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Nonlinear microwave and millimeter wave phenomena in high-temperature superconducting devices

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MICROFILMED 1993
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O rder N um ber 9403289
N on lin ear m icrowave and m illim eter w ave phenom ena in
h igh-tem perature superconducting devices
Kain, Aron Zev, Ph.D .
University of California, Los Angeles, 1993
UMI
300 N. Zeeb Rd.
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UNIVERSITY OF CALIFORNIA
Los Angeles
Nonlinear Microwave and Millimeter Wave Phenomena
High Temperature Superconducting D evices
A dissertation submitted in partial satisfaction of the
requirem ents for the degree
Doctor of Philosophy in Electrical Engineering
by
Aron Zev Kain
199 3
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The dissertation o f Aron Z ev Kain is approved
Tatsuo I ton
Eli Yablonovitch
R. Stan Williams
Harold R. Fetterman,
Com mittee Chair
University of California, Los Angeles
1993
11
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To m y wife Joyce, Children, Parents and In-Laws
But m ost im portantly to the
Rebono Shel Olam
W hose perfect work I am only rediscovering
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Table of Contents
Dedication..............................................................................................iii
List of Figures.......................................................................................vii
Acknowledgments................................................................................ xii
Vita........................................................................................................xiii
Abstract.................................................................................................xvi
1
Introduction
2
K in etic
1
In d u ctan ce
M easu rem en ts
and
M o d elin g
2.1
Introduction............................................................................................ 6
2 .2
Physical origin o f the kinetic inductance
2.3
Tem perature D ependence and Kinetic
effect......... 8
Inductance.................................................................................11
2 .4
W ave Propagation in Superconducting Guiding
Structures...................................................................................15
2.4 .A
London Equations............................................ 15
2.4.B
London Penetration Depth........................... 17
2.4.C
Wave Propagation........................................... 20
iv
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6
3
2.5
Kinetic Inductance Modeling...............................................25
2 .6
Experimental Design and Setup........................................ 31
2 .7
Experimental Results and Discussion................................ 36
2.8
Future Work............................................................................ 42
2 .9
References............................................................................... 49
HTS
D elay
Line
M easurem ents
U sing
Picosecond
Electrical Pulses
3.1
52
Introduction................................................................................ 52
3 .2 Frequency Characteristics of an HTS Delay Line
4
53
3 .3
Optoelectronic Switches......................................................... 57
3 .4
Experimental Design................................................................64
3 .4 .A
HTS Delay Line Fabrication..........................66
3.4.B
Photoconductive Switch Fabrication
3.4.C
Optical Path Layout......................................... 74
69
3 .5
Experimental Results and Discussion.................................77
3 .6
Summary.................................................................................... 103
3 .7
References.................................................................................105
Parametric Interactions
109
4.1
Introduction............................................................................. 109
4 .2
Physical Interpretation of Param etric
Amplification......................................................................... 110
v
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5
4.3
Manley-Rowe Relationships.............................................115
4 .4
Josephson Junction Inductance...................................... 121
4.5
Device Gain..........................................................................127
4 .6
Circuit Implementation and Gain.................................. 130
4 .7
Experimental Design.......................................................... 139
4 .8
Experimental Setup............................................................ 148
4 .9
Experimental Results and Discussion.......................... 151
4 .1 0
Future Work.........................................................................168
4.11
References............................................................................ 171
Conclusion
173
vi
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List Of Figures
2.3.1 Equivalent circuit of electronic conduction in a
superconductor.........................................................................13
2.3.2 Plot of equation 2.3.1 the Tc is 90K................................ 14
2.4.B.1 Geometry used for solving equation 2.4.B.3.............. 19
2.4.B.2 M agnetic field penetration depth from
equation 2.4.B.4....................................................................... 21
2.5.1 Geom etry under investigation; crossection view ....27
2.6.1 The layout o f the coplanar waveguide resonator.
This is a top down view.........................................................33
2.7.1 N orm alized Phase Velocity versus tem perature for
superconducting CPW using 100 nm thick YBCO
film............................................................................................. 37
2.7.2 Norm alized Phase Velocity versus tem perature for
superconducting CPW using 200 nm thick YBCO
film........................................................................................... 38
2.7.3 N orm alized Phase Velocity versus tem perature for
superconducting CPW using 300 nm thick YBCO
film........................................................................................... 39
2 .7 .4 Measured Q o f 5 GHz CPW resonator versus
temperature.............................................................................. 41
vii
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2 .7.5 Resonance frequency for 100, 200, and 300 nm
films at 20K and 77K...............................................................43
2.8.1 Typical resonance curve for narrow width 10
GHz CPW resonator................................................................. 46
2 .8 .2 Superimposed resonance curves of the 10 GHz
resonator displaying power dependence........................47
3.3.1 M icrostrip im plem entation of simple optical
switch......................................................................................... 58
3 .3 .2 Simple optical switch used
as sampling gate............. 60
3.3.3 Pictorial illustration of pulse sampling technique..61
3.4.1 Experimental layout of optical switches and HTS
CPW delay line......................................................................... 63
3.4.A.1 Layout of coplanar delay line for picosecond
measurements............................................................................68
3.4.B.1 Initial processing steps for the fabrication of
optical switches......................................................................... 71
3.4.B.2 Crucial processing steps for the fabrication of
optical switches......................................................................... 72
3.4.C.1 Layout and experim ental setup of picosecond
optical path................................................................................ 75
3.5.1 Sampled input pulse at 36K................................................. 78
3 .5 .2 Sampled output pulse at 36K.............................................. 79
3.5.3 Sampled output pulse at 56K.............................................. 83
3 .5 .4 Sampled output pulse at 76K.............................................. 84
3.5.5 Sampled output pulse at 86K..............................................85
viii
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3.5.6 Sampled output pulse at 88K..............................................86
3 .5 .7 Sampled output pulse at 89K............................................. 87
3.5.8 Sampled output pulse at 90K............................................. 88
3 .5 .9 Peak output pulse delay as a function of
increasing temperature.......................................................... 89
3.5.10 Phase velocity as a function of temperature
of
the delay line and a 10 GHz CPW resonator.............. 90
3.5.11 FFT of input pulse at 36K.................................................... 94
3.5.12 FFT of input pulse at 36K limited to 100 GHz........ 95
3.5.13 FFT of output pulse at 56K................................................96
3.5.14 FFT of output pulse at 76K................................................97
3.5.15
FFT of output pulse at 86K................................................. 98
3.5.16
FFT of output pulse at 88K.................................................99
3.5.17
FFT of output pulse at 89K...............................................100
3.5.18
FFT of output pulse at 90K...............................................101
4.2.1 C ircuit illustrating param etric am plification..........I l l
4 .2 .2 Output voltage of the circuit in figure 4.2.1.......... 114
4.4.1 Plot of equation 4.4.7 for varied critical
currents.................................................................................. 124
4 .4 .2 Plot of equation 4.4.7 in time. A 10 GHz
current signal is used......................................................... 125
4.6.1 RSJ equivalent circuit model...........................................132
4 .6 .2 Param etric am plifier equivalent circuit.................... 133
4 .6 .3 Simple param p reduced equivalent circuit............. 136
4 .6 .4 Plot of J liO L p )2 ............................................................... 140
ix
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4.6.5 Computer simulation of paramp gain, note the
device parameters............................................................... 141
4.6.6 Computer simulation of paramp gain, note the
device parameters............................................................... 142
4.7.1 Crossection view of step edge junction........................143
4.7.2 AFM of step edge junction. Step height is 130
nm.......................................................................................... 145
4.7.3 Typical I-V curve of step edge junction. Notice
the RSJ behavior..................................................................146
4 .7 .4 CAD layout o f 10GHz parametric amplifier................147
4.8.1 Experim ental setup to measure parametric
amplifier............................................................................... 149
4.9.1 25
junction array I-V curve at 4.2K.............................152
4 .9 .2 25
junction array I-V curve at 8K................................ 153
4.9.3 25
junction array I-V curve at 23K..............................154
4 .9 .4 25
junction array I-V curve at 45K............................. 155
4 .9 .5 25
junction array I-V curve at 60K..............................156
4 .9 .6 Reflected power of paramp with pump power
too low................................................................................... 158
4 .9 .7 Reflected power of paramp with activating
pump power. Notice the onset of parametric
interactions........................................................................... 160
4.9.8 Reflected pump power versus reflected signal
and idler power....................................................................162
4.9 .9 M agnified view of junction layout in paramteric
x
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amplifier................................................................................164
4.9.10 Internal signal gain versus tem perature...............166
4.9.11 Reflected pump power versus reflected signal
and idler power at 10K and 30K................................ 167
4.10.1 Layout of 60 GHz monolithic HTS parametric
amplifier............................................................................... 169
xi
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ACKNOWLEDGMENTS
F orem ost,
I
w ould
like
to
thank
my
th esis
advisor
P rofessor H arold R. Fetterm en for his unwavering enthusiasm ,
com m itm ent and
insightful teaching
through
out my Ph. D.
studies. His ability to clearly articulate and focus in on both the
problem at hand and the overall picture is a rare talent that I
deeply appreciate. I not only consider Prof. Fetterman a m entor
in my studies but a true friend.
I w ould also like to thank Professors Tatsuo Itoh, Eli
Y ablonovich, and R. Stan W illiams for their time and effort in
service on my doctoral committee.
I especially thank my colleagues and friends w orking in
Prof.
F etterm an's
and very productive
L aboratory
for their
rum p sessions
com m ents,
w ithout
suggestions
which my studies
would not have been as enjoyable as they were.
I am also very grateful to my colleagues at TRW too
num erous to enumerate here, for providing m e with samples for
th ese
e x p erim e n ts and
all
th eir
h elpful
d iscu ssio n s.
I'm
particularly grateful to Dr.'s Arnold Silver and Andrew Smith for
sponsoring me for my TRW Doctoral Fellowship.
xii
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V IT A
Aron Zev Kain
July 11, 1960
Born, Bridgeport, Conn. USA
198 2
B. A., Yeshiva University, New York
City
1984
M. S. E. E., Columbia University,
New
York City
1 9 9 0 -1 9 9 3
TRW Doctoral Fellowship, TRW Inc.,
Redondo Beach, California
1993
Ph. D., University of California at
Los
A ngeles
PUBLICATIONS
A. Z. Kain, G. S. Lee, " Demountable, High Density Package
for High-Speed Testing o f Superconducting
Electronics" Proceedings o f the ASM E Winter Annual
M ee tin g , AES-22, p39 (1990)
X lll
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A. Z. Kain, H.R. Fetterman, " Measurement of Parametric
Interactions in YBCO Step Edge Junctions", Appl. Phys.
L e tt , Sept. 1993
A. Z. Kain, J. M. Pond, H. R. Fetterman, "Impact of Kinetic
Inductance Effect in High-Tc Superconducting
Coplanar W aveguide Resonators" Microwave and
Optical Technology Letters, (Special Issue on
Superconductive Microwave Devices and Circuits),
Dec. 1993
A. Z. Kain, C.L. Pettiette-Hall, K.P. Daly,A.E. Lee, R.Hu, J.F.
Burch," Dielectric Properties o f SrTi03 Thin Films at
Low Temperature", IEEE Transactions on Magnetics,
M arch, 1993
A. Z. Kain, H.R. Fetterman, " Parametric Interactions in
High-Tc Superconducting Step Edge Junctions at XBand", Physica C, 209, p281 (1993)
C. M. Jackson, J.H. Kobayashi, A. Z. Kain, C. L. Pettiette-Hall,
A. E. Lee, R. Hu, J. F. Burch, " Novel Monolithic Phase
Shifter Combining Ferroelectrics and High
T em perature Superconductors", Proceedings o f the
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5th International Symposium on Integrated
F erro electrics, April 1993
J. Luine, K. Daly, R. Hu, A. Z. Kain, A. Lee, H. Manasevit, C.
Pettiette-Hall, R. Simon, D. St. John, M. Wagner, " 1 0
GHz Surface Impedance Measurements of (Y,Er)BaCuO
Films Produced by MOCVD, Laser Ablation, and
S puttering", IEEE Transactions on Magnetics, M a g -2 7 ,
p 1528 (1991)
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ABSTRACT OF THE DISSERTATION
N onlinear M icrowave and M illim eter W ave Phenom ena
in
High Tem perature Superconducting Devices
by
Aron Zev Kain
Doctor o f Philosophy in Electrical Engineering
University of California, Los Angeles, 1993
Professor Harold R. Fetterman, Chairman
N o n lin e a r
su p e rc o n d u c to rs
in d u c ta n c e
(H T S)
e ffe c ts
have
been
in
h ig h
te m p e ra tu re
in v e s tig a te d
at
b o th
m icrow ave and m illim eter wave frequencies. These effects can
be characterized as tem perature dependent effects such as the
k in e tic
in d u ctan c e,
and
device
dependent
e ffe cts
such
as
Josephson ju n ctio n s incorporated in param etric am plifiers. W e
have designed, m easured, m odeled, and analyzed structures that
ex h ib it these effects so as to better understand and provide
quantifiable design criteria for the developm ent of such devices
as
narrow band filters, resonators,
delay
iines and param etric
xvi
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am plifiers.
W e exam ined
coplanar w aveguide resonators at 5
GHz in order to m odel the nonlinear kinetic inductance effect.
W ith m easured Q ’s o f 16,000 we were able to model the devices
behavior w ith a " T squared" dependence which is drastically
d ifferent than the usual G orter-C asim ir dependence. Using this
model we designed a coplanar waveguide delay line. By optically
g en eratin g
a
p ico seco n d
exam ine the
electrical
p u lse
we
w ere
able
to
bandw idth and dispersion response o f the delay
line up to 100 GHz. The analysis o f this structure's perform ance
c o n firm s
th e
v e ra c ity
of
our
k in e tic
in d u ctan c e
m odel.
Investigating the nonlinear inductance device dependent effects,
we developed the first all HTS Josephson junction param etric
am plifier. W e have fabricated and tested series arrays o f HTS
e n g in eered
step
edge ju n ctio n s
fo r
four
photon
param etric
effects at 10 GHz. The series array o f 25 junctions shows a 10 dB
in c re ase
in
re fle c te d
signal
pow er
as
the
pum p
pow er
is
increased at 10 K. A t 50 K the reflected signal power shows a 3
dB
increase.
frequency
anom alous
w hereby
The reflected pow er
su g g ests
gain
the
four photon
satu ratio n
re fle c te d
characteristic
in te rac tio n .
effect
signal
at the
is
p o w er
In
addition,
experim entally
m ain tain s
a
id ler
an
observed
c o n stan t
increased value even as the reflected idler power decreases.
xvii
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CHAPTER 1
Introduction
The discovery
becom e
of ceram ic
sup erco n d u ctin g
at
m aterials w ith the ability
elevated
tem peratures
in
to
1987
sparked much renewed interest in the field of superconducting
electro n ics.
O ne
area
that
p articu larly
benefited
from
this
discovery was m icrow ave and m illim eter wave electronics. It
was quite natural to take these ceram ic m aterials and try and
p ro d u ce
thin
m icrow ave
film s
and
on
m illim eter
d iffe re n t
wave
su b stra te s.
passive
Since
com ponents
m any
require
only a single layer of patterned m etalization and a dielectric
substrate onto which the conductor resides, the effort was made
to
p ro d u c e
th ese
p a ssiv e
co m p o n en ts
u sin g
the
high
tem perature superconductors (HTS). The initial ceramic m aterial
1
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o f choice was yttrium
advances
allow
the
barium
use
of
copper oxide (YBCO). Recent
thallium
based
com pounds
that
exhibit higher transition temperatures than does YBCO.
At the time of their discovery and continuing today, the
potential
high
for device
tem perature
and com ponent developm ent using
superconductors
superconductors in thin film
m any
low
loss
w ide
form
bandw idth
is
vast.
High
these
tem perature
allow for developm ent of
p assive
m icrow ave
and
m illim eter wave components that would not be possible with
co n v en tio n al
m etals.
U ltra
w ide
bandw idth,
n o n-dispersive
delay lines using m onolithic topologies such as m icrostrip and
coplanar waveguide are presently being developed. 50 ns nondispersive delay lines with 100 GHz bandwidth are simply not
possible with normal metals. However, using HTS this sort of
device is realistically possible. Other devices such as coupled
line, delay line chirp filters have already been developed at MIT
Lincoln Labs using these high tem perature superconductors. In
addition,
the
ordinary
w orkhorse
m icrow ave
and
m illim eter
wave components such as resonators, filters, matching networks,
couplers and hybrids have all been designed, developed, and
fabricated using HTS on a variety of substrates.
In addition, because the materials exhibit very low surface
resistance at microwave frequencies, narrowband high Q filters
are also possible and are presently being developed. One key
2
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area in which these filters will find application is in the area of
m obile com m unications. As more and more cellular phones are
produced
the frequency spectrum
over w hich
com m unications
can occur becom es more finely divided up. Low loss, high Q,
narrow band
filters are essential for preserving com m unication
integrity and fidelity. Such filters, again, are not possible using
conventional norm al metal microwave technology.
A nother area in which
m icrowave and m illim eter wave
electronics can benefit from high tem perature superconductivity
is in the development of active devices. The development of the
SQUID and the Josephson junction in HTS allows for the exciting
p ossib ility
d etecto rs
o f the
and
devices can
developm ent o f param etric
m ixers,
and
active
phase
am plifiers,
shifters.
be m onolithicaliy produced which
All
RF
these
should greatly
influence both cost an reliability issues.
However, there is a price to pay for the vast benefits that
the high tem perature superconductor m icrowave and m illim eter
wave
com ponents
p rin c ip le s
th a t
prom ise.
govern
One
th e ir
of
the
b eh av io r
c ritical
is
th e ir
underlying
n o n lin e ar
inductance. T his nonlinear inductance m anifests itse lf in two
distinct ways. The first is the nonlinear effects that are produced
due to a change in temperature of the device. These effects have
been
ex p erim en tally
seen
course, when designing
but
not
adequately
m odeled.
Of
such devices as phase shifters, delay
3
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lines, filters
and resonators these effects m ust be taken into
account. Should they not be taken into account, these nonlinear
effects are so dom inant that some designs such as narrowband
filters can
be hopelessly invalidated.
The ability to quantify
these effects to the extent that design equations are developed
would be of great benefit.
The second m anifestation of the nonlinear inductance is
produced
not by tem perature changes, but rath er due to the
device itself. The Josephson junction is a prim e example of this.
It is the ju n ctio n itself that behaves as a lossless, nonlinear
inductor.
In fact, superconducting param etric am plifiers would
not be possible without this nonlinear effect. To date, no one has
experim entally
verified that these nonlinear inductance device
effects can be used to develop such an HTS amplifier.
This thesis therefore exam ines these nonlinear inductance
effe cts
p rovide
of
an
both
tem perature
overall
picture
and
of
device
these
design
effects
to
in
order to
the
design
engineers. The thesis will form a solid basis for understanding
th ese
fu n d am e n ta l
im p o rta n tly ,
p rin cip les
th ro u g h
in
a
con crete
e x p e rim e n ta tio n
and
fashion.
M ore
a n a ly sis
very
im portant principles that are unique to superconductivity will
be quantified.
