Propagation Characteristics of Coplanar Waveguides at Subterahertz Frequencies
by Jingjing Zhang
Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
Supervised by Professor Thomas Y. Hsiang
Department of Electrical and Computer Engineering The College School of Engineering and Applied Sciences
University of Rochester Rochester, New York
2007
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Curriculum Vitae
Bom in Hebei, China in 1973, the author attended the Beijing Institute o f Technology
from 1991 to 1995 and graduated with a Bachelor o f Science degree. She came to the
University o f Rochester in the Fall o f 2000 and began graduate studies in Electrical
Engineering, acting as a teaching assistant in 2000 and 2001. She received a Frank
Horton Fellowship from 2001 to 2006. Under the direction of Professor Thomas Y.
Hsiang, she pursued her research in propagation characteristics of coplanar
waveguides at subterahertz frequencies, receiving a Master of Science degree from
the University of Rochester in 2002.
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Acknowledgments
I would like to thank my thesis advisor, Professor Thomas Y. Hsiang, for his
guidance and support throughout my study and research at the University of
Rochester. I have greatly profited from his technical sharpness, intellectual creativity,
and many writing lessons.
I wish to express my thanks to Dr. William R. Donaldson for his invaluable
suggestions and publication support, Professor Marc J. Feldman for his illuminating
comments and advice on my research topic, and Professor Hui Wu for kindly
allowing me to access his computing resources. I am grateful to Dr. Ross A. Speciale
for his mentoring on my research and life in general. I also offer thanks to Dr.
Xuemei Zheng, Dr. Jianliang Li, and Yunliang Zhu for countless helpful discussions.
I thank the Laboratory o f Laser Energetics for awarding me a Frank Horton
Fellowship.
I dedicate this dissertation to my parents, my husband, my daughter, and my sister.
Their support and encouragement are always my source o f power. I particularly thank
my husband for his endless patience and constructive suggestions on my research and
my daughter for her love.
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Abstract
In this thesis research, I have developed and used numerical techniques to study and
model the propagation characteristics of coplanar waveguides (CPWs) over a broad
frequency range, from about 10 GHz to nearly 1 THz. This research is important as
CPWs gain broader applications in integrated circuits (ICs), both as interconnects and
as passive circuit-elements. My research is also timely because experimentally
measured waveguide properties have become available in recent years that allow
comparison with theoretical studies.
Two types of CPWs were investigated. The first type uses wide ground lines
(WG) and closely approximates an ideal waveguide, which contains semi-infinite
substrate thickness and ground-plane widths and has been analytically studied using
conformal mapping. These WG CPWs are thus important as a testing venue where
our numerically modeled waveguide characteristics are compared with both
experiments and closed-form analysis.
The second type o f CPWs uses narrow ground lines (NG) and is the practical
choice in IC applications that have severe “real estate limitation,” i.e. where minimal
chip area can be assigned to passive circuit components such as interconnects. These
waveguides have, until this work, only been qualitatively studied. In this thesis, their
properties will be thoroughly investigated and modeled.
In one particular group o f waveguides, made on a GaAs substrate, my work is
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compared with experiments for both the WG and NG waveguides. The close
comparison lends strong support to the validity of my approach. The modeling is
then extended to the practically important waveguides made on a silicon substrate. I
will detail how parameters such as waveguide ground-plane widths and lateral line
dimensions change the high-frequency characteristics and how they can be designed
to improve circuit performance. Finally, some directions for future studies are
discussed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents
Curriculum V itae ...... iii Acknowledgments...... iv Abstract...... v Table of Contents ...... vii List of Tables ...... ix List of Figures...... x Glossary of Symbols ...... xv 1 Introduction ...... 1 References...... 4 2 Coplanar Waveguide Theory ...... 7 2.1 Introduction ...... 7 2.2 Quasi-static Analysis ...... 9 2.3 High-frequency Propagation ...... 13 2.4 Distributed Circuit Analysis...... 16 2.5 Dispersion ...... 18 2.6 Attenuation ...... 20 References...... 26 3 Ultrafast Optoelectronic Characterization ...... 30 3.1 The Electro-optic Sampling System ...... 30 3.2 Fabrication and Measurement ...... 32 3.3 T ime-domain Analysis ...... 35 3.4 Frequency-domain Analysis ...... 38 References...... 40 4 Full-Wave Analysis ...... 42 4.1 Galerkin’s Method in the Spectral Domain ...... 42 4.2 Software Simulations ...... 44 4.3 Microwave Network Analysis ...... 50 References...... 55 5 Propagation Characteristics ...... 58 5.1 Attenuation ...... 59 5.1.1 The Effects of Ground-plane Width ...... 59 5.1.2 The Effects of Lateral Line Dimensions ...... 65 5.2 Dispersion ...... 72 5.2.1 The Effects of Ground-plane Width ...... 72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2.2 The Effects of Lateral Line Dimensions ...... 78 References...... 85 6 Distributed-Element Circuit Model ...... 86 6.1 Extraction of Frequency-dependent Transmission Line Parameters ...... 87 6.2 The Distributed-element Circuit Model of CPWs on GaAs ...... 90 6.2.1 Resistance ...... 91 6.2.2 Conductance ...... 92 6.2.3 Capacitance ...... 92 6.2.4 Inductance ...... 93 6.3 The Distributed-element Circuit Model of CPWs on Silicon ...... 99 6.3.1 Resistance ...... 99 6.3.2 Conductance ...... 105 6.3.3 Capacitance ...... 110 6.3.4 Inductance ...... 116 References...... 121 7 Summary...... 122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables
Table 4.1 Geometrical and electrical parameters of CPWs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures
Figure 2.1 Cross-sectional view o f the CPWs (a) with wide ground planes and (b) with narrow ground planes ...... 8 Figure 2.2 Conformal transformation planes for a wide-ground CPW with an infinitely thick dielectric substrate: (a) zy-plane and (b) z-plane ...... 11 Figure 2.3 Electromagnetic field distribution on the cross section of a CPW. Solid lines are electric field lines and dotted lines are magnetic field lines...... 14 Figure 2.4 Electromagnetic energy transmitted along a CPW ...... 15 Figure 2.5 Distributed circuit model of a CPW ...... 17 Figure 2.6 Effective permittivity of a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters ...... 19 Figure 2.7 Conductor loss o f a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters ...... 22 Figure 2.8 Radiation loss o f a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters ...... 24 Figure 3. 1 Schematic configuration of an electro-optic sampling system ...... 31 Figure 3. 2 Demonstration of subpicosecond electro-optic sampling on CPW 34 Figure 3. 3 Electrical pulses probed at 0.25, 1, 3, and 5 mm on two 50 -pm CPWs with ground-plane width of 500 (solid) and 50 / im (dotted) ...... 35 Figure 3. 4 Electrical pulses probed at 0.5, 1, 3, and 5 mm on a 10 -jum CPW with ground-plane width of 500 jum (solid) and electrical pulses probed at 0.25, 1, 3, and 5 mm on a 10 -fim CPW with ground-plane width of 10 fim (dotted) ...... 36 Figure 3. 5 Electrical pulses probed at 0.25, 1, 3, and 5 mm on \0-jum (solid) and 50- fim (dotted) CPWs with narrow ground planes ...... 37 Figure 3.6 A spectral representation of some time-domain data measured on a 10- jum CPW with narrow ground planes ...... 39 Figure 4. 1 Three-dimensional view of the CPW in the project editor window 45 Figure 4. 2 Equivalent two-port network of a CPW of length 1...... 51 Figure 5. 1 Simulated and experimental attenuation o f CPWs on a GaAs substrate with a 50-jum center conductor. Attenuation sim-wg refers to simulated attenuation of the CPW with wide ground planes. Attenuation exp-wg refers to experimental attenuation of the CPW with wide ground planes. Attenuation sim-ng refers to simulated attenuation of the CPW with narrow ground planes. Attenuation exp-ng refers to experimental attenuation of the CPW with narrow ground planes ...... 60 Figure 5. 2 Simulated and experimental attenuation o f CPWs on a GaAs substrate with a 10-jum center conductor. Attenuation sim-wg refers to simulated attenuation of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the CPW with wide ground planes. Attenuation exp-wg refers to experimental attenuation of the CPW with wide ground planes. Attenuation sim-ng refers to simulated attenuation of the CPW with narrow ground planes. Attenuation exp-ng refers to experimental attenuation o f the CPW with narrow ground planes ...... 61 Figure 5. 3 Simulated attenuation of aluminum CPWs on a HR Si substrate with a 10-pim and 20 -/um center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes ...... 63 Figure 5. 4 Simulated attenuation o f copper CPWs on a HR Si substrate with a 10- jxm and 20-jum center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes ...... 63 Figure 5. 5 Simulated attenuation o f aluminum CPWs on a LR Si substrate with a 10-jum and 20 -jum center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes ...... 64 Figure 5. 6 Simulated attenuation o f copper CPWs on a LR Si substrate with a 10 -/um and 20-/um center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes ...... 64 Figure 5. 7 Simulated and experimental attenuation o f CPWs on a GaAs substrate with narrow ground planes. The lateral line dimensions are 10 pm and 50 pm, respectively. Attenuation sim-10 refers to simulated attenuation of the CPW with a 10-pm center conductor. Attenuation exp-10 refers to experimental attenuation of the CPW with a 10-pm center conductor. Attenuation sim-50 refers to simulated attenuation of the CPW with a 50-pm center conductor. Attenuation exp-50 refers to experimental attenuation of the CPW with a 50-pm center conductor ...... 65 Figure 5. 8 Simulated attenuation o f aluminum CPWs on a HR Si substrate with wide ground planes ...... 66 Figure 5. 9 Simulated attenuation o f aluminum CPWs on a HR Si substrate with narrow ground planes...... 67 Figure 5. 10 Simulated attenuation o f copper CPWs on a HR Si substrate with wide ground planes ...... 68 Figure 5. 11 Simulated attenuation o f copper CPWs on a HR Si substrate with narrow ground planes ...... 68 Figure 5. 12 Simulated attenuation o f aluminum CPWs on a LR Si substrate with wide ground planes ...... 70 Figure 5. 13 Simulated attenuation o f aluminum CPWs on a LR Si substrate with narrow ground planes ...... 70 Figure 5. 14 Simulated attenuation of copper CPWs on a LR Si substrate with wide ground planes ...... 71 Figure 5. 15 Simulated attenuation o f copper CPWs on a LR Si substrate with narrow ground planes ...... 71 Figure 5.16 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with a 50-pm center conductor. e eff,Sim-wg and s eff,exp-wg refer to simulated and experimental effective permittivity of the wide-ground CPW, respectively. s eff,sim-ng and Seff,exp-ng refer to simulated and experimental effective permittivity o f the narrow- ground CPW, respectively ...... 75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5. 17 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with a 10-pm center conductor. seff,sim-wg and 8eff,exp-wg refer to simulated and experimental effective permittivity of the wide-ground CPW, respectively. seff,Sim-ng and Sen,exp-ng refer to simulated and experimental effective permittivity o f the narrow- ground CPW, respectively ...... 75 Figure 5.18 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with a \0-jum and 20-/um center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes...... 76 Figure 5. 19 Simulated effective permittivity o f copper CPWs on a HR Si substrate with a 10-jum and 20 -/urn center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers.to narrow-ground planes ...... 76 Figure 5. 20 Simulated effective permittivity o f aluminum CPWs on a LR Si substrate with a 10-jum and 20-jum center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes...... 77 Figure 5. 21 Simulated effective permittivity o f copper CPWs on a LR Si substrate with a 10-/um and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes ...... 77 Figure 5. 22 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with narrow ground planes. The lateral line dimensions are 50 pm and 10 pm, respectively. seff,sim-50 and £eff,exp-50 refer to simulated and experimental effective permittivity o f the CPW with a 50-pm center conductor, respectively. seff,Sim-io and seff exp-io refer to simulated and experimental effective permittivity o f the CPW with a 10-pm center conductor, respectively ...... 79 Figure 5. 23 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with wide ground planes ...... 80 Figure 5. 24 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with narrow ground planes ...... 80 Figure 5. 25 Simulated effective permittivity o f copper CPWs on a HR Si substrate with wide ground planes ...... 81 Figure 5. 26 Simulated effective permittivity o f copper CPWs on a HR Si substrate with narrow ground planes ...... 81 Figure 5. 27 Simulated effective permittivity of aluminum CPWs on a LR Si substrate with wide ground planes ...... 83 Figure 5. 28 Simulated effective permittivity of aluminum CPWs on a LR Si substrate with narrow ground planes ...... 83 Figure 5. 29 Simulated effective permittivity o f copper CPWs on a LR Si substrate with wide ground planes ...... 84 Figure 5. 30 Simulated effective permittivity o f copper CPWs on a LR Si substrate with narrow ground planes ...... 84 Figure 6. 1 The two-port network of a transmission line ...... 87 Figure 6. 2 The distributed-element circuit model of a transmission line ...... 89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6. 3 Unit-length resistance R o f CPWs on a GaAs substrate, extracted from S- parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions o f the CPWs ...... 95 Figure 6. 4 Unit-length conductance G o f CPWs on a GaAs substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions o f the CPWs ...... 96 Figure 6. 5 Unit-length capacitance C o f CPWs on a GaAs substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions o f the CPWs ...... 97 Figure 6. 6 Unit-length inductance L o f CPWs on a GaAs substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions o f the CPWs ...... 98 Figure 6. 7 Unit-length resistance R o f aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 p m , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs...... 101 Figure 6. 8 Unit-length resistance R o f copper CPWs on a HR Si substrate, extracted from S-parameters. w g and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs... 102 Figure 6. 9 Unit-length resistance R o f aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs...... 103 Figure 6. 10 Unit-length resistance R o f copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs...... 104 Figure 6. 11 Unit-length conductance G o f aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 106 Figure 6. 12 Unit-length conductance G o f copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs...... 107 Figure 6. 13 Unit-length conductance G o f aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs. 108 Figure 6. 14 Unit-length conductance G o f copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs. 109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x iv
Figure 6. 15 Unit-length capacitance C o f aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs...... 112 Figure 6. 16 Unit-length capacitance C o f copper CPWs on a HR Si substrate, extracted from S-parameters. w g and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 113 Figure 6. 17 Unit-length capacitance C o f aluminum CPWs on a LR Si substrate, extracted from S-parameters. w g and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 114 Figure 6 .1 8 Unit-length capacitance C o f copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 115 Figure 6. 19 Unit-length inductance L o f aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 117 Figure 6. 20 Unit-length inductance L o f copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions o f the CPWs. 118 Figure 6. 21 Unit-length inductance L o f aluminum CPWs on a LR Si substrate, extracted from S-parameters. w g and ng refer to wide ground and narrow ground, respectively. 5 pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 119 Figure 6. 22 Unit-length inductance L o f copper CPWs on a LR Si substrate, extracted from S-parameters. w g and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. 120
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Glossary of Symbols
a Attenuation constant
6Cc Conductor loss
ad Dielectric loss
ar Radiation loss
P Phase constant
y Propagation constant
A Cell size of CPW simulation
Ss Skin depth
£ Complex permittivity o f the dielectric material and s = s '- js "
£ Real part o f complex permittivity o f dielectric material
£ Imaginary part o f complex permittivity o f dielectric material
£o Permittivity of free space
£ eff Effective relative permittivity o f a CPW
eq Effective relative permittivity o f a CPW at the quasi-static limit
£r Relative permittivity of dielectric material
Xo Wavelength of the electromagnetic wave propagating along a CPW
M Complex permittivity of the dielectric material and fi = ju'-jju"
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H Real part of complex permeability of dielectric material
H Imaginary part o f complex permeability o f dielectric material
Ho Permeability of free space
Hr Relative permeability of dielectric material
n Complete elliptical integral o f the third kind
a Conductivity of conductor
Odieiec Conductivity o f dielectric material
if/ Radiation angle
io Angular frequency
A Element of the ABCD matrix o f a two-port network
A A constant in the conformal transformation
a Coordinate in z- plane corresponding to a/ in z/-plane
a ; =S/2
B Element of the ABCD matrix o f a two-port network
b Coordinate in z- plane corresponding to b] in z/-plane
b, =(S+2W)/2
C abcd Element of the ABCD matrix o f a two-port network
C Capacitance per unit length of a CPW
Cair Partial capacitance per unit length of a CPW in the absence o f the
dielectric substrate
Cdieiec Partial capacitance per unit length o f the substrate o f a CPW
c Speed of light in free space
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ci =(S+2W+2Wg)/2
D Element of the ABCD matrix o f a two-port network
E Electric field
Ex x-component of electric field
/ Frequency
f e Cut-off frequency of the lowest-order TE mode propagating on a CPW
G Conductance per unit length o f a CPW
H Magnetic field
h Thickness o f the substrate o f a CPW
Hy y-component of magnetic field
HZiair z-component of magnetic field in air
Hz,dieiec z-component of magnetic field in dielectric
I Current
j Imaginary unit
K(k) Complete elliptical integral of the first kind
k =S/(S+2W)
k' =Vl- k 2
-b\ ci F y q - a ^ x
V
kw Propagation constant of the guided CPW mode
ks Propagation constant of the substrate modes
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L Inductance per unit length of a CPW
I Length o f a CPW
m Coordinate in z- plane corresponding to m / in z/-plane
mi Coordinate corresponding to the left boundary of the cross section of a
wide-ground CPW in z/-plane
M i A constant in the expression for current o f a CPW
My A constant in the expression for voltage of a CPW
n Coordinate in z- plane corresponding to /?/ in z/-plane
nj Coordinate corresponding to the right boundary of the cross section of
a wide-ground CPW in z/-plane
Ni A constant in the expression for current o f a CPW
Nv A constant in the expression for voltage of a CPW
p A empirical parameter in the expression for £efjo fa CPW
P' A geometry dependent parameter in the expression for ac o f a CPW
q An empirical parameter in the expression for sejf of a CPW
R Resistance per unit length of a CPW
r A geometry dependent parameter in the expression for ar o f a
conductor-backed narrow-ground CPW
ri A geometry dependent parameter in the expression for ar of a
conductor-backed narrow-ground CPW
r2 A geometry dependent parameter in the expression for ar of a
conductor-backed narrow-ground CPW
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Rs Surface resistivity
S Width of the center conductor of a CPW
Si i Input port voltage reflection coefficient o f a two-port network
Sn Reverse voltage gain of a two-port network
5 21 Forward voltage gain o f a two-port network
5 22 Output port voltage reflection coefficient o f a two-port network
t Moment in time
tan<5 Loss tangent of the dielectric substrate of a CPW
tc Thickness of the conductors of a CPW
t0 Thickness of the oxide of a CPW
V Voltage
vi+ Incident wave in input port of a two-port network
v f Reflected wave in input port o f a two-port network
V2 Incident wave in output port o f a two-port network
V2" Reflected wave in output port of a two-port network
vp Phase velocity of the electromagnetic wave propagating along a CPW
W Width of separation between the center conductor and ground planes
o f a CPW
w Width of a conductor
Wg Ground-plane width of a CPW
Zc Characteristic impedance
Zo Reference input and output impedance of a CPW
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z Coordinate in z-plane, which is the objective plane of conformal
mapping
zi Coordinate in z/-plane, which is the original plane of conformal
mapping
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CHAPTER 1
Introduction
As a basic building block of monolithic microwave integrated circuits (MMICs),
transmission lines play a key role in microwave technology. A transmission line
structure is generally utilized as either the passive component or the interconnect in a
MMIC. Over the past decade, MMICs constructed with coplanar waveguides (CPWs)
have experienced a rapidly growing number o f applications in terrestrial
telecommunication systems, medical electronics, radar and remote sensing, intelligent
transportation systems, etc. CPWs have also attracted a significant amount o f
attention because of a number of advantages they offer over other transmission lines,
especially for MMICs operating at millimeter-wave frequencies. Instead of grounding
three-terminal and shunt-mounted components with via-holes, which introduce
inductance, CPW-based circuits can conveniently be grounded at the substrate edge to
improve circuit performance. CPWs also improve circuit performance by decreasing
the radiation loss and dispersion that dominate as the frequency increases. Being
more flexible to MMIC design, a CPW provides varying characteristic impedance by
keeping a constant cross section and changing the center conductor width. Moreover,
in the newly developed flip-chip interconnect system, CPW geometry most
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effectively minimizes the flip-chip mounting effect on the performance of the chip. In
addition, CPWs offer low fabrication cost, low assembly cost, simplified packaging,
and high circuit density.
