Fluxon Lattice Dynamics
in the Superconducting Mixed State
by
Ronald Richard Gans
B. A. University of Pennsylvania
(1989)
B. S. E. University of Pennsylvania
(1989)
submitted to the
Department of Physics
in Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
at the
Massachusetts Institute of Technology
February 1997
@ 1997 Ronald Richard Gans
All Rights Reserved
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or in part.
Signature of Author
Department of Physics
S.
Certified by
c/
.
rV
17 Dprecmh'r 1996
_
'Professor Robert M. Rose
Department of Materials Science and Engineering
Accepted by
FEB 1 1997
L0BRAA,!ES
Protessor George F. Koster
Chairman, Physics Graduate Committee
Fluxon Lattice Dynamics
in the Superconducting Mixed State
by
Ronald R. Gans
Submitted to the Department of Physics
1996 in partial fulfillment of the requirements
17
December
on
for the degree of Doctor of Philosophy,
Massachusetts Institute of Technology
Abstract
Steady state flux flow power losses in the mixed state of type-II superconductors
are reduced by the addition of an external flux flow damping mechanism. Flux flow
induced eddy currents in a nearby normal metal generate the additional damping.
Measurement of the resulting damping provides insights into the behavior of
fluxons during flux flow. A theoretical calculation of the eddy current damping is
proposed. An inverse linear relationship between the eddy current damping
coefficient and the spacing between the superconductor and the metal is predicted
for a restricted range of spacings. Experiments confirm the predicted damping
magnitude and magnetic field dependence.
Fluxon lattice shear is contrasted with uniform, rigid fluxon lattice motion.
Non-linear flux flow voltages measured in an indium-bismuth superconductor are
described in terms of lattice shear. Also, fluxon lattice shear is modeled using a
numerical simulation of fluxon flow. In the absence of pinning, a fluxon lattice
subjected to a small, non-uniform driving force will shear along easy directions
while maintaining long range order. At higher driving forces, the lattice structure
disappears and only very short range order remains.
Thesis Supervisor:
Robert M. Rose
Professor, Department of Materials Science and Engineering
Acknowledgments
This MIT doctoral thesis is the culmination of a seven year adventure. The
adventure encompassed many highlights and pitfalls, including side excursions into
the study of several related and unrelated physical phenomena. A doctoral thesis is
most certainly a "learning experience." As with any large endeavor, this thesis
could not have been completed without the technical, emotional, and financial
support of many people.
A research supervisor creates the environment and support structure for
productive research. The late Professor Margaret L. A. MacVicar believed that
students should be given responsibility and independence. Her students were
immediately put in charge of coordinating joint projects, writing technical reports,
creating funding proposals, and choosing their own research projects. This is
reflected in the wide variety of topics studied by her students.
Upon Margaret's passing, Professor Robert M. Rose took the responsibility of
guiding her students to the completion of their degrees. Students maintained their
freedom and independence while providing each other with daily support and
advice. The entire group met weekly with Bob over lunch for our infamous "goup
meeting."
My fellow graduate students under Margaret and Bob were a sounding board for
ideas and a source of technical support and research advice. I wish to acknowledge
Mira Misra, Lieutenant Colonel Bruce Jette, and especially Dr. Thomas W.
Altshuler.
When I began working with Bob, I had the pleasure of meeting Dr. Joseph Parse
and Dr. Kevin Rhoads. Both have provided key insights and guidance for the
theory and experiment of this thesis. Without their advice, the experiment might
not have been completed for another ten years.
I also wish to thank Leslie Lawrence, Bob's Senior Staff Assistant, who always
kept the paperwork moving and the helium flowing. She encouraged me through
the highs and lows. Professor Donald Sadoway graciously provided me laboratory
space when my research lab was converted into an administrative office. Dave
Robertson provided a home for my thermal evaporator and loaned me several
important pieces of laboratory equipment.
I would not have endured the entire graduate student experience without the
continuous moral and emotional support of my family and friends. I wish to thank
my parents, grandparents, siblings, and other family members. Special thanks to my
lovely and exceedingly patient wife and editor, Nina S. Gelman-Gans, who read
my entire document from cover to cover.
Funding was provided by M. L. A. MacVicar, the Le Petomane Memorial
Fellowship' (sponsored by R. M. Rose), and the MIT Physics Department though
the Karl Taylor Compton Memorial Fellowship and a lot of teaching assistantships.
J. Nohain and F Caradec, Le Petomane: 1857-1945, (Sherbourne Press, Inc., Los Angeles,
1967).
Preface
Chapter one is a general introduction to the history of superconductivity and the
theories of superconductivity. Type-I and type-II superconductors are defined and
the Meissner, mixed, and intermediate superconducting states are described.
Chapter two presents a description of the hydrodynamic models of flux flow and
historical experimental data. I introduce non-linear extensions to these theories. I
also discuss some theories of fluxon pinning and the Hall effect.
In chapter three I present my own theory for the conductor proximity effect,
whereby fluxon motion inside a superconductor is damped by the influence of eddy
currents in a nearby normal conductor. In this chapter, I predict the form of the
eddy current damping. I calculate in detail the effect of the eddy current damping
on the flux flow voltage observed for a superconducting film. The prediction and
measurement of this effect are the primary thrust of this thesis.
In chapter four I describe the design of the experiment to measure the conductor
proximity effect. Details on how I built the probe and prepared the substrates are
presented. I also discuss alternate designs and experiments that might be of interest
to future researchers.
Chapter five covers the fabrication and characterization of the superconductors
used in this thesis. I fabricated my own low temperature superconducting films and
I received several high temperature oxide superconductors from various sources.
This chapter covers my cryogenic probes which are used to examine the basic
properties of these superconductors. The results of basic measurements on these
superconductors are presented.
Chapter six contains detailed flux flow data for my indium-bismuth films.
Successful measurement of the conductor proximity effect is presented. The flux
flow voltage is shown to decrease due to the eddy currents in the normal
conductor. I examine how the flux flow voltage change depends on (1) the flux
flow voltage, (2) the gap between the normal conductor and the superconductor,
and (3) the magnetic field. All results are in qualitative agreement with my
predictions.
Chapter seven describes my computer model of flux flow and fluxon lattice
shear. I present results for circulating fluxons in a disk with a radial current
distribution. Two different types of circulating motion are observed. The type of
motion depends on the relative size of the Lorentz driving force versus the
repulsive force between neighboring fluxons.
In chapter eight I present my conclusions. I describe the flux flow voltages I
measured for my films. I discuss how the conductor proximity effect can offer
information on fluxon flow. Recommendations for future work are suggested.
Table of Contents
Abstract ....................................................................................................
Acknowledgments ..........................................................
................
2
Preface .................................................................................................
3
4
Table of Contents ...........................................................
5
...............
Figures .................................................................................................
Tables .....................................................................................................
8
10
1. Introduction .....................................................................................
11
1.1 History of Superconductivity .................................................................................. 11
1.2 Maxwell's Equations ....................................................................................................
15
16
1.3 Theories of Superconductivity ................................................................................
1.3.1 London Equations..............................................................................................16
1.3.2 Ginzburg-Landau Theory ................................................................................... 18
....... 19
1.3.3 Bardeen-Cooper-Schrieffer Theory ..................................... ....
1.3.4 Characteristic Lengths in Superconductors ........................................................... 19
1.4 The Mixed State ...................................................................................................... 21
.....
............... 21
1.4.1 Lower and Upper Critical Fields .....................................
1.4.2 London Model For Extreme Type-II Superconductor ......................................
22
1.4.3 Ginzburg-Landau Theory for Extreme Type-II Superconductor................................23
1.4.4 Hexagonal Lattice .............................................................................................. 24
1.5 Intermediate State Versus Mixed State ..............................................
26
2. Theory of Fluxon Behavior with Applied Current ........................ 28
2.1 Hydrodynamic Theories ..........................................................................................
2.1.1 M agnus Force ........................................................ ............................................
2.1.2 Driving Force ....................................................................................................
2.1.3 Bardeen-Stephen and Nozibres-Vinen Models ......................................
....
2.1.4 Viscous Coefficient..................................................................................................
28
28
29
30
31
2.2 Dirty Materials ........................................................................................................
2.2.1 Kim Voltage-Current Relation .....................................
..............
2.2.2 Fluxon Velocity Distributions ...............................................................................
2.2.3 Linear Damping ................................................................................................
2.2.4 Non-linear Damping ..........................................................
2.2.5 Static Versus Dynamic Pinning................................................
2.2.6 Static Pinning Mechanism .......................................................
2.2.7 Dynamic Pinning Mechanism .............................................
........
2.3 Hall Effect .......................................................................................................
33
33
35
36
37
38
38
39
41
3.Theory of the Conductor Proximity Effect...............................
3.1 Superconductor Multilayers ........................................................................
. 44
............44
3.2 Dipole Approximation - Electric and Magnetic Fields of Moving Fluxons ................. 45
... 45
3.2.1 Magnetic Dipole Moment of a Fluxon - Bulk .......................................
3.2.2 Magnetic Dipole Moment of a Fluxon - Surface .................................................... 46
3.2.3 Geometry of Experiment....................................................................................47
3.2.4 Induced Electric Field of Moving Dipole........................................................ 48
3.2.5 Induced Electric Field of a Dipole Lattice ............................................................ 50
3.3 Eddy Current Power Loss and Effective Viscous Coefficient .................................. 51
3.3.1 Eddy Current Power Loss - Thin Superconductor Approximation........................52
3.3.2 Electric Field and Power Density - General Case......................................53
3.3.3 V iscous Coefficient.................................................................................................. 58
60
3.4 Curved Film Geometries ................................................................................
3.5 Finite Penetration Depth - Fluxon Magnetic Fields...................................................62
3.5.1 Thin Superconductor - On Axis.................................................................... 62
3.5.2 Thick Superconductor - On Axis .......................................................................... 64
65
3.6 Fluxon Bundling Effects ................................................................................................
65
3.7 Effective Flux Flow Voltage With Eddy Current Damping ........................
4. Conductor Proximity Effect Experiment Design .......................... 68
4.1 Design Concepts ............................................................................................................
68
4.2 Description of the Bending Beam Experiment .....................................
69
4.2.1 Beam Bending Calculations ..................................................................................... 70
4.2.2 Beam Force - Mechanical Method Calculations........................... ......................... 71
4.2.3 Beam Force - Piezoelectric Calculations ............................................................... 73
4.3 Conductor Proximity Effect Probe Details ............................................ 74
75
4.3.1 Substrate Holder and Probe Tail ............................................................................
4.3.2 Probe Head and Feed-throughs...........................................................................77
4.3.3 Loading Procedure...................................................................................................77
78
4.4 Preparation of Substrates....................................................................
4.4.1 Photolithography ..................................................................................................... 79
4.4.2 Etching Procedures ......................................................................... ................... 80
81
4.4.3 Silicon Substrate Results - Surface Quality and Purity ....................................
5. Superconductor Fabrication and Characterization ...................... 82
5.1 Material Properties - In-BI Alloys .........................................................................
5.2 Superconductor Fabrication via Evaporation .....................................
82
....... 85
...... 85
5.2.1 Evaporation of Alloys - Raoult's Law .......................
5.2.2 Evaporation System and Evaporation Procedure .................................................... 87
87
5.2.3 Indium-Bismuth Films Produced..............................................
5.3 Superconductor Fabrication via Pressure Infiltration ............................................... 88
5.3.1 Pressure Vessel Design ..................................................................... ................. 89
5.3.2 Substrates and In-Bi Melt ................................................................. ................. 90
91
5.3.3 Results and D iscussion..........................................................................................
5.4 Characterization of Films ............................................................................................. 93
...94
5.4.1 Magnetic Critical Temperature Measurement Probe .................
95
5.4.2 Electrical Critical Temperature Measurement Probes .........................
95
.........
.....................................
Field
Magnetic
Versus
5.4.3 Measurements
5.5 Indium-Bismuth Films
............................................................................. 96
96
98
100
100
5.6 Lead Films ................................................................................................................... 102
5.7 YBCO Bulk and Film Samples .............................................................................. 103
......
5.5.1 Critical Temperature Measurements .........................................
.........
5.5.2 Critical Magnetic Field Measurements ............................ .....
5.5.3 Critical Current in Various Magnetic Fields ........................................................
5.5.4 Effect of Film Anneal on Temperature Transition and Critical Field..................
5.7.1 Magnetic Critical Temperature Measurements ....................................................... 103
5.7.2 Electrical Critical Temperature Measurements..................................................106
6. Conductor Proximity Effect Experiments ...................................
108
6.1 Flux Flow Measurements .........................................................................................
6.1.1 Voltage Versus Current at Constant Magnetic Field................................
6.1.2 Voltage Versus Magnetic Field at Constant Current..............................
6.1.3 Voltage Versus Driving Force................................................ .............................
6.1.4 Analysis of Flux Flow Data ...................................
6.1.5 Temperature Stability .......................................
6.2 Aluminum Film Conductivity ............................................
108
108
109
111
114
116
118
6.3 Gap Dependence of Flux Flow Data .................... .............
6.3.1 Voltage Versus Field Curves at Two Gaps .........................................................
6.3.2 Voltage Versus Current Curves at Two Gaps ....................................
6.3.3 Voltage Versus Current Curves at Many Gaps ..................................
6.3.4 Voltage Versus Gap at Constant Current and Field ........................................
6.3.5 Voltage Versus Capacitance Data at Various Magnetic Fields .............................
119
119
120
123
124
126
7. Computer Model of Lattice Behavior ...................
..................... 129
7.1 Model Parameters and Terminology ........................................................................ 130
7.1.1 Forces ......................... .....................................................................................
7.1.2 Boundary Conditions .......................................
7.1.3 Statistical Analysis ................................................................................................
7.2 Predictions and Results: No External Forces, No Pinning ....................................
7.3 Predictions: Externally Applied Radial Current, No Pinning ................................
130
130
131
132
133
7.4 Results: Externally Applied Radial Current, No Pinning .........................134
135
7.4.1 Visual Character................................
137
7.4.2 Voltage-Current Relation.................................
7.4.3 Lattice O rder ......................................................................................................... 138
8. Conclusions ......................................
140
8.1 Flux Flow in Indium-Bismuth Films ........................................................................ 140
......... 140
8.2 Conductor Proximity Effect ................................
141
8.3 Computer Model of Lattice Flow ...................................................
8.4 Recommendations for Future Work ........................................................................ 142
Appendix i: Constants and Major Symbols ....................................
Appendix II: Bessel Functions and Related Functions .................
Appendix III: Capacitance Position Sensing Circuit ......................
144
Appendix IV: Motor Timing Circuit .....................................
151
146
147
Figures
Figure 1-1 Equilibrium thermodynamic states of type-I and type-II superconductors ................ 14
..... 22
Figure 1-2 Magnetic flux versus applied field curves. ......................................
26
Figure 1-3 Unit cell of hexagonal lattice shown with basis vectors ...................................
Figure 2-1 Vortex and its circulation currents immersed in a uniform supercurrent. ............. 29
...... 33
Figure 2-2 Superconductor in a line pattern geometry ......................................
positive
of
case
the
in
behavior
showing
semiconductor
a
for
geometry
effect
Hall
2-3
Figure
42
.............................
charge carriers....................
47
....................................
calculations.
in
theoretical
used
Figure 3-1 Geometrical parameters
50
..
....................................
field
magnetic
dipole
Figure 3-2 Vertical component of fluxon
51
.....................................
lattice
fluxon
from
Figure 3-3 Shape of magnetic field near and far
Figure 3-4 Induced electric field - p component ................................................................... 54
Figure 3-5 Induced electric field - 0 component ............................................... 55
56
..............
Figure 3-6 Induced electric field - X component...........................
Y
component.....................................................................57
field
electric
Induced
3-7
Figure
Figure 3-8 Typical power density for single fluxon. ................................................................. 58
59
Figure 3-9 Eddy current power per unit length versus separation distance..................
Figure 3-10 Calculated rleddy versus insulating gap thickness. .................................................... 60
..... 61
Figure 3-11 Side view of curved film geometry used in calculations.....................
1 •
:......6:...... 62
Figure 3-12 Effect of film curvature on d dy ...........................................................
distribution
current
full
vs
approximation
dipole
fluxon
of
on-axis
field
Magnetic
3-13
Figure
- thin superconductor. ..................................................... ........................................... 63
Figure 3-14 Magnetic field on-axis of fluxon - dipole approximation vs full current distribution
- thick superconductor.................................................... ........................................... 65
Figure 3-15 Effect of eddy current damping on the ideal mixed state V-I curve......................66
Figure 4-1 Bending beam experiment design. ................................................ 69
Figure 4-2 Substrate holder - top view................................................................ ................ 76
Figure 4-3 Probe tail - side view in section ................................................... 76
Figure 4-4 Probe head sketch - side view in section .............................................. 77
Figure 4-5 Quartz substrate etching pattern. ............................................................................ 79
Figure 4-6 Aluminum mask for photoresist process. ............................................ 80
Figure 5-1 Critical temperatures of various indium-bismuth alloys................................82
83
.................. ............
....
Figure 5-2 Indium-bismuth phase diagram .....
90
.......
system.............................
infiltration
pressure
for
Figure 5-3 Pressure vessel
substrate....91
quartz
on
deposited
strip
spacer
of
aluminum
thickness
Figure 5-4 Profile showing
Figure 5-5 Superconducting critical transition temperatures for two indium-bismuth films. ...... 97
Figure 5-6 Second resistance transition vs temperature for two indium-bismuth films ............... 98
Figure 5-7 Resistance vs magnetic field for a mixed state indium-bismuth film ........................ 99
Figure 5-8 Dependence of the upper critical magnetic field on temperature for a mixed state
99
indium-bism uth film . .......................................................................................................
Figure 5-9 Dependence of the critical current density on magnetic field for two mixed state
100
indium-bismuth films..............................................................................................
102
.........
treatment.
temperature
after
film
in
In-Bi
observed
transition
critical
Wide
5-10
Figure
03
............
film
I
lead
a
type
for
current
on
field
magnetic
critical
Figure 5-11 Dependence of the
#1...........104
sample
YBCO
bulk
in
Figure 5-12 Magnetic measurement of critical temperature
Figure 5-13 Magnetic measurement of critical temperature in bulk YBCO sample #2...........105
Figure 5-14 Magnetic measurement of critical temperature in unpatterned YBCO film............106
Figure 5-15 Typical four-probe resistance measurement pattern used in NIST films. ............. 107
Figure 5-16 Transition temperature measurements showing disappearance of resistance upon
...... 107
cooling for two NIST patterned YBCO films .......................................
fields........................................109
magnetic
various
at
vs
current
Figure 6-1 Flux flow voltage
Figure 6-2 Log-log scale - flux flow voltage vs current at various magnetic fields. ...............109
Figure 6-3 Flux flow voltage vs magnetic field at various currents ........................................... 110
Figure 6-4 Log-log scale - flux flow voltage vs magnetic field at various currents. .................. 111
Figure 6-5 Log-log scale - flux flow voltage vs Lorentz driving force at various fields......... 112
Figure 6-6 Log-log scale - flux flow voltage vs Lorentz driving force at various currents......... 113
Figure 6-7 Log-linear scale - flux flow voltage vs Lorentz driving force at various fields ....... 113
Figure 6-8 Log-linear scale - flux flow voltage vs Lorentz driving force at various currents..... 114
Figure 6-9 Fraction of mobile fluxons as a function of applied stress........................................ 115
115
Figure 6-10 Fraction of mobile fluxons as a function of lattice strain ....................................
Figure 6-11 Voltage versus current curves over multiple days on the same film - various
118
cooling m ethods ............................................................................................................
120
Figure 6-12 Gap dependence of voltage versus field data. .....................................
121
Figure 6-13 Gap dependence of voltage versus current data. ........................................
Figure 6-14 Gap dependent voltage change versus current at various magnetic fields........... 122
Figure 6-15 Gap dependent voltage change versus voltage squared at various magnetic fields.. 122
Figure 6-16 Voltage change times magnetic field versus voltage squared at various magnetic
123
fields .... .......................................................................................................
124
Figure 6-17 Gap dependent voltage change versus current at various gaps ............................
125
Figure 6-18 Capacitance and flux flow voltage variation with beam force ..............
Figure 6-19 Flux flow voltage variation as a function of normal conductor to superconductor
gap size . ..................................................................................... 126
128
Figure 6-20 Change in flux flow voltage as a function of magnetic field. ..............................
130
Figure 7-1 Geometry of superconductor for computer model ......................................
Figure 7-2 Mean radial distribution function for complete disorder in a disc of radius R ......... 131
Figure 7-3 Triangular lattice resulting after 5000 iterations from random initial fluxon
positions. Distinct boundaries can be seen between the different grains that have formed.132
Figure 7-4 Radial distribution function for the fluxon lattice shown in Figure 7-3...............133
133
Figure 7-5 Hexagonal fluxon flow pattern due to lattice shear......................................
134
Figure 7-6 Circular flux flow pattern for large currents .....................................................
Figure 7-7 Overlays of 30 frames of fluxon positions in each of runs 6 (7-7a), 12 (7-7b),
18 (7-7c), and 24 (7-7d)..................................................................................................137
138
Figure 7-8 Flux Flow voltage vs current relation on log-log scale...................................
Figure 7-9 Radial distribution functions showing decreasing order range and intensity for
139
increasing applied current ...........................................................................................
148
ac
outputs..................................
with
circuit
sensing
distance
Figure III-1 Capacitance
149
.....................................................
Figure 111-2 Capacitance circuit ac to dc output converter.
150
Figure II1-3 Capacitance circuit power supply ...............................................
152
................
Figure IV-1 Frequency divide-by-six circuit.............................
Tables
1-1 Neighbor Spacings And Degeneracies For A Hexagonal Lattice.................................26
. .. . . . . 73
. .. .. . .. .. . . .. . . ..
4-1 Properties Of Selected Piezoelectric Material 70 ............
4-2 Properties Of Quartz ........................................................................ ................... 79
84
5-1 Properties O f Indium ..............................................................................................
5-2 Properties Of Bism uth................................................. ........................................ 85
5-3 Vapor Pressures For Pure Indium And Pure Bismuth ................................................. 86
5-4 Parameters For Evaporated Indium-Bismuth Films ................................................ 88
6-1 Flux Flow Voltage Change Observed For Various Magnetic Fields Substrate 11 Film 1 ..................................................... ............................................ 127
Table 6-2 Flux Flow Voltage Change Observed For Various Magnetic Fields 128
Substrate 17 Film 1 ...................................................................................................
