = ± 1) = (a + P) - j
(3.11)
z(w = ± B ) = a + p - j + j = a + p .
(3.12)
These conditions lead to determination of unknown constants A, B, and E.
The transformation
(3.13)
thus maps the trench geometry in the z-plane to the real axis and upper
half of the w-plane.
Closed form analytical solutions for arbitrary
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95
geom etries are difficult to invert, so a numerical technique was sought.
This strategy has been documented by other researchers in the field [Di
1986], [Co 1987], [Fo 1974], [An 1975]. The numerical solution too! used
was the FEM sim ulator, the Cornell University Model for Quantized
Transport in Semiconductors (CUMQUATS).
Created by Bob Burns of
Cornell University, CUMQUATS solves the Poisson equation for twodim ensional electrostatic problem s.
With m odifications, CUMQUATS
solves the required Laplace's equation in two dim ensions with the
geometry and boundary conditions relevant to Trench Waveguide.
Using the finite element method to solve electric field and potential
problem s is believed to have first been done by Courant in 1943 [Co
1943]. His paper was the first to set forth the solution of problems using
piecewise linear approximations resembling finite elements.
Duffin [Du
1959] is credited with developing first order triangular elements as they are
used today to approximate solutions.
Implementing the FEM requires
complex and extensive programming, if it is to treat a variety of geometrical
structures. Background required for pursuing such an effort is presented
elsewhere [Si 1983], [Zi 1973], [No 1978]. The following material sets forth
a brief description of the FEM for the purposes of the context of this work.
The problem to be solved was that of Laplace's equation in two
dimensions for a Trench Waveguide. That is, solve
V2U = 0
for the potential between the conductors shown in Figure 1.3.
(3.14)
The
boundary conditions to be met require that the value of the potential be
pre-set at the metal surfaces (Dirichlet boundary condition) and that there
be no electric field (normal derivative of the potential) at the metal surfaces
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96
(Neumann conditions).
The minimum-energy principle states that the
potential distribution, u, must take on a form that minimizes the stored field
energy per unit length of the transmission line. This stored energy can be
expressed as
(3.15)
to within a constant multiplier, er, the permittivity.
So, applying FEM to
Laplace's equation results in finding u that minimizes the energy function
W of (3.15). Note that this procedure is mathematically identical to solving
(3.14) because any u that satisfies (3.14) satisfies (3.15) also.
Solving a problem using the FEM involves six steps:
(i)
divide the region of the problem into triangles, i.e. create a finite
element mesh;
(ii)
define any sources and fix the boundary conditions;
(iii)
set down a m atrix representation for each separate triangle
(element);
(iv)
"join" all the elements via matrix operations and re-apply boundary
conditions to the connected mesh of elements;
(v)
solve the resulting system of algebraic equations;
(vi)
display the results in useful format.
In step (i) the physically com plex geom etry is discretized as an
unconnected set of elements.
The rest of the steps involve putting the
triangular pieces together again in a way that produces the desired
solution, u.
With the basic strategy followed by FEM's including CUMQUATS,
the problem geometry is divided into triangles or mesh elements, such as
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97
that depicted in Figure 3.3.
The desired potential, u, is approximated
inside each separate element.
The elements are connected and their
potentials linked such that they are continuous across the boundary
between two triangles, u is approximated by what is called a shape or trial
function.
In the simplest case of first-order elements the shape function
has the form
U = a + bx + c y .
(3.16)
So the true potential is represented by a piecewise-planar function. Even
though the estim ate of the true solution is piece-w ise planar, it is
continuous everywhere, since the surface, U(x,y), of (3.16) has no gaps.