4
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The thesis is divided into five chapters. The developm ent
of
the
tem perature
presented
effects
inductance
resonators
T h is
the
nonlinear inductance
a re
in chapters 2 and 3, while the device effects are
presented in chapter 4.
atypical
in
effect.
are
Chapter 2 is devoted to
M easurem ents
perform ed
and
the
tem perature dependence
e ffe c t
is
fu rth e r
and
m odeling
startling
of this
q u a n tifie d
of
high
conclusion
effect
to
the kinetic
is
allow
Q
of an
presented.
im m e d ia te
incorporation into the design of devices. Chapter 3 takes the
developm ents of chapter 2 and applies them to an ultra w ide
bandw idth delay line using coplanar waveguide geom etry and
optical switches. All the results o f chapter 2 are reconfirm ed in
this delay line experim ent. Chapter 4 provide the developm ent
of the "device caused" nonlinear inductance effects and the first
h ig h
te m p e ra tu re
s u p e rc o n d u c tin g
p a ra m etric
a m p lifie r
conclusion
for
an aly sis
of
the
th is
is
en tire
c ru c ial
p resen ted .
Josephson
C hapter
developm ent,
m icrow ave
5
ju n c tio n
provides
experim entation,
and
m illim eter
a
and
w ave
nonlinear inductive effect in high tem perature superconducting
d ev ices.
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CHAPTER 2
Kinetic Inductance Measurement and Modeling
2.1
INTRODUCTION
T hin
film
high
tem perature
superconductors
have
been
employed to design and develop passive microwave components
such as filters, delay lines, and resonators. Devices based on
coplanar w aveguide (CPW ) geom etries offer both an attractive
means of fabrication and ease o f experim ental im plem entation.
To accurately design superconducting m icrowave circuits it is
necessary to be able to predict the effects o f superconductivity
on
planar w aveguiding
structures.
In particular,
the
internal
inductance of the superconductor, consisting of both the internal
6
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m agnetic inductance and the kinetic inductance [1], results in a
te m p e ra tu re
dependence
te m p e ra tu re
dependence
superconducting
transition
of
the
is
p h a se
m o st
v e lo c ity .
p ro n o u n ce d
tem perature,
T his
near
the
TC- The essence of the
internal inductance effect is the tem perature dependence of the
superconducting penetration depth, X .
Recently there has been debate as to the actual functional
form
of this tem perature dependence for the high tem perature
s u p e rc o n d u c to rs
[2].
s u p e rc o n d u c to rs
U n lik e
w h ich
m any
e x h ib it
c o n v en tio n a l
a
type
G o rte r-C a s im ir
II
type
tem perature dependence, HTS data is better described using a
BCS
w eak-coupled
tem perature dependence or a "T squared"
dependence. This dependence is given by:
X = Xo / (1—(T/Tc)2)m
Xo is the zero tem perature penetration depth and Tc is
where
th e
(2.1.1)
c ritic a l
dependence
tem p eratu re
of
X
of
and hence
the
film .
the phase
T his
"T
squared"
velocity, becom es a
critical consideration when designing m icrowave components.
In conventional planar transm ission line geom etries using
ty p ic a l
d im e n sio n s
th e
p h a se
v e lo c ity
v a ria tio n
w ith
tem perature can be several percent. The prim ary benefit to be
7
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derived from the use o f HTS in microwave circuits is the high
quality factor that perm its the design and construction of high Q
resonators and narrow band filters which are not possible with
norm al
conductors.
O bviously it is
necessary
to be
able
to
p redict the phase velocity to within 0.1% in order to accurately
and reliably
design a m ultipole filter of 1% or less bandwidth.
T hese same
considerations are also true for delay lines as the
tim e delay of the m icrowave signal is an important param eter of
the
delay
lin e
d esign.
T herefore,
when
designing
c o p lan ar
w aveguide m icrow ave com ponents it is critical to be able to
m o d el
and
c h a ra c te riz e
th is tem p eratu re
d e p en d en t
phase
velocity effect.
2 .2
PHYSICAL ORIGIN OF THE KINETIC INDUCTANCE EFFECT
It is w ell know n that an electron gas with an applied
e lectric
field
E
w ill
behave
according
to the
characteristic
e q u a tio n
mv
1-------- — —e E
m
dr
(2.2.1)
t
w here YYl is the mass of the electron , V is the average velocity
o f th e electron
,
T
is
the
m om entum
relaxation
tim e.
The
average velocity is parallel to the direction of the applied field.
8
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This relaxation time is the time that is required for the average
velocity, after the electric driving field is removed, to decay. The
solution to equation (2.2.1) yields the average velocity to be
—( e t I m )E
V = — ---------------
(2.2.2)
1 + j(OZ
The ch aracteristic
current
density
associated
w ith this
driving field E is given by
J = nev
Yl is the
w here
(2.2.3)
num ber of electrons. Substituting (2.2.2)
into
(2.2.3) and rationalizing, the current density is now given by
(n e2 z / m ) E
,
J =
t
1+
CO
(n e2 z / m ) E
.
j 6 )7
T
1------------------------------- ( Z 2 A )
CD2 T 2
1+
The first term on the right hand side of equation (2.2.4) is
re a d ily
id en tifie d
B ecause the
term
This
w ith
the
m aterial's
norm al
relaxation time is usually very
conductance.
sm all the second
is neglected. This is the case
for normal metals.
Let us
now assume we have
a material such that T —> °°.
means
that the second term
in (2.2.4) can no longer be
9
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neglected. M ultiplying (2.2.4) by 1/T 2 /1 /T 2, i.e. 1, and taking
the lim it we see that equation (2.2.4) reduces to
(2.2.5)
This
new
equation
show s
th at
the
resu ltin g
current
density is in phase quadrature with the applied electric field. If
we
fu rth er
assum e
crossectional
area
a
A
physical
with
a
sam ple
uniform
of
length
current
1 and
density,
then
equation (2.2.5) is trivially transformed to relate the current to
the voltage. Realizing that
(2 .2 .6 )
we can im m ediately recognize that (2.2.5) simply represents an
inductive reactance. The inductance is proportional to
(2.2.7)
E q u a tio n
(2 .2 .5 )
rep re se n ts
u n d e rs ta n d in g
of
su p erco n d u cto r
w here
equation
the
the
k in e tic
we
assum e
b asis
fo r
in d u c ta n c e
the
e ffe c t
T —> °°.[3 ].
In
p h y sic al
in
a
essence,
(2.2.5) represents the electron's inertia opposing the
10
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applied electric field. It is also important to realize that equation
(2.2.7)
indicates
as the
num ber of superconducting electrons
goes down the value o f the inductance increases.
2.3
TEMPERATURE DEPENDENCE AND KINETIC INDUCTANCE
W e are now in a position to establish a very fundamental
u n d erstan d in g
of
the
k in etic
inductance
effect
in
a
real
superconductor. It is well established that the above argum ents
for a uniform m aterial in which T —> 00 is applicable only to the
condition where T=0 K. In reality, no superconductor is used at
this lim it. Therefore when a finite temperature below the critical
te m p e ra tu re
re a so n a b le
of
to
the
su p e rc o n d u c to r
assum e
th a t
both
is
co n sid ere d ,
n orm al
it
e le c tro n s
is
and
superconducting electrons coexist within the m aterial. This is the
cornerstone
m odel
of
assum ption
for the well established
su p erconductivity
[4].
W ith
this
"two fluid
assum ption
it
"
is
further reasonable to assum e that equation (2.2.4) governs those
electrons th at are non-superconducting
(2.2.5)
have
(norm al)
and equation
governs the superconducting electrons. N otice that we
now
e sta b lish e d
tw o
p a ra lle l
paths
fo r
e le c tro n ic
conduction in a superconductor. This is illustrated in figure 2.3.1
as
an
e q u iv a le n t
superconducting
c irc u it
electron
re p re se n tin g
cu rren t path)
an
in d u cto r
shunting
(
the
the resistiv e
norm al electronic conduction path. It is important to point out
11
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that we m ust now return to equation (2.2.5) and m odify Ttl
e
and
to be those values associated with superconducting electron
pairs as opposed to a single normal electron.
W e m ust now
exam ine
the
tem perature dependence
of
this mix of superconducting and normal electrons. It is obvious
from the
above
e le c tro n s
to
argum ents
n o rm al
that the ratio
e le c tro n s
of superconducting
d e c re a se s
w ith
in c re a s in g
tem perature. This is so since above Tc, the critical tem perature
at w hich the sam ple
ceases to
be a superconductor, all the
electrons behave as "normal". From the "two fluid" model it has
been established[4] that
(2.3.1)
n
Us is the density o f paired
w here
superconducting electrons.
Equation (2.3.1) is plotted in figure 2.3.2. As we will see later
this simple expression of "T to the fourth" is inadequate for the
High Tem perature Superconductors (HTS) and m ust be replaced
by a "T squared" dependence.
In summary, the kinetic inductance effect arises due to the
in ertia
o f the
superconducting
electrons
and
this
inductance
value increases as the tem perature approaches Tc.
12
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C u rren t Flow
Js
Jn
(inductance due
to superconducting
electrons)
Figure 2.3.1
(resistance due
to normal
electrons)
Equivalent circuit of electronic
conduction in a superconductor
13
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1.000
0.800
0.600
Tc is 90K
0.400
0.200
0.000
0
20
40
60
80
100
tem perature (K)
Figure 2.3.2
Plot of equation 2.3.1 the Tc is 90K.
14
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2 .4
WAVE PROPAGATION IN SUPERCONDUCTING GUIDING
STRUCTURES
2.4.A
London Equations
In
o rd er
incorporating
how
to
d esig n
the kinetic
m ic ro w a v es
m icrow ave
passiv e
com ponents
inductance effect, we m ust exam ine
p ro p a g a te
along
a
m icro w av e
guiding
structure. In order to do this we must first establish how electric
and magnetic fields behave in a superconductor. This was done
in
the
early
1930's
by
the
L ondon
brothers.
T heir
basic
assumption is that in equation (2.2.1) we can neglect the second
term on the left hand side from the outset. In view of our above
discussion this seem s reasonable since the relaxation
time is
infinite. This therefore reduces (2.2.1) to
dv
m — = —e E
dr
(2.4. A. l )
R ealizing that
J
V = ---ne
(2.4.A.2)
15
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from equation (2.2.3), and substituting this form into (2.4.A.1)
we arrive at the first London equation
A —- = £
dt
(2.4.A.3)
w h e re
m
A=
(2.4.A.4)
ne
w here
YYl
and e
superconducting
are
again
electron pairs.
those
values
If we now
associated
with
take the curl
of
(2.4.A.3) we get
A
(„
\
dJs')
V x “3 “
dt J
= (VxE)
(2.4.A.5)
realizing that
VxE =
dB
(2.4.A.6)
dt
we find that (2.2.A.5) becomes
A
(„
dJ s'
V x^ “
dt
V
dB
dt
(2.4.A.7)
16
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If we now integrate once with respect to time and realize that
the constant of integration appears on both sides of the equation
and therefore can be canceled out, we're left with the second
London equation
A ( V x A ) + Z? = 0
(2.4. A.8)
E quations (2.4.A .3) and (2.4.A.8) form the basis for the
b e h a v io r
of
e le c tro m a g n e tic
fie ld s
in te ra c tin g
w ith
a
su p e rc o n d u c to r.
2.4.B
London Penetration Depth
The consequences of equation (2.4.A.8) are crucial for the
u nderstanding
of how the kinetic inductance effect m anifests
itself in m icrowave guiding structures. Let us look at M axwell’s
equation
for currents
and
fields
neglecting
the
displacem ent
current term such that
V x B = JJ,0 J s
T aking
the
curl
o f this
equation
and
(2.4.B.1)
substituting
the usual
vector identities we are left with
V('V • B) - V2 B
= 11Q(V x /s)
(2.4.B.2)
17
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r e a liz in g
th a t
V *B
is
id e n tic a lly
z e ro
and
th a t
V x J s = - B / A we can establish that
V2B - ( 1 / A i 2) S = 0
(2.4.B.3)
2
w here X
l
= (A /
and is known as the London penetration
d e p th .
L et us now solve (2.4.B.3) in one dimension. It is easily
generalizable to three dim ensions. The geometry under analysis
is given in figure (2.4.B.1). If we assume a m agnetic field Bo
parallel to the superconductor-air interface i.e. in the z direction,
then the solution to (2.4.B.3) yields
(2.4.B.4)
B - = B 0e y a ‘
N o tic e
th a t
th e
m a g n e tic
fie ld
p e n e tra te s
in to
the
superconductor by a characteristic length of X l as is indicated in
figure 2.4.B.2. This is precisely the Meissner effect. The surface
current can be established from (2.4.B.1) so that
(2.4.B.5)
18
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SUPERCONDUCTOR
•<*
AIR
Figure 2.4.B.1
Geometry used for solving equation
2.4.B.3
19
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Since this current flows w ithin the superconductor it will
be
subject to the kinetic inductance effect as established
(2 .2 .5 ).
F u rth e r
recall
th at
by
defin itio n
Xi
is
in
in v ersely
proportional to the num ber o f superconducting electrons. The
num ber o f superconducting electrons in turn, are tem perature
dependent such that as the tem perature increases their num ber
d ecreases.
A
d ecreasing
num ber
of
electrons
in d icates
an
increasing penetration depth. More im portantly, the value of the
k in etic
in d u ctan c e
su p e rc o n d u c tin g
also
w ill
e le c tro n s
in crease
d e c re a se s
as
as
the
nu m b er
e q u a tio n
of
(2 .2 .7 )
indicates. This therefore shows that as the tem perature of the
superconductor approaches Tc
not
only does the penetration
depth increase but also the electrons become more sluggish and
hence the kinetic inductance effect is greatest.
We
penetration
have
therefore
established
that
by m onitoring
the
depth as a function o f increasing tem perature we
are actually looking at the kinetic inductance effect in action.
2.4.C
W ave Propagation
Starting from M axwell’s equations assuming an eJwt tim e
dependence we have
V x E = -j(O jJ ,oH
(2.4.C.1)
20
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AIR
SUPERCONDUCTOR
Bo
Figure 2.4.B.2
B = B o /e
M agnetic field penetration depth from
equation 2.4.B.4
21
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V x H = J + jc o sE
(2.4.C.2)
J= Jn+ Js
(2.4.C.3)
J n = (jn E
(2.4.C.4)
VxJs=-H/A,2
(2.4.C.5)
w h e re
on
is
the
norm al
m aterial
conductivity
and
all
other
variables have been previously defined. If we now proceed to
take the curl o f (2.4.C.2) and proceed in a m anner analogous to
the derivation of the London penetration depth we arrive at the
w ave
e q u atio n
fo r
a
superconducting
guiding
structure[5].
Specifically,
V 2H - ( P i 2+ j ( Q l l o O n - ( D 2lJLo £ ) H = 0
(2.4.C.6)
W e can therefore define a complex conductivity such that
&~G>n J & s c
(2.4.C.7)
22
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where
the
superconducting com ponent o f the conductivity
is
defined as
C7s c = y ( C Q ( l 0 ?l2)
(2.4.C.8)
The solution to equation (2.4.C .6) can be made slightly
more tractable if we assume a "z" dependence of the propagation
such that the dependence will have the form ePz . This will now
lead to
( ^ +
dx
^ ) H
- ( r 2+ j m 0< 7n-co2t i 0£ - p 2) H = o
oy
(2.4.C.9)
It is the solution to equation (2.4.C.9) that will ultim ately
yield
an accurate m odel for the observed kinetic inductance
effects that makes up part of this thesis.
Prior to indicating how one can go about solving this
equation we m ust specify the boundary conditions that apply to
a superconducting m icrowave guiding structure. Let us consider
the experim entally realizable situation where the thickness
of
the superconductor is sm aller than both the London penetration
depth and the usual skin depth at the frequency of interest. This
w ill lead to a condition where the current should be uniform ly
23
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distributed through out the sample thickness. The resistance of
such a device would then be
R = l!a t
(2.4 .C.10)
where t is the thickness of the superconductor. If we now let t
approach zero while the conductivity gets very large we m ust
insist that the resistance remains finite. This boundary condition
now allows us to relatively easily solve equation (2.4.C.9) for an
in fin ite ly
th in
su p e rc o n d u c to r w hile
im p o rtan t
featu res
of
m ain tain in g
superconductivity [6].
R ecall
a ll
the
th a t
the
conductivity we w ill use is a com plex quantity as given by
(2.4.C.7).
T he
rigorous
use
in
of
the
the
thin
film
su p e rco n d u c to r
is
H ow ever,
p o ssib le
it
is
resistiv e boundary
m uch
lim it where
less
to
than
extend
condition is
only
the thickness o f
the
the
the
p en etratio n
applicability
depth.
o f this
approach by using a generalized form of the boundary condition
into the regim e where the superconductor thickness is
on
the
order of a penetration depth. The use of a c o th (? /A ) (where t
is
the
thickness
of the
superconductor)
m odification
to
the
boundary condition allows for the internal damping of the fields
associated with the penetration depth [7,8]. This approxim ation
24
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can be used to successfully m odel planar transm ission lines of
m any
2 .5
geom etries.
KINETIC INDUCTANCE MODELING
T he
approach
used
in
this
thesis
to
solve
the
wave
equation is to em ploy a full wave spectral domain analysis of
the
m icrow ave
guiding
structure[9]. W e have
experim entally
investigated coplanar waveguide (CPW) as the guiding structure
because of the ease with which the device can be fabricated.
Because the spectral dom ain analysis development is beyond the
scope o f this thesis a brief description of it's salient points will
be discussed.
The geom etry under investigation is illustrated in figure
2.5.1. The key goal is to solve for the unknown propagation
constant /J over a range o f frequencies. Because we m ust solve
for
the fields w ithin this box and because the CPW runs
across
the width o f the box we will try and solve for the fields within
the gap, not the current distribution in the m etal. CPW
is the
dual of the case where we have two parallel m icrostrips side by
side. The space dom ain approach is to solve for the coupled
e q u a tio n s
j[Y z z ( x - x ' ,y)Ez(x' )+Yzx(x-x! ,y)Ex(x' )]dx' = J z(x )
25
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j[Y x z(x -x ' ,y)Ez(x' )+Yxx{x—x' ,y)Ex{x' )]dx = Jx(x)
’
(2 .5.1)
where the components Yzz, etc. are the Green's functions for this
structure and depend on /J . Ez and Ex are the fields within the
gaps and Jz and Jx are the current distributions in the metal.
Perform ing the integral results in the left hand side o f (2.5.1)
being equal to zero over the metal portion of the CPW
since the
parallel electric field must be zero. The right hand side of (2.5.1)
is zero when the integration is done over the gap region as no
current flows in the gap.
Ex, and /J)
and
only
Since there are three unknowns (Ez,
tw o
equations
we
have
to
som ehow
determ ine the functional form for Ez and Ex. This is done using
Galerkin's method whereby
the electric field is expanded in an
infinite series of orthogonal basis functions and inner products
are taken. Depending on the form of the basis functions few er or
m ore term s of the series are required for the solution for p to
co n v erg e.
N otice that (2.5.1) are two coupled equations that involve
convolution
integrals.