136
Table 7-1 Parameters Of Computer Simulation Runs ...................................
153
Table IV-1 JK Flip-Flop Logic Table .....................................
153
Table IV-2 State Table For Frequency Divide-By-Three Circuit...............................
153
Table IV-3 State Table For Frequency Divide-By-Two Circuit ................................... ..
Table
Table
Table
Table
Table
Table
Table
Table
1. Introduction
Superconductivity is a demonstration of quantum mechanics on the macroscopic
scale. Long range order in the quantum wave function results from interactions
among electrons at the Fermi surface. This long range order leads to remarkable
magnetic and electrical properties, including the expulsion of magnetic field, the
quantization of magnetic flux, the disappearance of electrical resistivity, and the
unusual behavior of weakly coupled junctions.
This chapter starts with a survey of the history of superconductivity. This is
followed by a discussion of the important theories of superconductivity: London
equations, Ginzburg-Landau theory, and the Bardeen-Cooper-Schrieffer theory.
The mixed state of type-II superconductors is then examined in detail, followed by
a brief outline of the intermediate state.
1.1 History of Superconductivity
Superconductivity was discovered in 1911 by H. Kamerlingh Onnes, 2 who
succeeded in liquefying helium and measured the low temperature resistance of
various materials. Below a transition temperature Tc the resistivity of mercury
suddenly dropped to an unmeasureably small value. It has since been shown that
superconductors in the Meissner state have a dc electrical resistivity of less than
10- 3 ohm-cm.3
When a magnetic field is applied to a perfect conductor, eddy currents form on
the perfect conductor's surface, preventing the field from penetrating the
material's interior. If a material becomes a perfect conductor in the presence of a
magnetic field, the field remains trapped in the material, even if the external
magnetic field source is removed. Therefore a perfect conductor's interior
magnetic field always remains constant.
In 1933, Meissner and Ochsenfeld 4 discovered that superconductivity is not
simply perfect conductivity. Superconductivity is a reversible, stable,
thermodynamic state. Magnetic field is reversibly expelled from the
superconductor on entering the Meissner state. The interior magnetic field of a
Meissner state bulk superconductor is always zero. The magnetic properties of a
superconductor are therefore more fundamental than its electrical properties.
For low magnetic fields, the magnetic field is expelled from the superconductor.
As the strength of the field is increased, however, it becomes energetically
2 H.
Kamerlingh Onnes, Communicationsfrom the Physical Laboratoryof the University of
Leiden 120b, 122b, and 124c (1911).
3 D J. Quinn, III and W. B. Ittner, III, "Resistance in a Superconductor", Journalof Applied
Physics 33, 748-749 (1962).
W. Meissner, and R. Ochsenfeld, "Ein Neuer Effekt bei Eintritt der Supraleitfahigkeit",
Naturwissenschaften21, 787-788 (1933).
unfavorable to expel the magnetic field. Two different types of behavior can occur,
depending on the material properties of the superconductor. For a type-I
superconductor, superconductivity is destroyed for magnetic fields larger than the
thermodynamic critical field Hc. At Hc, there is a transition between the Meissner
state and the normal state. For type-II superconductors, the behavior is more
complex. There are transitions at two critical magnetic fields. At the lower critical
field Hc1 , there is a transition from the Meissner state to the mixed state. At the
upper critical field Hc2, there is a transition from the mixed state to the normal
state. The parameters which determine if a superconductor is type-I or type-II will
be discussed below. It will also be shown how the thermodynamic critical magnetic
field provides information on the condensation energy of the superconducting
state.
In1935, F. and H. London 5"6 developed a theory of superconductivity
incorporating the results of Meissner's research. In their theory, the magnetic field
deep inside a bulk superconductor is zero. Magnetic fields applied to a
superconductor decay with a characteristic distance - the London penetration
depth XL. Equation (1) describes the decay of the magnetic field H at the surface of
the superconductor.
VH = H / 2
()
Ginzburg and Landau 7 developed a theory to describe the second order phase
transition between the normal and superconducting states. They introduced a
complex order parameter 4f. The local density of superconducting electrons is
proportional to jyI . An expansion of the superconducting state free energy near
the phase transition was performed in terms of i, resulting in a Schridinger-like
equation for 4y.
The Ginzburg and Landau order parameter must be single valued. It can be
shown that a quantity called the fluxoid must therefore be quantized around any
closed loop in a superconductor. The fluxoid, which will be discussed in more
detail below, is closely related to the magnetic flux threading the loop. Therefore,
the magnetic flux through a hole in a multiply-connected superconductor is
quantized.
Experiments in 1934 showed that magnetic flux could become irreversibly frozen
into certain simply connected superconductors."s' This behavior could not be
5F. London and H.London, 'The Electromagnetic Equations of the Supraconductor", Royal
Society of London Proceedings A149,71-88 (1935).
6F. London and H. London, "Supraleitung und Diamagnetismus", Physica 2,341-354 (1935).
7V. L. Ginzburg and L. D.Landau, "On the Theory of Superconductivity", Zhurnal
Eksperimentalnoii Teoreticheskoi Fiziki (Journalof Experimental and TheoreticalPhysics) 20,
1064-1082 (1950).
8K.Mendelssohn and J.D.Babbitt, "Persistent Currents in Supraconductors", Nature 133,
459-460 (1934).
described in terms of the Meissner state. The trapped flux phenomenon did not
occur in most pure elements, but occurred consistently in alloys. Superconductors
which can trap flux in this way form the class of type-II superconductors.
In 1953, Pippard'o introduced a non-local extension of London theory. The
supercurrent at a given point is related to the magnetic vector potential in a region
around that point. The size of the region is governed by a coherence length to,
which depends on the purity of the material. Changes in the superconducting order
parameter occur over a distance on the order of ýo. If the coherence length is less
than the magnetic penetration depth, the energy of the normal-superconductor
interface may become negative. In these cases, for magnetic fields greater than the
lower critical magnetic field, interfaces form and flux penetrates into the
superconductor.
In 1957, Abrikosov" used Ginzburg-Landau theory to show that
superconductors can be classified into those that have a positive surface energy
and those that have a negative surface energy at the interface between normal and
superconducting regions. Using the parameter K= X /5, Abrikosov showed that
the superconductor-normal interface energy is positive for K< 1 / 2 and negative
for K > 1 / F2. The coherence length 4 and the penetration depth X will be
discussed in more detail below. Superconductors with a positive surface energy are
type-I superconductors and those with a negative surface energy are type-II
superconductors.
Figure 1-1 shows the phase diagram for type-I and type-II superconductors. A
type-I superconductor is in the Meissner state below the Hc line and the normal
state above the Hc line. A type-II superconductor is in the Meissner state below
Hci, the mixed state between Hc and HC2 , and the normal state above HC2. In the
mixed state, the magnetic field penetrates the superconductor in the form of
quantized flux lines. Triuble and Essmann"'21 3 observed the flux-line lattice directly
via decoration of a superconductor with small ferromagnetic particles. An electron
9T. C. Keeley, K. Mendelssohn and J. R. Moore, "Experiments on Supraconductors", Nature
134, 773-774 (1934).
0o
A. B. Pippard, "An Experimental and Theoretical Study of the Relation Between Magnetic
Field and Current in a Superconductor", Proceedingsof the Royal Society of London A216,
547-568 (1953).
" A. A. Abrikosov, "On the Magnetic Properties of Superconductors of the Second Group",
Soviet Physics JETP 5 (1957) 1174-1182. A. A. Abrikosov, Akademia Nauk SSSR Doklady 86,
489-492 (1952).
12 U. Essmann and H. Trauble, "The Direct Observation of Individual Flux Lines in Type II
Superconductors", Physics Letters 24A, 526-527 (1967).
13H. Trauble and U. Essmann, "Flux-Line Arrangement in Superconductors as Revealed by
Direct Observation", Journalof Applied Physics 39, 4052-4059 (1968).
microscope is used to view the particles. More recently, the fluxon lattice has been
imaged with scanning tunneling microscopy. 14
Type-I
H5O)
Normal
o H (O)
H
HC(0)
00
..,,
,,=,,=
(O)
Meissner
TE
T
Temperature, TC
Temperature, T
Temperature, T
Temperature, T
Figure 1-1 Equilibrium thermodynamic states of type-I and
type-II superconductors.
Exploring the microscopic basis of superconductivity, Cooper' 5 showed in 1956
that if a pair of electrons has a net attractive interaction, they can form a bound
state, even though their total energy is larger than zero. Electron pairing reduces
the ground state energy and produces an energy gap between the ground state and
the first excited state. This energy gap results in a superconducting state. The
superconducting transition temperature was found to vary for different isotopes of
the same material. This led to examination of the role of the ion lattice in
superconductivity. Phonons can mediate an attractive interaction between pairs of
electrons at the Fermi surface. Pairs of electrons with equal and opposite momenta
can lower their energy by forming bound states called Cooper pairs. Bardeen,
Cooper, and Schrieffer' 6 (BCS) developed a microscopic theory of
superconductivity based on the theory of Cooper pairs. Later in that same year,
Gor'kov demonstrated that Ginzburg-Landau theory was derivable from BCS
theory. 17
In 1962, Josephson'8 presented a calculation showing that tunneling of Cooper
pairs could exist between two superconductors through an insulating gap. A finite
1' H.F. Hess, R. B.Robinson, R. C. Dynes, J. M.Valles, Jr., and J. V.Waszszak, "Scanning-
Tunneling-Microscope Observation of the Abrikosov Flux Lattice and the Density of States near
and inside a Fluxoid", Physical Review Letters 62, 214-216 (1989).
'5L. N. Cooper, "Bound Electron Pairs in a Degenerate Fermi Gas", Physical Review 104,
1189-1190 (1956).
and L. N.Cooper and J. R. Schrieffer, "Theory of Superconductivity", Physical
Review 108, 1175-1204 (1957).
17
L. P.Gor'kov, "Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of
Superconductivity", Soviet Physics JETP 36, 1364-1367 (1959).
18B. D.Josephson, "Possible New Effects in Superconductive Tunneling", Physics Letters 1,
251-253 (1962).
Bardeen
16 J.
dc supercurrent, less than some maximum critical current, could flow through the
gap at zero dc gap voltage. At finite dc gap voltage, a dc supercurrent and an ac
supercurrent flow across the gap. These effects were later shown to exist and have
been utilized to create a number of superconducting devices.
In 1986, Bednorz and Mtiller' 9 observed superconductivity in the Ba-La-Cu-O
system. A flurry of research on high Tc copper oxide based superconductors
followed, resulting in the development of oxide superconducting films 20 and the
discovery of superconductivity above the boiling point of liquid nitrogen in the YBa-Cu-O system. 2 Oxide superconductors are highly anisotropic, showing
reduced dimensionality. Besides having high superconducting transition
temperatures, oxide. superconductors have very short coherence lengths, making
them extreme type-II materials and giving them high upper critical fields. They also
have large critical currents in the mixed state. At these higher temperatures,
thermal fluctuations become much more important in describing the behavior of
oxide superconductors. The BCS microscopic model does not properly describe
the properties of these oxide superconductors.
In the Meissner state there is no electrical resistance to an applied current. In the
mixed state, electrical resistance remains zero, but the Lorentz interaction between
the current and the fluxons can cause fluxon motion and energy dissipation. This
results in the appearance of a voltage and is termed flux flow resistance. Fluxon
motion can be prevented by local effects which pin the fluxons, preventing them
from moving and dissipating energ. In this case there is no resistance. For current
densities exceeding a critical current density Jc, however, pinning is not capable of
preventing fluxon motion. For practical applications which depend on the zero-loss
properties of a superconductor, a large Jc is desired. The critical current density
depends on the microstructural properties of the material, including defects, grain
boundaries, impurities, and surface preparation. A greater understanding of the
voltage-current characteristics of mixed state superconductors is important for
practical applications as well as the fundamental understanding of
superconductors.
1.2 Maxwell's Equations
Maxwell's four electromagnetic equations play a key role in the study of
superconductivity. They are given below for the case of linear dielectric response,
no polarization, slowly changing electric fields, and no free charges. All equations
19 J.G. Bednorz and K.A.Miiller, "Possible High Tc Superconductivity in the Ba-La-Cu-O
System", ZeitschriftfiirPhysik B 64, 189-193 (1986).
20 H. Koinuma, M. Kawasaki, M. Funabashi, T. Hasegawa, K. Kishio. K. Kitazawa, K. Fueki,
and S.Nagata, "Preparation of superconducting Thin Films of (Lal.Sr•)CuO4 By Sputtering",
Journalof Applied Physics 62, 1524-1526 (1987).
21 M. K. Wu, J.R. Ashburn, C.J.Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J.Huang, Y. Q.
Wang, and C.W. Chu, "Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O
Compound System at Ambient Pressure", PhysicalReview Letters 58, 908 (1987).
in this thesis are presented in SI (MKS) units, following the current convention of
most publications.
VxH=J
VxE= at
(2)
(3)
(4)
E =O0
(5)
V.B=0
Here, the electric field E is related to the current density J by E = oJ, where a is
electrical conductivity. Similarly, the magnetic field H is related to the magnetic
induction B by B = gH, where g is the permeability of the material. Equation (5)
implies the existence of a magnetic vector potential A, defined by
B=VxA
(6)
To fully specify A, the gauge is chosen such that V . A = 0.
Following the practice in the literature, applied magnetic fields are given in Tesla
rather than in Ampere/meter. To convert from Ampere/meter to Tesla, the
permeability of free space is used: B = poH.
1.3 Theories of Superconductivity
1.3.1 London Equations
The London brothers created a theory of superconductivity based on two
postulated equations.""'22 First, they replaced Ohm's law with
H=
g-to
xJ
(7)
which says that the magnetic field H, rather than its time derivative, depends on
the curl of the current density J. The scaling constant is given by A = m / nse ,
where m, e, and n, are the electron mass, electron charge, and superelectron
number density, respectively. The permeability of free space is to. Secondly, they
concluded that the electric field E is related to the time derivative of the current
density rather than being proportional to the current density.
E = AdJ / dt
(8)
When equation (7) is combined with Ampere's Law (2) the result is the screening
of the magnetic field as given in equation (1) where
XL
m
(9)
ýIonse"
22 F.London, Superfluids: Volume I: Macroscopic Theory of Superconductivity, (John Wiley &
Sons, Inc., New York, 1950).
gives the characteristic penetration depth.
The London equation can also be derived from examination of the free energy of
a superconductor. The free energy of a superconductor is the sum of the normal
state free energy plus the volume integrals of the superconducting electron kinetic
energy density and the magnetic field energy density.
E nc =f mv2n,dV
2
(10)
fJH dV
Eo,,.nc = 2HdV
EMýO
The above expressions can be combined with the definition of current density
J = n,ev and Ampere's Law (2), to yield the free energy equation.
A.UPER
JO•MAL
-fH
+
2±
+
VI
')dV
(11)
The condition on H which minimizes this free energy expression is the London
Equation (1)."
The conservation of the fluxoid follows from Faraday's Law (3) and London's
second equation (8). Consider a contour C around a hole in a superconductor. The
contour lies entirely in superconducting material, while any open surface S defined
by that contour passes through both superconducting and normal regions. Apply
the integral form of Faraday's law to the contour C and surface S.
E.-dl = -
-B. dS
(12)
Since the contour lies entirely inside the superconductor, London's second
equation can be substituted.
A--•dl=
B•B
dS
(13)
The time derivative can be taken out of the integral, resulting in the conservation
equation for the fluxoid nfloid.
d
fluxoid
0
dt
(14)
nuxoid = AJ-dI+ I B dS
The first term is the contour integral of the screening current. For a sufficiently
large superconductor, this term can be made vanishingly small by taking the
contour deep within the superconductor.
P.G. de Gennes and J. Matricon, "Collective Modes of Vortex Lines in Superconductors of the
P3
Second Kind", Reviews of Modern Physics 36, 45-49 (1964).
London theory proves that the fluxoid is a conserved quantity. London also
predicted that a quantum condition would exist on the wave function of the
superconductor that would force the quantization of the fluxoid." Fluxoid
by Ginzburg-Landau theory.
quantization was later proven more rigorously
1 25
1961.24
in
followed
Experimental proof
1.3.2 Ginzburg-Landau Theory
The Ginzburg-Landau theory' of superconductivity is based on Landau's general
theory of second order phase transitions. In Landau's theory, there exists an order
parameter V which is zero at the transition and in the less ordered state. The free
energy of the system can be expanded in powers of xV where the expansion
coefficients are regular functions of temperature. Near the normal-superconducting
transition temperature Tc, Ginzburg and Landau took 4 to be a complex number
satisfying
n,-=lyj2
(15)
where n,is the local density of superconducting electrons. In the normal state 'gis
zero.
The free energy in the superconducting state near the normal-superconducting
transition is written as the sum of the normal state free energy, an expansion in
2,the superconducting electron kinetic energy, and the magnetic
terms of n,and n,
field energy.
PER =
AL.+
(T)I
1
12
I+-(T)I
2
+ 2m
2m i
- 2eA)
POHh-
(16)
+ 2
2
Here, h is the reduced Planck's constant h/2t. The temperature dependent
coefficients acand 0 are determined below. If there are no magnetic fields and the
superelectron density is constant, the last two terms above disappear. The
equilibrium state is determined by minimization of the free energy with respect to
V.P must always be positive, since v is finite. In the normal state V = 0 so ca must
be positive. In the superconducting state, yxis non-zero, so ca must be negative.
The condition
iHl
=
a
(17)
gives the free energy minimum in the superconducting state. Define Hc as the
temperature dependent critical magnetic field for the normal-superconducting
transition. Then Hc is related to the free energy difference between the normal and
superconducting states.
B. S. Deaver Jr. and W. M. Fairbank, "Experimental Evidence for Quantized Flux in
Superconducting Cylinders", Physical Review Leners 7, 43-46 (1961).
25 R. Doll and M. Nibauer, "Experimental Proof of Magnetic Flux Quantization in a
Superconducting Ring", Physical Review Letters 7, 51-52 (1961).
24
poH (T)
sUPER -- TORMAL
_cx
-
2
20
(18)
Minimization of free energy with respect to the order parameter and magnetic
vector potential yields the Ginzburg-Landau differential equations.
uay,+I
y+12 -
eh
J= xH =-i - ( *
- 2eA
- O
V =0
4e2 V--*VA = del vs
*
(19)
(20)
m
m
1.3.3 Bardeen-Cooper-Schrieffer Theory
The BCS theory16 explores the microscopic basis of superconductivity. The
exchange of virtual phonons can result in a net attractive interaction between
electrons at the surface of the Fermi sphere. The ground state in which electrons
form pairs with opposite spin can have a lower energy than the normal ground
state. The theory derives a second order phase transition at the critical
temperature, an energy gap for individual particle-like excitations, the Meissner
effect, zero electrical conductivity, and the isotope effect.
The BCS theory will not be explored in detail here. Several relations derived in
BCS theory will be used later, however, and are given here for reference. First, an
alternate expression for the London coefficient A is given in terms of the density of
states at the Fermi surface N(0) and the average Fermi velocity vF.
A-'= e2 N(O)v2
(21)
Second, the Pippard coherence length 4o is related to Fermi velocity, critical
temperature, and band gap at zero temperature 2Ao by
;o =
'
xd0
o
=
0.18
v
k Tc
where the band gap is related to the critical temperature.
Ao = L75k,Tc
(22)
(23)
Finally, the electronic specific heat of the normal state is given by CQ = yT where
the Sommerfield constant y is given in terms of the density of states.
y = j r 2 N(O)k2
(24)
1.3.4 Characteristic Lengths in Superconductors
There are many characteristics lengths that influence the properties of
superconductors. Many of them are closely related and the nomenclature can be
confusing. This section identifies notation used in this thesis when discussing
various lengths.
In any discussion of conductivity, the mean free path of electrons 1,, is
important. Clean and dirty limit superconductors are defined as materials having
long and short mean free paths respectively. Elemental materials such as niobium
can be very clean. Alloys, such as indium-bismuth, tend to be in the dirty limit. The
mean free path can be calculated from resistivity p and electronic density.26
,P= 9.2 nm x r.
1 ýL -cmJ
P
x(
ao
0
(25)
Values for rJao are given in Table 5-1 and Table 5-2.
The characteristic distance for magnetic field variation in London theory, XL, is
given by equation (9). The Ginzburg-Landau theory27 generalizes this to a
temperature dependent penetration depth X(T). Temperature dependence near the
critical temperature is given by
(T)=
XL
2 (1
IT-
(clean limit)
(26)
(dirty limit)
(27)
Tc)
4
0.61XL
1I-T Tc
1
MfP
with the Pippard coherence length 4o given by equation (22).
The range of variation of the superconducting order parameter is given by the
temperature dependence coherence length 4(T).
4(T) = 0.7440
11
(T) = 0.855 1TT
.1-F T/Tc
(clean limit)
(28)
(dirty limit)
(29)
Although both X(T) and 4(T) diverge at Tc, their ratio remains relatively
constant:
X(T) _0.96
K K
= 0.96
(clean limit)
(30)
0.715
(dirty limit)
(31)
Gorkov gives a formula for the variation of Kin terms of electronic specific heat,
normal state resistivity, and the limit for a pure material ro.
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New
York, 1976) 52.
27 For a more detailed discussion of the equations presented in this section see, for example, D.
Saint-James, E. J. Thomas, and G. Sarma, Type II Superconductivity (Pergamon Press, Oxford,
1969) 151-152.
26
K = Ko + 2.38 x 10- x
(P
I
)X(
m
3
(32)
This formula has been tested experimentally for indium-bismuth alloys and was
found to agree with the measured Kwithin a few percent. Additionally, using the
pure indium value of y introduced less than a five percent deviation from using the
true value of y.28
This experimental situation is complex and interesting for these superconducting
films. The geometrical distances involved are comparable to the temperature
dependent penetration depth and coherence length. Additionally, the fluxon lattice
parameter Iis comparable to these distances. The lattice parameter will be
discussed in more detail below.