The potential along size 1-2 of Figure 3.3, for example, is the average of
the potential values at vertex
1
(node
1)
and vertex
2
(node
2 ).
a, b and c
in (3.16) are found after fixing the potentials at the three nodes as U i, U2
and U3 . Doing so yields
U 1 = a + bx-j + cyi
U2 = a + bx 2 + cy2
(3.17)
U3 = a + bx 3 + cy 3
or casting it in matrix form
U ,'
U2
U3
=
■ 1
X,
Vt’
a
1
X2
y2
b
1
x3
y3
c
and more compactly
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98
io
■cn
*—
*
c
©
E
©
LU
•o
•*©
—
*
o
©
CM
d
c
o
O
_©
CO
CD
CO
d
©
"i_
H
sz
w
©
d
©
E
©
CO
LU
©
©
CO
CO
CO
©
©
CD
CM
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99
[Un] = [P][K],
(3.19)
where [P] is the 3 x 3 matrix of (3.18) and [K] is the column vector
containing the unknown coefficients a, b,c.
Solving for a, b and c, then
substituting in (3.16) yields
(3.20)
[K] = [PJ-1 [Un]
and
Ui = [P]’ 1 [Un] + M P ]-1 [U„] + y i[P ]-'[U n]
(3.21)
u 2 = [P]-1 [Un] + x2 [P]-1 [Un] + y 2 [P ] 1 [Un]
U3 = [PI' 1 [Un] + X3 [P] ' 1 [Un] + y a m U n ]
or
[U] =
[1
(3.22)
x y] [P]-1 [U n ].
Re-casting the problem in a more convenient form of coordinates
3
U = I U , a,(x,y)
(3.23)
j» 1
with
011
= 2 ^ { ( x2y3 • x3y2) + (y2-y3)x + (x3 -x 2 )y)
where A is the triangle surface area.
(3.24)
Next, using (3.15) and (3.23) the
energy of each element may be written
3
VU = X
U, V
(3.25)
«i
i *■ 1
so that
3
w (e) = i
f |VU |2 dS = i X
^ J
* i-1
3
X
u i fU, V
j-1
•V
a,
dS
J
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(3.26)
100
and (e) denotes that this is the energy for a particular element. Simplifying
notation via
s j e) = J V o ij* Vccj dS
(3.27)
allows the discretization (3.26) to be expressed as
W(e) = 1 UT S(e) U
(3.28)
where U is the column vector containing the potential values at the triangle
vertices.
Thus the FEM num erically approxim ates the energy of a
triangular element S2 1 , for example, as
S 2S1) = 4 X
{(y 2
‘ y3)(y3 ' y i> + (X3 ' X2 ^ X1 ' X3^
(3'29)
and likewise for other elements of the matrix S. Summing these individual
energies gives the total energy for a mesh built up by joining triangles one
at a time as
w =X
w (e)
(3-3°)
all e
Using techniques described elsewhere [No 1978] the energy is minimized
by evaluating
3W
aUk
„
’
3 ,3 1 )
where k corresponds to nodes in the resulting joined mesh.
The
approximated solution is a table of nodes and their potential values. The
node-potential pairs are sim ply a condensed representation for the
solution
surface,
U, w hich
yields
minimum
energy.
G raphically
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101
representing these piece-wise planar contours yields the equipotential
lines desired.
This num erical procedure, consisting of steps (i) through (vi)
outlined above, was carried out for Trench Waveguide on the Micro VAX II
Ultrix-32m 1.2 system. Figures 3.4-3.7 show the capacitance/length vs w
resulting from the simulations and the corresponding Zo
vs w
.
The values
derived from the Q-meter data are also shown, for comparison, and
reasonably.
Figures 3.8-3.11
a g re e
show typical m eshes generated by
CUMQUATS for a Trench W aveguide on silicon and "air" dielectric
substrates.
Similar mesh plots were produced for the silicon fabricated
Trench W aveguide whose s-param eters (measured with the Cascade
Microtech Probe™) are shown in Figures 2.48-2.51, as well as the large
scale models.
Note how the mesh clearly delineates the a ir-d ie le ctric
boundary at the trench sidewalls and bottom in Figures 3.10 and 3.11. The
resulting equipotential contours and electric field lines are shown in
Figures 3.12-3.19 for the above example. The dotted line shown in each
figure is the reference or "zero" potential contour. The plots show how the
trenches serve to confine the field. The same data for a large scale model
and a fabricated component appears in the Appendix. Capacitance/length
and Zo are summarized in Table 3.1.
Included are Zo values for a
conventional CPW of comparable dimensions. This is a useful comparison
because as the trench depth, b, and metal extension, d, shrink to zero,
Trench W aveguide becomes CPW.