The performance of MMICs is affected by the propagation characteristics of the
transmission lines. Indeed, due to the imperfection of materials—including the finite
conductivity o f the conductors and losses in the dielectric—and radiation o f energy
into the substrate, the amplitude o f microwave signals is always attenuated after
traveling any distance along practical CPWs. These losses must be controlled when
working with CPWs, especially those used as interconnects in MMIC. For transverse
electric (TE) and transverse magnetic (TM) wave propagation, the phase velocity is
different for different frequencies. The individual frequency components will not
maintain their original phase relationships as they propagate down the CPW, which
only supports TE and TM waves. This difference of phase velocities gives rise to
signal distortion, the so-called “dispersion effect” [1]. For MMICs operating at
relatively low frequencies up to several gigahertz, the effects of attenuation and
dispersion in the CPW used for interconnecting are negligible. With the rapid
progress in ultrafast components and devices, however, the operating frequency range
o f MMICs has been dramatically increased to subterahertz to fulfill the requirements
o f current microwave customers. In this frequency range, attenuation and dispersion
in CPWs should be studied for the purpose of fashioning designs that most closely
maintain the phase and shape o f the signal. In order to provide guidelines for practical
MMIC design, propagation models are needed for variable CPW structures. In
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particular, attenuation and dispersion at high frequencies up to subterahertz have
become an essential topic for facilitating more complete modeling efforts.
With the extensive exploitation of MMIC technology, CPWs have been widely
studied theoretically, experimentally, and numerically. Grischkowsky et al.
experimentally demonstrated that the radiation loss dominates over 200 GHz for a
coplanar transmission line [2]. Many of the subsequent studies have outlined the
propagation characteristics of CPWs up to subterahertz frequencies [3-6], while
previous studies were primarily limited to the low frequency range [7, 8]. However,
most research so far has been conducted on CPWs with wide ground planes.
Although CPWs with narrow ground planes provide important advantages for
microwave applications such as flexibility, low parasitics, and area saving, they have
not been sufficiently analyzed. A frequency-domain fmite-difference simulation [9]
has been carried out, and analytic formulae [10] have been derived to model radiation
and dispersion characteristics of conductor-backed CPW with narrow ground planes.
However, their results were not verified by experiments; furthermore, CPWs with a
finitely thick substrate but without back-side metallization still need to be
investigated. CPWs with narrow ground planes have also been studied elsewhere [11-
13], but either their measurement needed to be expanded to a higher frequency range,
or their dispersion and radiation characteristics were not fully considered.
This thesis represents an investigation of the basics of wave propagation in CPWs
with wide and narrow ground planes over a broad range of frequencies up to
subterahertz. Experimental results of propagation characteristics of CPWs have been
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previously obtained by using a subpicosecond electro-optic measurement technique
[14]. I have simulated the attenuation and dispersion of a variety of CPWs by a full-
wave analysis, and the simulation results are compared and verified by the
experimental data. I have also analyzed the effects o f ground-plane width and lateral
line dimensions on attenuation and dispersion characteristics in the subterahertz
frequency domain. Conductor loss dominates the attenuation of the CPWs at low
frequencies, while radiation loss becomes dominant at high frequencies. The CPWs
with wide-ground planes and larger lateral line dimensions suffer less conductor loss.
The CPWs with narrow-ground planes and smaller lateral line dimensions suffer less
radiation loss and dispersion. On the basis of the study of propagation characteristics
of CPWs, I also incorporated the results and analysis of simulation and experiments
into a distributed-element circuit model. For the purpose of providing guidelines for
MMIC designers, detailed design considerations for CPW-based MMICs are
presented.
References
[1] D. M. Pozar, Microwave Engineering. Hoboken, NJ: John and Wiley & Sons,
2005, ch. 3.
[2] D. Grischkowsky, I. N. Duling III, J. C. Chen, and C.-C. Chi, “Electromagnetic
shock waves from transmission lines,” Phys. Rev. Lett., vol. 59, no. 15, pp. 1663-
1666, Oct. 1987.
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[3] M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz
attenuation and dispersion characteristics of coplanar transmission lines”, IEEE
Trans. Microwave Theory Tech., vol. 39, no. 6, pp. 910-915, Jun. 1991.
[4] S. Gupta, J. F. Whitaker, and G. A. Mourou, “Subpicosecond pulse propagation
on coplanar waveguides: experiment and simulation”, IEEE Microwave and
Guided Wave Letters, vol. 1, no. 7, pp. 161-163, Jul. 1991.
[5] M. Tsuji, H. Shigesawa, and A. A. Oliner, “New interesting leakage behavior on
coplanar waveguides of finite and infinite widths,” IEEE Trans. Microwave
Theory Tech., vol. 39, no. 12, pp. 2130-2137, Dec. 1991.
[6] J. Zehentner and J. M achac,, “Properties o f CPW in the sub-mm wave range and
its potential to radiate,” in Microwave Symposium Digest., 2000 IEEE MTT-S
International, vol. 2, pp. 1061-1064, Jun. 2000.
[7] R. W. Jackson, “Considerations in the use of coplanar waveguide for millimeter-
wave integrated circuits”, IEEE Trans. Microwave Theory Tech., vol. 34, no. 12,
pp. 1450-1456, Dec. 1986.
[8] M. Riaziat, R. Majidi-ahy, and I. Feng, “Propagation modes and dispersion
characteristics of coplanar waveguides”, IEEE Trans. Microwave Theory Tech.,
vol. 38, no. 3, pp. 245-251, Mar. 1990.
[9] W. Heinrich, F. Schnieder, and T. Tischler, “Dispersion and radiation
characteristics of conductor-backed CPW with finite ground width”, in
Microwave Symposium Digest., 2000 IEEE MTT-S International, vol. 3, pp. 1663-
1666, Jun. 2000.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6
[10] F. Schnieder, T. Tischler, and W. Heinrich, “Modeling dispersion and
radiation characteristics of conductor-backed CPW with finite ground,” IEEE
Trans. Microwave Theory Tech., vol. 51, no. 1, pp. 137-143, Jan. 2003.
[11] N. S. Kuek and S. Uysal, “Investigation of discontinuous coplanar waveguide
lines with finite ground planes”, in 1997 Asia-Pacific Microwave Conference
Proceedings, vol. 3, pp. 969-972, Dec. 1997.
[12] G. Ghione and M. Goano, “The influence of ground-plane width on the ohmic
losses of coplanar waveguides with finite lateral ground planes”, IEEE Trans.
Microwave Theory Tech., vol. 45, no. 9, pp. 1640-1642, Sep. 1997.
[13] G. E. Ponchak, I. K. Itotia, and R. F. Drayton, “Propagation characteristics o f
finite ground coplanar waveguide on Si substrates with porous Si and polyimide
interface layers”, in 33rd European Microwave Conference Proceedings, vol. 2,
pp. 45-48, Oct. 2003.
[14] S. Alexandrou, “The bent coplanar waveguide at sub-terahertz frequencies,”
Ph.D. dissertation, Department of Electrical and Computer Engineering,
University of Rochester, Rochester, NY, 1994.
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CHAPTER 2
Coplanar Waveguide Theory
2.1 Introduction
Coplanar Waveguides (CPWs) are a family of transmission lines consisting of a
center conductor strip and two ground conductor planes with variable widths. All
three conductors are placed on the same side o f a dielectric substrate, as shown in
Figure 2.1.
The ideal CPW structure has a semi-infinitely thick substrate and two semi-
infinitely wide ground planes. The actual implementation of a CPW mainly modifies
the ideal structure in the following two ways. First, the substrate has a finite thickness
h, sometimes with back-side metallization. Second, the ground planes have a finite
width Wg and, although the conductors are very thin, their thickness tc is larger than
the skin depth. The width of the center conductor S, the width of separation between
the center conductor and ground planes W, and the thickness and relative permittivity
of the dielectric substrate er all determine the characteristic impedance, effective
dielectric permittivity, and attenuation of a CPW.
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w, w w Wg tc
h Dielectric substrate
(a)
wg w s w wg '____ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ fc
Dielectric substrate h
(b)
Figure 2.1 Cross-sectional view of the CPWs (a) with wide ground planes and (b) with narrow ground planes.
C. P. Wen first proposed the CPW and carried out a quasi-static analysis of it
using the conformal mapping method [1]. At low enough frequencies, the energy is
transmitted along the CPW with an electromagnetic field configuration closely
resembling a transverse electromagnetic (TEM) mode. Thus, the quasi-TEM mode is
assumed to propagate along the CPW. Under this assumption, Section 2.2 briefly
describes the conformal mapping method and presents the closed-form expressions
for phase velocity and characteristic impedance. As the frequency increases, the
quasi-static assumption becomes invalid, and higher-order modes propagate along the
CPW due to the non-zero longitudinal component of the magnetic field. Section 2.3
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discusses the mechanism of high-frequency propagation along a CPW. Section 2.4
addresses a distributed circuit analysis of the CPW. Section 2.5 and 2.6 discuss
dispersion and attenuation characteristics up to subterahertz frequencies respectively.
Section 2.6 addresses a distributed circuit analysis of the CPW.
2.2 Quasi-static Analysis
A quasi-static analysis of the CPW can be implemented by using a conformal
mapping method. Wide-ground CPWs, namely CPWs with semi-infinitely wide
ground planes, have been analyzed in [2]. The thickness of the dielectric substrate is
considered sufficiently large so that it takes the value of infinity. The thickness of
conductors is assumed to be zero, and we are concerned with the air-dielectric
interface. The CPW is then separated into an air half-plane and a dielectric half-plane,
and the electric field is confined within each partial region, as shown in Figure 2.2(a).
The air-dielectric interface in zy-plane (Figure 2.2(a)) is mapped into a rectangle in z-
plane (Figure 2.2(b)). The corresponding conformal transformation is given by
dz A' (2.1 ) dz ,
where A is a constant to be determined. The ratio alb in z-plane can be obtained by
integration in z/-plane and is given by
a _ K (k ) (2.2) b ~ K (k') ’
where
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£'= V i- k 2,
and K(k) is the complete elliptic integral o f the first kind. The ratio K(k)/K(k') can be
calculated by
0.707 < k < 1
(2.3) 0 < k < 0.707
First, the air half-plane in Figure 2.2(a) is mapped into the interior o f the rectangle
in Figure 2.2(b) to form a parallel-plate capacitor. Since the electric field equally
exists in each half-plane when the substrate is filled by air instead of dielectric
material, the partial capacitance per unit length in the absence of the dielectric
substrate is twice as much as that o f the capacitor:
(2.4)
where so is the permittivity of free space.
Secondly, the dielectric half-plane in Figure 2.2(a) is mapped into the interior of
the rectangle in Figure 2.2(b). The dielectric slab in Figure 2.2(b) has a relative
permittivity of er-\ so that its superposition with the air-filled substrate produces the
original dielectric substrate with a relative permittivity of er. The partial capacitance
per unit length of the dielectric substrate can be expressed as
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Cdidec=S ^ r - 1)X- (2.5) b
mi -bi -ai O ai bi ni ' • — ------
(a) zr plane
-a O a I • I
- - ..... -...... — .....- .J -a-jb m,n a-jb
(b) z-plane Figure 2.2 Conformal transformation planes for a wide-ground CPW with an infinitely thick dielectric substrate: (a) z/-plane and (b) z-plane.
Thus, the capacitance per unit length of the CPW is determined by the sum of two
items: the partial capacitance o f the dielectric half-plane and the partial capacitance of
the dielectric half-plane in the absence o f the dielectric substrate. The overall
capacitance per unit length of the CPW takes the expression
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C = C^+CM k = s0{s,+ l) y . (2.6)
Under quasi-static approximation, the phase velocity of wave propagation along
the CPW may be written as
( 2 J )
where L is the inductance per unit length of the CPW. Similarly, the speed of light in
free space may be written as
c = ~ r = > <2-8>
Using Eqs. (2.8), (2.7), (2.6) and (2.4), the effective relative permittivity of the
CPW is derived by
r \ 2
S eff ~ C . 2 \ VP J air
According to Eq. (2.9), the phase velocity can be evaluated by
V, = CJ——7 • (2-10) iK+i
Substituting Eqs. (2.7), (2.10), (2.6) and (2.2) into Zc=(LICcpw)m , the
characteristic impedance of the CPW is given by
_ 3 0 ^ ^ ) R T T m )
The application o f the conformal mapping method has been extended to narrow-
ground CPWs, namely CPWs with finitely wide ground planes [3]. Still under quasi-
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TEM approximation, the air-dielectric interface and dielectric substrate o f the narrow-
ground CPW on an original zy-plane are separately mapped into different structures
on an intermediate plane and then the same rectangle on an objective z-plane.
Similarly, the capacitance per unit length of the narrow-ground CPW is determined
by the sum o f two items: the partial capacitance o f the dielectric half-plane and the
partial capacitance of the dielectric half-plane in the absence of the dielectric
substrate. The overall capacitance per unit length o f the narrow-ground CPW is given
by
C = C * + Clk„, = 2 *„ (* ,+ 1 ) ^ 1 . (2.12) K (k {)
where
= £ l \b 2 - a 2
k ^ J T k 2.
The effective permittivity, phase velocity, characteristic impedance of narrow-
ground CPW take the same expressions as wide-ground CPW, that is, Eqs. (2.9),
(2.10), and (2.11).
2.3 High-frequency Propagation
As the frequency increases, higher-order modes propagate along a CPW due to the
non-zero longitudinal component of the magnetic field, as explained below. The
electromagnetic wave propagation along a CPW satisfies Maxwell’s equations. Its
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field distribution on the cross section of a wide-ground CPW is shown in Figure 2.3.
Assuming the dielectric substrate is lossless so that there is no charge within the
dielectric, then the curl of the magnetic field becomes
BF V x / / = £r£0-— , (2.13) ot
where E and H are electric and magnetic field and t is moment in time.
Dielectric substrate x Figure 2.3 Electromagnetic field distribution on the cross section of a CPW. Solid lines are electric field lines and dotted lines are magnetic field lines.
