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Transport and microwave properties of yttrium barium(2) copper(3) oxygen(7-x) superconducting films

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Order Number 9318782
Transport and m icrowave properties o f Y B a 2 Cu 3 0 7 _ 3
superconducting films
Jiang, Hua, Ph.D.
Northeastern University, 1992
Copyright �92 by Jiang, Hua. All rights reserved.
UMI
300 N. ZeebRd.
Ann Arbor, Ml 48106
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TRANSPORT A N D MICROWAVE PROPERTIES OF
YBa2Cu30 7_I SUPERCONDUCTING FILMS
A dissertation presented
by
Hua Jiang
to
The Department of Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the field of
Physics
Northeastern University
Boston, Massachusetts
August 1992
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NORTHEASTERN UNIVERSITY
Graduate School of Arts and Sciences
Dissertation Title:
Transport and Microwave properties of
YBa 2Cu3 0 7 _x Superconducting Films
Author:
Hua Jiang
Department:
Physics
Approved for Dissertation Requirements of the Doctor of Philosophy Degree
A,Vi/U ^AJ
a!b~
^ *?> (Prof. Carmine Vittoria)
(Prof. Allan Widom')
(Prof. Robert S. Markiewiczl
Thesis Committee
_________________
Head of Department
Date
(Prof. Stephen Reucroftl
Date
Graduate School Notified of Acceptance
KQ,il
Dean
Date
Z7
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�92
Hua Jiang
ALL RIGHT RESERVED
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TRANSPORT A N D MICROWAVE PROPERTIES OF
YBa2Cu30 7_I SUPERCONDUCTING FILMS
by
Hua Jiang
ABSTRACT OF DISSERTATION
Subm itted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the G raduate School of Arts and Sciences of
N ortheastern University, August 1992
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ABSTRACT
High quality YBCO c-axis oriented films were made by a laser ablation
technique. Josephson weak links were fabricated on a modified microbridge by
using bicrystal YBCO films. Shapiro steps were observed on modified micro�
bridge weak links.
Vortex ring nucleation by thermal activation and quantum tunneling pro�
cesses in high Tc superconductors is discussed. The nonlinear resistance scales
as
. Thermal activation yields a critical index of v = 1, while quantum
tunneling (without normal current dissipation) yields v = 2. I-V characteristics
were measured on both disordered (low J c), ordered (high J c) YBCO films and
microbridges below Tc. The data were examined in term s of both thermal acti�
vation and quantum tunneling models. For a disordered film, the u = 1 scaling
law is in reasonable agreement with the data, while u = 2 scaling is required to
explain the results of an ordered film at low tem peratures. For the microbridge,
the u = 2 scaling extended even to high tem peratures.
Intrinsic critical current densities in high Tc superconducting films are dis�
cussed. The enhancement of the quantum electrodynamic tunneling process
due to normal current dissipation is proposed to explain the measured J c re�
sults for YBCO films. The depairing current density calculated from vortex ring
model is in the order of 109A / c m 2 and an upper limit of J c is in the order of
107A / c m 2. Supercurrent was observed to flow across a bridge constriction with
a J c up to 1.3 x 109A /cm 2, the highest critical current density yet reported.
The limitation of J c appears to be due to vortex ring creation. We found that
studying J c of bridge constrictions was a way to approach the depairing current
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V
limit.
A novel microwave self-resonant (MSR) technique was developed to mea�
sure surface resistance, R s, of YBCO superconducting films directly and sur�
face reactance, X s, indirectly. R s at 21 GHz decreased by about three orders of
magnitude as the tem perature decreased from 90K to 80K. The smallest surface
resistance was 2.7
X
10-4 ft at 21 GHz and at 15K, and the surface reactance
was 0.0310. Using the MSR technique, we measured the London penetration
depth and coherence length, and found th a t both A and � were anisotropic,
their values depended on the direction of the microwave electric field relative
to the c-axis. We found th at A]|(0) (penetration depth in a-b plane at zero
tem perature) was about 1840A and Aj_(0) about 3640A, while |(0) ~ 25A
and l(0) ~ l l A . The Ginzburg-Landau param eters were /C|| in the range of
110 ?165 and k盻 in the range of 62 ?74, yield an anisotropic factor 7 about 2.
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ACKNOWLEDGEMENTS
I would like to express my deep gratitude to my advisors, Prof. Carmine
V ittoria and Prof. Allan W idom for their advcice, encouragement, and support.
They made my years at b o th Physics and Electrical Engineering Departm ents
at Northeastern University a rich and rewarding experience.
I would thank Prof. R obert. S. Markiewicz for being in my thesis commit�
tee, for his critical reading of this dissertation. Stimulating conversations with
him were special helpful.
My sincere thanks go to Drs. H. How, S. Oliver and Y. Huang, discussions
w ith them were highly appreciated.
Acknowlege and thanks go to all the members of Microwave Materials Lab�
oratory at Northeastern University, especially P. Dorsey, P. Kwan, S. Bushnell,
R. Seed, T. Yuan, S. Zhang, and D. Guan, for their help and friendship. Thanks
also go to S. Wang in Chemistry D epartm ent for his help.
I would thank R. Ahlquist, T. Hussey, and S. DiCiaccio for their technical
assistance through out this work.
I am grateful to Mechelle Hodges and K ate Workman for all the help and
assistance rendered while completing this work.
I would like to extend my thanks to the entire faculty, staff and graduate
students in the Physics and Electrical Engineering Departm ents at N ortheastern
University.
Finally, I would thank Prof. Z.X. Zhao in the Institute of Physics, Academy
of Sciences of China, who brought me into the field of superconductivity ten
years ago, for his continuing encouragement throughout this work.
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DEDICATION
This dissertation is dedicated to my wife, Xuesheng, my son, Michael,
and my parents, Mr. and Mrs. Zongmin Jiang, for th e ir unlimited love and
support.
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viii
TABLE OF CONTENTS
Copyright
ii
Abstract
iii
Acknowledgements
vi
Dedication
vii
Table of Contents
viii
List of Tables
xii
List of Figures
xiii
CHAPTER I
INTRODUCTION
References
1
6
C H A PT E R II
HISTORICAL REVIEW OF SUPERCONDUCTIVITY
2.1 Meissner Effect
9
9
2.2 Two Fluid Model
10
2.3 London Theory
12
2.4 Types of Superconductors
14
2.5 Ginzburg-Landau Theory
18
2.6 BCS Microscopic Theory
19
2.7 High Tc superconductors
21
References
29
C H A PT E R III
SAMPLE PREPARATION, CHARACTERIZATION
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AND EXPERIMENTAL SETUP
32
3.1 Bulk YBCO Samples
32
3.2 Pulsed Laser Ablation of YBCO Films
33
3.3 Submicron Fabrication Techniques
33
3.3.1 Photolithography and Chemical Etching
35
3.3.2 Ion Beam Milling and Deposition
36
3.4 Fabrication of YBCO Microbridges
37
3.4.1 Josephson Weak Link of Conventional Superconductors
37
3.4.2 Fabrication of Josephson Weak Link
42
(a) Microbridge on Single Grain Boundary
42
(b) Step Edge Junction
43
3.5 Characterization
3.5.1 Electrical Measurements
47
47
(a) Four-Probe Measurement
47
(b) Low Resistive Contact
47
(c) Electrical Measurements
50
3.5.2 Magnetic Characterization
53
3.5.3 Structural Characterization
56
3.6 Experimental Setup
56
3.6.1 Low Temperature Apparatus
56
3.6.2 Microwave Techniques
61
References
65
CHAPTER IV
VORTEX MECHANISM AND I-V CHARACTERISTICS
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66
4.1 Introduction
66
4.2 Vortex Ring Nucleation Theory
68
4.3 Experimental Detail
74
4.4 Results and Discussion
77
4.5 Conclusion
86
References
88
CHAPTER V
INTRINSIC CRITICAL CURRENT DENSITIES
90
5.1 Introduction
90
5.2 Depairing Current Density
92
5.3 Upper Limit of J c for YBCO Films
94
5.4 Microbridge W idth Dependence of J c
98
5.5 Observation of Ultrahigh J c in YBCO Bridge Constrictions
102
5.6 Conclusion
110
References
112
CHAPTER VI
SURFACE IMPEDANCE
115
6.1 Introduction
115
6.2 Calculation of Surface Impedance
116
6.3 Techniques of Surface Impedance Measurement
121
6.3.1 Early Surface Impedance Measurement
121
6.3.2 Strip Line, M icrostrip and Coplanar Resonantors
124
6.3.3 Cavity Technique
126
6.3.4 Parallel Plate Technique
128
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6.4 Development of Microwave Self-Resonant (MSR) Technique
131
6.5 Experimental Results and Discussion
139
6.6 Conclusion
149
References
154
CHAPTER VII
LONDON PENETRATION DEPTH AND COHERENCE LENGTH
157
7.1 Introduction
157
7.2 Anisotropic Lengths in YBCO Superconductor
159
7.3 London Penetration Depth Deduced from Surface Impedance
162
7.4 Determination of Coherence Length
166
7.5 Conclusion
171
References
174
APPENDIX
SOME EQUATIONS IN MKS UNITS
177
VITA
182
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LIST OF TABLES
2.1 Transition tem peratures for some high T c
superconductors
23
2.2 Characteristic parameters for Y B a 2 C u 3 0 i
and B i 2 S r 2 CaCuoOs
28
5.1 Summary of intrinsic criticalcurrent densities
111
6.1 Size effect corrections of R s, X s and L sof YBCO film
151
6.2 A comparison of our measured surface impedance
value with others? at 77K
152
6.3 A comparison of our measured surface impedance
value with others? at 4K
153
7.1 Summary of A, �, and k
173
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LIST OF FIGURES
Fig. 2.1 Diagram of Meissner effect.
11
Fig. 2.2 M agnetization of superconductors.
15
Fig. 2.3 Abrikosov lattice of type II superconductors.
16
Fig. 2.4 Structure of an isolated vortex.
(a) a single vortex, (b) order pa�
ram eter, (c) local field, (d) screening current density.
17
Fig. 2.5 Crystal structure of Y B a i C u z O i s -
25
Fig. 2.6 Crystal structure of B iiS r z C a C u z O i- s .
26
Fig. 3.1 Pulsed laser deposition system.
34
Fig. 3.2 Ion beam test milling process.
38
Fig. 3.3 Microbridge fabricated by using ion beam milling, width=4000A. 39
Fig. 3.4 Microbridge fabricated by using ion beam milling, width=500A. 40
Fig. 3.5 Microbridge fabricated on an artificial grain boundary, (a) artificial
grain boundary film, (b) finished microbridge.
44
Fig. 3.6 Current-voltage characteristics of a modified microbridge weak link,
(a) without microwave radiation, (b) with -34dBm microwave radia�
tion (power~0.0004mW).
45
Fig. 3.7 Side view of a step edge junction.
46
Fig. 3.8 Nonlinear response of a step edge junction. Input signals were sinu�
soidal.
Fig. 3.9 Four-probe and two-probe measurements.
48
49
Fig. 3.10 A schematic diagram of electrical measurements.
51
Fig. 3.11 A typical resistance versus tem perature of a YBCO film.
52
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XIV
Fig. 3.12 A typical current-voltage characteristic of YBCO film.
54
Fig. 3.13 Magnetic moments of a YBCO film as a function of magnetic field at
tem perature of 20K, 40K, 60K, 70K, 77K, and 84K, (using SQUID
magnetometer).
55
Fig. 3.14 Magnetic moment of a YBCO film as a function of magnetic field at
77K, (using VSM).
57
Fig. 3.15 X-ray diffraction pattern for a laser ablated YBCO film on MgO
substrate.
58
Fig. 3.16 X-ray diffraction pattern of orthorhombic structure c-axis oriented
YBCO film, (after M ahajan et al.)
59
Fig. 3.17 Low tem perature measure and control arrangem ent.
60
Fig. 3.18 Diagram of the CF1204 cryostat.
62
Fig. 3.19 Schematic diagram of the microwave experimental setup.
63
Fig. 4.1 Diagram of vortex ring nucleation.
71
Fig. 4.2 Vortex ring free energy as a function of vortex ring radius in normal�
ized units.
72
Fig. 4.3 Typical resistance as a function of tem perature for low J c and high
Jc films.
76
Fig. 4.4 I-V curves of high Jc films at tem peratures of 2.6K, 60K and 80K.
The stars are experimental data, dashed lines are fitting curve with
v = 1 scaling and solid lines are fitting curves w ith v = 2 scaling. 79
Fig. 4.5 I-lnV curves of high J c films at tem peratures of 2.6K, 60K and 80K.
The stars are experimental data, dashed lines are fitting curve with
u ? 1 scaling and solid lines axe fitting curves w ith v = 2 scaling.
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XV
The scales for 60Kand 80Kare not shown.
80
Fig. 4.6 Current-voltage characteristics of a microbridge at 77K near the crit�
ical current.
81
Fig. 4.7 Current-Voltage characteristics of a microbridge at 77K plotted in
I-lnV scale.
82
Fig. 4.8 The I-V d ata and fitting curves of a microbridge at 85K.
83
Fig. 4.9 The I-V d ata and fitting curves of a microbridge at 8 8 .5K.
84
Fig. 4.10 Low J c film I-V characteristics at 3K. The reader is referred to the
scale on the right side of the figurefor logarithmic scale.
Fig. 5.1 Schematic diagram of a bridge constriction.
85
99
Fig. 5.2 Current-voltage characteristics of a microbridge taken at 77K.
105
Fig. 5.3 Temperature dependence of resistance of three microbridges.
106
Fig. 5.4 Critical current densities as a function of tem perature for the three
microbridges. The scale for bridge (a) is on the right side of the
figure.
107
Fig. 5.5 Critical current densities as a function of tem perature in different
magnetic fields for bridge (b).
109
Fig. 6.1 (a) Diagram of two fluid model. <r? and as are connected in parallel.
(b) Modified two fluid model where crn and crs are connected in series.
120
Fig. 6.2 A resonator used for measuring surface impedance of superconductor.
( after A. Pippard)
Fig. 6.3 Coplanar resonator.
123
125
Fig. 6.4 A block diagram of cavity measurement, (after S. Sridhar et al.) 127
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xvi
Fig. 6.5 A schematic diagram of parallel plate resonator technique. (after R.
Taber)
Fig. 6.6 Microwave self-resonant measurement arrangement.
130
132
Fig. 6.7 (a) Diagram of microwave self resonator, (b) Equivalent circuit of
microwave self-resonator.
133
Fig. 6.8 IS21I as a function of frequency for a copper strip. The experimental
results are represented by points and fitting results by solid line. 138
Fig. 6.9 Microwave resonant spectra of a YBCO strip taken at 15K.
140
Fig. 6.10 Amplitude of transmission coefficient as a function of normalized fre�
quency at 8 6 K for a YBCO strip. The resonant frequency is 21GHz.
Solid line is fitting curve.
141
Fig. 6.11 Phase angle of transmission coefficient as a function of normalized
frequency at 86 K for YBCO strip.The resonant frequency is 21 GHz.
Solid line is fitting curve.
142
Fig. 6.12 Frequency shift of a YBCO strip with respect to the resonant fre�
quency at 4K.
143
Fig. 6.13 Surface inductance of a YBCO strip as a function of
tem perature.
145
Fig. 6.14 Surface impedance as a function of film thickness taken at 70K. 146
Fig. 6.15 Surface impedance as a function of film thickness taken at 8 6 .5K. 147
Fig. 6.16 Surface impedance of a YBCO strip as a function of temperature. In
the X s curve, there are few points not plotted around 60K, because
there were some frequency shift due to the system background at that
tem perature range.
148
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XVII
Fig. 7.1 London penetration depth as a function of tem perature.
Dashed
line is fitting with two fluid model, solid line is fitting with A(T) =
A(0)(1 - ( � ) 2) - / 2.
164
Fig. 7.2 London penetration depth as a function of tem perature deduced from
MSR and MMMA techniques and A(0) from CPW technique.
167
Fig. 7.3 Microwave transmission coefficient of the superconducting film at dif�
ferent magnetic fields.
168
Fig. 7.4 Change of resistivity of the superconducting film at 86.5K due to the
applied magnetic fields.
170
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l
CHAPTER I
INTRODUCTION
The discovery of high transition tem perature superconductors !1~3! has
brought us many challenges in physics as well as in technological applica�
tions. The high Tc material (for example, YBCO), unlike conventional su�
perconductors, has an unusually small coherence length (� in the order of
1 0 AM), and it leads to an extremely high upper critical magnetic field; the ionic
bonding!5! leads to a high transition tem perature; the structural anisotropy!6!
leads to anisotropic characteristic lengths and possibly multi-energy gaps!7?8!.
The mechanism for superconductivity in high Tc materials is not accounted for
by the well known BCS theory!9!, although its electrical properties are similar
to conventional metallic superconductors.
It is well known that vortex motion implies an induced electric field and
thereby resistance in otherwise superconducting materials. The vortex dynam �
ics has been extensively studied from a therm al activation point of view!10-18!.
The basic issue at hand is whether or not the thermal fluctuations are the only
means by which vortices are activated or set in motion. In this dissertation, a
vortex ring nucleation model is postulated in which the radius of a vortex ring
changes from a relatively small value (near the coherence length �, say) to i?0,
where R q is the threshold radius beyond which vortex rings are free to expand
and induce a voltage drop, by a quantum tunneling process.
Critical current density of high Tc superconductors is of interest from both
fundam ental and applied points of view. In the case of high Tc superconductors
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2
the promise of their dramatically increased transition tem perature (up to 125K,
compared w ith 23K for best conventional superconductors) was initially tem �
pered by the experimental observation of rather low critical current densities
in polycrystalline and even single crystal samples. Later experiments on epi�
taxial films)19" 21) and narrow constrictions)22?23) in high Tc m aterials (YBCO,
BSCCO), showed an improvement in the critical current density. We have
studied the intrinsic critical current densities of high Tc superconductors based
upon the vortex ring nucleation model. The depairing current density, which
is independent of material structure and extrinsic parameters, was discussed
in term of the vortex ring nucleation model. The upper limit of critical cur�
rent density of a YBCO superconducting film was estimated based upon the
vortex ring nucleation by a quantum tunneling process. The study of high Tc
superconducting film bridge constrictions was a way to approach the depairing
current limit. The depairing current density is higher than the critical current
density resulting from vortex motion. By making the cross sectional area of the
superconducting bridge constriction smaller, one may exploit an energy barrier
which prevents vortex creation in the superconducting channel. Therefore, the
exclusion of the vortex from the extreme bridge constriction then removes the
restriction on critical current due to flux flow, thereby allowing the depairing
limit to be approached)22,24).
High Tc superconducting films have lower loss and lower dispersion com�
pared to gold (and other good conventional conductors) up to a frequency of
about 100 GHz)25?26), which makes high quality superconducting films an ex�
cellent m aterial for the fabrication of microwave devices. On the other hand,
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3
microwave techniques are useful in measuring the electrical properties of high
Tc superconductors. Microwave surface impedance, for example, is very impor�
tant to measure in understanding the nature of superconductors, from which
the characteristic lengths^27?28} and energy gap^29! can be deduced. We have
developed a novel microwave self-resonant (MSR) technique to measure surface
impedance of high Tc superconducting films. This technique has an advantage
over conventional cavity perturbation techniques, since there are no background
contributions other th a n the superconducting sample. We used a modified two
fluid model to analyse our data.
London penetration depth and coherence length are fundamental charac�
teristic lengths. The London penetration depth, which measures the length over
which magnetic fields are attenuated near the surface of superconductor, repre�
sents the electromagnetic properties of the superconductor. Furthermore, the
zero-temperature value A(0) contains information about the effective mass and
density of superconducting pairs, while the coherence length measures the size
of the Cooper pair, or size of the vortex core. The London penetration depth
in the a-b plane of YBCO superconducting films was deduced from the surface
impedance measurements. By measuring the magnetic field dependence of the
surface resistance, the coherence length in the a-b plane was also deduced. We
find both characteristic lengths are anisotropic.
Superconducting weak links axe useful applications of superconductors.
Weak links are used for superconducting quantum interference devices (or so
called SQUID)f30?31], voltage standards^32!, mixers^33!, etc. There are usually
three ways to fabricate superconducting weak links: superconductor-insulator
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4
(or normal metal)-superconductor (SIS or SNS) junctions, point contact links,
and microbridges. Unlike conventional superconductors, the high Tc supercon�
ductors are difficult to be used for weak link fabrication. This is due to the fact
th at high Tc superconductors are ceramics. Machinable materials are required
to make a sharp tip; SIS or SNS junctions are also not the ideal ways for oxide
superconductor to make weak links. Normal-metal interfaces are not shaxply
defined in the cuprate superconductors. If the SIS or SNS is fabricated as weak
link somehow, the oxygen in the cuprate superconductor will react with the
metal. As such SIS (or SNS) weak links are degraded or damaged. Micro�
bridges seem to be a reasonable way to fabricate superconducting weak links.
Microbridges on the order of 500A were fabricated from a high quality YBCO
superconducting film, and no weak link was produced. It m ust be realized that
the small coherence length of a high Tc superconducting film (of the order of
10A), indicates th at the critical dimensionality or fabrication length needs to
be less than lOOAl
This dissertation includes seven chapters. In chapter II, a historical re�
view of superconductivity will be given briefly. In chapter III, the sintering of
bulk YBCO samples, laser ablation of YBCO films, fabrication of YBCO weak
links, and the experimental setup will be discussed. The vortex mechanism is
discussed in chapter IV, where a vortex ring nucleation model is postulated. In
chapter V, we discuss the intrinsic critical current densities of YBCO supercon�
ductors. Microwave properties of YBCO films are discussed in chapter VI, in
which a novel microwave self resonant technique is developed to measure surface
impedance of superconducting films. In chapter VII, the London penetration
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5
depth and coherence length of YBCO films are deduced and their anisotropic
properties are discussed.
CGS units are used throughout this dissertation. However, some of the
equations are also w ritten in MKS units and listed in the appendix for easy
conversion.
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6
REFERENCES
[1] J.G. Bednorz and K.A. Muller, Z. Phys. B, 64, 189, (1986).
[2] M.K. Wu, J. Ashburn, C. Torng, P. Hor, L. Gao, Z. Huang, and C.W. Chu,
Phys. Rev. Leit., 58, 908, (1987).
[3] Z.X. Zhao, L. Chen, Q. Yang, Y. Huang, G. Chen, R. Tang, G. Liu, C.
Cui, L.Wang, S. Guo, S. Li, and J. Bi, Kexue Tongbao, 6 , 421, (1987).
[4] T. W orthington, W. Gallagher, and T. Dinger, Phys. Rev. Lett., 59, 1160,
(1987).
[5] A. Widom, T. Yuan, H. Jiang, and C. Vittoria, subm itted to Phys. Rev.
B.
[6 ] C. Rao, in Chemical and Structural Aspects of High Temperature Superconducors, ed. C. Rao, (World Scientific, New Jersey, 1988).
[7] V.Z. Kresin and S. Wolf, Physica C, 169,496, (1990).
[8 ] C. Vittoria, A. Widom, and H. Jiang, J. Superconductivity, 4, 361, (1991).
[9] J. Bardeen, L.N. Cooper, and R. Schrieffer, Phys. Rev., 108, 1175, (1957).
[10] T.T.M . Palstra, B.Batlogg, R.B. Van Donver, L.F. Schneemeyer and J.V.
Waszczak, Appl. Phys. Lett., 54, 763, (1989).
[11] S.martin, A.T. Fiory, R.M. Fleming, G.P. Espinosa and A.S. Cooper, Phys.
Rev. Lett., 62, 677, (1989).
[12] M. Coffey, J. Clem, Phys. Rev. Lett., 67, 386, (1991).
[13] P.G. De Gennes and J. M antricon, Rev. Mod. Phys. 36, 45, (1964).
[14] M.V. Feigel?m an, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Phys.
Rev. Lett. 63,2303, (1989).
[15] M.P.A. Fisher, Phys. Rev. Lett. 62, 1415, (1989).
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7
[16] R.H. Koch, V. Fogliette, V.J. Gallagher, G. Koren, A. G upta and M.P.A.
Fisher, 63, 1511, (1989).
[17] N.C. Yeh, Phys. Rev. B, 40, 4566,(1989).
[18] H.J. Jensen and P.Minnhagen, Phys. Rev. Lett. 66, 1630, (1991).
[19] D. Fork, F. Ponce, J. Tramontana, N. Newman, J. Phillips, and T. Geballe,
Appl. Phys. Lett., 58, 2432, (1991).
[20] G. Jakob, P. Przyslupski, C. Slolzel, C. Tome-Rosa, A. Walenhorst, M.
Schmitt, and H. Adrian, Appl. Phys. Lett., 59, 1626, (1991).
[21] F.M. Sauerzopt, H.P. Wiesinger, H.W. Weber, G.W. Crabtree, and J.Z.
Liu, Physica C, 162, 751, (1989).
[22] H. Jiang, Y. Huang, H. How, S. Zhang, C. Vittoria, A. Widom, D.B.
Chrisey, J.S. Horwitz, And R. Lee, Phys. Rev. Lett., 66, 1785, (1991).
[23] I. Zitkovsky, Q. Hu, T. Orlando, J. Melngnilis, and T. Tao, Appl. Phys.
Lett. 59, 727, (1991).
[24] J. Gallop, Nature, 350, 465 , (1991).
[25] N. Klein, G. Muller, H. Piel. B. Boas, L. Schultz, U. Klein and M. Peiniger,
Appl. Phys. Lett., 54, 757, (1989).
[26] M. Namordi, A. Mogrom-Lampero, L. Turner and D. Hogue, IEEE T vans
M T T ., Sep. (1991).
[27] D. Wu, W. Kennedy, C. Zahopoulos, and S. Sridhar, Appl. Phy. Lett., 55,
698, (1989).
[28] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, and A. Drehman,
subm itted to Phys. Rev. B.
[29] R. Glover and M. Tinkham, Phys. Rev., 104, 844, (1956).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
[30] Y. Huang, H. Jiang, H. How, C. V ittoria, A. Widom, and R. Roerstler, J.
Superconductivity, 3, 441, (1990).
[31] M. Nisenoff, Principles & Applications of Superconducting Quantum Inter�
ference Divices, ed. A.Barone, (World Scientific Pub., Singapore, 1990).
[32] T. Finnegan et al. Phys. Rev. B , 4, 1487, (1971).
[33] Y. Yoshisato, M. Takai, K. Niki, S. Yoshikawa, T. Hirano, S. Nakano,
IEEE. Tras. Mag., 27, 3073, (1991).
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9
CHAPTER II
HISTORICAL REVIEW
OF SUPERCONDUCTIVITY
In 1908, H. Kamerlingh Onnes succeeded in liquefying helium gas and made
very low tem peratures (down to IK) available. Soon after that his assistant, G.
Holst observed that the resistance of pure mercury dropped to zero at tem per�
ature of 4.15K. After repeating the results, Onnes reported^ in 1911 th at the
mercury had ?passed into a new state, which on account of its extraordinary
electrical properties may be called the superconducting state?. This new phe�
nomenon of zero-resistance together w ith a perfect diamagnetic p ro p e rty ^ , the
so called Meissner effect, is called superconductivity, and was soon discovered in
many other metals and alloys. The tem perature at which the m aterial changes
from the normal conducting state to the superconducting state is the critical
tem perature, Tc. In a magnetic field, a superconductor can conduct normally
even below Tc. The minimum magnetic field th at destroys superconductivity is
called the critical magnetic field, H c. Similarly, the minimum current density
th at destroys superconductivity is called the critical current density, J c.
2.1 MEISSNER EFFECT
In 1933, Meissner and Ochsenfeld^ measured the flux distribution outside
tin and lead specimens which had been cooled below their superconducting
transition tem perature while in a weak magnetic field. They found th at at the
transition temperatures, the specimens spontaneously became perfectly dia�
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10
magnetic, cancelled all flux inside. Fig. 2.1 dem onstrates the Meissner effect.
The final state of diamagnetism was independent of whether the sample was
cooled through the critical tem perature and placed in a field and vice versa. A
superconductor never allows a magnetic flux density to exist in its interior. In
other words, inside a superconductor we always have
B = 0.
(2.1.1)
This is the so-called Meissner effect. To m aintain B = 0 inside the supercon�
ductor, screening currents arise on the surface and circulate so as to cancel
the flux density inside the superconductor. The screening surface currents, or
Meissner currents, can not decay, otherwise the field will penetrate into the
superconductor. The zero resistance property of the superconductor allows the
non-decaying screening current.
2.2
T W O F L U ID M O D E L
The two fluid model was a phenomenological model proposed by Gorter
and Casimirt3] in 1934. It assumed th at below the transition tem perature, a
superconductor appears to be perm eated by two electron fluids, consisting of
normal and super electrons. At the transition tem perature, all electrons are
normal electrons and, at OK, all electrons are super electrons. The normal
electrons will be scattered by phonons or im purities, and, therefore, resistance
exists. We can treat a superconductor as a combination of two conductors
in parallel: one has a normal resistance and the other zero resistance, the
super electrons electrically short circuit the normal electrons so th at in the
superconductor no electric field and no resistance exists. The ratio of super
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Fig. 2.1
Diagram of Meissner effect
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electrons to normal electrons w = ^
gives a measure of ordering. At T >
Tc, uj = 0; at T < Tc some electrons condense to be superelectrons; at T ?
0 all electrons condense to be superelectrons, w = 1. From thermodynamic
arguments, tu was determined as to be of the following form
u; = l - ( | - ) 4.
2.3
(2.2.1)
LONDON THEORY
In the previous section, we indicated th at in a superconductor, the electric
field vanished when the current is a constant. In 1935, the London brothers^
considered, otherwise, if a electric field E is maintained in a superconductor,
the superelectrons would be accelerated
dv
qE = m ~dt?
(2.3.1)
where v is the velocity of the super electrons, m and q are their mass and
charge, respectively (we will see in section 1.6 th at superelectrons are paired,
therefore, the mass and charge are twice of th at of electrons). We denote n 3 as
the density of superelectrons, hence, the supercurrent density is
Js = n sqv.
(2.3.2)
Substituting (2.3.2) into (2.3.1) one obtains
V - = ? E.
at
m
(2.3.3)
The implication of (2.3.3) is th at, at static conditions, the current is a constant,
and, therefore, E = 0. This is the condition for perfect conductivity, one of the
properties of a superconductor.