It is
well know n
that if the F o u rier
transform of a convolution integral is taken the result in Fourier
space is simply the two individual Fourier transformed functions
m ultiplied by each other. Therefore
26
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6.35mm
AIR ( E r = l)
0.364mm
2.54mm
0.364mm
►
►
0.208mm
CPW
Lanthanum Aluminate (Er=26)
Teflon (E r= 2.2)
Figure 2.5.1
0.5( ^9999999
12.7mm
Geometry under investigation;
crossection view
27
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j Y - J . X - it , y
)E M
= >
Y
<2 - 5 - 2 >
where we have used
< P ( a ,y ) = J ( p i x ^ e ^ d x
w here
(2.5.3)
0(x,y) represents any generalized function. Therefore in
the spectral dom ain (2.5.1) becomes
Yzz(oc,p,h)Ez(oc)+Yxz(cx,p,h)Ex(a)=Jz(oc)
Yxz(cc,p,h)Ez(oi)+Yxx(a,p,h)Ex(cc)=Jx(a)
(2.5.2)
w here the ~ over the variable represents a Fourier transform ed
quantity. A lthough we have elim inated the integral, the right
hand side o f these equations cannot be set equal to zero. This is
because when the Fourier transform is performed it is done over
the entire spatial variable, not only over the gap region. O f
course, outside of the gap there is a finite current therefore
Jz (a ) and Jx (a ) cannot be identically zero. Again, the Galerkin
approach
and
an infinite orthogonal series expansion for the
currents are used to solve for the propagation constant.
28
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It is w orthw hile to briefly outline the G alerkin m ethod
particularly for the spectral domain approach. L et us expand the
electric field in an infinite series of orthogonal basis functions
such th at
~
N
E x = J .C m E x m( 0 ‘ )
m=l
(2-5.3)
N
E z= 'LdmEzmW
m—\
(
2 54
.
.
)
substituting (2.5.3) and (2.5.4) into (2.5.2); m ultiplying by Ezk
and Exi and integrating i.e. taking the inner product, we arrive
at
N
N
X d m Akm+ X c / n5 fcn=J[£’Z)t(a )J z (a )]ja
m=1
m=1
N
N
'5Ldm Cim+ X c w^ = J [£ x /(a )/* (a )]fi? a
m=1
m=1
(2.5.5)
w h e re
Ahn=j[Ezk(a)Yzz(a,p,h)Ezm (cc)]da
(2.5.6)
B ^ iE z k W Y z x ia M E x m ia fld a
(2.5.7)
29
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C im =
(2.5.8)
\\_Exi{a)Yxz{ a,P,h)Ezm( a )] d a
(2.5.9)
D im =\[Exi(a)Yxx(a,f},h)Exm ( a ) \d i
It is important to realize that the right hand side of (2.5.5)
integrates out to be zero by virtue of Parseval's theorem. This is
because wherever E exist J is zero and visa versa since they are
m utually exclusive along the CPW at position h. T herefore we
are left with solving
A km
Bkm
dm
C lm
D im
cm
(2.5.10)
fo r the propagation constant.
Once the propagation constant has been determ ined then
the phase velocity can also be determined from
(2.5.11)
w h e re
f Q is
the
frequency
of
in te rest.
R ecall
th a t
the
propagation constant w ill be a function of tem perature as the
propagation constant is proportional to the penetration depth of
the
superconductor.
30
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The phase velocity is an important modeling param eter as
it
is
re la tiv e ly
easy
and
unam biguous
to
e x tra c t
from
experimental data. Hence it is the goal of this part of the thesis
to
obtain
experim ental data for the phase velocity and then
compare it to the aforementioned model.
2 .6
EXPERIMENTAL DESIGN AND SETUP
One
of the m ost straightforw ard m ethods
by which to
characterize the im pact of the kinetic inductance effect on the
phase velocity of a coplanar waveguide is to characterize a half
wavelength resonator as a function of tem perature. In order to
derive the phase velocity of this type of resonator we proceed as
follows. The phase velocity is defined as
V p h a se= c l 4 < ^ ff
( 2
' 6
' 1 )
where C is the speed of light in vacuum and eef f is the effective
dielectric constant of the planar structure. In addition
X = V phase1 F o
w here
X
(2.6.2)
is the wavelength of the guiding structure ( not to be
confused w ith the London penetration depth) and
p 0 is the
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
reso n an t frequency
of the
resonator.
Since
the
structure
is
designed to be a half-w avelength resonator the w avelength at
resonance is simply given by
A = 2 (l + K )
(2.6.3)
w here ^ is the physical length of the 1/2 wavelength resonator,
and K is a correction for end effects of the coplanar resonator.
K simply adds an additional effective length to the length of the
resonator.
T herefore
the
phase
velocity
of
such
a resonant
structure is given by:
Vphase ( T ) = 2 ( 1 +
T he
te m p e ra tu re
represents
d ep en d en ce
of
K )F o (T )
the
re s o n a n t
the k inetic inductance effect. T his
resonant frequency
(2.6.4)
freq u e n cy
is because the
is determ ined by the com bination
o f the
external m agnetic inductance due to the CPW and the internal
m agnetic inductance due to the inertia of the superconducting
electrons.
The
la tte r
is
of course
tem perature
dependent as
indicated in section 2.2.
The overall layout of the resonator is given in figure 2.6.1.
The 20 m il thick lanthanum aluminate substrate is 0.7" X 0.7" .
The resonator is designed to be an end coupled in line device
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
r, ^ x, x. > r r r m
Figure 2.6.1
•,> --■>
\ , \ . s ; s * S> S* ^
i '- " 'C ^ ^ <•., v v , * , ' - U k , ~ g
The layout of the coplanar waveguide
resonator. This is a top down view.
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
such that the transm ission characteristics, S21 can be measured.
C onstant 50Q im pedance coplanar waveguide tapers w ere used
to transform the center conductor width from 400 pm down to
200 pm . The 400 pm input width was used to facilitate the
connection
co n stan t
of the
resonator to
im pedance
taper
was
a W iltron
designed
K -connector.
using
w ell
The
know n
im pedance equations for coplanar w aveguide[10]. The 5
GHz
resonators were 8.54 mm long with a center conductor width of
200
pm
n o m in ally
and
50
a transverse
Q
on
a
gap
width
lanthanum
of 370
alum inate
pm .
T his
su b strate
is
of
dielectric constant 26. The coupling gap is 15 pm.
The YBCO was deposited using
a pulsed laser ablation
deposition technique [11]. After the YBCO was patterned, a layer
o f silver was deposited on the taper and the surrounding ground
plane to facilitate contact to the K-connector. A fter the silver
was deposited, the entire w afer was annealed at 450 C in an
oxygen rich atmosphere for 1 hr. This promotes excellent contact
re s ista n c e
betw een
the
HTS
and
the
silv e r
w hile
still
m aintaining the HTS integrity by reinjecting the oxygen into the
HTS that was lost during the preanneal processing steps.
For microwave testing the device was placed on a Teflon
block 0.50 inches thick and enclosed in an alum inum box to
which the K -connectors were attached. The center pin o f the
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
connector was attached to the center conductor of the CPW using
silv er
p ain t.
betw een
The
the
alum inum
block
backside of the
box
ch aracteristic
Teflon
to
enough separation
wafer and the
m inim ize
im pedance
provides
of the
the
e ffe ct
CPW
due
bottom
of
to
o f the
changing
ground
the
plane
proxim ity. A lid was then placed on top of the box such that a
channel
cutout
0.700" long
m easuring 0.250" wide by 0.100"
deep
by
was centered directly above the resonator. The lid
was then made to make contact to the CPW ground plane on
either side of the resonators close to the launch point of the Kconnector, through the use of spring loaded contact m achined
into the lid. In addition, a cryogenic therm om eter was inserted
into
the
Teflon block in
p e rfo rm a n c e
of
such a way as to not effect the
the re s o n a to r
w hile
y ie ld in g
a c c u ra te
tem perature data. The entire test structure was m ounted on a
coaxial probe
and inserted into a liquid helium dewar. The
coaxial leads were then connected to an HP 8720A netw ork
analyzer.
The
entire
length
of the leads
from
the
netw ork
analyzer to the device under test is approxim ately 2.5 meters.
Because of the length of the coax leads an accurate and stable
calibration of the reflection coefficient S n
T herefore, only S21
was not possible.
m easurem ents were conducted. The S 2 1
m easurem ents provides adequate data for m odeling purposes.
35
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2 .7
EXPERIMENTAL RESULTS AND DISCUSSION
F ig u res
2.7.1, 2.7.2,
and
2.7.3
show
the tem perature
variation in the v w ( T ) of the 100, 200 and 300 nm thick films,
respectively. The phase velocity is norm alized to the speed of
light in a vacuum, c. The data is represented by the open circles
while the solid and dashed lines represent the model fit. In view
of
the
debate
over the
functional
form
of
the tem perature
dependence o f the penetration depth, both a ”T squared" and a
G o rter-C a ssim e r
f it
are
represented.
The
zero
tem perature
penetration depths that provided the best model fits were 200,
250 and 225 nm for figures 2.7.1, 2.7.2, and 2.7.3 respectively.
The 100 and 300 nm film s were processed at the same time
w hile the 200 nm film was processed 6 m onths later using a
different YBCO target. It is im portant to note that no a priori
quantitative tem perature
dependence of the penetration depth
nor
zero
tem perature penetration depth values were assumed.
The
Tc of all the films with R=0 were measured to be 92 K
with
a transition region of 2K
To
establish
tem p eratu re
2.7.3.
the
dependence
validity
was
of such a m odel
used
The figures clearly indicate that
in
figures
2.7.1
the
same
through
a "T squared"
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.285
0.280
«
>
0.275
using h = \
(1- CT/T/)-1/2
using X = XQ (1 - (T/Tc)2)'1/2
measured
0.270
100
Temperature (K)
Figure 2.7.1
Norm alized Phase velocity versus
tem perature for superconducting CPW
using 100 nm thick YBCO film.
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.285
0.280
u
0.275
using ^ = ?io (l- (T/Tc)4)-1/2
using a = \ (i measured
0.270
100
Temperature (K)
Figure 2.7.2
Normalized Phase velocity versus
temperature for superconducting CPW
using 200 nm thick YBCO film.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.285
0.280
o
CO
0.275
u sin g x
=
cr/rc)4r1/2
using X = A>0 (1 - (T/T c)2)'I/2
measured
0.270
100
Temperature (K)
Figure 2.7.3
Normalized Phase velocity versus
temperature for superconducting CPW
using 300 nm thick YBCO film.
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tem p era tu re
accu rately
m odel
the
v ariatio n
of
with the film thickness as a controlled parameter. In all
V p h a s e (T )
cases
dependence
the
additional
m easured
vW;e(T)
had
to
be
3 % ( K = 0.03) to compensate for
m ultiplied
by
an
the end effects of
the resonator. This additional com pensation should be on the
order o f 1 to 2 linewidths o f the resonator as a rule of thumb.
Given the m easured dimensions o f the resonator, the K factor
should be on the order o f
2 - 5%; well within the predicted
m odel's com pensation value. The predicted value of
225 ±- 25
nm for the zero temperature penetration depth is also very close
to previous independently determined
tim e)
based on
values( done at the same
a parallel plate m ethod of determ ining the
penetration depth [12].
Figure 2.7.4 is a plot o f the m easured Q of the resonators
as
a function
of tem perature. The
first
thing
that becom es
apparent is the high Q o f the resonators; approxim ately 16,000
at 5 GHz. These are the highest known measured Q's for HTS
CPW end coupled resonators. Notice also that the m easured Q is
only a w eak function of the film thickness. In particular the
m easured Q of the 300 nm film degrades much m ore slowly
then does that o f the 100 nm film as T approaches Tc- This
d ifference
m ic ro strip
in Q degradation
re so n a to rs
of
is sim ilar to that predicted
d iffe re n t
film
th ick n esses
for
[13].
Furtherm ore, figure 2.7.4 indicates that all three thicknesses fall
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,5
o 100 nm film
♦ 200 nm film
□ 300 nm film
4
i3
40
20
60
80
100
T E M P E R A T U R E (K)
Figure 2.7.4
Measured Q of 5 GHz CPW resonator
versus
tem p eratu re.
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w ithin the m odel regim e where X is on the order of the film
thickness
hence
estab lish ed .
The
the
valid ity
shape
of the
of
the
curves
c o th (r/A
is
also
)
has
an
been
im portant
consideration. The m easured Q is inversely proportional to the
surface resistance of the HTS film . As one would expect, the
m odel surface resistance is low est at low T ( 128 (J.Q)
and
increase as T approaches Tc . The method used to calculate the
surface resistance is outlined in [14].
Finally figure 2.7.5 is a plot of the resonance frequency of
the resonator versus the film thickness at both 20 K and 77 K.
N otice in all cases the resonance frequency increases with film
thickness.
This indicates that for a given temperature the phase
velocity increases with film thickness. This implies
w ave
c o p la n a r
w a v e g u id e
s tru c tu re s
that slow-
re q u ire
th in n e r
superconductor film s. In addition the phase velocity decreases
w ith increasing tem perature for the same film thickness.
2 .8
FUTURE WORK
There
are
a num ber o f d ifferent
avenues
that present
them selves for future work. One area of particular interest is the
determ ination of the surface resistance of the films as a function
o f tem perature. The above m entioned m ethod
[14] was only
devised for normal m etals and will obviously need to be
42
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4.82
(3 4.81
4.8
O'
^
4.79
o T=20K
❖ T=77K
4.78
4.77
50
Figure 2.7.5
1 00
150 200 250
thickness (nm)
300
350
Resonance frequency for 100, 200, and
300 nm films at 20K and 77K.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
corrected for the superconducting case. The present m odel will
be able to extract surface resistance data by the addition of
some code. The general theoretical outline is as follows. It is well
known that the Q of a resonator is given by
Q = x l ( A a c)
(2.8.1)
w here
A is the guide w avelength and
is the
conductor
losses.
a c is related to the surface resistance by a
geom etric
p ro p o rtio n a lity
g e o m e try .
c o n stan t
T his
th a t
rela te s
p ro p o rtio n a lity
ac
to
c o n sta n t
the
actual
rem ain s
CPW
to
be
determ ined. Proceeding we realize that
P = 2n/A,
(2 .8 .2 )
a c = p /(2 Q )
(2.8.3)
therefore we see that
and
o f course
our m odel predicts
the propagation constant.
Therefore a self consistent m ethod for determ ining the surface
resistance of the CPW film is obtainable.
A nother area of particular interest for future study is the
pow er dependence of the CPW resonator associated w ith it's
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c ritical
current
re s o n a to r
density.
c o n sid e ra b ly
If we make
sm a lle r
the
than
dim ensions
the
of the
p re se n t
k in etic
inductance experim ents, we expect the critical current density of
the
su p erco n d u cto r
to
be
m uch
sm aller.
In
the
case
of
m icrostrip[15] a pow er dependence of the resonance curve was
observed for powers greater than 0 dBm. The explanation given
for this effect is associated with locally driving the HTS normal
along the edges of the m icrostrip. This effect, until now has not
been observed in CPW. In fact, CPW is probably more suited to
dem onstrating this effect as the currents along the edges of the
CPW conductors are much larger than those of m icrostrip. Hence
the
pow er
dependence
e ffect
should occur
at low er
pow er
In order to dem onstrate this effect we have m ade
narrow
values than in the m icrostrip case.
width 50 Q CPW resonators at 10 GHz. The topology is identical
to that presented in figure 2.6.1. A typical resonance curve is
shown in figure 2.8.1. The high Q of the measured device allows
us to observe the pow er dependence effect with certainty. A
num ber of resonance curve are superimposed in figure 2.8.2. As
the
input pow er is gradually increased from
w hile m aintaining
the
sam ple tem perature
-13 to +17 dBm
at 8K, the
power
dependence is clearly visible. It is interesting to note that not
only does the resonance frequency shift to lower frequency as
45
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CHI S 21
log M A G
10 d B /
Jj
RE F 0 dB
.000
BW:
i
pent:
.0166
dB
6 s 9 4 ^ 6 GHZ
i
|
!
Mix
a IREF-5|
1.002 5 7 9 7 2 6 GHz!
!
I
|
i
jlO .2 6 5 7 7 4 6 17 GHzj
3979.
CENTER
10.265
000
Figure 2.8.1
000
GHz
SPAN
3ss:
f 26. 3 6 2
.200
000
000
Typical resonance curve for narrow
width 10 GHz CPW resonator.
46
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d3
GHz
CHI S 2 i
CENTER
lOS M A G
10.265
5 dB/
000
000
Figure 2.8.2
RE F 0 dB
SPAN
GHz
05 0
OOO
000
GHz
Superimposed resonance curves of the
10 GHz resonator displaying power
dependence.
47
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we expect, but the curve becom es asym m etric. This has also
been observed in the m icrostrip case.
It will be critical to m odel this effect as it is crucial for
filter and delay line development.
48
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■
2 .9
REFERENCES
[1]
T. Van Duzer and C. W. Turner, Principles of
Superconductive Devices and C ircuits. Elsevier, New York,
Chapter 1
[2]
B. W. Langeley, S. M. Anlage, R. F. W. Pease, and M. R.
Beasley, " M agnetic Penetration Depth Measurements of
Superconducting Thin Films by a M icrostrip Resonator
Technique ", Rev. Sci. Instr., 62 (7), pp. 1801-1812, 1991
[3]
T. Van Duzer and C. W. Turner, Principles of
Superconductive Devices and C ircuits. Elsevier, New York,
p 26
[4]
T. Van Duzer and C. W. Turner, Principles of
Superconductive Devices and Circuits. Elsevier, New York,
p 124
[5]
J. M. Pond, C. M. Krowne, and W. L. Carter, " On the
Application of Complex Resistive Boundary Conditions to
Model Transmission Lines Consisting of Very Thin
Superconductors ", IE EE M icrowaves Theory and
T e c h n iq u e s, MTT-37, pp 181-190, Jan. 1989
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[6]
T. B. A. Senior, " Half Plane Edge Diffraction ", Radio Sci. ,
Vol. 10, pp.645-650, June 1975
[7]
D. Kinowski, F. Huret, P. Pribetich, and P. Kennis, " Spectral
Domain Analysis of Coplanar Superconducting Line Laid on
M ultilayered GaAs Substrate ", E lectronics Letters, 25 (12),
pp.
[8]
788-789,1989
N. Klein, H. Chaloupka, G. Muller, S. Orbach, H. Piel, B. Roas,
L. Schultz, U. Klein, and M. Peiniger, " The Effective
M icrowave Surface Impedance of High-Tc Thin Films ",
Journal o f A pplied Physics, 67 (11), pp. 6940-6945, 1990
[9]
T. Itoh, " Numerical Techniques for Microwave and
M illim e te r-W a v e Passive Structures ", W iley & Sons, New
York, 1989, Chapter 5
[10]
B. C. Wadell, Transmission Line Design Handbook. Artech
House, Boston, 1991, Chapter 3
[11]
J. Luine, J. Bulman, J. Burch, K. Daly, A. Lee, C. PettietteH all, and S. Schwarzbek, " Characteristics of High
Performance Y BA 2Cu307 Step-edge Junctions ", Appl.