The results indeed show that for
Trench Waveguide and CPW with the same dim ensions, the Trench
Waveguide Zo is significantly lower.
c o m p le m e n t
CPW
in
M M IC
So Trench Waveguide can feasibly
e n v iro n m e n ts
re q u irin g
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low
Zq.
102
5
•Q- C(MEAS)
C(SIM)
C/L(pF/cm)
4
3
2
1
0.1
0.2
0. 4
0.3
0. 5
0.6
W(cm)
Figure 3.4. Simulated and measured C/I vs W (er=2.76, s=0.325 cm,
d=1.27 cm).
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103
3
£ = 1.0
-o- C1 (MEAS)
C1 (SIM)
2
1
0
0.1
0.2
0 .4
0.3
0 .5
0.6
W(cm)
Figure 3.5. Simulated and measured C/I vs W (er=1.0, s=0.325 cm, d=1.27
cm).
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104
40 -i
e=2.76
-Q -
Z O (M E A S )
Z O (S IM )
ZO(fl)
30 -
20
-
0.1
0.2
0 .4
0.3
0.5
0.6
W(cm)
Figure 3.6. Simulated and measured Zo vs W (er=2.76, s=0.325 cm,
d=1.27 cm).
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105
60 -i
50 -
-o- zO(MEAS)
zO(SIM)
30 -
20
-
0.1
0.2
0 .4
0.3
0 .5
0.6
W(cm)
Figure 3.7. Simulated and measured Zo vs W (er=1.0, s=0.325 cm, d=1.27
cm).
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106
tren ch waveguide
50
i
100
i
i
i
i
100
i
I
I
in a i r
t
1----- 1----- 1----- r
-
-
mm
50 -
0
200
150
T'
- 50
-
-
w
-5 0 -
s fa ffm
m
ff^ s e m
m
^ k ^ g
a a o
i
-100
I
L
L _
I
L.
I
50
■
i
■
■_____ I_____ l _ J _____L _ _ J _____I_____ I-------- I_____I_____ L .
100
150
200
Figure 3.8. Typical mesh plot for Trench Waveguide in air.
(s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
- -5 0
-
-I
100
-100
trench waveguide
20
40
60
in a i r
80
100
120
40
20
0
-20
f- - 2 0
-4 0
-4 0
-6 0
-6 0
20
40
60
80
100
Figure 3.9. Mesh plot of Figure 3.8 magnified by 2X.
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120
108
t r e n c h w a v e g u i d e on S i
50
“i
1
i
i
100
150
2 00
1----- r— i---- 1-----1----- 1----- 1-----1-----1----- 1----- 1-----i-----1----- 1----- r— i---- 1----- 1-----r
100
100
50
50
VA
0
0
-5 0
-5 0
-100
-1 0 0
■j— i— i— i— i— i— i— i— i—
50
i__i
100
i
i
■ i
150
■__■ ■ ■ i
■
200
Figure 3.10. Typical mesh plot for Trench Waveguide on Silicon.
(s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
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109
t r e n c h w a v e g u i d e on S i
20
40
60
80
100
120
20
40
60
00
100
120
Figure 3.11. Mesh plot of Figure 3.10 magnified by 2X.
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110
trench waveguide
0
50
100
in a ir
150
200
100
100
50
- 50
-5 0
-1 0 0
-5 0
-100
-
0
50
100
150
200
Figure 3.12. Plot of equipotential contours for Trench Waveguide in air.
(s=50 pm, w=15 jim , b=50 pm, d=25 pm, t=5 um).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
111
tren ch
20
40
w aveg u id e
60
in
80
a ir
100
120
40
40
20
20
-20
-20
-4 0
-4 0
-6 0
-6 0
20
40
60
80
100
Figure 3.13. Plot of Figure 3.12 magnified by 2X.
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120
112
0
tr e n c h w a v e g u id e
50
100
in
a ir
150
200
100
100
50
50
0
0
-5 0
-5 0
-1 00
-100
0
50
100
150
200
Figure 3.14. Plot of equipotential contours and field lines for Trench
Waveguide in air. (s=50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 urn).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
40
60
80
100
Figure 3.15. Plot of Figure 3.14 magnified by 2X.