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To explain diamagnetism in superconductors, or the Meissner effect, Lon�
don brothers phenomenologically wrote another equation,
H =
7YIC
yVxJsn sq2
(2.3.4)
By combining (2.3.4) with one of Maxwell?s equations
V x H = ? J s,
c
(2.3.5)
1
V H = -y H ,
(2.3.6)
they obtained
XL
where
m e,2
= 4 ^ ? '
(2'3 J )
The solution of (2.3.6) is (for one dimensional)
H (x) = H 0 e~x^ L.
(2.3.8)
From (2.3.4) and (2.3.5) one can also obtain
V 2J s = ^ - J s ,
(2.3.9)
J a{x) = J 0e " l/A�.
(2.3.10)
with a solution
These equations show th at supercurrent is a surface current w ithin a depth of
Ai and the flux density decays exponentially inside a superconductor, falling
to 1/e of it value at a distance Al , the so called London penetration depth.
Applying n s = nu> and (2.2.1) to (2.3.7), we obtain the London penetration
depth as follows
Al (T ) = A(0)(1 - ( ^ f T 1' 2.
(2.3.11)
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14
2.4
TYPES OF SUPERCONDUCTORS
There are two types of superconductors w ith different magnetic properties.
They are called type I and type II superconductors. Type I superconductors
exhibit pure diamagnetism, or Meissner effect, when the applied magnetic field
is below the critical field, H c, see Fig. 2.2a. In type I superconductors, the
external field will decay and vanish w ithin a distance of
of the surface. In
type II superconductors, however, the diamagnetic effect is not complete when
the field exceeds a critical value called the lower critical field H ci . Diamagnetism
is then not destroyed until the field exceeds another critical value, the so called
upper critical field, H c2, as shown in Fig. 2.2b. This superconducting state is
called the mixed state. In the mixed state, the magnetic field can penetrate in
discrete vortices and each vortex is quantized in units of the flux quantum
<j>o = hc/q.
(2.4.1)
The flux lines are nucleated in a periodic structure called the Abrikosov la ttic e ^ .
The flux lines can be observed by the B itter p attern m e th o d ^ . Real supercon�
ducting materials always contain defects, and the flux lines can interact with
these defects. This interaction leads to an irreversible magnetization behavior,
see Fig. 2.2c. Superconductors w ith irreversible magnetization behavior are
called non-ideal type II superconductors. Fig. 2.3 shows flux lines in type II
superconductors. In the flux line, the superconducting order param eter (defined
in section 2.5) rises from zero to unity over a distance �, coherence length; the
local magnetic field penetrates over a distance \ i from the center of the flux
line; the screening current density rises from zero to maximum over a distance
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15
H
(A) type I
H
(B) type II
(C) non-ideal type II
Fig. 2.2 Magnetization of superconductors
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16
Fig. 2.3 Abrikosov lattice of type II superconductors
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17
(a)
1
' M S )I 2
i \
\
1
1
1
I
0
(b )
C(T)
R
A(T)
R
A(T)
Fig. 2.4 Structure of an isolated vortex
(a) a single vortex, (b) order parameter, (c) local field,
(d) screening current density.
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18
� and decays over a distance Al , see Fig. 2.4. All the elemental supercon�
ductors are type I superconductor except for V, Nb and Tc, which are type II
superconductors. Most of the superconducting alloys and compounds are type
II superconductors.
2.5
GINZBURG-LANDAU THEORY
Unlike the London theory, which is a purely classical theory, Ginzburg and
Landau used quantum mechanics to calculate and predict the electromagnetic
behavior of superconductors^. G-L theory is useful in strong external mag�
netic fields, where the London theory is not valid. It was proven in 1959 by
G orkov^ th at near Tc the G-L theory is in fact a limiting form of the micro�
scopic BCS th e o ry ^ , which was developed by Bardeen, Cooper and Schrieffer
in 1957. G-L theory is particularly useful in giving a clear grasp of the rela�
tionship between the various lengths (penetration depth and coherence length,
for example) involved in superconductivity.
The first assumption of G-L theory is th at the behavior of the supercon�
ducting electrons is described by an effective wave function (order param eter)
tpir) = y/n^exip.
(2.5.1)
and \i>(r) \ 2 = n s, where n a is the cooper pair density. The second assumption
is that the free energy of the superconducting state differs from th a t of the
normal state by an amount which is proportional to \ip\n, where n= 2, 4, ...
/ . = /? + c H
\2
+ J /3 M 1 + j j j j I H * V - ^ W > | 2 +
(2.5.2)
where a and j3 can be expressed as a power series in (T ?Tc), and furthermore,
a (T ) ? a(T - Tc) < 0,
when T < T C
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19
and
(constant),
/? > 0
where a and (3 can be determined experimentally.
The so called G-L equations axe
aif> + / 3 | # + ? ( - i h v ? ?) V = 0,
(2.5.3)
and
From the G-L equations, one can deduce many im portant quantities, for exam�
ple, the G-L param eter, k = j , where A is the penetration depth and � is the
coherence length, and thermodynamic critical field, H c. The significant results
ax e
( 2 - 5 -6 )
� T ) = %l(2m\a \ f l 2 = f(0 )(l - I ) ) - 1' 2,
=
(2.5.6)
Ia 2c2
(2.5.7)
and
A(T) - me HT W 2
�(T)
qft
V
2tt
Tic2
c?
( 5?8)
According to the G-L theory, the quantitative value of k gives different
superconducting b e h a v io rs^ . For k <
it leads to a positive surface energy,
hence, we have type I superconductors. For k > -^=, it leads to a negative
surface energy and type II superconductors.
2.6
B C S M IC R O S C O P IC T H E O R Y
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20
Frohlichj11! pointed out in 1950 th at the electron-phonon interaction is able
to couple two electrons together in such a way to produce a direct interaction
between them. Coopert1^ considered the electron-electron interaction in more
detail in 1956, and he was able to show th at if there is an attraction between
them, however weak, they are able to form a bound state. In 1957 Bardeen,
Cooper and Schrieffer (BCS) were able to show how Cooper?s simple result
could be extended to apply to many interacting electrons. The fundamental
assumption of the BCS theory is that the only interactions which m atter in
the superconducting state are those between any two electrons which happen
to make up a Cooper pair and that the effect on any one pair of the presence
of all the other electrons is to limit those states into which the interacting pair
may be scattered. Their total energy is less than 2ep, where e? is the Fermi
energy. The paired electrons are called a Cooper pair and are required to have
equal and opposite momenta as well as opposite spins. At zero temperature,
all the electrons near the Fermi surface are paired. At tem peratures below Tc,
some unpaired electrons exist near the Fermi surface. At Tc, all paired electrons
become normal electrons.
The BCS theory could explain almost all the experimental evidences at
th at time. It predicts th at there is an energy gap of magnitude E g = 2A in
the excitation spectrum of a superconductor, and A is the minimum energy
required to split up a pair into two single electrons. The value of A is
A(0) = 2
1
),
(2.6.1)
where u>p is the average phonon frequency, V is the m atrix element of the
scattering interaction, and N (ep) is the density of states for electrons at the
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Fermi energy of the normal state. A(0) may be determined experimentally from
single electron tunneling effects, measurement of critical fields, or ultrasonic
a b s o rp tio n ^ . According to the BCS theory Tc is
TC= l . l 3| W
^ L _ ) ,
(2.6.2)
where K b is the Boltzmann constant. By comparing (2.6.1) and (2.6.2) we
have
2A(0) = 3.53 K b Tc.
(2.6.3)
The measured ratio of ^H jr-for all the known conventional superconductors was
in excellent agreement with the prediction of BCS theory, and is equal to 3.5
(except Hg (3.95-4.6) and Pb (3.95-4.2)).
The phonon frequency u>p is proportional to M -1 /2, where M is the atomic
mass. Putting this into (2.6.2), we get
Tc oc M ~ xl2.
(2.6.4)
This is the so called isotope effect experimentally discovered by Maxwell^14) and
Reynolds et alS15^ independently in 1950, which confirmed th at the electronphonon interaction played an im portant role in superconductivity. BCS the�
ory also concludes th at materials will show superconducting behavior only if
the net interaction between electrons resulting from the combination of the
phonon-induced and coulomb interactions is attractive. This is why good con�
ductors, like Au, Ag and Cu, which have weak electron-phonon interaction, do
not exhibit superconductivity.
2.7
HIGH T c SUPERCONDUCTORS
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22
Since the discovery of superconductivity in 1911, thousands of materials
have been found to exhibit superconductivity from metallic elements, alloys,
compounds and organic compounds. Most applied conventional superconduc�
tors are metals and alloys such as Nb, Pb, N b T i, N b zS n and NbzGe. The
highest transition tem perature known prior to 1986 was 23.2K (NbzGe). The
first oxide superconductor was discovered in 1964 which was SrTiO z with Tc
about 0.3Kt16! and then NbO in 1965 with Tc around 1.2R!17!. Ten years later,
Sleight et alJ18! in 1975 discovered another oxide superconductor, BaPbBiOz,
with the relatively high transition tem perature of 13K. In 1979, the first organic
superconductor, ( T M T S F ) 2 PFq, was discovered^19!. Since then, there were
five organic superconducting families (TMTSF, B ED T-TTF, DMET, MDTT T F and DMIT) with more than 30 members discovered. Tc ranged from
below IK to H R !20!. The latest advance was reported by Williams et a/.!21! on
ft - (E T ) 2 C u [N (C N )2}Cl with Tc = 12.8K at 0.3Kbar.
One of the key limitations to the progress of applying superconductors had
been the low tem peratures at which most materials become superconducting.
In 1986, the discovery of the first high Tc cuprate La 2 - xB a xC u O ^ -y by Bednorz and Muller!22! opened a new world of high T c superconductors. The first
superconductor with high transition tem perature above liquid nitrogen temper�
ature was Y B a 2 C uz 0 7 ~x with Tc equal to 90K, reported by Wu et a!.!23! in
1987. The highest Tc reported so far is T l 2 B a 2 Ca 2 CuzO\z (125K)!24!. The
new high Tc superconductors are all oxides th at contain Cu and most of them
contain other transition elements or other rare earth elements such as Y, La,
and Sr. Table 1.1 list some of the high Tc superconductors and their transition
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23
T able 2.1
Transition Temperatures for Some High Tc Superconductors
Compounds
Tc (K)
L ai'SsB ciQ 'isC uO i
30
L a i'S s S r O 'iz C u O i
38
N d 1.6eSro.205C eo.1z5Cu.O4
28
SmLdo.75SrQ'25Cu )3.95
E r B d 2 C u 3 0 6.53
37
60
E r B d 2 C uz07
92
lo.8Cao.2Sa2Cu3O6.il
50
90
Y q.qC d o 'iB d 2C u i O s
Y B d 2 C u zO&.5
Y B d 2 C uz07
B i 2 S r 2 CuOo +6
B i 2S r 2C do.oY o.\C u2Oo.2i
B i 2 S r 2C dC u2 0s
B i 2 S r 2 Cd 2 CuzO\o+s
B i 2S r 2(Gdo,s2C e o .is )2 C u 2Oio.24:
(Bdo.&7Euo.zz)2(.Euo.o7Ceo.zz)2CuzO%jrs
(Cd0'5Ld0.5)(Bdi.25Ld0.75)CuzO6+S
60
93
80
93
85
110
40
48
Pb 2 S r 2 Y 0 .5 Cao.5Oz
90
83
Pbo. 5 Tlo. 5 S r 2 C dC u 20 7
110
Pbo.5Tlo,zSr2C d C u z O o
125
T l(B a o .o L d o A )2C u O z -8
T lB d 2 C d C u 2 0 7 - s
52
103
T lB d 2 Cd 2 CuzOg - 6
120
T l 2 B d 2 C uO &
T l 2 B d 2 C d C u 2 Os
87
110
T l 2 B d 2 Cd 2 C u zO 10
125
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24
tem peratures.
Y B a z C u z O i- x and B i 2 S r 2 CaCii 2 0 y systems are two good examples among
these high-Tc materials, and their crystal structures!25?26! are shown in Fig. 2.5
and Fig. 2.6. YBa, 2 C uz 0 7 - x (a; < 0.2) has an orthorhombic structure (high
Tc phase, Tc ~ 92K ) with unit cell parameters!27! a=3.8187A, b=3.8833A and
c = 1 1 .6 6 8 7 A ; when 0.2 < x < 0.5, the m aterial shows an order-disorder tran �
sition, and the transition tem perature drops to Tc ~ 60K.
when x > 0.5
YBa, 2 C u z 0 7 - x changes to a nonsuperconducting tetragonal phase!28,29! (see
Fig. 2.5). The C u 0 2 planes are primarily responsible for the electronic con�
duction, the CuO chain along the b-axis may be im portant in chemical and
structural stability. The superconducting transition tem perature strongly de�
pends on the oxygen composition in YBa, 2 C u z 0 7 - x. B i 2 S r 2 C aC u 2 0 y (see
Fig. 2.6) appears to have a relatively stable tetragonal crystal structure with
a = 3.814A, and c = 30.52A, there are no CuO chains in this compound. The
Transition tem perature of B i 2 S r 2 C aC u 2 0 y is less sensitive to oxygen stoich
than th at of Y B a 2 C u z O i- x .
The coherence length, �, is very small (typically of the order of 10A) in high
Tc superconductors, in comparison w ith conventional superconductors whose
� ~ 102 ?104A. The small coherence length leads to an extremely high value of
the upper critical field, H c2 , with potential applications. The coherence lengths
of high Tc superconductors are anisotropic. For example, the best available
data!26,27! are; the coherence length along a-b plane, &(0) ~ 15A and (0), the
coherence length along c-axis, is about 3A for Y B a 2 C u z 0 7 - x and &(0) ~ 27A
and (0) ~ 0.5A for B i 2 S r 2 C aCu 2 0 y, respectively. The penetration depths
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orthorhom bic
tetragonal
tetragon al
25
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26
I
CuO
Bond
?
Bi
�
Sr
�
Ca
?
Cu
O
o
Fig. 2.6 Crystal structure of B i2 S r 2C a C u 20 8 - s
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27
are anisotropic as well. Some measured values of superconducting parameters
and characteristic lengths are listed in table 1 .2 !26?27?30?31! for Y B a o C u ^ O j-x
and B i 2 S r 2 C a C u iO y.
YBCO m aterial has the highest critical current density in the high Tc
oxide superconducting family. Y B a 2 CuzOT~x film!32,33! can reach J c ~ 106 ?
107 A / c m 2 which is the same as conventional superconductors. Jiang et aZ.!34,35!
have observed ultrahigh critical current density of J c ~ 1.3 x 109 A / c m 2 on
YBCO bridge constriction w ith a width of 500A. Bulk YBcl 2 C u z O t- x has rela�
tively lower J c, since granular boundaries limit the transport current. The high�
est J c m easured in bulk m aterials (in YBCO) was in the order of 104A /cm 2
at 77K (7T magnetic field!36!). J c of YBCO coating on a dielectric fiber or
metallic ribbon has reached J c ~ 105A /cm 2!37!.
Although there is no complete theory to explain all the electrical proper�
ties of high Tc superconducting materials, there are several theories and mod�
els partially successful in explaining some experimental data. These theories
include: S-channel theory!38!, Van Hove excitons!39!, ionic bonding!40!, multi�
gap!41?42!, and theories of vortex mechanism, such as flux pinning!43!, vortex
glass!44?45!, vortex phase!46!, flux lattice melting!47-49!, two dimensional vortex
fluctuations!50! ? and unified theory of vortex pinning and flux creep!51!.
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28
T a b le 2.2
Characteristic Param eters for Y B a 2 C u 30 j and
Parameters
Y B a 2C u 30 7
Transition Temperature
Tc = 93
&
II
B i 2 S r 2 C aC u 2 0&
a = 3.8591
a = 3.814
00
Or
B i 2 S r 2C aC u 2 0 3
(K)
Unit Cell Constants
(A)
Coherence Length
(A)
Penetration Depth
(A)
6 = 3.9195
c = 11.8431
c = 30.52
i(0) = 15
b(0) = 27
&(0) = 3
(0) = 0.5
Aa6(0) = 1400
Aa6(0) = 2500
Ac(0) > 7000
Ac(0) > 10000
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29
REFERENCES
[1] H.K. Onnes, Leiden Comm. 119b, 122b, 124c, (1911).
[2] W. Meissner and R. Ochsenfeld,Naturwissenschaften 21, 787, (1933).
[3] C.J. Gorter and H.B. Casimir, Phys. Z. 35, (1934).
[4] F. London and H. London, Proc. Roy. Soc. A 149, 71, (1935).
[5] A.A. Abrikosov, JE T P 5. 1174, (1957).
[6] H. Trauble and U. Essmann, J. Appl. Phys. 39, 4052, (1968).
[7] V. Ginzburg and L.Landau, Zh. Eksperim. i Teor. Fiz, 20, 1064, (1950).
[8] L. Gorkov, JE T P (USSR), 9, 1364, (1959).
[9] J. Bardeen, L.N. Cooper, and R. Schrieffer, Phys. Rev., 108, 1175, (1957).
[10] D. Saint-James and P. de Gennes, Phys. Lett., 7 , 306, (1963).
[11] H. Frohlich, Phys. Rev., 79 , 845, (1950).
[12] L.N. Cooper,Phys. Rev., 104, 1189, (1956).
[13] R. Meservey and B.B. Schwartz, in Superconductivity vol.
I, ed. R.D.
Parks.
[14] E.M. Maxwell, Phys. Rev., 78, 477, (1950).
[15] C.A. Reynolds, Serin, W right, and Nesbitt, Phys. Rev., 78 , 487, (1950).
[16] J. Schooley, W. Hosier, and M. Cohen, Phys. Rev. Lett., 12, 474, (1964).
[17] R. Miller, R. Mein, C. Jones, Proc. LT9, 600, (1965), ed. J. Paunt, et al.
Plenum, New York.
[18] A. Sleight, J. L. Gillson, and P.E. Bierstedt, Solid State Comm., 17, 27,
(1975).
[19] D. Jerome, A. Mazaud, M. Ribault, K. Bechgaard, J. Phys. Lett. (France),
41, L95, (1980).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
[20] The Phys. Chem. Organic Superc. 1990, ed. G. Saito and S. Kagoshima,
Springer, New York.
[21] J. M. Williams, ei al, Inorg. Chem., 29, 3272, (1990).
[22] J.G. Bednorz and K.A. Muller, Z. Phys. B, 64, 189, (1986).
[23] M.K. Wu, J. Ashburn, C. Torng, P. Hor, L. Gao, Z. Huang, and C.W. Chu,
Phys. Rev. Lett., 58, 908, (1987).
[24] M.B. Maple, M R S Bulletine, 15, 13, (1990).
[25] J.D. Jogensen, M.A. Beno, D.G. Hinks, L. Soderholm, K .J. Volin, R.L.
H itterm an, J.D. Grace, I.K. Schuller, C.U. Segre, K. Zhang, and M.S.
Kleefisch, Phys. Rev. B, 36, 3608, (1987).
[26] S. M artin, A. T. Fiory, R.M. Fleming, G.P. Espinosa, and A.S. Cooper,
Appl. Phys. Lett., 54, 72, (1989).
[27] W. David, et al., Nature, 327, 310, (1987).
[28] C. Rao, in Chemical and Structural Aspects of High Temperature Superconducors, ed. C. Rao, (World Scientific, New Jersey, 1988).
[29] B. Raveau, C. Michel, M Hervieu, and D. Groult, Crystal Chemistry of
High-Tc Sperconducting Copper Oxides, (Springer-Verlag, New York, 1991).
[30] T.T.M . Palstra, B. Batlogg, L.F. Schneeneyer, R.B. VanDover, and J.V.
Waszczak, Phys. Rev. B, 38, 5102, (1988).
[31] U. Welp, W.K. Kwok, G.W. Crabtree, K.G. Vandervoot, J.Z. Liu, Phys.
Rev. Lett., 62, 1908, (1989).
[32] P. Vase, Y. Shen, and T. Frelloft, Physica C, 180, 90, (1991).
[33] D. Fork, F. Ponce, J. Tramontana, N. Newman, J. Phillips, and T. Geballe,
Appl. Phys. Lett., 58, 2432, (1991).
[34] H. Jiang, Y. Huang, H. How, S. Zhang, C. Vittoria, A. Widom, D.B.
Chrisey, J.S. Horwitz, And R. Lee, Phys. Rev. Lett., 66, 1785, (1991).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
[35] J. Gallop, Nature, 350, 465 , (1991).
[36] M. Lees, B. Bourgault, D. Braithwaite, P. de Rango, P. Lejay, A. Sulpice,
and Tournier, Physica C, 191, (1992).
[37] V. Pan, and V. Flis, Subm itted to Cryogenics.
[38] T.D Lee, In Symmetry in nature, (Scuola Normale superiore, Pisa, 1989).
[39] R.S. Markiewicz, Physica c, 168, 195, (1990); 169, 63, (1990); 177, 171,
(1991); 183, 303, (1991).
[40] A. Widom, T. Yuan, H. Jiang, C. Vittoria, subm itted to Phys. Rev. B.
[41] V.Z. Kresin and S. Wolf, Physica C, 169,476, (1990).
[42] C. V ittoria, A. Widom, H. Jiang, J. Superconductivity,
4, 361,(1991).
[43] A.P. Malozemoff, L. Krusin-Elbaum, D.C. Cronemeyer, Y. Yeshurum, and
F. Holtzberg, Phys. Rev. B , 38, 6490, (1990).
[44] M.P.A. Fisher, Phys. Rev. Lett., 62, 1415, (1989).
[45] R.H. Koch, V. Foglietti, W .J. Gallagher, G. Koren, A. G upta, and M.P.A.
Fisher, Phys. Rev. lett., 63, 1151, (1989).
[46] N.C. Yeh, Phys. Rev. B. 40, 4566, (1989).
[47] R.S. Markiewicz, Physica C, 171, 479, (1990).
[48] D.R. Nelson and H.S. Seung, Phys.
Rev. B, 39, 9153,
[49] A. Houghton, R.A. Pelcovits and A. Sudbo, Phys. Rev.
(1989).
B, 40, 6763,
(1989).
[50] H .J. Jensen and P. Minnhagen, Phys. Rev. Lett., 66, 1630, (1991).
[51] M.W. Coffey and J.R . Clem, Phys. Rev. Lett., 67, 386, (1991).
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32
CHAPTER III
SAMPLE PREPARATION, CHARACTERIZATION
AND EXPERIMENTAL SETUP
3.1
BULK YBCO SAMPLES
Bulk samples of polycrystalline YBa. 2 C uz 0 7 - x were prepared using solid
state reaction techniques. Our starting materials were high purity yttrium ox�
ide, Y 2 O 3 , barium carbonate, BaC O z, and copper oxide, CuO. The quantities
were determined to form the correct atomic ratio, i.e. Y :Ba:Cu=l:2:3. There
was no carbon left in the final m aterial although BaCOz was used, since carbon
was evaporated as CO 2 during the sintering processing. The starting materials
were mixed and ground by agate m ortar and pestle, and then were calcined
at high tem perature (900癈) heating in an oxygen atm osphere for solid state
reaction. After 10 hours heat treatm ent, the calcined powder was cooled down
to room tem perature with a rate of 150�/hour, re-ground and pressed with
200KPsi pressure into pellets of 20m m in diameter. The pellets were sintered
at 1000癈 for 15 hours, and slowly cooled over 4 hours to 450癈 and held there
for 5 hours. Finally the furnace was cooled down to room tem perature with
a cooling rate of 150�/hour. During the sintering, a flowing oxygen was ap�
plied with exhaust out through water. The finished bulk sample can be quickly
checked for Meissner effect by suspending it in liquid nitrogen and bringing a
magnet near it. High quality YBCO pellets could be used as targets for laser
ablation film deposition.
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33
3.2
PULSED LASER ABLATION OF YBCO FILMS
Pulsed laser ablation was used to make YBCO films. Fig.3.1 is a picture of
our system for the deposition. A Lambda Physik excimer laser was used whose
wavelength is 248 nm, output energy is 1.75 J/pulse. A vacuum system provided
a pressure of 10-6 to rr in the chamber. The substrate holder can. be heated
up to 800� C by electric resistive heater. The target is a stoichiometric YBCO
pellet. M gO , LaAIOz and SrTiO s substrate were used since their their crystal
lattices are closely matched with YBCO crystal lattice. An MgO substrate
was mostly used since its dielectric constant (e ~ 10) meets the requirement of
microwave applications. The focused laser produced a plume of ejected m aterial
deposited onto a heated substrate located about 5-8 cm from the target. The
target rotated at a rate of 14 cyc/m in to avoid the focused laser impacting
on the same spot, the deposition parameters we used were: laser voltage 24
KV, laser gas pressure 2500 mbar, pulse rate 6-8 Hz, chamber O 2 pressure
300-800 torr, and substrate tem perature 730-780癈. We found th a t the optimal
oxygen pressure was 500-600 torr and substrate tem perature was 750癈. Once
the deposition was completed, the chamber was filled with oxygen and the
film was held at the same tem perature as deposition in situ for 8-10 minutes
for the formation of orthorhombic structure. Finally the chamber was cooled
down to room tem perature and high quality YBCO superconducting films were
obtained. The finished films usually exhibited a transition tem perature around
90K and critical current density of the order of 106 A / c m 2.
3.3
SUBMICRON FABRICATION TECHNIQUES
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34
Fig. 3.1 Pulsed laser deposition system.
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35
In some of our experiments, samples needed to be fabricated into small di�
mensions. All weak link devices also involved submicron dimension fabrication.
A procedure was developed to routinely and reliably fabricate superconducting
microstructures. The fabrication includes two stages, photolithography and ion
beam direct milling. Photolithography followed by chemical etching provides
tolerances of about 2 ~ 20 fim. Smaller dimension needed to be accomplished
by ion beam direct milling. A tolerance of ~ 500A. was achieved.
3.3.1
Photolithography and Chem ical Etching
To process the photolithography, a 1:20 mask was needed. The mask could
be computer designed and printed on a transparency. The YBCO thin film
was precoated with Microposit S1813 photoresist before soft baking at 95癈
for 15 minutes. The thickness of the photoresist became im portant when small
tolerance was desired. The thiner the photoresist, the sharper the etched edge
that could be obtained. To get optim al photoresist coating, the photoresist
should be diluted with 1/10 m ethanol and spun at a speed of 5000 cyc/m in to
spin off the photoresist. After soft baking, the film was placed at the focus of
20 to 1 reduction from the mask and exposed to UV radiation for 30 minutes
at aperture 4. The exposed film was then developed in Microposit developer
MF-319 for about 2 minutes and rinsed with water, and afterwards hard baked
at 105癈 for 15 minutes. Now, the film was ready to be chemically etched.
Ethylenediamine Tetraacetic Acid (EDTA) solution was used for etching out
the unwanted YBCO film area. The average etching time was about 10 ~
20 minutes. By using the 20 to 1 reduction photolithography, 10 ~ 30\im
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tolerances could be obtained.
For more precise photolithography, a one-to-
one mask was needed. The one-to-one mask was made on a Nanofilm chrome
glass in the same m anner as 20 to 1 reduction photolithography but chromium
mask etchant instead EDTA was used as etching solution.
The one-to-one
photolithography is accomplished by a Cobilt 28A computervision, which allows
one to align the mask and superconducting film under a microscope.
The
advantage of one-to-one photolithography was th at one could precisely locate a
p attern on a desired position, such as to make a microbridge cross over a crystal
grain boundary. Whereas the quality of reduction photolithography depends on
the focus, one-to-one photolithography always produces high quality patterns
on the film. Since the ultra uniform UV source was located close to the film, the
exposive time was much shorter th an that in 20 to 1 reduction photolithography,
30 seconds instead 30 minutes as usually used. The developing and chemical
etching process were the same as in 20 to 1 photolithography. A tolerance as
small as 2 microns could be obtained by using one-to-one photolithography.
3.3.2
Ion Beam M illing and D eposition
A direct modification system, DMOD 907/908, was used for fine milling and
deposition on superconducting microbridges with a focused ion beam. Typical
beam energy was 25 KeV with a beam size of 500
20000A , Gallium liquid
metal was used as the ion beam milling source, while a tungsten source was
used for ion beam deposition. Secondary electron or ion imaging was used
to view the sample and milling process.
The processing was controlled by
computer. Before each milling, a test milling was performed to determine the
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37
milling parameters for which the beam just milled through the film to reach the
substrate. Fig. 3.2 shows a test milling process. The sudden change of the curve
slope indicates that the ion beam travel from one m aterial to another (from
the film to substrate). The x-axis is in dose/area, while y-axis is in arbitrary
units. Ion beam deposition was sometime used for malting SNS junctions. The
operation is the same as ion beam milling but a tungsten beam was used with
much smaller energy. The advantage of ion beam direct milling over chemical
etching is th at the photoresist is not needed and there is no chemical reaction
with the film. Later testing indicated th at the ion beam direct milling did not
degrade the superconductivity of the film. We have achieved a tolerance as
small as 500AW by using ion beam milling. Ideally the ion beam milling can
achieve 100A tolerance, and the ion beam deposition make 1000A tolerance.
Shown in Fig. 3.3 is an width~4000A microbridge and shown in Fig. 3.4 is a
500A microbridge made by using ion beam direct milling.
3.4
FABRICATION OF JO SEPH SO N W EA K LINK
3.4.1
Josephson Weak Link of Conventional Superconductors
A Josephson weak link can be viewed as two bulk superconductors con�
nected by a ?weak superconductor? . The ?weak superconductor? allows super�
conducting pair tunneling through the link (therefore supercurrent through the
link), but the critical current is very small, typically in the order of 10fiA ~
1mA. The superconducting wave functions on either side of the link have signif�
icant overlap. If a voltage is established across the link, there is a corresponding
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3S
Fig. 3.2 Ion beam test milling process.
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39
Fig. 3.3 Microbridge fabricated by using ion beam milling. width=4000A.
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40
gsaaaBM
~
Fig. 3.4 Microbridge fabricated by using ion beam milling, \vidth=500A.
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41
rate of phase slip between the two sides and this phase slip can synchronize with
an applied a.c. field. The ?weak superconductor? is not necessarily a supercon�
ductor, but can be an insulator or normal metal, if its thickness is sufficiently
small. The criteria for a Josephson weak link are first th at the coupling between
the superconductors should be strong enough for the phase of the wave functions
on either side to be related, and superconducting pairs have the opportunity to
tunnel through the barrier. Secondly, the coupling should be weak enough so
the superconducting pair can tunnel through instead conducting through the
barrier and so the system can be perturbed by applied radiation fields.