Phys. Lett., 61, pp 1128-1130, 1992
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[12]
J. Luine, K. Daly, R. Hu, A. Z. Kain, A. Lee, H. Manasevit, C.
Pettiette-Hall, R. Simon, D. St. John, and M. Wagner, " 1 0
GHz Surface Impedance Measurements of (Y,Er)BaCuO
Films Produced by MOCVD, Laser Ablation, and
Sputtering", IEEE Trans, on Mag., MAG-27, pp. 1528-1531,
1991
[13]
A. Fathy, D. Kalokitis, and E. Belohoubek, " Critical Design
Issues in Implementing a YBCO Superconductor X-Band
Narrow Bandpass Filter Operating at 77K ", IEEE
International M icrowave Symposium D igest, pp. 13291 3 3 2 ,.1 9 9 1
[14]
G. Ghione,C. Naldi, and R. Zich, " Q-Factor Evaluation for
Coplanar Resonators ", A lta Frequenza, 52 (3), pp. 191-193,
June
[15]
1983
D. E. Oates, P. P. Nguyen, G. Dresselhaus, M. S. Dresselhaus,
C. W. Lam, and S. M. Ali, " Measurement and Modeling of
Linear and Nonlinear Effects in Striplines ”, Journal o f
S u p e rc o n d u c tiv ity , Vol. 5, No. 4, pp 361-369, 1992
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
HTS
D elay
Line
Measurements
U sing
Picosecond
Electrical Pulses
3.1
INTRODUCTION
As
we
have
seen
in
the
p rev io u s
c h ap ter,
kin etic
inductance effects play an im portant role in determ ining device
characteristics. The experiments and modeling o f chapter 2 were
only with a single half wavelength resonator whose tem perature
dependence was examined at a single frequency. In this chapter
we
extend
inductance
the
e x p erim en tatio n
e ffect
to
include
of
the
a broader
n o n lin e ar
frequency
k in etic
range
by
employing a coplanar waveguide delay line. In order to obtain
the
m axim um
frequency
bandw idth
info rm atio n
fro m
52
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the
m ea su re m e n ts
we
w ill
use
p ico seco n d
e le c tric a l
p u lse s
generated by optical switches. The electrical pulse is sent along
the delay line and the output pulse is recorded. The Fourier
transform of this output pulse after having traveled the length
o f the delay line w ill yield the desired frequency perform ance
ch a ra c te ristic s
up
to
100
GHz.
O f course,
we
ex p ect
the
tem perature dependence that we saw in the resonator case to be
present in the delay line also.
3 .2
FREQUENCY CHARACTERISTICS OF AN HTS DELAY LINE
T he
freq u e n cy
c h a rac teristics
of
a
su p erco n d u ctin g
transm ission line can be categorized into at least two regim es.
T h ey
are
tem p eratu re
L et
te m p e ra tu re
dep en d en t
us
in d e p e n d e n t
c h a r a c te r is tic s
and
ch aracteristics.
e x a m in e
th e
te m p e r a tu r e
in d e p e n d e n t
characteristics first. It is well known that a pulse traveling along
a norm al m etal transm ission line will experience dispersion due
to the frequency dependent skin depth effect. The skin depth is
given by
1
8
(3.2.1)
53
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8 is the skin depth, f
w here
the
m agnetic
co n d u ctiv ity
in creases
is the frequency in question, f l is
p erm eability
o f the
o f the m aterial. W e
the
skin
depth
m aterial,
see
decrease.
and
(7
is the
th at as the frequency
In
addition,
the
surface
resistance o f the conductor is
„
1
R s= -^~
(3-2.2)
<5(7
T h ere fo re ,
as
the
re s is ta n c e
of
th e
com ponents
are
frequency
increases
c o n d u cto r
attenuated
and
so
h en ce
does
the
h ig h er
surface
frequency
m uch faster than low er frequency
com ponents. Therefore, a narrow pulse w ill result in a w ider
p u lse
afte r traveling a finite distance along a norm al m etal
c o n d u c to r.
This pulse dispersion due to surface resistance is largely
n eg ated
in
supercondu ctors.
R ecall,
that
the
electrom agnetic
field s only penetrate into the superconductor a finite am ount
know n
as the London penetration depth.
This was shown in
section 2.4.B. Furtherm ore, the London penetration depth is not
freq u en cy
dependent;
it is
only
a
function
of the
m aterial
characteristics. From the experim ental evidence of chapter 2 it
w as fu rth er show n that the penetration depth of the HTS was
a p p ro x im ate ly
0.2
p m . If we w ere to consider the case of a
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
normal conductor such as copper at 10 GHz we would find that
it’s skin depth in equation (3.2.1) would be 2 pm . However, in an
HTS transm ission line the 10 GHz signal is constrained to be
w ithin
0.2
pm
o f the
conductors
surface
by
virtue
o f the
penetration depth, rather than the usual 2 pm . This im plies that
all frequencies that have a normal skin depth greater than the
London penetration depth are forced to assum e a skin depth
equal
to
the penetration depth
o f the
superconductor.
Since
these frequencies are all effected in the same way, they travel
along
th e
sam e
path
w ith
id e n tic a l
p h a se
v e lo c ity
and
attenuation. Hence, the superconductor has the potential to be a
non-dispersive transm ission line up to very high frequencies.
In order to get a feel for the bandw idth o f this nondispersive delay line, w e can now estim ate the frequency at
which the norm al skin depth is equal to the penetration depth.
A t this crossover point, any higher frequency will be dispersive
as in an ordinary m etal. Substituting (3.2.2) into (3.2.1) and
solving for the frequency we find that
(3.2.3)
If
we
assum e
a
su rface
resistan ce
of
0.1
mQ
fo r
the
superconductor as found in section 2.7 o f chapter 2, we find that
55
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this crossover frequency is 810 GHz! It is for this non-dispersive
bandw idth that superconducting transm ission lines show one of
their greatest potentials. In sum m ary, because of the London
p enetration
depth
a
superconducting
delay
line
can
have
a
bandw idth approaching a Terahertz.
L et
us
now
c o n sid e r
th e
te m p e ra tu re
dependent
characteristics o f a pulse launched along an HTS delay line. We
know that
S= Vt
(3.2.4)
w here S is the distance the pulse has traveled, V is the velocity
the pulse travels with, and t is the time it takes for the pulse to
travel. It is im portant to note that since we are in the time
dom ain the velocity of the pulse is the group velocity of all the
frequency com ponents that make up the pulse. Therefore, if we
can m easure the time of flight of the pulse and the distance it
travels we can determ ine the group velocity. The time of flight
of the pulse is effected by the temperature. As we have seen in
ch apter 2, as
the
tem perature increases
so does
the kinetic
inductance. Since a superconducting transmission line consists of
both the m agnetic inductance and the kinetic inductance, we
expect the overall inductance of the transmission line to increase
as a function of tem perature. In addition, the surface resistance
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o f the superconductor increases as a function of tem perature
due to the decrease in available superconducting electron pairs
to carry energy.
T herefore,
as the
tem perature
increases we expect the
pulse time of flight to increase due to the increase in inductance,
and
the
p u lse
shape
to
broaden
and
attenuate
due
to
the
increase in surface resistance.
In order to verify this experim entally we m ust generate
very narrow pulses and record their time of flight as a function
o f tem perature. W e produce these pulses by optically pum ping a
sem iconductor photo switch.
3 .3
OPTOELECTRONIC SWITCHES
A ll
sem iconductors
have
the
ab ility
to
change
th eir
conductance by absorbing photons o f the proper energy. This
property can be used to m ake sem iconductor optical sw itchesfl].
The structure o f a simple optical switch is shown in figure 3.3.1
in
a
to p
dow n
view .
A
th in
la y e r
of
high
re s istiv ity
sem iconductor m aterial is grow n on an insulating m aterial. On
top o f the sem iconductor a thin layer of metal is deposited and
etched into a m icrostrip pattern. A narrow gap is etched into the
line. This gap effectively isolates the input DC voltage from the
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[sem ico n d u cto r o n in su la to r
I
x an sm issio n
line
DC lead
gap
Rload
Photons
Vdc
Figure 3.3.1
M icrostrip im plem entation o f sim ple
optical switch.
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
output transm ission line. The gap is the optical switch. As laser
light is focused onto the gap,
carriers are generated provided
the laser light photon energy is greater than the bandgap energy
o f the semiconductor. Therefore, the gap region of
flooded
with
figure 3.3.1 is
extra electrons. These electrons now
provide a
conductive path for the electrons in the DC lead. Therefore, while
the laser light is impinging on the gap, electrons from the DC
source can flow into the microstrip transmission line. As soon as
the laser light is switched off, as in a pulsed laser, the gap
re tu rn s
to
it's
high
re sistan c e
state
and
re -iso la te s
the
transmission line from the DC voltage source. Hence, an electrical
pulse can be generated based on the duration of the laser light
p u ls e .
It is very im portant to realize that the optical switch can
also be used as a sampling gate. This is illustrated in figure 3.3.2.
Suppose a repetitive electrical pulse is propagating along the
microstrip. If the laser light is shown on the gap at precisely the
time that the pulse arrives at
the gap then part of the pulse will
propagate through to the voltm eter. Only the part o f the pulse
that arrives at the switch during the time that the laser light is
activating the switch, will get through to be observed. Therefore,
if a narrow laser pulse is used such that the timing between the
laser light pulse and the electrical pulse to be observed can be
tuned, the potential exist to be able to sample the entire
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
se m ic o n d u c to r on in su lato r
T
x an sm issio n
line
DC lead
\
m e ta liz a tio n
Photons
Pulse In
Figure 3.3.2
Simple optical switch used as sampling
gate.
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Sampling gate time on
incoming pulse
0 dt
T1
mm.
sampled pulse output
Figure 3.3.3
Pictoral illustration
of pulse sampling
technique
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
electrical pulse. In other words, if an optical time delay can be
incorporated in the operation o f the switch, the laser pulse can
be swept through the electrical pulse in the time domain and the
entire electrical pulse can be sampled. This is shown in figure
3.3.3. N otice the sampling gate time is continuously delayed by a
factor dt that allows for the sampling gate to sweep through the
input pulse
duration.
There are a num ber of factors that influence the electrical
pulse width. The speed of the electrical pulse generated depends
on the w idth o f the optical excitation, the capacitance of the
sw itch gap,
the carrier lifetim e in the sem iconductor, and the
characteristic im pedance of the m icrostrip transm ission line. The
recom bination
tim e of norm al sem iconductors used in device
fabrication are on the order of nanoseconds.
To generate picosecond electrical pulses two qualities m ust
be optim ized. The first is to start off with an optical source that
can provide
picosecond optical
the
lifetim e
c a rrie r
of
pulses. The second,is to reduce
the sem iconductor.
One
m ethod of
achieving shorter carrier lifetim es is by heavy ion im plantation
of
the
se m ic o n d u cto r!^ ].
se m ic o n d u c to r
w ith o u t
H eavy
an n ealin g
ion
im p lan tatio n
c re ate s
dam age
of
the
to
the
crystalline structure. This damage creates traps, dislocation sites,
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vout2
Vgeneration
photons
photons
generation
gap
sampling
gap
H i- N
w
photons
delay
line
bondwires
sampling
ggp_
V o utl
Figure 3.4.1
Experimental layout of optical switches
and HTS CPW delay line
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
surface interfaces, and other non crystalline structures that can
reduce the carrier lifetim es to subpicosecond levels[3].
T h e re fo re ,
sw itc h es,
c o m b in in g
u ltra fa s t
p ico seco n d
e le c tric a l
p u lse s
la sers
can
be
and
o p tical
g e n erate d
to
characterize devices.
3.4
EXPERIMENTAL DESIGN
In order to m easure the pulse propagation on the HTS
co p lan ar
w aveguide
delay
lin e
a m inim um
o f three
optical
sw itches are required. A typical experimental layout is shown in
figure 3.4.1. Since the
optical sw itches are
produced
by a
sem iconductor m anufacturing process and the CPW delay line is
produced by an HTS m anufacturing process, a total of three
in d ep en d en t
su b stra te s
su b stra tes
are
are
e le c tric a lly
used.
T he
co n n ected
circuits
through
on
the
the
three
u se
of
b o n d w ire s .
In order to accurately quantify the kinetic inductance
effect in the HTS delay line, there are two pulse shapes that
m ust be determ ined. They are the input pulse and the output
pulse. The input pulse is required in order to establish the input
pulse frequency bandwidth. This bandwidth will be used as the
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
baseline to compare to the output pulse bandwidth to determine
pulse dispersion.
The input pulse shape is determ ined as follow s in figure
3.4.1. The laser light is focused on the generation lead gap. As
the pulse propagates tow ard the HTS delay line the sampling
gate
1 is illum inated. The tim e that the sam pling gate 1 is
switched "on" is varied with respect to the generation gate such
that it
sw eeps
through
the
entire
generation
pulse
duration.
Hence, a DC voltage is read at the sampling gate while it is "on"
and the entire input pulse is sampled.
The output pulse is sam pled the same way as the input
pulse, except sam pling gate 2 is used. By doing so, the pulse
m ust travel
along
the entire HTS
delay line p rio r to being
sampled at gate 2. In this way, by looking at V outl we can see
the input pulse shape, and by looking at Vout2 w e can see the
output
p u lse
from
the
delay
lin e.
By
com paring
the
time
difference between V outl and Vout2 we can determ ine the time
o f flight o f the pulse and hence the delay of the line. Taking the
F ourier transform o f the tim e dom ain pulses w ill reveal their
frequency
content.
It is also very im portant to note that the central microstrip
line over which the pulse travels is intentionally grounded in
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fig u re
3.4.1.
generation
This
and
potential forces
is
to
provide
sam pling gates
the carriers
a
poten tial
betw een
the
and the m icrostrip line. This
generated
w ithin the gap
to be
swept into the m icrostrip line as opposed to being swept into the
m icrostrip groundplane. This in essence "guides" the pulse to
propagate along the m etalization and into the HTS delay line.
Had the m icrostrip line been left at a floating potential, the
carriers would be sw ept into the underlying ground plane and
no pulse would be sampled.
3 .4 .A
HTS Delay Line Fabrication
The layout of the coplanar waveguide delay line is shown
in figure 3.4.A .I. The layout is very similar to the layout of the
CPW resonator used in chapter 2 with the notable exception of
the central line width. The delay line linewidth is 10 pm and the
gapwidth on either side is 19 pm . This is a 50 Q configuration.
Typical linew idth tapers are used as in chapter 2 in order to
m aintain 50 £2 com patibility. The overall chip dim ensions are
1.27 cm long by 0.5 cm wide by 508 pm thick.
The reason for using such narrow linewidths in the central
region is as follows. Recall that the total line inductance is made
up of the com bination o f the kinetic and m agnetic inductances.
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
kinetic
inductance
effect prim arily
is
seen
through
the
increase in the penetration depth as a function of tem perature.
If the
transm ission
line is m ade sm all enough
so that the
penetration depth and it’s tem perature dependent increase is a
reasonable fraction of the transm ission line's actual geometry,
then
the
kinetic
inductance
effect w ill
m arkedly
effect the
overall inductance o f the transm ission line. Since the kinetic
inductance will increase the overall inductance of the line, there
w ill
also
be
a noticeable drop
in
the phase
velocity
with
increasing tem perature. Therefore, it is advantageous to m ake
the physical size of the transmission line geometry as narrow as
possible in order to increase the relative effect of the kinetic
inductance. In fact, slow wave kinetic inductance transm ission
lines are usually m ade using a m icrostrip configuration whereby
the
in te rv en in g
propagation
d ielec tric
m aterial
th at
supports
the
wave
has a thickness on the order o f the penetration
d e p th [4].
The processing of the YBCO film is identical to that of the
YBCO processing in chapter 2 section 2.6 and w ill not be
repeated
here.
67
with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3 .4 .A .I.
Layout of coplanar delay line for
p ico seco n d
m easurem ents
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.4.B
Photoconductive Switch F abrication[5]
T he
op tical
sw itches
w ere
fab ricated
using
silico n -o n -
sapphire (SOS) substrates. SOS substrates are very com m only
used for fabrication of optical switches [2,6,7,8,9]. The advantage
of SOS as an optical switch material is that it has a well defined
photoconductive layer on a very low loss substrate. The SOS
substrates used (3” Union Carbide SOS wafer) consisted of a thin
intrinsic epitaxial layer of silicon (about 0.5 pm ) on a 425 pm
thick sapphire substrate. The initial processing steps are shown
in figure 3 .4 .B .I. First the wafers are cleaned to rem ove any
organic residues from the surface and then etched with buffered
HF
solution
to
rem ove
any
S i0 2
layer. This step is very
im portant in achieving good ohmic contact to the silicon surface.
A num ber o f different m etals have been used to m ake ohm ic
contacts to silicon [10,11,12]. In this case a Cr-Au system was
selected because o f its ability to m ake good ohm ic contact to
intrinsic silicon. A very thin layer of chromium (less than 200 A)
follow ed by a 1000-2000 A layer o f gold was evaporated onto
the silicon side of the SOS wafer. The thin layer of chromium is
used to im prove
adhesion
betw een th e gold and the silicon
su rfa ce .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A plating-up process was used for fabricating the switches.
A layer of photoresist (American Hoechst AZ 4620) about 5 pm
thick
is
applied
p ho to resist
to
the
surface
spinner (5000
RPM
o f the
for 30
w afer
by
seconds).
using
(AZ
a
4620
photoresist is used because it is a thick photoresist suitable for
up-plating
purposes.) Then
the
substrate
is pre-baked
in
an
oven at about 70 C for 15-20 m inutes to dry the photoresist.
Then the photoresist covered substrate is exposed through the
m ask using a m askaligner (K asper). W hen the photoresist is
developed (Am erican H oechst AZ 400 K developer diluted 1:3)
photoresist covers only those areas which will be etched free of
m etal.
Figure 3.4.B .2 shows the crucial steps in the fabrication of
the optical switch. The metal covered areas of the wafer that are
exposed are then electroplated with approxim ately 3 pm
thick
layer of gold (Sel-Rex Pur-A-Gold 125 gold-plating solution). The
rem aining
photoresist is
stripped
off by rinsing
in
acetone,
which exposes the areas with a thin layer of gold on them. The
thin layer o f gold is etched using an iodine and potassium iodide
solution (10 g I and 20 g KI in 100 ml of de-ionized water). The
chrom ium
lay er
is
then
rem oved
by
using
a
potassium
ferricyanide and sodium hydroxide solution (18 g K3Fe(CN)6 and
9 g NaOH in 100 ml of de-ionized water). The up-plating
70
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Gold
Silicon
S ilico n ’
S opphire
Sopphire<
•C r
2 —Deposit Cr and Au
1 - Cleon t h e SOS waf er
P hotoresist
G old—*
Silicon »
Cold
Silicon
Sapphire-
Sopphire
3 —Apply photoresist
Figure 3.4.B .I.
4—Pattern photoresist
Initial processing steps for the fabrication
of optical switches
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C o ld Silicon ♦
•Cr
Silicon
M
M
V
/A
S o p p h ire-
S a p p h ire
6— Etch the thin Cr—Au layer
5— Electroplate gold
Ion implant
m rn ssm
mii
Silicon
SappM reSopphlr*
8— Lap the sapphire wafer
7— Ion implant the wafer
Figure 3.4.B.2.