Considering the tangential component of an electric field is continuous across the
air-dielectric interface, Eq. (2.13) gives rise to
a n ,* , , ,d H y dy dy dz
where the first subscript of the magnetic field denotes the direction of the field
component, and the second one denotes the location of the field component. Since the
right side of Eq. (2.14) is not zero, the longitudinal component of the magnetic field
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exists. The transverse plane cannot contain the magnetic field and, thus, the
conformal mapping fails. The CPW does not support a pure TEM mode but rather
hybrid modes that can be considered as superpositions of TE and TM modes. The
higher-order modes propagating along a CPW with a dielectric mismatch are called
“surface-wave modes” [4],
The CPW mode, referring to the dominant mode propagating along a CPW,
approximates a TEM mode. Since only transverse components of electric and
magnetic fields exist, the CPW mode leaks relatively little electromagnetic energy.
For the surface-wave modes, longitudinal components of electric and magnetic fields
are present and make the energy leak into the substrate [4].
Energy y Dielectric substrate
Figure 2.4 Electromagnetic energy transmitted along a CPW.
Due to the existence of the longitudinal field component, the magnetic field leans
toward the direction of propagation so that the Poynting vector leans toward the
substrate, as shown in Figure 2.4. Hence, the electromagnetic energy radiates from
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the CPW to the substrate. When the CPW does not support the pure TEM mode, a
full-wave analysis is needed, which is described in Chapter 4. The full-wave analysis
can be implemented by Galerkin’s method in the spectral domain [5, 6], by a
variational method [7], or by non-uniform discretization o f integral equations [8].
2.4 Distributed Circuit Analysis
In addition to electromagnetic field methods, the propagation of electromagnetic
waves along a CPW can also be analyzed in terms of a distributed circuit model as
long as the lateral dimensions of the CPW are much smaller than the wavelength of
propagating signals [9]. Considering a short length of a uniform CPW with the
resistance per unit length R, inductance per unit length L, conductance per unit length
G and capacitance per unit length C, as shown in Figure 2.5, the voltage across R and
L and current through G and C can be expressed as
- d V = (R + jcoL)dl ■ I (2.15) -d I = (G + jo)C)dl ■ V
where co is the angular frequency. Then the second-order differential equation for the
voltage is given by
^ = (R + jcoL)(G + jcoC)V = y 2V. (2.16) dz
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I+ d l
V+dVV
l+ dl
Figure 2.5 Distributed circuit model of a CPW.
Thus, the propagation constant y o f a CPW is introduced by
r = ^(R + jcoL \G + j o C ) = a + j/3 , (2.17)
where a is the attenuation constant and ft is the phase constant. Two important
properties o f CPW, attenuation and dispersion, are determined by a and [i.
The general solution of Eq. (2.16) is
V = M v cosh (yl) + N v sinh(^), (2.18)
where M y and Ny are constants to be determined. Similarly, we obtain
I = M , cosh(^) + N j sinh(^/), (2.19)
where M/ and TV) are constants to be determined.
Substitution o f Eqs. (2.18) and (2.19) for Eq. (2.15) gives rise to
M v cosh(^) + N v sinh(y/) = -Z 0 [m, cosh(jl) + N, sinh(?7)], (2.20)
where we can find the characteristic impedance of the CPW
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The analysis of CPWs by distributed circuit models is not distinct from the
analysis by electromagnetic field methods because the circuit elements of CPWs are
calculated on the basis o f voltages and currents associated with electromagnetic
fields.
2.5 Dispersion
When an electromagnetic wave propagates on a CPW, the electric fields above
conductors experience the permittivity of the air, while those below conductors
experience the permittivity of the substrate. The effective permittivity should thus
become a value between that o f the air and the substrate. When the frequency o f
propagating wave increases, the effective permittivity approaches that of the
substrate, as the density o f electric field lines below conductors is getting higher and
higher. Using a full-wave analysis, G. Hasnain computed the effective permittivity of
a CPW [10], converting the numerical results to the following formula:
\2
Sr +■ (2.22) l + P
where eq is the effective permittivity of the substrate at the quasi-static limit, / is the
frequency,/) and q are empirical parameters,^ is the cut-off frequency of the lowest-
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order TE mode defined by
f = ------7------
The effective permittivity o f a CPW with wide ground planes up to 1 THz is
shown in Figure 2.6.
13 —
Frequency(GHz)
Figure 2.6 Effective permittivity of a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters.
Since signal components with different frequencies experience different
permittivities, they will not propagate at the same phase velocity. This effect is the so-
called “modal dispersion” [11-13], which distorts any signal propagating on a CPW,
as given by a frequency-dependent phase factor:
P = 2jrf-\tf (2.23) c
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With the signal frequency increasing, the effective permittivity increases and a
steep step is located at the position corresponding to the cut-off frequency of the first
TE mode. This implies that the lowest-order TE mode and inclusively higher-order
modes both contribute to the increase of dispersion. Dispersion can be alleviated by
reducing the thickness of the substrate, as the cut-off frequency of the lowest-order
TE mode increases. Another way to alleviate the dispersion of CPWs is to reduce the
lateral dimensions. Compared to other types of transmission lines, CPWs operate with
relatively more dispersion.
2.6 Attenuation
When an electromagnetic wave propagates on a CPW, it suffers three types o f
attenuation: conductor loss, dielectric loss, and radiation loss.
Due to the imperfection of the dielectric substrate, a shunt conductance G is
introduced to describe the effect of dielectric loss, as shown in Figure 2.5. The
expression for dielectric loss is reproduced below [2]:
s r £eff tan£ a d = 27* (2.24)
where tand is the loss tangent o f the substrate and Xo is the wavelength of the
electromagnetic wave propagating along the CPW. Dielectric loss is proportional to
the loss tangent of the substrate, which is strongly influenced by the dielectric
relaxation and the conductivity o f the dielectric substrate. In high-quality
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semiconductor substrates that have dielectric relaxation in the range of several
terahertz, the loss tangent has a small value as long as the conductivity is kept low by
avoiding high doping. In practical applications, the above conditions are true; as
compared to conductor loss and radiation loss, dielectric loss is very small and can be
neglected [12, 14].
The second mechanism of attenuation is due to the finite conductivity of the
metallization. A series resistance per unit length R is introduced to describe this
effect, as shown in Figure 2.5 [9]. Under the high-frequency approximation, surface
resistivity of the CPW conductors is determined by conductivity and skin depth as
Rs=1/gSs. The resistance R, in turn, can be expressed in terms o f the surface resistivity
and geometric parameters of the conductors:
(2.25)
where a is the conductivity of the conductor, Ss is the skin depth, I is the length of the
conductor, and w is the width of the conductor. The conductor loss ac is evaluated in
terms of the surface resistivity. In the case of CPWs with wide ground planes, ac is
given by [2]:
tt 1.25,1 « 4 S i , 1 . 1.25/ (dB/unit length)
(2.26)
where Rs is the surface resistivity of the conductors and has a square-root frequency
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dependence as long as the skin depth is much smaller than the thickness of the
conductors, and P ’ is a geometry dependent parameter. The conductor loss o f a CPW
with wide ground planes is demonstrated in Figure 2.7.
0.6
E 0.5
3 0.3
o 0.2
Frequency(GHz)
Figure 2.7 Conductor loss of a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters.
The surface resistivity accounts for the conductor loss, and ac increases as the line
dimensions become smaller. At low frequencies, the skin depth can become
comparable or even larger than the thickness o f the conductors, and in this case
conductor loss tends to saturate to a constant value. The above discussion is based on
the CPW operating under room temperature, that is, in the skin-effect domain, where
the mean free path for normal electrons is short compared to the thickness o f the
conductor. In the anomalous skin-effect domain, where the CPW operates under
extremely low temperature, the surface resistivity can be more than two orders of
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magnitude lower for a normal metal at liquid helium temperature, as compared to one
at room temperature [15]. Conductor loss will be much lower than as indicated by the
skin effect only.
When the CPW mode propagates along the transmission line at a velocity faster
than that o f the substrate mode, the coupling between the CPW mode and the
substrate mode forces energy to radiate from the transmission line into the substrate
and radiation loss arises [16]. The energy radiates into the substrate at an angle if/ ,
which is given by
cos(y/) = ^= p-, (2.27) K v e r
where kw and ks are the propagation constants of the guided CPW mode and the
substrate modes respectively. The electromagnetic energy leaking into the substrate is
a form of radiation loss. For the case of wide ground planes, the loss is given by [17]
r7/\\^ -sefflsr} ( s + lW fe3/2 - 2 v . eir r’ ■ f (mrn') • (2.28) 2 ) ^ 7 J r c K' (k)K(k)
For the CPW with a semi-infmitely thick substrate, the energy radiates into the
substrate from /=0 to high frequencies. For the CPW with an infinitely thick substrate,
the radiation occurs above the cutoff frequency of the lowest-order surface-wave
mode. If the back plane of the substrate is metallic or high-conductivity dielectric, the
parallel-plate mode, similar to the microstrip mode, exists as the second dominant
mode [18].
Equation (2.28) is valid when 0.1<5/IF<10 and h>3W, as well as f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 4 high frequencies where the value of seff starts to increase, the angle y is reduced so that the rate o f increase o f radiation loss is limited below the f dependence. Equation (2.28) also suggests that CPWs with smaller lateral dimensions suffer reduced radiation loss while this reduction inevitably leads to an increase in the conductor loss. The radiation loss of a CPW with wide ground planes up to 1 THz is demonstrated in Figure 2.8. Frequency(GHz) Figure 2.8 Radiation loss of a CPW with wide ground planes. See Chapter 3 for geometry and electrical parameters. The above analysis of conductor and radiation losses applies strictly to ideal CPWs only (in our case, approximately the CPWs with wide-ground planes). For CPWs with narrow-ground planes, much less is known and no closed-form analysis has ever been carried out. One focus of my work is then to numerically compute the characteristics o f these CPWs. Qualitatively, there have been descriptions o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 5 physics of these lines [4, 19, 20], which I will review below. In the case o f CPWs with wide ground planes, the central guiding region is the substrate with ground planes on its top surface, and the lowest substrate mode that is supported is the TM0 mode. In the case o f CPW with narrow ground planes, the central guiding region is the substrate without ground planes on its top surface, and the lowest substrate mode that CPWs can support is the TEo mode [4]. Considering the effect o f ground-plane width on high-frequency current distribution o f the CPW, conductor loss of CPWs increases with the ground-plane width decreases, while other geometry parameters stay constant [19]. Due to the reduction in coupling between the CPW mode and the substrate mode, CPWs with narrow ground planes suffer much less radiation loss and dispersion than CPWs with wide ground planes. The narrower the ground planes are, the less the radiation loss and dispersion arise. When the ground-plane width is fixed, reducing the lateral dimensions of the CPW also reduces the radiation loss and dispersion, whereas the ground-plane width is a more important factor. To be noted, the combination of reduction in center-conductor width and spacing between center conductor and ground planes plays a more obvious role than reduction in these two widths respectively in reducing the radiation loss and dispersion. In addition to previous study on the conventional CPW, closed-form expressions for dispersion and radiation characteristics of conductor-backed CPWs with narrow ground planes have been derived and implemented into an existing quasi-static CPW model [20]. The CPW high-frequency dispersion is obtained by revising the quasi- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 6 static value with fitting full-wave simulation data. The expression for radiation loss of a conductor-backed CPW with narrow ground planes is given by: W w ^(y)-ri(ri,r)-n(r2>r) + s = ^Xes k - g, K ^ ~ 2-75) K (r) (2.29) where /iq is the permeability of free space, fl(r/, r) is the complete elliptical integral o f the third kind, r}, r2, and r are geometry dependent parameters. It can be seen that radiation loss of a conductor-backed CPW with narrow ground planes shows a f dependence in Eq. (2.29), while radiation loss of a conventional CPW with wide ground planes shows a f dependence in Eq. (2.28). References [1] C. P. Wen, “Coplanar waveguide: a surface strip transmission line suitable for non-reciprocal gyromagnetic device application,” IEEE Trans. Microwave Theory Tech., vol. 17, no. 12, pp. 1087-1090, 1969. [2] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1979, ch. 7. [3] C. Veyres and V. Fouad Hanna, “Extension of the application of conformal mapping techniques to coplanar lines with finite dimensions,” Int. J. Electronics, vol. 48, no. 1, pp. 47-56, 1980. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 [4] M. Tsuji, H. Shigesawa, and A. A. Oliner, “New interesting leakage behavior on coplanar waveguides of finite and infinite widths,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 12, pp. 2130-2137, 1991. [5] J. B. Knorr and K. D. Kuchler, “Analysis of coupled slots and coplanar strips on dielectric substrate,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 7, pp. 541-548, 1975. [6] Y. Fujiki et. al., “Higher-order modes in coplanar-type transmission lines,” Electronics and Comm, in Japan, vol. 58-B, pp. 74-80,1975. [7] R. Pregla and S. G. Pintzos, “Determination o f the propagation constants in coupled microslots by a variational method,” in Proc. V Colloquium Microwave Comm., Budapest, 24-30, pp. MT-491-500, 1974. [8] E. Yamshita and K. Atsuki, “Analysis o f micro strip-like transmission lines by nonuniform discretization of integral equations,” IEEE Trans. Microwave Theory Tech., vol. 24, no. 4, pp. 195-200, 1976. [9] F. A. Benson and T. M. Benson, Fields Waves and Transmission Lines. London: Chapman & Hall, 1991, ch. 2. [10] G. Hasnain, A. Dienes, and J. R. Whinnery, “Dispersion of picosecond pulses in coplanar transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 34, no. 6, pp. 738-741, 1986. [11] M. Riaziat, R. Majidi-ahy, and I. Feng, “Propagation modes and dispersion characteristics of coplanar waveguides,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 3, pp. 245-251, 1990. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 8 [12] D. S. Phatak, N. K. Das, and A. P. Defonzo, “Dispersion characteristics of optically excited coplanar striplines: comprehensive lull wave analysis,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 11, pp. 1719-1730, 1990. [13] J. F. Whitaker, R. Sobolewski, D. R. Dykaar, T. Y. Hsiang, and G, A, Mourou, “Propagation model for ultrafast signals on superconducting dispersive striplines,” IEEE Trans. Microwave Theory Tech., vol. 36, no. 2, pp. 277-285, 1988. [14] C. Shu, X. Wu, E. S. Yang, X. C. Zhang, and D. H. Auston, “Propagation characteristics of picosecond electrical pulses on a periodically loaded coplanar waveguide,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 6, pp. 930-935, 1991. [15] J. F. Whitaker, “Ultrafast electrical signals: transmission on broadband guiding structures and transport in the resonant-tunneling diode,” Ph.D. dissertation, Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY, 1988. [16] D. S. Rutledge, D. P. Neikirk, and D. P. Kasilingham, Infrared and Millimeter Waves. N ew York: Academic Press, 1983, ch. 1. [17] M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 6, pp. 910-915, 1991. [18] W. Heinrich, F. Schnieder, and T. Tischler, “Dispersion and radiation characteristics of conductor-backed CPW with finite ground width”, in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Microwave Symposium Digest., 2000 IEEE MTT-S International, vol. 3, pp. 1663- 1666, Jun. 2000. [19] G. Ghione and M. Goano, “The influence of ground-plane width on the ohmic losses o f coplanar waveguides with finite lateral ground planes,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 9, pp. 1640-1642, 1997. [20] F. Schnieder, T. Tischler, and W. Heinrich, “Modeling dispersion and radiation characteristics of conductor-backed CPW with finite ground,” IEEE Trans. Microwave Theory Tech., vol. 51, no. 1, pp. 137-143, 2003. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 0 CHAPTER 3 Ultrafast Optoelectronic Characterization My research on the propagation characteristics of transmission lines is conducted with a variety of CPWs, including wide-ground and narrow-ground waveguides, over a broad frequency range, from about 10 GHz to nearly 1 THz. This work is validated by the previous experiments from our group [1]. In this chapter, I will review the experimental setup and the measured data that provide verification for my research. At the beginning of this chapter, the electro-optic sampling system, used for the ultrafast characterization of microwave devices and circuits, is briefly reviewed. Section 3.2 describes the fabrication and measurement of the CPW in my study. The time-domain and frequency-domain analysis o f the measured data are explained in Section 3.3 and 3.4 respectively. 