There are usually three types of Josephson weak link systems in conven�
tional superconductors^2!. They are:
(a) A point contact between two superconductors. Usually a supercon�
ducting wire is ground to a point and then the point is pressed against a piece
of bulk superconductor. The optimal superconductor for this purpose is Nb.
(b) Superconductor-insulator (or normal metal)-superconductor junction
(SIS or SNS). Typically an oxide barrier between evaporated superconducting
thin films. The oxide barrier thickness is critical, usually less than 30A.
(c) A microbridge formed by etching a superconducting film to a bridge.
The dimension of the bridge constriction is the same order as the coherence
length of the superconducting film.
One way to test whether the link is a Josephson weak link or not is by
shining microwave radiation onto the link and measuring the V-I characteristics.
If it is a weak link, or Josephson junction, the so called Shapiro step will be
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42
observed and the height of the step is ^
W hen a Cooper pair tunnels through the barrier, it will absorb microwave
energy fao, where oj is the frequency of the the microwave radiation.
The
energy is equal to 2 eV, where 2 e is the charge of the superconducting pair.
3.4.2
Fabrication of High T c Josephson Weak Links
U sing YBCO Superconducting Film
Since the high Tc superconducting m aterials are ceramics, it is very difficult
to fabricate point contact weak links with such material. Machinable material
is required to make a sharp tip. SIS or SNS junctions are also not the ideal
ways to make weak links w ith oxide superconductors. If the SIS or SNS is
fabricated as a weak link, the oxygen in the cuprate superconductor will react
w ith the metal or intermediate layer which leads the SIS (or SNS) weak link to
be degraded or damaged. Microbridges seem to be a reasonable way to fabricate
superconducting weak links. However, fabrication of microbridges down to 500A
produced no weak links. We realized th at we needed microbridge structures in
the order of 10-lOOA to produce weak links. We propose an alternative approach
to fabricate weak links on modified microbridges.
(a) Weak Links on Single Grain Boundaries
We used artificial single grain boundary YBCO films, which were produced
at C o n d u c tu sM , to fabricate modified microbridge weak links. The deposition
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43
procedure of single grain boundary film is briefly as follows. A 1000A seed
layer of epitaxial MgO was laser deposited on an 7 -plane sapphire substrate,
and removed from part of the substrate by photolithographic patterning and Ar
ion milling. Subsequently, a 1000A S rT iO z buffer layer was grown epitaxially
on both the sapphire and the patterned MgO, but with orientations which were
separated by a 45� grain boundary located at the edge of the MgO. W ithout
breaking vacuum, a 2000A YBCO film was deposited epitaxially on the SrTiO z
buffer.
Therefore, an artificial single grain boundary was produced on the
YBCO film, see Fig. 3.5a. One-to-one photolithography allowed us to make
a microbridge across the grain boundary of the film. Shown in Fig. 3.5b is a
finished microbridge with w idth of 24[im across a single grain boundary. The
critical current of the bridge was about 50^A. By shining microwave radiation
onto the link, we observed the Shapiro step. Fig. 3.6 shows the voltage-current
characteristics of the microbridge, where curve a is without microwave radiation
while curve b is under a 0.0004 mW (-34dBm) microwave radiation. The height
of the step is in agreement with (3.3.1).
(b) Step Edge Junction
Shown in Fig 3.7 is a side view of a step edge junction. A single crystal
(100) MgO substrate was cleaved to a sharp step with a height about 1p,m. A
3000A thick YBCO film was deposited on the cleaved substrate. A microbridge
was fabricated across the step by using photolithography and ion beam milling.
The finished dimension of the bridge was 1fj,m(width) x 10fj,m(length). An
a.c. current from a function generator was applied to the step edge junction.
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44
[010 ]
[100]
- [ 010]
[100] -
45 degree grain boundary
I
V
V
I
Fig. 3.5 Microbridge fabricated on an artificial grain boundary, (a) artificial
grain boundary film, (b) finished microbridge.
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45
0.20
Microwave pow er = 0
0.15 ?
>
s
0.10
?
0.05 ?
-J 1 i I i ' ' '
0.00
0.2
0.4
o.a
0.6
I (mA)
"I
I
I '
I
I
I
I 1
I ' ' I
I
I
I ""
I
I 1 1I
I
I '
I 1
0.20
Microwave pow er = 0.0004mW
0.15
%
0.10
0.05 ?
0.00
0.2
0.4
0.6
0.8
I (mA)
Fig. 3.6 Current-voltage characteristics of a modified microbridge weak link.
(a) without microwave radiation,
(b) with microwave radiation (power~0.0004mW).
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46
Step edge junction
YBCO
YBCO
MgO substate
MgO substate
Fig. 3.7 Side view of a step edge junction.
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47
Nonlinear response was observed in the frequency range of 2MHz~10MHz. Fig.
3.8 shows the nonlinear response at 4MHz. All the input signals were sinusoidal.
The output nonlinearity was observed when the inputs were larger than 2V.
3.5
CHARACTERIZATION
3.5.1
Electrical M easurem ents
(a) Four probe m easurem ents
Electrical measurements include current-voltage characteristics, critical cur�
rent densities and electrical resistivity versus tem perature. All the measure�
ments were based upon the four-probe method which eliminates the contact
resistance. The difference between two-probe and four-probe methods is dis�
cussed in Fig. 3.9. In the four-probe measurements current does not transport
through the voltage pick-up contacts; therefore, there is no voltage drop at those
contacts other than from the sample. The contact resistance can be determined
from
K = R2? - R i' ,
(3.5.1)
where f?2p and R眝 are the measured resistances from two-probe and four-probe
methods, respectively.
(b) Low Resistive contacts
Although the values of contact resistances are excluded from the fourprobe measurements, the quality of contacts affects the electrical measurements.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 3.8 Nonlinear response of a step edge junction. Input signals were
sinusoidal.
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49
Two-probe
Four-probe
h?
t
<� 1
v
V I
I
1> i 1
V
V
m
I
n m
v = y1+vr2+ vs
R ?Ri + R2 + R3
R ? R�
Fig. 3.9 Four-probe and two-probe measurements
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50
Highly resistive contacts introduce a noise level as high as 10 fxV ~ Im V ,
which makes a small signal measurement impossible (in some measurements on
microbridges, we need to measure 10 nV). On the other hand, the poor contact
will heat up the sample when measuring the critical current density of a high
J c film. A low resistive contact technique was employed to minimize the lead
contact resistance in order to avoid the heating effects and high noise. A layer
of silver bars was evaporated onto the contact area of a superconducting film.
A post annealing of the silver bars evaporated on a YBCO film was carried out
in an oxygen atmosphere at 450� C for 2 hours (this post-annealing should not
apply to YBCO microbridge or weak link samples). Gold wires were bonded
onto the silver bars by using an EMB1100 ultrasonic wire bonder, or silver wires
were soldered with pure indium onto the silver bars. Low contact resistivities
were found to be in the order of 10-5 ftcm 2 at room tem perature.
(c) Electrical m easurem ents
Fig. 3.10 is a schematic diagram of electrical measurements. A Keithley
224 programmable current source was employed to provide current in the range
5n A ~ 101mA. Although the current value was displayed on the current source,
a standard resistor (1ft or lAfft, depending on the current range) was connected
in the current supply circuit w ith an HP voltmeter across the resistor to measure
the voltage. A Keithley 182 nanovoltmeter was employed to measure the voltage
across the superconducting samples. The measured d ata were collected and
processed by a computer. In the critical current density measurement, the J c
was determined by applying an electrical field of 1\iVjcm . Fig. 3.11 shows a
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51
Work Station
Keithley 182
HP
Nanovoltmeter
Voltmeter
Keithley 224
Current source
Sample
Fig. 3.10 A schematic diagram of electrical measurements.
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52
R esistance
(m fi)
200
150
100
100
150
200
250
300
Temperature (K)
Fig. 3.11 A typical resistance versus temperature of a YBCO film.
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53
typical resistance versus tem perature of a YBCO film and in Fig. 3.12 a typical
current-voltage characteristic is plotted.
3.5.2
Magnetic Characterization
Magnetic properties of the superconducting films were measured by us�
ing a superconducting quantum interference device (SQUID) at Hanscom Air
Force Base, as well as by using a vibrating sample magnetometer (VSM). In
the SQUID measurements, the SQUID unit was linked via a superconducting
circuit to a pair of counter-wound balanced pick-up coils located outside the
sample. This helped to cancel extraneous magnetization measurements, reduce
instrum ent drift, background noise and consequently increased the signal to
noise ratio. The SQUID had a very high sensitivity (10~n e m u /g m ) of about
two orders higher than VSM?s. In the operation mode, the sample was cycled in
and out between the pick-up coils. The change of the magnetic flux enclosed in
the coils, therefore the change of current in the coils which was detected by the
SQUID, was proportional to the magnetic moment of the sample. The SQUID
produced an output voltage proportional to the current and therefore to the
magnetic moment of the sample.
The sample magnetic moments were measured at different tem peratures
with magnetic fields applied parallel to the c-axis of the sample. After each
measurement was completed at a certain tem perature, the sample was warmed
up above Tc to completely remove the trapped flux inside the sample. Shown in
Fig 3.13 are magnetic moments as a function of magnetic field at tem peratures
of 20K, 40K, 60K, 70K, 77K, and 84K of a YBCO film deposited on an MgO
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54
3 00
I
200
100
100
150
Current (mA)
Fig. 3.12 A typical current-voltage characteristic of YBCO film.
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55
0.4
???
"O'"? ??
?a ~
?
0.3
?r'
0.2
77
60
84
70
40
20
K
K
K
K
K
K
0.0
-
0.1
-
0.2
\
- 0 .3 -
-0 .4
-5 0 0 0
-3 0 0 0
-1 0 0 0
1000
3000
5000
Magnetic field (G)
Fig. 3.13 Magnetic moments of a YBCO film as a function of magnetic field at
tem perature of 20K, 40K, 60K, 70K, 77K, and 84K, (using SQUID
magnetometer).
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56
substrate by laser ablation.
In Fig. 3.14, the magnetic m oment at 77K is
plotted using a vibrating sample magnetometer.
3.5.3
Structure Characterization
The crystal structures of our YBCO films were examined by x-ray diffrac�
tion. A GE XRD-6 diffractometer was used. The wavelength of the Cu ?K a
radiation is 1.542A. Shown in Fig 3.15 is an x-ray diffraction p attern for a laser
ablated YBCO film on a MgO substrate. All the peaks were identified in an
orthorhombic single phase YBCO s tru c tu re ^ , and all the peaks w ith index h
and k disappeared comparing w ith a standard x-ray powder spectrum of or�
thorhombic YBCO p h ased . The (00/) orientation diffraction peaks indicated
that our laser ablated YBCO thin films were oriented. The c-axes were perpen�
dicular to the surface of the films. The twin peaks at 43�, in Fig. 3.16, were
from the MgO substrate. For comparison purposes, a published x-ray spectrum
of a c-axis oriented YBCO film is shown in Fig. 3.16^.
3.6
EX PER IM EN TAL SE T U P
3.6.1
Low Tem perature Apparatus
For electric and microwave measurement at low tem peratures, a CF1204
(or CF1200) Oxford instrum ents continuous flow cryostat was used. The tem �
perature capability of the cryostat was 2.6K to 300K (2K to 300K for CF1200
cryostat).
Shown in Fig. 3.17 is the low tem perature measure and control
arrangement. Liquid helium was drawn in through the innermost tubing of a
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57
0.02
3
s0)
.4J
eV
G
0
S
o
0.00
-
0.02
-600
-400
-2 0 0
0
200
400
600
Magnetic field (G)
Fig. 3.14 Magnetic moment of a YBCO film as a function of magnetic field at
77K, (using VSM).
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58
o
IO
IO
Tt<
o
IO
co
Tt<
o
o
eo
10
OJ
for a laser ablated
o
iO
pattern
10
diffraction
CO
Fig. 3.15 X-ray
o
10
YBCO
film
o
on MgO
CD
CO
o
o
^isua^uj
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
>V)
z
UJ
i0 -
20 .
90.
40 .
50 .
60 .
70.
2e->
Fig. 3.16 X-ray diffraction pattern of orthorhombic structure c-axis oriented
YBCO film, (after Mahajan et al.)
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60
Temperature controller.
Gas flow controller
Arm adjuster nut
Sample access
port_________
10 pin seal___
Exchange
gas valve____
GFS Transfer tube,
Vacuum
valve_______ _
Flow adjuster^-"
'
TTL Transfer tube.
Helium dewai
Fig. 3.17 Low temperature measure and control arrangement.
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61
highly efficient three layered GFS300 transfer tube into the cryostat and the
returned cold gas was sucked by the vacuum created by a GF2 pump through
the second jacket of the transfer tube. The outer jacket of the GFS300 tube was
highly evacuated for low loss. Temperature was measured by using a rhodiumiron therm al sensor for the CF1204 cryostat and an AuFe/chromel therm al
couple with a reference point at liquid nitrogen for the CF1200 cryostat.
An Oxford intelligent electronic tem perature controller ITC4 was used to
control the temperature. For good tem perature stability and a fast cooling rate,
good vacuum in the cryostat jacket was required. A diffusion pum p was utilized
for this purpose achieving a pressure of approximately 5 x 10-5 torr. Fig. 3.18
shows a diagram of the CF1204 cryostat.
The sample chamber of CF1204
cryostat was isolated from the liquid helium flow; therefore, the chamber needs
to be filled with 0.5 torr helium gas for good tem perature control. A carbon
glass thermometer was used to measure tem perature under magnetic fields,
whose magnetoresistance is negligible^.
3.6.2
M icrow ave S e tu p
Fig. 3.19 is a schematic diagram of the microwave experimental setup. The
main part of the setup is a HP 8510C network analyzer. W ith an HP 8516A
S-parameter test set and an HP 85105A Millimeter wave controller, it operates
from 0.045 GHz to 110 GHz. The network analyzer can measure the amplitude
and phase of transmission or reflection coefficients of microwave signal through
a sample or device simultaneously. It can operate in either frequency or time
domain and measure microwave signals as small as -90dB (10-12 watts). Wave
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
rftanga gas vaive
?Top plata
Syphon entry arm.
Access screws
.Return gas
Radiation shield
Helium feed.
Heat exchanger &
temperature sensor.
Sample space
Sample position
Windows
Fig. 3.18 Diagram of the CF1204 cryostat.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
of the copyright ow ner.
HP 8510C
Vector Analyzer
HP Printer
Further reproduction
HP 8350B
oscillator
HP 8516A
S-Parameter Set
0.045-40GHz
HP 85105A
mm-Wave Controller
40-llOGHz
Work
Station
HP 8341B
Sweeper
Variable
Attenuator
prohibited without p e rm issio n .
Dewar
Gaussmeter
Sample
x
Fig. 3.19 Schem atic diagram of the microwave experim ental setup.
cn
co
64
guide or ridged coaxial cable was used to guide the microwave signal to and
from the sample. At the high frequency bands, (26 < / < 110G Hz), we used
stainless steel waveguide, whose inner surface was silver coated, fit into the
cryostat (stainless waveguide has lower therm al loss over copper waveguide).
At lower frequencies, ( / < 26GH z), it is better to use ridged coaxial cable
instead of waveguide to fit into the cryostat. This is because the waveguide at
this frequency band are too big to fit into the cryostat and the therm al loss
will be significant, while the thin coaxial cable introduces less therm al loss at
low tem peratures. Two wave guide-coaxial adapters were used to place the
superconducting sample for microwave self-resonant measurement. A variable
attenuator (0-40 dB) was connected in the circuit to vary the incident microwave
power for low power measurements. An Alpha adjustable electromagnet was
located outside the cryostat to provide variable magnetic fields. Experimen�
tal data can be printed out through an HP printer and can be stored in the
computer.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
[1] H. Jiang, Y. Huang, H. How, S. Zhang, C. Vittoria, A. Widom, D. Chrisey,
J. Horwitz, and R. Lee, Phys. Rev. Lett., 66, 1785, (1991).
[2] D.R. Tilley and J. Tilley, Superfluid and Superconductivity, (Van Nostrand
Reinhold Company, New York,1974)..
[3] S. Shapiro et al., Rev. Mod. Phys., 36, 223, (1964).
[4] A.Miklich, J. Kinston, F. Wellstood, J. Clark, M. Colclough, K. Char, and
G. Zaharchuk, Appl. Phys. Lett., 59, 988, (1991).
[5] R. Cava, B. Batlogg, R. van Dover, D. Murphy, S. Sunshine, T. Siegrist,
J. Remeika, K. Rietman, S. Zahurak, and G. Espinosa, Phys. Rev. Lett.,
58, 1676, (1987).
[6] S. M ahajan, R. Cappelletti, R. Rollins, and D. Ingram, in Superconductivity
and Its Applications, ed. Y.H. Kao et al., (Buffalo, 1990).
[7] H. Jiang, C.G. Cui,and S.Q. Guo, Cryogenics, 27, 90, (1987).
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66
CHAPTER IV
VORTEX MECHANISM
AND I-V CHARACTERISTICS
4.1
INTRODUCTION
Vortex dynamics is of interest from a fundam ental as well as an applied
point of view. There are several models which have discussed the vortex dynam�
ics, and have been referred to in the literature as the flux creep!1-4!, unified the�
ory of vortex pinning and flux creep!5!, collective flux creep!6'7!, vortex glass!8,9!,
vortex phase!10!, 2-D vortex fluctuations!11!, and flux lattice melting!12-14! mod�
els. The basic issue at hand is whether or not therm al energy is the only means
by which vortices are activated or set in motion. It is well known that vor�
tex motion implies an induced electric field and thereby resistance in otherwise
superconducting materials. The threshold currents at which vortices induce
voltages represents a limitation on potential applications of high Tc materials.
In this chapter, we discuss a mechanism for vortex dynamics applicable to
type II superconductors. Specifically, we dem onstrate both theoretically and
experimentally th at vortex motion can also be induced quantum mechanically
for tem peratures well below Tc. This is true for both superconducting films
and extremely narrow bridge constrictions. The flux flow resistivity, p, based
on therm al activation of the vortex ring motion is theoretically given by!15!
p ~ exp(?J i / J ) .
(4-1.1)
Experimentally, we find (at low tem peratures) th at p obeys the following expo�
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67
nential power law,
p ~ e x p [ -( J 2/ J ) 2].
(4.1.2)
We have postulated a quantum tunneling model, in which the radius of
a vortex ring changes from a relatively small value (near the coherence length
�) to i?o, where R q is the threshold radius beyond which vortex rings are free
to expand and induce a voltage drop. The vortex ring quantum nucleation
transition probability is governed by a tunneling probability through an energy
barrier. The resultant resistivity from this model obeys (4.1.2).
In section 4.2 we formulate the model on the basis of a phenomenological
free energy. The free energy consists of the vortex ring self energy, and the
interaction energy of the vortex ring and the driving current (equivalent to the
Lorentz force). The free energy is a maximum at R n =
where r is the
?tension? of the vortex ring viewed as a closed string. Hence, the transition is
defined between values of R below R n, say (R ~ �) and R greater than R n, i.e.
(Ro = 4^j). The WKB m ethod (without normal current dissipation) is used to
calculate the transition probability. In section 4.3 the experimental details are
presented. In section 4.4 the d ata is presented in which the tunneling model is
fitted to the film d ata at low tem peratures and microbridge d a ta over a wide
tem perature range. D ata characteristic of low Jc films are in good agreement
with the exponential linear power law. We are not discussing the values of Ji
and J2 here, since they will be addressed in chapter V, where quantum tunneling
with dissipation is included in the theory. Finally, conclusions are discussed in
section 4.5.
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68
4.2
VORTEX RING NUCLEATION THEORY
In the Ginzburg-Landau model of type II superconductors, the vortex line
can be viewed as a string with tension (energy per unit length), and mass
density.
If there is a straight vortex with its axis (say) along the z-axis, then
the velocity of the electron pairs at position r = (x, y) in the planes normal to
the vortex axis is given by^16!
mvs =
where
m
(4.2.1)
is the mass of electron pair, and
tp
is the phase of G-L wave function.
The kinetic energy of the flow gives rise to an energy per unit length (i.e.
tension) given by
r = ^ n 3m
J
d3r|vs |2,
(4.2.2)
where n s is the number of electron pairs per unit volume. The integral has to
be cut off at the lower end (at �) and at the upper end (at A)
(2 ? fr )i,
(4.2.3)
to get the usual formula for the tension
7T1%Tl o
t
=
m
. A.
�
ln{-r)-
(4.2.4)
'
Rewriting the above equation in terms of more fundam ental parameters,
T = ( i f e )2/nK?
(4'2'5)
where k = j , A is the London penetration depth, � is the vortex core coherence
length (or equivalently the size of superconducting pair) and the magnetic flux
quantum is
~ with q = 2e.
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69
Now let us suppose th at the vortex moves with a velocity V in a direction
normal to the vortex axis. The electric field will then be given by
e
= M
-
(4-2-6>
or equivalently
E = ^V ((V v> )-V ).
(4.2.7)
The magnitude of E is then
|e |2 = ( ^
)2i
q
(428)
r%
so th at the electric field energy stored per unit length of the vortex is
K =
A | E | 2.
(4.2.9)
The integral only needs to be cut off at the lower length �, and leads to
(4.2.10)
The electric field energy can then be viewed as a ?kinetic energy per unit length?
K = ^ V 2,
(4.2.11)
by comparing (4.2.10) and (4.2.11), the mass per unit length of vortex is thent17^
If a current is flowing in a superconducting film (and the film is not ex�
tremely thin), then it is possible th a t vortex rings will be nucleated in the plane
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70
normal to the direction of current flow, see Fig. 4.1. The free energy of the
vortex ring in the presence of an applied current is
U (R ) = 2-k R t ? ? J n R 2,
(4.2.13)
where R is the radius of the vortex ring, J is the current density flowing through
the superconductor. The first term in (4.2.13) is the self-energy of a vortex
ring, while the second term is the interaction energy of vortex ring and applied
current which plays a role equivalent to the Lorentz force. A sketch of U(R) is
shown in Fig. 4.2.
A nucleation radius R n exists for the ring such th at rings of radius R < R n
have not the classical energy to grow ever larger, and rings w ith radius R > R n
find it classically favorable (with respect to energy) to grow ever larger giving
rise to electric fields. R n may be determined by setting
dU(R)
= 0,
dR
(4.2.14)
or
R* =
= I S w ' " ' 5-
(4'2' 15?
The nucleation energy, therefore, is
O
7TCT
Un = U ( R n) = TirRn = ?? .
<PoJ
(4.2.16)
From a classical point of view, only vortex rings w ith radius larger than
R n will be energetically free to expand or break up into vortex lines. However,
the vortex ring with radius R < R n can escape from being pinned by thermal
activation over the energy barrier given in (4.2.13). The therm al nucleation
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71
>
J
>
>
Fig. 4.1 Diagram of vortex ring nucleation.
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72
0.3
u/u0
0.1
TUNELLING
-
0.1
-0.3
0
0.3
0.6
0.9
1.2
R/Rq
Fig. 4.2 Vortex ring free energy as a function of vortex ring radius in
normalized units.
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73
rate is dominated by the activation factor
r = e~Un/ KBT,
(4.2.17)
where K b is the Boltzmann constant. If we denote the critical escape current
density as
Ji =
(4-2-18)
then we have
T=
(4.2.19)
From a quantum mechanical point of view, on the other hand, the radius of
the vortex ring can change from a very small value, say (�), to i?o by a quantum
tunneling process. The transition probability through the energy barrier of Fig.
4.2 may be calculated by using the WKB approximation,
2 ffRo
Ro
P0 = e x p ( - ~ J
y/\2M (E-U (R))\dR),
?S
(4.2.20)
where the mass of the ring is given by
M = 2irRp,
(4.2.21)
and R q is the w idth of the energy barrier,
(4.2.22)
For spontaneous tunneling, we take E = 0, therefore, (4.2.20) becomes
Po = exp[?? j
AttRii{2ttRt ? ? JirR2)dR}.
(4.2.23)
To simplify the integration, we change the lower limit of the integral from � to
0, which only affects the result a little. Hence, (4.2.23) yields
P0 = e- ^ /J)2,
(4.2.24)
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74
where
?,2 = ^ l f f t )1,2(2T5'?)1/4'
(4-2-25)
High Tc superconductors are anisotropic. The London penetration depth
depends upon whether the screening currents move in the a-b plane (A||) or
whether the screening currents flow normal to the a-b plane (Aj_). The anisotropy
param eter 7 = (Aj_/A||) is about a factor of three for the samples here of
interest^18,19]. However, the coherence length anisotropy ratio 7
=
(� ||/� _ l )
inversely proportional to the London penetration depth ratio so th a t
is
A_ll =
A|||. Thus, the quantum tunneling current J 2 does not depend very strongly
on anisotropy effects apart from the Inn factor which varies by 20%. On the
other hand, the thermal activation current J\ depends much more strongly on
anisotropy effects, decreasing by a factor of ~ 7 2, i.e. 10 , since in this case the
coherence length enters weakly into the problem.
The vortex rings nucleated by either therm al activation or quantum tunnel�
ing processes produce electric fields as a current is driven through the supercon�
ductor. The resulting voltage is proportional to the probability of free vortex
ring production. The signature for such effects is a nonlinear current-voltage
characteristic,
E = pJe~{J^ j y ,
(4.2.26)
where p is the normal flux flow resistivity. W hen v = 1, (4.2.26) represents
the therm al activation process while v = 2 represents vortex ring nucleation by
quantum tunneling.
4.3
EXPERIMENTAL DETAILS
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Several Y B a v C u z O j - x films prepared by either laser ablation or ion beam
sputtering were used in our experiments. High quality YBCO thin films were
prepared (using the pulsed laser deposition techniques) with a thickness of
~ 5000A. Films were deposited on polished MgO(lOO) substrates at 750癈, and
then quenched in situ. X-ray diffraction measurements indicated the samples
were c-axis oriented perpendicular to the film plane. Pole figure analysis of
the (0,1,2) peak indicated some misorientation in the a-b plane peaked strongly
at 90� and to a much lesser extent at 45�. The thin films studied had room
tem perature resistivity about 300jiQ. ? c m , and showed sharp superconducting
transitions (A Tc ~ 0.5K ) at Tc ~ 92K. The critical current densities at 77K
were typically from 2 ~ 4 x 106A / c m 2. Disordered low J c films were made
by ion beam sputtering on YSZ(100) substrates. The target was first heated
to 950癈, and then slowly cooled at a rate of 60癈 /h. The sputtering rate
was ~ 2.8 A/sec. All as-deposited films were insulators before post-annealing.
The films were thermally annealed in a furnace tube at an oxygen flow rate of
1.2 liters/m in. The annealing tem perature was increased linearly (360 癈 /h)
from room tem perature to 700癈, held at th a t tem perature for ~40 min, then
increased to 870癈 in 1 h, held at 870癈 for 1 h, and then slowly cooled to
room tem perature in 15 h. For comparison purposes, we chose a sputtered film
with very low J c (103 ~ 104A / c m 2) and broad transition tem perature. Typical
R-T curves of high and low J c films are shown in Fig. 4.3.
The YBCO strips were fabricated w ith widths of 500A to 1mm by pho�
tolithography, chemical etching and ion beam direct milling, as described in
chapter III.
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76
0.8
o
0.6
Low Jc film
High Jc film
0 .4
0.2
0.0
50
75
100
125
150
Temperature (k)
Fig. 4.3 Typical resistance as a function of tem perature for low J c and
high J c films.
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77
Standard four-point probe DC measurements were performed on these
strips. To obtain high quality leads on the films, the films were first cleaned
with methanol, and then a layer of silver bar was evaporated onto the contact
area. Indium soldering was employed to connect silver wires to the silver bar.
Typical contact resistivity was ~ 10-5 D ?cm2 at room tem perature. A Keithley
224 programmable current source was used. A Keithley 182 nanovoltmeter was
employed for voltage measurements, w ith a resolution of ~ 10-9 V. An Oxford
CF-1204 continuous liquid helium flow cryostat was used to cool the samples,
in the tem perature range 2.6K~300K. An Oxford ITC4 tem perature controller
with a Rh-Fe therm al sensor was employed to control the tem perature with an
error < 0.1K.
4.4.
RESULTS AND DISCUSSION
In Fig. 4.4, current-voltage characteristics are plotted for a high J c film
(3 x 106A/cm? at 77K) at tem peratures of 2.6K, 60K and 80K, respectively.
The stars are experimental data, the dashed lines are curves fitted to (4.2.26)
with v = 1 and the solid lines are fitting curves with v = 2. Fig 4.5 shows
the same d ata but plotted in logarithmic scale. The scales for 60K and 80K
are not shown. At low tem peratures, 2.6K, the fit to the experimental d ata
is vastly improved, if v = 2 instead of v = 1. In the fitting process, we also
used the combination of two models
A/-02) to fit the data.
The confidence level was highest when A ~ 0 and R ~ 1 at low temperatures.
At 80K the v = 1 scaling is in b etter agreement with experimental data. We
interpret the d a ta in term s of a quantum tunneling process at low tem peratures.
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W hen the tem perature is near Tc, the therm al energy of vortex rings increases
sufficiently to overcome the energy barrier. Therefore, the therm al activation
process appears to be dominant at high temperatures.
The I-V characteristics were measured on a microbridge w ith a w idth of
oOOA. The critical current density of this bridge was 3.6 x 108A /cm 2 at 77K.
Experimental d ata and fitting curves at 77K , 85K and 88.5K are shown in Fig.
4.6 to Fig. 4.9. In Fig. 4.6, the I-V curve obtained near the critical current
and the experimental points fit the v = 2 scaling quite well. In Fig.
4.7,
the InV versus current at 77K are shown w ith fitting curves. The dashed line
corresponds to v ? 1 scaling, while the solid line has v = 2. Clearly, at 77K the
quantum nucleation of vortex rings is still playing a major role in microbridges.
This is not very surprising to us. First of all, J c of the microbridge is extremely
high (more than two orders higher th an th at of films). This implies a very
small vortex ring (R < lOOA). The size of the ring is in the same order of the
bridge width, therefore, the edge pinning is very important. If we consider the
contribution of edge pinning, the effective energy barrier will be very high and
narrow, and the vortex ring would require additional energy beyond (4.2.13) to
overcome the barrier by therm al activation. Quantum tunneling through the
barrier allows the vortices to move. At 85K, both models approximately fit
the experimental d ata (see Fig. 4.8), but the agreement is poor at T=88.5K,
see plots in Fig. 4.9. The models here considered are not valid near Tc, since
fluctuations near the critical tem perature have not been taken into account.