1.4 The Mixed State
A type-II superconductor has a negative surface energy." When a normal region
forms in a superconductor, magnetic energy decreases over the penetration depth
range and the state energy increases over the coherence length range. Since the
penetration depth is larger than the coherence length, there is a net decrease in
energy at the interface. For applied magnetic fields greater than a lower critical
field HcI, it becomes energetically favorable to form normal-superconductor
interfaces and allow the field to penetrate the superconductor. The superconductor
enters the mixed state.
Due to the quantization of the fluxoid, the minimum magnetic flux in each
normal region is one flux quantum Oo = h /2e, where h is Planck's constant. The
energy of the flux line goes as the square of flux contained in that line. This results
in single quantization of fluxons. As will be shown below, fluxons repel each other.
The central repulsive force tends to distribute the fluxons into a hexagonal lattice
configuration.
1.4.1 Lower and Upper Critical Fields
The lower critical magnetic field Hc, is the externally applied, axial, uniform
magnetic field necessary to introduce the first fluxons into a long, cylindrical,
type-II superconductor. Typical magnetization curves for type-I and reversible
type-II superconductors are shown in Figure 1-2. At the lower critical field, the
Gibbs free energy density of the Meissner state GCImss,,- is equal to the Gibbs free
energy density of the mixed state GC~n. Since the fluxon density approaches zero
at Hcj, assume the flux lines are non-interacting. The mixed state free energy
density is therefore the Meissner state free energy plus the formation energy of the
flux lines minus the magnetic energy of the flux line penetration.
2sT. Kinsel, E. A. Lynton, and B. Serin, "Magnetic and Thermal Properties of Second-Kind
Superconductors. I. Magnetization Curves", Reviews of Modern Physics 36, 105-109 (1964).
GVMXED = GMEISSN•R +
nE - BH
(33)
where Eis the energy of formation per unit length of a flux line, n is the number of
flux lines per unit area, and B is the local flux density. Since B = noo, taking
29
GMIXED = GM·EssE,,R yields the lower critical field Hc1 .
(34)
Hc = 00
C7
0r
SType-II
HC HeHc
Magnetic Field, H
Figure 1-2 Magnetic flux versus applied field curves.
As the applied magnetic field is increased, the fluxon spacing will decrease. At
the upper critical field He, fluxon cores overlap and the material undergoes a
second order phase transition to the normal state. Near the upper critical field, the
order parameter Wis small. This allows the Ginzburg-Landau differential equations
to be linearized and an expression found for the upper critical field. 30
Hc2 = 0 /2r, 0 • 2 (T)
(35)
The upper critical field can be related to the thermodynamic critical field via the
Ginzburg-Landau parameter K,defined in equation (30) and (31).
HC2 = KvHc
(36)
1.4.2 London Model For Extreme Type-II Superconductor
The London Model discussed in section 1.3.1 assumes no flux penetration into
the superconductor. This model can be modified to account for fluxons in the case
of an extreme type-II material. 29 An extreme type-II superconductor has a
coherence length much shorter than its penetration depth so x goes to infinity.
Fluxons can be treated as rigid, straight lines with no thickness. They are
29 A. L. Fetter and P. C. Hohenberg, "Theory of Type II Superconductors", in Superconductivity
(In Two Volumes), edited by R. D. Parks, Vol. 2, Chap. 14, (Marcel Dekker, Inc., New York,
1969) 817-923.
30 See, for example, D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, 2nd edition,
(Adam Hilger Ltd, Bristol and Boston, 1986) 303-306.
two-dimensional delta functions. For a single fluxon in the z direction at the origin,
equation (1) can be modified to yields
H
o0
(37)
()_
)
H-H --Z8xy o-,
where &yis a two-dimensional delta function in x and y and p is the cylindrical
radius vector.
The solution for H involves the zero order Hankel function, and the solution for
J, using Ampere's Law (2) and equation (11.7), involves the first order Hankel
function. The Hankel functions are described in detail in Appendix II.
H(p) - 2,t7to
i Ko(p/L)
(38)
(38)
J(p)
o= 8K(p/ X)
2nrtoL.
Approximation (1.6) shows that the magnetic field and current density decay
exponentially to zero far away from the fluxon.
H(p)
lim
-lim J(p) =
I'-
2xt
0=z " e-'AL
2p L
OL
o. 0
(39)
-e-PL
2p
2rotX3
Near the fluxon, but outside the core, approximation (11.5) can be used.
ln(
lim H(p) =
lim J(p)=
4o<<p<<"L
S
2
/ p)
(40)
o
2oXv2oP
For the case of N fluxons, a delta function is used to represent each fluxon, and
equation (37) becomes
V2H- H
_0o
xy=
-
)
(41)
where the solution for the magnetic field and current density is the superposition of
the solutions for N individual fluxons.
1.4.3 Ginzburg-Landau Theory for Extreme Type-II Superconductor
Ginzburg-Landau theory generalizes XL to X(T) for the fields and currents around
a fluxon. The above fields and currents calculated from London theory are then
correct for bulk specimens. For thin films or surface layers of bulk specimens the
current and field distributions are different. Pearl uses GL theory to calculate the
current distribution within a distance X(T) of the metal-air interface of an extreme
type-II superconductor. 31
Consider a single fluxon with a narrow core in a superconductor which fills the
half-space z>0.Cylindrical coordinates are used. The GL order parameter modulus
•o is taken to be constant. The phase 6 is single valued, satisfying a quantum
"°.The GL
condition and allowing the order parameter to be written as 4 = ioe
equation for the current density (20) results in the vector potential A.
VxVxA+(1/X(T))A
=
2np%(T)
VxVxA=0
z>0
(42)
(42)
z<0
By symmetry, A is a function of r and z only and is in the e direction. The solution
for the current densities was calculated for the bulk z--oo and the surface z---. In
the bulk limit, Ginzburg-Landau theory gives (38) in agreement with London
theory. In the surface limit, the current distribution agrees with the London result
(40) near the fluxon. Far from the fluxon, the current density follows a p-2 rather
than an exponential dependence.
o 2
lim J(p) = 2xn(T)!itoP
p--+
(43)
The range of the fluxon-fluxon interaction force is long range at the surface
while it is short range in the bulk. Young's modulus is therefore higher on the
surface than in the bulk and the lattice becomes stiff near the surface. Although the
lattice is stiff with respect to compression, the shear modulus vanishes. This has
implications for the lattice order, as discussed below.
1.4.4 Hexagonal Lattice
Free energy minimization determines the shape of the fluxon lattice. The detailed
calculation is available in the literature 32"3~ 4 so it will not be repeated here. The
hexagonal lattice (also called triangular) has a slightly lower free energy than the
square lattice. Consider the hexagonal lattice.
3
J.Pearl, "Structure of Superconductive Vortices Near a Metal-Air Interface", Journalof
Applied Physics 37, 4139-4141 (1966).
W. H.Kleiner, L. M.Roth, and S.H. Autler, "Bulk Solution of Ginzburg-Landau Equations
for Type II Superconductors: Upper Critical Field Region", Physical Review 133, A1226-1227
(1964).
33J. Matricon, "Energy and Elastic Moduli of a Lattice of Vortex Lines", Physics Letters 9,289291 (1964).
A.L. Fetter, P. C. Hohenberg, and P.Pincus, "Stability of a Lattice of Superfluid Vortices",
3
Physical Review 147, 140-152 (1964).
Take the distance between nearest neighbors in a hexagonal lattice as the fluxon
lattice spacing 1.To form a two dimensional hexagonal lattice, as shown in
Figure 1-3, use two independent lattice basis vectors R1 and R2.
R, =li
RI)
+
)(44)
The linear combinations of these basis vectors form a one-to-one correspondence
with the fluxon positions in a perfect lattice. The area per fluxon is •12 / 2. This
leads to a relation between lattice parameter and magnetic induction.
I
(45)
21=
For example, a 2.3 mT or 23 Gauss magnetic induction gives a lattice parameter of
1 pm. Table 1-1 shows the first eighteen neighbor distances and degeneracies for a
perfect hexagonal lattice.
Since the fluxon lattice is only two dimensional, the detailed nature of lattice
defects is expected to be different from a crystal lattice. However, the fluxon
lattice, analogous to a crystal lattice, is expected to contain lattice defects. 35 Point
and line defects were directly observed by Trtiuble and Essmann using decoration
techniques on lead-indium single crystals.36 The dynamics of flux lattice
dislocations influence flux flow behavior just as the dynamics of crystal lattice
dislocations influence mechanical properties.
In addition to hexagonal lattices, fluxons may form square lattices. The small free
energy difference between the hexagonal and square lattice structures corresponds
to a small lattice shear modulus. As discussed above, the shear modulus vanishes
near the surface of an extreme type-iI superconductor. Therefore there is no
particular lattice structure near the surface in this case. For a hexagonal lattice, the
shear modulus c6a has been calculated by Labusch.37
S=
C
66
0.24goH,2 (2K2 -1)2
1))
(1+L16(2K 2 1)2
H
1-
2
(46)
HC2
For example, with an upper critical field of 35 mT and Ic= 3, the shear modulus is
(9 Pa)(1 - H / Hc2 )2.
35 R.
Labusch, "Dislocations in the Flux Line Lattice", Physics Letters 22, 9-10 (1966).
36 H. Trauble and U. Essmann, "Fehler im FluBliniengitter von Supraleitern zweiter Art",
Physica Status Solid 25, 373-393 (1968).
37 R. Labusch, "Elastic constants of the Fluxoid Lattice Near the Upper Critical Field", Physica
Status Solid 32, 439-442 (1969).
Figure 1-3 Unit cell of hexagonal lattice shown with basis vectors.
Table 1-1 Neighbor Spacings And Degeneracies For A Hexagonal Lattice
Neighbor
#
Distance in
lattice
1
2
3
4
5
6
7
8
9
spacings
1.00
1.73
2.00
2.65
3.00
3.46
3.61
4.00
4.36
Degeneracy
6
6
6
12
6
6
12
6
12
Neighbor Distance
in lattice
#
spacings
10
11
12
13
14
15
16
17
18
4.58
5.00
5.20
5.29
5.57
6.00
6.08
6.25
6.56
Degeneracy
12
6
6
12
12
6
12
12
12
1.5 Intermediate State Versus Mixed State
The above discussions of critical magnetic fields assume that the applied
magnetic field is uniform at the superconductor surface. This is strictly correct only
for the case of a long, narrow superconducting cylinder in a magnetic field parallel
to its axis. For other geometries, the demagnetization factor must be taken into
account. The demagnetization factor causes a magnetic field nonuniformity at the
superconductor surface. Minimization of the sum of condensation energy and
magnetic field energy will occur by the formation of alternating normal and
superconducting regions in the material.
The demagnetization effect is smallest for a long, narrow superconducting
cylinder in a magnetic field parallel to its axis and largest for a thin, flat
superconductor in a magnetic field perpendicular to its surface. For a thin film
type-II superconductor, the demagnetization effect permits magnetic flux
penetration at fields lower than Hcj.
Bulk type-I materials do not enter the mixed state. Since the intermediate state is
a purely geometrical effect, however, bulk type-I materials can enter the
intermediate state. Very thin film type-I materials can enter the mixed state for
some values of kappa and the magnetic field. In fact, they can form a triangular
lattice of singly quantized fluxons, form a honeycomb lattice of multiply quantized
fluxons, or enter the intermediate state. The state depends on K, the film thickness,
and the magnetic field.38
38G.
Lasher, "Mixed State of Type-I Superconducting Films in a Perpendicular Field", Physical
Review 154, 345-348 (1967).
2. Theory of Fluxon Behavior with Applied Current
This chapter explores the various theories of flux flow in a mixed state
superconductor. First, the hydrodynamic theories of clean superconductors are
reviewed. This includes a description of the Magnus, Lorentz, and viscous forces.
Pinning is then introduced, leading to a description of the critical current density
and an exploration of static and dynamic pinning mechanisms. The experimental
status of the Hall effect is summarized.
In the Meissner state there is no electrical resistance or voltage for small applied
currents. Large applied currents destroy the Meissner state. In the mixed state,
electrical resistance remains zero, but losses can occur through a mechanism called
flux flow resistance. Applied electrical currents can cause fluxons to move.
Viscous damping of fluxon motion produces power dissipation, causing the
appearance of a voltage drop across the superconductor. In this case, the
superconductor has a finite resistance.
2.1 Hydrodynamic Theories
The two major theories of fluxon motion are the Bardeen and Stephen (BS)
model and the Nozibres and Vinen (NV) model. These are hydrodynamic theories
of clean, extreme type-II materials. They examine the interaction of externally
applied currents with individual fluxons, thereby ignoring most of the
fluxon-fluxon interactions. Clean materials are homogenous, providing no pinning
sites to trap fluxons. Therefore, all fluxons are freely mobile.
2.1.1 Magnus Force
The BS and NV models differ in their assumptions about the existence of a
Magnus force on moving fluxons in a superconductor. In superfluid helium, a
Magnus force results from the linear motion of vortex circulation currents.
Consider a vortex moving through a superfluid. In the vortex reference frame
shown in Figure 2-1, there are two currents, the circulation current and the
superfluid current. On one side of the vortex (region Q), these two currents sum
together, while on the opposite side of the vortex (region P) these two currents
oppose each other. According to Bernoulli's theorem, the fluid pressure is greater
on the side where the net current is smaller (region P). The vortex therefore
encounters a force transverse to the direction of vortex motion, in the plane of the
vortex circulation currents.
hýho
10
.
b.
r
Force
Figure 2-1 Vortex and its circulation currents immersed in a
uniform supercurrent.
2.1.2 Driving Force
The applied transport current and the magnetic flux of the fluxon interact
through a Lorentz force. For a uniform transport current density J, the Lorentz
force per unit length is given by Jx~o or (VxH)x 0o.
Friedel 39 derives the driving force based on a thermodynamic approach for rigid
flux lines. For a constant number of fluxons, small changes in the flux density B
and the area A are related by dB/ B = dA / A. The fields B and H are parallel to
the fluxon direction. The pressure is calculated from the free energy as
r " AL?
(d(Ay) )) r==- f
P= dA
dA
P=--"
dB
and the force per unit volume on the vortices follows as dP /dx.
9 J. Friedel, P. G. De Gennes, and J. Matricon, "Nature of the Driving Force in Flux Creep
Phenomena", Applied Physics Letters 2, 119-121 (1963).
(
(47)
dP
=
dx
f
dJ" dB
ddJ
d
T)d
+
+B
dB
dB2
dB )dx
=BdB d (d
(48)
dx dB dB
dB dH
dx dB
=BdH
dx
The driving force per unit length depends on the gradient of the total magnetic
field.
Drivi dH
(49)
fx• = - -o
dx
2.1.3 Bardeen-Stephen and Nozibres-Vinen Models
The Bardeen and Stephen 40 (BS) model is based on a local generalization of
London theory. A flux line 0o is treated as a circulation current flowing around a
normal core of radius 4. The direction of the fluxon line is given as the direction of
the magnetic field, normal to the plane of the circulation current. A uniform
transport current J is applied perpendicular to the flux line direction. The current
produces a Lorentz force on the fluxon, causing the fluxon to move perpendicular
to J and o. Electric fields caused by the motion of the flux line drive a dissipative
normal current through the fluxon core.
The derivation of the Magnus force for a moving fluxon assumes translational
invariance. The stationary fluxon reference frame must be as valid as the stationary
superfluid reference frame. The background of positive ion cores differentiates a
superconductor from superfluid helium. In considering the Magnus force, the BS
model assumes that electron-lattice scattering dominates electron-electron
scattering. The ion lattice balances out the Magnus force on any particular
fluxon.4 1
Under the BS model, therefore, the only forces acting on a flux line are the
Lorentz force and the viscous force. The Lorentz force per unit length acts
perpendicular to the applied current
f or
= JX 90
(50)
4o J. Bardeen and M. J.Stephen, 'Theory of the Motion of Vortices in Superconductors",
Physical Review 140, A1197-A1207 (1965).
J.Bardeen, "Motion of Quantized Flux Lines in Superconductors", Physical Review Letters 13,
J4
747-748 (1964).
The viscous drag force per unit length is assumed to be proportional to the fluxon
velocity.
(51)
fv Ou= -T77v
where rl is the viscosity coefficient The viscosity coefficient will be discussed in
more detail below. The steady state fluxon velocity is given by balancing the
Lorentz and drag forces, yielding v = J.ý/ r. If the Hall effect is neglected, the
fluxon velocity, fluxon orientation, and applied current density are found to be
perpendicular.
The BS model 42 derives an expression for the viscous coefficient by considering
energy dissipation of electrons flowing in the normal core as well as energy
dissipation of normal electrons outside the core. An analysis is made in terms of
the circulation current momentum of a moving vortex. The electron velocity in the
core is equal to the transport current velocity, which implies no backflow. The
viscous coefficient is determined by equating the viscous power dissipation of a
fluxon rlv to the calculated power dissipation inside and outside the fluxon core.
The BS result was found to match the Kim empirical relation discussed below in
section 2.1.4.
The Nozibres and Vinen 43 (NV) model considers the motion of fluxons to be
analogous to vortices in a classical uncharged fluid. Vortex motion is determined
by the balance between a Magnus force and a friction force. In a perfect crystal
with no defects, the frictional force vanishes and the fluxon moves at the superfluid
velocity. This is the major difference between the NV and BS models. The NV
model disputes the BS assumption that electrons in the fluxon core are in a local
equilibrium with the ion lattice. The mean free path of the electron in a clean
material is much larger than the fluxon core size. It is questioned how an
equilibrium distribution can be established with such a rapidly varying order
parameter.
2.1.4 Viscous Coefficient
The viscous coefficient 1rhas been studied since 1964, when Kim" and Stmad,45
measured flux flow in low Kalloys Nb-Ta and Pb-In and in high Kalloys Nb-Zr,
Nb-Ti, and V-Ti. The slope of the voltage versus current curve above the
depinning threshold is defined as the dynamic flux flow resistance dV/dl. In the
presence of a magnetic field, the flux flow resistivity persists down to zero
42
M. J. Stephen and J. Bardeen, "Viscosity of Type-II Superconductors", Physical Review Letters
14, 112-113 (1965).
" P. Nozibres and W. F. Vinen, "The Motion of Flux Lines in Type II Superconductors",
PhilosophicalMagazine 14, 667-688 (1966).
" Y.B. Kim, C.F. Hempstead, and A.R. Stmad, "Flux-Flow Resistance in Type-II
Superconductors", PhysicalReview 139, A1163-1172 (1965).
"5A.R. Strnad, C. F. Hempstead, and Y.B. Kim, "Dissipative Mechanism in Type-II
Superconductors", Physical Review Letters 13, 794-797 (1964).
temperature. The resistivity was found to increase linearly with magnetic field for
very low temperatures, obeying a law of corresponding states.
(52)
Pf /P, = H/Hc
pf is flux flow resistivity and p, is normal resistivity. Although this equation does
not hold for all materials or all temperatures, this assumption will be used to define
the intrinsic viscous coefficient.
In steady state motion, the mean Lorentz force per unit length f= Joo on an
isolated fluxon is equal to the viscous force per unit length f= rlv. The power
density dissipated by the viscous force, nf', is equal to the power density provided
by the current, pfJ2, assuming all fluxons move with same velocity v. Combining
these equations, the law of corresponding states, and definition of the flux
quantum yields an empirical relation for the intrinsic viscous coefficient rjo.
(53)
/e
ro0 = xhl oHc,.,
0
It has been assumed that n4o = B = toH. The normal state conductivity o, is given
by a, = ne 21•, / mvF, where n is the density of electrons, l, is the electron mean
free path at the Fermi surface, and vF is the average electron Fermi velocity. Using
equation (21), the conductivity can be written as o, = 2vFN(O)e lP / 3.
Alternate expressions for the viscous coefficient can be found by inserting
equation (35) and the above conductivity relation into equation (53).
1o =
r0
h v, N (O)l,,)
(T)
3
34?(T)
(54)
12 will be made. Using equation
In the dirty limit, the approximation 4(T)=( o•,)"
(22) and ignoring a constant factor close to one, the viscous coefficient in the dirty
limit equals
(55)
1o0 = xthN(O)Ao
with 2Ao equal to the band gap at zero temperature. An alternate form can be
found using the BCS equations (23) and (24).
(56)
7io = 0.87hlTc/kB
Alternately, using the BCS relation (22) and the density of states
N(O) =m v
(57)
yields two more expressions for rlo.
mo
m
t(8
1o =
3.06m k BT 4o
h3
(58)
2.2 Dirty Materials
The BS and NV models are applicable primarily to the case of ideal clean
superconductors where there are no pinning effects. In these materials, any applied
transport current will cause the fluxons to move and dissipation to occur. Thus,
zero resistance behavior does not occur with these ideal materials for magnetic
fields greater than Hci.
Dirty materials with pinning sites have more useful electrical characteristics in
the mixed state. Impurities, as will be shown below, are energetically preferred
locations for fluxons and therefore serve as pinning sites. If a sufficient number of
fluxons are pinned, then all fluxons will be pinned via their mutual interactions.
This effect is a result of the fluxon lattice rigidity.
4
P,
Ht
Figure 2-2 Superconductor in a line pattern geometry.
The following sections use the superconducting line geometry shown in
Figure 2-2. The line pattern has length 1,, width w,, and thickness h,. An applied
current I produces a uniform current density J = I / wh,. The film is in a magnetic
field perpendicular to its surface so that fluxons penetrate parallel to h,. Assume
there are N total fluxons for a fluxon density n = N / wj,.
2.2.1 Kim Voltage-Current Relation
The pinning force competes with the Lorentz force. For low current densities,
the fluxon lattice remains pinned. Since fluxons are not moving, there is no energy
dissipation and the effective resistance is zero. For high currents, the pinning is
insufficient to overcome the Lorentz force. The fluxons break free from the
pinning sites and energy dissipation occurs. This behavior was measured by Kim.4
The empirical voltage-current relation found by Kim can be written as
V=O
V=R(I-I
I<I c
c)
I>>I c
(59)
where Ic, I'c, and R are considered here to be empirical constants. This behavior
was interpreted as uniform, rigid fluxon lattice motion at velocity v.