Crucial processing steps for the
fabrication of optical switches
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
technique used above m inim izes under-cutting that accom panies
etching a thick layer of metal.
The w afers were then ion im planted to shorten the carrier
lifetim e.
A
num ber o f d ifferen t ions
can
be
used
for
ion
im plantation. In this case four different energies o f silicon ions
w ere used to achieve uniform damage through the silicon epilayer. The im plant dosage is also an im portant factor. If the
dosage is too small the carriers will have a long lifetim e. B ut if
the
dosage
is to o
high
the resp o n siv ity of the
sw itch
w ill
decrease w ithout too m uch im provem ent in speed. An im plant
dosage
betw een
about
10*4 to 10^5 Cm 2
gives
satisfacto ry
results in our case (the im plant dosage depends on many factors,
among them , the ions being im planted and the sem iconductor
s u b s tra te ).
T he
w idth
of
the
m icrostrip
lin e s
(fig u re
3.4.1)
on a 125 p m
are
designed
to have an im pedance of 50 Q
thick
sapphire
substrate. A thinner substrate is necessary to increase
the frequency at w hich the higher order m odes w ill start to
propagate in the substrate. A fter ion im plantation, the sapphire
substrates
boron
w ere lapped
carb id e
lap p in g
down from 425 p m to 125 p m
pow der
(the
thickness o f the substrate was about +/steps o f
the
process
accuracy
in
th e
using
fin a l
12 pm ). In the final
a layer of chrom ium and then gold was
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
evaporated on the back side (sapphire side) of the wafer. The
thin layer of gold on the back was then up-plated with about 5
p m of gold to form the ground plane of the m icrostrip. The 3”
SOS w afers were then diced using a dicing saw with diamond
pow der tipped carbon blades. Finally, the optical sw itches are
attached to a gold-plated brass fixture using silver epoxy.
3.4.C
Optical Path Layout[5,13]
The picosecond optoelectronic experim ental setup used in
this research is shown in figure 3.4.C .I. The picosecond laser
so urce
co n sists
of
an
actively
m ode-locked
NdrY AG
laser
(C oherent A ntares) operating at a wavelength of 1.06 p m with
output pulses o f approximately 100 ps FWHM (full width at half
m axim um), average power of 20 W, and a repetition rate of 76
MHz. A KTP crystal is used to double the frequency of these
pulses to 532 nm with output power of 2 W and 70 ps pulses.
A bout 800 mW of this output pow er is used to synchronously
pum p a dual je t dye laser (Coherent 702-1). The dye laser uses
Rhodamine 6G (R6G) dye as the gain medium and DODCI as the
saturable
absorber. T he dye laser is equipped with
a cavity
dum per w hich allow s the repetition rate o f the pulses to be
varied. The dye laser is typically operated at a w avelength of
600 nm with repetition rate of 7.6 MHz and average power of
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OSCILLOSCOPE
COMPUTER
LOCK-IN
AMPLIFIER
AUTOCORRELATOR
STEPPER MOTOR
DRIVER
Ref. in
S ig n a l in
Translation Slage
With S tepper
'Motor
I
S am p lin g Boa]
B eam
S p li tte r
CLOSED-CYCLE
HELIUM
REFRIGERATOR
Lens
ir
DC
POWER SUPPLY
C hopper
M ir r o r
MODE-LOCKED
FREQUENCY-DOUBLED
Nd: YAG LASER
PICOSECOND DYE
LASER
Figure 3.4.C .I.
Layout and experimental setup of
picosecond optical path.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100 mW. T he optical pulses
have a pulse width
o f 1.8 ps
m easured using an optical autocorrelator (Inrad model 5-14B).
R eferring to figure 3.4.C.1, the train o f picosecond laser
pulses from the dye laser is split into two parts . One beam
passes through an optical chopper and is focused onto the pulse
g eneration
sw itch
w ith
the c hopper
reference
fed into
the
reference input o f the lock-inam plifier. The second beam travels
a path
w ith
p recisely
by
a
v ariable
length,
m ovem ent of a
stage. The pathlength of
w hich
be
varied
com puter controlled
the second
that it arrives at the sampling
can
switch,
very
translation
beam can be varied such
before, during, or after the
arrival of the optical pulse at the generation switch. The output
from the sampling switch is fed into the input of the lock-in
a m p lifie r.
T he
p h o to c u rre n t
q u a n tity
p ro d u c e d
m e a su re d
at th e
by
o u tp u t
th e
LIA
is
the
of
th e
seco n d
photoconductor as a function of relative delay between the two
optical pulses.
In order to m easure the response of the HTS delay line a
closed cycle refrigerator (CTI-Cryogenics model 350 cp) is used.
The delay line is mounted on the closed cycle refrigerator’s cold
finger.
T his
low
tem p eratu re
dew ar
has
about
14
coaxial
electrical feed throughs which can be isolated from the ground.
The isolation
o f the electrical coaxial wires is im portant in
*T
S’
/O
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
certain cases when the signal to be m easured is very sm all, of
the order of a few jxV. The optical beams enter the dewar from a
large window at the top, then are focused via lenses onto the
photoconductive switches
and the device w hich are inside the
dewar. The electrical lines are fed to the device from the sides of
the dewar. A tem perature controller and a heater are used to
m o n ito r
and raise/lo w er
the tem perature
electrical and optical response
in
steps, w ith
the
m easurem ents obtained also as a
function of different tem peratures.
3 .5
EXPERIMENTAL RESULTS AND DISCUSSION
Figures 3.5.1 through 3.5.8 show the resulting picosecond
pulse propagation m easurem ents of the optical switches and the
HTS
coplanar w aveguide delay
lin e at various
tem peratures.
Figure 3.5.1 is the sampled input pulse at 36K. The amplitude of
the pulse is in pV . The optical delay path w as adjusted to start
the sw eep before the generation pulse arrived at the sw itch.
T herefore, figure 3.5.1 reveals no detected signal at the input
sw itch
sam pling gate until 35 picoseconds have elapsed. The
pulse centered at 40 pS is the generation pulse sent
from
the
laser. It’s full wave half maximum (FW HM ) is approxim ately 10
pS wide. Figure 3.5.1 also shows that approxim ately 20 pS later
the reflected pulse from the bondw ires that attach the optical
switch to the delay line arrive at the sampling gate. The end of
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IN P U T
P U LS E
3S K
248.5
177.5
C
D
142. 0
35.5
_1
CL
z:
CE
0. 0
-35.5
-71.0
-106.5
-142.0,
DELRY
Figure 3.5.1.
CPSEC)
Sampled input pulse at 36K
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OUTPUT P U LS E
36K
129.G
C
3
100.8
SB.4
72.0
57.G
_J
Q.
.4.
CE
43.2
28.8
-14.4
DELRY
Figure 3.5.2.
CPSEC)
Sampled output pulse at 36K
r\
/y
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
bondw ire reflection
occurs approxim ately 40 ps after the
generation pulse arrives at the sampling gate. It is im portant to
note that this delay is twice the true delay since it is a reflected
signal. Hence the length from the sampling gate to the HTS delay
lin e is approxim ately 20 pS. K eeping the start point o f our
o ptical d elay
sw eep
constant so that w e m aintain the 0 pS
location, we then sweep a full 300 pS and sam ple the pulse on
the sam pling gate after the pulse has traversed the HTS delay
line. This output pulse is shown in figure 3.5.2. The sampled
p ulse is at 36 degrees Kelvin (36K). N otice the output pulse’s
peak occurs at
228 pS and has broadened to 20 pS FWHM.
There are two im portant observations that can be m ade
from figures 3.5.1 and 3.5.2. The first is the overall time of flight
in the HTS delay line. This value is calculated by subtracting
tw ice the bondw ire length and the initial 40 pS zero offset from
the output pulse. This results in a time of flight of approximately
150 pS.
W ith this time o f flight data we can now check the validity
o f the experim ent. O ur objective is to be able to back out the
dielectric constant of the substrate m aterial from this tim e of
flig h t data. T his value will be compared to the published value
of
betw een
24
and
26[14,15,16].
To
the
e x te n t
that
our
calculated value o f dielectric constant agrees with the published
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
data, we can establish the veracity of the experim ental setup.
The way in which we do this is as follows. By virtue of equation
3.2.4 and realizing that
V=
w here
Co
effective
Co
(3.5.1)
is the speed o f light in vacuum and £ e^
d ielectric constant of the guiding
is the
structure, we can
solve for the effective dielectric constant such that
Cot
i2
(3.5.2)
£eff:
w here
I is the length of the delay line and T is the delay in
seconds. Using the value of 1.27 cm for I and 150 pS for T, we
find that the effective dielectric constant is 12.56. Because the
d ielectric constant o f the lanthanum alum inate substrate is so
high
and
the linew idth
is
so narrow ,
w e can
represent the
crossection
of the transm ission line as
a point
source. In so
doing, we
can also estim ate the effective dielectric constant as
£ r+ 1
£e f f = 2
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3-5 *3)
w here £ r is the relative dielectric constant of the material. From
equation 3.5.3 and the extracted value from equation 3.5.2 we
d e te rm in e
alum inate
the
to
re la tiv e
be
24.12.
d ie le c tric
T his
c o n sta n t
com pares
very
of
lanthanum
well
with
the
published value of 24.5. H ence our m easurement system is self
c o n siste n t.
The second im portant observation from figure 3.5.2 is the
broadening o f the output pulse. Recall that the optical switches’
substrate was purposefully lapped down in order to allow for
high frequency perform ance. Unfortunately, the HTS CPW delay
line
su b strate
could
no t
be
lapped
dow n.
T herefore,
at
a
thickness of 508 p m , the substrate w ill cause an appreciable
degradation of the high frequency components of the pulse and
hence the pulse will broaden. This will be discussed at length
fu rth er
below
in
the
frequency
com ponent
section
of
the
experim ental results. Future experim ents will include a lapped
down version of the delay line to increase bandwidth.
Figure 3.5.3 through 3.5.8 show the pulse as a function of
tem perature. The initial zero point has now been shifted so that
only a 135 pS sweep is needed to capture the output pulse. The
zero pS point in these figures is the 185 pS point in figure 3.5.2.
Notice as the tem perature approaches the Tc o f the material the
pulse not only attenuates in amplitude but also broadens and
82
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CPW P U LS E
5GK
210.0
183.0
168.0
147.0
C
3
128. 0
105.0
84.0
_l
CL
63.0
2:
IT
0.0
DELAY
Figure 3.5.3.
CPSEC)
Sampled output pulse at 56K
83
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CPW P U L S E
7SK
201.5
173.2
15 S. B
134.4
83.8
CL
67.2
44.8
22.4
0.0
DELRY
Figure 3.5.4.
CPSEC)
Sampled output pulse at 76K
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CPW P U L S E
8SK
162.0
145.8
C
IS
S7.2
SI
s_
10
64 . 8
48.6
32.4
16.2
0.0
-16.2
DELRY
Figure 3.5.5.
(P S E C )
Sampled output pulse at 86K
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CPW P U L S E
88K
197.0
177.3
157. B
•P
137.9
C
ZS
98.5
78.8
DL
2:
cc
59. 1
39.4
19.7
0.0
-19.7
DELRY
Figure 3.5.6.
(P S E C )
Sampled output pulse at 88K
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CPW P U LS E
89K
B5.0
78.5
£ 8.0
59.5
42.5
34.0
25.5
17.0
8.5
0.0
-8.5
0.0
DELAY
Figure 3.5.7.
CPSEC)
Sampled output pulse at 89K
87
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CPW 'P U LS E
90K
27 .2
23 . 8
20.4
C
3
-Q
10.2
<0
_]
CL
2:
CE
-3.4
-B.B
-
10.2
0.0
13.5
DELPY
Figure 3.5.8.
(P S E C )
Sampled output pulse at 90K
88
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•
PEAK DELAY
185
180
175
170
165
160
155
150
145
0
20
40
60
80
100
T E M P E R A T U R E (K)
Figure 3.5.9.
Peak output pulse delay as a function of
in creasin g
tem perature
89
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° 10 GHz resonator (freq)
♦ picosecond pulse (time)
0 .2 90
0 .2 8 0
0 .2 7 0
♦o
0 .2 6 0
Tc is 90K
0 .2 5 0
0 .2 4 0
0 .2 3 0
0
20
40
60
80
100
TEMPERATURE
Figure 3.5.10.
Phase velocity as a function of
temperature of the delay line and a 10
GHz CPW resonator.
90
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increases it's delay
as predicted in section 3.2. The peak output
pulse delay time as
a function of temperature is plotted in figure
3.5.9.
N otice
tem p eratu re
as
of
the
the
tem perature
m aterial
approaches
(90K )
the
the
delay
transition
d ram atically
in c re a se s.
From
fig u re
3 .5 .9
com parison. This is
we
can
d evelop an in te re s tin g
shown in figure 3.5.10. As has been shown
earlier the kinetic inductance effect is responsible for the time
delay increase as a function of temperature. Let us assume that
the dispersive nature of the pulse in figures 3.5.2 through 3.5.8
is solely due to the increase in surface resistance and dielectric
substrate properties. Hence, if we neglect the surface resistance
and dielectric effects we are left with a non-dispersive pulse
th at
only
fncreases
it's
tim e
o f flight
due
to
the
kinetic
inductance effect. Such a non-dispersive pulse would have, by
definition, a constant phase velocity. We can estim ate this phase
velocity by virtue of equation 3.2.4 where the group velocity is
th e
sam e
as the
phase
velocity
because
the
pulse
is
non-
dispersive. Figure 3.5.10 compares this resulting phase velocity
versus tem perature with the phase velocity of a high Q 10 GHz
resonator of the same geom etry (the resonator's phase velocity
is m easured in the same way as the experiments in chapter 2).
It is im m ediately noticeable that the two curves overlap and
com plem ent each other quite nicely. This suggests that we may
91
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estim ate the kinetic inductance effect in the tim e dom ain by this
method of only looking at the peak pulse delay.
Let us look at figure 3.5.10 in more detail. W e can compare
the general
shape of fig u re 3.5.10 and figures
2.7.1 through
2.7.3. The shapes are very similar, as we expect. In all figures,
both
the
pulse
delay
and
the
reso n ato r
fig u res,
the
low
tem perature phase velocity is approxim ately 0.282 o f the speed
o f lig h t
in vacuum
w hich
again,
is
expected.
The notable
difference is in the m agnitude of the change in phase velocity.
A t tem peratures approaching the transition tem perature of the
m aterial the figures in chapter 2 only exhibit a 4% change while
figure 3.5.10 shows an 18% change. This large percentage change
in the phase velocity is due to the sm aller dim ensions of the
coplanar waveguide delay lin e as indicated in section 3.4.A.
In sum m ation, we see that all o f our assum ptions have
been essentially correct. T hese are: a narrow transm ission line
g eom etry
produces
a
g rea ter
overall
change
in
the
phase
velocity, a thick substrate w ill broaden the pulse, and the kinetic
inductance
effect
sim ply
induces a further tim e
delay
function of tem perature.
92
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as a
Let us now focus our attention on the frequency content of
the output pulses. Figure 3.5.11 through 3.5.18 show the Fourier
transform s
of
the corresponding
preceding
in p u t and
output
pulse figures. Each of these figure indicates the dispersion o f the
pulse at the m easured tem perature. A ll the frequency d ata in
these figures has been derived using an FFT on the raw pulse
data. No pre data processing was em ployed. Figure 3.5.11 is the
FFT o f the input pulse at 36K w ith the effect of the bond wires
rem oved. N o tice
the frequency
co n ten t severely
rolls o ff at
greater than 50 GHz, with the frequency content not being able
to be quantified due to the experim ental noise, at greater than
100 GHz. H ence, we expect the dispersion o f the output pulse
from the HTS delay line to be no better than this. Figure 3.5.12
is that of 3.5.11 plotted only up to 100 GHz. Figure 3.5.13 is the
FFT of the output pulse at 56K. W e can see that the frequency
content starts to severely roll off at greater than 20 GHz. This
change in dispersion characteristic from that of the input pulse
can
be
attrib u ted
to
a num ber o f effects.
The
first is
the
substrate thickness. If the substrate is thick enough, particularly
in CPW [17], energy can be launched into the substrate rather
than into the CPW . We can estim ate the frequency where the
substrate can have energy traverse in it as opposed to in the
c o p la n a r
w a v eg u id e
tran sm issio n
lin e ,
by
looking
at
the
frequency a t which the substrate thickness looks like a quarter
w avelength. This relationship is given by
93
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FFT
IN P U T
P LS
3GK
59.4
52.8
48.2
39.6
33 . 0
0.0
10
FREQ.
Figure 3.5.11.
20
50
100
CG H z )
FFT of input pulse at 36K
94
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250
FFT
IN P U T
P LS
3GK
59.4
52.8
4S.2
33.0
10
FREQ.
Figure 3.5.12.
20
50
100
( GHz )
FFT of input pulse at 36K limited to 100
GHz
95
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FFT
CPW P U LS E
56K
74.0
66.S
59.2
8
44 .4
37.0
22.2
B
4
0.0
1
2
5
10
FREQ.
Figure 3.5.13.
20
50
(GHz)
FFT o f output pulse at 56K
96
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100
FFT
CPW P LS
7S K
74.0
66.6
59.2
37.0
29. B
0.0
20
FREQ.
Figure 3.5.14.
50
100
CG H z )
FFT of output pulse at 76K
97
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FFT
CPW P U LS E
86K
84.6
57.6
50.4
43 .2
36.0
26.8
21.6
14.4
7.2
0.0
10
FREQ.
Figure 3.5.15.
20
50
(GHz)
FFT of output pulse at 86K
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
FFT
CPW P U LS E 8 8 K
74.0
65 . 6
59.2
44.4
37.0
26. 6
22.2
0.0
10
FREQ.
Figure 3.5.16.
20
50
(GHz)
FFT of output pulse at 88K
99
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100
FFT
CPW P U LS E B9K
S3.9
5S.B
49.7
DQ 42.S
•C
35.5
28.4
CC
0.0
10
FREQ.
Figure 3.5.17.
20
50
( GHz )
FFT o f output pulse at 89K
100
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100
FFT
CPW P L S
S0K
63.0
56.7
50.4
44. 1
37.8
25.2
12.6
0. 0
20
FREQ.
Figure 3.5.18.
50
CG H z )
FFT of output pulse at 90K
101
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100
Co
=^
w here
X/4
(3.5.1)
/ 4
is now the thickness of the substrate and £ ^
in
this case is just the £ r of the substrate. The resulting frequency
at which energy is launched into the substrate is 30 GHz. Hence
w e can expect the roll
off of the frequency content of the
transm ission line to be severely degraded at 30 GHz since some
o f the input energy will go into a substrate mode and not along
the CPW. A second reason for the severe rolloff in figure 3.5.13
is due to the bondw ire effect. U nfortunately, this effect cannot
be quantified w ithin the context of this experiment as a full two
p o rt an aly sis
m ust be conducted
in
order to
de-em bed
the
bondw ire effect. Future experim entation will have to take the
dispersion effects of the bondwire into account.