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120
114
tren ch
0
i
|
0
w a v eg u id e
50
i
1
i
i
1
I
I—.1
|
50
on
Si
100
1
1—
1
|
150
200
i
i
i
i
1
i
1
1
I i. -1—..I--- 1 I
1
I
I
I
I
I
I
100
150
|-----r
I
I
L
200
Figure 3.16. Plot of equipotential contours for Trench Waveguide on
Silicon. (s=50 pm, w *15 pm, b=50 pm, d=25 pm, t=5 urn).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t r e n c h w a v e g u i d e on S i
20
40
60
80
100
120
40
40
20
20
-20
-2 0
-4 0
-4 0
-6 0
-6 0
20
40
60
80
100
Figure 3.17. Plot of Figure 3.16 magnified by 2X.
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120
116
tren ch
,
0
|
1
I
0
50
( —r— «——i-f "T ' i
1
1
1
1
* *
50
w aveg u id e
i
1
1
100
i | i
1
1
100
on
1i
Si
150
i - j 11 i
■ ■ * * I
150
i
200
i......... t |-r
■ ■ ■ * I
■
200
Figure 3.18. Plot of equipotential contours and field lines for Trench
Waveguide on Silicon. (Ss50 pm, w=15 pm, b=50 pm, d=25 pm, t=5 um).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
117
t r e n c h w a v e g u i d e on S i
20
40
60
80
100
120
40
40
20
20
-2 0
-20
-4 0
-4 0
-6 0
-6 0
20
40
60
80
100
Figure 3.19. Plot of Figure 3.18 magnified by 2X.
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120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE 3.1
TRENCH WAVEGUIDE LARGE SCALE MODEL
CHARACTERISTIC IMPEDANCE
SIMULATED
Z0 (diel)
MEASURED
Z0 (air)
SIMULATED
ZO (air)
5.7 0.16 12.6349674
5.7
0.3 22.3743883
32.18697
5.7 0.445
5.7 0.559 34.8889545
11.9928564
21.1469415
29.7525104
36.0414356
20.8333333
34.5423143
46.6200466
52.4934383
18.4102051
31.7605509
43.8365772
52.4257389
87.53
105.06
117.65
125.37
120.01
144.05
161.31
171.91
7.6 0.16 11.9433095
7.6
0.3 21.7744485
7.6 0.445 28.8925351
7.6 0.559 39.2852215
11.9928564
21.1469415
29.7525104
36.0414356
16.6666667
35.1851852
40.8602151
56.0471976
18.4102051
31.7605509
43.8365772
52.4257389
87.53
105.06
117.65
125.37
120.01
144.05
161.31
171.91
9.6 0.16 11.3599696
9.6
0.3 21.1880575
9.6 0.445 31.4970394
9.6 0.559 36.9800131
11.9928564 16.2601626 18.4102051
21.1469415 30.3030303 31.7605509
29.7525104 47.6190476 43.8365772
36.0414356 51.2820513 52.4257389
87.53
105.06
117.65
125.37
120.01
144.05
161.31
171.91
11.9928564
21.1469415
29.7525104
36.0414356
87.53
105.06
117.65
125.37
120.01
144.05
161.31
171.91
length w
cm
cm
MEASURED
Z0 (diel)
11.7 0.16 11.2785924
11.7
0.3
11.7 0.445 29.1666667
11.7 0.559 33.6010753
19.4444444 18.4102051
31.7605509
43.75 43.8365772
50 52.4257389
CPW
ZO (diel)
CPW
ZO (air)
119
3.2
Summary, Conclusions and Recommendations for Future
W ork
An overall summary of the data and evaluation of the results will
now be presented. Relevant data to be discussed was gathered from five
sources:
1.
Q-m eter capacitance/length measurements for large scale Trench
Waveguide models leading to
Zq
2. RF measurements of s-n and s 2 ifo r large scale model leading to Zo
3.
De-imbedded RF m easurements of s-n and S21 leading to Zo for
Silicon fabricated component.