The induced longitudinal emf in the strip is given by the differential form of
Faraday's Law (3), V= -d4 / dt, where 4 is the total flux in the strip. Since each
flux line has one quanta of flux 0o, the induced voltage is V = n4ovl, = Bvl,. The
electric field induced by moving fluxons is simply the magnetic field of the fluxons
times their velocity. Assuming all fluxons move with the same velocity v, then the
empirical voltage-current curve can be written in terms of the critical current
density Jc as a force balance
v=0
J < Jc
60
v= -(J-J")
J >> Jc
(60)
where 4oJc is the maximum static pinning force and Ool'c has been classified as a
mean 'frictional force' associated with the pinning sites. Jc and J'c are
approximately equal in magnitude. Note that at a given current density, the fluxon
velocity is inversely proportional to the viscous coefficient. This will be important
later when the conductor proximity effect is modeled as an additional viscous
coefficient term.
Dynamic resistance R varies inversely with the viscous coefficient.
dV
R
dl
B-oo- I_-
rnwl h)
(61)
The power supplied by the current is IV.
Power = (B-Jwh ) I(I-Ic)
71
w,h,
(62)
For this linear, uniform motion model, the flux flow voltage is proportional to the
flux flow velocity which is proportional to the applied current density. Therefore
the voltage-current curve is linear above the critical current and the dynamic
resistance is a function of B and r1 only. The power is inversely proportional to the
viscous damping coefficient. A small fractional increase in the viscous coefficient
causes an equal fractional decrease in the fluxon velocity for a given current
density. An equal fractional decrease in the flux flow voltage and power occurs as
well.
In the general case, the fluxons do not all move with the same velocity. The
lattice may shear, allowing some sections of lattice to remain pinned while others
move. There may be dislocations which move through the lattice while the bulk of
the lattice remains motionless, analogous to motion of dislocations in crystalline
solids. Finally, when all or nearly all fluxons are moving, inhomogeneities in the
material may cause different fluxons to move at different speeds. The above
relation is linear in v. It is therefore valid if v is taken to be the weighted average
fluxon velocity even if the fluxon velocities are distributed. However, complete
theories of Jc and Ic must account for the possible distributions of fluxon
velocities. Below, the calculation of the flux flow voltage is generalized for various
non-uniform fluxon velocity distributions as well as for non-linear damping
mechanisms.
2.2.2 Fluxon Velocity Distributions
A more general consideration of fluxon motion allows for a distribution in the
number of moving fluxons and their velocities. The fluxons are assumed to move
perpendicular to the applied current density. Factors which would cause a
deviation in the flux flow direction are neglected. The Hall effect and flux flow
channeling by defects are therefore ignored. The distribution of fluxon velocities
N(V') depends only on fluxon speeds. The distribution is defined by summing over
all N fluxons:
N
N() = 18(7 - v,)
(63)
i=1
where vi is the speed of fluxon i. The normalization condition, average speed, and
average speed squared are defined below.
N= I N(PV)dd
(v) - N
(V2=1
_.
N(v)dv
(64)
j2N(V)dv
In order to determine the average speed of the moving fluxons, the stationary
fluxons are not considered. The average speed is
f N(V')dV
(v),
=
(65)
f N(V)dv +f N(C)dv
for an arbitrarily small, positive A.
In general, the fluxon velocity distribution can not be found from voltagecurrent-field relations. There are simply too many adjustable parameters in a
general distribution. However, velocity distributions models may be designed with
only one or two adjustable parameters. The predictions of these models may then
be compared with the data.
Uniform Lattice Motion Model
All fluxons move at same velocity v.
Adjustable parameter v.
N(V) = NS(V - v)
(v) =
() =v2
= V
Fractional Motion Model
Fraction 0 of fluxons move at same velocity v.
Adjustable parameters 3 and v.
N(V) = N[P3(i7 - v) + (1- P)8()]
(v) = v
(v2) =
v'
(v),n, = v
Gaussian Model
Velocities distributed in Gaussian packet.
Adjustable parameters vo and v,.
N exp(-(~- vo / V)
N()
.N(J)
(>=Vo
(v?)
- VVo:+
(V).•.V,-o
2.2.3 Linear Damping
Various models of N(') can be fit to measured voltage-current curves. Assume
that Faraday's Law always determines the voltage generated by a moving fluxon.
The voltage is therefore proportional to the average fluxon velocity.
V=Bl,(v)
(66)
The fluxon velocity is determined by a force balance between the Lorentz driving
force and the viscous damping force. For linear damping, the velocity of moving
fluxons is specified by the current density and the damping coefficient.
I - I, = w,h, Trl < v >,,
/o00
(67)
This equation assumes that the driving force on the fluxons is determined by the
Lorentz force minus a non-dissipative dynamic pinning force.
The power dissipation per fluxon is the product of the viscous coefficient per
unit length, the fluxon length, and the velocity squared.
P=Bwhl, ()
(68)
Conservation of energy requires that this dissipated power corresponds to the
input power IV.
P
0hl q(vXv),,
+ Bl,Ic(v)
(69)
For linear damping, <v><v>,~ogmust equal <v2> well above the critical current
Ic. Therefore the fluxons must move with characteristic velocity. Uniform lattice
motion and fractional motion models are consistent with linear damping.
Linear damping results in a linear voltage-current relationship when the uniform
lattice model is used. It can also account for non-linear voltage-current
relationships under the fractional motion model, if the fraction of moving fluxons is
a function of the applied field and current. A linear relationship between flux flow
velocity and applied current is supported by direct velocity measurements. 4
Jones47 explained some non-linearity of voltage-current curves by a critical
current variation along the film length. The films used in these experiments had a
varying cross section. For a given current, the current density varies in different
sections of the film. The critical current, therefore, varies with position. This can
not account for observed non-linearities, however. First, the observed non-linearity
is much sharper than could be explained by a geometrical approach. Second, the
geometrical model predicts a linear relationship at high currents. This linear
relationship is not observed.
2.2.4 Non-linear Damping
Non-linear damping models can quickly become more detailed and complex. The
form of the velocity-current relationship has to be chosen. Here, assume that the
velocity of moving fluxons is proportional to some power x of the driving force:
v= vo(
(70)
where fo and vo are constants. Non-linear damping reduces to linear damping for
x= 1 and 1r =fo/ vo.
6H. Meissner, "Flux Flow Velocity in Superconducting Films by Measurement of Passage
Time", American Physical Society Bulletin 14, 112 (1969).
R. G.Jones, E. H. Rhoderick, and A. C. Rose-Innes, "Non-linearity in the Voltage-Current
"47
Characteristic of a Type-2 Superconductor", Physics Letters 24A, 318-319 (1967).
Consider non-linear damping and the fractional motion model. Assume the
current is much larger than the critical current. The driving force is given by
f = WoJ. The induced voltage is V= 3Blv.
V = B1lv( o 'J
(71)
Non-linear voltage-current relations can be explained using this model in two
ways: (1) 3 independent of J or (2) 3 some specified function of J. Method (1)
requires fewer assumptions as to the form of P(J). In practice, however, it is
expected that 3 is some increasing function of J.
2.2.5 Static Versus Dynamic Pinning
Experimentally it is found that as materials are modified so that Jc changes, J'c
changes in a corresponding manner. These two parameters are correlated and
nearly equal. There is no a priori reason why this is true. A static pinning attraction
does not necessarily imply a velocity independent motion impeding force.
Generally, it has been assumed that J'c and Jc depend on the static interaction of
individual fluxons with individual pinning sites. It is possible that they both depend
on some other underlying properties as well, such as fluxon line tension, fluxon
lattice compressibility, or geometry of the current distribution in the material. A
theory which accurately derives the values and dependencies of J'c and Jc does not
exist.
In summary, the Lorentz force drives fluxon motion. There is a velocity
dependent damping force rjv and a velocity independent damping force oJ'c. The
effect of the velocity independent damping force is to slow the motion of the
fluxons and thereby decrease the power loss for a given applied current.
Alternately, for a given applied voltage, a larger driving current is required to
maintain that voltage. The power loss is larger in this case. The slope of the
voltage-current curve depends on the magnetic field and the viscous parameter.
Thus the dynamic resistivity is independent of the critical current density for any
given material.
2.2.6 Static Pinning Mechanism
A pinning site is a local variation that couples to the order parameter and forms
an energetically preferred location for a fluxon. The simplest example of a pinning
site is a non-superconducting region imbedded in the superconductor. A volume
h ' 2 of material is normal at the fluxon core. Since the superconducting state has a
lower free energy than the normal state below Tc, there is a condensation energy
cost associated with the normal core of a fluxon. This condensation energy equals
the difference in free energy per unit volume of the normal and superconducting
states multiplied by the volume of normal material at the core. If the fluxon can
move to a location in the material which is already normal, then the total free
energy of the system is reduced.
Consider a spherical impurity of radius a, where a is of the order of 4. Using
equation (18), the free energy of a fluxon which moves to a spherical pinning site
can be reduced by as much as 27ra 3glic 2 / 3. For a line defect parallel to the fluxon
of length h, and radius a, the free energy can be reduced by as much as
7ta 2hs.toHc2 / 2.
If the fluxons are non-interacting, then all of the fluxons must be individually
pinned to prevent dissipation. Depinning will occur when the Lorentz force on any
particular fluxon exceeds the pinning force on that fluxon. This is analogous to
breaking the weakest link in a chain.
Fluxons interact strongly if their density is high. Pinning a fraction of the fluxons
can then indirectly pin all fluxons. The lattice structure will be warped as the
fluxons try to position themselves as close as possible to the pinning sites. The
lattice order range will depend on fluxon density, as well as the strength and spatial
distribution of pinning sites.
The fluxon lattice is strong under compression, but weak when subjected to
shear. Consider a macroscopically inhomogeneous pinning site distribution.
Confine all pinning sites to a small local area. The fluxons in that local area will be
pinned. They will be able to weakly pin the rest of the fluxons. For some Lorentz
force and above, however, the locally pinned region will not be able to support the
shear needed to pin the rest of the lattice. An uneven fluxon velocity distribution
will occur.
Consider a rigid lattice of rigid straight flux lines and assume a random
distribution of pinning sites in the superconductor. Since the lines are completely
rigid, it is irrelevant whether these are line pins or point pins. Since the fluxons can
not move independently, each fluxon is not able to adjust its position to be on a
pinning site. The pinning force is therefore randomly oriented for each fluxon.
These vector pinning forces cancel each other out and the total net pinning
approaches zero, independent of the pinning site density. If the fluxon lattice is not
completely rigid, pinning will occur.
The total static pinning force, in general, is therefore a complex summation over
the individual pinning sites. The fluxon lattice coefficients play a key role in that
summation.
2.2.7 Dynamic Pinning Mechanism
The static pinning force can be calculated by looking at the static interaction
between a single fluxon and a pinning site. Using this method when the fluxoid is
moving, however, is incorrect in principle." Yamafuji points out that the dynamic
pinning should depend on the local deformations of the fluxon lattice. Yamafuji
assumes the fluxon lattice suffers small deformations due to pinning sites. A fluxon
interacting with a pinning site undergoes a Hooke's law restoring force -kAx. Ax is
the deviation of the fluxon from its mean position in the steady lattice flow. The
fluxons have a mass per unit length M and the usual viscous coefficient r7o. The
equation of motion for the fluxon becomes
MM3 + rAi + k&A = -dU,, / dx
(72)
where U¢p is the pinning site potential. The fluxon deformation energy transforms
into lattice vibration energy, which dissipates by viscous damping. For constant
force pinning sites of spatial periodicity dp and magnitude fpf, Yamafuji finds a net
dynamic pinning force which is always positive.
0o0Jc
3f
2
3f
kd,
(73)
This result does not agree with experiments,"9 however, which show that the
dynamic pinning force varies asfpf rather than f-.
Lowell5o examines the dynamic pinning based on the elastic constants of fluxons
and the fluxon lattice. Consider an x directed, isolated fluxon moving in the +y
direction. It encounters a small, point defect pinning site which holds a portion of
the flux line stationary, until the line tension is sufficient to release the fluxon. The
equation for the line shape and motion is
y
+ r2 + oJ = 0
(74)
where the line tension y is taken to be much larger than the pinning strength,
implying a small fluxon curvature. The current is assumed to be large, so that the
interaction time with the pinning site is short. The inertial mass of the fluxon is
neglected, which is justified by the short electron collision time of 10-'2 seconds.
The lattice is taken into account as in the Yamafuji method. The effect of nearby
fluxons is to impose a Hooke's law restoring force on the fluxon proportional to
its deviation from the mean motion. The Hooke's law spring constant is k. The
mean fluxon position is vt, resulting in the new equation of motion
2Y
(75)
y
+'-+ +oJ
- 2k(y - vt)= O
K. Yamafuji and F. Irie, "On The Concept of Pinning Force in Type II Superconductors",
Physics Letters 25A, 387-388 (1967).
*9D. D. Morrison and R. M. Rose, "Controlled Pinning in Superconducting Foils by Surface
Microgrooves", PhysicalReview Letters 25, 356-359 (1970).
50 J. Lowell, "The Frictional Force on a Moving Fluxon", Journalof Physics C: Solid State
Physics 3, 712-721 (1970).
48
(75)
which is solved by neglecting the viscous force at large values of t.
Considering fluxon lattice deformations as a basis for dynamic pinning is
fundamentally more reasonable than the application of standard static pinning
methods. It also provides a means of justifying the approximate equality of Jc and
J'c in terms of an underlying principle. Lowell's specific solution, however, does
not accurately predict the behavior of the voltage-current curve. In the case of line
pinning sites the first term in equation (75) becomes zero. This would indicate that
isolated fluxons have no dynamic pinning effect. In practice, line defects have
strong pinning. 5'
2.3 Hall Effect
The Hall effect for a semiconductor in a uniform magnetic field occurs when a
current is applied to the material. A voltage appears across the semiconductor that
is transverse to both the magnetic field and the applied current. Consider Figure 23, a semiconductor in a uniform -z oriented magnetic field. A current Jy is applied
in the +y direction. Charge carriers are either holes +q moving in the +y direction
or electrons -q moving in the -y direction. Magnetic force qvxB pushes either
type of charge carrier in the -x direction. The charge carrier will move until the
separation of charge produces an electric force qvyB in the +x direction which
opposes and balances the magnetic force. The direction of the electric field will be
different for positive and negative charge carriers. The sign of the resulting
transverse voltage Vy gives the charge carrier sign. The electric field magnitude is
Ex = vyB, = JyBz / qn = aEyBz / qn, where n is the number density of charge carriers
and a is the conductivity of the semiconductor. Using a = -zq 2 / m, where t is the
charge carrier lifetime and m is the electron mass 5" gives the Hall angle
tan 8H = Ey / Ex = tqpoH / m.
Under BS theory for a superconductor, the Hall effect is taken into account by
assuming that the electric field produced in the core is at the Hall angle relative to
Jr. This Hall angle is the same as that of a normal metal in a field equal to that in
the core. No viscous drag on the Hall component is considered.
5'L. Civale, A.D. Marwick, T. K.Worthington, M. A.Kirk, J. R. Thompson, L. KrusinElbaum, Y.Sun, J. R. Clem, and F. Holtzberg, "Vortex Confinement by Columnar Defects in
YBa2 Cu30 7 Crystals: Enhanced Pinning at High Fields and Temperatures", Physical Review
Letters 67, 648-651 (1991).
52 B. G. Streetman, Solid State Electronic Devices, 2 nd ed. (Prentice-Hall, Inc., Englewood Cliffs,
New Jersey, 1980).
I
x
Figure 2-3 Hall effect geometry for a semiconductor showing
behavior in the case of positive charge carriers.
The BS calculation leads to fluxon motion at Hall angle 0Oequal to (H / H2)
times the Hall angle in a normal metal at Hc2. H is the magnetic field in the normal
core and r is the electron lifetime.
tan
eP H
m
(76)
The NV calculation leads to a larger, constant Hall angle at all fields.
tanH = e=t Ha2
(77)
For normal metals, the Hall angle is linear with magnetic field. The experimental
results for the Hall effect in the mixed state are inconclusive. In niobium, for
example, Hall angles have been found which are constant with field, corresponding
to the normal metal at Hc:.53 Some researchers have found a decreasing Hall angle
with decreasing field, smaller than the angle seen in the normal metal. 54 Other
measurements show a rapidly decreasing Hall angle with a decreasing magnetic
field" or a Hall angle which changes sign.56
53 A. T. Fiory and B. Serin, "Resistivity and Hall Effect in Superconducting Nb", Physical
Review Letters 21, 359-361 (1968).
54 W. A Reed, E. Fawcett, and Y. B. Kim, "Observation of the Hall Effect in Superconductors",
Physical Review Letters 14, 790-792 (1965).
55 K.Noto, S.Shinzawa, and Y. Muto, "Hall Effect in Intrinsic Type II Superconductors Near the
Lower Critical Field", Solid State Communications 18, 1081-1084 (1976).
56 H. Van Beelen, J. P. Van Braam Houckgeest, H. M. Thomas, C. Stolk, and R. De Bruyn
Ouboter, "Some Measurements on the Effective Resistance and the Hall Angle in Type II
Superconductors", Physica 36, 241-253 (1967).
The Hall angle in the mixed state is difficult to measure for comparison with
theory. First, the magnitude of the effect is small, so exact probe alignment must
be maintained and careful measurements must be made. Second, the hydrodynamic
theories describe clean superconductors, so homogenous materials with minimal
pinning must be used. Inhomogeneities in even the best samples may still cause
fluxons to move along tracks, upsetting the Hall measurement. Third, the Hall
effect assumes that the magnetic field and current distributions are uniform and
perpendicular to the sample surface. Flat, mixed state superconductors carrying
electric currents generally have very complex magnetic field and current patterns
due to the demagnetizing factor, the magnetic field from the applied current, and
the magnetic field from shielding currents.5 7 These complex distributions must be
considered in the evaluation of experimental measurements.
'7 E. H. Brandt
and M.Indenbom, "Type-II-Superconductor Strip with Current in a
Perpendicular Magnetic Field", Physical Review B 48, 12893-12906 (1993).
3. Theory of the Conductor Proximity Effect
This chapter presents the development of the conductor proximity effect. A
normal metal conductor is placed in close proximity to, but not in direct contact
with, a mixed state superconductor. The effect of this metal on fluxon flow in the
superconductor is examined. The separation distance between metal and
superconductor will be large enough to prevent tunneling effects, but small enough
to avoid overlap in the magnetic fields from adjacent fluxons. This chapter begins
by introducing multilayer structures involving superconductors from the literature.
Then it derive the normal conductor eddy current losses using a magnetic dipole
approximation of a fluxon. The dependence on various geometrical parameters is
considered. Modifications to this result are calculated for substrates with
curvature. Then finite penetration depths are considered. Their influence on the
eddy current damping is calculated.
The prediction and measurement of the additional eddy current loss mechanism
are the primary thrust of this thesis. This mechanism provides additional
stabilization to a composite superconductor and normal conductor matrix. This
effect is small, however, and is measurable primarily in low magnetic fields, for
reasons that will be discussed below. The conductor proximity effect does provide
an additional means, however, of measuring the flux flow velocity distribution.
The voltage vs current and voltage vs field curves are modified by eddy current
damping. The observed changes in these curves depend on the detailed behavior of
the fluxons during flux flow. If the fluxons move uniformly at constant velocity, for
example, then the voltage change at constant current is smaller than if only a small
fraction of the fluxons move at higher speed. The final sections of this chapter will
discuss various models of flux flow and calculate the resulting eddy current
damping voltage change.
3.1 SuperconductorMultilayers
Proximity effects in multilayers involving mixed state superconductors have been
studied for some time. In an elegant demonstration of flux flow, Giaever5 8
deposited a multilayer with a tin primary several hundred nanometers thick, an
SiO2 insulator 10 to 20 nm thick, and a tin secondary 50 to 100 nm thick. A dc
current was passed through the primary. Voltages across the primary and
secondary were measured. Voltages were detected across the secondary only when
both films were superconducting. The moving fluxons in the primary dragged
fluxons through the secondary. This fluxon motion induced a flux flow voltage in
the secondary even in the absence of a current in the secondary. This
superconductor sandwich geometry is known as a dc flux transformer.
Giaever, "Magnetic Coupling Between Two Adjacent Type-Il Superconductors", Physical
Review Letters 15, 825-827 (1965).
581.
More recently, multilayers have been studied which replace the secondary with a
two-dimensional electron gas.59.60 A six probe setup has also been used to study
two-dimensional vortex behavior 6ý in Bi2Sr2CaCuO0 and both two- and threedimensional behavior6' in YBazCu 307-.. It has also been proposed that the dc flux
transformer behaves analogously to a resistively shunted Josephson junction. A
corresponding ac Josephson effect is predicted.6 3
This work will use a normal conductor as a secondary. The dc motion of fluxons
will produce local ac eddy currents in the conductor. The losses due to the eddy
currents cause an additional drag on the fluxons, slowing their velocity and
reducing the voltage induced in the primary.
3.2 Dipole Approximation- Electric and Magnetic Fields of Moving
Fluxons
The dipole source approximation of a fluxon is used to calculate the induced
electric field in a nearby conductor. First, the magnetic dipole moment of a fluxon
is derived. Then, the corresponding static magnetic field of a fluxon is calculated.
The time derivative of a moving fluxon's magnetic field is related to the spatial
derivative of the static field. The induced electric field in a plane conductor is
determined. This enables calculation of the power loss in a plane conductor.
3.2.1 Magnetic Dipole Moment of a Fluxon - Bulk
To calculate the fluxon magnetic field outside the superconductor, fluxons will
be treated as magnetic dipole sources. This approximation is valid as long as the
penetration depth is small with respect to the other distances in the problem. The
dipole moment of a fluxon is calculated directly from its circulation currents. The
magnetic dipole is defined in terms of the current density. "
m=
x'xJ(x')d 3x'
(78)
H. Kruithof, P.C. van Son, and T. M. Klapwijk, "Superconducting Vortices and the TwoDimensional Electron Gas", Helvetica Physica Acta 65, 488-489 (1992).
60 J.
Rammer and A. L. Shelankov, "Non-local Effects in Transport Properties of Macroscopic
Conductors", Superlarticesand Microstructures 11, 65-68 (1992).
61 H. Safar, E. Rodriguez, F. de la Cruz, P. L. Gammel, L. F. Schneemeyer, and D. J. Bishop,
59 G.
"Observation of Two-Dimensional Vortices in Bi2Sr2CaCu 2O,", Physical Review B 46, 14238-
14241 (1992).