Figures 3.5.14 through 3.5.18 show the rem ainder FFT's of
th e
o u tp u t
freq u en cy
tem perature
p u lse
at
h ig h er
ro llo ff rem ains
tem peratures.
c o n stan t
until
N o tice
86K.
th at
the
Beyond
this
the frequency ro llo ff severely degrades. A t 90K,
w hich is very close to the norm al transition tem perature, the
rollo ff is severe at greater than 10 GHz. This of course is the
frequency dual o f the tim e dom ain data culm inating in figure
3 .5 .1 0 .
102
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3.6
SUMMARY
In conclusion, we have seen from chapters 2 and 3 the
m anifestation,
n o n lin e a r
experim entation,
in d u ctan ce
of
m odeling
high
and
tem p eratu re
analysis
of the
superco n d u ctin g
passive devices. The nonlinear inductance is termed the kinetic
inductance effect and arises due to the change in tem perature of
the m aterial and hence the content of superconducting electron
pairs. Starting from a single frequency device, that of a high Q
coplanar
w aveguide
resonator,
we
were
able
to
definitively
dem onstrate and m odel the overall behavior of these devices.
W ith this know ledge of the establishm ent that the penetration
depth of the HTS has a "T squared" dependence as opposed to
the usual G orter-Cassim er "T to the fourth" dependence, we are
in a position
re s o n a to rs .
to design very narrow band filters and high Q
A m p lify in g
on
th ese
d e v elo p m e n ts
we
then
proceeded to look at wide bandwidth devices such as coplanar
w aveguide delay lines. W e used optically generated picosecond
electrical pulses to probe the non linear inductance workings of
these structures. This powerful experim ental technique revealed
the basic assum ptions of the workings of this device explicitly
including the kinetic inductance effect to be essentially correct.
A lth o u g h
c e rta in
m icrow ave
d ev ice
p a ra m e te rs
such
103
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as
substrate
thickness
w ere beyond
the experim ents
control,
the
overall perform ance of this technique is quite sound.
At this point we have established in the last two chapters
a w ealth
of experim ental
and
analytical
data
to
allow
the
m icrowave design engineer to begin the task of producing useful
passive m icrowave and m illim eter wave devices that can benefit
from this nonlinear tem perature dependent effect.
W e are now in a position to extend this experim entation
and
analysis
to in clu d e n o n lin ear inductance
effects due
to
device param eters. Specifically the developm ent of an all HTS
param etric
am plifier base on the nonlinear inductance of the
Josephson junction. T his developm ent w ill be covered in the
next chapter.
104
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3 .7
REFERENCES
[1]
D. H. Auston, “Picosecond Optoelectronic Switching and
Gating in Silicon,” Appl. Phys. Lett., vol. 26, no. 3, pp. 101—
1 0 3 ,1 9 7 5 .
[2]
P. A. Smith, D. H. Auston, A. M. Johnson, and W. M.
A ugustyniak, “Picosecond P hotoconductivity in R adiationDam aged Silicon-on-Sapphire Film s,” Appl. Phys. Lett., vol.
38, no. 1, pp. 47—50, 1981.
[3]
M. B. Ketchen, D. Grischkowsky, T. C. Chen, C-C. Chi, I. N.
Duling III, N. J. Halas, J-M. Halbout, J. A. Kash, and G. P. Li, "
G eneration o f Subpicosecond Electrical Pulses on Coplanar
Transm ission Lines", Appl. Phys. Lett., vol. 48, no. 12, pp.
7 51-753,
[4]
1986
J. M. Pond, J. H. Claassen, and W. L. Carter, " Kinetic
Inductance Microstrip Delay Lines", IEEE Trans. Mag., vol.
M AG-23, pp. 903-907, 1987
[5]
M. Matloubian, " Optical Network Analyzer for
Characterization of M illim eter W ave Sem iconductor
Devices and Integrated Circuits", D octoral Thesis,
University of California at Los Angeles, 1989
10 5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[6]
A. B. Hammond, N. G. Paulter, and A. 5. Wagner, “Observed
Circuit Limits to Time Resolution in Correlation
M easurem ents with Si-on-Sapphire, GaAs, and InP
Picosecond Photoconductors,” Appl. Phys. Lett., vol. 45, no.
3, pp. 289— 291,
[7]
19 8 4 .
D. E. Cooper and 5. C. Moss, “Picosecond Optoelectronic
M easurem ent of the High-Frequency Scattering
Parameters of a GaAs FET,” IEEE J. Quantum Electron., vol.
QE22, no. 1, pp. 94— 100, 1986.
[8]
J.-M. Halbout, P. G. May, and M. B. Ketchen,
“ C haracterization of Logic Devices with Photoconductively
Generated Picosecond Pulses,” Characterization o f Very
High Speed Semiconductor D evices and Integrated Circuits,
pp. 247-254, SPIE, 1987.
[9]
D. Krokel, D. Grischkowky, and M. B. Ketchen,
“ Subpicosecond Electrical Pulse Generation Using
Photoconductive Switches with Long Carrier Lifetim es,”
Appl. Phys. Lett., vol. 54, no. 11, pp. 1046— 1047, 1989.
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[10]
J. J. Bart, “The Analysis of Chemical and Metallurgical
Changes in M icrocircuit Metallization Systems,” IEEE Trans.
Electron D evices, vol. ED-16, no. 4, pp. 351—360, 1969.
[11]
L . E. Terry and A. W . Wilson, “Metallization Systems for
Silicon Integrated Circuits,” Proc. IEEE, vol. 57, no. 9, pp.
1580— 1586, 1969.
[12]
P. H. Holloway, “Gold/Chromium Metallizations for
Electronic Devices,” Solid State Technol., pp. 109— 115,
1980.
[13]
M . Martin, " Optical and Electrical Response of Three
Terminal High Frequency Semiconductor Devices ",
D octoral Thesis, University of California at Los Angeles,
1992
[14]
W . G. Lyons, R. S. Withers, J. M. Hamm, A. C. Anderson, P.
M. Mankiewich, M. L. O'Malley, R. E. Howard, R. R. Bonetti,
A. E. Williams, and N. Newman, " High-Temperature
Superconductive Passive Microwave Devices", IEEE Int.
M icrowave Sym. D igest, MTT-S, pp. 1227-1230, 1991
[15]
Z. Y. Shen, P. S. W. Pang, W. L. Holstein, C. Wilker, S. Dunn,
D. W. Face, and D. B. Laubacher, " High-Tc Superconducting
107
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Coplanar Delay Line with Long Delay and Low Insertion
Loss", IEEE Int. Microwave Sym. D igest, MTT-S, pp. 12351238, 1991
[16]
W. Chew, L. J. Bajuk, T. W. Cooley, M. C. Foote, B. D. Hunt, D.
L. Rascoe, and A. L. Riley, " A Coplanar Waveguide Filter
using Thin-Film High Tem perature Superconductor[sic.]",
IEEE Int. Microwave Sym. D igest, MTT-S, pp. 1333-1336,
1991
[17]
M. Riaziat, R. Majidi-Ahy, and I-J. Feng, " Propogation
Modes and Dispersion Characteristics of Coplanar
W aveguides", IE EE Trans. Microwave Theory and
T e c h n iq u e s, MTT-38, pp. 245-251, 1990
108
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CHAPTER 4
Parametric
4.1
Interactions
INTRODUCTION
As we have seen from the last two chapters there has
been a growing interest in exploiting the nonlinear tem perature
dependent
in d u c tiv e
c h a ra c te ris tic s
of
h ig h
te m p e ra tu re
superconducting (HTS) transm ission lines for filters, delay lines
and other passive m icrowave components. It is also possible to
exploit the nonlinear device dependent inductive characteristics
o f Josephson ju nctions
for use at m icrowave
and m illim eter
wave frequencies [1,2] for the use in active components. One of
the exciting possibilities is the developm ent o f a param etric
109
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am plifier (param p) using all HTS com ponents. Such a device
w ould operate over a broader range o f tem peratures than the
previous low tem perature superconducting param ps [3,4] while
m a in tain in g
Jo sep h so n
the
unique
ju n c tio n
a m plifier relies
c a p a b ilities
based
param ps.
and
c h a rac teristics
B ecause
the
an
in h e re n tly
param etric
on a nonlinear inductive elem ent it does
dissipate energy within it's internal structure and
low
noise
d evice.
Such
a
of
not
therefore it is
device can
be
m onolithically incorporated as a pream plifier for phased array
ra d a r a t 60
GHz to im prove system
perform ance and noise
figure. The first step in developing such an am plifier in HTS is
th e
d e m o n stra tio n
of
p aram etric
in te ra c tio n s
due
to
the
nonlinear Josephson junction element.
4.2
PHYSICAL INTERPRETATION OF PARAMETRIC
AMPLIFICATION
The basic principle on which param etric am plification is
founded
is the
transference of energy from
a signal at one
frequency to that of a signal at a different frequency. Because
the nonlinear device that accomplishes this is non dissipative by
nature this energy transfer is ideally lossless. To illustrate this
transference o f energy im agine we have the circuit illustrated in
figure 4.2.1. The circuit consists of a sim ple capacitor and an
inductor[5]. Im agine further that we allow the capacitor plates
110
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movable
plates
Inductor
Capacitor
V sin wt
Figure 4.2.1. Circuit illustrating parametric am plification
111
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to be pulled
apart and pushed
back together again at will.
Suppose we now impose a sinusoidal voltage across the capacitor
at
a
frequency
0 )$ .
As
the
sinusoidal
voltage
reaches
its
maximum we suddenly pull apart the capacitor plates. Of course
work m ust be done in order to do this and the resulting energy
is im pinged across the capacitor plates. Because the voltage is
related to the capacitance by the charge on the plates
V = Q /C
(4.2.1)
and the capacitance, C is inversely proportional to the distance
betw een
the
plates,
the
overall
voltage
will
discontinuously
increase by an am ount proportional to how far the plates are
pulled apart. We assum e, fo r the sake of illustration, that the
plates are pulled
apart instantaneously so that the charge Q
rem ains
C ontinuing fu rth er in the sinusoidal
constant.
cycle,
there w ill be a point in tim e at which the voltage goes from
p o sitiv e
voltage
to
negative
in
m agnitude. A t the point w here
the
across the capacitor is identically zero we push the
plates back together again to their original position. We do no
work in doing this since there is no charge induced on the plates
at this tim e by virtue of (4.2.1). At the point further on in the
cycle w here the negative m agnitude is at a maximum we again
pull the plates apart. Again a discontinuous negative voltage
jum p results in an increase in m agnitude of the signal. Further
112
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on in the cycle when the voltage across the plates is zero we
push the plates together again. And so the cycle repeats itself
c o n tin u o u sly .
Let us see what we have accom plished by this. A t each
quarter cycle of CQS the voltage increases a proportional amount
to
the
distance
the
plates
are
pulled
apart resulting
in
an
increase or am plification of the signal. Every half cycle o f (Og
the p lates are pushed together again w ith no expenditure of
energy. The resulting
signal am plification
is shown in figure
4.2.2. The X 's represent the point in tim e when the plates are
pulled apart and the open circles represent the point in tim e
when the plates are pushed together.
T here
are
still
som e
further points
that
need
to
be
com m ented on. A capacitor was chosen to illustrate this device
because it does not dissipate power, only transfers it. Hence the
energy transfer should be quite efficient and also not introduce
resistive noise into the overall device. Also, the analogy rem ains
equally valid if the inductor is substituted for the capacitor as
the nonlinear elem ent.
T he
action
o f p u lling/pushing
the
plates
apart/to g eth er
requires energy and is done with a periodicity o f (Op. We call
this action the " pump" action. This is the source of energy from
113
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t
V
Figure 4.2.2.
Output voltage of the circuit in figure
114
.............................................................................................
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I
_
J
which the device is able to amplify the signal. The energy from
the pump is transferred to the signal and results in amplification
o f this signal. It is important to note that there has to be a signal
there
in
the
first place for energy to be transferred
to it.
T herefore we require at least two signals Q)p and COs to be
p resent in
p ushing
any param etric am plifier. Furtherm ore, the act of
or pulling
n o n lin ear
p roducts
process.
oth er
the
It
than
is
capacitor
plates
reasonable
(Op
(0 $
and
to
is
a discontinuous,
assum e
th at
m ixing
should result from
this
process. If so how do they relate to (Qp and (Qs ? In order to
answ er this question
energy
is
transferred
we need
from
to explore quantitatively how
one
frequency
to
another
in
a
nonlinear device such as this one.
4.3
MANLEY-ROWE RELATIONSHIPS [6]
We will now briefly outline the general energy relations in
nonlinear reactances. The beauty of this derivation requires only
the single assum ption that the characteristic equation describing
the nonlinear elem ent is single valued. For illustrative purposes
let us
use
equation
(4.2.1) as the relationship
betw een
the
voltage and the charge by way of a nonlinear capacitor. Since we
req u ire
tw o
frequencies,
(Op
and
0 )$
we
can
expand
unknow n charge quantity as
115
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the
q=
S
I
m=—oon=—co
QmneKm°>s'+™Pt)
(4.3.1)
since the charge is a real measurable quantity
<<■
^
^/72« Q -m —niQ—m—n Qm,n
(4.3.2)
the current flowing in the nonlinear capacitor is just
dt 7
J = -f=
dt
°°
m
£
— — 0 0
°°
(4.3.3)
Z I mneKma^ naPt)
n=—°°
w h e re
Imn=j(mCOs+n(Op)Q mn
(4.3.4)
Imn~ I-m -n ’I-m,-n=I m,n
(4.3.5)
since the voltage is a single valued function of the charge we
have
v= I
(4.3.6)
m=—°° n=—°°
(4.3.7)
The coefficients of V mn are unknown and are given
by the
fa m ilia r
116
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2K
i
2 it
Vmn=-7-z Id (o s IdcopF(a>sM p)e~J(mo,s‘+nc’, p’)
*** o
w here
F{(os,(Op)
(4.3.8)
o
is
the
nonlinear relationship
betw een
the
voltage and the charge of the capacitor. If we multiply (4.3.8) by
jmQ*m,n„ and sum over m and n,’ we °get
jjj
j
2n
2k
OO
OO
2
1 V mn<
™J m Q m n = ^ 2 IdtOs \dco/((Ds.cop)
m = —o on= —oo
I
I
j m Q*
e-K m^ +^ ) ( A 3 . 9 )
m=—°°n=—e»
By virtue of (4.3.1) and (4.3.2), the summation on the right hand
side of (4.3.9) is imm ediately recognized as -d q ld co s . In addition
substituting in the left hand side of (4.3.9) from (4.3.4) we are
left with
OO
OO
m'Vrr.J*
i
2*
2
dq
(4 .3 .1 0 )
We im m ediately recognize
that Vmnlmn *s Just l^e Power in the
circuit elem ent and th at
F{a)s ,(O p)^ F(q) . T herefore
(4.3.10)
b eco m es
117
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00
00
1
X
mP
i
mm ™
2z ^
A 7T 0
= _ o o „ = -o o "I 6 ) 5 + « £ y ^
x
SdO)p {F(.q)dq (4.3.11)
= -7 -2
p 0
A sim ilar derivation would hold if we had multiplied by jn(?m n
hence an equally valid equation would be
00
o°
nP
X
X
/~ 7
2%
1 2?r
Id0)s I F(q)dq
= - — s
m = —0 0 n = —o o m COs + n 6 ) p
A TT
0
(4 .3 .12 )
0
Since the voltage is sinusoidal and single valued the charge must
also
be. T herefore
(4.3.12)
the charge integral in
m ust be taken
rep re se n ts
the
"
over a full
area
under
the
both
(4.3.11) and
period. But the integral
curve"
and
hence
for
a
sinusoidal variation the total area under the curve identically
reduces to zero.
In other words, since the total charge at time
zero is zero, then it is also zero at 271. Hence we are left with the
fam iliar M anley-R ow e relationships
rr
2
I
m
r\
-------------— = 0
(4.3.13)
m=—°° n=—oo m (Os + n COp
00
oo
1
I
n
p
-----------=
m ——o o 22=—OO m COs + n (Op
0
(4.3.14)
118
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W e now m ust ask how can these relationships help us in
determ ining the characteristics of a param etric am plifier? It is
apparent
from
the above equations
that there
would
infinite num ber of resulting powers and frequencies
to us from simply using (Qp and Q)s
be an
available
unrestricted.
Let us restrict the total number of available frequencies in
the param etric am plifier circuit to only three. They are (Op ,
(O s and COi and are related by
(4.3.15)
2 ( 0 p = ( 0 s+C0i
In
oth er
w ords,
the
three
frequencies
can
be very
closely
spaced. As we look at the Josephson junction we will see the
justification and reasons for choosing (4.3.15). But for now let’s
ju st assume we w ant to look at this case vis-a-vis the ManleyRowe relationships. Applying (4.3.15) to (4.3.13) and (4.3.14)
and assigning
Pol
PrPio
P s ' P - 1,2
Pi
(4.3.16)
we arrive at
(4.3.17)
CDs
COi
119
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from equation (4 .3 .1 3 ) and
(Op
(4.3.18)
CQi
from equation (4.3.14). By virtue of (4.3.17) we can transform
(4.3.18) to be
s +. P
LPp
JL+ P
r jL
n . =0
(Op (Os COi
(4 .3 .1 7 )
sim ply
states
that
the
(4.3.19)
energy
transferred
from
the
pump equally distributes itself between the signal and the idler.
(4.3.19) indicates conservation of energy. Hence if I define a
p o sitiv e
value
of pow er as flow ing
in to
the
device
and
a
negative value flowing out of the device then if I apply C0p and
(0 § to the param etric am plifier I will get pow er out at a new
frequency
(Of.
In sum m ation, we expect to see three frequencies from
the device, 0 )p , CDs anc* (Of
CO$ anc* COi sym m etrically about
the pum p frequency (Qp , and also that C0S and (01 have equal
power m agnitudes. A s we shall see in the experimental section
of this chapter this is precisely what occurs.
120
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4.4
JOSEPHSON JUNCTION INDUCTANCE
The task that rem ains before us is to dem onstrate that the
Josephson junction used in this device behaves like a single
valued nonlinear elem ent. If we can dem onstrate this then the
M anly-Rowe relationships will be applicable.
W e start with the canonical relationships that describe a
Josephson junction ;
/
= / csin0
2 71
dt
where
I
ju n ctio n
(4.4.1)
V
(4.4.2)
<&0
is the current flowing through the junction,
c ritic a l
w avefunction
cu rren t, (j)
across
is
the
the junction,
phase
Oo
is
Ic
d ifferen ce
the
flux
is the
in
the
quantum
defined as h / 2 e and V as the voltage across the junction. Taking
the tim e derivative of (4.4.1) and substituting in (4.4.2)
we
arrive at
dl
2n
dt
O0
— = —
v / c COS0
121
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(4.4.3)
R ealizing that v = L ^
w e can define a Josephson inductance as
L=-
O
(4.4.4)
2;r/c COS0
W e can
m ake this
expression
m ore experim entally palatable.
Rearranging (4.4.1) to be
I
=sin
(4.4.5)
VIcJ
and using the trigonom etric identity
cosx=Vl-sin2*
(4.4.6)
we get
L= -
(4.4.7)
27TlcJ l \*cj
Equation (4.4.7) is plotted in figure 4.4.1.