3.1 The Electro-optic Sampling System First developed in the 1980s, the ultrafast electro-optic sampling technique is able to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 characterize electrical signals with subterahertz bandwidth [2]. Figure 3.1 illustrates the schematic configuration of such an electro-optic sampling system. A mode-locked laser emits a pulse train o f light that is divided into two beams by a splitter: a switching beam and a sampling beam. Splitter Mode-locked laser 'Optical delay Detector Polarizer Compensator Detector Acousto-optic Analyzer Differential modulator ■Lens amplifier Crystal Lock-in Computer Optical fiber amplifier DUT Switching beam' Sampling beam Figure 3.1 Schematic configuration of an electro-optic sampling system. The switching beam is first passed through an acousto-optic modulator to facilitate lock-in amplification, which reduces noise. It is then directed by an optical fiber or focused by a lens (not shown in Figure 3.1) to generate electrical signals in the device under test (DUT) by exciting a photosensitive device. The sampling beam is first passed through an optical delay to adjust the optical path difference between the two beams and then focused to an electro-optic crystal. The electric field in the DUT produces birefringence proportional to the strength of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 the electric field in the electro-optic crystal. This is so-called Pockels effect [3]. The change in birefringence leads to a change in the polarization of the sampling beam. After being reflected by the electro-optic crystal, the sampling beam is guided through an analyzer, where its change in polarization is converted to intensity modulation and detected by two detectors, such as photodiodes. The polarizer and compensator further adjust the phase o f the sampling beam and help to improve the linearity and sensitivity of the light intensity modulation. Then the differential detector outputs are processed by a differential amplifier and a lock-in amplifier to enhance signal-to-noise ratio. At the final stage, the amplified detector data are recorded and displayed by a computer. Since the light intensity detected by the analyzer is proportional to the electric field in DUT, measuring light intensity modulation characterizes the electrical signals in the time domain. 3.2 Fabrication and Measurement In a previous experiment, S. Alexandrou measured the propagation of picosecond step-like transients on a range o f CPWs that were fabricated on a semi-insulating GaAs substrate with a thickness o f 500 fxm and a high resistivity o f more than 10 Q cm [1], The electron mobility was -5000 cm2/V-s and the substrate’s orientation was (100). The transmission lines were patterned with evaporated gold using a “lift o ff’ process; the thickness of the gold film was 290 nm. Two separate sets o f transmission lines were fabricated: CPWs with wide ground Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 planes and CPWs with narrow ground planes (see Figure 2.1). All transmission lines had the dimension S the same as W with a value o f 10 or 50 /urn. CPWs with wide ground planes had a ground-plane width Wg=h=500 pim, while for CPWs with narrow ground planes, the value of Wg was reduced to the size of the line dimensions. The transmission lines were incorporated into two chips. The first chip included the above two types of CPWs with lateral dimensions of 50 /um. The second chip included the smaller version o f the same CPWs with lateral dimensions o f 10 /um. Each CPW included a gap in its center conductor that formed a photoconductive switch in series with the transmission line. An optical beam from a femtosecond Ti:sapphire laser illuminated the switches, generating subpicosecond electrical pulses that propagated on the CPW. The electrical pulses were generated by the nonuniform gap illumination method, which permits the generation of subpicosecond pulses even on a semi-insulating GaAs substrate that has a relatively long carrier lifetime [1,4]. With this method, the laser only illuminates the positive edge of the photoconductive switch instead of the whole gap. Since most o f the switch gap is never illuminated, drift currents are suppressed. Thus, the generation of transients with long fall times due to the long carrier lifetime o f substrate is avoided. Under these conditions, it is possible to generate electrical pulses with ultrashort fall times by utilizing the effect of field screening, which does not depend on carrier lifetime. A detailed experimental and theoretical characterization of this effect has been presented elsewhere [5-7]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 4 S. Alexandrou probed the generated electrical pulses with a subpicosecond electro-optic sampling system [1]. In this system, light from a femtosecond laser is divided into two beams, the first of which is used to generate the electrical pulses. The second beam is reflected through a LiTaC >3 crystal positioned at consecutive positions along the CPW and senses the electric field of the propagating pulses as a change in polarization. In this manner, the electrical profile of the propagating waveforms was mapped as a function of time at consecutive positions along each CPW. This measurement is non-intrusive, thus ensuring an accurate study of the CPWs themselves unintruded by line discontinuities. Probing beam Center conductor Ground planes Illuminating beam LiTaO, crystal Bias GaAs substrate Photoconductive switch Figure 3. 2 Demonstration of subpicosecond electro-optic sampling on CPW. The geometry of the above sampling process is demonstrated in Figure 3.2. The evolution of the electrical signals was first studied in the time domain and then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 5 analyzed in a broad frequency domain. 3.3 Time-domain Analysis In the time domain, it is easy to give a qualitative evaluation of the relative contributions of attenuation by focusing our attention on the pulse ringing and the peak-amplitude reduction and broadening. In this section, some measured data from S. Alexandrou’s experiments will be analyzed in the time domain [1]. O.S 0.0 0 20 4 0 6 0 8 0 Time (ps) Figure 3.3 Electrical pulses probed at 0.25, 1, 3, and 5 mm on two 50 -fim CPWs with ground-plane width of 500 (solid) and 50 fim (dotted). The waveforms o f electrical pulses probed on two CPWs with lateral dimensions o f 50 f.im are shown in Figure 3.3. One CPW had wide ground planes with a width of 500 fim and the other had narrow ground planes with a width of 50 jum. It is worth Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 noting that at the first measurement position, the signals are essentially identical even though they were generated by different switches on separate lines. At this position, the main pulses in both CPWs have a temporal duration of about 1 ps. The first observation about the evolution of these signals is that as the pulses propagate along CPWs, they become broader and their peak amplitude is reduced. A closer look at the experimental data reveals some differences between the propagation characteristics of the two CPWs. The signals on the CPW with wide ground planes suffer a larger reduction in peak amplitude, have a broader temporal width and show reduced “ringing.” This is most noticeable in the last set of traces, measured at the propagation distance of 5 mm. These observations are consistent with a greater loss of high-frequency components in CPWs with wide ground planes, which are associated with broader pulses and reduced ringing. 1.0 o.s 0.0 0 20 4 0 6 0 8 0 Time (ps) Figure 3. 4 Electrical pulses probed at 0.5, 1, 3, and 5 mm on a 10-fim CPW with ground- plane width of 500 fim (solid) and electrical pulses probed at 0.25, 1, 3, and 5 mm on a 10-///M CPW with ground-plane width of 10 fim (dotted). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 These findings are in agreement with predictions of [8, 9], where it was suggested that a reduction of the ground-plane width limits the leakage from the CPW mode to the substrate modes and reduces radiation loss. A similar comparison between the waveforms propagated on 10 -pm CPWs with wide and narrow ground planes, as shown in Figure 3.4, indicates that, again, the loss is substantially lower in CPWs with narrow ground planes. Our results show that a reduction o f the ground-plane width improves the attenuation characteristics of these CPWs. This observation will be further reinforced with the results of our frequency-domain analysis. »• o .s o .o o 20 4 0 6 0 8 0 Time (ps) Figure 3. 5 Electrical pulses probed at 0.25, 1, 3, and 5 mm on 10-fim (solid) and 50 -fim (dotted) CPWs with narrow ground planes. The lateral dimensions o f CPWs have a strong effect on their characteristics. The influence of this parameter is investigated with the results in Figure 3.5, which shows a comparison o f the electrical waveforms propagated on CPWs with narrow ground Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 planes with lateral dimensions o f 10 and 50 /um. The initial pulses, both of which were probed at 0.25 mm from the switch, have strong similarities, with the waveform from the 10 -/um CPW being slightly narrower. At longer distances, the signals propagated on the 10 -/um CPW evolve in a distinctly different way compared to the pulses on the 50 -/um CPW. In sharp contrast to the pulses on the 50-/um CPW, which are heavily distorted, the waveforms from the narrow transmission line are only marginally broadened and show very little ringing, indicating that high frequency loss is limited. On the other hand, the pulses from the 10 -/um CPW have progressively smaller peak amplitudes: a clear indication of increased low-frequency attenuation dominated by conductor loss. With time-domain analysis, we have been able to present a qualitative description of the attenuation characteristics of CPWs. When the data are analyzed in the frequency domain, they pave the way for quantitative interpretation and verification of theory, as discussed in the next section. 3.4 Frequency-domain Analysis In order to facilitate frequency-domain characterization of CPWs, the time-domain data from S. Alexandrou’s experiments were converted to the frequency domain by the Fourier transform [1], An example of such a transformation is seen in Figure 3.6. The pulses measured on a 10 -/um CPW with narrow ground planes, also shown in Figure 3.5 in the time domain, are displayed as frequency-dependent functions of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 amplitude and phase. As the signals propagate on a CPW, their phase increases and their amplitude is reduced because o f attenuation. The spectral distribution of the waveforms is altered in a way that is directly related to their complex propagation factor y. The factor y can give a complete description o f the CPW, with its real part a representing the attenuation suffered by the signal as it travels along the CPW, and its imaginary part p being inversely proportional to the phase velocity. At a moment t and a propagation distance /, the initial pulse V(t,0) is modified to V(t,l). The spectral distribution of V(t,l) is related to the initial pulse by V(f,l) = V(f,0)e^f)l, (3.1) where r ( f ) = <* (/) + M f ) - 2000 .§ 20 1S00 I0 increasing z 2 1000 £ increasing z 1t 5 0 0 s o 200 4 0 0 6 0 0 6 0 0 1000 Frequency (GHz) Figure 3. 6 A spectral representation of some time-domain data measured on a lQ-fim CPW with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 By expressing the input and the propagated signals, V(f,0) and V(J'J), as functions of frequency as shown in Figure 3.6, the attenuation a can be evaluated. Since time- domain data are readily available not only in two but several measurement positions, we can use a least-square fit for the calculation of a to improve the accuracy of the measurement. Frequency-domain data will be further studied in Chapter 5. References [1] S. Alexandrou, “The bent coplanar waveguide at sub-terahertz frequencies,” Ph.D. dissertation, Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY, 1994. [2] J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett., vol. 41, no. 3, pp. 211-212, 1982. [3] C. R. Pollock, Fundamentals of Optoelectronics. Boston, MA: Richard D. Irwin, Inc., 1995, ch. 17. [4] S. Alexandrou, C.-C. Wang, R. Sobolewski, and T. Y. Hsiang, “Generation of subpicosecond electrical pulses by nonuniform illumination of GaAs transmission line gaps,”IEEE J. Quantum Electron., vol. 30, no. 5, pp. 1332-1338, 1994. [5] D. Krokel, D. Grischkowsky, and M. B. Ketchen, “Subpicosecond electrical pulse generation using photoconductive switches with long carrier lifetimes,” Appl. Phys. Lett., vol. 54, no. 11, pp. 1046-1047, 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 [6] C.-C. Wang, M. Currie, R. Sobolewski, and T. Y Hsiang, “Subpicosecond electrical pulse generation by edge illumination of silicon and indium phosphide photoconductive switches,” Appl. Phys. Lett., vol. 67, no. 1, pp. 79-81, 1995. [7] X. Zhou, S. Alexandrou, and T. Y. Hsiang, “Monte Carlo investigation o f the intrinsic mechanism of subpicosecond pulse generation by nonuniform illumination,” /. Appl. Phys., vol. 77, no. 2, pp. 706-711, 1995. [8] M. Tsuji, H. Shigesawa, and A. A. Oliner, “New interesting leakage behavior on coplanar waveguides of finite and infinite widths,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 12, pp. 2130-2137, 1991. [9] J. Zhang, “Propagation characteristics o f coplanar waveguides at subterahertz frequencies,” Ph. D. dissertation, Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY, 2007, ch. 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 2 CHAPTER 4 Full-Wave Analysis 4.1 Galerkin’s Method in the Spectral Domain When the electrical signals propagate along a CPW, the longitudinal component of the magnetic field is present due to dielectric mismatch (see Chapter 2). Instead o f a pure TEM mode, hybrid modes are supported by the CPW configuration. Under quasi-static approximation, static capacitance derived from the electrostatic field is utilized to depict the quasi-TEM mode. When the frequency increases, the quasi static approximation becomes invalid and quasi-static analysis has to be replaced by a full-wave analysis. In order to fully describe wave propagation by full-wave analysis, a time-varying electromagnetic field is introduced and electric current density takes the place of electrostatic charge density. As a result, frequency-dependent propagation characteristics of CPWs are determined by Galerkin’s method in the spectral domain [1, 2], by a variational method [3], or by non-uniform discretization of integral equations [4]. Most of recent analyses are based on these techniques. In this section, we briefly review the basic outline of Galerkin’s method in the spectral domain, which is employed by the software that we use to investigate CPWs. Section 4.2 represents software simulations of CPWs. Section 4.3 furnishes an equivalent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 3 network analysis of simulation results. Although CPWs do not support a pure TE or TM mode, the electric and magnetic fields can be considered as superpositions o f TE and TM modes and expressed as a weighed sum of these modes [5]. TE and TM modes are chosen to be orthogonal sets, which form a basis for the expansion of the electric and magnetic fields. Components of the electric and magnetic fields can be conveyed in terms of electric and magnetic potentials. A Fourier transform of the potentials is taken along the direction parallel to the air-dielectric interface and perpendicular to the wave propagation vector. In the Fourier transform domain, the transformed electric and magnetic potentials satisfy the transformed wave equation and take different expressions in different regions. The continuity conditions are also applied at the air-dielectric interface and introduce the longitudinal and transverse components of electric current density. Substituting the above potential expressions and continuity conditions into the boundary conditions at the air-dielectric interface produces equations o f current distribution on the CPW. T. Itoh and J. B. Knorr used the Method o f Moments with Galerkin’s approach to solve these equations [6, 7]. In this method, the components o f electric current density are expanded as a weighted sum of chosen basis functions and substituted into the current distribution equations. Thus, a matrix equation o f current distribution is obtained. Setting the determinant of the matrix equation to zero and solving the matrix equation give rise to the propagation constant. Consequently, the current distribution and field distribution, in turn, are determined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 4 4.2 Software Simulations For analyzing the electromagnetic fields in CPWs by full-wave analysis, we used Sonnet Suites [8, 9], a high-frequency electromagnetic software, to simulate the scattering matrix of the microwave network consisting of CPW. The electromagnetic analysis module of the software, em, employs a modified method of moments based on Maxwell’s equations to perform a three-dimensional full-wave analysis of predominantly planar structures [10]. I entered and edited the circuit geometry of the CPW in the project editor, xgeom. Although the CPW in the project editor window is edited as a two-dimensional planar structure, it is actually analyzed in a three-dimensional box, as shown in Figure 4.1. The CPW is enclosed by six grounded metal walls, which construct the three- dimensional box. All the conductors touching one or more of the six walls are also grounded unless specified by assigning ports on the walls. In Figure 4.1, the two ports labeled with “1” and “2” are standard box-wall ports. They can be considered as two- terminal devices, with one terminal connecting the center conductor of the CPW and the other connecting to the ground. The de-embedding algorithm is required to eliminate the undesirable effects of port discontinuity and fringing fields. The Thru- Reflect-Line (TRL) calibration has been developed by measuring a thru connection, a reflect termination, and a line connection [11]. Based on the three measured S- matrices, the error matrices can be derived and then de-embedded from the S-matrix of the device under test. It assumes a single-mode propagation. However, we are concerned with the propagation of higher-order modes so that the multi-mode Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 5 operation invalidates the de-embedding calibration. Furthermore, the six metal walls are artificially created in the simulation to enclose the CPW and, therefore, bring in box resonances. These unwanted artificial box resonances introduce error into the calculation of S-parameters even if they can be manually removed in part [8, 9], xgeoni 10. ri I Lite f«vg rpw.son:^ i>P)j &tit jftew £ffcUt Anatvss Project Window yrip lE n n a w M Figure 4.1 Three-dimensional view of the CPW in the project editor window. In order to sufficiently implement the summation of waveguide modes to approximate the radiation, the lateral substrate dimensions are set to be greater than at least two times of the wavelength corresponding to the operating frequency. The sidewalls of the boundary are far enough from the center conductor strip such that they have no effect on the radiating structure. By adding air layers above and below the substrate and setting both the top cover and bottom cover o f the boundary as free space (they are metal in default), the radiation loss is fully evaluated in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 6 simulation. The CPWs studied in this thesis are analyzed at high frequencies up to subterahertz; therefore, a consideration o f the effects of radiation into air and substrate needs