4. Electrostatic FEM simulations for large scale models
5. Electrostatic FEM sim ulations for small scale Silicon fabricated
component.
Correlating results for Zo will verify that Trench W aveguide is, in fact,
suitable for making low characteristic impedance transmission lines.
Some com m ents regarding the m easurem ent techniques and
strategies are in order. First, recall that, impedance of planar transmission
lines is not defined by the absolute widths of conductors and gaps, but
usually by their ratios. Therefore, transmission lines of different sizes and
dimensions may have identical Zo if their conductor and gap relative sizes
are identical. So a Trench Waveguide with
s=0.325 cm, w=0.160 cm, b=1.27 cm
has the same Zo as one with
s=.325 mm, w=.160 mm, b=1.27 mm
since all dimensions have been scaled equally (by a factor of 10).
Large
scale models of Trench Waveguide were examined because they were
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120
convenient and useful in determining capacitance/length and eventually
Zo-
These models were expected to behave like the actual fabricated
components because of the scaling just described.
Secondly, Trench W aveguide was m easured as a resonant
structure on the 8510A Network Analyzer. As seen from the plots of s-n in
F igures
3 .3 8 -3 .4 7
transmission line is
resonances
rO J2 )
(frequencies w here
length
of the
occur at evenly spaced frequencies in both the
dielectric-filled and "air" dielectric cases.
Over this region then, there is
virtually no dispersion and it is valid to extract the transm ission line
parameters from the resonant structure data.
Finally, simulations were done for convenient large scale Trench
Waveguide models. CUMQUATS produced results of capacitance /length
leading to Zo that were confirmed by two other independent means. As a
result, there is every confidence that it analyzed the small scale Si
structure just as accurately.
Trench W aveguide has been proposed as a means of producing
low Zo transmission lines. Motivation for the concept has been presented,
the fabrication described and the feasibility of producing low characteristic
impedances demonstrated.
In order to evaluate the Trench Waveguide
fairly, however, it is necessary to cite considerations to be weighed when it
is the transmission line of choice. Resolving these concerns is the goal of
continuing future work on Trench Waveguide.
Regarding processing concerns, the availability of <110> Si on
which to fabricate Trench Waveguide is an important matter.
The real
problem is finding the right combination of appropriate crystal orientation
and resistivity. To my knowledge, semi-insulating <110> Si wafers of any
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121
diameter are not available as of the date of this writing. Until such material
is available, it is necessary to use a doping compensation technique such
as that described in this work or some similar scheme. A possible solution
to this deficit could be to intentially grow material with enough fixed
im purities to render the resulting Si boule sem i-insulating.
Another
approach would be to build the "inverse" of Trench Waveguide. That is,
instead of fabricating a structure extending down into the bulk of a silicon
wafer, build one rising above the surface of the substrate.
This idea is
illustrated in Figure 3.20, where very thick metal electrodes sit atop a semiinsulating substrate or an insulating film like Si 0
2
-
Sufficiently tall
"pedestals" with air gap trenches between them can also provide the
parallel plate regions required to reduce the characteristic impedance.
Another area of concern may be com patibility of <11 0 > Si with
standard processing techniques.
W hether it is feasible to design a
fabrication process for microwave circuits completely produced on < 1 1 0 >
Si is an area worth examining.
Would FETs, amplifiers, oscillators and
other circuits producd on <110> Si perform as well as those fabricated on
<100> material? Or would Trench Waveguide find broader application in
hybrid circuits where active devices would be produced as normal and
then bonded to a substrate of another type containing Trench Waveguide?
These questions provide several technical challenges worth pursuing.
The last consideration in the processing area is whether a surface
topography problem is created by the presence of deep trenches. Surface
planarization is routinely required in order for subsequent levels of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
122
s
IV
Metal
s\w^
yr*<*+ ' i ^
Insulator
Substrate
Figure 3.20. Cross-section of alternate version of Trench Waveguide
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
processing to be successful.
present a difficulty.
Refilling deep (<70 pm) trenches may
New thick polyim ides and spin-on oxides are
available, however, as a possible means of overcoming this problem.