H. Safar, P. L. Gammel, D. A. Huse, S. N. Majumdar, L. F. Schneemeyer, D. J. Bishop, D.
L6pez, G. Nieva, and F. de la Cruz, "Observation of a Nonlocal Conductivity in the Mixed State
Experimental Evidence for a Vortex Line Liquid", PhysicalReview Letters 72,
of YBa 2Cu 3 .07:
1272-1275 (1994).
63 A. Gilabert, I. K. Schuller, V. V. Moshchalkov, and Y. Bruynseraede, "New Josephson-like
Effect in a Superconducting Transformer", Applied Physics Letters 64, 2885-2887 (1994).
64 J. D. Jackson, ClassicalElectrodynamics, 2" ed. (John Wiley & Sons, New York, 1975) 181.
62
The bulk current density around a fluxon is given by equation (38). The
cylindrical symmetry of the current results in a magnetic dipole moment in the z
direction. Treating the film momentarily as a bulk sample, the dipole moment
becomes
m=
go
f K,(p/•(T))2xp dpjdz,
(79)
where h, is the fluxon length which equals the superconducting film thickness.
Using equation (1.8) with g = 2, v = 1, a = 1, and u = p / %(T), the radial integral
is found to equal 4nk(T)3 . The dipole moment per unit length inside the
superconductor is therefore
dm
0
(80)
z
dz' tLo
where 0 < z' < h,. The total dipole moment of a thin superconductor is
(81)
m=
Y0
when h, is much smaller than the other geometrical length scales in the problem.
3.2.2 Magnetic Dipole Moment of a Fluxon - Surface
For the above derivation, the current density satisfied in the bulk is used to
calculate the dipole moment. The current density near the surface is given by
equation (43). When this current density is plugged into the integral for the
magnetic dipole moment, the integrand is independent of p. A cutoff distance must
be chosen so that the radial integral does not diverge. It is reasonable to expect the
cutoff to be a constant of order unity times the square root of the area per fluxon
(o / B. The result for a thin superconductor is
m -
Oohs
go
0o
T -2
4X(T) B
(82)
which is equal to the bulk result times the correction factor in the square root The
correction factor is of order unity for reasonable choices of the magnetic field and
penetration depth. Note that the surface dipole moment is a function of the
magnetic field. Specifically, the square of the dipole moment varies inversely with
magnetic field.
Pearl,65 however, does not make use of this cutoff and calculates that the
magnetic moment of each fluxon is proportional to the sample radius. For a
rectangular film,
Pearl, "Distinctive Properties of Quantized Vortices in Superconducting Films", Low
Temperature Physics: LT-9: Proceedings of the IXth InternationalConference on Low
65 J.
m=
go,
(83)
zo
where w,is the film width. This magnetic moment is much larger than expected.
Allowing the current distribution from each fluxon to extend to the sample edges,
however, does not appear to be justified for a superconductor containing a large
number of fluxons.
The calculations below use the bulk magnetic dipole moment. Corrections due to
surface current distributions will be discussed later.
3.2.3 Geometry of Experiment
Figure 3-1 shows the experimental geometrical parameters. The normal metal
conductor of thickness ho is separated by a gap hi from a superconductor of
thickness h,. The distance from the top of the superconductor to a small volume
dV of normal conductor is z. The distance from the top of the superconductor to
the element of magnetic dipole moment dm is z'. The distance from dV to dm is
given by s = z + z'. For a thin superconducting film,
S -- Z
(84)
h'dm,
0oh,
/
0S,C*dz
0
o
-
as z' - 0.
Normal
.I
h
-
d
zS
uperconductor
S-p
r
o
c -o
-----------
Figure 3-1 Geometrical parameters used in theoretical
calculations.
TemperaturePhysics, edited by J. G. Daunt, D. O. Edwards, F. J. Milford, and M. Yaqub
(Plenum Press, New York, 1965) 566-570.
3.2.4 Induced Electric Field of Moving Dipole
Consider a z oriented magnetic dipole dm on the z axis, as shown in Figure 3-1.
This dipole moves at constant non-relativistic speed v in the x direction. The
moving magnetic dipole field dB induces an electric field dE according to
Faraday's Law (3). The induced electric field will cause current flow and energy
dissipation in the normal conductor.
For a conducting sheet parallel to the xy plane, any z directed induced electric
field will cause surface charges to appear on the top and bottom of the sheet.
These surface charges will set up static electric fields that oppose the induced
electric field. As a result, continuing z directed currents will not exist. Therefore
assume that the z component of the induced electric field is zero, restricting
consideration to induced circulating currents in the xy plane. The total induced
electric field E is found by superposition of the electric fields dE due to each
dipole dm.
Start with the well known general solution for the magnetic field of a stationary
dipole 66
dB(r) =
4
(3(dm.r)r
5
r
(85)
r3
where r is the vector from dm to the point of interest. By cylindrical symmetry, a z
oriented magnetic dipole has field components in the z and p directions only. Since
only the circulating currents in the xy plane are considered, only the z component
of the dipole magnetic field needs to be calculated.
2
dB, = 0odm, 2s - x'- y
4r (x2 +y +s)25
=
0dm
4x
2s 2 - p2
(p 2 +s2)'
2
(86)
An example of the z component magnetic field of a fluxon is shown in Figure 3-2.
The time dependence of dB, is contained in the time dependence of x and
therefore of p. Since x = vt, therefore dx /dt = v.
a(dB:)
-
at
a(dB.)
v-
Dx
-
.tdm. 3vpcos(p -4s)
(p2+s) 7/ 2
4x
(p'+s )7/
(87)
(87)
The components of dE are Ep, Ee, and Ez. As mentioned above, take Ez = 0.
Gauss's Law (4) is applied to the two remaining components.
18
1DE
V -(dE) =pE )+
=0
p Do
P ap
Extract the angular dependence by the substitution
(88)
" J. D. Jackson, ClassicalElectrodynamics, 2nd ed. (John Wiley & Sons, New York, 1975) 182.
E, (p,
e)-E)(p)
sin e
(89)
Ee(p,8) - Eo (p)
cose
which leads to a condition relating EO and E.
dEo
dp
E° = EO + pd
(90)
Consider the z component of the left side of Faraday's Law (3) in cylindrical
coordinates.
xcdE)
=i
(pE,)
(91)
P
Apply Faraday's Law and substitute equations (87), (89), and (90) to yield a
differential equation for E.O
d2E+
dE°
dp 2
dp
3o0dm.v p(4s-2)
41r
(p2 +s2) ?7
(92)
Then, solve this differential equation to get the electric field of dipole element dmn,.
dE=
po9dma v
4r
sine
.cos(s
+s
(p
(p +s2)
2
32
/
2 -2p 2
2
+
s
)
5/
This electric field was calculated only for the case of a fluxon on the z axis. If the
system considered is translationally invariant with respect to the direction the
dipole is moving, namely the x direction, then this is the electric field at all times,
where the p and e coordinates are measured relative to the moving dipole.
Using the approximation (84), the total electric field E is
ohv
for a thin superconductor.
sin
cos 8(z2
2pz)
(94)
hs
=1 micron
z = 1 micron
Fluxon Magnetic Dipole Field
(z component)
1.20E-04
--....
1.00E-04
B.OOE-Os
"'C
Q)
u::
6.00E-Os
(.)
+oJ
(1)
c
C)
ro
:2
4.00E-05
2.00E-Os
-2.00E-Os
1.5
0.5
-0.5
-1.5
111111
l{')
•
IIllj I
l{')
C\J,....
l{')
•
9
,...
Y (micron)
I
X (.
micron )
-2.5
I
Figure 3-2 Vertical component of fluxon dipole magnetic field.
3.2.5 Induced Electric Field of a Dipole Lattice
Before calculating the power loss in the metal, it is important to understand
quaUtatively the shape of the fields from a single fluxon and a lattice of fluxons.
The spatial derivative of the magnetic field due to stationary dipoles with respect
to x is closely related to the time derivative for dipoles moving in the x direction.
The induced electric field and eddy current losses depend on the time derivative.
To maximize these losses, it is desirable to have a spatially strongly varying
magnetic field.
As the distance from the superconductor increases, two effects conspire to
reduce the eddy current losses. First, the absolute magnitude of the magnetic field
produced by a dipole drops approximately as 1/ S3f2, where s is the normal
conductor to dipole spacing. The power loss therefore drops at 1 / S3. Second, as
the separation distance becomes comparable to or larger than the lattice spacing,
the effects of the fluxon lattice must be taken into consideration. The contributions
of neighboring fluxons a lattice spacing I away diminishes the gradient of the
magnetic field due to a single dipole. For separation distances s > > I, the magnetic
50
field becomes uniform and eddy current losses go to zero. Consider the field of a
fluxon at the origin and its twelve nearest neighbors, as shown in Figure 1-3. The
shape of the static magnetic field with respect to the x direction is shown in
Figure 3-3. The lattice spacing is taken to be 1 gm. The magnetic field for each
is
separation
orppa
ximately
to
normalized
1
/
s"'-
is
its
on
value
not
therefore
the
z
axis.
The
overall
field
dependence
en
shown
smooth out for larger separations.
Theoretical Magnetic Field Shape versus Separation
for 1 micron lattice parameter
o
U-
CD
V
o0
a:
I
0
o
0o
X Position (micron)
---
s=0.2 micron -21--s=0.5 micron --4--s=l.0 micron
-w--s=5 micron
Figure 3-3 Shape of magnetic field near and far from fluxon lattice.
3.3 Eddy Current Power Loss and Effective Viscous Coefficient
With the calculation of the electric and magnetic fields completed, it is now
possible to calculate the Joule heating in the normal metal due to induced eddy
currents. This power loss can be treated as adding an additional viscous damping
term to the fluxon motion thereby giving a new effective viscous coefficient.
3.3.1 Eddy Current Power Loss - Thin Superconductor Approximation
The following is a discussion of the eddy current power loss using a thin
superconductor approximation. Consider a sheet conductor parallel to the xy
plane, with conductivity a. The Joule heating per unit volume is given by
1
1
riE:, =(ohv
a ohv
4a
3+
(95)
(95)
3p (p -2 z)cs
+z2)J
(p-
where the electric field is given by equation (94).
Take the conducting sheet to have thickness ho as shown in Figure 3-1. The
radial extent of the dissipation due to an individual dipole is limited to a distance
p,, which is taken to be one-half of the lattice spacing 1.The total power loss in
the metal is a volume integral of the power loss density.
s+ho p, 2x
Power= f f JoEl2pdedpdz
(96)
z=s p=O 8=0
Perform the 6 integral first.
Power =
4r
+
2
2
=, O (P2 + z
2
rp2(p
2
(97)
) pdpdz
(p +z2)
Then perform the p integral.
Power = ar
4
47
2ov
f
3z4 + 4z2p + 10p~
3
(
-
8(p2 +Z2)
8'
4
(98)
Jdz
For large pm /z the second term in the integral becomes small with respect to the
first term in the integral. This corresponds to a large lattice spacing or low
magnetic field. Fluxons dipole fields do not overlap. Using this low field
approximation, the z integral is solved.
Power =-v
8 4n-
)23
7
-
(99)
(s + h 0 )3
(s+h
For a very thick normal conductor, the second term is small with respect to the
first. The power is
Power -
4=
for a thick normal conductor and thin superconductor in a low magnetic field.
(100)
3.3.2 Electric Field and Power Density - General Case
Now consider the more general case where the thicknesses of the normal
conductor and superconductor are comparable to their separation distance. The
dipole moment distribution must be taken into account. The total induced electric
field is calculated by integrating equation (93).
E=
00V
p
cos9(s
-2p2)
2)5/2
(2
sine
+s)3/2
(101)
can p. solved (be
This integral
for finite
This integral can be solved for finite p.
ov
/
E= 47r
z+ h,
z
psin
(p2++(z+h,)2)
2 +z2) 2 )
p (p
(102)
S (-(z+h,)(2p2 +(z+h,)2) z(2p 2 +z 2)
2 +p2 )3/2
p2 ((z+h$)
P2(z 2 +P 2)3/2
cos
In this form, E is not well defined as p--O. The electric field and Joule heating
on the z axis are handled as a special case.
(103)
' ds ~o v
o
x.f
2
3Y
12z)2
h,2 (1+ h, / 2z)2
ov)
81r
8n y(z(+hs
z4
Z
(1+hs / Z)4
(104)
Calculating the off-axis eddy current power analytically becomes cumbersome at
this point. It is instructive to calculate the electric field numerically and to plot the
electric field shape. The author wrote software to calculate the electric field. The
components of the electric field are calculated as a function of p and 0 for a given z
and h,.The exact fluxon velocity v is unknown and will not be needed to calculate
the viscous coefficient. Therefore, the velocity is normalized out of the calculation.
The p, 0, x, and y components of the induced electric field from an isolated
fluxon are given in Figure 3-4, Figure 3-5, Figure 3-6, and Figure 3-7. These
figures are generated from equations (102) and (103). The field is calculated one
micron above a one micron thick superconductor. Normalization with respect to
the fluxon velocity gives the dimensions indicated in the figures.
The corresponding power density, also normalized to the fluxon velocity, is
given in Figure 3-8. The power is sharply peaked near the axis above the fluxon, as
expected. The conductivity of 99.996% pure aluminum at room temperature is
-
- - - -
---~~~
3.77xI0 7 n- 1m- 1.67 The 4.2 K conductivity of pure aluminum is about twenty-one
times the room temperature conductivity. Therefore, for this power density
calculation, the nonnal metal conductivity is assumed to be 7.9xI08 n-1m- 1•
Electric Field - Rho Component
hs = 1 micron
z = 1 micron
S.OOE-OS
-E
4.00E-OS
en
3.00E-OS
a..
Q)
2.00E-OS
E
--....
1.00E-OS
a.
-...
>
~
0')
c
...
Q)
a..
(J)
'C
Q)
u:
-1.00E-OS
-2.00E-OS
-3.00E-OS
-4.00E-OS
X (micron)
Y (micron)
Figure 3-4 Induced electric field - p component.
CRC Handbook of Chemistry and Physics, 63 rd edition, edited by R. C. Weast, (CRC Press,
Inc., Boca Raton, Florida, 1982), page F-133.
61
54
Electric Field - Theta Component
h s = 1 micron
z = 1 micron
6.00E-OS
-E
tJ)
4.00E-OS
10-
(1)
c.
2.00E-OS
E
>
---...
.c
C)
c
...
(1)
10-
CJ)
-2.00E-OS
'0
(1)
u::
-4.00E-OS
-1
X (micron)
Y (micron)
Figure 3-5 Induced electric field -
55
e component
Electric Field - X Component
h s = 1 micron
z = 1 micron
6.00E-OS
_
4.00E-OS
(/)
E
'-
Q)
c.
E
2.00E-OS
-3>...
J::.
C)
c
...
Q)
'-
(J)
-2.00E-OS
'0
Q)
u:
-4.00E-OS
-1
X (micron)
Y (micron)
Figure 3-6 Induced electric field - X component
56
Electric Field - Y Component
h s = 1 micron
z = 1 micron
4.00E-05
-E
en
~
3.00E-05
2.00E-05
Q)
1.00E-05
E
......
O.OOE+OO
. !:
-1.00E-05
a.
-...
>
C)
c:
...
Q)
-2.00E-05
~
CJ)
"'C
Q)
u::
-3.00E-05
-4.00E-05
-5.00E-05
X (micron)
Y (micron)
Figure 3-7 Induced electric field - Y component
57
Power Density
hs = 1 micron
z = 1 micron
-
3
e:-.I
en
~
E
2.5
'Q)
a.
M
2
E
-~
>.
1.5
+J
en
c
Q)
C
1
'-
Q)
~
0
0.5
c.
X (micron)
-1
1
Y (micron)
Figure 3-8 Typical power density for single fluxon.
3.3.3 Viscous Coefficient
Total power can be calculated from power density, by integrating over the
coordinates p, 8, and z. Define the eddy current damping viscosity per unit length
2
11eddy by Power =11eddyhsv , where hs is the superconductor thickness and v is the
average fluxon velocity. Use this normalization factor so that direct comparisons
may be made between 11eddy and the intrinsic fluxon viscosity per unit length 110.
Integration of the power density over p and 8 gives the power per unit length as
a function of distance from the superconductor. The integration is performed
numerically by calculating the power density on a 5 Jlm x 5 Jlm grid with a grid
spacing of 5 nm. Twenty-five values of z were chosen from 0.1 Jlm to 2.5 Jlm. The
resulting data are shown in Figure 3-9. Next, a straight line was fit to the data on a
log-log scale. The numerical fit
58
Power = a (2.51 x 10-37 )z2- 73
[Power] = Ws2 Im ;[z] = m; [0]==
(105)
-1m-'
is also displayed in Figure 3-9.
To calculate the viscous coefficient, a normal conductor thickness must be
chosen. The above relation is then integrated with respect to z over the thickness
of the normal conductor. The velocity factors cancel out and the result is
normalized to the superconductor thickness. The resulting relationship is shown in
Figure 3-10. This equation is calculated using the film thicknesses and normal
metal conductivity given inthe figure.
rl, = 1.15 x 10-2(h;1
" 73
' )
- (hi + 10) -) -1n
kiuu)
[1,4 ] = kg/ ms ; [h] = m
Eddy Current Power Loss
versus Separation Distance
1.00E-08
4 1.00E-09
E
S1.00E-10
E
L. 1.00E-11
) 1.OOE-12
1.00E-13
0.0
0.5
1.0
1.5
2.0
z (micron)
Figure 3-9 Eddy current power per unit length versus
separation distance.
2.5
Viscous Coefficient
1.OE-09
.
....
Superconductor Thickness: 1 micron
aE 1.0E-10
Conductor Thickness: 1 micron
Conductivity: 7.9x10 (ohm-m)"
.I.
> 1.0E-11
1.0E-13
0.0
0.5
1.0
1.5
2.0
2.5
Gap Thickness hi (micron)
Figure 3-10 Calculated 1,ddy versus insulating gap thickness.
3.4 Curved Film Geometries
In equation (106) an effective rlady as a function of hi was determined for a
specific choice of film thicknesses. If the superconductor and normal conductor are
flat, parallel films, this will be the measured eddy current viscous coefficient.
However, for films which are not flat or parallel, the measured eddy current
viscous coefficient will be an area average of the local coefficients. Specifically,
TI
= 1 f , • (h,)dA
(107)
where the hi is the local gap distance for area element dA.
Deviations from parallelism result in a linear variation of hi. This can be
represented as
hi (x,y) = h, +ax+a,y
(108)
where the constants a, and ay are the relative slopes between the two films. This
becomes important if the film spacers are uneven or non-uniform or if the beam
does not sit evenly on the spacers. This case has not been considered in great detail
since the etching process produces spacers of uniform thickness.
The next possible deviation from flatness is curvature. For small curvatures, a
curved surface can be treated as parabolic. Consider the case of a flat surface
opposite a surface which is parabolic along one dimension. For this discussion, it
does not matter which of the two films is on the flat surface and which is on the
curved surface. The beam is chosen to be curved across its width. Note that
curvature of the beam along its length causes a smaller effect, since the
superconducting line width is much smaller than its length.
60
The two cases of interest are concave and convex curvatures. The geometries
are shown in Figure 3-11. In both cases, the separation distance hi is independent
of position along the width of the substrate, but varies along the length of the
substrate. The curvature is assumed to be symmetric over the center of the
superconducting line, therefore the point of closest approach for the convex
(concave) case is the center (edge) of the beam. The gap is
4Ah
x
h;6 (x)
8 + ! r = ho
2
convex
(109)
concave
,--x-
h, (x) = h + Ah
Wb
where hio is defined as the closest approach distance, Ah defines the magnitude of
the curvature, and x is given in the figure. The beam width wý is equal to the
superconductor line length I,. Noting that dA = wdx and A = w,wb, the average
eddy current viscous coefficient can be found directly. The results for a few
curvatures are given in Figure 3-12. The results are independent of the
superconductor line width. They also do not depend on the line length directly.
However, for a given factor Ah, the absolute curvature is a function of the line
length.
Ah
__t ...
Height
Length
X
*T
Ah.
Concae ..
i
hi
Figure 3-11 Side view of curved film geometry used in calculations.
Viscous Coefficient Vs Closest Approach
For Films With Curvature
1E-10
.....
_
Curvature in microns
Superconductor: 1 gm thick
Conductor: 1p•m thick
..
-
Conductivity: 7.9x10 8 (ohm-m) "
"
E 1E-11 - ----
.
flat films
ncave
1.0
-
corncave 0.3
..........
cor
.......-
ncave 3.0
nvex 0.3
-co
nvex 1.0
0..
-12
-
-- - †E
I
1
--- ---
-
co
nvex 3.0
._____
-...
12-13
0.0
0.5
1.0
1.5
2.0
2.5
Closest Approach Distance (micron)
Figure 3-12 Effect of film curvature on 1leddy.
For a given curvature magnitude, concave films show a much larger deviation
from flat films than convex films. This is a result of the slope of the film at the
point of closest approach. Since the viscosity is highly dependent on the local gap
size, most of the contribution to the overall eddy current viscosity comes from the
region where the two films are closest together. For convex curvature, the slope is
zero at the point of closest approach. The gap is close to the minimum value over a
large area of the films. For concave curvature, on the other hand, the slope is a
maximum at the point of closest approach. There is a very small area over which
the gap is small.
3.5 Finite Penetration Depth - Fluxon Magnetic Fields
All the calculations above have approximated the fluxon by its dipole moment.
This approximation is valid as long as the penetration depth is much smaller than
the other length scales in the problem. In this section, the dipole approximation is
relaxed. The fluxon magnetic field is calculated directly using the full bulk current
distribution around the fluxon, equation (38).
3.5.1 Thin Superconductor - On Axis
Compare the on-axis magnetic field calculated using the dipole approximation
with the field due to the full current distribution. For a thin superconductor, the
dipole moment is equation (81). The magnetic field on the axis of the dipole is
oriented in the z direction.
S2tz
3
(110)
Next, calculate the magnetic field due to the current distribution. Consider the
magnetic field of a radius p current ring with a circulation current density Je(p).
The on-axis solution is easily found by integrating the Biot and Savart law.
dBuhi
= oh , r2 Je (p)dp2
2 (p2 +z2 )3/
(111)
Plug in the current density distribution and integrate it over p from zero to infinity.