For sputtered films characterized by low J c ( J c = 5 x 103A /cm 2 at 77K),
the v = 1 scaling agrees with experimental d ata at all tem peratures below Tc.
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79
15 ?
?
- -
E x p e r im e n t a l D a t a
V = [ R e - J: / J
VolLage
(fi V)
V = [R e-^JJ)2
80K
60K
100
200
2 .6 k
300
400
500
600
700
Current (mA)
Fig. 4.4 I-V curves of high J c films at tem peratures of 2.6K, 60K and 80K.
The stars are experim ental data, dashed lines axe fitting curve with
v ? 1 scaling and solid lines are fitting curves with v ? 2 scaling.
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80
+ Experimental Data
- V = I Re ~ Ji l J
? V = IRe-V*
60K
2.6K
-2
300
400
500
600
700
Current (mA)
Fig. 4.5 I-lnV curves of high J c films at temperatures of 2.6K, 60K and
80K. The stars are experimental data, dashed lines are fitting
curve with u = 1 scaling and solid lines are fitting curves with
v = 2 scaling. The scales for 60K and 80K are not shown.
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81
Voltage
(/xV)
4
3
2
1
0
60
70
80
90
100
Current (mA)
Fig. 4.6 Current-voltage characteristics of a microbridge at 77K near the crit�
ical current.
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82
+ Experimental Data
-
- ~ V = l R e - Ji/J
V = IRe-(J*/Jf
I
77K
-2
60
80
100
120
140
Current (mA)
Fig. 4.7 Current-Voltage characteristics of a microbridge at 77K plotted
in I-lnV scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<1
s
U
Sh
o
O
O
O
o
CO
o
o
o
(yirf) 93'B^OA
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 4.8 The I-V data and fitting curves of a microbridge at 85K.
83
Reproduced
with permission
of the copyright owner.
150
Experim ental D ata
V = I R e ~ Ji f J
=1
100
V = IRe-V.2/J)2
Further reproduction
faO
-+-3
r-H
50
prohibited without p e r m is s io n .
0.5
1
1.5
2
2.5
Current (m A)
Fig. 4.9 The I-V data and fitting curves of a microbridge at 88.5K.
00
Reproduced
with permission
t
i
I
i
|
i
i
i
I
|
i
i
i
I
I
i
i
i? i? |? i? i? i? i? i? r
of the copyright owner.
Experim ental D ata
V = I
V = I R e- ( Jo/J)a ,
Further reproduction
prohibited without p e r m is s io n .
0
0
0.2
0.4
0.6
0.8
Current (m A)
Fig. 4.10 Low J c film I-V characteristics at 3K. T he reader is referred to the scale
on the right side of the figure for logarithm ic scale.
1
86
The results at 3K are plotted in both I-V and I-lnV scale in Fig. 4.10. Again
solid lines are fitted with u = 2 and dashed lines are fitted w ith v = 1. The
logarithmic scale is shown on the right side of the figure. For the low Jc films,
the vortex ring model is inappropriate. We can estimate the radius of the ring
by using (4.2.25), where we chose k ~ 100 and A ~ 1500A, to be in the order
of a few thousand microns, which is much bigger th an the thickness of the
film (5000A ). Hence, this can be viewed as a vortex-antivortex pair excitation.
At low tem peratures films characterized by low J c also exhibit u = 1 scaling.
The tunneling model can be made to fit to a v = 1 scaling law for vortexantivortex nucleation^20! (instead of vortex rings) where the therm al activation
model predicts scaling. Hence, at low tem peratures and for low Jc films it is
inconclusive to choose one model over the other. For films characterized by
high J c the d ata only fit a v = 2 scaling law at low tem peratures, which our
tunneling model predicts.
4.5
CONCLUSION
We have dem onstrated th at the vortex dynamics could be either thermally
activated or quantum mechanically induced15^. In high J c superconducting
films, the quantum nucleation of vortex rings plays a m ajor role when the tem�
perature is substantially below Tc. In high J c microbridges, quantum tunneling
of vortex rings occurs even at high temperatures (about 10K below Tc). For low
Jc superconducting films, the quantum tunneling or therm al activation models
may be both invoked to explain our experimental results at low temperatures.
We have not included the normal current dissipation in our calculation of the
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87
tunneling processing, which would tend to increase the tunneling rates.
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88
REFERENCES
[1] P.W. Anderson and Y.B. Kim, Rev. Mod. Phys., 36, 39, (1964).
[2] M.R. Beasley, R. Labusch and W .W . Webb, Phys. Rev. 181, 682, (1969).
[3] T.T.M . Palstra, B.Batlogg, R.B. Van Donver, L.F. Schneemeyer and J.V.
Waszczak, Appl. Phys. Lett., 54, 763, (1989).
[4] S. M artin, A.T. Fiory, R.M. Fleming, G.P. Espinosa and A.S. Cooper,
Phys. Rev. Lett., 62, 677, (1989).
[5] M. Coffey, J. Clem, Phys. Rev. Lett., 67, 386, (1991).
[6] P.G. De Gennes and J. Matricon, Rev. Mod. Phys. 36, 45, (1964).
[7] M.V. Feigel?man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Phys.
Rev. Lett. 63,2303, (1989).
[8] M.P.A. Fisher, Phys. Rev. Lett. 62, 1415, (1989).
[9] R.H. Koch, V. Fogliette, V.J. Gallagher, G. Koren, A. G upta and M.P.A.
Fisher, 63, 1511, (1989).
[10] N.C. Yeh, Phys. Rev. B, 40, 4566,(1989).
[11] H.J. Jensen and P. Minnhagen, Phys. Rev. Lett. 66, 1630, (1991).
[12] R.S. Markiewicz, Physica C, 171, 479, (1990).
[13] D.R. Nelson and H.S. Seung, Phys. Rev. B , 39, 9153, (1989).
[14] A. Houghton, R.A. Pelcovits and A. Sudbo, Phys.
Rev.
B, 40, 6763,
(1989).
[15] H. Jiang, A. Widom, Y. Huang, T. Yuan, C. V ittoria, D.B. Chrisey, and
J.S. Horwitz, Phys. Rev. B, 45, 3048, (1992).
[16] E.M. Lifshitz and Pitaevskii, Statistical Physics, p art II, (Pergamon Press,
Oxford, 1980).
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89
[17] A. Widom, Y.N. Srivastava, C. Vittoria, H. How, R. Karim, and H. Jiang,
Phys. Rev. B, 46, xxxx, (1992).
[18] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, J. Appl. Phys.,
(1993).
[19] A. Widom, T. Yuan, H. Jiang, and C. Vittoria, subm itted to Phys. Rev.
B.
[20] J. H. Miller, private communication.
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90
CHAPTER V
INTRINSIC CRITICAL CURRENT DENSITIES
5.1
INTRODUCTION
Current transport in superconductors has long been one of the most im�
portant issues both from fundam ental and applied points of view. The critical
current density of a superconductor is defined as the highest current density
that the m aterial can carry w ithout loss. In type I superconductors the crit�
ical current arises when the total magnetic field due to the transport current
exceeds the critical field of the superconductor. Since the critical fields of type
I superconductors are very low, the critical current densities, in turn, are in�
trinsically low. In type II superconductors, however, besides the higher critical
tem peratures, the critical fields and critical current densities are much higher
than those of type I superconductors. Although the mechanism of the critical
current densities of type II superconductors are not yet fully understood at the
microscopic level, it has been known th at the critical current density is very
sensitive to the m icrostructure of the materials. Flux lines in a superconductor
will move due to a Lorentz force when current passes through a superconductor,
and their motion will generate an electric field. Hence, the m aterial will lose
its superconductivity. Pinning centers, due to defects, will prevent the motion
of a flux line. The critical current density, thus, is strongly dependent on the
pinning strength.
The newly discovered high Tc superconductors are type II superconductors.
The extremely small coherence length (of the order of 10A) implies a very high
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91
upper critical field H c2 (more than 100T at 4K and more th an 30T at 77K for
YBCO, for example), and a large G-L parameter k (in the order of 100) gives
a big depairing current density, which will be discussed in detail in Section 5.2.
Bulk high-Tc superconductors have very low critical current densities, and the
critical current densities strongly depend on the external magnetic field. By
applying a field, for example, the critical current densities drop significantly in
bulk materials. This is a strong evidence for the existence of weak links in the
material. The measured critical current densities are a collection of Josephson current densities through the grain boundaries. It has been observed that,
the intergranular critical current density is much smaller th an an intragranular
oneW. On the other hand, high critical current densities measured on epitaxial
filmst2-8! dem onstrated th a t the low values obtained in poly crystalline samples
were not intrinsic to this class of materials. The structural anisotropy is another
issue affecting the electrical and magnetic p ro p e rtie s^ . In particular, the crit�
ical current density along the c-axis is almost three orders of magnitude lower
than the one measured in the a-b planes (C u-0 planes) for B i S r C a C u O t10l and
one order for YBCO.
The critical current density of type II superconductors is an extrinsic
property, since it is affected by many external factors. Several models have
discussed the critical current density in terms of flux creep^11-13), interplane
coupling^14?15!, edge barrier^16!, and flux lattice melting^17-19}.
In this chapter, we propose a vortex ring model to discuss the critical cur�
rent densities. The depairing current density is an intrinsic current density
which separates superconducting pairs into single charge carriers. It is inde�
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pendent of m aterial structure and extrinsic parameters. The critical current
density of a superconductor is not necessarily the same as the depairing cur�
rent density. Actually, it is usually much smaller th an the depairing current
density since it is governed by flux flow in the superconductor. From vortex
ring nucleation by a quantum tunneling process, we have derived the upper
limit of critical current density of YBCO films and compared it with reported
experimental results. There is resonable agreement between our predictions
and measurements. In addition, we have studied critical current density as a
function of strip width (L ef f ) in the regime of � < L e/ / < A, where � is the co�
herence length and A is the London penetration depth. We find th a t the critical
current density is inversely proportional to the width L ef f . O ur experimental
results are consistent with a theoretical calculation based upon the theory of
Onsager and Feynman. It should be pointed out that, although the discussions
are concentrated on YBCO films, the model used here is valid for all type II
superconductors.
5.2
DEPAIRING CURRENT DENSITY
Before the vortex ring model is used to deduce the depairing current den�
sity, we use London theory to derive it. The definition of depairing current
density is th at value of current density for which the local kinetic energy density
of the paired charge carriers equals the condensation energy density, expressed
in terms of the thermodynamic critical field, H c,
1
2
2
1
=
71 87
?
d O 11
( 5 '2 ' 1 )
where m and n s are the mass and density of superconducting pairs, respectively,
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93
Vd is the maximum velocity a pair can ever have, beyond which the pair will be
depaired. The depairing velocity is
= S
r
( 5 -2 ' 2 )
The depairing current density is therefore
Jd. = nsqvd.
(5.2.3)
The London penetration depth can be w ritten in term of n 3
,
/
m
c
,
A= V w
(6-2-4)
Putting (5.2.2) and (5.2.4) into (5.2.3) and eliminating n s, we get
J ? = -Ct y
^
Writing (5.2.5) in terms of more fundamental parameters we have
* = iv fe v
( 5 -2 -6 )
and
t
<t>oc
_ 8v ^ 7r2^A2
CK
8>/5^ A � ?
1
J
G-L theory!20] contains also this relation except for a factor of -y/8/27.
Next, the vortex ring nucleation model is used to deduce the depairing
current density. As described in previous chapter, If a current is flowing in a
superconducting film, vortex rings will be nucleated in the plane normal to the
direction of current flow. The vortex ring is the initial stage of flux inside the
superconductor when a current pass through without external magnetic field.
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94
A nucleation radius R n exists such that rings of radius R < R n do not have
sufficient energy to grow larger (although there is a possibility to grow larger by
quantum tunneling through the energy barrier, discussed in chapter IV and, in
more detail, in the next section), and rings with radius R > R n find it classically
favorable to grow larger. We call the ring w ith radius R > R n a free vortex
ring. Free vortex ring m otion will give rise to an electric field, and therefore
resistance, in the superconductor. The nucleation radius is
= IS&7'"*-
<5'2-8>
The nucleation radius cannot be less th an the size of the vortex core (i.e.
the thickness of the vortex ring). By demanding th at i?n/� be at least unity in
(5.2.8), we obtain an absolute maximum current density flow in the supercon�
ducting film,
CtbnK
Ji =
(5-2-9 )
Since the size of the vortex core is equal to �, the size of superconducting pair,
(5.2.9) is, therefore, defined as depairing current density. This is the same as
(5.2.7) deduced from London theory except for a factor of
Putting A = 1000A and k = 100, which are appropriate for YBCO films,
the depairing current density turns out to be 6 x 109A /cm 2.
5.3
U P P E R LIMIT OF Jc FOR YBCO FILM
From a quantum mechanical point of view, the radius of a vortex ring can
change from a very sm all value, (say, �, th e size of vortex core) to R q by a
quantum tunnelling process. In this section, we consider a small vortex ring
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95
which ?grows? into a free vortex ring by tunneling through the energy barrier
given in (4.2.13). The transition probability, or the quantum production rate,
may be calculated by using WKB approximation^21! to reasonable accuracy (but
without normal current dissipation)
PQ = e x p ( -U n/hQ 0).
(5.3.1)
The mass of the ring with R = R n is given by
M? = 2irRnp,
(5.3.2)
which via (5.2.9) and p = ( ^ ^ ) 2 reads
=
S(T
<5 ' 3 ' 3 >
The ratio of the nucleation energy, or the height of the energy barrier Un =
U (R n)
Un = U (R n) = TTTil n =
to the energy
167rA
(5.3.4)
J
k
where
fto = y jU " (r )/M n =
J
A
(5.3.5)
Jd
gives the quantum tunneling factor for the potential barrier maximum of ( 5 .3 .4 ).
Therefore,
p =
)2)-
(5-3-6)
This free vortex ring production rate leads to a nonlinear current-voltage char�
acteristic reported elsewhere!22!.
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It is well known that the resistivity of a superconductor below the transi�
tion tem perature equals zero. However, the normal resistivity of a supercon�
ductor below Tc is not necessarily zero. A superconductor can be treated as two
conductors in parallel. Below Tc one conductor has zero resistance, the super�
current short circuits the normal conductor. Therefore, the normal resistivity
is no longer observed. It is reasonable to estimate the normal resistivity below
Tc from the extrapolation of resistivity above Tc.
At tem perature well below Tc, the tunneling rate is given by (5.3.6). At
relatively high tem perature, but still below Tc, we introduce normal current
dissipation in the tunneling. The ?damped harmonic oscillator? in the course
of its motion will decay in amplitude. This is conventionally described by a
complex frequency <� which obeys the equation
C2 + *7C - wo = 0,
(5-3.7)
where loq is the resonant frequency and 7 is the damping strength. For the prob�
lem at hand we do not have an oscillator represented by a parabolic potential.
W hat we do have is an ?inverted? parabolic potential which can be represented
by an imaginary frequency (wo ???i0) yielding a damping renormalization
(( ???iti) which obeys (5.3.7) in the form
Q2 - 7 ft - a 20 = 0.
(5.3.8)
W ith normal current dissipation described by 7 = 47r<7 n, (5.3.8) yields the
solution
= 27r<7? + a/(27r<rn )2 +
S l20 .
(5.3.9)
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97
where an is the normal conductivity of the superconductor. Thus, the quantum
nucleation probability with normal current dissipation is now enhanced
P = eXp(^ ~ M ^ '
(5.3.10)
The enhancement is quite large in the regime of strong dam ping a n �
in
which
P = exp(-y ) ,
(5.3.11)
where
猾*<*? .
is the critical current density of high Tc superconductors.
��? )
In fact, (5.3.12)
does not take defect, grain boundary, strong anisotropy and inter-layer effects
into account. The Jc calculated here is the intrinsic critical current density
resulting from vortex ring nucleation by quantum tunneling, or the upper limit
of critical current density. In ceramic samples of YBCO the grain-boundaries
are dominant. Therefore, the critical current is the to tal Josephson current
which is significantly smaller than the intrinsic critical current.
For YBCO films, the normal conductivity, <r?, at 77K is in the order of
104 (f7cm ) -1 while flo is in the order of 102 (flcm )- 1 . Thus, the strong damping
assumption is satisfied. Equation (5.3.12) may be used to estim ate the upper
limit of Jc for a YBCO film at 77K. The London penetration depth A is temper�
ature dependent. For best YBCO film, \( 7 7 K ) is around 1500A, and k = 100.
The intrinsic critical current densities turn out to be 2 x 107A /cm 2 at 77K.
Experimentally, P. Vase et alW reported Jc = 8 x 106/ c m 2 for a YBCO
film at 77K, we have measured J c in the range of 106 ~ 107A / c m 2, and many
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98
groups^5-8] have measured J c to be greater th an 106 A /cm 2 at 77K. These re�
sults are very close to the upper limit predicted by (5.3.12). This implies th a t
grain boundaries or defects are not dominant in films in comparison to bulk
samples.
5.4
M IC RO BR ID G E W ID T H D E P E N D E N C E OF J c
The close analogy between superconductor and super-fluid flow allows one
to carry over many results from one problem to the other. The system of in�
terest in this section is two ?bulk superconductors? (viewed as reservoirs of
superconducting pairs), connected by a narrow bridge (viewed as a tube which
superconducting pairs travel through), see Fig. 5.1. The bridge has a w idth
(effective length scale of the tu b e?s cross section) in the regime � < L ef f < A,
where � is the coherence length and A is the penetration depth of superconduc�
tor.
We consider the Onsager-Feynman superfluid mechanism for critical veloc�
ities of superconducting pair flow. In this model, we assume th a t if there is
not enough kinetic energy in the fluid to nucleate a vortex, no resistance will
appear. Once the vortices can be formed they are ultim ately dissipated as heat.
Most probably vortices will be created at the jet of the tube if the velocity of
the flow is sufficiently high. However, it is not necessary th at all the loss occurs
at the exit end of the tube. The wall of the pipe are irregular, so a vortex
may be created inside the tube also. There exists a critical super-fluid velocity
beyond which dissipation will appear.
In the case of a superconductor, the critical velocity of superconducting
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99
Fig. 5.1 Schematic diagram of a bridge constriction.
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flow is due to the creation of vortex rings, and is given ast23^ (the same manner
as a superfluid)
(5.4.1)
where m is the mass of superconducting pair and L ef f is the effective length
scale of the microbridge cross section. The critical current density is
J c ? QY13Vc
(5.4.2)
where n s is the density of superconducting pairs. The London penetration
depth A is determined in (5.2.4) as A = J
so th at (5.4.2) reads
(5.4.3)
The critical current density determined this way depends on the usual two
characteristic lengths, but it also varies with the width of the bridge. If the
bridge is very narrow the vortex ring will be attracted by the edge of the bridge
(wall of the tube, in the case of superfluid), it can never get away from the
wall. Even if started somehow, it will fall back into the wall unless the current
density suffices to keep it in the flow. Therefore, the smallest bridge has the
highest critical current density. The extreme limit is if the vortex ring is not
created bu t reaches a critical current value. That value will be the absolute
maximum current density, the depairing current density.
Likharev^24} theoretically derived the microstrip width dependence of crit�
ical current from G-L theory. He obtained the following result.
(5.4.4)
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101
where
Aj_ =
----- ?
(5.4.5)
e//
If we treat the cross section of the strip as L ^ , and use <j>o =
the critical
current density will be
Jc
1
ftc2 , . L ef f .
= 2n q \ 2L e ff
~4
,
(5-4?6)
which is the same as (5.4.3) that we deduced from superfluid model, except for
a prefactor of | and a factor of \ in the logarithm function.
If we chose A = 1000A, � = 10A (for Y B C O film),(5.4.3)(deduced from
superfluid theory) gives J c 108A /cm 2 for L ef f > 500A
for
L ef f
and J c ~
109A /cm 2
< 500A.
We have fabricated narrow bridge constrictions on high quality Y B C O
films by ion-beam-milling techniques. The smallest bridge has w idth L = 500A.
In total three microbridges were fabricated with the following cross sections:
(a) L =4000A , t= 5000A ; (b) L = 500A , t=5000A ; (c) L =500A , t= 1000A . where
t is the film thickness. The critical current densities measured on the three
microbridges at 77K were 5 x 107, 3.6 x 108, and 1.3 x 109A /cm 2, respectively.
They are in good agreement with our theoretical model and will be addressed
in more detail in next section. Tahara et alM6^ reported the enhancement of
critical current density of YBCO microbridges. When they reduced the width of
the microbridge to 8500A, they found that the J c increased to 2.4
X
10 7A /cm 2,
which is significantly higher than th a t of films.
Zitkovsky et alS25^ observed J c of a B iS rC a C u O microbridge fabricated
by ion beam milling to be much higher than th at of planar films. They reported
that Jc of a microbridge with w idth of 8000A at 8 K is more than 106A /cm 2
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102
which is about 2 orders of m agnitude higher than th at of planar films. Gavaler
et alJ26I reported high J c results on N b N strip constrictions. On a N b N strip
with width of 600A, they observed J c
>
3 x 107A /cm 2, compared to bulk
J c ~ 5 x 105A / c m 2.
Let us use (5.4.3) to calculate J c to compare with the experimental results.
The London penetration depth A of N b N film at 4.2K is about 2700A while
�(4.2) ~ 32A. Taking L ef f = 600A, (5.4.3) yields <7C ~ 7 x 107^4/cm2, which is
in very good agreement w ith Gavaler et aVs experimental result. The London
penetration depth and coherence length of a B iS rC a C u O film at 8K are A ~
2500A and � ~ 27A, respectively. Taking L ef f = 8000A, (5.4.3) gives J c ~
1.2 x 107A / c m 2 (compared to J c > 106 of Zitkovsky et aVs result). Taking the
same characteristic lengths of YBCO as before, using L e/ / ~ 8500A, (5.4.3)
yields J c ~ 8.3 x 107A / c m 2 (compared to Tahara et a V s measurement of J c ~
2.4 x 107A /cm 2). The agreement is good, but the discrepancies are obvious.
This is not surprising since the widths of the bridges in the latter cases are big,
L ef f > A, which is beyond the range of validity of our formula for J c.
Macroscopically, critical current density is independent of the cross section
normal to the current flow. We have shown, however, th a t this is no longer true
if the cross section is extremely small, say, when the width of a microbridge
L ef f is in the regime � < L ef f < A. In this regime, the critical current density
is inversely proportional to the w idth of the microbridge.
5.5
OBSERVATION OF ULTRAH IG H J c
IN YBCO BR ID G E C O N STR IC TIO N S
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103
The study of critical current densities in restricted geometries for high T c
superconductors is both of technical importance (for the fabrication of super�
conducting devices) and of intrinsic physical interest (for the understanding of
superconducting transport through the material). A particularly interesting
geometry consists of a narrow ?superconducting bridge? of width L connecting
two ?bulk superconductors? . The natural length scales of the bulk supercon�
ductor are the London penetration depth A and the coherence length � (roughly
the size of an electron pair or equivalently the size of the core of a supercon�
ducting vortex). High T c superconductors, e.g. the Y B C O films of interest in
this work, have large values of
Recently, S. Tahara et
k
= A/�.
reported critical current densities for super�
conducting bridges, fabricated from Y B C O films, having widths in the range
L > A , and found that the critical current density J c increased as L is de�
creased. Their highest value was J c
2.4 x 101A / c m 2, and they theoretically
predicted th at an order of magnitude higher J c could be achieved if L were
brought down significantly below A. In this section, we experimentally confirm
this theoretical prediction. Other experimental groups discussing the problems
of achieving high critical current densities include B. Oh et al\21\ J. M annhart
et a P s\ B. Roas et alS291 and P. Chaudhari et a/.W for Y B C O , I. Zitkovsky et
a
l in B S C C O films and J.R. Gavaler et alS2^ in N b N films.
We report ultra-high critical current densities, with values up to Jc ~
1.3 x 109A / cm2, for Y B C O microbridge constrictions studied in the regime
� -C L < A. The large critical current densities are consistent with the OnsagerFeynman vortex ring nucleation model as a limiting factor for superconducting
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104
flow and the highest critical current density is also consistent with the critical
depairing mechanism.
We fabricated several superconducting microbridges (using ion beam milling)
from L ~ 500 A to L ~ 4000 A with accuracy better than 10%. The milling was
done in several stages, between which measurements were made to assure that
(at each stage of milling) the bridge retained its superconducting integrity.
Shown in Chapter III Fig. 3.3a is an L ~ 4000A microbridge, and shown in
Fig. 3.3b is an L ~ 500A microbridge. In total three microbridge junctions
were fabricated with the following cross sections: (a) L = 4000A , t=5000A ; (b)
L =500A , t=5000A ; (c) L =500A , t=2000A, where t is the film thickness.
From the current-voltage curve in Fig. 5.2, one can see th at the voltage
increase rapidly when the current change a little around the critical current.
We have tested that, by changing the electric field criterion one order, (say,
from lOOnV to liW ), the critical current density will be differed within 10%.
However, since the w idth of the bridges we investigated are extremely small,
the electrical field criterion for critical current density is improper. Instead, we
use the resistivity criterion,
V A
p= yy,
(5.5.1)
where V is the voltage between the electrical probes, I is the current passing
through the bridge, A is the bridge cross section, and I is the length of the
microbridge. For the smallest bridge, we used a criterion of p = 10-13ficm.
Figure 5.3 shows the tem perature dependent resistance of microbridge junc�
tion (a), (b) and (c), indicating the transition to the superconducting state. In
Fig. 5.4 critical current densities, J c, as a function of T are plotted for the three
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105
250
200
3
150
<D
bJO
cd
100
1 .5
0 .5
J /J c
Fig. 5.2 Current-voltage characteristics of a microbridge taken at 77K.
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106
25
15
r
(n)
20
10
5
0
80
90
100
110
120
130
Tem perature (K)
Fig. 5.3 Tem perature dependence of resistance of three microbridges.
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107
'I ?
125
12.5
Bridge (a)
(M
Bridge (b)
100
10.0
Bridge (c)
a
o
7.5
i?I
50
5.0
25
2.5
+?!>... *
...
80
90
0.0
Tem perature (K)
Fig. 5.4 Critical current densities as a function of tem perature for the three
microbridges. The scale for bridge (a) is on the right side of the
figure.
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108
microbridge junctions. Finally, in Fig. 5.5 typical critical current densities as
a function of tem perature are plotted for different magnetic fields. The other
junctions exhibit the same behavior. The critical current densities measured on
(a), (b) and (c) microbridges at 77 K were 5 x 107A /c m 2, 3.6 x 108A / c m 2 and
1.3 x 109.A/cm2, respectively. In arriving at the current density, we divided the
total current by the geometrical cross sectional area of the microbridge (L x t)
with no assumptions made about the distribution of the current.
To understand the large magnitude of critical current density we examined
several theoretical models, each of which had some partial success, (i) The flux
creep model gives a reasonable tem perature dependence for J c as the critical
tem perature is approached, but the electric field dependence of current density
was not in very good agreement with the notion of a pinning therm al activation
energy. In order for a flux pinning therm al activation energy to explain the
observed current-voltage characteristics, the activation energy itself would have
to depend on the square of the current density. Presently, this is beyond the
scope of the proposed flux pinning models, (ii) It is difficult to separate out the
contributions to the critical current due to edge barrier effects as opposed to
bulk effects, (iii) The critical ?depairing? current density (discussed in section
5.2, Jd = 16�2A2^ n/c) ves < A ~ 6 x 10 9A /cm 2 for A ~ 1000A and � ~ 10A in
good agreement with our highest critical current density sample (with smallest
cross sectional area), which indicated th at our sample with smallest cross section
was approaching the absolute value, the depairing current density. However,
this leaves unexplained the somewhat lower values also observed in the thicker
samples, (iv) The critical current density deduced from the Onsager-Feynman
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109
40
?
H=0 G
H=600 G
H=3000 G
30
(N
O
hO
r-H
-*
20
77.5
80
82.5
85
87.5
90
Temperature (K)
Fig. 5.5 Critical current densities as a function of tem perature in different
magnetic fields for bridge (b).
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110
mechanism for critical velocity of a superfluid (see section 5.4) explains our data
better. This theory gives J c ~ 108A / c m 2 for Le/ / > 500A and J c ~ 109A /c m 2
for L ef f < 500A. This is in reasonable agreement with our results. The effective
length scale is reduced whenever the microbridge cross sectional area is made
smaller.
By careful fabrication of superconducting microbridges, we have shown
that it is possible to obtain (in YBCO) critical current densities ~ 109A / c m 2,
(but still below the depairing current density). The limitations on the critical
current densities are not at present fully understood, but appear to be due to
vortex ring nucleation.
5.6. CONCLUSION
By using a vortex ring nucleation model, we have shown th a t the depairing
current density in Y B C O superconductor is about 6 x 109A / c m 2. The intrinsic
(or upper limit of) critical current density of Y B C O films is of the order of
101A /c m 2 at 77K. Finally, the critical current density is increased significantly
when the bridge w idth is decreased to the regime of � < L ef f < A. When L ef f
o iY B C O film bridges are reduced to 500A, J c can be as high as 109A /cm 2, but
still smaller than the depairing current density. A summary of our theoretical
predictions and experimental results as well as other?s are listed in table 5.1.
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Ill
Table 5.1 Summary of intrinsic critical current densities.
Theory
Experiment;
Prediction |
Depairing Current Density
J
ii
i
6
r. _
,
4 16jr*A2
x
10*A / c m - j
A ~ 1000A
:
K~ 100
5K
NHL. Washington DC
> 10 7A / c m 2
SRL. ISTC. Japan
i
77K
Critical Current Density
j _
8 X 106 A /crn2
Y B C O / S r T i O t , Laser
P. Vas� et al., NKT. Danmark,
P hysicaC 90 180(1991)
c<b2ln2K
2.3 xlO7A / c m 2
4<7n/i(47rA)4
6 X 106 A / c m 2
A ~ 1500A
K
Y B C O / L a A l O z , Laser
D. Chrisey et al., NRL. Washington
~ 100
<7~ 104(ncmi-1
4.6 x 106A / c m 2
YBCO/YSZ/Si
D.K. Fork et al. Xerox.
j
Appi. Phys. Lett.. 58 2432(1991) I
4
106A / c m 2
X
Y B C O / L a A I O z , L10CVD
R. Hiskes et ai.. Hewlett Packard
Appl. Phys. Lett.