From a circuits point of view, the main concern with Trench
W aveguide is how to remove heat from the substrate when there is no
metallization on the backside.
Particularly in power applications, drawing
heat away from the active devices is crucial.
In MICs with no backside
metallization, contact to a heat sink may be difficult. However, the utility of
the Trench Waveguide is in no way compromised by depositing a backside
ground piane.
As is the case for CFW, mis aouiiionai ground plane is
believed only to simply further reduce the characteristic impedance.
The final area for continued research is an exhaustive study of the
effect on Zo of trench depth, b, and d, the extent of the metal along the
trench sidewall.
Design curves are required in order to determ ine the
Trench Waveguide geometry needed to realize a particular desired ZoFor planar transmission lines Zo is usually specified relative to a geometric
aspect ratio.
Similarly for Trench Waveguide.
Additional considerations
are involved, however, because not only are there a conductor width, s,
and gap spacing, w, as in CPW, but trench depth, b, and the metal
coverage,d. These four dimensions, s,w, d, and b, are sure to have varied
effects on the Zo of the Trench Waveguide.
As with CPW, a horizontal
shape factor,
k = s / s+ 2 w
has been adopted.
Simulations could be done over a useful range of k.
To avoid cumbersome design curves, a different set could be produced for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
each new fixed trench depth and metal coverage.
This com bination is
defined by a new, additional vertical shape factor,
g = d / b.
Preliminary considerations suggest that Zo will be sensitive to d. This is
because increasing d increases capacitance, thus lowering the associated
impedance. The same effect is achieved by narrowing the trench width or
decreasing w. Families of curves for er =1.0, 10.0, and 11.8 corresponding
to air, alumina and silicon, respectively, would provide Trench Waveguide
design information in a format consistent with that currently used by MMIC
designers. Analyzing how sensitive Zo is to trench shape (straight versus
sloped sidewalls) will be another goal of on-going work.
Also to be
pursued is the effect of these parameters as well as metal thickness, t, on
attenuation, dispersion and field confinement.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A P PE N D IX
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
Table A.1. Trench Waveguide Silicon Fabricated Component Data.
10:46:10
FREQUENCY
MHz
100.0000
600.0000
1100.0000
1600.0000
2100.0000
2600.0000
3100.0000
3600.0000
4100.0000
4600.0000
5100.0000
5600.0000
6100.0000
6600.0000
7100.0000
7600.0000
8100.0000
8600.0000
9100.0000
9600.0000
10100.0000
10600.0000
11100.0000
11600.0000
12100.0000
12600.0000
13100.0000
13600.0000
14100.0000
14600.0000
15100.0000
15600.0000
16100.0000
16600.0000
17100.0000
17600.0000
18100.0000
TRENCH WAVEGUIDE
RETURN-LOSS-IN
SI 1
DB
ANG
9.88
10.11
10.80
11.86
12.84
12.52
13.34
14.38
15.26
15.98
16.29
16.39
16.11
15.73
15.31
14.88
14.62
14.24
13.96
13.80