to occur— implemented by selecting “Free Space” from the Top Metal and Bottom Metal drop list in the Box Settings dialog box. Once a box wall is set to “Free Space,” the characteristic impedance of the free space, 377 Q, is assigned to it. Setting “Free Space” removes the box wall and approximates a termination to TE modes and TM modes [8, 9]. The cell size is also determined in the Box Settings dialog box. The plane where the CPW is located is divided into a number o f small squares and rectangles. These squares and rectangles, which are called “cells,” are the basic elements of current calculation. In order to guarantee the calculation accuracy, the cell size is generally not greater than l/20th o f the wavelength corresponding to the highest operating frequency. The quasi-static effective permittivity o f the CPW on a GaAs substrate is 6.95, given by Eq. (2.9). Provided the highest operating frequency is ITHz, the cell size is determined by c A = — 6jum , (4.1) 20 where c is the speed of light in free space, seff is the effective relative permittivity of the CPW, and/is the operating frequency. For the 10 -/um and narrower lines, the cell size is set to 2/um for the accurate subsectioning of narrow center conductors. Although the calculation accuracy will increase with the cell size decreasing, it may take more or even an unacceptable amount of time to complete the simulation. In the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Box Settings dialog box, the cell size in x- and ^-direction is entered in X and Y text entry box respectively. It should be noted that the cell sizes in x- and y-direction are not required to be the same. In the Dielectric Editor dialog box, the substrate material is selected by entering “Gallium Arsenide” in the Mat. Name text entry box; the thickness o f the substrate is set to 500.0 (in /um) in the Thickness text entry box; the relative permittivity is set to 12.9 in Erel text entry box; the dielectric loss tangent is set to 0.006 in the Dielectric Loss Tan text entry box; the dielectric conductivity is set to 0 in the Diel Cond text entry box; the relative permeability is set to 1 in the Mrel text entry box; and the magnetic loss tangent is set to 0 in the Mag Loss Tan text entry box. The above settings also can be loaded from the Global Dielectric Library by clicking the Select Dielectric From Library button. The materials in the Global Dielectric Library come from David Pozar’s book [12]. The choice of the conductors also affects the performance of the CPW. The properties of conductors are determined in the Metal Types dialog box. For the gold CPW on a GaAs substrate, its metal properties are entered in the Metal Editor dialog box. Sonnet Suites models the normal-type metal conductor by using bulk conductivity, the thickness o f the conductor and the ratio o f the currents on the top and bottom of the conductor. Both the direct current resistivity and the skin-effect surface impedance have been taken into account in the calculation of conductor loss. The geometrical and electrical parameter settings of CPWs to be analyzed are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 listed in Table 4.1, including the thickness of the conductors tc, conductivity of the conductors a, thickness o f the substrate h, relative dielectric constant o f the substrate er, dielectric loss tangent of the substrate e / e , dielectric conductivity o f the substrate Odieiec, relative magnetic permeability of the substrate and magnetic loss tangent of the substrate jj. /ju . The dielectric loss tangent e /e comes from the complex permittivity o f the dielectric substrate e and e = e'-js". (4.2) Similarly, the magnetic loss tangent /i /fi comes from the complex permeability of the dielectric substrate /i and ju = ju'-jju". (4.3) Sonnet Suites calculates the total effective dielectric loss tangent tan S for the dielectric material as below: tan<5 = — + ^ e c _ } (4.4) s' 0)SrSQ The dielectric loss is then determined by the dielectric loss tangent. In order to predict the subterahertz propagation characteristics of silicon-based CPWs, by using Sonnet Suites I also simulated aluminum lines on high-resistivity silicon (HR Si), aluminum lines on low-resistivity silicon (LR Si), copper lines on HR Si, and copper lines on LR Si. Since standard silicon wafer is low-resistivity, microwave transmission lines on a silicon substrate endure more loss, thus limiting the performance of MMIC. This loss can be alleviated by either using HR Si wafers or inserting a layer o f insulator between the conductors and substrate [13, 14]. Even Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 in MMIC on HR Si, it is necessary to insert a layer o f insulator as passivation to avoid low reliability or high leakage. A 7-^m-thick layer of Si02 is inserted between the conductors and silicon substrate. The relative permittivity o f the S i0 2 layer is set to 3.78, the dielectric loss tangent of it is set to 0.0004, and the dielectric conductivity of it is set to 0. HR Si wafers are fabricated by a pure float zone (FZ) process or by high doping only at device locations in a low-doped substrate. For the substrate material, HR Si has a dielectric conductivity o f 10 S/m, corresponding to a doping density of l x 1015 cm ; while LR Si has a dielectric conductivity of 100 S/m, corresponding to a doping density o f 1 x 1016 cm'3. Table 4.1 Geometrical and electrical parameters o f CPWs. Gold A1CPW A1CPW Cu CPW Cu CPW CPW on on HR Si on LR Si on HR Si on LR Si GaAs tc (pm) 0.29 2 2 2 2 a (S/m) 4.09x107 3.72*107 3.72x107 5.8X107 5.8X107 h (jim ) 500 500 500 500 500 cr 12.9 11.9 11.9 11.9 11.9 e /e 0.006 0.004 0.004 0.004 0.004 @dielec (S/fTl) 0 10 100 10 100 flr 1 1 1 1 1 H /pi 0 0 0 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 The geometrical and electrical parameters o f CPWs to be analyzed are listed in Table 4.1. After entering and editing the geometry and properties of the CPW, I invoked the em module to simulate the S-parameters and other related parameters. It should be pointed out that the box resonances introduce artificial error into the simulated S- parameters. Although the top and bottom cover have been set to free space, the side walls of the enclosed box produce unwanted fake resonances. Since the CPW operates at frequencies as high as subterahertz, a number of fake resonances have come into their presence. We have to consider the effect of these resonances on the simulation results during data analysis. Spreadsheet is selected as the format of the output file so that the simulation results can be directly read and processed by MATLAB [15]. MATLAB codes have been written to analyze the propagation characteristics and extract the distributed- element circuit model o f the CPWs on a GaAs and a silicon substrate. The results and models are described in details in Chapter 5 and 6. 4.3 Microwave Network Analysis I have calculated 5-parameters for CPWs on a GaAs and a silicon substrate by using Sonnet Suites. In this section, I will derive propagation characteristics of CPWs from its 5-parameters, based on K. C. Gupta and P. M. Chirlian’s equivalent network analysis [16, 17]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Figure 4.2 shows a CPW of length I, which can be regarded as a two-port network. The incident and reflected waves in port 1 and 2 are vi+, vf and v i , v { respectively. The 5-matrix equation can be written as Onc o12 e (4.5) VV2 J V 21 ° 2 2 VV2 J Port 1 CPW Port 2 Load ► - / 0 z Figure 4. 2 Equivalent two-port network of a CPW of length /. For the case o f a symmetric network, the corresponding S-parameters are interchangeable, that is, The expression for propagation constant of a CPW in terms of 5-parameters can be drawn from the relationship between its 5-matrix and ABCD matrix [16, 17]. The voltage and current at the location z along the CPW are expressed as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 2 V(z) = Axe ” + V I(z) = ^-e-rz ~ ^ e rz' ^ Z„ Z At the location z=0, consequently, Eq. (4.7) is simplified as v(0) = 4 + 4 4 4 . (4.8) 7(0) = zc z c While at the location z=l, Eq. (4.7) turns into V{l) = Axe rl + A 1erl m = A e-n + l L e«- (4'9) By combining Eq. (4.8) and (4.9), we obtain 4 = z, 7(/)-/(0)erf 0 e „~rl —e „rl (4.10) A ^ m - m e -rl ~ -rl rl e - e Substitution o f Eq. (3) into Eq. (4.8) and (4.9) yields (4.11) or v( sinh 4 sinh 4 According to the definition of ABCD matrix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 3 ' v ( oy (4.13) J ( O b A m , the elements of the matrix can be written as V(0) A-- v ( i ) /(/)=0 V (0 ) 1(1) V{1)= 0 (4.14) 7(0) c V(l) 7(0=0 7(0) D = 1(1) r(i)=o Assigning 7(/)=0 and then dividing V(l) by V(0) in Eq. (4.12) yielding A = cosh yi. (4.15) Assigning V(l)=0 in Eq. (4.12) and then simplifying it into 7(0) = cosh y l. (4.16) 1(1) Dividing V(0) in Eq. (4.12) by 1(1) and then synthesizing Eq. (4.16), another element o f ABCD matrix can be calculated as cosh2 y l- 1 B = Z, (4.17) sinh yl Considering cosh2 x - sinh2 x = 1, Eq. (4.17) takes the following expression B = Z sinh y l. (4.18) Assigning l(l)=0 in Eq. (4.12) and then simplifying it gives Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assigning F(/)=0 in Eq. (4.12) and then simplifying it gives D = cosh y l. ( 4 .2 0 ) Therefore, the ABCD matrix of a transmission line in Figure 4.2 can be expressed as cosh yl Zc sinh yl A B \ (4.21) VC D; — sinh yl cosh;/ \ZC The detailed derivation of the propagation constant from the relationship between the 5-matrix and ABCD matrix refers to Chapter 6. The expression is given by 25. y = a + j/3 = - j In 21 (4.22) 2 1-S,2, +S12,±i!(l + S;,-S1J-4 S11 l Based on the simulated S-parameters, attenuation constant and phase constant of a CPW is calculated by (4.23) P = — Im / 1 - Sl + S22, ± V(l + Sl-Slf-4Sl For each frequency component of the electrical signal, its amplitude is suppressed by attenuation. If different frequency components propagate at different phase velocities, the signal is further broadened by dispersion. The phase velocity of each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 5 frequency component is determined by v,=~. (4.24) = P References [1] J. B. Knorr and K. D. Kuchler, “Analysis o f coupled slots and coplanar strips on dielectric substrate,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 7, pp. 541-548, 1975. [2] Y. Fujiki, et. al., “Higher-order modes in coplanar-type transmission lines,” Electronics and Comm, in Japan, vol. 58-B, pp. 74-80, 1975. [3] R. Pregla and S. G. Pintzos, “Determination o f the propagation constants in coupled microslots by a variational method,” in Proc. V Colloquium Microwave Comm., Budapest, 24-30, pp. MT-491-500, 1974. [4] E. Yamshita and K. Atsuki, “Analysis of micro strip-like transmission lines by nonuniform discretization of integral equations,” IEEE Trans. Microwave Theory Tech., vol. 24, no. 4, pp. 195-200,1976. [5] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1979, ch. 7. [6] T. Itoh and R. Mittra, “Spectral-domain approach for calculating the dispersion characteristics o f microstrip lines,” IEEE Trans. Microwave Theory Tech., vol. 21, no. 7, pp. 496-499,1973. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 6 [7] J. B. Knorr and A. Tufekcioglu, “Spectral-domain calculation o f microstrip characteristic impedance,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 9, pp. 725-728, 1975. [8] Sonnet User’s Guide Release 9, Sonnet Software Inc., Syracuse, NY, 2003. [9] Sonnet User’s Guide Release 10, Sonnet Software Inc., Syracuse, NY, 2005. [10] J. C. Rautio and R. F. Harrington, “An electromagnetic time-harmonic analysis of shielded microstrip circuits,” IEEE Trans. Microwave Theory Tech., vol. 35, no. 8, pp. 726-730, 1987. [11] E. F. Engen and C. A. Hoer, ““Thru-reflect-line”: an improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microwave Theory Tech., vol. 27, no. 12, pp. 987-993, 1979. [12] D. M. Pozar, Microwave Engineering. New York: Wiley, 1998, Appendices F and G. [13] C. Schollhom, W. Zhao, M. Morschbach, and E. Kasper, “Attenuation mechanisms of aluminum millimeter-wave coplanar waveguides on silicon,” IEEE Trans. Electron Devices, vol. 50, no. 3, pp. 740-746, 2003. [14] V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broad-band transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 5, pp. 632-640, 1998. [15] http://www.mathworks.com/products/matlab/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 [16] K. C. Gupta, R. Garg, and R. Chadha, Computer-aided Design of Microwave Circuits. Artech, 1981, pp. 25-43. [17] P. M. Chirlian, Basic Network Theory. Mcgraw-Hill, 1969, pp. 506-508. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 CHAPTER 5 Propagation Characteristics In my thesis, the propagation of electrical signals along CPWs has been investigated by using ultrafast optoelectronic measurement and full-wave analysis, as described in previous chapters. Chapter 3 illustrates S. Alexandrou’s experiments, in which subpicosecond electrical pulses are generated and propagate along wide-ground and narrow-ground CPWs on a GaAs substrate. S. Alexandrou measured the subpicosecond pulses by an electro-optic sampling system in the time domain. Then he converted the time-domain data to the frequency domain via the Fourier transform. In Chapter 4 ,1 simulated S-parameters of wide-ground and narrow-ground CPWs on a GaAs and a silicon substrate by a full-wave analysis. Based on the simulated S- parameters, attenuation and dispersion are calculated up to subterahertz frequencies. In this chapter, the experimental data and the simulation results of the propagation characteristics o f CPWs obtained from previous chapters are presented. Section 5.1 and 5.2 discuss the attenuation and dispersion of CPWs respectively. I also analyze the effects o f ground-plane width and lateral line dimensions on subterahertz propagation of CPWs. In different frequency ranges, different attenuation mechanisms dominate while dispersion shows different behavior as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 5.1 Attenuation When an electromagnetic wave propagates on a CPW, it suffers three types o f attenuation: conductor loss, dielectric loss, and radiation loss. In this section, we present the experimental data and simulation results o f subterahertz attenuation of CPWs on a GaAs and a silicon substrate. The effects o f ground-plane width and lateral line dimensions on attenuation characteristics o f CPWs are analyzed in Section 5.1.1 and 5.1.2 respectively. 5.1.1 The Effects of Ground-plane Width Figure 5.1 shows the attenuation o f CPWs on a GaAs substrate with a 50-pm center conductor as a function of frequency [1, 2]. The experimental attenuation curve for the wide-ground CPW, which is roughly the same as the simulated attenuation curve, is given to assess the validity of the full-wave analysis. We present the simulated and experimental attenuations o f CPWs with both wide and narrow ground planes for comparison. As expected, the attenuation of CPWs with narrow ground planes is much lower than that of CPWs with wide ground planes over 200 GHz, where radiation loss dominates. The reduction of radiation loss arises from the reduction of overlap between the quasi-TEM CPW mode and the substrate modes. Therefore, reducing the width of ground planes leads to an obvious decrease of radiation loss, and consequently this causes the decrease o f total attenuation at subterahertz frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 0 — Attenuation sim-wg — Attenuation exp-wg — Attenuation sim-ng Attenuation exp-ng E 10‘ 10 Frequency(GHz) Figure 5.1 Simulated and experimental attenuation of CPWs on a GaAs substrate with a 50-fim center conductor. Attenuation sim-wg refers to simulated attenuation of the CPW with wide ground planes. Attenuation exp-wg refers to experimental attenuation of the CPW with wide ground planes. Attenuation sim-ng refers to simulated attenuation of the CPW with narrow ground planes. Attenuation exp-ng refers to experimental attenuation of the CPW with narrow ground planes. In addition, the subterahertz attenuation o f CPWs on a GaAs substrate with a 10- pm center conductor has also been simulated and compared to the corresponding experimental data. Figure 5.2 demonstrates that, again, CPWs with narrower ground planes have a strongly lower attenuation over the high frequency range. On the other hand, Figure 5.2 shows a higher attenuation for CPWs with narrow ground planes over the low frequency range [1, 2]. In this region, conductor loss is a more important factor than radiation loss, so that the attenuation characteristics are different than at high frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 Attenuation sim-wg Attenuation exp-wg — Attenuation sim-ng 10,o Attenuation exp-ng 10'.3 Frequency(GHz) Figure 5. 2 Simulated and experimental attenuation of CPWs on a GaAs substrate with a 10-/WM center conductor. Attenuation sim-wg refers to simulated attenuation of the CPW with wide ground planes. Attenuation exp-wg refers to experimental attenuation of the CPW with wide ground planes. Attenuation sim-ng refers to simulated attenuation of the CPW with narrow ground planes. Attenuation exp-ng refers to experimental attenuation of the CPW with narrow ground planes. Besides CPWs on a GaAs substrate, we also investigate CPWs on a silicon substrate with aluminum or copper conductors. The effects of ground-plane width and lateral line dimensions on subterahertz attenuation and dispersion of CPWs on a silicon substrate are examined in Section 5.1 and 5.2 respectively. If a n-type GaAs and a n-type silicon wafer are equivalently doped, then the former has electric charge carriers with much less effective mass and thus much higher electron mobility than the latter. This fundamental property means GaAs is a suitable substrate material for high-frequency MMIC while the silicon substrate is dominantly employed in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 2 relatively low-frequency applications for it is a mature technology. Therefore, we observe propagation characteristics o f CPWs on a silicon substrate up to 100s of GHz. As explained in Chapter 4, two kinds o f substrates, HR Si and LR Si, have been analyzed. Figures 5.3 and 5.4 illustrate simulated attenuation o f CPWs on a HR Si substrate. We consider two sets of CPWs, one with a 10 -jum center conductor and the other with a 20-jum center conductor. It demonstrates that CPWs on a HR Si substrate show similar attenuation characteristics to CPWs on a GaAs substrate. Given the same lateral line dimensions, the attenuation o f narrow-ground CPWs is higher than that of wide-ground CPWs below 200 GHz, where the conductor loss dominates. As the frequency increases above 200 GHz, the effect of radiation loss prevails and it makes narrow-ground CPWs operate with lower attenuation. Comparing Figure 5.3 to 5.4, it can be noted that the attenuation of CPWs with copper conductors is slightly lower than that of CPWs with aluminum conductors. This difference arises because of the higher conductivity o f copper and, in turn, its lower conductor loss. Figures 5.5 and 5.6 illustrate the simulated attenuation of CPWs on a LR Si substrate. Again, narrow-ground CPWs operate with higher attenuation due to the effect of conductor loss at the low frequency range, while they operate with lower attenuation due to the effect of radiation loss at the high frequency range. As in the case of HR Si substrate, CPWs on a LR Si substrate also show lower attenuation with copper conductor due to its higher conductivity and lower conductor loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 10 10 8 2 Frequeney(GHz) Figure 5.3 Simulated attenuation of aluminum CPWs on a HR Si substrate with a 10-ftm and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide- ground planes while ng refers to narrow-ground planes. 