YBCO
1.3 x 109A / c m 2
Size dependent J r
A ~ 1000A. ( ~ 10A
Lef f
he2
Jc = ?q \ 2 L ( j f
M
$
1.3
X
109-4/cm 2
oOOA
NbN
1 x 108A /cm 2
A ~ 2700A. � ~ 32A
L. j i ~ 6 0 0 A
J*anS et a'- Northeastern
Phys. Rev. Lett. 66 1785(1991)
> 3
X
10 A / C m
J.R . Gavaler et al., Westinghous
IEEE Tran. Mag. 17 573(1981)
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112
REFERENCES
[1] P. Chaudhari, D. Dimos, J. M annhart. IB M J. Res. Develop., 33, 299,
(1989).
[2] P. Vase, Y. Shen, and T. Freltoft, Physica C, 180, 90, (1991).
[3] H. Jiang, Y. Huang, H. How, S. Zhang, C. Vittoria, A. Widom, D. Chrisey,
J. Horwitz, and R. Lee, Phys. Rev. Lett., 66, 1785, (1991).
[4] D. Fork, F. Ponce, J. Tramontana, N. Newman, J. Phillips, and T. Geballe,
Appl. Phys. Lett., 58, 2432, (1991).
[5] G. Jakob, P. Przyslupski, C. Slolzel, C. Tome-Rosa, A. Walenhorst, M.
Schmitt, and H. Adrian, Appl. Phys. Lett., 59, 1626, (1991).
[6] H. Jiang, A. W idom, Y. Huang, T. Yuan, and C. Vittoria, IEEE. Tran.
Mag., (1992).
[7] F.M. Sauerzopt, H.P. Wiesinger, H.W. Weber, G.W. Crabtree, and J.Z.
Liu, Physica C, 161, 751, (1989).
[8] R.B. Van Dover, E.M. Gyorgy, L.F. Schneemeyer, J.W . Mitchell, K.V. Rao,
R. Puzniak and J.V. Waszczak, Nature, 431, 55, (1989).
[9] T. W orthington, W. Gallagher, and T. Dinger, Phys. Rev. Lett. 59, 1160,
(1987).
[10] S. M artin, A. Fiory, R. Fleming, G. Espinosa, and A. Cooper, Appl. Phys.
Lett., 54, 72, (1989).
[11] M. Tinkham, Introduction to Superconductivity, (McGraw-Hill; New York,
1975).
[12] M. Beasley, R. Labusch, and, W. Webb, Phys. Rev., 181, 682, (1969).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
[13] T. Palstra, B. Batlogg, R. Van Dover, L. Schneemeyer, and J. Waszczak,
Appl. Phys. Lett., 54, 763, (1989).
[14] H. Jensen, P. Minnhagen, Phys. Rev. Lett., 66, 1630, (1991).
[15] L. Glazman and A. Koshelev, Sov. Phys. JETP, 70, 774, (1990).
[16] S. Tahara, S. Anlage, J. H albritter, C. Eon, D. Fork, T. Geballe, and M.
Beasley, Phys. Rev. B, 41, 11203, (1990).
[17] R.S. Markiewicz, Physica C, 171, 479, (1990).
[18]
D.R. Nelson and H.S. Seung, Phys. Rev. B, 39, 9153, (1989).
[19]
A. Houghton, R.A. Pelcovits and A. Sudbo, Phys.
Rev. B, 40,
6763,
(1989).
[20] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2, (Pergamon
Press, Oxford, 1980).
[21] A. Widom and T. Clark, Phys. Rev. Lett., 48, 63, (1982).
[22] H. Jiang, A. Widom, Y. Huang, T. Yuan, C. V ittoria, D. Chrisey, and J.
Horwitz, Phys. Rev. B 45, 3048, (1992).
[23] R. Feynman, in Progress in Low-Temp. Phys., edited by C. Gorter, (NorthHolland, Amsterdam, 1955).
[24] K: K. JT h x a p e n. H3n. nucui. yi. 3ao.?Pajino(jiii3iiKa, 14, ,Nb fi, !)19. (1071).
[25] I. Zitkovsky, Q. Hu, T. Orlando, J. Melngailis, and T. Tao, Appl. Phys.
Lett. 59, 727, (1991).
[26] J. Gavaler, A. Santhanam , A. Braginski, M. Ashkin, and M. Janocko, IEEE
Trans. Magn., 17, 573, (1981).
[27] B. Oh, M. Naito, S. Arnason, P. Rosanthal, R. B arton, M.R.Beaseley,
T.H. Geballe, R. Hammond and A. Kapitulik, Appl. Phys. Lett., 51, 11,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
(1987).
[28] J. M annhart, P. Chaudhari, D. Dimos, C.C. Tsuei and T.R . McGuire,
Phys. Rev. Lett. 61, 2476, (1988).
[29] B. Roas, L. Schultz and G. Endres, Appl. Phys. Lett. 53, 1557, (1988).
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115
CH APTER VI
SURFACE IM PEDANCE
6.1
IN T R O D U C T IO N
One of the most im portant applications of high Tc superconductors is the
microwave applications. YBCO films are now comparable to, or even better
than, conventional superconducting films in terms of their respective surface
resistance at microwave frequencies. High Tc superconducting materials have
lower loss and lower dispersion when compared to gold (and other good conven�
tional conductors) up to lOOGHzt1?2!. This makes high Tc superconducting films
excellent m aterials for the fabrication of microwave devices^3-14!, such as res�
onators (cavity, stripline, coplanar, ring), circulators, filters, mixers, antennas,
and SQUIDs. For such applications, the most im portant m aterial param eter
is the surface impedance. On the other hand, microwave techniques are useful
techniques to investigate the physical properties of high T c superconductors.
T he microwave surface impedance, for instance, is a very im portant quantity
to characterize for superconductors.
Since the discovery of superconductivity in high T c m aterials of YBCOt1^,
the measured value of the surface resistance, R 3, at microwave frequencies has
decreased steadily^16-21!. The reduction in R s may be a reflection on the im�
proved quality of films or perhaps due to the improved measurement techniques.
This implies th at we may be approaching th e intrinsic limit of R s for YBCO. In
this chapter we employ a novel microwave measurement technique to measure
R s directly and X s, the surface reactance, indirectly. In section 6.2 the surface
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116
impedance of a superconductor is calculated in the cases where where conven2
tional two fluid model is assumed (J = (an ?i incUJ^2 )E, where an is the normal
conductivity and A is the London penetration depth), and a modified two fluid
model is assumed (E = (p +
where p is the normal resistivity of the
material). In section 6.3 techniques for the measurement of surface impedance
are reviewed. In section 6.4, our MSR (microwave self-resonant) technique is
discussed in detail. The experimental results are discussed in section 6.5 fol�
lowed by conclusions in 6.6.
6.2
C A L C U L A T IO N O F S U R F A C E IM P E D A N C E
Before going to the superconductor case, it is worthwhile to review the
surface impedance of normal metals. In normal metals, the electrical behavior
is described as
J = crE.
(6.2.1)
The surface impedance is defined as
Z s = R s + i X 3,
(6.2.2)
where R s is the surface resistance and X 3 is the surface reactance, and
RS= XS=
(6.2.3)
where the skin depth is defined as
6= \
V 27TW(7
(6.2.4)
In the superconducting case, there are two types of carriers, normal elec�
trons and superelectrons. Normal electrons will be scattered by phonon or
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117
impurity while superelectrons are nondissipative. We firstly consider an elec�
trodynamic problem of superconductors in which the effects due to the presence
of normal and superelectrons will be taken into account in a two fluid model.
We calculate the field distribution for a plane surface incident normal to the xaxis, taking E and H along the y and z axes, respectively. Therefore, Maxwell?s
equations can be w ritten as follows
dE
dx
dH
dt ?
(6.2.5)
and
dH
4x
dx
& - c2 '
<6-2-6>
The time-harmonic uniform field can be w ritten as
H = H Qeiut,
(6.2.7)
so that (6.2.5) becomes
dE
- f a
( 6 .2 . 8 )
=
In the two fluid model,
j = Jn + Js =
(<7n ?icra)E,
(6.2.9)
where J n and J s are the normal and supercurrent, respectively, a n is the normal
conductivity and a s is defined as
"? = S ? '
( 6 -2 1 0 )
This configuration is shown in Fig. 6.1a wherean and as are connected in
parallel. (6.2.9) can be rew ritten as
J = aef f E ,
(6.2.11)
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118
where
c2
Combining (6.2.6), (6.2.8) and (6.2.12) we have
d2E
i&TTuicref f ?
w = ?
(6-2-i 3>
E = E 0e kx,
(6.2.14)
1
2?47ro;CTn
k= y j2 + ??
(S'2-13)
and (6.2.13) yields a solution
where
is the wave vector constant of the electromagnetic disturbance. The surface
impedance of a material is defined as
-E(O)
= 'r鞍 tJ �
6-2' 16)
Jo ^
and (6.2.16) yields
Putting (6.2.15) into (6.2.17) we get
_
i4 7 ro ;A .,
i47ro;crn A2 .
, /9
Z 3 = ? ? (! + ------ Y ? )~112.
c*
,
(6.2.18)
In order to separate Z s into real and imaginary parts, we write the denominator
as
1+
= y
+
(6.2.19)
with
eIV5/ 2 = c o s~ + isin^-,
�
(6.2.20)
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119
where
co s% = � ( 1 +
2
^
= ) } 1/ \
(6.2.21)
= ) ] a/ 2.
(6.2.22)
' v/l + (l2a^Ai)2'
and
s i V = [ I ( ! -----
2
?
}
v l + (i�!T ^ 2
W riting Zs as Z s = R s + i X 3 with R a being the surface resistance given by
_
R ? =
and
1
- M ?
47TU;A
f
=
I
d '
V 2 c2 J l + ( l H ^ ) 2
ATCCTnCvX2
(v 1+ (
V
A
C
) ~ 1)
.
(6.2.23
being the surface reactance by
/ 4T A ?
d l + F f Z - r + lt'* .
v 2 c2 ^ 1 + ( 4 * 2 ^ 1 ^ V
c
Experimentally, we found th at
(6.2.24)
was roughly independent with tem per�
ature when T < 80JsT, (which will be seen in section 6.5, Fig. 6.16). This can
not be explained by two fluid model, where a superconductor is represented by
two conductors in parallel, we consider now a superconductor represented as
in Fig. 6.1b. One can see th a t on and cr3 are connected in series here instead
of in parallel in the two fluid model. The constitutive equation now becomes
T 4ttA2 83
E = ',J + ?
ap
(6.2.25)
E = (p + ^ ! ) J ,
(6.2.26)
or
where p is the normal resistivity and equal to l/<7?. The effective conductivity
reads
^ / = ( p + ^ ) - ?.
(6-2.27)
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120
Jn
Fig. 6.1a Diagram of two fluid model. a n and cra are in parallel connection.
J
On
Fig. 6.1b An alternative assum ption where crn and <r3 are in series connection.
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121
(6.2.26) along with (6.2.6) and (6.2.8) implies that the electromagnetic distur�
bances within the superconductor propagate with a complex wave vector k
,i47cw
ic2p
* = \ / ^ r - - W / = \ A 2 - z47^u;,
�
(6.2.28)
and this translates into a surface impedance of
_
i47rw
H ttujX .
.
C2 p
,1
Z- = 1 5 F = ?
,
(� * � >
The real and imaginary parts of (6.2.29) are respectively
and
<6-2-3�
6.3
TECH NIQ UES OF SURFACE IM PED A N C E
M EA SU REM EN T
6.3.1
Early Surface Im pedance M easurem ent
In 1940, London^22! measured the non-zero resistance of a tin ellipsoid in
the superconducting state, when a microwave field (1.5 GHz) was applied to
the sample. This implied that the resistance he measured was different from
DC resistance. Probably, it was the first observation of surface resistance of a
superconductor. P ip p a r d ^ carried out an experiment to systematically study
surface impedance of superconductors in 1947. The superconducting specimen
was in the form of a narrow loop of wire attached to a distrene rod, and may be
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considered to be a section of a transmission line, open-circuited at the bottom
and short-circuited at the top, so th at it resonated when it was approximately
a quarter wave length long, see Fig. 6.2. The measurement of surface resistance
was carried out by determining the frequency band width at resonance, Aw;
therefore the Q was determined from
<3=
(6.3.1)
where w0 = 2tt/ o and /o is the resonant frequency of the resonator. The Q
value is roughly inversely proportional to the total resistance of the resonator
circuit. For theirl23! system,
Q= ^
(6.3.2)
where L is the inductance and R the resistance of the whole circuit. If the
only source of power dissipation was the resistive loss in the superconductor,
the value of Aw would be directly proportional to the surface resistance of the
material, which was the quantity to be measured. In fact, however, there were
a number of other sources of power loss, which might be regarded as resistances
in series with the surface resistance of the superconductor, and which would
limit Q to a finite value even if the superconductors were perfect. To find the
intrinsic Qo, or unloaded Q, which depended only on the superconductor and
dielectric losses in the resonator, Pippard used a linear equation
^
+ t = l,
(6.3.3)
where Q is the measured value and t is the voltage transmission coefficient.
When t = 0 the measured value of Q is the ideal Q of the system. Pippard used
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123
German silver tube
Coaxial line
Sample
Sheath
Fig. 6.2 A resonator used for measuring surface impedance of superconductor.
(after A. Pippard)
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124
this m ethod to measure the surface impedance of P b and Hg. Although the
accuracy could not be high, b u t, nevertheless, the measurement was initiated
by Pippard?s group.
6.3.2
Strip Line, M icrostrip, and Coplanar Resonators
Strip line, microstript2^ and coplanar^8! resonators fall under the same
analysis developed by Pippard^23!. In these techniques, the superconducting
film under investigation needs to be patterned into different configurations.
One of the advantages of a coplanar resonator over a microstrip or strip line is
that both the conducting and ground planes are in the same film plane. The
others have the ground plane physically removed from the conducting plane.
Fig. 6.3 shows a coplanar resonator.
In the coplanar resonator technique, one measures the reflection coefficient
5 n from the resonant circuit shown in Fig. 6.3. The amplitude of 5 n is given
as follows![S1.
|<? , _ r( ^ - l ) 2 + 4 Q o ( / / / o - l ) 2 1/2
1 111 [( K + l)2 + 4 Q 0( / / / o - 1)2] ?
(6?3?4)
where
/o = resonant frequency of resonator,
K = coupling coefficient,
Qo= unloaded Q.
The change in the inverse of Q q is related to the total resistance, R t ,
A (-r?) oc R s + R ( + R r + R w = R t -,
Q
(6.3.5)
o
where
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125
Fig. 6.3 Coplanar resonator.
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126
R s = surface resistance of superconductor,
R e = related to loss tangent of substrate,
R r = related to radiation resistance, and
R w = related to surface wave losses.
Usually, in order to measure R s reliably and accurately w ith this technique
one needs to minimize R e + R r + R w. Although this technique can be used to
measure the surface impedance of a superconducting film, its preferred use is
to determine the penetration d e p th ^ .
6.3.3
Cavity Technique
Cavity resonator techniquest25-28] are most often used for surface impedance
measurements. Different from the linear resonator described in (b), a cavity
resonator has an advantage of not patterning the sample. The fundamental
resonance depends on the geometry of the cavity itself. The sample placed
in the cavity only perturb the fundamental resonance. A cavity can be used
for measuring the surface impedance of a superconductor of any shape. If the
entire cavity is built from superconducting material, it will give the greatest sen�
sitivity achievable, limited by the properties of the superconducting material
only. To overcome the lim itation of copper absorption background, an improved
Pb-plated high Q copper cavity operated at an ambient tem perature of 4.2K
was reported^26]. The superconducting sample, m ounted on a sapphire rod, was
placed at the center of the cavity, and thermally insulated from the cavity walls,
enabling variability of sample tem perature. A diagram of the measurement is
shown in Fig. 6.4.
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127
SYNTHESIZER
PIN DIODE
PULSER
PULSE
GENERATOR
FAST SCOPE i
DETECTOR
COUFUMC UXD
n ru n s
CAVITY
He,
> is m i.
TE011 = 8 .5 6 7 GHz
TM
9 61 7 GHz
Cu
mm
Fig. 6.4 A block diagram of cavity m easurem ent. ( after S. Sridhar et al.)
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128
Basically, the loaded Q and change in resonance frequency, A /, is mea�
sured. The loaded Q may be related to the unloaded Q by determining the
coupling, K, in a similar manner as described in (b). In the cavity technique^26!
R r = R w = 0 so that
yo
oc
R s + R c + Rh,
(6.3.6)
where R c is the surface resistance of the cavity walls and Rh is the contribution
from the sample holder. Presently, superconducting cavities are used so that
(R c -f-Rh) ~ R a- From fo it may be possible to deduce L s + L c, where L s is the
surface inductance of the superconducting specimen and L c the inductance of
the cavity walls and self inductance. Usually, L c is bigger than L a by as much
as 103.
The limitation of this technique is the Q of the background. It is possible
th at the losses in such a cavity would be dominated by resistance encountered
by current flow around the joints connecting the separate pieces of walls. It
is technically impossible to machine the entire cavity from bulk superconduc�
tors under investigation, which would give higher Q values and eliminate the
contribution of background.
6.3.4
Parallel Plate R esonator Technique
Taberl17! proposed an improved technique over the cavity technique. In
this technique, a parallel plate resonator was formed with a thin electric spacer
placed between two flat superconducting surfaces. The plates were intended
to be congruent but their exact shape was not crucial. Square or rectangular
pieces were convenient. No electrical contact was made to the edges of the
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129
plates, so the edges presented open circuit boundary conditions for transverse
electromagnetic modes that can be excited between the plates. A schematic
diagram is shown in Fig. 6.5. The samples were pressed together using dielectric
posts and placed approximately in the center of the test chamber. The test
chamber was made from brass and was gold plated to provide low surface loss.
For this technique again the unloaded Q and resonant frequency was mea�
sured. Here, R c = Rh = 0, but one needs to contend with the dielectric loss
tangent. Specifically, Taber^17^ found that
1
? = tanS + /3Rs/ S + aS,
Vo
(6.3.7)
where S was the spacing between the parallel plates, a and /? were coefficients
depending on the geometry of the resonator and tanS = e"/e' (e = e' ?ie").
As stated by Taber, S and tanS could be purposely chosen to make the second
term in (6.3.7) the dominant term. Thus, the lower limit of R s depended on
the selection of substrate material. It might be possible to deduce imaginary
parts of the surface impedance by using:
(6-3-8)
where To is the characteristic admittance. L is the length of transmission line
and v is the phase velocity which can be expressed in terms of A (penetration
depth) as
I 2 A d
V = vQ\ l l + - c o t h ( - ) ,
(6.3.9)
where d is the thickness of the film.
This method can measure surface resistance as low as 10
at 10GHz.
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130
SHORT MICROSTRIP LINES
ATTACHED TO SEMLRI01D
COAX LINES
DIELECTRIC
SPACER
DIELECTRIC
SU PPO RT POST
-SUPERCONDUCTING
SURFACES
Fig. 6.5 A schematic diagram of parallel plate resonator technique.
( after R� Taber)
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131
6.4
DEVELOPMENT OF MICROWAVE SELF-RESONANT
(MSR) TECHNIQUE
We have developed a novel microwave self-resonant (MSR) technique^29?30!
to measure surface resistance, R s, directly and surface reactance, X a, indirectly.
Fig. 6.6 is a schematic diagram of the measurement. In this method, a super�
conducting thin film strip (4.24x0.9mm) of YBCO was placed in a waveguide
whose cross section was 10.67x4.32mm. The strip was placed at the center of
the waveguide as shown in Fig. 6.7a. The transition tem perature of the film
was about 90K and the critical current density was of the order of
10� A / c m 2
at
77K. The films were c-axis oriented and the thickness was typically 0.5 f i m . A
HP8510B vector network analyzer was employed to measure the transmission
coefficient (am plitude and phase) when a microwave was propagating through
the superconducting film strip. The equivalent c ir c u it^ of strip and waveguide
transmission line is shown in Fig. 6.7b. The capacitance, C , was due to the
gap between the waveguide and the strip. A typical gap value was 0.04mm
and it corresponded to C ~ 10-12/ . No systematic study was made to relate
the gap width w ith C. However, besides the gap the dielectric constant of the
substrate on which the YBCO film was deposited affected the value of C. We
empirically varied the width of the gap until transmission microwave resonance
was observed between 18 and 26 GHz - the fundam ental band of propagation
in the waveguide.
The inductance L was a combination of the superconductor intrinsic in�
ductance, L s, and the self inductance, To, calculated approximately asf32J
_
Zq a
f
r . 37r, s i n x . , 9 .
" SA I77?T7?lsmT (?
u>.
. , .
(6-4-1)
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132
RF
DETT
RF
[SOURCE
VECTOR
ANALYZER
H PS510C
DEW AR
LHe
S A M PL E
PRINTER
WORK
STA TIO N
Fig. 6.6 Microwave self-resonant measurement arrangem ent.
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Fig. 6.7a Diagram of microwave self resonator.
Fig. 6.7b Equivalent circuit of microwave self-resonator.
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134
where
37T w
and
w = 0.9mm,
a = 10.67mm,
b = 4.32mm.
fo is the transmission resonant frequency, c is the speed of light and Z 0 is the
characteristic impedance of the waveguide and was determined to be equal to
43 Q from our calibration runs. A typical value of L 0 was ~ 10-11 h. L s was
the inductance due to the superconducting screening currents. In comparison
to L 0, L s was about a factor of 100 lower. R s was the surface resistance of
the superconducting film.
From the equivalent circuit of Fig. 6.7b, the reflection, S n , and transm is�
sion, S 21, coefficients may be calculated as follows,
(6.4.2)
and
(6.4.3)
where (j>n and <f>2i are the phase angles of S \\ and Shi- For the equivalent
circuit of Fig. 6.7b,
Z = R s + i(ujL - ^ ) ,
(6.4.4)
putting (6.4.4) into (6.4.3), we get
(6.4.5)
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135
where
w0 = -� =
Vlc
(6.4.6)
is the transmission resonant frequency. Letting
UJ
U)o
? =
(6.4.7)
(6.4.8)
and
b2 = � ( f ) 2,
(6.4.9)
_
ax + i(x2 ?1)
21 = (a + 6)x + f(a:2 - l ) '
,
(6.4.10)
(6.4.5) can be written as
The amplitude and phase are respectively (we used IS21I2 instead of IS21I for
simplifying the fitting procedure) and
(5 A 1 1 )
<j>2i = ta n -1 ( - ? ? ) - tan~1( 1 X ).
ax
(a + o)x
(6.4.12)
By fitting the experimental curves, the parameters a and 6, and therefore R s
and L, can be deduced. At resonant frequency, the imaginary p art vanishes;
hence, Z = R s, the surface resistance. R s may be determined from (6.4.3)
directly,
R
�
=
2 [1/|� 2i | ?1]'
(6A13)
It is noted th at in (6.4.13) there is no other loss param eter in the expression.
Loss tangents of the substrate have minimal effects on the measurement of R 3.
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136
For example, we have included realistic loss tangents for a typical substrate
m aterial of MgO or SrTiO s, etc.. and calculated jS'21 J. The correction to R s
(using (6.4.13)) from the loss tangent contribution was calculated to be about
0.04% at 80K (using MgO as an example).
Since most of the samples we measured are thin films, the size (thickness)
effect should be taken into account.
The transformation m atrix of a wave
propagation through a film medium is
f &i \ _ f coskd
\h i J
\-^sin k d
where
i Z s i n k d \ ( e2 \
coskd / \ ^ 2 /
.
.
? ?
and e<i are the surface electric fields, while hi and fi2 are the surface
magnetic fields at the two film surfaces, respectively, k is the wave vector of the
propagation, d is the thickness of the film. (6.4.14) can be w ritten as
Denote
= Z j and
ei
^-coskd + iZ sin k d
hi
� j^ s in k d + coskd
(6.4.15)
= Z q, the measured and characteristic waveguide
impedances, respectively. Rewriting Z =
Zj,
Zoo
(6.4.15) becomes
Zocoskd + iZoosinkd
iZosinkd + Zoocoskd'
(6.4.16)
It can be proven th a t for a thin film (6.4.16) becomes approximately
Zd = -iZooCot(kd) = ?iZooCot(
47vuid
C
).
(6.4.17)
Z /Q Q
The subscript 00 is to denote th at as d ?*? 00 , Z ?>?Z,OO*
Z 00 = R s +j{uJL - Z - ) .
(6.4.18)
As in any development of a new technique, we calibrated the measurement
with standard materials, such as films of copper, silver, etc.. We placed strips
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137
of copper, silver, aluminum and bulk YBCO in the waveguide for the purpose of
determining Zo and to measure the conductivity of each material. Conceivably,
I/o could have been determined from the calibration runs, since L q was not
sensitive to whether or not the m aterial was superconducting. However, this
required identical strip dimensions and substrate m aterial for both YBCO and
copper films, for example. We did not exercise this option. We determined
L q from /o and lineshape fitting of IS21I versus frequency for each run under
consideration.
In Fig. 6.8 we plot IS21 I as a function of frequency for a copper strip
of 4.24 x 0.9mm. The thickness was about 0.05mm. A least squares fit was
obtained and the deduced param eters were as follows,
I/0 = 5.38 x 10~n h,
and
C = 1.08 x 10~12/.
For Cu, Ag and Al, the surface resistance was determined from
R * = V CTC4
and it agreed with expectations.
( 6 -4 -1 8 )
Since u> was determined precisely, cr, the
conductivity was measured accurately. Thus, Z q may be finally determined
from (6.4.10) and was equal to 4312. It changed little w ith tem perature. It is
worthwhile to point out th at the Zo calculated from (6.4.10) is in very good
agreement w ith the result from fitting the amplitude and phase curves of the
transmission coefficient, S 21?
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138
ap
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139
6.5
EXPERIM ENTAL RESULTS A N D DISCUSSION
We have measured R s and X 3 by using the MSR technique^29!. Fig. 6.9
shows typical amplitude and phase of transmission resonance curves as a func�
tion of frequency observed at 15K for YBCO film strip. In Figs. 6.10 and 6.11
we plot measured IS21 I and, fax, as a function of normalized frequency at a
temperature near Tc. The d ata are represented by points and the least square
fit by a solid line in both Figures. The following parameters were deduced from
the fit at T=86K: L = 2.8 x 10~ u h, C = 2.05 x 10~12/ , and R s = 0.052S2. We
can deduce R 3 from either curve fitting or using (6.4.10), but the determ ination
of X 3 is nontrivial. To deduce X 3, we need to separate L s from L. Assuming
the frequency shift with respect tem perature is due to the change of the surface
inductance of the superconducting film strip, it is approximated th at
ui = ?^ - -= = ~ ioo ?
y/{L + A L)C
2 L
(6.5.1)
v
'
Rewriting (6.5.1) we have
L S(T + A T ) - L S( T ) = - 2 ^ - L .
Jo
(6.5.2)
The valuesof L a were deduced from (6.5.2). Frequencyshifts weremeasured
with respect to the resonant frequency at 4K, see Fig. 6.12. L 3 at 4K can also
be deduced from the surface resistance, R 3. When T < < Tc,
R s = 4-7r2<7?u;2A3,
c4
(6.5.3)
X 3 = u L 3 = -^o;7rA.
c2
(6.5.4)
and
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140
S 2 1/ M 1
REF 0.0
7 .0
lag
MAG
S2 j/Ml
REF 0.0 �
5 0 . 0 �/
dB
dB/
START
STOP
16.000000000
26.000000000
GHz
GHz
Fig. 6.9 Microwave resonant spectra of a YBCO strip taken at 15K.
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141
-2 0
-3 0
-4 0
0 .8 5
0 .9
0 .9 5
1
1 .0 5
1.1
1 .1 5
a /u Q
Fig. 6.10 Amplitude of transm ission coefficient as a function of normalized fre�
quency at 86K for a YBCO strip. The resonant frequency is 21GHz.
Solid line is fitting curve.
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142
0 (H ad.)
0.5
-0 .5
-
1.0
0.9
1
1.1
1.2
w / cjq
Fig. 6.11 Phase angle of transm ission coefficient as a function of normalized
frequency at 86K for YBCO strip. The resonant frequency is 21GHz.
Solid line is fitting curve.
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143
0.0
?t?i
1V
I ' 11
1 1 1 1 1 1 1 1 i i i j i i ?i ??i?
-
-
+
-0 .1
+
? -
(G H z)
-
-
_
+
+
+
-0 .2 --
f0= 2 1.31 GHz (4K)
-
+
?
+
+
-0 .3 --
?
+
-
-0 .4
?F
0
?
I * i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i
20
40
60
60
T (K)
Fig. 6.12 Frequency shift of a YBCO strip w ith respect to the resonant fre�
quency at 4K.
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144
Therefore,
L s = ( - ^ ^ - ) 1/ 3.
u>l cza n
(6.5.5)
K
'
The 4K point served as a reference point for the rest of the data of L s at higher
tem peratures. A plot of L s versus tem perature is shown in Fig. 6.13.
Size effects of R 3 and X s were calculated by using (6.4.16). Demonstra�
tions of the thickness dependent on R s and X a are shown in Fig. 6.14 (for
tem perature at 70K) and Fig. 6.15 (for T = 86.5K), respectively. One can see
that the corrections get more im portant when d (the thickness of the sample)
get smaller. Our samples were about 5000A thick, so it was necessary to take
the size effect into account. We have included the measured and corrected val�
ues of R s and X 3 in table 6.1 for different tem peratures. One can see th at
the correction of surface resistance, R s, is negligible in the whole range of tem �
perature below Tc, but the size effect of surface reactance X 3 is significant,
especially when T ?> Tc. The correction of X 3 was as large as a factor of 3,
since at high tem peratures the microwave penetration depth was comparable
to, or even greater than, the thickness of the films.
In Fig. 6.16, R 3 and X 3 are plotted as a function of tem perature. One
can see th at R 3 decreased by about 3 orders of m agnitude as the tem perature
decreased from 90K to 80K reaching a low value of 4.9 x 10_4O. At 15K,
R s was 2.7 x 1CT40 , the lowest value measured using the MSR method. If
we extrapolated^19] R s to lower frequencies we found th at R s was equivalent
to 100/rfi at 10 GHz and 8.7fiO, at 2.95 GHz. The tem perature was assumed
to be equal to 77K. However, at 15K, the extrapolated values would be 61 /j.Q,
at 10 GHz and 5.3
/ J .Q ,
at 2.95 GHz. A comparison w ith other measurements
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145
30
20
+ ++
0
20
40
60
80
T (K)
Fig. 6.13 Surface inductance of a YBCO strip as a function of tem perature.
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146
Solid line : R
Dashed line: X
70K
x & r (n)
i- l
i-2
i-3
0
0.2
0.1
Fig. 6.14 Surface impedance
e ls
0.3
d (fim)
0.4
0.5
a function of f i l m thickness taken at 70K.