13.55
13.40
13.29
13.16
13.09
13.06
12.87
12.87
12.82
12.84
12.88
12.87
12.88
12.92
13.03
13.14
13.27
178.4
169.8
163.9
160.6
164.6
162.9
159.6
160.4
163.8
169.0
174.4
-179.6
-174.3
-170.5
-167.5
-166.4
-165.7
-16 5 . 0
-165.1
-165.4
-16 5 . 6
-166.0
-166.4
-166.4
-167.1
-16 7 . 6
-16 8 . 5
-16 9 . 0
-17 0 . 2
-17 0 . 9
-17 1 . 8
-172.4
-17 3 . 0
-17 4 . 7
-17 5 . 7
-17 7 . 2
-177.9
LOSS-■FORWARD
S21
DB
ANG
20. 80
17.12
14.18
12.21
11. 75
12. 23
11.15
10. 53
10.19
9. 94
9.97
9.91
9. 98
10.22
10.39
10.86
11.15
11.09
11.31
11.59
11.61
12.24
12. 26
12.32
12.53
12. 14
12.22
12. 11
12.48
12.70
12.43
11.77
11.61
1 1 . 70
12.23
12.22
11 .94
9. 1
31. 7
33.2
27. 6
17.4
20.6
20.8
17.3
13.9
10.3
7.3
4.8
1. 8
-.8
-2.2
-3.9
-2.2
-2.9
-4.5
-2.9
-4.0
-3.6
-2. 1
-.1
1.9
4. 8
1.5
2. 7
3.3
5.2
9.0
7.7
7.2
4.2
4.8
7.0
8.6
#4 — 2
2 Sep
LOSS -REVERSE
S12
DB
RNG
20.71
17.06
14.12
12. 16
1 1. 70
12.19
11.16
10. 49
10. 08
9. 87
9. 85
9. 85
9. 98
10.16
10. 37
10. 67
10. 90
11.01
11. 33
11 .57
11 ,65
12.01
12.04
12.22
12.39
12.31
12.23
12.28
12.12
12.27
12. 16
11 .97
12.01
11.80
12.00
11.94
11.85
9. 1
31. 9
33.4
27.9
17.8
20.9
21.4
13.2
14.6
10.9
3.0
5. 1
2.4
.5
-1 . 1
-2.3
-2. 1
-2.9
-3. 1
-2.4
-2.6
-2.1
-.3
-. 1
1. 7
3.3
3.7
5.4
5.2
5.8
7.2
7.8
7.3
7.7
5.8
5. 7
3.2
1987
RETURN-LOSS
S22
DB
ANG
1.10
1. 40
1. 93
2. 64
3.17
2. 84
3.21
3. 66
3. 92
4. 08
4.18
4.21
4. 22
4.16
4.07
4. 05
3. 95
3.98
4.01
3.97
4.09
4. 16
4.22
4.43
4.53
4.81
5.17
5.36
5.71
5.94
6.20
6.68
6.96
7.27
7.31
7. 12
7.28
178. 9
1 74. 0
171.0
169. 1
172. 8
172. 5
170.6
171.1
172. 1
174. 1
175.3
176.6
177. 9
178. 1
178.9
178. 8
178. 7
179. 0
177. 9
177.7
177. 5
176. 1
176.2
175.4
174.5
175. 1
174.0
174. 1
174.4
174.6
177.3
179. 1
-179.3
-177.8
-176.6
-172.8
-168.8
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
l a r g e s c a l e w a v e g u i d e model i n a i r
0.0
I
I
|
0.5
I
1 —I—1■I
0.0
I
I
l-l
I
1.0
1.5
2.0
|
I
I
I
I
|
I
I
(
t
j
I
I
II
I
I
1
I
I
I
I
I i
0.5
1.0
I T "l
1.5
I
I
I
|
I
i I
I
I
I
2.0
Figure A.1. Plot of equipotential contours for large scale Trench
Waveguide in air. (w=0.16 cm, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
l a r g e s c a l e w a v e g u i d e model
0.0
0.2
0.4
0.6
0.8
in a ir
1.0
.5
0.5
.0
0.0
-0.5
-0.5
.0
-
.5
-1.5
0.0
0.2
0.4
0.6
0.8
1. 0
1.0
Figure A.2. Plot of Figure A.1 magnified by 2X.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
l a r g e s c a l e w a v e g u i d e model i n a i r
0.0
0.5
1.0
1.5
2.0
1
0
-1
-I
-2
-2
0. 0
0.5
1.0
1.5
2. 0
Figure A.3. Plot of equipotential contours and field lines for largescale
Trench Waveguide in air. (w=0.16 cm, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
l a r g e s c a l e w a v e g u i d e model
0.0
0.2
0.4
0.6
0.8
in a i r
1.0
0.5
0.5 -
0.0
-
-
- -0.5
-0.5 -
-
1.0
-
-
\
-1.5
0.0
0.0
0.2
0.4
0.6
-
1.0
- -1.5
0.8
1.0
Figure A.4. Plot of Figure A.3 magnified by 2X
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l a r g e s c a l e w a v e g u i d e model w i t h d i e l e c t r i c
0.0
0.5
1.0
1.5
2.0
-1
-1
-2
-2
0.0
0.5
1. 0
1.5
2.0
Figure A.5. Plot of equipotential contours for large scaleTrench