10 f* J 10 '20nJT)-wg •2 10 2 Frequency(GHz) Figure 5. 4 Simulated attenuation of copper CPWs on a HR Si substrate with a 10 -fim and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 4 > 1 1 1 l""l lOum-wg 10(xm-ng 20urrvwg 2Q|Am-ng i i i i i in 10 Frequency(GHz) Figure 5. 5 Simulated attenuation of aluminum CPWs on a LR Si substrate with a 10-fim and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide- ground planes while ng refers to narrow-ground planes. 10 10.0 10 ■2 10' .2 Frequeney(GHz) Figure 5. 6 Simulated attenuation of copper CPWs on a LR Si substrate with a and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 5 A comparison of Figures 5.3 and 5.4 to 5.5 and 5.6 illustrates that the attenuation curves o f wide- and narrow-ground CPWs on a HR Si substrate intersect at a frequency higher than that where the attenuation curves of CPWs on a LR Si substrate do. Furthermore, the CPWs on a HR Si substrate show lower attenuation than the CPWs on a LR Si substrate. This difference is determined by the higher resistivity o f HR Si and, in turn, its lower dielectric conductivity. 5.1.2 The Effects of Lateral Line Dimensions 10 Attenuation sim-10 Attenuation exp-10 — Attenuation sim-50 Attenuation exp-50 "E 10 5 10' Frequency(GHz) Figure 5. 7 Simulated and experimental attenuation of CPWs on a GaAs substrate with narrow ground planes. The lateral line dimensions are 10 pm and 50 pm, respectively. Attenuation sim-10 refers to simulated attenuation of the CPW with a 10-pm center conductor. Attenuation exp-10 refers to experimental attenuation of the CPW with a 10-pm center conductor. Attenuation sim-50 refers to simulated attenuation of the CPW with a 50-pm center conductor. Attenuation exp-50 refers to experimental attenuation of the CPW with a 50-pm center conductor. The study so far reveals that CPWs with narrow ground planes suffer lower attenuation than those with wide ground planes. On the basis of the obtained results, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 we proceed to further investigate the effect o f lateral dimensions on the performance o f CPWs. First, CPWs with narrow ground planes on a GaAs substrate are investigated. The lateral dimensions of both center conductors and ground planes are chosen to be 10 pm or 50 pm. Figure 5.7 shows their attenuation in detail [1, 2]. At low frequencies up to 200 GHz, the CPW with larger lateral dimensions yields lower attenuation mainly arising from lower conductor loss. As the frequency increases up to 1 THz, the CPW with smaller lateral dimensions performs better with lower attenuation due to lower radiation loss. 10 '5|im '10p.m '50|j.m 10 2 10 Frequency(GHz) Figure 5. 8 Simulated attenuation of aluminum CPWs on a HR Si substrate with wide ground planes. In the following part o f this section, we investigate the attenuation characteristics of CPWs on a silicon substrate. As compared to the case of CPWs on a GaAs substrate, dielectric loss plays an important role in CPWs on a silicon substrate because of the comparatively large dielectric conductivity. Figure 5.8 shows the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 7 attenuation o f aluminum CPWs on a HR Si substrate with wide ground planes. When the separation between conductor lines is comparable to the thickness of the oxide, the electric field lines penetrate into the silicon substrate [3]. With the lateral line dimensions increasing, dielectric loss increases and gives rise to increasing attenuation. Figure 5.9 shows the attenuation of aluminum CPWs on a HR Si substrate with narrow ground planes. At the frequencies below 50 GHz, CPWs with wider line dimensions suffer from higher attenuation. With the further increase in frequency, CPWs with wider line dimensions operate with lower attenuation due to lower conductor loss below a few 100s of GHz, while higher attenuation due to higher radiation loss above a few 100s of GHz. 10 V 1^' - - a '10pm 10 2 10 Frequency(GHz) Figure 5. 9 Simulated attenuation of aluminum CPWs on a HR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 10 '5pm '10pm '2Qpm '50pm ■2 10 1 .2 Frequency(GHz) Figure 5.10 Simulated attenuation of copper CPWs on a HR Si substrate with wide ground planes. 10 '20pm '50pm ,2 Frequency(GHz) Figure 5.11 Simulated attenuation of copper CPWs on a HR Si substrate with narrow ground planes. Figures 5.10 and 5.11 illustrate the attenuation of copper CPWs on a HR Si substrate with wide and narrow ground planes respectively. Similarly, with the lateral Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 line dimensions increasing, dielectric loss increases and gives rise to increasing attenuation of wide ground CPWs. For the case of narrow ground CPWs, CPWs with wider line dimensions show higher attenuation below 50 GHz and above a few 100s o f GHz and lower attenuation in the intermediate frequency range. If we compare Figures 5.10 and 5.11 to 5.8 and 5.9, it can be noted that the attenuation of CPWs with copper conductors is slightly lower than that o f CPWs with aluminum conductors due to the higher conductivity of copper. Figures 5.12 and 5.13 illustrate the attenuation of aluminum CPWs on a LR Si substrate with wide and narrow ground planes respectively. Since the dielectric conductivity of LR Si is much higher than that of HR Si, the dielectric loss affects the overall attenuation o f CPWs on a LR Si substrate in a more apparent manner. Throughout the frequency range from 10 GHz to 500 GHz, CPWs with wider line dimensions suffer from higher attenuation due to a larger penetration o f the electric field into the silicon substrate. This effect is mitigated if the separation between conductor lines is smaller to the thickness of the oxide, referring to attenuation curve o f CPWs with a 5-pim center conductor in Figure 5.13. Figures 5.14 and 5.15 illustrate the attenuation of copper CPWs on a LR Si substrate with wide and narrow ground planes respectively. Again, the attenuation of CPWs with wider line dimensions is higher owing to higher dielectric loss. Comparing Figures 5.14 and 5.15 to 5.12 and 5.13, we notice that copper CPWs exhibit lower attenuation than aluminum CPWs because of their higher metal conductivity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 0 The drop in attenuation curves at the high frequency end, shown in Figure 12-15, conies from the interpolation error of Sonnet Suites. 10 o 10 E E 10 '20pm '50pm •2 10 ,2 Frequency(GHz) Figure 5.12 Simulated attenuation of aluminum CPWs on a LR Si substrate with wide ground planes. 10 '5pm ,0 10 « m -w" £S 1 10 ■2 10' 1 ,2 Frequency(GHz) Figure 5.13 Simulated attenuation of aluminum CPWs on a LR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 10 10nm 10° a 1 10 “1 ■2 10 1 7 Frequency(GHz) Figure 5.14 Simulated attenuation of copper CPWs on a LR Si substrate with wide ground planes. '5Q|im Frequeney(GHz) Figure 5.15 Simulated attenuation of copper CPWs on a LR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 5.2 Dispersion Modal dispersion occurs when signal components with different frequencies propagate at different phase velocities. When electrical signals propagate along a CPW, the effective permittivity, as a function of frequency, contributes to the phase constant and influences dispersion characteristics of the CPW. In this section, we present the experimental data and simulation results o f subterahertz dispersion o f CPWs on a GaAs and a silicon substrate. The effects o f ground-plane width and lateral line dimensions on dispersion characteristics of CPWs are analyzed in Section 5.2.1 and 5.2.2 respectively. 5.2.1 The Effects of Ground-plane Width Figure 5.16 shows the subterahertz effective permittivity of CPWs on a GaAs substrate with a 50-pm center conductor as a function of the frequency [4, 5]. We present the simulated and experimental effective permittivity of CPWs with both wide and narrow ground planes for comparison. There is a good agreement except for the several peaks on the simulated curves. These peaks are remnants of the poles in the Green’s function used in the Sonnet Suites and correspond to the sequential entry of the surface-wave modes. Although the poles are removed one by one in the final results, some oscillations remain and are an unavoidable artifact. According to Eq. (2.9), the static value of the effective permittivity of a wide- ground CPW on a GaAs substrate is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is verified in Figure 5. 16. The static value of effective permittivity of a narrow-ground CPW on a GaAs substrate is slightly lower than that o f a wide-ground CPW but still approximates 6.95. An important observation o f Figure 5.16 is that the CPWs with narrow ground planes return a lower effective permittivity and reduced dispersion in both the experimental and simulated data. This reduction in dispersion is due to the reduction in coupling between the CPW mode and surface modes when the wide-ground planes are trimmed to narrow-ground planes. The propagation of surface modes gives rise not only to radiation loss but also to dispersion. With the frequency increasing, the interaction between the CPW mode and surface-wave modes modifies the phase constant with an increasing mode coupling coefficient [6, 7], For the case o f narrow- ground CPWs, the diminished ground planes weaken the mode coupling coefficient and, consequently, improve the dispersion by reducing the phase constant. We also investigated dispersion characteristics of CPWs with narrower lines. Figure 5.17 shows the subterahertz effective permittivity of CPWs on a GaAs substrate with a 10-pm center conductor [4, 5], The static value of effective permittivities of both wide- and narrow-ground CPWs are in the region of 6.95, as derived above. Again, it can be seen that narrow-ground CPWs encounter lower effective permittivity than wide-ground CPWs. Dispersion is slightly improved in the CPWs with narrow ground planes. In the remaining part of this section, we investigate the effects of ground-plane Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 4 width and lateral line dimensions on subterahertz dispersion o f CPWs on a silicon substrate. The combination o f two kinds o f substrates, HR Si and LR Si, and two kinds of conductors, aluminum and copper, have been examined. Figures 5.18 and 5.19 illustrate simulated effective permittivity of CPWs on a HR Si substrate. Since a layer o f SiC >2 with permittivity of 3.78 is inserted between the conductors and silicon substrate with permittivity of 11.9, the static value of effective permittivity of CPWs on a silicon substrate is between (11.9+l)/2=6.45 and (3.78+l)/2=2.39, as shown in Figure 5.18 and 5.19. We consider two sets of CPWs, one with a 1 0-/um center conductor and the other with a 20-/j.m center conductor. CPWs on a HR Si substrate show similar dispersion characteristics to CPWs on a GaAs substrate. Given the same lateral line dimensions, narrow-ground CPWs operate with slightly lower effective permittivity than wide- ground CPWs. We should notice that, unlike CPWs on a GaAs substrate, the effective permittivity of CPWs on a HR Si substrate monotonically decreases as the frequency increases. The higher the frequency is, the more the effect of Si02 prevails. Since the layer of SiC >2 reduces the coupling between the CPW mode and substrate modes, the effective permittivity of the CPW gradually declines. Comparing Figurse 5.18 to 5.19, we observe that there is not a noticeable difference between the dispersion curves of the CPWs on a HR Si substrate with copper and aluminum conductors. The selection of conductor material does not play a significant role in reduction in dispersion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 9.5 ’eff,sim-wg 'eff,exp-wg 'eff,sim-ng 3= 8.5 'eff,exp-ng a> w 7.5 6.5 100 150 200 250 300 Frequency(GHz) Figure 5.16 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with a 50-pm center conductor. £efr,Sim-wg and £eff!exp_wg refer to simulated and experimental effective permittivity of the wide-ground CPW, respectively. seff,sim-ng and eeff,exp-ng refer to simulated and experimental effective permittivity of the narrow-ground CPW, respectively. 8.5 eff,sim-wg 'eff,exp-wg eff,exp-ng ® 7.5 150 200 250 300 Frequency(GHz) Figure 5. 17 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with a 10-pm center conductor. £eff,Sim-wg and £eff,exp-wg refer to simulated and experimental effective permittivity of the wide-ground CPW, respectively. £eff>Sim-ng and £eff,exp-ng refer to simulated and experimental effective permittivity of the narrow-ground CPW, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 6 eff,10|im-ng MB I MB I • 2 eff,20|im-wg ""“"’s eff,20|ifn-rtg W % m!Srf!frcvs6, Frequency(GHz) Figure 5.18 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with a 10-fim and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes. eff.10tjm-ng st= 'eff,2Qjim-ng _ J 1 _ /' • ! to l'“* ~ ;*■««....>■ ■ - 10‘ Frequency(GHz) Figure 5. 19 Simulated effective permittivity of copper CPWs on a HR Si substrate with a 10-fim and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 w «B Frequency(GHz) Figure 5. 20 Simulated effective permittivity of aluminum CPWs on a LR Si substrate with a 10-fim and 20-fim center conductor respectively. For the subscript in the legend, wg refers to wide-ground planes while ng refers to narrow-ground planes. to mt\ Frequency(GH z) Figure 5. 21 Simulated effective permittivity of copper CPWs on a LR Si substrate with a 10- fim and 20-//am center conductor respectively. For the subscript in the legend, wg refers to wide- ground planes while ng refers to narrow-ground planes. Figure 5.20 and 5.21 illustrate simulated effective permittivity of CPWs on a LR Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Si substrate. As in the case o f HR Si substrate, narrow-ground CPWs suffer lower dispersion than wide-ground CPWs. The effective permittivity of CPWs on a LR Si substrate has a static value located between 6.45 and 2.39, and it declines with the frequency increasing. There is not a noticeable difference between the effective permittivities of CPWs with aluminum and copper conductors. Comparing Figures 5.18 and 5.19 to 5.20 and 5.21, we notice that CPWs on a HR Si substrate suffer lower dispersion than CPWs on a LR Si substrate. Dispersion is related to the conductance, as shown in Eq. (2.17). Thus, the HR Si substrate with higher resistivity and lower conductivity gives rise to lower dispersion. 5.2.2 The Effects of Lateral Line Dimensions In the previous section, we explored the effect of ground-plane width on dispersion characteristics o f CPWs up to sub terahertz frequencies. It shows that narrow-ground CPWs produce lower effective permittivity than wide-ground CPWs. In this section, we proceed to investigate another factor influencing the performance of CPW— lateral line dimensions. To examine the effect o f lateral line dimensions, we combine the narrow-ground CPW data into Figure 5.22 [4, 5]. The static values of effective permittivities of CPWs on a GaAs substrate are around 6.95, which is calculated in Section 5.2.1. It is clearly seen that the effective permittivity of the CPW with a 10-pm center conductor is lower than that of the CPW with a 50-pm center conductor, and the overall dispersion is much less. With the frequency increasing, the enhancing coupling Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 between the CPW mode and surface-wave modes results in growing dispersion. The narrower lines diminish the coupling effect so that the dispersion is alleviated. eff,sim-50 8.5 eff,exp-50 eff,sim-10 eff,exp-10 Jt= 0 to 7.5 150 200 250 300 Frequency(GHz) Figure 5. 22 Simulated and experimental effective permittivity of CPWs on a GaAs substrate with narrow ground planes. The lateral line dimensions are 50 pm and 10 pm, respectively. £eff,sim-50 and £cfr,exp-so refer to simulated and experimental effective permittivity of the CPW with a 50-pm center conductor, respectively. £eff,Sim-io and £eff,exp-io refer to simulated and experimental effective permittivity of the CPW with a 10-pm center conductor, respectively. Finally, the effect of lateral line dimensions on the dispersion characteristics of CPWs on a silicon substrate is studied. Figures 5.23 and 5.24 illustrate the effective permittivity o f aluminum CPWs on a HR Si substrate with wide and narrow ground planes respectively. Since a layer of SiC >2 with permittivity of 3.78 is inserted between the conductors and silicon substrate with permittivity of 11.9, the static value of effective permittivity of CPWs on a silicon substrate is between (11.9+l)/2=6.45 and (3.78+l)/2=2.39, as shown in Figures 5.23 and 5.24. The two figures also illustrate that the dispersion is improved with the lateral line dimensions decrease. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 10 ‘eff,5|im 8 6 $= 0 w 4 MUM * * » < •« » ______. 2 010 1 10 2 Frequency(GHz) Figure 5. 23 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with wide ground planes. < * £ V , ^ ** t Matt * * * * * * "+ ■ Frequency(GHz) Figure 5. 24 Simulated effective permittivity of aluminum CPWs on a HR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 % w * ***1 -, a# > m i *»' •* *> Frequency(GHz) Figure 5. 25 Simulated effective permittivity of copper CPWs on a HR Si substrate with wide ground planes. % m . Frequency(GHz) Figure 5. 26 Simulated effective permittivity of copper CPWs on a HR Si substrate with narrow ground planes. Figures 5.25 and 5.26 illustrate the effective permittivity of copper CPWs on a HR Si substrate with wide and narrow ground planes respectively. The static value of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 effective permittivity is between (11.9+l)/2=6.45 and (3.78+l)/2=2.39. It is shown that, with the lateral line dimensions diminishing, the effective permittivity is decreasing. Comparing Figures 5.23 and 5.24 to 5.25 and 5.26, we observe that there is not a noticeable difference between effective permittivities of CPWs with aluminum and copper conductors. Figures 5.27 and 5.28 illustrate the effective permittivity of aluminum CPWs on a LR Si substrate with wide and narrow ground planes respectively. The static value of effective permittivity is between (11.9+l)/2=6.45 and (3.78+l)/2=2.39. The effective permittivity is reduced for CPWs with narrowed lateral line dimensions. It should be pointed out that the effective permittivity monotonically declines as the frequency increases. This correlation is caused by the insertion of SiC> 2, which plays a more obvious role at higher frequencies because it reduces the coupling between the CPW mode and substrate modes and, consequently, alleviates the dispersion. Figures 5.29 and 5.30 illustrate the effective permittivity of copper CPWs on a LR Si substrate with wide and narrow ground planes respectively. The static value of effective permittivity is between (11.9+l)/2=6.45 and (3.78+l)/2=2.39. It can be seen that narrower lines suffer from lower dispersion. Furthermore, the effective permittivity monotonically declines as the frequency increases. Comparing Figures 5.27 and 5.28 to 5.29 and 5.30, we notice that the selection of conductor material does not have a significant influence on the dispersion characteristics of CPW s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 eff,5M.m ” “ "Seff,10Mm “ '“ ,,£ eff.20Mm St: tMlMllg eff.SOMm 0 W Frequency(GHz) Figure 5. 