T heoretically, the
inductance should go to infinity when the current through the
122
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device is identically equal to the critical current. In reality, it
assum es
a
fin ite
value
due
to
experim ental
noise
in
the
m easurem ent o f the inductance. In other words one cannot hold
the current to within an arbitrary deviation from the critical
current. O f course, if the current exceeds the critical current the
Josephson junction inductance is no longer defined. Hence we
have found a single valued nonlinear inductance device that has
the potential to work as a parametric amplifier.
It is im portant to get a physical feel for w hat equation
(4.4.7) really m eans. Figure 4.4.2 is a plot of (4.4.7) but the x
axis is now time. In other words we now let the current through
the device vary sinusoidally. Since
E =^LjU )I2
(4.4.8)
the Josephson inductor is really functioning as a variable energy
storage device. Now let us follow the change in inductance over
one cycle of the current through the device. Prior to reaching the
critical current value (point A in figure 4.4.2) the inductance is
m onotonically increasing. This means that the device can store
more and more energy. After reaching and passing through the
critical current (point B in figure 4.4.2) the inductance now
decreases since the device current is also decreasing. This means
the device can store less and less energy. Hence all the energy
123
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-®— lc = 1 0 0
-•— lc = 5 0
pA
j iA
L
-*— lc = 2 5 p.A
vs
I (equation 4.4.7)
120
■A—lc = 1 0 p.A
100
80
60
40
20
0
0
0.2
0 .4
0.6
0.8
I/Ic
Figure 4.4.1.
Plot of equation 4.4.7 for varied critical
c u rre n ts
124
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0(0.1 l c )
40
120
80
160
TIME |pS)
lc = 150 uA
L _ _____ Oo_____
J
2 t c Ic \ / 1 " d / i c ) 2
I = I COS27l/t
/ = 10 GHz
Figure 4.4.2.
Plot of equation 4.4.7 in time. A 10 GHz
current signal is used.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
that the device has stored up in the positive going half of the
current cycle
now must be dispersed
in the negative going half
o f the current cycle. W here does this energy
now go? Of course,
this is precisely what the M anley-Row e relationships tell us.
Since the energy is not dissipated in the device because it lacks
any resistance, it m ust be distributed among all the allowable
frequencies of device operation. It is precisely because o f this
th at
and
id ler
frequency
is
created
and
is
a necessary
by
product of a param etric amplifier.
A nother im portant point when looking at figure 4.4.2 is
the shape o f the inductance curve. N otice
in ductance
is
an
even
function.
This
can
that the Josephson
also
be
seen
by
exam ining equation (4.4.4). The voltage across the device is on
the other hand an odd function by virtue of v = L — . Hence if we
J
dt
apply
voltages
across the
signal
frequencies
device
we should
at the required
only expect to see
pum p
odd
and
order
frequencies output from the device. In other words frequencies
such as cos ±(Oi and a DC term are not allowed. This is distinctly
d ifferent from what one expects from a typical sem iconductor
d io d e.
Now that we have developed an intuitive feel for how the
Josephson junction param etric
am plifier works we can
L Z, \J
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begin
quantitatively determ ining how we can get gain at the signal
frequency from its operation.
4.5
DEVICE GAIN
Let us now exam ine what would happen if voltages at (Op
, (O s and COi w ere allowed to exist across the bare Josephson
junction. R epresenting the voltages as
v=£vtCOs ((okt + e k)
w here
k=p,s,i
(4.5.1)
represents arbitrary phase differences betw een
the
different frequencies. Substituting (4.5.1) into (4.4.2) and then
integrating we arrive at
(pk( t ) = a ksm(cokt + 6 k)
(4.5.2)
w h e re
(4.5.3)
and o>£ = 2 ^ / ^ . This then recasts (4.4.1) as
/=/csin(0p+<^+01.)
127
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(4.5.4)
In order to solve this equation we will require that
a s, a i « a p
In other
then
that
words, the signal and idler powers are much
the pump pow er. W ith this assum ption we may
(4.5.5)
sm aller
assume
are small perturbations on the pump am plitude and
frequency and therefore we can expand (4.5.4) in a Taylor series
expansion. N eglecting all higher order term s greater than the
first derivative term in the expansion, we get
/ = / c s i n ^ i, + / c( ^ J+ ^ i) c o s ^ p
(4.5.6)
U sing the identity
ejapSino)pt-
J
J k( a p)ejko)Pt
(4.5.7)
w here J k ( ( X p ) are the Bessel functions of the first kind, and the
previous relation
o f 2 (0 p = cos + o>i » the currents at the three
frequencies of interest are
Ip
2j
I CJ i((X p)
128
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(4.5.8)
■2x-(r,
r /
Joiccn)
J o ( a p)
T f ^v* ^
M ap)—
Vi
(Oi
\
(4.5.9)
(Q iJ
CDs
\
.271 _ (
Jiiotp)—
(4.5.10)
(O s)
where w e've used com plex notation. W e may represent (4.5.9)
and (4.5.10) in m atrix form so that
y*
~Yss
7 /
1 si
Vs
y*
1 a ~ _V / _
j
_ /* _
(4.5.11)
*
.J o((Xp)
Y ss= ~J
Y si= - J
(4.5.12)
LjCOs
. J 2 ((Xp)
(4.5.13)
L jm
Y is——j
Y u= —j
J i { a P)
(4.5.14)
Lj(os
J 0(&p)
(4.5.15)
LjCOi
129
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The im portant thing to recognize from (4.5.11) is that the
signal current, I s , is no longer composed of a single voltage
com ponent as in a linear device, but rather is com prised of
contributions from both the signal and the idler voltages. Hence
gain is possible from the device at the signal frequency provided
the idler voltage is greater than zero. Notice also that only over a
specific range of pump powers will the device realize gain. This
can
be
seen
d ependent
positive
on
from
the fact that the m atrix elem ents are all
the
B essel
functions
which
can
have
eith er
or negative m agnitudes. Therefore, for exam ple, gain
will only be realized when both J o ( ( X p ) and J 2 ( 0 C p )
have a
negative magnitude. Again, it is important to stress that the gain
is realized through the cross-coupling terms in the adm ittance
jjj
m atrix for the bare Josephson junction, Y s[ and Y is-
4.6
CIRCUIT IMPLEMENTATION AND GAIN
The foregoing analysis has been carried out for the bare
Josephson
ju n ctio n .
In
o rder
to
m ake
a
useful
device
the
junction must be inserted into a circuit. Because we are using
HTS junctions a resistively shunted junction model[7] will be
used in the analysis. The equivalent circuit m odel is shown in
figure 4.6.1. The justification for using this model will be given
in
th e
ex perim ental
equivalent circuit
p o rtio n
of
this
chapter.
The
overall
used in the gain analysis of the param etric
130
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am plifier is given in figure 4.6.2. We assume N equivalent series
ju n ctio n s
addition
are put
an
in
arbitrary
shunt across
term ination
the transm ission
Yt is provided.
line.
T he
In
only
stipulation is that Yt is purely imaginary. It is apparent that if
we consider N identical junctions in this configuration that the
admittance matrix [Y] will be modified to be
[Y/N], this is because the Lj goes to NLj which is present in each
of the denom inators o f the individual matrix elements. Similarly
Rj goes to NRj and therefore Gj goes to Gj/N. The characteristic
adm ittance o f the transm ission line is G o=l/Zo. Because we are
considering the case where all frequencies of interest are very
close to each other (2 o)p = £0 s + 0)i) we will assume that both Zo
and Yt are constant over the frequency range of interest. This
consideration greatly reduces the complexity of the analysis.
B efore
we
p roceed
it
is
im portant
to
d e fin e
som e
dim ensionless param eters [3]
(4.6.1)
COoL/j
(Qo^&o
Rj
2 k I cR j
131
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(4.6.2)
DEVICE MODEL (RSJ)
c
Figure 4.6.1.
RN
RSJ equivalent circuit model
132
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Yt
Figure 4.6.2.
Param etric am plifier equivalent circuit
133
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where we have substituted the maximum of (4.4.4) into (4.6.2),
and
(4.6.3)
where (0 o tacitly assumes that all frequencies will be very close
to each other i.e.(0o = G)p= <os = G)i- (4.6.1) indicates how well the
junctions are m atched to the transmission line, (4.6.2) describes
the quality of the individual junction and (4.6.3) compares the
term ination to the characteristic im pedance of the transm ission
line.
From figure 4.6.2 it is evident that we will be interested in
analyzing a 1 -port netw ork. This necessarily entails having the
p aram etric
T herefore
am plifier operate
the
gain
o f the
in a reflection
device
w ill
be
am plifier mode.
achieved
if
the
reflection coefficient from the device is greater than one. This
can only be achieved if the load impedance has a negative value.
Hence we m ust look to solve
2
r = Ys_Go
Y s+ G 0
134
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(4.6.4)
w here Y s is the admittance of the circuit in figure 4.6.2 at the
signal frequency. We can immediately write down that
Is = Y xV s
(4.6.5)
in addition from inspection of the circuit and using (4.5.11)
, _ Y SSV S . Y l - v :
Is~
N
G iV s
N
N
Y tV ’
(
}
We can w rite a sim ilar equation for the current at the idler
frequency but here we must be careful. Recall that there is no
incident pow er at the idler frequency only power that will be
gotten from the energy exchange with the pump. Therefore we
can view the entire right hand side of the circuit that includes
the ju n ctio n s
and
the term ination
as a sim ple
generator as
indicated in figure 4.6.3. This therefore leads to
I * = ~ G 0V *
(4-6.7)
Again, from inspection of the circuit and using (4.5.11)
t =m
+ X i,! U + G jV L + T * v *
N
N
N
135
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(4A 8)
Parametric
amplifier
Josephson
junction
signal generator
Figure 4.6.3.
Sim ple paramp reduced equivalent
c irc u it
136
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We can now solve for
then
su b stitu ted
into
V *
using (4.6.7) and (4.6.8). The result is
(4 .6 .6 ).
T his
fu rth e r
re su lt
is
then
substituted into (4.6.5) to yield
Y s= —
+Y
t
+ ^-l —
N
If
N
;—
Y s , Y ‘s^ N -------------
(4.6.9)
Y u/ N + Y T+ G j / N + G 0
we now substitute (4.6.9) using the explicit forms of [Y], i.e.
equations (4.5.12) -(4.5.15), and (4.6.1) - (4.6.3), into (4.6.4) we
get after considerable algebra
T = ( { F - g 2¥ + [ M c c P) - g ! ; b T ? - M a P) 2f
+ 4 g 2? l J o ( a P) - g Z b Tf )
x
{ f ( l + g ) 2+[ J 0( a p) - g £ b r f - J 2 ( a p)2f 2
(4 .6 .1 0 )
Let us now examine some of the salient points represented
by (4.6.10) and (4.6.9). N otice that (4.6.10) depends only on the
dim ensionless
the m agnitude
variables
C X p , ^ , § , and b T • We can determ ine
of the negative conductance o f the device by
exam ining (4.6.9) under the conditions where
137
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_Jo^CCp)
(4.6.11)
~
i r
U sing this value of the term ination
equation (4.6.9)
, the im aginary part of
reduces identically to zero and we are left with
_ G ; __________J j i a p f
s
N
( Na>0L j ) 2( G o+ G j / N )
O f course, because of the
(4'612>
in (4.6.12) the real conductance can
take on a negative value. Hence the parametric amplifier acts as
a negative resistance amplifier. It is important to realize that the
num erator
of the
second
term
in
(4.6.12)
can
assum e both
positive and negative values. A plot of Iii-C C p)
is given in
figure 4.6.4. Therefore one can expect that the gain of the device
w ill be strongly pow er dependent. In other words, only those
values of (Xp that allow (4.6.12) to be negative will allow the
Josephson junction to exhibit gain. It is further important to note
that the term ination, b j , does not necessarily have to assume
the form o f (4.6.11) in order for the device to exhibit gain. The
device w ill exhibit gain on the condition that ReY is negative
regardless o f how the term ination is specified. In fact, figures
4.6.5-6 are plots of (4.6.10) assuming a termination o f an open
ended
transm ission
line
stub
of length
A /1 2
and
different
138
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values for N and the critical current. The specific stub length
was chosen so that at 3 (0 o this stub will look like a short. Recall
that the next higher harmonic that we have to concern ourselves
with is 3 C0o as was indicated in section 4.4. By shorting this
harm onic out we force all the energy that will be transferred
from the pum p to the signal and idler to be dumped into the
fundam ental
harm onic.
H igher
order
harm onics
greater
than
3 CD0 are neglected in these calculations.
4 .7
EXPERIMENTAL DESIGN
The heart of the param etric am plifier is the Josephson
ju n ctio n .
The Josephson junctions used in
these
experim ents
were m icrobridges formed by using a step edge junction process
[8]. A crossectional view o f a typical junction is given in figure
4.7.1. The process for forming the m icrobridge junction is as
follow s. An initially bare lanthanum alum inate w afer 20 mils
thick, is masked with niobium to form the junction etch pattern.
The substrate is then ion milled to form pits approxim ately 100300 nm in depth. Because the niobium is deposited much thicker
than the intended depth of the etch pit, the covered part of the
substrate is not etched at all and is fully protected
by the
niobium. The niobium is then stripped off and a 250 nm layer of
the HTS m aterial, YBCO, is deposited over the step edge in the
substrate. Because the film is thinned over the top edge and also
139
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0 .2 5
0.20
0 .1 5
0 .1 0
0 .0 5
0.00
20
Figure 4.6.4.
Plot o f J l { ( X p )
140
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40
20
Z
<
o
-5
-8 0
-4 0
-6 0
-20
PUM P POW ER (dB)
I c = 24 (lA
R j = 250 Q.
N = 5
Figure 4.6.5.
Com puter simulation of param p gain,
note the device param eters
141
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20
Z
/"^si W lr>
-5
-8 0
-4 0
-6 0
PUM P PO W ER (dB)
I c = 14 |iA
R j = 250 Q
N = 5
Figure 4.6.6.
Computer simulation of paramp gain,
note the device param eters
142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 0
HTS
JUNCTIONS
LANTHANUM ALUMINATE
Figure 4.7.1.
Crossection view of step edge junction
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
at the bottom corner, the superconductive m aterial is electrically
w eaker and form s the junction. In fact there is one junction on
top in series w ith the junction on the bottom of the etch pit.
Silver contacts are then deposited and the junction is complete.
A typical atom ic force m icrograph of a step edge junction is
indicated in figure 4.7.2. The resulting I-V characteristics of a
typical junction fashioned in this way is given in figure 4.7.3.
B ecause the junction is fashioned as a m icrobridge there is no
junction capacitance and the device exhibits classical RSJ model
c h a ra c te ristic s.
An array
experim ent.
o f 25 series junctions were designed
The Rj
of these devices
for this
were anticipated
to be
approxim ately 2 £2 per junction. This therefore facilitates the
im pedance m atching of the array to the transmission line, as the
series array looks like 50 £2. The device could also work well if
th e
ju n c tio n
transm ission
array
line
as
had
is
a
h ig h e r
evid en t
from
im p ed an ce
equation
than
the
(4.6.10)
and
(4.6.12). The layout of the param etric am plifier is given in figure
4.7.4. The 50 £2 m icrostrip line is approxim ately 150 p.m wide.
This line feeds the sym m etrically located X / 1 2
open circuited
shunting stubs. The stubs are located directly in front of the
array
to
m axim ize
th eir
harm onic, since at 3 (Q0
e ffect
of
shorting
out
the
third
stubs are a quarter wavelength
long. The junctions themselves are located along the center axis
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.7.2.
AFM of step edge junction. Step height
130 nm.
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
300.0
200.0
100.0
i
0.0
-
100.0
-
200.0
-
300.0
-
5.0
-
3.0
-
1.0
1.0
3.0
5.0
Volts (mV)
Figure 4.7.3.
Typical I-V curve of step edge junction.
Notice the RSJ behavior.
146
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.7.4.
CAD layout of 10 GHz parametric
a m p lifie r.
147
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
o f the transm ission line and are 2 p.m wide w here they cross
over the etch pit. The series array is terminated in a large HTS
patch that is hard wired to a copper block to form an RF and DC
ground. The overall length of the device is 7 mm.
4.8
EXPERIMENTAL SETUP
A fter the device was
specially
sim ilar
constructed
to the
copper
type used
in
fabricated
it was
m ounted
m ounting
block.
A
on a
K -connector
the previous kinetic
inductance
experim ents was mounted to the block and silver pasted to the
50 Q m icrostrip of the param etric am plifier. The device under
test was then m ounted on a probe, also sim ilar to that in the
k in etic
inductance
experim ents,
and
inserted
in to
a
liquid
helium dewar. Because the param etric am plifier uses HTS the
device under test was never actually inserted into the liquid
helium . The param etric am plifier never went below
10 K. A
therm om eter was mounted to the copper block m ount to verify
the param etric
am plifier's tem perature.
The entire experimental setup is illustrated in figure 4.8.1.
The signal, at 10.303 GHz ,and the pump, at 10.300 GHz, were
generated by two HP8350B sources. Isolators were attached at
the
so u rce's
o u tp u t
to
prevent
unw anted
in te rfe re n c e
coupling from one source to the other. After the isolators
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and
PUMP = 10.300 GHz
ISOLATOR
ATTENUATOR
HP 8350 B
SWEEP
POWER
COMBINER
HP 8350 B
/V W
SWEEP
ISOLATOR
v
ATTENUATOR
SIGNAL = 10.303 GHz
22GHz
8592 A
CIRCULATOR
D. C. BLOCK
LHe
DEWAR
ARRAY
SM A
SHORTING CAP
Figure 4.8.1.
Experim ental setup to measure
param etric
am plifier
1 49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MAX
variable wide bandwidth attenuators were placed so as to allow
the
pum p
and
signal
incident
pow ers
to
be
independently
adjustable. The output from the attenuators were then input to a
pow er
com biner.
The
pow er
com biner
fed
p o rt
1 of
the
circulator shown in figure 4.8.1. Port 2 was directly connected to
the cabling assem bly that led to the device under test in the
dewar. In between the circulator and the device under test was
a full DC block. This means that both the center conductor and
the outer conductor of the coax leading to the device under test
rem ained DC isolated. This prevented unwanted ground currents
from biasing the param etric am plifier. This is im portant because
only the four photon process given by equation (4.3.15) allows
the device to be operated w ithout a DC bias current[9]. The
reflected pow er from the device under test was then brought
out from
port 3 of the circulator and entered
the spectrum
analyzer via a low noise amplifier.
A shorting cap was used in place of the device under test
for calibration purposes. Using the shorting cap, only the pump
and the signal w ithout the idler would be reflected back with a
maximum reflection coefficient of 1. This would then form the
b aseline
pow er
referen ce
for
gain
determ in atio n
w hen
param etric am plifier is inserted instead of the shorting cap.