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147
10.0
Solid line : R
5.0
Dashed line: X
T = 86.5K
?
�
X
1.0
0.5
0
0.1
0.2
0.3
d (//m)
0.4
0.5
Fig. 6.15 Surface impedance as a function of film thickness taken at 86.5K.
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14S
10�
& r ? (n)
-w-
++
++
0
20
40
60
80
T (K)
Fig. 6.16 Surface impedance of a YBCO strip as a function of tem perature. In
the X 3 curve, there are few points not plotted around 60K, because
there were some frequency shift due to the system background at that
tem perature range.
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149
of R s is given in table 6.2 and 6.3. We noticed that our measured values of
R s exhibited an oscillating behavior with tem perature, although L s did not.
Bonnl19! et al. observed an oscillatory behavior of R s at tem peratures between
10-60K, and they claimed that the behavior was from the nonlinear conductivity
of the superconducting film at tem peratures below Tc. We don?t attribute the
oscillatory behavior of R a to an intrinsic process at this stage, since lower values
of R s have been measured at lower frequencies (lOGHz)!17?18^ and no oscillatory
behavior of R 3 was observed.
6.6
CONCLUSION
We have calculated the surface impedance by using both two fluid and mod�
ified two fluid models. We have developed a novel microwave technique to mea�
sure surface impedance. This technique has an advantage over other techniques
for th at there are no background contributions to the surface impedance mea�
surement. The sensitivity achieved by this technique is better th an the linear
and cavity resonators and comparable to th at of the parallel plate resonator^17!.
We have used the microwave self-resonant (MSR) method to measure both real
and imaginary parts of the surface impedance of YBCO superconducting films.
The lowest value of surface resistance was 2.7 x 10- 4 fl at 21 GHz and 15K.
The extrapolated values would be Qlfj.0, at 10GHz and 5.3jiCl at 2.95GHz, com�
pare to published values^17?19! of 56^0, at 10GHz and
at 2.95GHz. By
measuring resonance frequency shift with respect to the resonance frequency at
4K, we measured the surface reactance indirectly, the lowest value of surface
reactance was 0.031 fh We also estim ated the size effect of surface impedance.
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We found that when the thickness of the measured film was comparable to
the microwave penetration depth, the size effect correction of surface reactance
became im portant, while the size effect of surface resistance was less significant.
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151
T able 6.1
Size effect corrections of R a, X s and L 3 of YBCO film
Results without correction
W ith size effect correction
T(k)
L s( 10~13H )X3()
Rs(tt)
X .(Q )
L S(10~13H
4.00
2.30002
0.0310
0.00048
0.000481
0.0312
2.330
15.0
2.59596
0.0350
0.00027
0.000271
0.0353
2.640
20.0
2.73390
0.0366
0.00054
0.000543
0.0370
2.766
27.0
2.89102
0.0387
0.00070
0.000704
0.0393
2.939
30.0
2.99004
0.0399
0.00076
0.000765
0.0406
3.037
34.0
3.08806
0.0413
0.00088
0.000887
0.0422
3.157
40.0
3.16697
0.0423
0.0C110
0.001110
0.0433
3.241
43.0
3.28497
0.0439
0.00020
0.000202
0.0451
3.376
49.0
3.38400
0.0452
0.00048
0.000486
0.0466
3.489
52.0
3.45231
0.0492
0.00051
0.000520
0.0502
3.610
72.0
4.13670
0.0571
0.00049
0.000492
0.0588
4.321
77.0
4.46797
0.0592
0.00043
0.000441
0.0639
4.797
80.0
5.05897
0.0673
0.00048
0.000496
0.0752
5.653
82.0
5.65097
0.0750
0.00090
0.000937
0.0869
6.542
84.5
6.43900
0.0853
0.00280
0.002937
0.1040
7.845
86.5
7.14320
0.0926
0.05210
0.059750
0.1420
10.78
88.0
7.72002
0.1020
0.09800
0.107200
0.1880
14.22
90.0
8.80399
0.1160
0.30800
0.330100
0.3910
29.66
91.2
10.1840
0.1340
0.31000
0.331000
0.4130
31.43
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152
T ab le 6.2
A comparison of our measured surface impedance value with others?
at 77K
Technique
441
X a(Sl)
f (GHz)
0.063
21
Sample
Ref
our
100 *
10
8.7 *
2.95
B
1500
18.7
film
[21]
C
720
10
film
[17]
D
22
2.95
crystal
[19]
A
film
MSR
tech.
* We have scaled R s as oc f 2
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153
T able 6.3
A comparison of our measured surface impedance value with
others?at 4K
Technique
480*
A
f (GHz)
Rs{fJ-tt)
0.031
Sample
21
Ref
our
108**
10
film
MSR
9**
2.95
B
60
18.7
film
[21]
C
56
10
film
[17]
D
15 (1.7K)
2.95
crystal
[19]
E
<400
10
crystal
[16]
F
12~180
10
film
[20]
G
20~ 43
10
film
[18]
tech.
* We measured R s ? 270[id at 15K and at 21GHz
** We have scaled R s as a f 2
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154
REFERENCES
[1] N. Klein, G. Muller, H. Piel, B. Boas, L. Schultz, U. Klein and M. Peiniger,
Appl. Phys. Lett., 54, 757, (1989).
[2] M. Namordi, A. Mogrom-Lampero, L. Turner and D. Hogue, IEEE Trans
M T T ., Sep. (1991).
[3] R. S. W ithers, A. Anderson, and D. E. Ontes. Solid State Technology,
83-87, Aug. (1990).
[4] E.F. Belohoubek, Defense Electronics 82-86, Jan. (1990).
[5] J. Bybokas, Supercurrents, 72-76, July (1990).
[6] P. A. Ryan, J. of Electronic defense, 55-59, May (1990).
[7] D. E. Oates, A. Anderson and P. Mankiewich, J. Supercontivity 3, 251,
(1990).
[8] H. How, R. Seed, C. Vittoria, D. Chrisey, J. Horwty. C. Carosella, and V.
Folen, IEEE Trans M T T, 4 0 (8 ), (1992).
[9] Y. Huang, H. Jiang, A. Widom, and C. Vittoria, IEEE Trans. Mag., 1992).
[10] C. Zahopoulos, S. Sridhar, J. Bautista, G. Ortig, and M Laragan, Appl.
Phys. Lett., 58, 977, (1990).
[11] Y. Yoshisato, M. Takai, K. Niki, S. Yoshikawa, T. Hirano, S. Nakano,
IEEE. Tras. Mag., 27, 3073, (1991).
[12] Y. Huang, M. Lancaster, T. Maclean, Z. Wu, N. McN.Alford, Physica C,
180, 267, (1991).
[13] Y. Huang, H. Jiang, H. How, C. Vittoria, A. Widom, and R. Roerstler, J.
Superconductivity, 3, 441, (1990).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
[14] M. Nisenoff, in Principles & Applications of Superconducting Quantum In �
terference Divices, ed. A.Barone, (World Scientific Pub., Singapore, 1990).
[15] K.W. Blazey, K.A. Muller, J.G. Bednorz, and W. Berlinger, Phys. Rev.
B
, 36, 7241, (1987).
[16] D. Wu, W. Kennedy, C. Zahopoulos, and S. Sridhar, Appl. Phy. Lett., 55,
698, (1989).
[17] R.C. Taber, Rev. Sci. Instrum., 61, 2200, (1990).
[18] P. Merchant, R. Jacowitz, K. Tibbs, R. Taber, and S. Laderman, Appl.
Phys. Lett., 60, 763, (1992).
[19] D.A. Bonn, P. Dosanjh, R. Liang, and W.N. Hardy, Phy. Rev. Lett., 6 8 ,
2390, (1992).
[20] D. Miller, L. Richards, S. Etemad, A. Inam, T. Vankatesan, B.D utta, X.
Wu, C.Eom, T. Geballe, N. Newman, and B. Cole, Appl. Phys. Lett., 59,
2326,(1991).
[21] N. Klein, U. Dahne, U. Poppe, N. Tellmann, K. Urban, S. Orbach, S.
Hensen, G. Muller, and H. Piel, J. Superconductivity, June, (1992).
[22] H. London, Proc. Roy. Soc., A 176, 522, (1940).
[23] A. Pippard, Proc. Roy. Soc., A 191, 370, (1947).
[24] S. Anlage, H. Sze, H. Snortland, S. Tahara, B. Langley, C.Eom, M. Beasley,
and R. Taber, Appl. Phys. Lett., 54, 2710, (1989).
[25] N. Klein, In High Temperature Superconductors, J. Pouch et al.
(Trans. Tech. Pub., Switzerland, 1992).
[26] S. Sridhar, W. Kennedy, Rev. Sci. Instrum., 59, 531, (1988).
[27] A. Portis, D. Cooke, and F. Gray, J. Superconductivity, 3, 297, (1991).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ed.
156
[28] J. Marteus, V. Hietaln, D. Ginley, T. Zipperian, and G. Hohenwarter, Appl.
Phy. Lett., 58, 2543, (1991).
[29] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, Proc. M RS, San
Francisco, (1992).
[30] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, and A. Drehman,
subm itted to Phys. Rev. B.
[31] M.K. Skrehot, K. Chang, IEEE Trans. Microwave Theory Tech., 38, 434,
(1990).
[32] R.L. Eisenhart, IEEE Trans. Microwave Theory Tech., 24, 987, (1976).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
CHAPTER VII
LONDON PENETRATION DEPTH
AND COHERENCE LENGTH
7.1
IN TR O D U C T IO N
The London penetration depth and coherence length are some of the most
im portant param eters in a superconductor. The London penetration depth
which measures the length over which magnetic fields are attenuated near the
surface of a superconductor, contains information about the effective mass and
density of superconducting pairs, while the coherence length measures the size
of Cooper pairs, or size of the vortex core. They are referred in the literature
as the characteristic lengths of superconductors. One of the key reasons why
high Tc superconductors are different from conventional superconductors is that
the high Tc superconductors have a small coherence length]1_4^ (of the order of
10A) comparing to conventional superconductors^ (of the order of 1000A).
Both lengths cannot be directly measured. There are many techniquest6_14]
for measuring the penetration depth, such as muon-spin-rotation (j i +SK), po�
larized neutron reflectometry, kinetic inductance, AC susceptibility, DC magne�
tization and microwave techniques. The measured values of penetration depth
varies from sample to sample and sometimes m ethod to method. One of the rea�
sons is th at high Tc superconductors are highly anisotropic, and the orientation
dependence should be taken into account. Uncertainties about local demagne�
tizing factors, surface and grain boundary effects of polycrystalline samples also
lead to inaccurate results.
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158
Measurements of the coherence length usually are deduced from the mea�
surement of the upper critical magnetic field^2?3], H c2 . However, the H c2 of
YBCO is extremely high (> 100 Tesla at 4.2K^15^), the measurement requires
special and difficult technique. Many measurements of the cohernence length
are measuring the upper critical field at high tem perature (say, near Tc). �(0)
may be extrapolated from
�(T) = �(0)(1 -
(7.1.1)
In this chapter, we deduce the London penetration depth from the mea�
surements of surface impedance by using the microwave self-resistant (MSR)
technique^16?17!. We use both the two fluid model and modified two fluid model
to deduce the penetration depth. We use these relationships in determining A.
and
A=
P - 1-3)
respectively. At low tem peratures, X a �
R s, the two expressions converge to
the same result
c2
A=-L?
(7.1.4)
O ther than measuring the upper critical magnetic field, we developed a
m ethod based upon our microwave self resonant technique to measure the co�
herence length near Tc.
YBCO is an anisotropic material, and both the London penetration depth
and coherence length are anisotropic. Their values depend on the direction of
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159
the microwave electric field relative to the c axis. From our microwave measure�
ment, we find th at A||(0) is about 1840A, where Ay is defined as the penetration
depth in the a-b plane, and
= 129A at 86. 5K, and |(0) extrapolated from
86.5K is about 25A, where | is defined as the coherence length applicable for
microwave induced electric field (or current) applied along the a-b plane. To�
gether w ith the E PR measurement results by our group!18?4!, we have London
penetration depth and coherence length values both in the a-b plane and in
the c direction. The values in c-axis direction were 3640A and 12A at ~0K ,
respectively.
Widom!19! et al. have calculated A and � from a microscopic ionic bonding
model and general agreement was found between the calculated values and our
experimental results.
7.2
ANISO TR O PIC LENG TH S OF YBCO FILM
To understand the anisotropic property of YBCO m aterial, it is worth�
while to review its crystal structure. From Fig. 1.4 we can see th a t the C u-0
bond form a plane, called the a-b plane, which is responsible for the elec�
tronic conduction. Along the b-axis, there is a C u-0 chain, which may be also
conductive. There is no C u -0 chain along c-axis. Vortex-lattice decoration
experiments!20,21! have shown th at in YBCO the penetration depths are ap�
proximately in the ratios Aa : A& : Ac = 1.2 : 1 : 5.5. It is im portant to realize
th a t the penetration depth depends on the direction of the screening current
density, not upon the direction of the local magnetic field. The penetration
depths Aa and Ab are much smaller than Ac since the screening currents in the
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160
a-b planes (in the a or b direction) flow much more easily th an along the c-axis
direction. Moreover, A& is somewhat smaller than Aa because current parallel
to the a-b planes flows a little easier along the C u-0 chains (b-axis direction)
than perpendicular to these chains (in the a-direction).
There are several factors which will affect the measured value of penetration
depth. Beside the local uncertainties, such as surface effect, grain boundary
effect and local demagnetizing factors, the anisotropic property, therefore the
orientation, will affect the result a lot. Measurement on YBCO single crystals
gives a clear orientation dependence while polycrystalline sample average out
the anisotropy and give very different results from single crystals. However, we
noticed th at the anisotropy along the a and b axes is negligible compared to
the c-axis result. Therefore, measuring c axis oriented films will yield results
approximately similar to single crystal samples.
For anisotropic material, the average coherence length and penetration
depth are defined ast22^
� = (6 � c ) 1/3,
(7.2.1)
A = (AaA&Ac)1/3.
(7.2.2)
and
For the case ( a ~ �& and Aa ~ A&, (YBCO, for example), we can write (7.2.1)
and (7.2.2) as
� = (�!� j. ) 1/3,
(7.2.3)
A = (A|A� )1/ 3,
(7.4.2)
and
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161
where
and Ay are the coherence length and penetration
depth in the a-b
plane and _and Aj_ in the c-axis direction, respectively.They can
be defined
a s [23,24]
| = �/\/? ib
(7.2.5)
U =
(7.2.6)
Ay =
(7.2.7)
A_l = Ay/m �,
(7.2.8)
and
where m\\ and mj_ are the effective mass in the a-b plane and along the c-axis
direction. The anisotropy factor, 7 , is defined as
7 = a/ ? ,
V m l
(7.2.9)
so that
| = 7
(7.2.10)
A,| = ? Aj_.
(7.2.11)
and
For homogeneous superconductors, the Ginzburg-Landau param eter k is defined
as k = A/�. For anisotropic superconductors, however, the situation is more
complicated, and one must introduce the following definitions^23,24!,
kj .
=
(7.2.12)
�
and
/cii = 4 / ^ � ,
(7.2.13)
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162
and the relation between them is
K|| = 7玧..
7.3
(7.2.14)
L O N D O N P E N E T R A T IO N D E P T H
D E D U C E D F R O M S U R F A C E IM P E D A N C E
In chapter VI, we discussed the surface impedance of superconducting films.
The expressions of surface impedance are
Z, =
+
?
iAnusX .
(7.3.1)
and
. c2p . i
.
(7.3.2)
using two fluid model and modified two fluid model, respectively. We can solve
for the London penetration depth and obtain
c2
X 2 4- R 2
>?= Anu { X I - R2)i/2 ?
p.3.3)
and
c2
x =
<7-3-4)
for each case. Here
Ra =
1
1 _
^
^
=
1+ ( " 1 ) 2
(J l + ( "
_ ,)! /* ,
(7.3.5)
! ) 2 + 1)1/*,
(7.3.6)
> / 2 c2 ^ 1 + ( i E 2 ^ A i ) 2
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163
for (7.3.3) in which the two fluid model is assumed; and
_
47TU>A, /
R- = ^
r
. c2p
+ ^
si
) - 1)!-
(7'3'7>
and
x *= ^
(V1 + (4 ^ ) 2+1)?-
p -3-8>
for (7.3.4), where the modified two fluid model is used. At low tem peratures,
the surface reactance is dominant; X a �
R s, therefore, (7.3.3) and (7.3.4)
converge to the same result
A = -? X 3 = ? L a.
47rw
47r
T �
Tc
(7.3.9)
v
?
We applied the two fluid model to deduce A and we found A ~ 18424A in
the a-b plane at 88K (the Tc was 92K), which gave a value of A(0) ~ 7435A.
This value is in disagreement with values A(0) ~ 1400A reported by others^13?14!.
Hence, we use (7.3.4), the modified two fluid model, to calculate the London
penetration depth by using the surface resistance and surface reactance values
deduced from our microwave self-resonant measurement. In Fig. 7.1 we plot A
as a function of tem perature. Typical values of A are 1840A at 4K and 8000 A at
86.5K, for example. We have fitted A to the Gorter-Casimir two fluid modelf25!,
A(r) = A ( 0 ) ( l - ( | ) 4) - 1/2.
(7.3.10)
The results are shown in Fig. 7.1 w ith a dashed line. If we change the power
factor in (7.3.10) from 4 to 2,
A(r) = A ( 0 ) ( l - ( ^ ) 2r ' / 2,
(7.3.11)
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164
Experimental Data
M7' ) = A ( 0 ) ( l - ( � ) ' ) - ' / 2
*c
A(T) = A(0)(l-( � ) ?)->/?
(uni) y
0.5
0.0
0
20
40
60
80
T (K)
Fig. 7.1 London penetration depth aa a function of tem perature.
Dashed
line is fitting w ith two fluid model, solid line is fitting with A(T) =
A ( 0 ) ( l - { � ) 2) - ?/ 2.
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165
we obtain a better fitting curve, see solid line in Fig. 7.1.
The penetration depth, A, was also measured in our group by using E P R or
the so called MMMA technique developed by Kariml26! et ah. Before we display
the results, it is worthwhile to review the technique briefly. Normally in an E P R
measurement
is measured, where P is the microwave power absorbed by the
sample in question and H is the m agnetic field. The E P R technique was used
in the past to measure
low H fields on superconducting high Tc material.
Karim et alS261 introduced this technique to measure j j f in 1987 in order to
measure Tc of very fine particles of YBCO crystal. In th at experiment j j j was
measured as a function of tem perature. The analysis for A was firstly developed
by K arim et a/J18] and it was based upon the Bardeen-Stephens model for
fluxoid nucleation at low magnetic fields. The basic assumption of the analysis
is th a t the microwave absorption scales as the B-field or the volume of the
normal region. From the fit to the lineshape the value of A_l / � jl was deduced.
An alternative approach in analyzing Aj_ is to define the magnetic field at which
was maximum as H ci, the lower critical field. To obtain Aj_ the following
relationship was used,
Ba=^ k lnK-
(7-3-n)
This type of measurement complements the MSR technique in the following
sense. W hereas in the waveguide technique (MSR technique) the microwave
electric field, E, is applied in the plane of the film, in the E PR experiment, E
is applied normal to the film plane. The c-axis for all of our films is aligned
perpendicular to the film plane. By combining these two microwave techniques,
we are able to deduce the penetration depth along two directions relative to the
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166
c-axis.
In Fig. 7.2 we have included A values deduced from MSR and MMMA
data. We also have superimposed the value of A(0) obtained from the shift in
the resonant frequency with tem perature using a CPW resonator. However, a
clear distinction is to be pointed out here. D ata for A generated in waveguide
and coplanar resonator measurements assume the electric microwave field to be
parallel to the film plane or perpendicular to the c-axis. This d ata is designated
in Fig. 7.2 as Ay. For the MMMA case the d ata is designated as Aj_, since the
electric field is parallel to the c-axis.
7.4
DETERM INATIO N OF C O H ERENCE LENGTH
We have measured the coherence length of YBCO films by means of the
MSR technique. Again, the films were laser ablated with transition tem perature
Tc ~ QOK and the critical current density was in the order of 106A /cm 2 at
77K. The c-axis was normal to the film plane. From the surface impedance
given in (7.3.2), (7.3.5) and (7.3.6), we can calculate the norm al resistivity of
the superconducting film and it yields
In deducing
from the MSR technique we needed to apply a magnetic field, H,
parallel to the c-axis or perpendicular to the film plane. From the microwave
transmission coefficient data, we measured only the peak values at resonance.
The surface resistance R s changed as H was varied. The surface inductance
L s, remained constant for fields up to 3KG, see Fig. 7.3, since /o remained
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167
1 1? I
1
' r
'
1
1
1
1
| 1 1 1 1 | 1 1?1?1?|?r <*, 1 "I?
2.0
1.5
-
+ f0=21GHz, MSR
-
0 f0=9.32GHz, EPR
?
0 f0= 14.5GHz, CPW
�
X
{fj)
_
+
0
+
0
-
?
-
1.0
-
o
-
0
.
o
-
0.5
_
-
? o +
0.0 ; i n
+
+
+
- J ___1.
+
_J---1.
20
+ +
J
o o o o
+ ++
---L
1 J
I
1
40
0
O
*
60
+
+
_
+
+
-
+
1 .. 1
1 .1
60
!
1
1
1
I
I
I -J__
80
100
T (K)
Fig. 7.2 London penetration depth as a function of tem perature deduced from
MSR and MMMA techniques and A(0) from CPW technique.
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168
S21
REF - 2 4 . 2 9
lo g HAG
dB
3000G
2000G
1000G
?
START
STOP
19.850000000 GHz
20.350000000 GHz
Fig. 7.3 Microwave transm ission coefficient of the superconducting film at dif�
ferent magnetic fields.
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169
constant with H. Hence, the change of resistivity due to the applied magnetic
field may be expressed as follows
c2
A p = ? L 3A R 3.
(7.4.2)
K
Using this equation, we can calculate the change of resistivity of the supercon�
ducting film through the measured values of surface resistance changes due to
the applied external magnetic fields. The results are plotted in Fig 7.4. Using
the Bardeen-Stephens relationship
AP = ^ ?
y O&ti
and (7.4.2) we can determine the coherence length. At 86.5K we find
(7.4.3)
= 129A,
where we treat B as approximately equal to H (this is approximately true near
Tc). At 80K we find R s to be less field dependent, which implied a reduction of
magnetic flux penetration in the superconducting film. Again, using (7.4.2) and
(7.4.3) we deduced | ~ 60A at 80K. Below 77K, we were not able to measure
significant changes in R s with respect to the application of a magnetic field. At
lower tem peratures, the magnetic field penetrated only a small distance into
the superconducting film (as seen in Fig. 7.1, A is more flat at tem peratures
below 80K). Therefore, less microwave absorption occured below 80K. Hence,
no estimate of | was obtained. If we extrapolate our results to T ?>OK by
using the relation
f = � 0 ) ( l - | ) - 1/ 2 ,
(7.4.4)
we find that <fy(0) to be 25A. The coherence length in the c-axis direction
can be deduced from E P R measurement, where the microwave induced electric
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170
T = 06.5K
(ixro?(J7Y) udy
8
?
*+
0
2000
1000
3000
H (G)
Fig. 7.4 Change of resistivity of the superconducting film at 86.5K due to the
applied magnetic fields.
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171
field was along the c-axis. Their values were �(86.5Iv) = 47*4. and �(0 ) = llA .
Together with the London penetration depths in section 7.3, we can calculate
the Ginzburg-Landau parameters, and
k
�
is in the range of 62 ~ 74 while rey
in the range of 110 ~ 165. In table 7.1 we report our deduced values of ,
A||, Aj_, K||, and n� from our MSR and E PR measurements.
7.5
D ISC U SSIO N A N D CONCLUSION
Based upon our measurements of surface impedance using a microwave
self-resonant technique, we have deduced the London penetration depth in the
a-b plane of a superconducting film. Application of a magnetic field allowed us
to deduce the coherence length in the a-b plane as well. E P R results provided
the London penetration depth and coherence length in the c-axis direction. The
values of A and � are anisotropic and depend on the direction of electric field
or current relative to the c-axis. At 86.5K, =129A and Ay = 8000A while
l
= 47A and
A jl
= 18000A. At OK, (0)=25A and Ay(0 )
=
1840A while
= llA and Aj_(0 ) = 3640A. The Ginzburg-Landau param eters are /ey in
l ( 0 )
the range of 110 ~ 165 and kj_ in the range of 62 ~ 74, respectively. These
yield a anisotropy factor of 7 about 2 .
We don?t believe that the above values of �s represent their intrinsic limit,
for example, we measured Ay ~ 1840A at 4K. However, others^13,14! have re�
ported Ay (0) as low as 1400A. If we assume the latter value of Ay to be intrinsic,
we extrapolate the following values for  and _, respectively, at OK: 19A and
8.3A.
Widom et
theoretically calculated the London penetration depth, co�
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172
herence length, energy gap and Tc for YBCO material from a microscopic ionic
bonding model. The results of the calculation^19] tu rn out to be |(0)=36A and
A||( 0 ) = 3194A while .( 0 ) = llA and Aj_(0 ) = 9283A, yielding an anisotropy
factor of 7 ~ 3. They are in general agreement with our measured values.
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173
Table 7.1
Summary of A, � and k
T (K)
a ||(A)
*ll(A)
KJL
a x (A )
e�(A)
-Wf-JL
K||
86.5
8000*
129*
62
18000**
47
380**
110
80
4498*
*
oD
C
74
9000**
24
370**
165
0
1840*
25+
74
3640**
lit
330
156
ot
3194
36
88
9283
12
773
260
80J
5677
76
74
16543
26
636
216
* MSR result
** E P R result
] extrapolated from 86.5K with �(T) = �(0)(1 ? f r ) -1 /2
| calculation by Widom et al.
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174
REFERENCES
[1] A. Umezawa, G. Crabtree, J. Liu, T. Moran, S. Malik, L. Nunez, W. Kwok,
and C. Swoers, Phys. Rev. B , 38, 2843, (1988).
[2] T. W orthington, W. Gallagher, and T. Dinger, Phys. Rev. Lett., 59, 1160,
(1987).
[3] U. Welp, W. Kwok, G. Crabtree, K. Vandervoort, and J. Liu, Phys. Rev.
Lett., 62, 1908, (1989).
[4] H. Jiang, T. Yuan, H. How, A. Widon, C. V ittoria, and A. Drehman,
Subm itted to Phys. Rev. B.
[5] A.C. Rose-Innes and E.H. Rhoderick, Introduction to Superconductivity,
(Pergamon Press. Oxford, 1978).
[6] G. Aeppli, R .J. Cava, E.J. Anasaldo, J.H. Brewer, S.R. Kreitzman, G.M.
Luke, D.R. Noakes and R.F. Kiefl, Phys. Rev. B, 35, 7129, (1987).
[7] D.W. Cooke, R.L. Hutson, R.S. Kwok, M. Maez, H. Rempp, M.E. Schillace,
J.L. Smith, J.O. Willis, R.L. Lichti, K-C.B. Chan, C. Boekema, S.P. Weathersby, J.A. Flint, and J. Oostens, Phys. Rev. B, 37, 9401, (1988).
[8] E.M. Jackson, S.B. Liao, J. Silvis, A.H. Swihart, S.M. Bhagat, R. Critten�
den, and R.E. Glover, Physica C, 152, 125, (1988)
[9] A. Fiory, A. Hebard, P. Mankiewich, and R. Howard, Phys. Rev. Lett.,
61, (1988).
[10] R. Cava, B. Batlogg, R. van Dover, D. Murphy, S. Sunshine, T. Siegrist,
J. Remeika, E. Rietman, S. Zahurak, and G. Espinosa, Phys. Rev. Lett.,
58, 1676, (1987).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
[11] H. How, R. Seed, C. Vittoria, D. Chrisey, J. Horwitz, C. Carosella, and V.
Folen, IE EE M T T , 40(8), (1992).
[12] D. Wu, W. Kennedy, C. Zahopoulos, and S. Sridhar, Appl. Phy. Lett., 55,
698, (1989).
[13] D. Harshman, L. Schneemeyer, J. Waszczak, G. Aeppli, R. Cava, B. Batlogg, and L. Rupp, Phys. Rev. B, 39, 851, (1989).
[14] L. Krusin-Elbaum, R. Greene, F. Holtzberg, A. Malozemoff, and Y. Yeshurun, Phys. Rev. Lett., 62, 217, (1989).
[15] J.L. Smith and F.M. Mueller, APS March Meeting, (1992).
[16] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, Proc. MRS, San
Francesco, (1992).
[17] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, J. Appl. Phys.,
(1993).
[18] R. Karim, C. Vittoria, A. Widom, D. Chrisey, and J. Horwitz, J. Appl.
Phys., 69, 4891, (1991).
[19] A. W idom, T. Yuan, H. Jiang, and C. V ittoria, subm itted to Phys. Rev.
B.
[20] G. Dolan, F. Holtzberg, C. Feild, and T. Dinger, Phys. Rev. Lett. 62,
2184,(1989).
[21] L. Vinnikov, I. Grigoreva, L. Gurevich, and Y. Osip?Yan Sov. Phys. JE TP
Lett., 49, 99, (1989).
[22] J. Clem, Supercond. Sci. Tech., 5, s33, (1992).
[23] P. de Trey, and Suso Gygax, J. Low Temp. Phys., 11, 421, (1973).
[24] D. Tilly, Proc. Phys. Sco., (London), 85, 1177, (1965).
[25] M. Tinkham , Introduction to Superconductivity, (McGraw-Hill, New York,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
176
1975.)
[26] R. Karim, S. Oliver, C. Vittoria, G. Balestrino, S. Barbanera, and P. Paroli,
J. Superconductivity, 1, 81, (1988).
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177
A PPEN D IX
SOME EQ UATIONS IN MKS UNITS
H = "
^
V xH=
VXJs
(2'3-4)
moJs
(2.3.5)
m
AI =
j
Mon sq2
h
h = ~
fa = fn + o # | 2 + ^ /? M 4 +
(2.3.7)
v
'
(2.4.1)
v - ? A ) ^ |2 + - J - B 2
(2.5.2)
Oil/) + /3\^\2xp + 2~ ( - * f r v -q A .)2if>= 0
(2.5.3)
i2 m ^ * W ? - V ? V </>*)- ^ M 2A
(2-5.4)
A(T) = ^
)
' /2=
w
- | ) ' 1/2
<2-5-5>
Q!2
fl' ( T ) = V ^
(2 '5 J )
*= If (T)
n = lqhhY/ M
^o = ^ R a2j??