27 Simulated effective permittivity of aluminum CPWs on a LR Si substrate with wide ground planes. ’eff,5M.m 'eff.SOMm % 03 mimt Mtta* JUUUMC Frequency(GHz) Figure 5. 28 Simulated effective permittivity of aluminum CPWs on a LR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 % to jsms. Frequeney(GHz) Figure 5. 29 Simulated effective permittivity of copper CPWs on a LR Si substrate with wide ground planes. to Frequency(GHz) Figure 5.30 Simulated effective permittivity of copper CPWs on a LR Si substrate with narrow ground planes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 References [1] J. Zhang and T. Y. Hsiang, “Subterahertz attenuation in coplanar waveguides,” Microwave Symposium Digest, 2005 IEEE MTT-S International, Long Beach, CA, pp. 1905-1908, 2005. [2] J. Zhang, S. Alexandrou, and T. Y. Hsiang, “Attenuation characteristics of coplanar waveguides at subterahertz frequencies,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3281-3287, 2005. [3] B. Kleveland, T. H. Lee, and S. S. Wong, “50-GHz interconnect design in standard silicon technology,” Microwave Symposium Digest, 1998 IEEE MTT-S International, vol. 3, pp. 1913-1916, 1998. [4] J. Zhang and T. Y. Hsiang, “Dispersion characteristics of coplanar waveguides at subterahertz frequencies,” Proc. PIERS 2006, Progress in Electromagnetics Research Symposium 2006, Cambridge, MA, pp. 232-235, 2006. [5] Jingjing Zhang and Thomas Y. Hsiang, “Dispersion characteristics of coplanar waveguides at subterahertz frequencies,” Journal of Electromagnetic Waves and Applications, vol. 20, no. 10, pp. 1411-1417, 2006. [6] M. Riaziat, R. Majidi-Ahy, and I. J. Feng, “Propagation modes and dispersion characteristics of coplanar waveguides,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 3, pp. 245-251, 1990. [7] W. M. Louisell, Coupled Mode and Parametric Electronics. New York: Wiley, 1960. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 CHAPTER 6 Distributed-Element Circuit Model An accurate circuit model is essential to the computer-aided design (CAD) of integrated circuits (IC). For the sake of signal integrity and reliability, transmission lines used for an MMIC interconnect should be appropriately modeled. The selection o f an inappropriate interconnect model in IC analysis and simulation may lead to signal delay, distortion, power dissipation, and reliability concerns. With the operating frequency increasing and the feature size decreasing, the propagation characteristics of transmission lines considerably impact the performance of MMICs. IC designers must employ an accurate transmission line model for each critical interconnect with circuit design and simulation tools. In the interconnect-aware design flow, the transmission line model is extracted in the physical design and included in the schematic design. Chapter 5 presents experimental and simulation results from the attenuation and dispersion of CPWs up to subterahertz frequencies. I have analyzed the effects of ground-plane width and lateral line dimensions on the propagation characteristics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 After understanding the degradation and distortion mechanisms of signal propagation, we are able to provide guidelines for CPW-based circuit design and simulation. The distributed-element circuit model of CPWs as a function of frequency is extracted and discussed in this chapter. Section 6.1 describes the extraction method of frequency- dependent transmission line parameters on the basis of S-parameters obtained in Chapter 5. The distributed-element circuit models o f CPWs on GaAs and silicon substrate are extracted in Section 6.2 and 6.3 respectively. From the circuit-design perspective, the effects o f ground-plane width and lateral line dimensions on R, L, G, C parameters of CPWs are evaluated. 6.1 Extraction of Frequency-dependent Transmission Line Parameters 7 2««**« . T Figure 6.1 The two-port network of a transmission line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 A transmission line of length /, characteristic impedance Zc, and propagation constant /can be modeled as a two-port network, as shown in Figure 6.1. The ABCD matrix for the network is given as [1] cosh yl Z c sinh yi A B sinh yi , , (6 .1) — cosh yl Due to the symmetry of the transmission line, the scattering matrix can be simplified with Su = S22 and SI2 = S2j. Provided that both the input and output impedances equal the reference impedance Z0, the relationship between the S- parameters and the ABCD matrix becomes ^ 1 - ^ + 5 ’ 25, (l + 2 5i i+ 5 12,- 5 221)Zq B = 25. 21 (6.2) _ l-2 5 n +5„ -5 21 ABCD 2521Z0 D j_-sfL+s^L 2 S2l Substituting A, B, C a b c d , and D into Eq. (6.1), the propagation constant and characteristic impedance are expressed in terms o f S-parameters, as below: 2 5 0-* - . 21 l-S,!,+S22,± 1/(l + S,!,-5 |,)!-4S l (6.3) /(1 + 5 . J - 5 '(1 -5 J -5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Experimentally, y and Z are the measured quantities and are directly compared with the computed S-parameters [2, 3]. R L ^ V W ---- C G A i Figure 6. 2 The distributed-element circuit model of a transmission line. In an alternative model, Figure 6.2, which is more useful for the circuit designer, the transmission line can be described with a distributed-element circuit model: r = -\I(R + jcoL \G + ja C ) = a + j p R + jcoL (6.4) Z. = G + jcoC where R, L, G, C are the unit-length resistance, inductance, conductance, and capacitance; a and /? are the attenuation and phase constants o f signal propagation, respectively. Solving Eq. (6.4) yields the circuit-element parameters of the transmission line: z = 3{rzJA» (6.5) G = SR{r/Zj C = 3{y/ZJ/ffl Since the S-parameters are readily obtained from our measurements or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 simulations, Eqs. (6.3-6.5) then provide the algorithm from which the circuit-model parameters, shown in Figure 6.2, can be extracted. 6.2 The Distributed-element Circuit Model of CPWs on GaAs In our previous work [2, 3], two classes of CPW lines (made of gold on a GaAs substrate) were studied experimentally and numerically. The first class contains ground planes at least 10 times wider than the center conductor or the conductor spacing (the “wide-ground lines”), closely approximating an ideal CPW, while the second class uses narrow ground planes with the same width as the center electrode (the “narrow-ground lines”). Simulations of signal propagation that make use of full- wave analysis were compared with and verified by experimental data. For computing the distributed-element circuit parameters, we chose to use the numerically simulated S-parameters as the starting point and solve Eq. (6.3) and (6.5). The transmission line parameters R, G, C, L o f CPWs on a GaAs substrate are extracted from S-parameters and, as shown in Figures 6.3-6.6, correspond to the CPWs previously studied [2]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 6.2.1 Resistance The strongest frequency-dependent parameter is the resistance term, R. At very low frequencies, R approaches a constant, corresponding to the quasi-static limit where the resistance per unit length of the CPW can be calculated by where cr is conductivity of the conductor metal, while W and t are the width and thickness of the center conductor. For the 10-pm CPW, R is 7.63 P /mm and, for the 50-pm CPW, it is 1.53 P/mra: both closely matching the values in Figure 6.3. With the frequency increasing (up to 100 GHz), the skin depth becomes smaller than the conductor thickness and surface resistance increases as the square-root of frequency. In this region where conductor loss dominates, wider lines would have lower resistance and lower loss compared with narrower lines, which can be seen from Figure 6.3. At still higher frequencies, above 100 GHz, radiation loss becomes dominant. In this case, narrower lines would have lower resistance and lower loss. The cross-over of the value of Rs for different line-widths as frequency is increased is prominently shown in the data. The width of the ground plane also has an effect on the value of R. Comparing Figures 6.3(a) and 6.3(b), we can see that at higher frequencies, the narrow-ground lines have a lower value of R , as a result of the reduced coupling of surface-wave modes with the ground conductors [2, 4], This effect shows that the narrow-ground lines are preferred over the wide-ground lines for high-frequency applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 6.2.2 Conductance A secondary loss component for the CPW is its conductance term, G. In the quasi static limit, G arises when the dielectric contains a finite loss tangent. The value of G increases with decreasing line separation. For each of the examples shown in Figure 6.4, the line separation equals the line width of the center conductor, thus giving rise to a higher conductance for the narrower lines. In the frequency range of a few 100s GHz, we observe an increase in the value of G, followed by a cross-over in frequency for the two line widths shown, similar to that observed for R. A plausible explanation is that, in this region, part o f the radiation loss contributes to the coupling between the conductors and increases G; since the loss due to the conductance is two orders o f magnitude lower than that from the resistance for the samples considered, a small part of the radiation loss dominates the dielectric loss at high frequencies. We also investigate the effect of ground-plane width on the conductance per unit length o f CPWs on a silicon substrate. Figures 6.4(a) and 6.4(b) illustrate that G o f the wide-ground lines is lower than that of the narrow-ground lines from the smaller area o f air-dielectric interface. 6.2.3 Capacitance The other two terms are the capacitive and inductive terms. In the low-frequency region and for the wide-ground lines, capacitance is determined by [5] C = (sr + l)s 02 ^ ~ , (6.7) r 0 K'(k) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 where s0 is the vacuum permittivity, sr is relative permittivity o f the substrate, and K (k)/K ’(k) is a geometry coefficient that depends only on the line separation-to-width ratio (see Eq. (2.3)). Using this expression, the capacitance o f wg CPWs is found to be 0.157 pF/mm and closely approximates the quasi-static values in Figure 6.5. We also observe that the narrow-ground lines have an almost identical capacitance to that o f the wide-ground lines, showing that Eq. (6.7) is applicable irrespective o f the ground-plane widths. Further, the capacitance for the 10-pm lines remains nearly constant to the highest frequency considered, showing that the surface-wave modes that dominate the high-frequency losses have only a small effect on the overall field distribution. For the 50-pm lines, we observe a drop in C at higher frequencies: this is attributed to a computational artifact. 6.2.4 Inductance Inductance is proportional to the magnetic flux contained in the CPW. Its dominant component is the external inductance, associated with the magnetic flux between the electrodes, and can be expressed as: where c is speed of light and Ca is the capacitance with the substrate replaced by air. Using Eq. (6.8), we obtain L = 0.49 nH/mm, in agreement with the quasi-static values in Figure 6.6. Secondary (and frequency-dependent) components in L include the magnetic flux in the conductor and the surface-wave modes that modify the field distribution. These two effects appear to be negligible, as evidenced by the nearly flat Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 frequency dependence. The drop in L at higher frequencies for the 50-pm lines is attributed to a computational artifact only. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 E E E sz O cc Frequency(GHz) (a) E m E E O cc Frequency(GHz (b) Figure 6. 3 Unit-length resistance R of CPWs on a GaAs substrate, extracted from S- parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 6 m e 10 o *o 00 rr> 10 Frequency(GHz (a) e 10 Frequency(GHz) 0>) Figure 6. 4 Unit-Length conductance G of CPWs on a GaAs substrate, extracted from S- parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and SOpm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 7 10 « * * • * 10 CL (J 10 ,2 .3 10 10‘ 10' Frequency(GHz) (a) 10 10 10 .2 ,3 10‘ 10' Frequency(GHz) (b) Figure 6. 5 Unit-length capacitance C of CPWs on a GaAs substrate, extracted from S- parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 8 10 * 4k * *a*i 10 ,2 .3 10 Iff 10' Frequency(GHz) 10 i f 11 r |------1— i—rr n n[ * ng-10|im o ng-SOpm t # m # x c 0 % % - 1 % 10 ______1___ 1__ L i ■«■ ■ ______i i___ i 111mir 10 10 10 Frequency(GHz) (b) Figure 6. 6 Unit-length inductance L of CPWs on a GaAs substrate, extracted from S- parameters. wg and ng refer to wide ground and narrow ground, respectively. 10pm and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 6.3 The Distributed-element Circuit Model of CPWs on Silicon In this section, we establish the distributed-element circuit model of CPWs on a silicon substrate with aluminum or copper conductors. As in Section 6.2, two classes of CPW lines are studied. The first class contains ground planes at least 10 times wider than the center conductor or the conductor spacing (the “wide-ground lines”), while the second class uses narrow ground planes with the same width as the center electrode (the “narrow-ground lines”). Based on the numerically simulated S- parameters, we extract transmission line parameters R, G, C, L of CPWs by solving Eq. (6.3) and (6.5). The extracted parameters are plotted and discussed. 6.3.1 Resistance Comparing to R o f CPWs on a GaAs substrate, which is direct-bandgap material and has better high-frequency properties, we observe that the resistance per unit length of CPWs on a silicon substrate cannot be predicted by Eq. (6.6) at around 10 GHz. Looking into the effect of lateral line dimensions on the unit-length resistance of aluminum CPWs on a HR Si substrate, as shown in Figure 6.7, we found that the resistance increases with the widened lateral lines, which are equal to the separation between conductor lines in my study. When the separation between conductor lines is comparable to the thickness of the oxide, the electric field lines penetrate into the silicon substrate [6]. Due to the comparatively large dielectric conductivity of the silicon substrate, dielectric loss plays an important role and contributes to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 increasing unit-length resistance. The 5-pm line is exclusive because its separation between conductor lines is small, as compared with the thickness of the oxide. Considering copper CPWs on a HR Si substrate (see Figure 6.8), aluminum CPWs on a LR Si substrate (see Figure 6.9), and copper CPWs on a LR Si substrate (see Figure 6.10), similar behaviors will be discovered. The selection of the conductor or the doping density of the silicon substrate does not markedly determine the unit-length resistance of CPWs. We also investigated the effect of ground-plane width on the resistance per unit length o f CPWs on a silicon substrate. As Figures 6.7(a) and 6.7(b) illustrate, the wide-ground lines operate with lower unit-length resistance at frequencies below 100 GHz, while the narrow-ground lines operate with lower unit-length resistance at frequencies above 100 GHz. This phenomenon can be explained by the reduced coupling of surface-wave modes with the ground conductors within the high frequency range [3, 5]. The comparison between Figures 6.8(a) and 6.8(b), 6.9(a) and 6.9(b), and 6.10(a) and 6.10(b) draws the same conclusion. The drops in R at the high-frequency end are attributed to a computational artifact only. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 10 i ~ i rT T 'i" • w g-5|im a wg-10|im « w g-20|im 10 * wg-50pm E I 10' osz cc 10 10' J I ...... J I L 10 10 Frequency! GHz) (a) 1 1 1 1 ng-5|im *> ng-10iim ng-20um ng-50|j,m a » 10 Frequency(GHz) (b) Figure 6. 7 Unit-length resistance R of aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 2 icr "1 i ...r i n |— ------1------1— r ~ • wg-5g.m o w g - 10|im ft w g -20p,m 10 ft wg-50|o,m 4ftftMee#4 m W 1 I 10' a 1 * * * * 4 1= ft I a 8 t a i l ” o fi CL t 10° — ■ i ...... 1 ■ i i 10' o1 102 Frequency(GFIz) (a) 10 i i i i n t— r « ng-5um 9 ng-IOpm ft n g -20|im 10 ft ng-50y,m E 10 x: a a O a * CL 10 -1 10 ■ « i i i i i J__ L 10 10 Frequency(GHz) (b) Figure 6. 8 Unit-length resistance R of copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 10 I r T T T T * w g-5|im o wg-IOpm * wg-2Q|im 10 * w g-50|im E * & a*** I 10' A s z I i I I f t f t i o 1 O ft cc 10° 10 J I ■ ■ ■ i i ■ J I L 10 10 Frequency(GHz) (a) 10 I | | IT t— r • ng-5p.m o ng-10y,m » ng-20g,m 10 * ng-50|im E 10 I I •ftSioS** s z & S ft O i QC 10° -1 10 J ...... J___ L 10 10 Frequency(GHz) (b) Figure 6. 9 Unit-length resistance R of aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 10J l i i i i i • w g-5|im o wg-10pm *■ w g-20um 10 * wg-50|im E E £ 10 4 4 * * A * n 10 - 1 10 J I I I I I 11 J I L 10 10 Frequency(GHz) (a) 10 I I I I I I t— r « ng-5|am o ng-10|im 9 ng-20|im 10 4 ng-50g.m E ,®a§1 I 101 a a it i • Aa SZ i a O a ® cc 1 10° -t 10 _i_L J L 10 10 Frequency(GHz) (b) Figure 6 .10 Unit-length resistance R of copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 6.3.2 Conductance Figures 6.11-6.14 demonstrate the conductance per unit length of CPWs on a silicon substrate. It has been noticed that the narrower lines suffer from lower unit-length conductance, G. As explained in Section 6.2.2, part o f the radiation loss contributes to the coupling between the conductors and increases G; since the loss due to the conductance is two orders of magnitude lower than that due to the resistance for the samples considered, a small part of the radiation loss dominates the dielectric loss at high frequencies. Comparing to G o f CPWs on a GaAs substrate, which has better high-frequency properties, the conductance per unit length of CPWs on a silicon substrate shows high-frequency behavior at around 10 GHz. We also investigated the effect of ground-plane width on the conductance per unit length o f CPWs on a silicon substrate. Comparing Figures 6.11(a) and 6.11(b), we can see that G of the wide-ground lines is lower than that of the narrow-ground lines due to the smaller area of air-dielectric interface. The comparison between Figures 6.12(a) and 6.12(b), 6.13(a) and 6.13(b), and 6.14(a) and 6.14(b) draws the same conclusion. Considering Figures 6.11-6.14, we observe that the selection of conductor and substrate resistivity does not apparently change the G of the CPWs on a silicon substrate. The drops in G at the high-frequency end are attributed to a computational artifact only. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 -1 10 I I I I I • wg-5pm a wg-IOpm -2 » wg-20(j,m 10 * wg-50nm t A *9 A* E * * E t i l l l GO 9 9 C -3 O O Q a 9 CD 10 E go W 0 -4 10 J ...... 1 J L 10 10 Frequency(GHz) (a) 10 I I I I I I t------1— r • ng-5u.m o ng-10(j,m « ng-20um 10' * ng-50|am A A A E A A A A 9 E 9 9 » 9 "go O a a a 9 J ■ i i i i i J L 10 10 Frequency(GHz) (b) Figure 6.11 Unit-Iength conductance G of aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 E E 1% c o> E GO C5 .2 10 10' Frequency(GHz) (a) 10"' T—I I I II i— r • ng-5pm a ng-10|im s ng-20|am 10 * ng-50|j,m & 4 Ik & E g i « E . 'co a ® 10 E 30 mm CO cu 10 J » I I » I I J L 101 102 Frequency(GHz) (b) Figure 6.12 Unit-length conductance G of copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 10' ■ i i i i i • w g-5|im o wg-IOpm -2 « wg-20(j.m 10 * wg-50|i.m 4 * & A 44 E A 4 k E « § i t S S w -3 ® 10 E C5 10' J L 10 10 Frequency(GHz) (a) -i 10 t i i i ri" t------1— r * ng-5M,m a ng-10|im -2 * ng-20um 10 * ng-50|im E 4 4 E Vl o a a a g C -d 10 ft * i * E go CO <3 10' J ...... J__ L 10 10 Frequency(GHz) (b) Figure 6.13 Unit-length conductance G of aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 9 10"' I I I I I I | • wg-Spm o wg-IOpm 0 wg-20|im 10'2 A wg-50pm * A A AAA E A A E • ftiSS. 0 • 0 0 0 - a a a J g 10 E • • GO < 03 10'4 J I I 1 I I I __1 J I L 10 102 Frequency(GHz) (a) 10" t------1— r * ng-5|im M o ng-10|im * ng-20um 10 * ng-50|im * * 4 & l E i * | E » * $ 0 "oo Q O a a g S 10 » * £ at CO 03 10 J » i i i n J L 10 10 Frequency(GHz) (b) Figure 6.14 Unit-length conductance G of copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 6.3.3 Capacitance Figures 6.15-6.18 demonstrate the capacitance per unit length of CPWs on a silicon substrate. In the low-frequency region and for the wide-ground lines, the unit-length capacitance, C, determined by Eq. (6.7) is 0.146 pF/mm, which closely approximates the quasi-static values in Figures 6.15-6.18. We observe that the unit-length capacitance remains nearly constant to the highest frequency considered, showing that the surface-wave modes that dominate the high-frequency losses have only a small effect on the overall field distribution. We also observe that the narrow-ground lines have almost identical capacitance to that of the wide-ground lines, showing that Eq. (6.7) is applicable irrespective of the ground-plane widths. For the CPWs in my study, Eq. (6.7) does not reflect the effect of lateral line dimension because the width of the center conductor equals to the spacing between conductors. The layer of Si (>2 between the electrodes and the substrate has a permittivity much lower than the silicon substrate. Thus, the capacitance of the CPW can be considered as that of the parallel-plate capacitor formed by the center conductor and the high-permittivity silicon substrate. The unit-length capacitance of the CPW based on SiC>2 and silicon substrate is computed by C = srs0—, (6.9) K w here Er is relative permittivity o f SiC> 2, S is the width of the center conductor, and tQ is the thickness of the SiC>2 layer. For the 5-, 10-, 20-, and 50-pm lines, their unit- length capacitances are 0.024, 0.048, 0.096, and 0.239 pF/mm and approximate the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I l l values o f capacitance in Figures 6.15-6.18. The errors in 5- and 10-pm lines are larger than those in 20- and 50-pm lines because their widths of center conductor are small, as compared with the thickness of the SiC >2 layer. Therefore, the wider lines have higher capacitance because their larger metal area results in higher substrate capacitance. Considering Figures 6.15-6.18, we observe that the selection of conductor and substrate resistivity does not apparently change the C of the CPWs on a silicon substrate. The drops in C for wide-ground lines at the high-frequency end are attributed to a computational artifact only. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 2 10 I I I I I I i r • wg-5|im o wg-10|am » wg-20u.m * wg-50[j,m i iiiii 10-1. LL C l o -2 10 J I I I I » I 10 10 Frequency(GHz) (a) 10° i — r • ng-5pm a ng-10|am * ng-20[j,m a ng-50g,m E A A A A A A A AAA E g | I a iiiieg LL. CL O 10 I I M i l J L 10 10 Frequency(GHz) (b) Figure 6.15 Unit-length capacitance C of aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 3 10 ~l 1 11 M | t — r • w g-5|im a wg-10g,m « wg-20um A wg-SOpm tA* I I itU ii I 10',J Q_ o 10" J ...... J L 10 10 Frequency(GHz) (a) 10° l l l l l l • ng-5p.m a ng-1 Oum a ng-20|am & ng-50|am E A A AAA E I I a ii i “ ' in Q_ u 10 j « i » 11 ■ 10 10 Frequency(GHz) (b) Figure 6.16 Unit-length capacitance C of copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 102 Frequency(GHz) (a) 10° I I I I ng-5nm ng-IOpm a ng-20|im ng-50|am E E a aa ii LL Q_ o 10 10 Frequency(GHz) (b) Figure 6.17 Unit-length capacitance C of aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 5 10 I I I I I I t------1— r * w g-5|im o wg-10um * wg-20pm * wg-50|im .1 10"'* (J -2 10 J I ...... J I L 10 10 Frequency(GHz) (a) 10 I I I I II t r * ng-5|im a n g - 1 0 m ,iti * ng-20(j.m * ng-50|im *ft ft4 ft i ia ii I 10' 1 CL -2 10 j ...... 10 10 Frequency( GHz) (b) Figure 6.18 Unit-length capacitance C of copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 6.3.4 Inductance Figures 6.19-6.22 demonstrate the inductance per unit length of CPWs on a silicon substrate. The unit-length inductance, L, is proportional to the magnetic flux contained in the CPW. As stated in Section 6.2.4, L consists of both the external inductance, associated with the magnetic flux between the electrodes, and the internal inductance, associated with the magnetic flux in the conductor and the surface-wave modes that modify the field distribution. According to Eq. (6.8), the external inductance is 0.49 nH/mm, in agreement with the quasi-static values in Figure 6.19- 6.22. If we compare the unit length inductance o f CPWs on a silicon substrate to that of CPWs on a GaAs substrate, we can see the effects of the magnetic flux in the conductor and the surface-wave modes that modify the field distribution cannot be neglected for CPWs on a silicon substrate. As shown in Figure 6.19, the wider lines with a wider metal area operate with higher inductance. Further, L slightly decreases as the frequency increases. The same observations can be seen in Figures 6.20-6.22. Figures 6.19(a) and 6.19(b) illustrate that wide-ground lines and narrow-ground lines show similar behaviors except for the drops in L at the high-frequency end. We can draw the same conclusion from the comparison between Figures 6.20(a) and 6.20(b), 6.21(a) and 6.21(b), and 6.22(a) and 6.22(b). The drops in L at the high- frequency end are attributed to a computational artifact only. Considering Figures 6.19-6.22, we observe that the selection of conductor and substrate resistivity does not apparently change L o f the CPWs on a silicon substrate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 7 10u T l I I I I t— r • wg-5jim a wg-IOpm * wg-20|im 4 wg-50|im 4 4 E 4 4 4 4 4 4 4444rtMft«arf -i 10 » I » I I I J L 10 10 Frequency! GHz) (a) 10 I I I I I I t— r • ng-5u,m a ng-10|im « ng-20|am 4 ng-50pm E 4 4 A 4 A 4 A E i I § 2 « « » » aO®, X c ■ 9 f i « » 10 Frequency(GHz) (b) Figure 6.19 Unit-length inductance L of aluminum CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 8 10 I I II t— r • wg-5pm a wg-10um * wg-20|im a wg-50g.m E A A A A A A AA £ § ft 9 0 0 *•* ' “ X § c I i S S * 1 - 1 10 J I I 11 11 J L 10 10 Frequency(GHz) (a) 10 I I I I I I • ng-5nm a ng-10|im « ng-20(j,m * ng-50|im & & & it m * x 2 0 0*000* c f 99«« - 1 10 J ■ i i i i i 10 10 Frequency GHz) (b) Figure 6. 20 Unit-length inductance L of copper CPWs on a HR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 9 10 ■ i l l • w g-5pm 0 wg-IOpm & wg-20pm A wg-50pm A A E A A A A A E ft ft A « » * • X c 8 8888 - 1 10 J ■ i i i i i J I L 10 10 Frequency(GHz) (a) 10 T ' l'l I I' t— r • ng-Sp-m a ng-10pm « ng-20pm A ng-50p,m E A A E 1 ft ft * A A A &AA fl 9 X sz 10' J ...... J L 10 10 Frequency(GHz) (b) Figure 6. 21 Unit-length inductance L of aluminum CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 0 10 i 1 1 1 • wg-5|am a wg-10pm « w g-20|im A w g-50|im E -S. * A A A & A E a » « » #»; X c * 8S**1 10 Frequency! GHz) (a) 10 • ng-5um o ng-10um « ng-20|im * ng-50|im E A E A * * A A AAAAA** a * P » »»»«, X c 9 9i9* -i 10 J I...... 1 1.1,111 J I 10 10* Frequency(GHz) (b) Figure 6. 22 Unit-length inductance L of copper CPWs on a LR Si substrate, extracted from S-parameters. wg and ng refer to wide ground and narrow ground, respectively. 5pm , 10pm, 20pm, and 50pm indicate the lateral line dimensions of the CPWs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 References [1] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1979. [2] J. Zhang, S. Alexandrou, and T. Y. Hsiang, “Attenuation characteristics of coplanar waveguides at subterahertz frequencies,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3281-3287, 2005. [3] J. Zhang and T. Y. Hsiang, “Dispersion characteristics o f coplanar waveguides at subterahertz frequencies,” Journal of Electromagnetic Waves and Applications, vol. 20, no. 10, pp. 1411-1417, 2006. [4] F. Schnieder, T. Tischler, and W. Heinrich, “Modeling dispersion and radiation characteristics of conductor-backed CPW with finite ground width,” IEEE Trans. Microwave Theory Tech., vol. 51, no. 1, pp. 137-143, 2003. [5] K. C. Gupta, R. Garg, and R. Chadha, Computer-aided Design of Microwave Circuits. Dedham, MA: Artech House, 1981. [6] B. {Cleveland, T. H. Lee, and S. S. Wong, “50-GHz interconnect design in standard silicon technology,” Microwave Symposium Digest, 1998 IEEE MTT-S International, vol. 3, pp. 1913-1916, 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 CHAPTER 7 Summary In this dissertation, I investigated the propagation characteristics of CPWs with wide and narrow ground planes over a broad range o f frequencies up to subterahertz. Wave propagation on CPWs was simulated by a full-wave analysis and compared with experimental results previously obtained by using a subpicosecond electro-optic measurement technique. Considering the imperfection of materials and the presence of the substrate, I described the mechanism of conductor loss, dielectric loss and radiation loss. I also analyzed the effects o f ground-plane width and lateral line dimensions on the attenuation and dispersion characteristics in the subterahertz frequency domain in detail. On the basis of the study of propagation characteristics of the CPW, the results and analysis o f simulation and experiments were incorporated into a distributed-element circuit model. For the purpose of providing guidelines for MMIC designers, detailed design considerations for CPW-based MMICs were presented. The attenuation and dispersion of wide- and narrow-ground CPWs with gold conductors on a GaAs substrate and aluminum or copper conductors on a HR Si or LR Si substrate have been simulated and validated by the good agreement with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 experimental results. The CPWs with narrow ground planes have lower attenuation than the CPWs with wide ground planes over a few 100s GHz, where the radiation loss dominates. Therefore, reducing the width of ground planes leads to an obvious decrease of radiation loss and, consequently, this causes the decrease o f total attenuation at subterahertz frequencies. On the other hand, the CPWs with narrow ground planes suffer higher attenuation up to 200 GHz. As for the frequency range below 200 GHz, conductor loss is a more important factor than radiation loss so that the attenuation characteristics are different than at high frequencies. In addition to ground-plane width, the lateral dimensions also affect the attenuation of the CPWs. For the CPWs on a GaAs substrate, the wider lines suffer lower attenuation, mainly arising from lower conductor loss at low frequencies up to 200 GHz. As the frequency increases up to 1 THz, the narrower lines perform better with lower attenuation due to lower radiation loss. For the CPWs on a silicon substrate, the narrower lines yield lower attenuation at frequencies as low as 10 GHz. The attenuation o f CPWs with copper conductors is slightly lower than that o f CPWs with aluminum conductors. This lower attenuation is determined by the higher conductivity of copper and, in turn, its lower conductor loss. The CPWs on a HR Si substrate show lower attenuation than the CPWs on a LR Si substrate. This lower attenuation is determined by the higher resistivity of HR Si and, in turn, its lower dielectric conductivity. The quasi-static effective permittivity of wide-ground and narrow-ground CPWs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 both approximate the empirical estimation of their static value. The CPWs with narrow ground planes return a lower effective permittivity and reduced dispersion up to subterahertz frequencies. This reduction in dispersion owes to the reduction in coupling between the CPW mode and surface-wave modes when the wide-ground planes are trimmed to narrow-ground planes. The propagation of surface-wave modes gives rise not only to radiation loss but also to dispersion. With the frequency increasing, the interaction between CPW mode and surface-wave modes modifies the phase constant with an increasing mode coupling coefficient. For the case of narrow- ground CPW, the diminished ground planes weaken the mode coupling coefficient and, consequently, improve the dispersion by reducing the phase constant. The effective permittivity o f narrower lines is lower than that o f wider lines and the overall dispersion is much less. With the frequency increasing, the enhanced coupling between the CPW mode and surface-wave modes results in growing dispersion. The narrower lines diminish the coupling effect so that the dispersion is alleviated. While CPWs on a GaAs substrate yield a monotonically increasing effective permittivity, the effective permittivity of CPWs on a silicon substrate is monotonically decreasing as the frequency increases. Since the layer of SiC >2 between the conductors and silicon substrate reduces the coupling between the CPW mode and substrate modes, the effective permittivity of the CPWs on a silicon substrate gradually declines. The selection of conductor material does not play a significant role in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 5 reduction o f dispersion. The CPWs on a HR Si substrate suffer lower dispersion than CPWs on a LR Si substrate. Dispersion is related to the conductance. Thus, the HR Si substrate with higher resistivity and lower conductivity gives rise to lower dispersion. Based on the subterahertz attenuation and dispersion characteristics o f CPWs studied, the distributed-element circuit model of CPWs on a GaAs and a silicon substrate is extracted as a function of frequency. The strongest frequency-dependent circuit-model parameter is the resistance term, R. At very low frequencies, R approaches a constant, corresponding to the quasi-static limit where the resistance per unit length of the CPW can be calculated by its closed-form expression. As the frequency increases, the skin depth becomes smaller than the conductor thickness, and surface resistance increases as the square-root of frequency. In this region where conductor loss dominates, wide-ground and wider lines would have lower resistance and lower loss. At still higher frequencies, radiation loss becomes dominant. In this case, narrow-ground and narrower lines would have lower resistance and lower loss. This effect shows that the narrow-ground lines are preferred over the wide-ground lines for high-frequency applications. The cross-over of the value of Rs for CPWs on a GaAs substrate is located at a higher frequency than that for CPWs on a silicon substrate. A secondary loss component for the CPW is its conductance term, G. In the quasi- static limit, G arises when the dielectric contains a finite loss tangent. G of the wide- ground lines is lower than that of the narrow-ground lines due to the smaller area of the air-dielectric interface. Furthermore, narrower lines show higher G at low Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 6 frequencies, while lower G at high frequencies. The cross-over of the value of Gs for CPWs on a GaAs substrate is located at a higher frequency than that for CPWs on a silicon substrate. The other two terms, capacitance and inductance, are both frequency-insensitive. Their quasi-static values can be approximated by the closed-form expressions. The narrow-ground CPWs have an almost identical capacitance to that of the wide-ground CPWs so the closed-form expression is applicable irrespective of the ground-plane widths. The capacitance for the narrower lines remains nearly constant to the highest frequency considered, showing that the surface-wave modes that dominate the high- frequency losses have only a small effect on the overall field distribution. Further, the wider lines have a higher capacitance, which can be attributed to the higher substrate capacitance due to the larger metal area, as shown more obviously for CPWs on a silicon substrate. Inductance is proportional to the magnetic flux contained in the CPW. The wide- ground and narrow-ground CPWs show similar behaviors so that the closed-form expression for L applies for both of them. As compared to the unit length inductance of CPWs on a GaAs substrate, the effects of the magnetic flux in the conductor and the surface-wave modes that modify the field distribution cannot be neglected for CPWs on a silicon substrate. The wider lines with a wider metal area operate with higher inductance, as shown more obviously for CPWs on a silicon substrate. The selection of conductor and substrate resistivity does not apparently change R, L, G or, C o f the CPWs on a silicon substrate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 7 In summary, this dissertation presents the attenuation and dispersion characteristics and distributed-element circuit model of CPWs with different ground- plane width, conductor, and/or substrate material. The investigation is conducted up to subterahertz frequencies, while most o f the research so far has focused on wide- ground CPWs over a relatively low frequency range. After understanding the degradation and distortion mechanisms of signal propagation, we were able to provide guidelines for the CPW-based circuit design and simulation. The frequency- dependent transmission line parameters are extracted on the basis o f S-parameters. In the future, more efforts can be made to establish empirical equations of R, L, G, C parameters based on current work. Curve-fitting techniques and numerical methods may be employed to achieve this purpose. The frequency-dependent circuit model can be further incorporated into currently available software packages to implement device and circuit design and simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.