Let u = pk(T).
u K,(u)du
4. 0oh, 1 f
3
2 )3
2ntz 2J (1+(X(T)ulz)
(112)
where K,(u) is the first order Hankel function of imaginary argument as defined in
Appendix II. This integral can be solved numerically. The dipole and full current
fields are plotted for a thin superconductor in Figure 3-13. Consider the
dependence of B on z. For the dipole approximation, B Cz3 for all z. For the full
current distribution, the dependence varies. When z is one-half micron, the
dependence is B oc z-9 .
Field On-Axis of Fluxon - Thin Superconductor
Superconductor
Field On-Axis of Fluxon - Thin
Dipole Approximation
-
I C+V
I
•l
"
A
------.
-----
1E-1
Full Current Distribution
-
Assume = 390 nm
rUr,
S1E-2
Thin superconductor I..
• 1E-3Cn
S1E-4 -
...
...
..............
...........
---.
-----
1E-5 -
1
0.1
10
z (micron)
Figure 3-13 Magnetic field on-axis of fluxon - dipole approximation vs full
current distribution - thin superconductor.
63
3.5.2 Thick Superconductor - On Axis
Generalization to a finite thickness superconductor is straightforward. Replace z
in equations (110) and (112) with s = z + z', where z' is the distance from the top of
the superconductor to the plane of integration inside the superconductor. z is the
distance from the top of the superconductor to the point of interest above the
fluxon (see Figure 3-1). Replace h, with dz'. Integrate over z' from 0 to the
superconductor thickness hs.
+ z')d
B~·2n f-(z
o
(113)
u2 K
0
h'
S41cfo o((z+z')
1(u)dudz'
2 +X(T)u 2
) 3/2
The z' integrals can be solved in closed form. The u integral was performed
numerically. The results of the z' integrals are given here.
Bd
0
ole
1
1
2 - z2 (z+h,)2J
(114)
Bf
0O
S4n
K (u)
(1+ (T)2U2 /
(Z
+h)
2 )1
K1(u)
2
(1+ (T)2
/ z2 )
d
0
These results are graphed in Figure 3-14. Again consider the dependence of B on
z. For both curves, the dependence of B on z varies with z. Consider z = 0.5 Ptm
again. For the dipole approximation B c z"2 2. For the full current distribution
B 0Cz' 4.
The eddy current viscous damping coefficient for the dipole approximation is
plotted in Figure 3-10. At z = 0.5 .tm, when B cc z2.2, the viscous coefficient rleddy
is proportional to z'". It is expected, therefore, that the full current distribution,
which gives B cc z 1 4 , should give Teddy roughly inverse linear in z.
Field On-Axis of Fluxon - Thick Superconductor
1E+O
-.- Dipole Approximation
Full Current Distribution
. ... .
..
.. ..
S
- - -.---- -----
1 E-2
.....
. ..
......
u 1E-3
•h= 31 j m
...
................
... .. .... i
0.1
1
10
z (micron)
Figure 3-14 Magnetic field on-axis of fluxon - dipole approximation vs full
current distribution - thick superconductor.
3.6 Fluxon Bundling Effects
If fluxons were bundled, rather than evenly spaced, this would serve to increase
the eddy current damping coefficient. In an extreme case, assume bundled fluxons
are so close that they can be treated as a single, multiply-quantized fluxon. Assume
there are Nb fluxons per bundle in a superconductor containing N total fluxons.
The number of bundles is therefore N / Nb.
Using the magnetic dipole approximation for a fluxon, the dipole moment of a
fluxon bundle is directly proportional to the number of fluxons in the bundle. The
eddy current damping coefficient for a single bundle is proportional to the square
of the dipole moment. Therefore the eddy current damping coefficient for a single
bundle is proportional to the number of fluxons in the bundle squared. Since the
number of bundles varies inversely with the fluxons per bundle, the eddy current
damping coefficient per fluxon is directly proportional to the number of fluxons per
bundle:
Tieddy
c Nb.
It is also important to note that if the fluxons are arranged in perfect bundles,
then the spacing between bundles increases as the square root of Nb. Therefore the
magnetic fields of nearby bundles overlap less than for the singly-quantized fluxon
case.
3.7 Effective Flux Flow Voltage With Eddy Current Damping
The eddy current damping force on a fluxon is proportional to fluxon velocity.
Therefore, the eddy current power loss in the metal is proportional to the fluxon
velocity squared. However, the additional retarding force of eddy current damping
slows down the fluxon velocity at constant applied current density. This results in a
net drop in the flux flow power dissipation. This result is sketched in Figure 3-15.
xX
0 0n
IIru
Applied Current Density
Figure 3-15 Effect of eddy current damping on the ideal mixed
state V-I curve.
Consider an intrinsic viscous coefficient rio independent of current density and
field. This corresponds to Kim's linear damping and a linear voltage-current
relationship. The equation of motion for flux flow (60) becomes (7ro + rleddy)V =
co(J - J'c), where rleddy is the effective viscosity due to eddy current damping
losses. For a fixed current density, fluxons will move slower in the presence of a
normal conductor. Total power loss due to flux flow and eddy current damping
combined will be less than the power loss due to flux flow alone. The power
dissipation is
Power =
-_Booo
11 + 71cidy
iJ I(I Id)
c
wh, )
(115)
using equation (62).
For a constant applied current and magnetic field, the flux flow voltage with
eddy current damping will be Vorioi (r0o+ tieddy), where Vo is the flux flow voltage
in the absence of eddy current damping. When 1eddy <<rio, the change in voltage
AV = V - Vo is equal to -Voleddy/ r0o and the fractional change is --Tcddy / Tro.
Although rio can and will be calculated for these films, it is instructive to
eliminate it and write the fractional voltage change in terms of the directly
measurable parameters of current, field, and voltage. Assume the fractional motion
model from section 2.2.2 and the flux flow voltage relationship (66). Eliminating
rio with the equation of motion for flux flow, the fractional voltage change is
found.
AV_
V
Vwh - ld (linear damping)
o0(I-Ic)Bl1 P
(116)
Next, consider the non-linear voltage-force relation
V = V0
(117)
where fi,e is the net driving force on all fluxons, fo and Vo are constants to be
determined later, and x is the non-linearity factor. Let the driving force be
described byfdi,, = NJ4o - Np3rIddyv. The total Lorentz force per unit length on all
fluxons is NJ•o. The damping force per unit length on a single fluxon is vIeddy and
the number of fluxons moving at speed v in the fractional motion model is N3.
AV
_
Vw,h,
AV- =V
xrl,y
C)BlW
(non-linear damping) (118)
4. Conductor Proximity Effect Experiment Design
The previous chapter discussed the theoretical model of the conductor proximity
effect. It described how the flux flow resistance of a mixed state superconductor is
modified by the presence of a nearby normal metal. This chapter presents the
design of experiments to demonstrate the conductor proximity effect via its effect
on the flux flow viscous coefficient.
4.1 Design Concepts
The simplest experiment one might design would be to fabricate a set of
superconducting films. These films would be covered with insulators of various
thicknesses and coated with normal conductors. A comparison of the viscous
coefficients of these samples would yield a relationship between viscous coefficient
and insulator thickness. This relationship could be compared directly to theory.
There are a number of difficulties associated with this experiment. First, the
intrinsic viscous coefficient strongly depends on magnetic field and temperature.
Therefore, one would need to be very careful to measure all the samples under the
same conditions. Second, the properties of superconducting films vary. Variations
in thickness, composition, dimensions, or transition temperature effect ro. This
variation, even for films fabricated together, could easily dominate the properties
of the films and prevent any correlation from being measured.
A design which can measure the reddy directly for a single sample at fixed
temperature would be much more likely to succeed. Therefore, the spacing
between the normal conductor and the superconductor will be varied dynamically.
The flux flow voltage at constant applied current varies with reddy, which strongly
depends on the spacing. Measurement of the flux flow voltage as a function of
spacing will reveal the conductor proximity effect.
The eddy current viscous coefficient Tledd is small with respect to the intrinsic
viscous coefficient ro0 . Since leddy is a strong function of the superconductor and
normal conductor spacing, the normal conductor must be positioned within a
fraction of a micron from the superconducting film. Additionally, the spacing must
be small with respect to the fluxon lattice parameter. To maintain a low density of
fluxons in the superconductor, the experiment is performed in a low magnetic field.
For a four-probe superconducting line pattern, the fractional area of the line near
Therefore, the
the normal conductor will control the overall ratio of eddy / 0ro.
normal conductor needs to approach the superconducting line uniformly. The
requirement of a line contact rather than a single point contact distinguishes this
experiment from scanning probe microscope designs.
Additionally, the normal conductor must be electrically insulated from the
superconductor so as not to provide an alternate current path. Any tunneling or
conduction between the normal conductor and the superconductor would produce
changes in the superconductor voltage. These changes might interfere with the
measurement of Tieddy.
Several other factors are important in the experimental design. A superconductor
with a low critical current is desired. The superconducting film thickness should be
about equal to the minimum expected gap size. The normal conductor thickness
must be large with respect to the gap size, superconductor thickness, and fluxon
lattice spacing.
4.2 Description of the Bending Beam Experiment
The above requirements are incorporated into a design termed the bending beam
experiment Figure 4-1 shows the geometry of the bending beam setup. A
patterned superconducting film is deposited into a two micron deep valley on a
quartz optical flat (designated as the substrate). A normal conducting film is
deposited on a second quartz optical flat (designated as the beam). The beam is
placed across the substrate, supported by spacer bars above the valley. This forms
a gap between the normal conductor and superconductor equal to the space bar
thickness (or valley depth) minus the superconductor film thickness.
.....
.j,~::~.::.~;i~
.....
~:·:;;·;·-:;
·
·
....
...
. ......
........ :lx:::~:~
.....
...
.....
.....
..
.
.~:·~·~~·~::-·::·:
-iii
i
.
........
i
............
........
:::jj·~::::~jj~::~:5::_g
.........
...
g
A.K...
-----ý111.......
..
..
.......
.....
...
.
..
....
....
..
;:·:~::~::'·'
.
·
.....
.
........
.......
....
.....
......
.....
........
...
**'
*...
.......
**'*"'...
;;f····;::::~~:~~~~
..,*-"
...............
-""~""
::::::j~~.
.
.:···::·'
..
.........
...
%
...
............
1.I:ii:··
.
.....
.......
.·-----------....
...
..
...
....
....
....
.........
...........
:::~~:'
..
....
~.~Y_~:~;~::~
~~~-·~~--~-·---
·
....
.
..............
:·.:,
...................
-i ..
...........................
....
.. ··..
..
.
...
.. ........
............ .. ......
~
~.:-
~·.·..·..~..~.
Figure 4-1 Bending beam experiment design.
A downward force can be applied to the top of the beam, causing the beam to
bend. This reduces the size of the gap between the normal conductor on the beam
and the superconductor on the substrate. By modulating the beam force, the gap
size may be modulated. Proper alignment is required so that the beam bends at the
desired location, just above the superconductor line pattern.
The substrate is a 38 mm x 19 mm x 4 mm quartz optical flat and the beam is a
25 mm x 12 mm x 2 mm quartz optical flat. The superconductor is patterned for a
four-probe resistance measurement. Two contacts are used to apply a fixed dc
current to the superconducting line. Two contacts are used to measure the
resulting voltage across the superconducting line. Electrical contacts to the
superconductor are produced by silver-painting copper wires from the
superconductor leads to strain relief pads.
An electrical contact is also made to the aluminum film on the beam.
Measurement of the gap size between the aluminum film and the superconducting
film can be made using capacitive distance sensing. Custom circuitry was built to
measure the capacitance and conductance between the beam film and the substrate
film. This circuit is described in more detail in Appendix m.
The entire apparatus may be sealed in a vacuum tight copper can. The copper
can may be evacuated so that there is no intervening fluid or frozen solid in the gap
to impede the bending of the beam. The substrate is thermally connected to the
copper can, which is submerged in liquid helium. Good thermal contact is
important since the critical temperature of the superconductor is not much higher
than the boiling point of liquid helium.
For improved thermal contact between the film and the helium bath the copper
can may be left open. This permits the flow of liquid helium directly over the film.
The presence of liquid helium in the can is undesirable for dynamic measurements,
when the beam is modulated at higher frequencies. For static measurements,
however, the helium liquid does not influence the measurement of the conductor
proximity effect.
4.2.1 Beam Bending Calculations
The initial gap distance is equal to the spacer thickness minus the
superconducting film thickness. An insufficiently large initial gap distance will limit
the range of beam motion while an insufficiently small initial gap distance will
prevent the normal conductor from being brought close to the superconductor.
The maximum initial gap distance is therefore limited by the maximum possible
beam deflection. The beam deflection is determined by the force versus
displacement curve for the beam and the maximum possible beam deflection force.
The force versus displacement curve for the beam depends on beam shape,
support points, loading, and material. Young's modulus for clear fused quartz at
room temperature is" Y= 7.17x100 N/mr. In this design, the beam has an
effective length lb = 17 mm, width Wb = 12 mm, and thickness hb = 2 mm. Consider
a line force F applied to the center of a rectangular solid beam, supported at two
edges. The displacement of the center of the beam Ahi is given by69
Ah,
Fl3
b
48 Y
(119)
Handbook of Tables for Applied EngineeringScience, 2nd edition, edited by R. E. Bolz and G.
L. Tuve (CRC Press, Cleveland OH, 1973) 187.
69 Mark's StandardHandbookfor Mechanical Engineers, 9 h edition, edited by E. A. Avallone
and T. Baumeister III (McGraw-Hill Book Company, New York, 1987) chapter 5 pages 23, 29.
68
where the moment of inertia I in this case equals wbhb3 / 12.
1
F13
Ah =
b
F
4Ywb h
k,,m
(120)
The behavior of the beam can therefore be described in terms of the effective
spring constant ka,, which has dimensions of force per unit length. For the above
parameters, k.. = 5.6x106 N/m = 5.6 N/gm.
4.2.2 Beam Force - Mechanical Method Calculations
One method for applying a force to the beam is to mechanically push on the
beam with a lever, which is controlled from the probe head using a mechanical
feed-through. Figure 4-3 shows the basic layout. A string under tension is used to
apply an upwards force to a force lever, pivoting the lever and applying a
downwards force on the beam. The string tension is provided by a spring which is
stretched to a known displacement. The displacement is provided by a string
wrapped around a shaft. The shaft is rotated from outside the vacuum space,
changing the length of the string. A spring in the system couples the change in
string length into a force on the beam. For experiments in which the beam is
vibrated by an ac driving force, it is important that the natural oscillation
frequencies of all parts of the mechanical system lie well above the selected
operating frequency of the experiment.
The spring constant for the quartz beam was calculated above. Allow for a
maximum static beam force of 15 N (2.7 pm beam displacement) and consider ac
beam modulations as small as 0.56 N (0.1 mrn beam displacement). The resonance
frequency of the quartz beam is much higher than all of the other resonance
frequencies in the system so it will not be a limiting factor.
First consider the force lever. In general, 69 a cantilever of uniform rectangular
cross-section has a spring constant
Ybh'
k
v -
41'
(121)
where Y is Young's modulus and the beam has length 1,width b, and thickness h.
The resonance frequency of the cantilever is
f
7 1
=-21,e,
(122)
where the mass m is given in terms of the density p by m = pblh. The strain E on
the lever at any given point is a function of the radius of curvature r, at that point.
The relationship is E= h / re. The maximum radius of curvature is r, = Ybh 3 / 12W1,
where W is the maximum expected load (15 N). The maximum strain
6W1
E= Y- bh
(123)
SYbh 3
is constrained to be less than 0.1 %.
Once the lever material, shape, and maximum loading (W) are chosen, there are
six remaining variables: h, 1,b, Emax,fir,and kiver. Only three of these variables
are independent. The constraints on the maximum strain, minimum resonance
frequency, range of desirable spring constants, and size of the system further
restrict the problem. Choose, for example, the three independent variables E.,
ki,,e,, and b. The dependent variables can be found in terms of the independent
variables.
fr'Let - k 61rW
,,,
47r2 fur
2
Y
J
1/6
3 4kr
h
(124)
pbh
For example, choose brass and let E, = 10-3 (@15 N), b = 1.25 cm, and
kkver = 3x10 4 N/m.This choice of kv,,,r makes the lever stiff with respect to the
spring but flexible with respect to the quartz beam. Using the above relations,
flve = 380 Hz, m = 5.3 g, h = 1.5 mm, and I = 3.3 cm. The lever displacement at
15 N is 0.5 mm and the displacement at 0.56 N is 18 gLm.
For dynamic measurements, the natural frequency of oscillation of the string
becomes important The natural frequency is
1
-1
fv,.
T
(125)
where the spring length is L, tension T, and mass per unit length g. For example,
consider "seven pound test trout fishing line." It has a mass per unit length
g = 0.017 g/m. The string tension must be sufficient to maintain the desired
minimum string natural frequency. For a string length L = 1 m and tension T= 1 N,
the natural frequency is f,,,= 120 Hz.
Now consider the spring and mechanical feed-through shaft. Chose an off-theshelf brass spring with a rated spring constant krA,i = 1.3x10' N/m. The actual
spring constant for each spring is different and needs to be individually calibrated.
The spring mass is about 2.4 g. Therefore the spring has a resonance frequency of
115 Hz. For a 15 N load, the spring displacement is 1.2 cm. For a 0.56 N load, the
displacement is 0.45 mm. The feed-through shaft has a diameter of 3.18 mm. A
0.45 mm spring displacement corresponds to a 160 shaft rotation. The 1.2 cm
displacement corresponds to a 3600 + 720 shaft rotation.
4.2.3 Beam Force - Piezoelectric Calculations
Another method for applying a force to the beam is to use a piezoelectric device.
Although this method was not actually used, calculations are presented here for
comparison and for future experiment designs. In the following section,
displacement and force relation for a piezoelectric will be calculated. Typical
piezoelectric properties at room temperature are given in Table 4-1. The piezo is
polarized across its thickness. Leads are attached to opposite faces of the piezo so
that a voltage may be applied parallel or anti-parallel to the polarization. For a
voltage V applied across the thickness T, the corresponding electric field E = V/T.
Motor relations govern the change in size of an unconstrained piezo for a given
applied voltage. The thickness of the piezo changes by AT = Vd33, where d33 is the
on-axis piezoelectric strain coefficient. The length L will change by the relation
ALo I L = Vd31 I T, where d31 is the off-axis piezoelectric strain coefficient.
Generator relations give the forces applied by a fully constrained piezo for a
given applied voltage. The force parallel to the length of the piezo is Fo = WV/g31,
where g31 is the off-axis piezoelectric voltage coefficient and W is the piezo width.
This force is independent of the length and thickness of the piezo.7 0
Table 4-1 Properties Of Selected Piezoelectric Material70
PSI-5A-S2 Lead Zirconate Titanate
Piezoelectric strain coefficients:
Piezoelectric voltage coefficient:
2
d33: 4.50x10 "'0 mV
g31: -1.15x10- V-m/N
d 3 1: -1.80x10-' 0 m/V
Compressive strength: 5.2x10' N/m 2
Thermal expansion: 0.25 p.m/m-K
Initial depolarization field: Edepd = 5x10 5 V/m
For a partially constrained piezo, the true length change AL and force Fwill
depend on the manner of constraint. The true force will be less than the force of a
fully constrained piezo Fo and the true length change will be less than the
unconstrained length change ALo. The operating point which balances the forcedisplacement relation therefore satisfies
F
AL
F0
A,
-+A
=1
(126)
where the force-displacement relation is assumed to be a straight line from (Fo,0)
to (0,ALo).
It is important to know the limits of any particular piezoelectric's capabilities.
Both Fo and ALo are linear in piezo voltage. Since the applied electric field must be
kept less than the initial depolarization field, the maximum applied voltage is
E&poT.For a 0.19 mm thick sheet (7.5 mils), this corresponds to 95 volts. The
70 Piezo Systems, Inc. Product Catalog (Piezo Systems, Inc., Cambridge, MA, 1993).
maximum imparted force and maximum displacement may now be calculated.
Select a piezo with length L = 6.4 cm (2.5 inch) and width W= 3.8 cm (1.5 inch).
The maximum unconstrained displacement is AL
0 .95vo, = 5.8 1tm. The
corresponding strain of 9.0x10 is well within the elastic limit. The maximum fully
constrained imparted force is Fo.9ss5,0 = 314 N. The corresponding stress of
4.3x107 N/m2 is much less than the compressive strength. Define the voltage
independent kpj,,EFo I ALo = 5.4x10' N/m.
The piezo force-displacement relation can be combined with the beam relation to
determine the actual beam deflection resulting from a given applied piezo voltage.
Combine equations (120) and (126), noting that the piezo length change AL is
equal to the beam deflection distance Ahi.
F
F= 1+ (kpi=
/kbhm)
(127)
ALo
1
S1+(kb
/kpic)
The dependence on the voltage applied to the piezo can be made explicit using
the motor and generator relations.
F
AhI
kbeam
kbm +ki=o
=
kPi
beam
-t
V
31
(128)
(_L31')V
piczo
The disadvantage of using a piezoelectric is the coupling of the piezo driving
voltage into the signal leads from the superconductor. Any such coupling could
lead to false signals that are a function of the beam position. These signals could be
measured by the use of a blank beam (one which has no conducting film layer).
They could also be reduced by careful design of the lead wires and the
experimental apparatus. Piezoelectrics are therefore a possible option in future
conductor proximity effect experiments.
4.3 Conductor ProximityEffect Probe Details
The vacuum tight cryogenic probe consists of three main sections. The top, or
probe head, remains at room temperature. It contains electrical and mechanical
feed-throughs, vacuum ports, and a motor assembly. The probe body consists
primarily of a stainless steel tube 70 cm long, which connects the head to the tail.
Next to this vacuum tight tube is an open tube which contains a helium bath
temperature gage and the solenoid electrical leads. The probe tail is a complex
assembly which contains the superconductor substrate and all the associated
hardware for the conductor proximity effect measurement. A solenoid on the
probe exterior is used to apply a magnetic field to the tail.
4.3.1 Substrate Holder and Probe Tail
Substrate holders physically support the substrate and beam, permitting substrate
alignment and providing a thermal sink. Electrical leads connect the films to strain
relief pads on the substrate holders. The walking beam holder was mounted and
aligned on the substrate holder. Three substrate holders were machined by the
author. This enabled a quick turn-around time between liquid helium runs.