(2-5-8)
f = ( ^ ) 2
(4.2.12)
U(R) = 2nRT ? (poJnR2
(4.2.13)
R n = f o J = lQn2\ 2J lnK
(4.2.15)
= U(.R?)
7TT2
= ??
(poJ
(4.2.16)
J > = ( 4 A ] ^ T < l f > 3,??2K
<4-2'18)
=
T 7 r i? ra
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178
*� = W o
2
f R�
P0 = exp[?? J
(4 2 '22)
r ________________________
\Z4:-k R ij,(2ttR t ? cj)0jTrR2)dR]
(4.2.23)
h = h
(4.2.25)
/< = ( ^ ) 2
(4.2.12)
t/(i2) = 27riZr ?<f>0J n R 2
(4.2.13)
fi" - S J ?
<4'2' 15)
2
I7? = U (Rn ) = TizRn =
(4.2.16)
(poJ
Jl = ( S ) ^ f (i r )3'n2't
* =
<4-2-18)
ik
f4-2-22)
2 Z^0 >-------------------------------P0 = e x p [ - - J
\f4TTRp(2TrRT - <f)0J n R 2)dR]
(4.2.23)
= ^ (I i � )1/2(2TV)I/*
<4-2'25?
A= J - = - 1
(5.2.4)
V Z ^a ?
=
v f e
<5-2'5)
^ = 4 ,/S w r f A 2 = 4 V 5 5 S W
= j j = i
d
^5'2'7'
k
(5-2-8)
= z � % - ^ lrui
(5-2'9)
M? =
(5.3.3)
= ?\fU"{r)/Mn
=
A
(*L
i/rf y Mo
( 5 .3 . 5 )
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179
_ / W o ( ^ )2 )
p = e x p ( - l 2- \ = e x p t
Mlo
167tkS V
47r
J
(5.3.6)
y
c
(5.3.12)
2<7?Ti(47rA)4
Jc = - r , %
In (^f-),
q \ 2HoLeff
f
(< L.? < A
in ( h � L )
(5.4.4)
4tt2Aj_ 1 4� J
(5.4.6)
Jc = ?
w ref~f l< ^ 4�
r '>
Hoq^2L
& 7
R3- Xa- ] / ^ ~
(6.2.3)
(6.2.4)
(iQUJcr
dH
1
a T = ,?0'7
_
S
(6.2.6)
1
fiQUjX2
(6.2.10)
(6.2.12)
<Te f f =a n ~ i ^
d 2E
~fa2 = W o W e f f E
(6.2.13)
k = \ J ^ 2 + *>oWCTn
(6.2.15)
?a ?
&
< je / /
inoutJnX2
?
j/ioW
(6.2.17)
,
A;
i n o u a n X 2) - 1 / 2
(6.2.18)
= (y'l + (/J,ou;crn \ 2)2 e't>p) 1l 2
(6.2.19)
= z /u q o ;A (1 +
\/l +
(5.4.3)
? s | = [ j t 1 + t /1/ ,+, , 1
. , , ? )i1/2
( f l 0 U>CTn
\
s in ^ = [ ^ ( 1 ------ ?............ ?
2
2
2 ) 2
( 6 . 2 . 21 )
'
)]1/ 2
(6.2.22)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ISO
R
?
=
0 ^
x *= ^ 7T +
(
+ {^
V
aX2)2 - 1)1/2
1
+
(6-2-23)
+ 1)1/2
<6-2-24)
C?J
E = pJ + /x0A2?
(6.2.25)
E = (p + ^ 0wA2)J
(6.2.26)
癡.f = 0� + �owA2)-1
(6.2.27)
k = y/ijiQUj(jt f f = y A2 -
(6.2.28)
_
ZUqLO
%.
0
i
Zs = ?
= i/i0wA(l - l ^ ? ^ ) 2
(6.2.29)
i??
(6-2'30)
^
' \ / 1 + !' f.?(iA'-)2
=^
(\ / 1
+0
2
Zd = ?iZooCotikd) = - i Z
1)3
+
o
o
^OO
= V l7
/?. = i(i?a ?w 2As
Xs ?
= Po^ A
. , _ ( ^ ) -
'-
1
(6 -2 '31)
1 ) 2
C
o
t (6.4.17)
(6'4'18)
(6.5.3)
(6.5.4)
(6 .5 .6)
*?+
flow ( X | - B 2) l/2
A = jlQUJ
'
(7'L3)
A= - l l .
(7.1.4)
Po
Za = i/i0o;A(l + z>ow<7nA2)_1/2
(7.3.1)
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181
Z s = ifj,0u \ ( l - i -
1
W
(7.3.2)
X * + R?s
(7.3.3)
(x * - R i y / 1
(7.3.4)
figUJ
1
R- -
X, ?
jUqwA
1)112
V5 7 T
+ ( iM,v?wX?Y ( ^ \ + (n0anu>\2)2 + l ) 1^'
(7.3.5)
(7.3.6)
(7.3.7)
x * = ^ v 1+<^
)2+1)?
A = ? X s = ? La
Hou
fj,o
p?
2
fi ow
X 3R 3 ?
2
2
p0
L 3R 3
Ap = ? L 3A R S
^0
(7.3.8)
(7.3.9)
(7.4.1)
(7.4.2)
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182
VITA
Hua Jiang was born on november 20, 1956 in Sichuan, China. He gradu�
ated from W eiyuan middle school in 1975. He was adm itted to the University
of Science and Technology of China in March, 1978 and gradated with Bachelor
of Science degree in Physics in 1982. In the same year, he was employed in the
Institute of Physics, Chinese Academy of Sciences. During the four years in
the Institute, he conducted his research in the superconductivity. He was one
of the members worked in the N b^Sn superconductor group which received a
third prize of National Science and Technology Development in 1986, awarded
by Chinese Academy of Sciences. He began graduate studies in Physics D epart�
ment at Fisk University and Vanderbilt University in August 1986, and received
his first M aster degree in July, 1987. He entered Physics D epartm ent at North�
eastern University in September, 1987, passed the doctoral qualifying exam in
September, 1988, and received his second M aster degree in June, 1989. Since
the end of 1988, he has joined the Microwave Materials Laboratory in Electrical
Engineering D epartm ent as an interdisciplinary Ph.D candidate (Physics and
Electrical Engineering), working on the transport and microwave properties of
YBCO high Tc superconducting films. He is a member of American Physical
Society, of American Material Research Society and of Chinese Physical Society.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
, Phys. Rev. B, 39, 9153, (1989).
[19]
A. Houghton, R.A. Pelcovits and A. Sudbo, Phys.
Rev. B, 40,
6763,
(1989).
[20] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2, (Pergamon
Press, Oxford, 1980).
[21] A. Widom and T. Clark, Phys. Rev. Lett., 48, 63, (1982).
[22] H. Jiang, A. Widom, Y. Huang, T. Yuan, C. V ittoria, D. Chrisey, and J.
Horwitz, Phys. Rev. B 45, 3048, (1992).
[23] R. Feynman, in Progress in Low-Temp. Phys., edited by C. Gorter, (NorthHolland, Amsterdam, 1955).
[24] K: K. JT h x a p e n. H3n. nucui. yi. 3ao.?Pajino(jiii3iiKa, 14, ,Nb fi, !)19. (1071).
[25] I. Zitkovsky, Q. Hu, T. Orlando, J. Melngailis, and T. Tao, Appl. Phys.
Lett. 59, 727, (1991).
[26] J. Gavaler, A. Santhanam , A. Braginski, M. Ashkin, and M. Janocko, IEEE
Trans. Magn., 17, 573, (1981).
[27] B. Oh, M. Naito, S. Arnason, P. Rosanthal, R. B arton, M.R.Beaseley,
T.H. Geballe, R. Hammond and A. Kapitulik, Appl. Phys. Lett., 51, 11,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
(1987).
[28] J. M annhart, P. Chaudhari, D. Dimos, C.C. Tsuei and T.R . McGuire,
Phys. Rev. Lett. 61, 2476, (1988).
[29] B. Roas, L. Schultz and G. Endres, Appl. Phys. Lett. 53, 1557, (1988).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
CH APTER VI
SURFACE IM PEDANCE
6.1
IN T R O D U C T IO N
One of the most im portant applications of high Tc superconductors is the
microwave applications. YBCO films are now comparable to, or even better
than, conventional superconducting films in terms of their respective surface
resistance at microwave frequencies. High Tc superconducting materials have
lower loss and lower dispersion when compared to gold (and other good conven�
tional conductors) up to lOOGHzt1?2!. This makes high Tc superconducting films
excellent m aterials for the fabrication of microwave devices^3-14!, such as res�
onators (cavity, stripline, coplanar, ring), circulators, filters, mixers, antennas,
and SQUIDs. For such applications, the most im portant m aterial param eter
is the surface impedance. On the other hand, microwave techniques are useful
techniques to investigate the physical properties of high T c superconductors.
T he microwave surface impedance, for instance, is a very im portant quantity
to characterize for superconductors.
Since the discovery of superconductivity in high T c m aterials of YBCOt1^,
the measured value of the surface resistance, R 3, at microwave frequencies has
decreased steadily^16-21!. The reduction in R s may be a reflection on the im�
proved quality of films or perhaps due to the improved measurement techniques.
This implies th at we may be approaching th e intrinsic limit of R s for YBCO. In
this chapter we employ a novel microwave measurement technique to measure
R s directly and X s, the surface reactance, indirectly. In section 6.2 the surface
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116
impedance of a superconductor is calculated in the cases where where conven2
tional two fluid model is assumed (J = (an ?i incUJ^2 )E, where an is the normal
conductivity and A is the London penetration depth), and a modified two fluid
model is assumed (E = (p +
where p is the normal resistivity of the
material). In section 6.3 techniques for the measurement of surface impedance
are reviewed. In section 6.4, our MSR (microwave self-resonant) technique is
discussed in detail. The experimental results are discussed in section 6.5 fol�
lowed by conclusions in 6.6.
6.2
C A L C U L A T IO N O F S U R F A C E IM P E D A N C E
Before going to the superconductor case, it is worthwhile to review the
surface impedance of normal metals. In normal metals, the electrical behavior
is described as
J = crE.
(6.2.1)
The surface impedance is defined as
Z s = R s + i X 3,
(6.2.2)
where R s is the surface resistance and X 3 is the surface reactance, and
RS= XS=
(6.2.3)
where the skin depth is defined as
6= \
V 27TW(7
(6.2.4)
In the superconducting case, there are two types of carriers, normal elec�
trons and superelectrons. Normal electrons will be scattered by phonon or
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117
impurity while superelectrons are nondissipative. We firstly consider an elec�
trodynamic problem of superconductors in which the effects due to the presence
of normal and superelectrons will be taken into account in a two fluid model.
We calculate the field distribution for a plane surface incident normal to the xaxis, taking E and H along the y and z axes, respectively. Therefore, Maxwell?s
equations can be w ritten as follows
dE
dx
dH
dt ?
(6.2.5)
and
dH
4x
dx
& - c2 '
<6-2-6>
The time-harmonic uniform field can be w ritten as
H = H Qeiut,
(6.2.7)
so that (6.2.5) becomes
dE
- f a
( 6 .2 . 8 )
=
In the two fluid model,
j = Jn + Js =
(<7n ?icra)E,
(6.2.9)
where J n and J s are the normal and supercurrent, respectively, a n is the normal
conductivity and a s is defined as
"? = S ? '
( 6 -2 1 0 )
This configuration is shown in Fig. 6.1a wherean and as are connected in
parallel. (6.2.9) can be rew ritten as
J = aef f E ,
(6.2.11)
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118
where
c2
Combining (6.2.6), (6.2.8) and (6.2.12) we have
d2E
i&TTuicref f ?
w = ?
(6-2-i 3>
E = E 0e kx,
(6.2.14)
1
2?47ro;CTn
k= y j2 + ??
(S'2-13)
and (6.2.13) yields a solution
where
is the wave vector constant of the electromagnetic disturbance. The surface
impedance of a material is defined as
-E(O)
= 'r鞍 tJ �
6-2' 16)
Jo ^
and (6.2.16) yields
Putting (6.2.15) into (6.2.17) we get
_
i4 7 ro ;A .,
i47ro;crn A2 .
, /9
Z 3 = ? ? (! + ------ Y ? )~112.
c*
,
(6.2.18)
In order to separate Z s into real and imaginary parts, we write the denominator
as
1+
= y
+
(6.2.19)
with
eIV5/ 2 = c o s~ + isin^-,
�
(6.2.20)
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119
where
co s% = � ( 1 +
2
^
= ) } 1/ \
(6.2.21)
= ) ] a/ 2.
(6.2.22)
' v/l + (l2a^Ai)2'
and
s i V = [ I ( ! -----
2
?
}
v l + (i�!T ^ 2
W riting Zs as Z s = R s + i X 3 with R a being the surface resistance given by
_
R ? =
and
1
- M ?
47TU;A
f
=
I
d '
V 2 c2 J l + ( l H ^ ) 2
ATCCTnCvX2
(v 1+ (
V
A
C
) ~ 1)
.
(6.2.23
being the surface reactance by
/ 4T A ?
d l + F f Z - r + lt'* .
v 2 c2 ^ 1 + ( 4 * 2 ^ 1 ^ V
c
Experimentally, we found th at
(6.2.24)
was roughly independent with tem per�
ature when T < 80JsT, (which will be seen in section 6.5, Fig. 6.16). This can
not be explained by two fluid model, where a superconductor is represented by
two conductors in parallel, we consider now a superconductor represented as
in Fig. 6.1b. One can see th a t on and cr3 are connected in series here instead
of in parallel in the two fluid model. The constitutive equation now becomes
T 4ttA2 83
E = ',J + ?
ap
(6.2.25)
E = (p + ^ ! ) J ,
(6.2.26)
or
where p is the normal resistivity and equal to l/<7?. The effective conductivity
reads
^ / = ( p + ^ ) - ?.
(6-2.27)
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120
Jn
Fig. 6.1a Diagram of two fluid model. a n and cra are in parallel connection.
J
On
Fig. 6.1b An alternative assum ption where crn and <r3 are in series connection.
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121
(6.2.26) along with (6.2.6) and (6.2.8) implies that the electromagnetic distur�
bances within the superconductor propagate with a complex wave vector k
,i47cw
ic2p
* = \ / ^ r - - W / = \ A 2 - z47^u;,
�
(6.2.28)
and this translates into a surface impedance of
_
i47rw
H ttujX .
.
C2 p
,1
Z- = 1 5 F = ?
,
(� * � >
The real and imaginary parts of (6.2.29) are respectively
and
<6-2-3�
6.3
TECH NIQ UES OF SURFACE IM PED A N C E
M EA SU REM EN T
6.3.1
Early Surface Im pedance M easurem ent
In 1940, London^22! measured the non-zero resistance of a tin ellipsoid in
the superconducting state, when a microwave field (1.5 GHz) was applied to
the sample. This implied that the resistance he measured was different from
DC resistance. Probably, it was the first observation of surface resistance of a
superconductor. P ip p a r d ^ carried out an experiment to systematically study
surface impedance of superconductors in 1947. The superconducting specimen
was in the form of a narrow loop of wire attached to a distrene rod, and may be
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considered to be a section of a transmission line, open-circuited at the bottom
and short-circuited at the top, so th at it resonated when it was approximately
a quarter wave length long, see Fig. 6.2. The measurement of surface resistance
was carried out by determining the frequency band width at resonance, Aw;
therefore the Q was determined from
<3=
(6.3.1)
where w0 = 2tt/ o and /o is the resonant frequency of the resonator. The Q
value is roughly inversely proportional to the total resistance of the resonator
circuit. For theirl23! system,
Q= ^
(6.3.2)
where L is the inductance and R the resistance of the whole circuit. If the
only source of power dissipation was the resistive loss in the superconductor,
the value of Aw would be directly proportional to the surface resistance of the
material, which was the quantity to be measured. In fact, however, there were
a number of other sources of power loss, which might be regarded as resistances
in series with the surface resistance of the superconductor, and which would
limit Q to a finite value even if the superconductors were perfect. To find the
intrinsic Qo, or unloaded Q, which depended only on the superconductor and
dielectric losses in the resonator, Pippard used a linear equation
^
+ t = l,
(6.3.3)
where Q is the measured value and t is the voltage transmission coefficient.
When t = 0 the measured value of Q is the ideal Q of the system. Pippard used
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123
German silver tube
Coaxial line
Sample
Sheath
Fig. 6.2 A resonator used for measuring surface impedance of superconductor.
(after A. Pippard)
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124
this m ethod to measure the surface impedance of P b and Hg. Although the
accuracy could not be high, b u t, nevertheless, the measurement was initiated
by Pippard?s group.
6.3.2
Strip Line, M icrostrip, and Coplanar Resonators
Strip line, microstript2^ and coplanar^8! resonators fall under the same
analysis developed by Pippard^23!. In these techniques, the superconducting
film under investigation needs to be patterned into different configurations.
One of the advantages of a coplanar resonator over a microstrip or strip line is
that both the conducting and ground planes are in the same film plane. The
others have the ground plane physically removed from the conducting plane.
Fig. 6.3 shows a coplanar resonator.
In the coplanar resonator technique, one measures the reflection coefficient
5 n from the resonant circuit shown in Fig. 6.3. The amplitude of 5 n is given
as follows![S1.
|<? , _ r( ^ - l ) 2 + 4 Q o ( / / / o - l ) 2 1/2
1 111 [( K + l)2 + 4 Q 0( / / / o - 1)2] ?
(6?3?4)
where
/o = resonant frequency of resonator,
K = coupling coefficient,
Qo= unloaded Q.
The change in the inverse of Q q is related to the total resistance, R t ,
A (-r?) oc R s + R ( + R r + R w = R t -,
Q
(6.3.5)
o
where
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125
Fig. 6.3 Coplanar resonator.
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126
R s = surface resistance of superconductor,
R e = related to loss tangent of substrate,
R r = related to radiation resistance, and
R w = related to surface wave losses.
Usually, in order to measure R s reliably and accurately w ith this technique
one needs to minimize R e + R r + R w. Although this technique can be used to
measure the surface impedance of a superconducting film, its preferred use is
to determine the penetration d e p th ^ .
6.3.3
Cavity Technique
Cavity resonator techniquest25-28] are most often used for surface impedance
measurements. Different from the linear resonator described in (b), a cavity
resonator has an advantage of not patterning the sample. The fundamental
resonance depends on the geometry of the cavity itself. The sample placed
in the cavity only perturb the fundamental resonance. A cavity can be used
for measuring the surface impedance of a superconductor of any shape. If the
entire cavity is built from superconducting material, it will give the greatest sen�
sitivity achievable, limited by the properties of the superconducting material
only. To overcome the lim itation of copper absorption background, an improved
Pb-plated high Q copper cavity operated at an ambient tem perature of 4.2K
was reported^26]. The superconducting sample, m ounted on a sapphire rod, was
placed at the center of the cavity, and thermally insulated from the cavity walls,
enabling variability of sample tem perature. A diagram of the measurement is
shown in Fig. 6.4.
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127
SYNTHESIZER
PIN DIODE
PULSER
PULSE
GENERATOR
FAST SCOPE i
DETECTOR
COUFUMC UXD
n ru n s
CAVITY
He,
> is m i.
TE011 = 8 .5 6 7 GHz
TM
9 61 7 GHz
Cu
mm
Fig. 6.4 A block diagram of cavity m easurem ent. ( after S. Sridhar et al.)
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128
Basically, the loaded Q and change in resonance frequency, A /, is mea�
sured. The loaded Q may be related to the unloaded Q by determining the
coupling, K, in a similar manner as described in (b). In the cavity technique^26!
R r = R w = 0 so that
yo
oc
R s + R c + Rh,
(6.3.6)
where R c is the surface resistance of the cavity walls and Rh is the contribution
from the sample holder. Presently, superconducting cavities are used so that
(R c -f-Rh) ~ R a- From fo it may be possible to deduce L s + L c, where L s is the
surface inductance of the superconducting specimen and L c the inductance of
the cavity walls and self inductance. Usually, L c is bigger than L a by as much
as 103.
The limitation of this technique is the Q of the background. It is possible
th at the losses in such a cavity would be dominated by resistance encountered
by current flow around the joints connecting the separate pieces of walls. It
is technically impossible to machine the entire cavity from bulk superconduc�
tors under investigation, which would give higher Q values and eliminate the
contribution of background.
6.3.4
Parallel Plate R esonator Technique
Taberl17! proposed an improved technique over the cavity technique. In
this technique, a parallel plate resonator was formed with a thin electric spacer
placed between two flat superconducting surfaces. The plates were intended
to be congruent but their exact shape was not crucial. Square or rectangular
pieces were convenient. No electrical contact was made to the edges of the
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129
plates, so the edges presented open circuit boundary conditions for transverse
electromagnetic modes that can be excited between the plates. A schematic
diagram is shown in Fig. 6.5. The samples were pressed together using dielectric
posts and placed approximately in the center of the test chamber. The test
chamber was made from brass and was gold plated to provide low surface loss.
For this technique again the unloaded Q and resonant frequency was mea�
sured. Here, R c = Rh = 0, but one needs to contend with the dielectric loss
tangent. Specifically, Taber^17^ found that
1
? = tanS + /3Rs/ S + aS,
Vo
(6.3.7)
where S was the spacing between the parallel plates, a and /? were coefficients
depending on the geometry of the resonator and tanS = e"/e' (e = e' ?ie").
As stated by Taber, S and tanS could be purposely chosen to make the second
term in (6.3.7) the dominant term. Thus, the lower limit of R s depended on
the selection of substrate material. It might be possible to deduce imaginary
parts of the surface impedance by using:
(6-3-8)
where To is the characteristic admittance. L is the length of transmission line
and v is the phase velocity which can be expressed in terms of A (penetration
depth) as
I 2 A d
V = vQ\ l l + - c o t h ( - ) ,
(6.3.9)
where d is the thickness of the film.
This method can measure surface resistance as low as 10
at 10GHz.
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130
SHORT MICROSTRIP LINES
ATTACHED TO SEMLRI01D
COAX LINES
DIELECTRIC
SPACER
DIELECTRIC
SU PPO RT POST
-SUPERCONDUCTING
SURFACES
Fig. 6.5 A schematic diagram of parallel plate resonator technique.
( after R� Taber)
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131
6.4
DEVELOPMENT OF MICROWAVE SELF-RESONANT
(MSR) TECHNIQUE
We have developed a novel microwave self-resonant (MSR) technique^29?30!
to measure surface resistance, R s, directly and surface reactance, X a, indirectly.
Fig. 6.6 is a schematic diagram of the measurement. In this method, a super�
conducting thin film strip (4.24x0.9mm) of YBCO was placed in a waveguide
whose cross section was 10.67x4.32mm. The strip was placed at the center of
the waveguide as shown in Fig. 6.7a. The transition tem perature of the film
was about 90K and the critical current density was of the order of
10� A / c m 2
at
77K. The films were c-axis oriented and the thickness was typically 0.5 f i m . A
HP8510B vector network analyzer was employed to measure the transmission
coefficient (am plitude and phase) when a microwave was propagating through
the superconducting film strip. The equivalent c ir c u it^ of strip and waveguide
transmission line is shown in Fig. 6.7b. The capacitance, C , was due to the
gap between the waveguide and the strip. A typical gap value was 0.04mm
and it corresponded to C ~ 10-12/ . No systematic study was made to relate
the gap width w ith C. However, besides the gap the dielectric constant of the
substrate on which the YBCO film was deposited affected the value of C. We
empirically varied the width of the gap until transmission microwave resonance
was observed between 18 and 26 GHz - the fundam ental band of propagation
in the waveguide.
The inductance L was a combination of the superconductor intrinsic in�
ductance, L s, and the self inductance, To, calculated approximately asf32J
_
Zq a
f
r . 37r, s i n x . , 9 .
" SA I77?T7?lsmT (?
u>.
. , .
(6-4-1)
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132
RF
DETT
RF
[SOURCE
VECTOR
ANALYZER
H PS510C
DEW AR
LHe
S A M PL E
PRINTER
WORK
STA TIO N
Fig. 6.6 Microwave self-resonant measurement arrangem ent.
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Fig. 6.7a Diagram of microwave self resonator.
Fig. 6.7b Equivalent circuit of microwave self-resonator.
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134
where
37T w
and
w = 0.9mm,
a = 10.67mm,
b = 4.32mm.
fo is the transmission resonant frequency, c is the speed of light and Z 0 is the
characteristic impedance of the waveguide and was determined to be equal to
43 Q from our calibration runs. A typical value of L 0 was ~ 10-11 h. L s was
the inductance due to the superconducting screening currents. In comparison
to L 0, L s was about a factor of 100 lower. R s was the surface resistance of
the superconducting film.
From the equivalent circuit of Fig. 6.7b, the reflection, S n , and transm is�
sion, S 21, coefficients may be calculated as follows,
(6.4.2)
and
(6.4.3)
where (j>n and <f>2i are the phase angles of S \\ and Shi- For the equivalent
circuit of Fig. 6.7b,
Z = R s + i(ujL - ^ ) ,
(6.4.4)
putting (6.4.4) into (6.4.3), we get
(6.4.5)
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135
where
w0 = -� =
Vlc
(6.4.6)
is the transmission resonant frequency. Letting
UJ
U)o
? =
(6.4.7)
(6.4.8)
and
b2 = � ( f ) 2,
(6.4.9)
_
ax + i(x2 ?1)
21 = (a + 6)x + f(a:2 - l ) '
,
(6.4.10)
(6.4.5) can be written as
The amplitude and phase are respectively (we used IS21I2 instead of IS21I for
simplifying the fitting procedure) and
(5 A 1 1 )
<j>2i = ta n -1 ( - ? ? ) - tan~1( 1 X ).
ax
(a + o)x
(6.4.12)
By fitting the experimental curves, the parameters a and 6, and therefore R s
and L, can be deduced. At resonant frequency, the imaginary p art vanishes;
hence, Z = R s, the surface resistance. R s may be determined from (6.4.3)
directly,
R
�
=
2 [1/|� 2i | ?1]'
(6A13)
It is noted th at in (6.4.13) there is no other loss param eter in the expression.
Loss tangents of the substrate have minimal effects on the measurement of R 3.
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136
For example, we have included realistic loss tangents for a typical substrate
m aterial of MgO or SrTiO s, etc.. and calculated jS'21 J. The correction to R s
(using (6.4.13)) from the loss tangent contribution was calculated to be about
0.04% at 80K (using MgO as an example).
Since most of the samples we measured are thin films, the size (thickness)
effect should be taken into account.
The transformation m atrix of a wave
propagation through a film medium is
f &i \ _ f coskd
\h i J
\-^sin k d
where
i Z s i n k d \ ( e2 \
coskd / \ ^ 2 /
.
.
? ?
and e<i are the surface electric fields, while hi and fi2 are the surface
magnetic fields at the two film surfaces, respectively, k is the wave vector of the
propagation, d is the thickness of the film. (6.4.14) can be w ritten as
Denote
= Z j and
ei
^-coskd + iZ sin k d
hi
� j^ s in k d + coskd
(6.4.15)
= Z q, the measured and characteristic waveguide
impedances, respectively. Rewriting Z =
Zj,
Zoo
(6.4.15) becomes
Zocoskd + iZoosinkd
iZosinkd + Zoocoskd'
(6.4.16)
It can be proven th a t for a thin film (6.4.16) becomes approximately
Zd = -iZooCot(kd) = ?iZooCot(
47vuid
C
).
(6.4.17)
Z /Q Q
The subscript 00 is to denote th at as d ?*? 00 , Z ?>?Z,OO*
Z 00 = R s +j{uJL - Z - ) .
(6.4.18)
As in any development of a new technique, we calibrated the measurement
with standard materials, such as films of copper, silver, etc.. We placed strips
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137
of copper, silver, aluminum and bulk YBCO in the waveguide for the purpose of
determining Zo and to measure the conductivity of each material. Conceivably,
I/o could have been determined from the calibration runs, since L q was not
sensitive to whether or not the m aterial was superconducting. However, this
required identical strip dimensions and substrate m aterial for both YBCO and
copper films, for example. We did not exercise this option. We determined
L q from /o and lineshape fitting of IS21I versus frequency for each run under
consideration.
In Fig. 6.8 we plot IS21 I as a function of frequency for a copper strip
of 4.24 x 0.9mm. The thickness was about 0.05mm. A least squares fit was
obtained and the deduced param eters were as follows,
I/0 = 5.38 x 10~n h,
and
C = 1.08 x 10~12/.
For Cu, Ag and Al, the surface resistance was determined from
R * = V CTC4
and it agreed with expectations.
( 6 -4 -1 8 )
Since u> was determined precisely, cr, the
conductivity was measured accurately. Thus, Z q may be finally determined
from (6.4.10) and was equal to 4312. It changed little w ith tem perature. It is
worthwhile to point out th at the Zo calculated from (6.4.10) is in very good
agreement w ith the result from fitting the amplitude and phase curves of the
transmission coefficient, S 21?
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138
ap
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139
6.5
EXPERIM ENTAL RESULTS A N D DISCUSSION
We have measured R s and X 3 by using the MSR technique^29!. Fig. 6.9
shows typical amplitude and phase of transmission resonance curves as a func�
tion of frequency observed at 15K for YBCO film strip. In Figs. 6.10 and 6.11
we plot measured IS21 I and, fax, as a function of normalized frequency at a
temperature near Tc. The d ata are represented by points and the least square
fit by a solid line in both Figures. The following parameters were deduced from
the fit at T=86K: L = 2.8 x 10~ u h, C = 2.05 x 10~12/ , and R s = 0.052S2. We
can deduce R 3 from either curve fitting or using (6.4.10), but the determ ination
of X 3 is nontrivial. To deduce X 3, we need to separate L s from L. Assuming
the frequency shift with respect tem perature is due to the change of the surface
inductance of the superconducting film strip, it is approximated th at
ui = ?^ - -= = ~ ioo ?
y/{L + A L)C
2 L
(6.5.1)
v
'
Rewriting (6.5.1) we have
L S(T + A T ) - L S( T ) = - 2 ^ - L .