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
4.9
EXPERIMENTAL RESULTS AND DISCUSSION
Figures 4.9.1-5 show the I-V characteristic curve of the 25
series array o f junctions at different tem peratures. N otice the
structure of the I-V curve in figure 4.9.1. Since there are 25
junctions in series we can expect the critical currents of all the
junctions to vary som ewhat. This is indicated by the lumpiness
above the critical current. Since only when all the junctions are
below the critical current, i.e. the lowest common denom inator
o f critical current values, will the I-V curve trace out a current
at zero voltage. T herefore the lum piness is due to junctions
sw itching to th e voltage state, i.e. exceeding their individual
critical currents, at different values of critical current. As the
tem perature increase tow ards the transition tem perature of the
superconductor, Tc, the critical currents of the devices decrease
and
we w ould
tend
to
see both
the overall critical current
decrease and a sm oothing of the lum piness. This is precisely
w hat is shown in these figures.
A nother im portant param eter that we can extract from the
I-V curves is shown in figure 4.9.2. At this tem perature the
average critical current appears to be 30 pA . This is a very
im p o rtan t
num ber.
R ecall
that
the
greatest
effe ct
of
Josephson inductance occurs when the current through the
151
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the
V ert
Figure 4.9.1.
20 fiA/div.
H oriz.
2 mV/div.
25 junction array I-V curve at 4.2K
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V ert
Figure 4.9.2.
20 (lA/div.
H oriz.
2 mV/div.
25 junction array I-V curve at 8K
153
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V ert
Figure 4.9.3.
20 jiA/div.
Horiz.
2 mV/div.
25 junction array I-V curve at 23K
154
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V ert
Figure 4.9.4.
20 jxA/div.
Horiz.
2 mV/div.
25 junction array I-V curve at 45K
15 5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vert.
Figure 4.9.5.
20 |j.A/div.
Horiz.
2 m V/div.
25 junction array I-V curve at 60K
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
device is approxim ately equal to the critical current. We can
estim ate the RF pow er required to swing the junction to reach
the critical current by
P = I 2C
Z0
A ssum ing
a characteristic
(4.9.1)
im pedance of 50
Q
for the
param etric am plifier, (4.9.1) tells us that the required power is
ap p ro x im ately
-43
dBm .
T his
th erefo re
m eans
that
the
param etric effect will occur when the pump power swings the
junction
closest to the critical current in order to m axim ally
swing the Josephson inductance. This occurs at about -43 dBm.
From figure 4.9.5 we estim ate that we should no longer
see any param etric effects above 60K as the critical current has
been com pletely suppressed at this tem perature.
We are
now
in
a position
am plifier results. Figure 4.9.6
to exam ine the param etric
shows the param etric am plifier
and the shorting cap with both the pump and the signal incident
at 10K. N otice the incident pow ers are both the same at -65
dBm .
By
virtue
o f the
foregoing
argum ent concerning
the
required pump power, the incident pump power seen here is too
weak to swing the inductance Lj to produce parametric effects.
Therefore, no apparent idler frequency is observable and for all
157
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-40
ARRAY
-45
10°K
SHORTING NUT
-50
5 . -55
E
ca
5
-60
LU
g
-65
Uf -70
uLU
-75
-85
-90
“s
FREQUENCY (2 MHz/Div)
Figure 4.9.6.
Reflected power of param p with pump
power too low
158
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intents and purposes the device behaves as a linear one. This is
because the inductance changes very little until an appreciable
amount o f critical current has been achieved as equation (4.4.7)
indicates. N otice the 10 dB difference between the array and the
shorting cap. Since the frequency spectrum analyzer is set up to
look at the reflected signal from the device under test, lower
reflected pow er indicates more power is going into the device.
This therefore means that the device in figure 4.9.6 is well
m atched
to the 50 £2 coax verifying the device's microwave
d esign.
F igure
4.9.7
show s
the
param etric
am plifier
and
the
shorting cap with both the pump and the signal incident at 10K,
but with the incident pum p power increased to -45 dBm. Notice
the signal incident pow er, calibrated through the shorting cap,
has not changed from figure 4.9.6. Since -45 dBm is close to the
magic num ber o f -43 dBm, we expect to see parametric effects.
In fact, this is precisely what figure 4.9.7 portrays. Notice the
advent of the idler at precisely 10.297 GHz. This is exactly what
we expect to see from equation (4.3.15)! In addition, and more
im portantly, notice the signal has increased by +7.5 dBm from
that in figure 4.9.6. Although it has not exceeded the calibration
cap , indicating that there is no overall system gain, we can
conclude that there is an internal gain o f +7.5 dBm. In other
words the reflection coefficient of the "black box" is still less
159
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-40
ARRAY
SHORTING
NUT
10°K
-45
-55
-60
-
-65
-70
-75
REFLECTED POWER (5 dBm /D iv)
-50
-80
-85
-90
COj
FREQUENCY (2 MHz/Div)
Figure 4.9.7.
Reflected power of paramp with
activating pump power. N otice the onset
o f param etric interactions.
160
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than 1, yet the internal guts of the box indicate a gain process
with internal losses. Notice also that the values predicted by the
I-V characteristics, equation (4.9.1) and figure 4.9.7 agree quite
well. Of course, (4.9.1) was used to calculate the point at which
maximum param etric effect was to occur. Clearly the onset of
this effect occurs at a low er incident power value. Figure 4.9.8
is a plot
of the reflected pump pow er versus the reflected signal
and idler
power at 10K. Notice the extremely low level of pump
power (-63 dBm) and input signal power (-73 dBm) at which the
device exhibits an increase in reflected signal power, i.e. the
onset of param etric effects in the series array. The results of
figures 4.9.7 and 4.9.8, while encouraging, seem to indicate a
com peting process occurring within the junction array. Classical
Josephson junction param etric am plifier theory [3] predicts that
over a range of incident pum p pow er both the signal gain and
the idler
to w ard s
should increase, reach
the
non
p a ra m etric
a m aximum, and then decrease
in te rac tio n
state
as
eq u atio n
(4.6.10) indicates. Figure 4.9.8 clearly shows that this occurs for
the idler but not for the signal. The signal rem ains high in a
saturated
state. Further evidence fo r a com peting process is
shown through the asym metry in the idler response, contrary to
classical Josephson junction param etric am plifier theory.
161
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25 SERIES JUNCTIONS HTS PARAMP (10 K)
- 6 0 .0
- 6 5 .0
- 7 0 .0
SIGNAL POWER
IDLER POWER
- 7 5 .0
- 8 0 .0
SPECTRUM
- 8 5 .0
A N ALY ZER -------------NOISE FLOOR
- 9 0 .0
-8 0
-7 0
-6 0
-5 0
-4 0
-3 0
-2 0
REFLECTED PUMP POWER (dBm)
Figure 4.9.8.
Reflected pump pow er versus reflected
signal and idler power
162
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W e can
speculate as to
the nature of
these com peting
processes by looking at figure 4.9.9 more closely. Figure 4.9.9 is
sim ply a m agnified view of the junction area from figure 4.7.4.
The junctions occur along the step edge where the very narrow
lines cross the
step edge pit. The wider lines at either end of the
junctions w ere
made so that the processing yield would be high
for
m anufacturing
trad e o ff
had
to
the
be
devices.
m ade.
U nfortunately
T he
w ide
lines
an
do
engineering
provide
the
probability o f an increase in m anufacturing yield, but at the
same tim e allow fo r m icrow ave energy to be coupled to the
adjacent
w ide
line
w ith
out
passing
through
the ju n ctio n .
(Because a series array of 25 HTS junctions had never been
processed before we opted for the form er benefit.) Therefore, in
reality we have two m icrow ave conduction paths through the
junctions. The
first path forces the microwave
energy to flow
through the junctions in series. All our above analysis is based
on this assum ption. The second path is where energy is coupled
to the ju n ctio n s in a parallel
way for which no theoretical
analysis has been done. It is im portant to realize that because
we are dealing with an inherently nonlinear inductor both paths
result in nonlinear circuit behavior.
W e therefore speculate as follows. When the pump power
is
low
enough
the
m icrow ave
energy
predom inately
travels
along the series array and yields classic parametric am plifier
163
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25 SERIES
JUNCTIONS
Figure 4.9.9.
Magnified view of junction layout in
param etric
am plifier
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
behavior. As the pump power is increased the coupling between
adjacent junctions becom es im portant and the nonlinear effects
due to this conduction path influence the overall behavior of the
device. Hence it is possible that as the idler frequency power
d ecreases
due
to
the
param etric
am plifier effect
the
signal
pow er still rem ains high due to the nonlinear parallel junction
array
effect
since
it too
is
governed
by
the
M anley-R ow e
re la tio n s h ip s .
F u rth e r
e v id e n c e
fo r
p ro p er
p a ra m e tric
in te ra c tio n
behavior can be seen from figure 4.9.10. This figure indicates
the internal gain versus the temperature. We expect by virtue of
figures 4.9.1-.5
that the param etric interaction
effects
should
decrease and be come negligible at about 60 K. This is precisely
what figure 4.9.10 indicates. The device exhibited at least a 3 dB
increase in reflected signal power up to as much as 50K.
A dditionally
we
would
expect the
onset
of param etric
interactions to occur at lower power levels as the tem perature
rises, also. This is because as the critical current decreases with
tem perature increase,
the Josephson
inductance
also increase.
This means that the device achieves the proper inductance value
for param etric interactions at a lower power level as seen from
figure 4.6.5-7. A gain, this is precisely what we see in figure
4.9.11. where we've essentially plotted figure 4.9.8 with
1 65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25 series junction HTS paramp
(T vs peak Gain)
12.0
10.0
pum p power = -46 dBm
i Fs;Fp;Fi = 10.3 Ghz
8.0
6.0
4.0
2.0
0.0
0
10
20
30
40
50
60
70
T e m p e ra tu re
Figure 4.9.10. Internal signal gain versus tem perature
166
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25jjparamp
-
6 0 .0
-
6 5 .0
-
7 0 .0
•—signal power(10K)
•— idler power(IOK)
-
7 5 .0
-
8 0 .0
-
8 5 .0
-
9 0 .0
►• signal power(30k)
-■idler power(30K)
-80
-70
-60
-50
-40
-30
-20
pum p power
Figure 4.9.11.
Reflected pump power versus reflected
signal and idler power at 10K and 30K
167
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tem perature as the param eter. N otice the overall shift of both
the
id le r
and
the
signal
to
low er
pum p
values
as
the
tem perature increases between 10 and 30K.
In
su m m atio n ,
we
have
d e m o n stra te d
p a ra m e tric
interactions in a series array of Josephson step edge junctions.
W e have confirm ed these results by exam ining the junction’s
behavior
quite
and
w ell
p rev io u sly
extracted
w ith
other
unobserved
key
predictive
quantities
experim ental evidence.
gain
saturation
In
phenom ena
that
agree
addition,
has
a
been
seen which needs further investigation.
4.10
FUTURE WORK
W ith the clear indication
that we are seeing
verifiable
param etric interaction with the series array we have proceeded
to design a monolithic param etric am plifier to work at 60 GHz.
The device incorporates the series array in conjunction with an
an tipodal
finline
that
allow s
for
w aveguide
to
m icrostrip
transition. The layout o f the device is shown in figure 4.10.1.
The antipodal finline is constructed such that m etalization occurs
on both the front and backside of the chip. The chip is mounted
in a waveguide such that the dom inant TE10 m ode's electric
field lines run parallel to the short side of the chip. The E field
then couples into the finline and is guided towards the junctions
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AU CONTACT
BACKSIDE AU
^
26ffiRIE
WJUNCTION$
M ICROSTRIP^^^
mmd
RF CHOKES 2 % %
1.0 ^.AT 60 GHz
Figure 4.10.1.
Layout of 60 GHz monolithic HTS
param etric
am plifier
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w hile
being
spatially
rotated
90
degrees
to
conform
to
a
m icrostrip E field configuration. The 1/4 w avelength stubs on
the finline prevents unw anted
stray current from propagating.
The large circular aperture allows for a uniform transition from
the waveguide m ode to the m icrostrip mode of a characteristic
im pedance
o f 20
O hm s. 20
Ohms
was chosen
as
the
best
com prom ise fo r fabricating a m icrostrip transmission line on this
lanthanum alum inate substrate. By virtue of our discussion in
c h ap ter
3
c o n ce rn in g
thin
su b stra tes
for
high
frequency
m easurem ents, the lanthanum alum inate is 125 pm thick. The
20
£2 m icrostrip
line
feeds directly
into
the ju n ctio n s. The
frequency discrim inating elem ent is the open circuited stub of
1/4 wavelength at 60 GHz at the back end of the junction array.
This stub forces a virtual ground at 60 GHz and hence our model
in section 4.6 is applicable. It must be noted that the stub makes
this param p a narrow band device. The RF chokes are a simple
alternate
arrangem ent
of
1/4 w avelength
high/low
im pedance
lines. The chokes will allow us to monitor the I-V curves of the
junction array during RF operation. It is to be noted that the I-V
curve
cannot
be
m onitored
d u rin g
param etric
am plification
functionality as the current used to generate the I-V curve will
also
inadvertently
bias
the device and
the param p
w ill not
function properly.
170
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4.11
REFERENCES
[1]
J. H. Takemoto-Kobayashi, et a
l Monolithic High-Tc
Superconducting Phase Shifter at 10 GHz", IEEE MTT-S
S y m p o s i u m 1992
[2]
C. M. Jackson, and D. J. Durand, "10 GHz High Temperature
S uperconductor
Phase
S hifter",IEEE
M TT-S
Symposium
1991
[3]
M. J. Feldman, P. T. Parrish, and R. Y. Chiao, " Parametric
Am plification by Unbiased Josephson Junctions", J. Appl.
P h y s .,V ol. 46, No. 9, pp4031-4042, Sept 1975
[4]
A. D. Smith, et a l . " Low Noise Microwave Parametric
A m p lifier",
ppl 022-
[5]
1028,
IEEE
on M agnetics,
M AG-21,
1985
H. Heffner and G. Wade, " Gain, Banwidth, and Noise
C h a rateristics
Appl.
Transactions
o f the
V ariable-P aram eter
A m plifier",
P h y s .J, vol. 29, no. 9, pp. 1321-1331, 1958
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
J.
[6]
J. M. Manley and H. E. Rowe, " Some General Properties of
Nonlinear Elements: Part I - General Energy Relations",
Proceedings o f the IRE, vol. 44, no. 7, pp. 904-913, 1956
[7]
T.
Van
D uzer
Superconductive
and
C.
W.
T u rn er,
P rin cip les
of
Devices and Cicuits, Chapter 5, Elsevier,
New York, 1981
[8]
J.
L uine,ef al . "C h a ra c te ristic s
YBa2Cu307
of
H igh
P erfo rm an ce
Step-edge Junctions", Appl. Phys. Lett. , vol.
61,No. 9, pp. 1128-
1130,
1992
[9]
P. T. Parrish and R. Y. Chiao, " Amplification of Microwaves
by
Superconducting
M icrobridges in a Four-w ave Param etric
Mode", Appl. Phys. Lett., vol. 25, no. 10, pp. 627-629, 1974
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 5
Conclusion
In this dissertation we have dem onstrated, measured, and
a n aly ze d
th e
in d u c ta n c e
m icro w av e
e f f e c ts
superconducting
devices. W e have
can
e ffe c t
a
d e v ic e
d ep en d en t
effe ct
is
m icrow ave
and
d ep en d e n t
p a rtic u la rly
m illim e te r
a s s o c ia te d
in ductance
and
and
be characterized
m illim eter
e ffe c ts are
w ith
h ig h
shown
that this nonlinear
by
d e p en d e n t
p rim arily
e ffect.
w hen im p lem en ted
w ith
in
a
The
w ith
devices,
asso ciated
te m p e ra tu re
a tem perature dependent
associated
w ave
w ave n o n lin e a r
w hile
tem p era tu re
HTS
p assiv e
the
device
Josephson ju n c tio n s
p aram etric a m p lifie r
c o n fig u ra tio n .
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We
in itia lly
n o n lin e a r
in v estig ate d
in d u ctan ce
e ffe cts.
the
tem p era tu re
U sing
the
dependent
theory
of
kinetic
inductance we designed and fabricated HTS coplanar waveguide
re so n a to rs
in
o rd e r
to
m easure
and
m odel
the
th eo ry ’s
intricacies. The CPW resonator topology was chosen due to it's
h ig h er
cu rren t
d en sity
along it's edges
and
for
ease
of
fabrication. The reso n ato r’s frequency was centered around 5
GHz with a m easured loaded Q of 16,000 at 10K. This is the
highest known value of Q m easured for such a device. The
m easured Q also indicates that the surface resistance of the HTS
is less than 200 |J.Q. A model was developed based on the
assum ed
tem perature
dependence
depth and a full wave
of
the
L ondon
penetration
spectral dom ain analysis of the CPW
topology. We found excellent agreem ent between a " T squared"
tem p eratu re
data.
dependence
T his
confirm s
profile and the
w hat many
people
accurately
have
m easured
suspected;
the
tem perature dependence in HTS is not typical Gorter-Cassim er.
In
a d d itio n ,
we
dem onstrated
reso n ato r characteristics
a
pow er
dependence
com m ensurate with
the
in
choice
the
of a
CPW topology. The accuracy of the model is sufficient to allow
the design of narrowband filters and delay lines.
B ased
bandw idth
on
th ese
re s u lts ,
we extended
of our experim entation
and focused
the
frequency
on m easuring
the kinetic inductance effect in an HTS CPW delay line. We used
174
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a very pow erful technique o f optically generating and sampling
a picosecond electrical pulse in a sem iconductor sw itch, after
having traversed the delay line. This probing technique allowed
us to exam ine the tem perature dependent nonlinear inductance
characteristics of the delay line up to 100 GHz. Not surprising,
based on our experim entation with the resonators, we were able
to confirm all of our assum ptions in the design and predicted
behavior o f the delay line through this technique. This adds
fu rth er
im petus
tem p eratu re
passive
to
our
dependent
m odel's
veracity
n o n lin ear
for
predicting
inductive behavior
the
o f HTS
com ponents.
O ur focus then shifted to exam ine the device dependent
nonlinear m icrow ave and m illim eter wave inductive effects in
HTS
d ev ices.
We
em ployed
step
edge
Josephson
junctio n s
incorporated in a m icrow ave layout of a parametric am plifier to
ex p lo re
th ese
effe cts.
T he
resu lts
indicate
that
we
have
definitively seen param etric interactions that can only be due to
the
device
p a ra m etric
dependent nonlinear
a m p lifie r
co u ld
not
inductance.
be
A lthough
dem onstrated
a true
due
to
excessive internal resistance o f the junctions, ample evidence
w as
p ro v id ed
inductance
to
in d ic a te
of the device.
the
behavior
of the
A crucial com parison betw een the
critical current of the device and
the predicted onset
p aram etric
the
in teractio n s
no n lin ear
confirm s
role o f the
of these
nonlinear
1 75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in d u c to r.
tem perature
w hich
In
a d d itio n ,
dependence
fu rth er
d e p en d e n t
am plifier's
confirm ed
n o n lin e a r
c ritic a l
of
the
the
in d u c tiv e
ev id e n ce
nonlin earity
in te g ral
e ffe c t
c o n c e rn in g
was
ro le
of
in
the
the
p resen ted
the
d ev ice
p a ra m e tric
perform ance.
176
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