The basic substrate holder design is given in Figure 4-2. An oversized 38.9 mm
long by 20.1 mm wide by 3.6 mm deep channel is milled in the holder for the
substrate. Originally, substrates were secured with a set screw in a tight fitting
channel. Concern that the differential thermal contraction would cause the
substrate to warp and crack led to elimination of the set screw and widening of the
channel. Pins are used to guide the beam into the proper position and orientation,
perpendicular to the substrate. The beam and substrate are secured into position by
the walking beam (not shown), which is mounted so as to apply an initial loading
force. Three horizontal screws (not shown) give position and rotation alignment
control. One vertical bolt clamps the substrate holder bottom to the walking beam
holder.
Four strain relief pads are aligned with the four-probe leads on the
superconductor. A fifth strain relief pad is used for the electrical contact to the
beam. Strain relief pads are attached to the substrate holder with super-glue.
Heavy gage wires from the strain relief pads are held in place at the strain relief
terminal and then connected to a 9-pin plug.
The walking beam holder and probe tail design are given in Figure 4-3. The
stainless steel tube connects the probe tail to the probe head. At the tail, it is
soldered to the upper brass collar, which in turn is soldered to the cylindrical
copper can. A solenoid around the copper can is used to apply a vertical magnetic
field to the superconducting film. The bottom of the copper can is soldered to the
lower brass collar, against which the removable aluminum walking beam holder
makes an indium o-ring seal.
The walking beam holder supports the substrate holder (not shown) and the
walking beam assembly. The walking beam height can be adjusted by its two
mounting screws. An upwards vertical force applied with a string to the force lever
causes the walking beam to rotate around its pivot point and apply a downwards
force on the beam (not shown). The string passes through a glass tube to protect it
from the electrical leads (not shown), which also pass up through the stainless steel
tube.
strain relief
terminal
strain relief
Dads
~sel screw 51011
vertical
mounting
bolt hole
Figure 4-2 Substrate holder - top view.
~ upper brass collar
I
copper can
force lever
solenoid
pivot point
walking beam
substrate holder
goes here
\iiMiii;r======;;;;;;w.
'---1 lower brass collar I
Figure 4-3 Probe tail - side view in section.
76
4.3.2 Probe Head and Feed-throughs
The probe head is sketched in Figure 4-4. The stainless steel probe body tube
contains the force application string (inside a glass tube) and electrical wires. The
wires exit the vacuum space through an electrical feed-through in copper and brass
top fittings. The string connects to a shaft in the probe head cap. A mechanical
feed-though is used to rotate the shaft, taking up string slack and applying a force
through the string. The probe head cap is sealed to the probe body with a Viton
o-ring. A motor can be mounted on the probe head cap exterior and connected to
the shaft. Running the motor produces an ac shaft rotation.
Figure 4-4 Probe head sketch - side view in section.
4.3.3 Loading Procedure
The loading procedure is typically as follows:
* The substrate holder is securely mounted using the horizontal and vertical
screws to the walking beam holder.
* The prepared substrate and beam are loaded into the walking beam holder
and the various electrical leads are attached. Silver-paint connects thin
copper wires to the substrate and beam. These copper wires were previously
soldered to the strain relief pads.
* The walking beam assembly is mounted to apply an initial downwards force
on the beam. This secures the substrate and beam in place.
* In mounting the walking beam holder into the probe tail, the string is
attached first. A short length of string from the force lever passes through a
red and white striped drinking straw and terminates with a small brass hook.
The hook is easily attached to the brass spring at the end of the string coming
from the probe head. The straw guides the hook and spring into the glass
tube when the string is retracted. The string is kept taut at all times so that is
wraps neatly around the mechanical feed-through shaft.
* Next, the electrical leads are connected by mating the 9-pin plug with the
corresponding receptacle in the probe body. As the walking beam holder is
lifted into the probe body, the slack in the electrical leads is pulled into the
stainless steel tube by a small additional string at the probe head. This keeps
the electrical leads away from the string.
* The probe tail is fully secured by tightening socket screws at the lower brass
collar. To seal the probe can, an indium o-ring is inserted between the collar
and the walking beam holder. To leave the probe can open, spacers add a
gap between the collar and the walking beam holder.
* At the probe head, any remaining slack in the electrical leads and/or string is
removed. The probe head assembly is secured to the top of the probe head
with /4-20 socket screws, and sealed with a Viton o-ring.
* If the probe can is to be left open in the liquid helium, it must be temporarily
sealed for liquid nitrogen precooling. In this case, the tail is covered by a
rubber balloon.
* At this point the probe can be evacuated and cooled. The experiment may
begin.
4.4 Preparation of Substrates
The primary substrates were made from rectangular 38 mm x 19 mm x 4 mm
quartz optical flats. A masking and etching process was used to introduce the
spacers (shaded areas) shown in Figure 4-5. The spacers support the bending beam
a few microns away from the substrate. This section describes the
photolithography and etching process.
First, a mask was machined from 0.5 mm aluminum sheet. Photoresist was
deposited on the substrate in the desired pattern and then developed. The substrate
was wet etched with a dilute hydrofluoric acid etch. When 38 mm x 19 mm x
2 mm silicon optical flats were used, the same photolithography procedure was
followed. For the silicon substrate, a dry plasma etch was necessary to produce the
spacers.
Table 4-2 Properties Of Quartz7
Softening Point: 16650C
Young's Modulus = 10.4 Mpsi
Anneal Point: 11400C
Strain Point: 10700C
4.4.1 Photolithography
All photolithography processing was performed by the author in the MIT
MicroLab Facility. Futurrex brand NR-8-3000 positive photoresist was used. The
mask pattern is shown in Figure 4-6. The four circles are for alignment with the
substrate corners. The photolithography procedure is
* Clean substrates: trichloroethane, followed immediately by acetone, followed
immediately by methanol. Dry with clean nitrogen gas. Prebake at 2000 C for
at least 30 minutes to evaporate organics.
* Apply photoresist using spin station: Spin substrate and blow clean with
nitrogen gas. Apply photoresist to substrate. Spin off photoresist, typically for
30 sec at 3000 RPM.
* Softbake substrate: Put on hot plate at 900-1300 C (varied for different runs)
for one minute. This prepares photoresist for exposure.
* Expose photoresist: Use desired pattern mask. Exposure times varied due to
intensity of bulb in aligner and humidity. Varied from 10 to 50 seconds.
* Develop photoresist: Use Futurrex brand RD-2 developer. Developing times
varied with exposure time and lab humidity. Typically 4 to 6 minutes.
* Hardbake: Heat to 1300 C for 30 minutes. This hardens photoresist against the
etching procedure.
r19.0 mm.1
38.0 mm.
-T
I
15.9 mm.
i
i
lO mm.
-
__ 2 gm
(not to scale)
4.0
mm.
Figure 4-5 Quartz substrate etching pattern.
1CRC Handbook of Chemistry and Physics, 63" edition, edited by R. C.Weast, (CRC Press,
Inc., Boca Raton, Florida, 1982), page F-65.
0.732". -------
---
0.000"-.......
.----
.. ...
-- ------
0.000" 0.466" 1.030" 1.496"
1/16' end mill cuts shown in black.
Dashed lines are guides for the eye only.
Figure 4-6 Aluminum mask for photoresist process.
4.4.2 Etching Procedures
The quartz substrates were etched by soaking them in a buffered oxide etch
(BOE) solution under mild agitation. BOE is seven parts ammonium fluoride to
one part hydrofluoric acid. All parts of the quartz substrate were etched except for
the spacer bars. The etching rate was calibrated by performing Dektak
profilometry on the substrates before and after etching. The etching rate was
approximately one micron per ten minutes. Typical etching times were therefore
twenty to twenty-five minutes. Once the substrates were etched to the desired
depth, they were loaded into a mask holder for vapor deposition of the
superconducting films.
A wet etch of silicon substrates, using potassium hydroxide for example, is not
effective. Photoresist dissolves in the etching solution. To perform a wet etch, a
stronger mask layer would be required. Instead, a dry etch was performed using
the MicroLab Day Etcher, an rf plasma etching system. This system is set up to
etch with 02 gas and with SF6 gas. The 02 etches photoresist while the SF6 etches
silicon. The plasma etch only removes material from one side of the substrate. The
etching procedure is straightforward:
* Set proper tuning parameters to produce a good plasma.
* Load substrate into etcher and evacuate system.
* Fill the chamber with 02 etchant gas to desired pressure.
* Turn on plasma briefly to remove any stray photoresist monolayers.
* Evacuate 02 gas. Reset system for SF6.
* Fill chamber with SF6 etchant gas to desired pressure.
* Turn on plasma for desired etching time.
The etching rate depends strongly on plasma quality. The etch rate was about one
micron per minute for the power level and tuning parameters used.
4.4.3 Silicon Substrate Results - Surface Quality and Purity
Surface stress variance on the silicon substrate causes a spatial variation in
etching rates. The plasma etch reveals small surface scratches. These scratches
indicate surface stress variations that occur during the substrate polishing process.
Visually, the etched surface is dull while the masked region remains shiny.
A second effect of etching was an overall warping of the substrate. Dektak
profiles were made of the substrate's etched side before, midway through, and
after processing. The overall convex curvature along the substrate length increased
from about one half micron to one micron to one and a half microns, respectively.
This behavior is consistent with the removal of surface material under tension.
The impurity levels in silicon substrates were sufficiently low so that room
temperature resistivity of the silicon matched expected intrinsic behavior. At low
temperature, however, the impurity level was high enough to cause the silicon to
be extrinsic. The residual conductivity at low temperature was sufficient to disrupt
capacitance measurements between the superconductor and the normal conductor.
Since these substrates could not be treated as a perfect insulator, they were not
acceptable for use in the conductor proximity effect measurements.
5. Superconductor Fabrication and Characterization
This chapter presents the superconducting specimens used in the electrical
measurements. Superconducting In-8.7% Bi (Ino.95Bio.os) films and pure lead films
were successfully fabricated by thermal evaporation. Bulk and thin film yttriumbarium-copper-oxide samples from various sources were also used. Additionally,
an unsuccessful attempt to produce thin films via pressure infiltration is described.
The superconducting samples are characterized by critical temperature, critical
field, and voltage-current measurements.
5.1 Material Properties - In-Bi Alloys
An indium-bismuth ca phase alloy was chosen for the superconductor. The
experimentally determined critical temperatures of various indium-bismuth alloys
from multiple researchers are shown in Figure 5-1. The data from various
researchers is not completely in agreement. The current work uses Ino.95Bio.o5 with
a critical temperature of 4.7 Kelvin. For thermal stability, operating in boiling
liquid helium was desired. Therefore, the a phase Ino.s9 Bio.o5 alloy was selected.
Critical Temperatures of Various In-Bi Alloys
O.U 1
i
T-BA
5.5
5.0
4.5 -
B
Heiu....
............
.
E 4.0I-
3.5
3.0
0% 5%
0% 5%
10%
10%
15% 20% 25%
25%
15% 20%Percent
Atomic
Atomic Percent
30% 35% 40% 45% 50%
Bismuth
30% 35% 40% 45% 50%
Bismuth
Figure 5-1 Critical temperatures of various indium-bismuth alloys.72
In equilibrium at room temperature, the selected alloy is not single phased.
According to Hansen, 73 the room temperature solubility limit of bismuth in indium
j.V. Hutcherson, R. L. Guay, and J. S. Herold, "Superconducting InsBi 3", Journalof the LessCommon Metals 11, 296-298 (1966). T. McConville and B. Serin, "Specific Heat of Type II
Superconductors in a Magnetic Field", Physical Review Letters 13, 365-367 (1964). T. Kinsel, E.
A. Lynton, and B. Serin, "Magnetic and Thermal Properties of Second-Kind Superconductors. I.
Magnetization Curves", Reviews of Modern Physics 36, 105-109 (1964).
73M. Hansen, Constitution of BinaryAlloys, 2nd edition, (McGraw-Hill Book Company, New
York, 1958) 313-314.
72
is 7.1 weight % (4.0 atomic %), with the precipitate In2Bi forming at higher
bismuth concentrations. Therefore, the selected In-8.7% Bi alloy at room
temperature equilibrium will form an In-7.1% Bi matrix with an imbedded In2Bi
precipitate. Using the lever rule, four percent of the total mass will be in
precipitate, while the remainder will make up the ct phase matrix.
A more recent phase diagram is shown in Figure 5-2. According to this phase
diagram, the solubility of bismuth in indium at 25 TC is almost exactly
8.7 weight %. The solubility limit drops to 5.5 weight % bismuth at 0 TC. At the
latter temperature, almost eight percent of the total mass of In-8.7% Bi will be in
precipitate. Experimental data presented below seems to be more consistent with
Hansen than with this more recent phase diagram.
Atomic Percent Indium
0
10
20
30
40
50
60
70
80
90
100
Q
0
1
I-
Weight Percent Indium
In
Figure 5-2 Indium-bismuth phase diagram.7 4
The indium-bismuth alloy can be described in terms of the number relationship
of the components IntxBix or the mass relationship In-m Bi, where m is the mass
fraction of bismuth. Since the atomic masses of indium and bismuth are
114.82 grams/mole and 208.980 grams/mole respectively. The relationship
between x and m is given here.
114.82m
208.980x
(129)
x 2=
and m=
208.98 - 94.16m
7
114.82 + 94.16x
ASM Handbook: Volume 3 Alloy Phase Diagrams,edited by Hugh Baker, (ASM International,
Materials Park, Ohio, 1992) page 2.100.
Properties of the elements indium and bismuth will be relevant to the
experimental design. Properties of indium are given in Table 5-1 and those of
bismuth in Table 5-2. Using equation (58), the critical temperature from
Figure 5-1, and the coherence length from Table 5-1, the intrinsic viscous
coefficient may be calculated. For Ino. 95Bio.os, the result is rlo = 4. 1x10 -9 kg/ms.
Normal resistivity at low temperature is measured below. The value is found to
be p, = 11 Lgft-cm. With these values of p, and rio, equation (53) is used to
calculate the upper critical magnetic field. The predicted tiH/c 2 = 220 mT is about
five times the experimentally measured value. Therefore, the viscous coefficient
appears to be overestimated here.
Measured resistivity may also be used to determine the mean free path. Using
equation (25) and electronic density of pure indium, the mean free path is found to
be 4.8 nm. The alloy is in the dirty limit. Using equation (32) and the Sommerfield
constant for indium, it is found that ic= 2.8.
The critical temperature of this alloy is approximately 4.7 K. In liquid helium, the
reduced temperature ratio T / Tc = 0.9. At this temperature, the penetration depth
and coherence length may be found from equations (27) and (29). It is found that
4(4.2 K) = 0.12 gm and X(4.2 K) = 0.39 gm.
Table 5-1 Properties Of Indium7 5
Atomic mass: 114.82 g/mol
Atomic number: 49
Thermal conductivity:
Crystal structure: tetragonal at 250 C
83.7 W/m-K
Melting Point: 155 OC
Volume change on freezing: 2.5%
contraction
Specific heat: 233 J/kg-K at 250 C
264 J/kg.K at 156.630 C (solid)
257 J/kg.K at 156.630 C (liquid)
Latent heat of fusion: 28.47 kJ/kg
Resistivity: 8.4 g2-cm at 200 C
Pippard coherence length: 440 nm
Density: 7.30 g/cm'
Superconducting at 3.38 K
Penetration depth: XL = 21 nm
Electronic density: rjao = 2.41
where ao is the Bohr radius
Sommerfield constant:
y = 1.7 mJ/mole-deg 2
Sy= 1.07x10 2 J/m 3 K '
Thermal expansion: 24.8 tm/m-K
Fermi velocity: vF = 1.74x10 6 m/s
75 Phase Diagramsof Indium Alloys and Their EngineeringApplications, edited by C. E. T.
White and H. Okamoto (ASM International, Metals Park, Ohio, 1992). Sommerfield data from
G. Gladstone, M. A. Jensen, and J. R. Schrieffer, "Superconductivity in the Transition Metals" in
Superconductivity, edited by R. D. Parks (Marcel Dekker, Inc., New York, 1969), 734.
Superconducting data from T. Van Duzer and C. W. Turner, Principlesof Superconductive
Devices and Circuits (Elsevier, New York, 1981), inside back cover. Electronic density and
Fermi velocity data from N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart
and Winston, New York, 1976) 5, 38.
Table 5-2 Properties Of Bismuth7
Atomic mass: 208.98 gmol
Atomic number: 83
Latent heat of fusion:
Melting Point: 271 oC
10.8 kJ/mol
Volume change on freezing: 3.35% contraction
Electronic density: r,/ao = 2.25
Fermi velocity:
where ao is the Bohr radius
Thermal expansion: 13.4 tm/m-K
vF = 1.87x106 m/s
Density: 9.8 g/cm 3
5.2 SuperconductorFabrication via Evaporation
Superconducting films were successfully fabricated using vapor deposition in the
Specialty Materials Laboratory7 thermal evaporator. The source material, or
charge, is placed in a tungsten boat within a high vacuum chamber. A large electric
current is passed through the boat, heating and vaporizing the source material.
This vapor strikes a target substrate oriented in the line-of-sight of the source. A
film is formed on the target by the adherence of vapor molecules. The vacuum in
the deposition system is sufficient such that most vapor molecules do not undergo
collision with residual gases before striking the target.
Film patterning was performed by inserting a mechanical mask between the
source and the target. The finite area of the source results in a thickness variation
of the resulting pattern close to the mask edges. The size of the affected region is
governed by the source size and mask thickness. Large featured masks were
machined into the desired shape from a piece of 0.020" aluminum shim stock.
Small features, such as thin test lines, were added by attaching pieces of 0.002"
shim stock to the larger aluminum mask with superglue.
Source temperature is controlled by varying the current passing through the
boat. Deposition time is regulated by a shutter between the source and the target.
Substrate cleanliness is very important in the production of high quality
superconducting films.
5.2.1 Evaporation of Alloys - Raoult's Law
Evaporating an alloy is more complex than evaporating a single element. The
objective is to produce a film with the desired composition and thickness.
However, the components of an alloy tend to have different vapor pressures.
Comprehensive Inorganic Chemistry, vol. 2, edited by J. C. Bailar Jr., H. J. Emeleus, Sir R.
Nyholm, and A. F. Trotman-Dickenson (Pergamon Press, Oxford, 1973) 555. Electronic density
and Fermi velocity data from N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt,
76
Rinehart and Winston, New York, 1976) 5, 38.
77 The former Specialty Materials Laboratory of the MIT Physics Department was the research
group of the late Professor M. L. A. MacVicar.
Raoult's law gives the partial vapor pressure at equilibrium for the various
components.7 8 For an ideal system,
(130)
PA =xAP ; PB = x P
where Pi is the partial vapor pressure of component i in solution and Pio is the
vapor pressure of the pure element i. xi is the mole fraction of component i in
solution. For a non-ideal system, the mole fractions are replaced by the
temperature and concentration dependent activities ai.
PA= aAPA
; PB =aPf
(131)
The activities can be determined experimentally but are frequently unknown.
The deposited film will have a composition equal to that of the vapor, assuming
an equal sticking probability. If the vapor does not have the same relative
component concentrations as the source, then the composition of the -source will
change with time, resulting in a film composition which varies with depth. The
composition of the vapor equals the composition of the source if PA / PB = XA / XB.
This only occurs for very specific combinations of elements in specific proportions.
Po = PO
aPO/ x = aP / xB
ideal solution
non-ideal solution
(132)
Raoult's law assumes that the source is in equilibrium with the vapor. If this is
not the case, than the vapor composition may be closer to the source composition
than predicted by Raoult's law.
The properties of indium and bismuth are of interest here. The melting point of
pure indium is 157 OC, which is lower than the 271 oC melting point of pure
bismuth. However, the vapor pressure of bismuth is higher than that of indium, as
shown in Table 5-3. Therefore, when an indium-bismuth source is heated, the
initial vapor is bismuth rich with respect to the source. After the source becomes
depleted of bismuth, the vapor is indium rich with respect to the initial source
composition. The composition of deposited films will therefore depend on the
timing of the shutter with respect to the vapor's composition variation.
Table 5-3 Vapor Pressures For Pure Indium And Pure Bismuth79
Temperature (OC) at Vapor Pressure
Element
10 4 Torr
10- Torr
108 Torr
742
597
487
Indium
520
410
330
Bismuth
See, for example, Vapor Deposition, edited by C. F. Powell, J. H. Oxley, and J. M. Blocker Jr.,
(John Wiley & Sons, Inc., New York, 1966) 242-244.
79 Thin Film EvaporationSource Reference (R. D. Mathis Company, Long Beach, California,
1987).
78
Deposition of a desired composition alloy is difficult when the vapor
composition varies with time. The solution is to start with a source charge of the
desired composition and then deposit all of the charge onto the target. The average
composition of the resulting deposited film will match the initial charge
composition. Additionally, the film thickness is related to the total charge mass,
assuming the geometry is kept constant. By proper charge preparation, therefore,
both the film thickness and average composition may be controlled.
5.2.2 Evaporation System and Evaporation Procedure
The evaporation system consists of a bell jar main chamber which is evacuated
by means of a roughing pump and a water-cooled vapor diffusion pump. Pressure
in the system is monitored with thermocouple and ionization gauges. A liquid
nitrogen cold trap between the diffusion pump and the bell jar prevents diffusion
pump oil from entering the chamber. An additional liquid nitrogen vapor trap in the
main chamber is used to adsorb residual gases before the vapor deposition begins.
An electrical pass-through allows for application of large currents through the
source boat. Mechanical pass-throughs are used for shutter control and
occasionally for target rotation.
To keep the system clean, the bell jar is usually evacuated for at least one day
before a deposition is begun. The bell jar is then backfilled with nitrogen gas and
vented so that the source and target can be loaded into the system. Once the
system is closed and evacuated again with the roughing pump, the liquid nitrogen
trap is filled and the gate valve and backing value are opened. When the pressure
drops below 1 microTorr, the large vapor trap is filled with liquid nitrogen,
dropping the pressure to about 0.6 microTorr. The evaporation is then performed.
A shutter is used to shield the target from the source during outgassing and to
control deposition time. Film thickness depends on deposition time and current
applied through the source boat. Film thickness and uniformity are monitored
during the deposition by visually observing the simultaneous formation of a film on
a glass slide next to the target.