Jo
(6.5.2)
The valuesof L a were deduced from (6.5.2). Frequencyshifts weremeasured
with respect to the resonant frequency at 4K, see Fig. 6.12. L 3 at 4K can also
be deduced from the surface resistance, R 3. When T < < Tc,
R s = 4-7r2<7?u;2A3,
c4
(6.5.3)
X 3 = u L 3 = -^o;7rA.
c2
(6.5.4)
and
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140
S 2 1/ M 1
REF 0.0
7 .0
lag
MAG
S2 j/Ml
REF 0.0 �
5 0 . 0 �/
dB
dB/
START
STOP
16.000000000
26.000000000
GHz
GHz
Fig. 6.9 Microwave resonant spectra of a YBCO strip taken at 15K.
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141
-2 0
-3 0
-4 0
0 .8 5
0 .9
0 .9 5
1
1 .0 5
1.1
1 .1 5
a /u Q
Fig. 6.10 Amplitude of transm ission coefficient as a function of normalized fre�
quency at 86K for a YBCO strip. The resonant frequency is 21GHz.
Solid line is fitting curve.
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142
0 (H ad.)
0.5
-0 .5
-
1.0
0.9
1
1.1
1.2
w / cjq
Fig. 6.11 Phase angle of transm ission coefficient as a function of normalized
frequency at 86K for YBCO strip. The resonant frequency is 21GHz.
Solid line is fitting curve.
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143
0.0
?t?i
1V
I ' 11
1 1 1 1 1 1 1 1 i i i j i i ?i ??i?
-
-
+
-0 .1
+
? -
(G H z)
-
-
_
+
+
+
-0 .2 --
f0= 2 1.31 GHz (4K)
-
+
?
+
+
-0 .3 --
?
+
-
-0 .4
?F
0
?
I * i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i
20
40
60
60
T (K)
Fig. 6.12 Frequency shift of a YBCO strip w ith respect to the resonant fre�
quency at 4K.
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144
Therefore,
L s = ( - ^ ^ - ) 1/ 3.
u>l cza n
(6.5.5)
K
'
The 4K point served as a reference point for the rest of the data of L s at higher
tem peratures. A plot of L s versus tem perature is shown in Fig. 6.13.
Size effects of R 3 and X s were calculated by using (6.4.16). Demonstra�
tions of the thickness dependent on R s and X a are shown in Fig. 6.14 (for
tem perature at 70K) and Fig. 6.15 (for T = 86.5K), respectively. One can see
that the corrections get more im portant when d (the thickness of the sample)
get smaller. Our samples were about 5000A thick, so it was necessary to take
the size effect into account. We have included the measured and corrected val�
ues of R s and X 3 in table 6.1 for different tem peratures. One can see th at
the correction of surface resistance, R s, is negligible in the whole range of tem �
perature below Tc, but the size effect of surface reactance X 3 is significant,
especially when T ?> Tc. The correction of X 3 was as large as a factor of 3,
since at high tem peratures the microwave penetration depth was comparable
to, or even greater than, the thickness of the films.
In Fig. 6.16, R 3 and X 3 are plotted as a function of tem perature. One
can see th at R 3 decreased by about 3 orders of m agnitude as the tem perature
decreased from 90K to 80K reaching a low value of 4.9 x 10_4O. At 15K,
R s was 2.7 x 1CT40 , the lowest value measured using the MSR method. If
we extrapolated^19] R s to lower frequencies we found th at R s was equivalent
to 100/rfi at 10 GHz and 8.7fiO, at 2.95 GHz. The tem perature was assumed
to be equal to 77K. However, at 15K, the extrapolated values would be 61 /j.Q,
at 10 GHz and 5.3
/ J .Q ,
at 2.95 GHz. A comparison w ith other measurements
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145
30
20
+ ++
0
20
40
60
80
T (K)
Fig. 6.13 Surface inductance of a YBCO strip as a function of tem perature.
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146
Solid line : R
Dashed line: X
70K
x & r (n)
i- l
i-2
i-3
0
0.2
0.1
Fig. 6.14 Surface impedance
e ls
0.3
d (fim)
0.4
0.5
a function of f i l m thickness taken at 70K.
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147
10.0
Solid line : R
5.0
Dashed line: X
T = 86.5K
?
�
X
1.0
0.5
0
0.1
0.2
0.3
d (//m)
0.4
0.5
Fig. 6.15 Surface impedance as a function of film thickness taken at 86.5K.
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14S
10�
& r ? (n)
-w-
++
++
0
20
40
60
80
T (K)
Fig. 6.16 Surface impedance of a YBCO strip as a function of tem perature. In
the X 3 curve, there are few points not plotted around 60K, because
there were some frequency shift due to the system background at that
tem perature range.
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149
of R s is given in table 6.2 and 6.3. We noticed that our measured values of
R s exhibited an oscillating behavior with tem perature, although L s did not.
Bonnl19! et al. observed an oscillatory behavior of R s at tem peratures between
10-60K, and they claimed that the behavior was from the nonlinear conductivity
of the superconducting film at tem peratures below Tc. We don?t attribute the
oscillatory behavior of R a to an intrinsic process at this stage, since lower values
of R s have been measured at lower frequencies (lOGHz)!17?18^ and no oscillatory
behavior of R 3 was observed.
6.6
CONCLUSION
We have calculated the surface impedance by using both two fluid and mod�
ified two fluid models. We have developed a novel microwave technique to mea�
sure surface impedance. This technique has an advantage over other techniques
for th at there are no background contributions to the surface impedance mea�
surement. The sensitivity achieved by this technique is better th an the linear
and cavity resonators and comparable to th at of the parallel plate resonator^17!.
We have used the microwave self-resonant (MSR) method to measure both real
and imaginary parts of the surface impedance of YBCO superconducting films.
The lowest value of surface resistance was 2.7 x 10- 4 fl at 21 GHz and 15K.
The extrapolated values would be Qlfj.0, at 10GHz and 5.3jiCl at 2.95GHz, com�
pare to published values^17?19! of 56^0, at 10GHz and
at 2.95GHz. By
measuring resonance frequency shift with respect to the resonance frequency at
4K, we measured the surface reactance indirectly, the lowest value of surface
reactance was 0.031 fh We also estim ated the size effect of surface impedance.
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We found that when the thickness of the measured film was comparable to
the microwave penetration depth, the size effect correction of surface reactance
became im portant, while the size effect of surface resistance was less significant.
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151
T able 6.1
Size effect corrections of R a, X s and L 3 of YBCO film
Results without correction
W ith size effect correction
T(k)
L s( 10~13H )X3()
Rs(tt)
X .(Q )
L S(10~13H
4.00
2.30002
0.0310
0.00048
0.000481
0.0312
2.330
15.0
2.59596
0.0350
0.00027
0.000271
0.0353
2.640
20.0
2.73390
0.0366
0.00054
0.000543
0.0370
2.766
27.0
2.89102
0.0387
0.00070
0.000704
0.0393
2.939
30.0
2.99004
0.0399
0.00076
0.000765
0.0406
3.037
34.0
3.08806
0.0413
0.00088
0.000887
0.0422
3.157
40.0
3.16697
0.0423
0.0C110
0.001110
0.0433
3.241
43.0
3.28497
0.0439
0.00020
0.000202
0.0451
3.376
49.0
3.38400
0.0452
0.00048
0.000486
0.0466
3.489
52.0
3.45231
0.0492
0.00051
0.000520
0.0502
3.610
72.0
4.13670
0.0571
0.00049
0.000492
0.0588
4.321
77.0
4.46797
0.0592
0.00043
0.000441
0.0639
4.797
80.0
5.05897
0.0673
0.00048
0.000496
0.0752
5.653
82.0
5.65097
0.0750
0.00090
0.000937
0.0869
6.542
84.5
6.43900
0.0853
0.00280
0.002937
0.1040
7.845
86.5
7.14320
0.0926
0.05210
0.059750
0.1420
10.78
88.0
7.72002
0.1020
0.09800
0.107200
0.1880
14.22
90.0
8.80399
0.1160
0.30800
0.330100
0.3910
29.66
91.2
10.1840
0.1340
0.31000
0.331000
0.4130
31.43
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152
T ab le 6.2
A comparison of our measured surface impedance value with others?
at 77K
Technique
441
X a(Sl)
f (GHz)
0.063
21
Sample
Ref
our
100 *
10
8.7 *
2.95
B
1500
18.7
film
[21]
C
720
10
film
[17]
D
22
2.95
crystal
[19]
A
film
MSR
tech.
* We have scaled R s as oc f 2
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153
T able 6.3
A comparison of our measured surface impedance value with
others?at 4K
Technique
480*
A
f (GHz)
Rs{fJ-tt)
0.031
Sample
21
Ref
our
108**
10
film
MSR
9**
2.95
B
60
18.7
film
[21]
C
56
10
film
[17]
D
15 (1.7K)
2.95
crystal
[19]
E
<400
10
crystal
[16]
F
12~180
10
film
[20]
G
20~ 43
10
film
[18]
tech.
* We measured R s ? 270[id at 15K and at 21GHz
** We have scaled R s as a f 2
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154
REFERENCES
[1] N. Klein, G. Muller, H. Piel, B. Boas, L. Schultz, U. Klein and M. Peiniger,
Appl. Phys. Lett., 54, 757, (1989).
[2] M. Namordi, A. Mogrom-Lampero, L. Turner and D. Hogue, IEEE Trans
M T T ., Sep. (1991).
[3] R. S. W ithers, A. Anderson, and D. E. Ontes. Solid State Technology,
83-87, Aug. (1990).
[4] E.F. Belohoubek, Defense Electronics 82-86, Jan. (1990).
[5] J. Bybokas, Supercurrents, 72-76, July (1990).
[6] P. A. Ryan, J. of Electronic defense, 55-59, May (1990).
[7] D. E. Oates, A. Anderson and P. Mankiewich, J. Supercontivity 3, 251,
(1990).
[8] H. How, R. Seed, C. Vittoria, D. Chrisey, J. Horwty. C. Carosella, and V.
Folen, IEEE Trans M T T, 4 0 (8 ), (1992).
[9] Y. Huang, H. Jiang, A. Widom, and C. Vittoria, IEEE Trans. Mag., 1992).
[10] C. Zahopoulos, S. Sridhar, J. Bautista, G. Ortig, and M Laragan, Appl.
Phys. Lett., 58, 977, (1990).
[11] Y. Yoshisato, M. Takai, K. Niki, S. Yoshikawa, T. Hirano, S. Nakano,
IEEE. Tras. Mag., 27, 3073, (1991).
[12] Y. Huang, M. Lancaster, T. Maclean, Z. Wu, N. McN.Alford, Physica C,
180, 267, (1991).
[13] Y. Huang, H. Jiang, H. How, C. Vittoria, A. Widom, and R. Roerstler, J.
Superconductivity, 3, 441, (1990).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
[14] M. Nisenoff, in Principles & Applications of Superconducting Quantum In �
terference Divices, ed. A.Barone, (World Scientific Pub., Singapore, 1990).
[15] K.W. Blazey, K.A. Muller, J.G. Bednorz, and W. Berlinger, Phys. Rev.
B
, 36, 7241, (1987).
[16] D. Wu, W. Kennedy, C. Zahopoulos, and S. Sridhar, Appl. Phy. Lett., 55,
698, (1989).
[17] R.C. Taber, Rev. Sci. Instrum., 61, 2200, (1990).
[18] P. Merchant, R. Jacowitz, K. Tibbs, R. Taber, and S. Laderman, Appl.
Phys. Lett., 60, 763, (1992).
[19] D.A. Bonn, P. Dosanjh, R. Liang, and W.N. Hardy, Phy. Rev. Lett., 6 8 ,
2390, (1992).
[20] D. Miller, L. Richards, S. Etemad, A. Inam, T. Vankatesan, B.D utta, X.
Wu, C.Eom, T. Geballe, N. Newman, and B. Cole, Appl. Phys. Lett., 59,
2326,(1991).
[21] N. Klein, U. Dahne, U. Poppe, N. Tellmann, K. Urban, S. Orbach, S.
Hensen, G. Muller, and H. Piel, J. Superconductivity, June, (1992).
[22] H. London, Proc. Roy. Soc., A 176, 522, (1940).
[23] A. Pippard, Proc. Roy. Soc., A 191, 370, (1947).
[24] S. Anlage, H. Sze, H. Snortland, S. Tahara, B. Langley, C.Eom, M. Beasley,
and R. Taber, Appl. Phys. Lett., 54, 2710, (1989).
[25] N. Klein, In High Temperature Superconductors, J. Pouch et al.
(Trans. Tech. Pub., Switzerland, 1992).
[26] S. Sridhar, W. Kennedy, Rev. Sci. Instrum., 59, 531, (1988).
[27] A. Portis, D. Cooke, and F. Gray, J. Superconductivity, 3, 297, (1991).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ed.
156
[28] J. Marteus, V. Hietaln, D. Ginley, T. Zipperian, and G. Hohenwarter, Appl.
Phy. Lett., 58, 2543, (1991).
[29] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, Proc. M RS, San
Francisco, (1992).
[30] H. Jiang, T. Yuan, H. How, A. Widom, and C. Vittoria, and A. Drehman,
subm itted to Phys. Rev. B.
[31] M.K. Skrehot, K. Chang, IEEE Trans. Microwave Theory Tech., 38, 434,
(1990).
[32] R.L. Eisenhart, IEEE Trans. Microwave Theory Tech., 24, 987, (1976).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
CHAPTER VII
LONDON PENETRATION DEPTH
AND COHERENCE LENGTH
7.1
IN TR O D U C T IO N
The London penetration depth and coherence length are some of the most
im portant param eters in a superconductor. The London penetration depth
which measures the length over which magnetic fields are attenuated near the
surface of a superconductor, contains information about the effective mass and
density of superconducting pairs, while the coherence length measures the size
of Cooper pairs, or size of the vortex core. They are referred in the literature
as the characteristic lengths of superconductors. One of the key reasons why
high Tc superconductors are different from conventional superconductors is that
the high Tc superconductors have a small coherence length]1_4^ (of the order of
10A) comparing to conventional superconductors^ (of the order of 1000A).
Both lengths cannot be directly measured. There are many techniquest6_14]
for measuring the penetration depth, such as muon-spin-rotation (j i +SK), po�
larized neutron reflectometry, kinetic inductance, AC susceptibility, DC magne�
tization and microwave techniques. The measured values of penetration depth
varies from sample to sample and sometimes m ethod to method. One of the rea�
sons is th at high Tc superconductors are highly anisotropic, and the orientation
dependence should be taken into account. Uncertainties about local demagne�
tizing factors, surface and grain boundary effects of polycrystalline samples also
lead to inaccurate results.
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158
Measurements of the coherence length usually are deduced from the mea�
surement of the upper critical magnetic field^2?3], H c2 . However, the H c2 of
YBCO is extremely high (> 100 Tesla at 4.2K^15^), the measurement requires
special and difficult technique. Many measurements of the cohernence length
are measuring the upper critical field at high tem perature (say, near Tc). �(0)
may be extrapolated from
�(T) = �(0)(1 -
(7.1.1)
In this chapter, we deduce the London penetration depth from the mea�
surements of surface impedance by using the microwave self-resistant (MSR)
technique^16?17!. We use both the two fluid model and modified two fluid model
to deduce the penetration depth. We use these relationships in determining A.
and
A=
P - 1-3)
respectively. At low tem peratures, X a �
R s, the two expressions converge to
the same result
c2
A=-L?
(7.1.4)
O ther than measuring the upper critical magnetic field, we developed a
m ethod based upon our microwave self resonant technique to measure the co�
herence length near Tc.
YBCO is an anisotropic material, and both the London penetration depth
and coherence length are anisotropic. Their values depend on the direction of
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159
the microwave electric field relative to the c axis. From our microwave measure�
ment, we find th at A||(0) is about 1840A, where Ay is defined as the penetration
depth in the a-b plane, and
= 129A at 86. 5K, and |(0) extrapolated from
86.5K is about 25A, where | is defined as the coherence length applicable for
microwave induced electric field (or current) applied along the a-b plane. To�
gether w ith the E PR measurement results by our group!18?4!, we have London
penetration depth and coherence length values both in the a-b plane and in
the c direction. The values in c-axis direction were 3640A and 12A at ~0K ,
respectively.
Widom!19! et al. have calculated A and � from a microscopic ionic bonding
model and general agreement was found between the calculated values and our
experimental results.
7.2
ANISO TR O PIC LENG TH S OF YBCO FILM
To understand the anisotropic property of YBCO m aterial, it is worth�
while to review its crystal structure. From Fig. 1.4 we can see th a t the C u-0
bond form a plane, called the a-b plane, which is responsible for the elec�
tronic conduction. Along the b-axis, there is a C u-0 chain, which may be also
conductive. There is no C u -0 chain along c-axis. Vortex-lattice decoration
experiments!20,21! have shown th at in YBCO the penetration depths are ap�
proximately in the ratios Aa : A& : Ac = 1.2 : 1 : 5.5. It is im portant to realize
th a t the penetration depth depends on the direction of the screening current
density, not upon the direction of the local magnetic field. The penetration
depths Aa and Ab are much smaller than Ac since the screening currents in the
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160
a-b planes (in the a or b direction) flow much more easily th an along the c-axis
direction. Moreover, A& is somewhat smaller than Aa because current parallel
to the a-b planes flows a little easier along the C u-0 chains (b-axis direction)
than perpendicular to these chains (in the a-direction).
There are several factors which will affect the measured value of penetration
depth. Beside the local uncertainties, such as surface effect, grain boundary
effect and local demagnetizing factors, the anisotropic property, therefore the
orientation, will affect the result a lot. Measurement on YBCO single crystals
gives a clear orientation dependence while polycrystalline sample average out
the anisotropy and give very different results from single crystals. However, we
noticed th at the anisotropy along the a and b axes is negligible compared to
the c-axis result. Therefore, measuring c axis oriented films will yield results
approximately similar to single crystal samples.
For anisotropic material, the average coherence length and penetration
depth are defined ast22^
� = (6 � c ) 1/3,
(7.2.1)
A = (AaA&Ac)1/3.
(7.2.2)
and
For the case ( a ~ �& and Aa ~ A&, (YBCO, for example), we can write (7.2.1)
and (7.2.2) as
� = (�!� j. ) 1/3,
(7.2.3)
A = (A|A� )1/ 3,
(7.4.2)
and
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161
where
and Ay are the coherence length and penetration
depth in the a-b
plane and _and Aj_ in the c-axis direction, respectively.They can
be defined
a s [23,24]
| = �/\/? ib
(7.2.5)
U =
(7.2.6)
Ay =
(7.2.7)
A_l = Ay/m �,
(7.2.8)
and
where m\\ and mj_ are the effective mass in the a-b plane and along the c-axis
direction. The anisotropy factor, 7 , is defined as
7 = a/ ? ,
V m l
(7.2.9)
so that
| = 7
(7.2.10)
A,| = ? Aj_.
(7.2.11)
and
For homogeneous superconductors, the Ginzburg-Landau param eter k is defined
as k = A/�. For anisotropic superconductors, however, the situation is more
complicated, and one must introduce the following definitions^23,24!,
kj .
=
(7.2.12)
�
and
/cii = 4 / ^ � ,
(7.2.13)
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162
and the relation between them is
K|| = 7玧..
7.3
(7.2.14)
L O N D O N P E N E T R A T IO N D E P T H
D E D U C E D F R O M S U R F A C E IM P E D A N C E
In chapter VI, we discussed the surface impedance of superconducting films.
The expressions of surface impedance are
Z, =
+
?
iAnusX .
(7.3.1)
and
. c2p . i
.
(7.3.2)
using two fluid model and modified two fluid model, respectively. We can solve
for the London penetration depth and obtain
c2
X 2 4- R 2
>?= Anu { X I - R2)i/2 ?
p.3.3)
and
c2
x =
<7-3-4)
for each case. Here
Ra =
1
1 _
^
^
=
1+ ( " 1 ) 2
(J l + ( "
_ ,)! /* ,
(7.3.5)
! ) 2 + 1)1/*,
(7.3.6)
> / 2 c2 ^ 1 + ( i E 2 ^ A i ) 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
for (7.3.3) in which the two fluid model is assumed; and
_
47TU>A, /
R- = ^
r
. c2p
+ ^
si
) - 1)!-
(7'3'7>
and
x *= ^
(V1 + (4 ^ ) 2+1)?-
p -3-8>
for (7.3.4), where the modified two fluid model is used. At low tem peratures,
the surface reactance is dominant; X a �
R s, therefore, (7.3.3) and (7.3.4)
converge to the same result
A = -? X 3 = ? L a.
47rw
47r
T �
Tc
(7.3.9)
v
?
We applied the two fluid model to deduce A and we found A ~ 18424A in
the a-b plane at 88K (the Tc was 92K), which gave a value of A(0) ~ 7435A.
This value is in disagreement with values A(0) ~ 1400A reported by others^13?14!.
Hence, we use (7.3.4), the modified two fluid model, to calculate the London
penetration depth by using the surface resistance and surface reactance values
deduced from our microwave self-resonant measurement. In Fig. 7.1 we plot A
as a function of tem perature. Typical values of A are 1840A at 4K and 8000 A at
86.5K, for example. We have fitted A to the Gorter-Casimir two fluid modelf25!,
A(r) = A ( 0 ) ( l - ( | ) 4) - 1/2.
(7.3.10)
The results are shown in Fig. 7.1 w ith a dashed line. If we change the power
factor in (7.3.10) from 4 to 2,
A(r) = A ( 0 ) ( l - ( ^ ) 2r ' / 2,
(7.3.11)
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164
Experimental Data
M7' ) = A ( 0 ) ( l - ( � ) ' ) - ' / 2
*c
A(T) = A(0)(l-( � ) ?)->/?
(uni) y
0.5
0.0
0
20
40
60
80
T (K)
Fig. 7.1 London penetration depth aa a function of tem perature.
Dashed
line is fitting w ith two fluid model, solid line is fitting with A(T) =
A ( 0 ) ( l - { � ) 2) - ?/ 2.
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165
we obtain a better fitting curve, see solid line in Fig. 7.1.
The penetration depth, A, was also measured in our group by using E P R or
the so called MMMA technique developed by Kariml26! et ah. Before we display
the results, it is worthwhile to review the technique briefly. Normally in an E P R
measurement
is measured, where P is the microwave power absorbed by the
sample in question and H is the m agnetic field. The E P R technique was used
in the past to measure
low H fields on superconducting high Tc material.
Karim et alS261 introduced this technique to measure j j f in 1987 in order to
measure Tc of very fine particles of YBCO crystal. In th at experiment j j j was
measured as a function of tem perature. The analysis for A was firstly developed
by K arim et a/J18] and it was based upon the Bardeen-Stephens model for
fluxoid nucleation at low magnetic fields. The basic assumption of the analysis
is th a t the microwave absorption scales as the B-field or the volume of the
normal region. From the fit to the lineshape the value of A_l / � jl was deduced.
An alternative approach in analyzing Aj_ is to define the magnetic field at which
was maximum as H ci, the lower critical field. To obtain Aj_ the following
relationship was used,
Ba=^ k lnK-
(7-3-n)
This type of measurement complements the MSR technique in the following
sense. W hereas in the waveguide technique (MSR technique) the microwave
electric field, E, is applied in the plane of the film, in the E PR experiment, E
is applied normal to the film plane. The c-axis for all of our films is aligned
perpendicular to the film plane. By combining these two microwave techniques,
we are able to deduce the penetration depth along two directions relative to the
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166
c-axis.
In Fig. 7.2 we have included A values deduced from MSR and MMMA
data. We also have superimposed the value of A(0) obtained from the shift in
the resonant frequency with tem perature using a CPW resonator. However, a
clear distinction is to be pointed out here. D ata for A generated in waveguide
and coplanar resonator measurements assume the electric microwave field to be
parallel to the film plane or perpendicular to the c-axis. This d ata is designated
in Fig. 7.2 as Ay. For the MMMA case the d ata is designated as Aj_, since the
electric field is parallel to the c-axis.
7.4
DETERM INATIO N OF C O H ERENCE LENGTH
We have measured the coherence length of YBCO films by means of the
MSR technique. Again, the films were laser ablated with transition tem perature
Tc ~ QOK and the critical current density was in the order of 106A /cm 2 at
77K. The c-axis was normal to the film plane. From the surface impedance
given in (7.3.2), (7.3.5) and (7.3.6), we can calculate the norm al resistivity of
the superconducting film and it yields
In deducing
from the MSR technique we needed to apply a magnetic field, H,
parallel to the c-axis or perpendicular to the film plane. From the microwave
transmission coefficient data, we measured only the peak values at resonance.
The surface resistance R s changed as H was varied. The surface inductance
L s, remained constant for fields up to 3KG, see Fig. 7.3, since /o remained
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167
1 1? I
1
' r
'
1
1
1
1
| 1 1 1 1 | 1 1?1?1?|?r <*, 1 "I?
2.0
1.5
-
+ f0=21GHz, MSR
-
0 f0=9.32GHz, EPR
?
0 f0= 14.5GHz, CPW
�
X
{fj)
_
+
0
+
0
-
?
-
1.0
-
o
-
0
.
o
-
0.5
_
-
? o +
0.0 ; i n
+
+
+
- J ___1.
+
_J---1.
20
+ +
J
o o o o
+ ++
---L
1 J
I
1
40
0
O
*
60
+
+
_
+
+
-
+
1 .. 1
1 .1
60
!
1
1
1
I
I
I -J__
80
100
T (K)
Fig. 7.2 London penetration depth as a function of tem perature deduced from
MSR and MMMA techniques and A(0) from CPW technique.
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168
S21
REF - 2 4 . 2 9
lo g HAG
dB
3000G
2000G
1000G
?
START
STOP
19.850000000 GHz
20.350000000 GHz
Fig. 7.3 Microwave transm ission coefficient of the superconducting film at dif�
ferent magnetic fields.
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169
constant with H. Hence, the change of resistivity due to the applied magnetic
field may be expressed as follows
c2
A p = ? L 3A R 3.
(7.4.2)
K
Using this equation, we can calculate the change of resistivity of the supercon�
ducting film through the measured values of surface resistance changes due to
the applied external magnetic fields. The results are plotted in Fig 7.4. Using
the Bardeen-Stephens relationship
AP = ^ ?
y O&ti
and (7.4.2) we can determine the coherence length. At 86.5K we find
(7.4.3)
= 129A,
where we treat B as approximately equal to H (this is approximately true near
Tc). At 80K we find R s to be less field dependent, which implied a reduction of
magnetic flux penetration in the superconducting film. Again, using (7.4.2) and
(7.4.3) we deduced | ~ 60A at 80K. Below 77K, we were not able to measure
significant changes in R s with respect to the application of a magnetic field. At
lower tem peratures, the magnetic field penetrated only a small distance into
the superconducting film (as seen in Fig. 7.1, A is more flat at tem peratures
below 80K). Therefore, less microwave absorption occured below 80K. Hence,
no estimate of | was obtained. If we extrapolate our results to T ?>OK by
using the relation
f = � 0 ) ( l - | ) - 1/ 2 ,
(7.4.4)
we find that <fy(0) to be 25A. The coherence length in the c-axis direction
can be deduced from E P R measurement, where the microwave induced electric
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170
T = 06.5K
(ixro?(J7Y) udy
8
?
*+
0
2000
1000
3000
H (G)
Fig. 7.4 Change of resistivity of the superconducting film at 86.5K due to the
applied magnetic fields.
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171
field was along the c-axis. Their values were �(86.5Iv) = 47*4. and �(0 ) = llA .
Together with the London penetration depths in section 7.3, we can calculate
the Ginzburg-Landau parameters, and
k
�
is in the range of 62 ~ 74 while rey
in the range of 110 ~ 165. In table 7.1 we report our deduced values of ,
A||, Aj_, K||, and n� from our MSR and E PR measurements.
7.5
D ISC U SSIO N A N D CONCLUSION
Based upon our measurements of surface impedance using a microwave
self-resonant technique, we have deduced the London penetration depth in the
a-b plane of a superconducting film. Application of a magnetic field allowed us
to deduce the coherence length in the a-b plane as well. E P R results provided
the London penetration depth and coherence length in the c-axis direction. The
values of A and � are anisotropic and depend on the direction of electric field
or current relative to the c-axis. At 86.5K, =129A and Ay = 8000A while
l
= 47A and
A jl
= 18000A. At OK, (0)=25A and Ay(0 )
=
1840A while
= llA and Aj_(0 ) = 3640A. The Ginzburg-Landau param eters are /ey in
l ( 0 )
the range of 110 ~ 165 and kj_ in the range of 62 ~ 74, respectively. These
yield a anisotropy factor of 7 about 2 .
We don?t believe that the above values of �s represent their intrinsic limit,
for example, we measured Ay ~ 1840A at 4K. However, others^13,14! have re�
ported Ay (0) as low as 1400A. If we assume the latter value of Ay to be intrinsic,
we extrapolate the following values for  and _, respectively, at OK: 19A and
8.3A.
Widom et
theoretically calculated the London penetration depth, co�
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172
herence length, energy gap and Tc for YBCO material from a microscopic ionic
bonding model. The results of the calculation^19] tu rn out to be |(0)=36A and
A||( 0 ) = 3194A while .( 0 ) = llA and Aj_(0 ) = 9283A, yielding an anisotropy
factor of 7 ~ 3. They are in general agreement with our measured values.
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173
Table 7.1
Summary of A, � and k
T (K)
a ||(A)
*ll(A)
KJL
a x (A )
e�(A)
-Wf-JL
K||
86.5
8000*
129*
62
18000**
47
380**
110
80
4498*
*
oD
C
74
9000**
24
370**
165
0
1840*
25+
74
3640**
lit
330
156
ot
3194
36
88
9283
12
773
260
80J
5677
76
74
16543
26
636
216
* MSR result
** E P R result
] extrapolated from 86.5K with �(T) = �(0)(1 ? f r ) -1 /2
| calculation by Widom et al.
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174
REFERENCES
[1] A. Umezawa, G. Crabtree, J. Liu, T. Moran, S. Malik, L. Nunez, W. Kwok,
and C. Swoers, Phys. Rev. B , 38, 2843, (1988).
[2] T. W orthington, W. Gallagher, and T. Dinger, Phys. Rev. Lett., 59, 1160,
(1987).
[3] U. Welp, W. Kwok, G. Crabtree, K. Vandervoort, and J. Liu, Phys. Rev.
Lett., 62, 1908, (1989).
[4] H. Jiang, T. Yuan, H. How, A. Widon, C. V ittoria, and A. Drehman,
Subm itted to Phys. Rev. B.
[5] A.C. Rose-Innes and E.H. Rhoderick, Introduction to Superconductivity,
(Pergamon Press. Oxford, 1978).
[6] G. Aeppli, R .J. Cava, E.J. Anasaldo,
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