Waveguide on Silicon.(w=0.16 cm, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
l a r g e s c a l e w a v e g u i d e model w i t h d i e l e c t r i c
0.0
0.2
0.4
0.6
0.8
1.0
0.5
0.0
-0.5
-
1.0
-0.5
-
-
-1.5
1.0
-1.5
0.0
0.2
0.4
■■■i ‘ » ■i ■ ■■
0.6
0.8
1.0
Figure A.6 . Plot of Figure A.5 magnified by 2X.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
large s c a le
0.0
w a v e g u i d e model w i t h d i e l e c t r i c
0.5
1.0
1.5
2.0
-1
-2
-1
-2
-
0.0
0.5
1.0
1.5
2.0
Figure A.7. Plot of equipotentiat contours and field lines for large scale
Trench Waveguide on Silicon. (w=0.16 cm, s=0.325 cm, d=1.27 cm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
large scale waveguide model with dielectric
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0 .5
0. 0
-0.5
-
1. 0
-1.5
Figure A.8 . Plot of Figure A.7 magnified by 2X.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
o
o
o
o
o
o
OJ
o
0
OJ
1
o
o
o
o
o
o
o
o
CD
(JO
o
o
'T
o
o
T
o
o
o
o
OJ
OJ
w aveguide
tre n ch
o
o
o
OJ
o
o
o
o
0
OJ
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A.9. Plot of equipotential contours for Trench Waveguide in
air.(s=250 pm, w=37 pm, b=95 pm, d=40 pm).
00
in
a ir
00
136
0
in
1
o
in
o
o
i
o
in
o
o
in
o
o
in
o
o
m
o
o
m
o
o
cu
o
o
o
o
o
o
OJ
o
in
o
o
in
i
o
o
in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A.10. Plot of Figure A.9 magnified by 2X.
tre n c h
w aveguide
in
a ir
o
o
Figure A.11. Plot of equipotential contours and field lines for Trench
Waveguide in air. (s=250 pm, w=37 pm, b=95 pm, d=40 pm
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
o
o
m
ID
o
o
''T
o
o
•<r
O
O
cn
o
o
on
(0
03
■a
3
O)
03
>
It)
2
U
c o
o
03
c. OJ
-M
o
o
o
o
o
o
OJ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A. 12. Plot of Figure A.11 magnified by 2X.
o
o
139
o
0
OJ
1
o
o
OJ
I I I
i "i i 1r
i i i i
I
I" I T
o
o
O
o
CO
CO
o
o
o
o
CD
CD
CO
c
o
(D
TD
•H
D
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(U o
O
O
rr
>
to
2
U
c
(D
C_
o
o
O
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OJ
OJ
J -J -
o
o
OJ
o
o
I I 1.
o
o
I I I I 1 I
o
0
OJ
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A. 13. Plot of equipotential contours for Trench Waveguide on
Silicon.(s=250 pm, w=37 pm, b=95 pm, d=40 pm).
o
o
o
o
140
o
o
o
in
o
o
in
o
o
in
o
o
■'T
o
o
O
o
■O m
•H
D
O)
o
o
m
CO
c
o
0
0
>
0
2
.C
o
c
0
c
-p
o
o
o
o
OJ
CM
o
o
t1
o
o
o
in
0
in
1
o
o
o
in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A.14. Plot of Figure A.13 magnified by 2X.
o
in
0
in
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
waveguide
on
Si
Figure A.15. Plot of equipotential contours and field lines for
TrenchWaveguide on Silicon. (s=250 pm, w=37 pm, b=95 pm, d=40 pm)
trench
141
142
0
in
1
o
in
o
o
o
in
o
o
in
o
o
■'T
co
c
0
(U O
o
X3
•cl
O
o
00
00
3
01
CD
>
(0
3
.C
u
c o
0) 0
c_ 01
o
o
OJ
-p
o
o
o
o
o
in
o
o
in
i
o
o
in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A. 16. Plot of Figure A. 15 magnified by 2X.
o
o
•<T
LITERATURE CITED
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
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