The mass of the initial charge is chosen so that deposition of the entire charge
results in a film of the desired thickness and average composition. The In-8.7% Bi
films deposited are good superconductors for the conductor proximity effect
measurements. The lead films deposited form type-I superconducting films with
much higher critical currents and fields than the indium-bismuth alloy.
5.2.3 Indium-Bismuth Films Produced
Superconducting films of various composition, quality, thickness, and mask
pattern were evaporated onto a variety of substrates. Standard composition and
deposition techniques were ultimately chosen to produce consistent films with
known properties. Several different four-probe mask designs were employed to
deposit films with different line lengths and widths. Substrates were occasionally
reused. However, thorough substrate cleaning ensured that film quality remained
high.
The process of choosing deposition parameters and determining optimum
substrate cleaning procedures, although time consuming, is not particularly
interesting. Detailed deposition parameters are useful only on the SML evaporator.
Substrate cleaning techniques involved ultrasonic cleaning with standard acid and
organic cleaners. Therefore, this section will summarize only those films for which
actual data is presented in this document.
films. A Dektak profilometer was used to
Table 5-4 describes the Ino.95Bio.os
measure film width and thickness parameters. The film thickness was measured
either directly on the film (usually after taking superconducting measurements) or
on a glass slide film which was deposited at the same time. Profiles on glass slides
deposited at the same time as the film are only a film thickness estimate. The film
width was measured directly on the film or was assumed to be the same as other
films deposited with the same mask. Again, the assumed width should only be
considered an estimate. The line width actually varied over the line length, since
the masks were made by hand. The average line width is given in the table. The
line length was measured on the mask directly.
Table 5-4 Parameters For Evaporated Indium-Bismuth Films
Geometry
Substrate Film DepositionDate
11
10
17
18
19
1
2
1
1
1
31-May-96
3-July-96
23-July-96
23-July-96
23-July-96
Length
Width
Thickness
12.7 mm
9.0 mm
12.7 mm
9.0 mm
12.7 mm
150 pm
51 gm (est.)
140 glm
73 lpm
140 ptm
0.6 p.m
0.8 pLm (est.)
0.77 pLm
0.9 gm
0.7 ptm
5.3 SuperconductorFabrication via Pressure Infiltration
Indium-bismuth micron thick film fabrication was attempted via pressure
infiltration into a capillary between two optical flats. Morrison and Rose sOcast
127 p.m indium-bismuth films between two glass plates via pressurization with a
syringe. Here, since much thinner films with better surface flatness were required,
fabrication took place inside a high pressure vessel.
The approach was to sandwich optical flats of sodium chloride and quartz
together. Micron thick aluminum strips on the quartz acted as spacers. This
sandwich was then submerged in an In-Bi melt of the desired composition which
so D.D.Morrison and R.M.Rose, "Quantitative Control of Surface Pinning ina Low-ic Type II
Alloy", JournalofApplied Physics 42, (1971) 2322-2327.
was put inside a pressure vessel. The vessel was evacuated to remove air from the
gap and then pressurized with nitrogen gas up to 750 pounds per square inch
(5.2 MPa ). The melt was allowed to cool and solidify under pressure. The sodium
chloride counter-face could then be dissolved in warm water leaving an optically
flat film of In-Bi on the quartz substrate.
Below, the design of the pressure infiltration system is discussed, along with the
first attempts to use this system. An analysis is made of why this method proved to
be unsuccessful. Design improvements to make this method successful are
suggested.
5.3.1 Pressure Vessel Design
The pressure vessel design in shown in Figure 5-3. The body of the pressure
vessel is a 2" IPS pipe nipple five inches long connected to a 2" IPS by 1"IPS
reducer. Various fittings, as shown, connect the base to a 1/2" IPS tee, providing
ports for vacuum and high pressure. The pressure port is connected through a
shutoff value to a high pressure regulator on a nitrogen gas cylinder. Thick-walled
3/8" stainless steel pipe and Swagelock® fittings were used. The vacuum port splits
to an air vent valve and a roughing pump port. All valves on the system are rated
to 3000 psi. All pipes and pipe fittings are rated to 3000 psi and are either 316
stainless steel schedule 80 or forged carbon steel. Pipe threads are sealed with
teflon tape. The 2" IPS by 1"IPS reducer is welded to support braces which are
bolted to a lab bench. This holds the pressure vessel body firmly in place, allowing
the various parts of the pressure vessel to be tightened with a large pipe wrench.
Inside the pressure vessel, ceramic fiber insulation is used to keep the pressure
vessel cool while keeping the melt crucible hot. The hot melt crucible is placed
inside the pressure vessel and allowed to cool under pressure. In this design, there
is no heating or cooling control of the melt crucible while it is inside the pressure
vessel. As seen below, this leads to a problem in the fabrication process.
2" IPS cap
2" IPS nipple (5" length)
2"-1" IPS reducer
1" IPS close nipple
1"-0.5" IPS reducer
0.5" IPS close nipple
To vent and vacuum
To pressure regulator
0.5" IPS tee
Figure 5-3 Pressure vessel for pressure infiltration system.
5.3.2 Substrates and In-Bi Melt
A detailed presentation of the 20 March 1995 pressure infiltration run is given
here. This run provides all the information needed for discussion and analysis
below. Quartz and sodium chloride 25 mm x 12 mm x 2 mm optical flat substrates
were purchased from Spectral Systems, Incorporated. Aluminum strips 25 mm
long by 2 mm wide were electron-beam deposited on the long edges of the quartz
substrate. During processing, a small piece of the quartz substrate was broken off.
This did not prevent the use of the substrate in the pressure infiltration run.
Figure 5-4 shows profilometry data taken with a Dektak 3. The data confirms that
the aluminum strips are just over one micron thick. The use of aluminum for strips
and for all machined components submerged in the indium-bismuth melt is
motivated by the phase diagram between indium and aluminum. The solubility of
aluminum in liquid indium is less than 0.05% by weight at 3000 C.81
Profile of Aluminum Strip on Quartz Substrate
.
.
.A _
aluminum strip
0.02
1-
quartz surface
0.01
0
"
rl
4
_
0.8-0.01
S0.6
-0.02
r-
S0.4 -
0 .
0.5
0.0
..,
.
-
1.0
02 -
quartz substrate
.• • ,,
00
d
0.5
j_.
,.
1
J_
I_* *..
.
1.5
2.0
. ,, . .
2.5
3.0
2
3
Position (mm)
Figure 5-4 Profile showing thickness of aluminum spacer strip
deposited on quartz substrate.
Substrates are held together by an all aluminum clamp which is inserted into a
glass test tube. The clamp is fabricated by turning a rod on a lathe until its outer
diameter equals the test tube inner diameter. The rod is cut in half along its axis.
The halves are milled such that when they are screwed together there is a 5 mm
gap between them and they just fit into the test tube. Two 6-32 countersunk holes
are tapped on one half, while corresponding pass holes are drilled in the other half.
The facing clamp surfaces can either be milled flat or guide rails can be milled into
them to ease alignment of optical flats. This is useful since optical flat substrates
are very slippery and difficult to keep aligned.
The chosen composition of the melt was Ino.9sBio.o 75 or In-13% Bi. While
mixing the melt, some of material spilled before a uniform composition was
achieved. Therefore, the actual melt composition was not precisely known. This
has no effect on the results, however, since finished films were never ultimately
produced from this particular melt.
5.3.3 Results and Discussion
The pressure infiltration procedure was straightforward. Substrates were loaded
into the clamp, which was then secured into the melt crucible, a glass test tube.
"' M. Hansen, Constitutionof Binary Alloys, 2nd edition, (McGraw-Hill Book Company, New
York, 1958) 100-102.
The clamp required anchoring, since otherwise it would float in the higher density
indium-bismuth melt. Some graphite was applied to the outside of the clamp to
prevent the melt from sticking to the clamp, but the amount applied seemed to be
insufficient to have any effect. The crucible and a solid piece of In-13% Bi were
heated in a box furnace set to 200 oC. When the In-13% Bi melted, it was poured
into the melt crucible such that the clamp and substrates were fully submerged.
The crucible was then returned to the box furnace which was set to 240 oC. At this
point the pressure vessel insulation was preheated by putting a 300 OC brass rod
into the pressure vessel. After one hour total heating time (10 minutes at the higher
set point), the crucible was moved to the pressure vessel. The pressure vessel was
evacuated to about 600 millitorr and then slowly pressurized to 750 psi over the
course of one minute. The crucible was allowed to cool for one hour and was then
removed from the pressure vessel and examined.
To access the substrates, the test tube was cut away. The layer of indiumbismuth that solidified between the test tube inner wall and the outside of the
clamp was peeled off. One of the two clamp screws was removed, allowing the
clamp to slide apart, revealing the substrates. There was no need to dissolve the
sodium chloride substrate, since it had broken into three pieces and the quartz
substrate had broken into four. Most likely they shattered when the clamp was
being taken apart or during cooling because of the differences in the SiO 2, NaC1,
In-Bi, and Al coefficients of thermal expansion. The substrate pieces were
sufficient for an analysis of the infiltration process, however.
The In-Bi was almost completely absent from the NaCl substrate. On the quartz
substrate it formed dendrites near the substrate edges but was absent from the
center. In addition, there appeared to be multiple phases of indium on top of the
aluminum strips, some of which had been dissolved away in places. A more
detailed elemental analysis of one of the quartz pieces was made using EDXA on
an SEM.
In the aluminum strip region, a variety of phases were present. In one area
aluminum, silicon, and chlorine were found, while in another aluminum, indium,
and chlorine were found. This suggests that the chlorine from the NaCl substrate
was interacting with the aluminum strips, etching them away and forming an
aluminum chloride compound. The presence of silicon indicates thin or bare
regions of aluminum, exposing the SiO, substrate. Detection of bismuth in the
presence of chlorine was difficult since the primary bismuth peak was small and
overlapped the chlorine peak.
In the indium-bismuth film region there were also multiple phases present. There
was an area of indium-bismuth film in which EDXA identified only those two
elements present. There were voids in the shape of dendrites extending into the
indium-bismuth film. This seems to indicate that infiltration initially occurred, but
then the In-Bi left the capillary during solidification. An additional phase formed in
cross-like patterns scattered throughout the In-Bi film regions. EDXA revealed
that this phase consists of indium and chlorine, with possibly some bismuth. This
cruciform phase is apparently a chloride of indium.
The above examination revealed three problems in the deposition process. First,
the interaction of aluminum and chlorine damages the substrate spacers. Second,
dendrite patterns indicate a problem with the cooling process. Third, the formation
of an undesired indium-chlorine alloy occurs.
These three issues are addressed as follows. First, the aluminum-chlorine
interaction could be avoided by finding an inert material for a spacer. The only
spacer requirements are that it can be formed into one micron films and that it does
not react with indium, bismuth, or chlorine. Alternatively, the spacers could be part
of the quartz substrate itself. They would be formed by masking and etching the
substrate. This method was actually used for spacers in the final conductor
proximity effect experiments. Details were discussed in section 4.4.
Second, the cooling process needs to be modified. When the melt cools, it
solidifies and contracts. Material in the capillary is still sufficiently fluid to flow out
of the capillary. This occurs because this material is no longer under pressure. To
produce a quality film, the region in the capillary must solidify before the melt or
the melt and capillary regions must solidify simultaneously. In any case, cooling
has to be controlled such that the capillary remains under positive pressure until
after it solidifies.
Third, the indium-chlorine reaction remains problematic. Although the chlorides
of indium are soluble in water, their removal would leave holes in the indiumbismuth film. The objective is to get a film with a uniformly flat surface. A process
is needed to avoid or minimize the formation of this phase. Finding an alternate
substrate, rather the NaC1, is difficult, since the substrate would still have to be
removable in a way that does not disturb the indium-bismuth film surface. The
selection of NaCl was motivated by its solubility in water.
5.4 Characterization of Films
Films were characterized in terms of their physical and geometrical properties as
well as their electrical properties. Knowledge of the surface smoothness,
composition, thickness, and quality of the films is important in giving a full
description of the experimental data. The quality of the masked pattern and the
basic surface quality of the film can be visually observed with an optical
microscope using up to 1000x magnification. The film thickness, line width, and
surface roughness are measured using a Dektak profilometer. High resolution
microscopy and elemental analysis are performed with an SEM equipped with
EDXA.
A number of superconducting properties are of interest. The most basic property
in the critical temperature Tc, below which the material is superconducting in zero
magnetic field. This was measured via four-probe resistance as a function of
temperature. In the mixed state, the critical current density Jc is defined as the
current density at which a specified electric field (voltage per unit length) appears.
The critical current was measured as a function of magnetic field. The voltagecurrent relation above the critical current provides information on some of the
intrinsic properties of the superconductor. This is also the regime in which the
conductor proximity effect occurs.
5.4.1 Magnetic Critical Temperature Measurement Probe
One of the most important distinguishing properties of a superconductor is
perfect diamagnetism. The magnetic critical temperature probe makes use of the
change in a material's magnetic susceptibility when it undergoes a superconducting
transition. This susceptibility change is used to identify and characterize the
superconducting critical temperature of the material.
The superconducting transition temperatures for YBCO samples were measured
magnetically using a pancake coil probe. The probe consists of two pancake coils,
a primary and a secondary. An ac voltage applied to the primary induces a
corresponding ac voltage in the secondary. The coupling into the secondary
depends on the magnetic susceptibility of the sample adjacent to the pancake coil.
When the sample undergoes a superconducting transition, its susceptibility
changes. This results in a sudden measurable change in ac voltage across the
secondary coil.
The voltage in the secondary is measured as the sample is cooled in liquid
nitrogen and then warmed to room temperature. Temperatures less than 77 Kelvin
are achieved by using a roughing pump to lower the nitrogen gas pressure.
Lowering vapor pressure can reduce the boiling point of liquid nitrogen by over
10 degrees. For these measurements, the sample temperature is monitored by a
platinum resistance temperature device 82 (RTD). Thermal lag between the RTD
and the sample appears as hysteresis in the secondary voltage versus temperature
curves.
This pancake coil system has not been calibrated to give the exact susceptibility
of samples with various geometries. It is therefore not proven that perfect
diamagnetism is being realized in these experiments. The purpose of these
measurements is not to prove superconductivity, but rather to measure the location
of the superconducting transition. The measured susceptibility change is assumed
to be due to a superconducting transition, rather than some other effect.
82 The RTD is type 1 PT 100, K 20 15. See Omega Complete Temperature Measurement
Handbook and Encyclopedia, Vol. 27 (New York: Omega Engineering, Inc. 1989) page E-13.
5.4.2 Electrical Critical Temperature Measurement Probes
The second basic experimental test of superconductivity is four-probe resistance
measurement. This experiment seeks to measure the disappearance of electrical
resistivity in the superconducting state. Electrical and magnetic transition
experiments measure two different aspects of the superconductor. The magnetic
experiments measure the bulk effect of diamagnetism. They characterize the point
at which the bulk of the material becomes superconducting. A superconductor of
sufficient size to affect the probe susceptibility is required. The electrical
experiments measure resistance between two fixed points. They therefore
characterize the single path of least resistance between those points, a
determination of local superconductivity. A patterned superconducting film is
required. For a non-homogeneous superconductor, magnetic and electrical
measurements can produce very different results for the same material.
The probe used for liquid nitrogen four-probe measurements is similar to the
magnetic pancake coil probe. The primary and secondary coils are eliminated and
electrical contacts are made directly to the superconductor. A current source is
used to apply a constant dc current at the current leads. A nanovoltmeter is used to
measure the corresponding voltage at the voltage leads. As above, an RTD is used
to monitor the temperature as the probe is cooled and warmed.
In principle, the probe used for liquid helium measurements is similar. It uses a
Lake-Shore thermometer, which is thermally well connected to the substrate and
film. Much better thermal contact is required to get precise results for critical
transition temperatures. Four electrical contacts to the film provide for current
sourcing and voltage sensing.
It is important to point out that the Lake-Shore thermometer used in all the
liquid helium tests was very precise, but not extremely accurate. The thermometer
was never precisely calibrated near 4.2 Kelvin. As a result, when the thermometer
is submerged in liquid helium, the temperature typically reads about 4.28 or
4.30 Kelvin, rather than 4.23 Kelvin. The thermometer calibration was left
unchanged so that data taken over several months on various films could be
compared directly. When comparisons are made between the data here and the
literature data, the 0.1 degree variance must be considered.
5.4.3 Measurements Versus Magnetic Field
A large copper solenoid was wound to facilitate application of magnetic fields to
the superconducting films. It was designed to be compatible with both the liquid
helium critical temperature measurement probe and the conductor proximity effect
probe. The solenoid fits outside the helium dewar and inside the nitrogen dewar.
Cooling the solenoid in liquid nitrogen significantly reduces its electrical resistance
and allows for the production of large magnetic fields. The applied magnetic field
is always perpendicular to the film surface.
The solenoid may be used in conjunction with the temperature measurement
probe to generate critical temperature vs magnetic field curves. It may also be used
at a fixed temperature to produce voltage vs magnetic field curves at constant
current or voltage vs current curves at constant magnetic field. The fixed
temperature curves can be measured with either the temperature measurement
probe or the conductor proximity effect probe.
5.5 Indium-Bismuth Films
The primary thrust of this thesis involves conductor proximity effect
measurements on In-8.7% Bi films. The basic properties of the In-8.7% Bi films
were therefore characterized in detail over a number of films. Some of the basic
properties are listed here:
* Critical temperature measurements showed a transition from finite resistivity to
superconducting state at 4.7 + 0.2 Kelvin.
* Flux flow voltage vs applied current and applied magnetic field data fell on or
near a common curve when plotted as flux flow voltage vs Lorentz force.
* The observed voltage at a given field and current density varied slightly with
thermal cycling. However, curve shapes and trends remained consistent.
* Behavior remained consistent for a number of similar composition films.
The basic film behavior was interesting and unusual in several ways:
* Critical temperature measurements show a resistivity jump at 5.7 Kelvin,
indicating phase separation. The resistivity vs temperature transition changed
when films were annealed immediately prior to testing.
* Voltage vs Lorentz force curves were highly non-linear above the critical
current Voltage-current curves were not consistent with simple flux flow
models.
Below the measurements of Tc, HC2 , and Ic are presented. Detailed flux flow data
is presented in the next chapter.
5.5.1 Critical Temperature Measurements
Critical temperature measurements were performed with the liquid helium fourpoint temperature probe described in section 5.4.2. This data was taken on asdeposited films which had been stored at room temperature for days or weeks. A
constant current was applied to the film. Voltage and temperature were measured
vs time as the probe was thermally cycled several times around the critical
temperature. Hysteresis in the resulting resistance vs temperature curves is a
function of the warming and cooling rates. Over repeated thermal cycling, typical
hysteresis is about 0.2 K. Data for two samples around the critical transition
temperature is given in Figure 5-5.
Superconducting Critical Transition Temperature
,,/'
(n
E 15
o0
0
U
Data taken 9/17/96
Substrate 10 Film 2
In-8.7%Bi Film
l......
10
C
adw
......
5
a0
0
.
4.2
4.4
4.6
4.8
5.0
5.2
5.6
5.8
6.0
5.6
5.8
6.0
Temperature (K)
20
=E 15
%ft..
CD
C
-
5
cc A
0
I
uk•
4.2
4.4
4.6
4.8
5.0
5.2
5.4
Temperature (K)
Figure 5-5 Superconducting critical transition temperatures for
two indium-bismuth films.
A close look at these resistance vs temperature plots reveals interesting
transitions around 5.7 or 5.8 K. A closer examination reveals that there is a
resistance transition here as well. The resistance drops by approximately 4.5% to
6%. This second transition was always seen infilms stored at room temperature.
Data which clearly highlights this transition is shown in Figure 5-6. This transition
temperature corresponds to the superconducting transition temperature of In 2Bi.
Therefore, a phase separation is occurs at room temperature between the In-Bi
alpha phase and the In 2Bi phase. The In 2Bi phase appears as unconnected islands
imbedded in the alpha matrix. When the islands become superconducting, they
short out sections of the matrix, reducing the observed resistance. These islands
also act as pinning centers and may account in part for some of the electrical
behavior described below. This volume fraction is consistent with the maximum
room temperature bismuth solubility limit given by Hansen and discussed in section
5.1.
Just above the critical transition temperature, the resistance of substrate 11 film 1
is 15.5 £. This corresponds to a resistivity p, = 11 g.Q-cm.
Second Resistance Transition
71
I
.
______________
.5
Data taken
Substrate 1
E
S17.0
3
16.5
n16.0
-i
16.0
5.2
5.4
5.6
5.8
6.0
6.2
Temperature (K)
.4-7
"ft'
III.V
1E
E
16
5
~
Data taken 9/20/96
.
e I 1"11im1 IT
0
......
-
.... ...........
.............................
YoBi Film
u 16.0
5.5
S...
.........................
15.0
5.2
5.4
5.6
5.8
6.0
6.2
Temperature (K)
Figure 5-6 Second resistance transition vs temperature for two
indium-bismuth films.
5.5.2 Critical Magnetic Field Measurements
Critical magnetic field measurements can be made with the transition
temperature probe or the conductor proximity effect probe. The transition
temperature probe can apply magnetic fields up to 40 mT. The conductor
proximity effect probe has an extra solenoid and is capable of reaching magnetic
fields 4 mT higher than the transition temperature probe.
The shape of the resistance versus magnetic field data at very low current
densities was consistent from film to film and independent of annealing procedures.
The magnitude of the upper critical field, however, varied from film to film and
with annealing. The shape of the resistance versus field curve is presented in
Figure 5-7. This particular data is for an as-deposited film stored for several weeks
at room temperature. Since the curve flattens as the resistivity ratio approaches
one, it is hard to identify the upper critical field from the data. Resistance versus
field curves are therefore most easily compared by examining the magnetic field
which produces a resistivity ratio of 0.5.
Resistance vs Field at Low Current Densities
16
_
E 12 -
Data taken 10/12/96
Substrate 18 Film 1
In-8.7%Bi Film
8 (in liquid helium)
--0
cc
--- --- ------- . ........
A0 4 -- --,
0
__.
. I
I
10
0
I
....................
I
20
40
30
Magnetic Field (mT)
Figure 5-7 Resistance vs magnetic field for a mixed state
indium-bismuth film.
The dependence of the upper critical magnetic field on temperature was
determined by measuring the resistance versus temperature at various applied
magnetic fields. An example of the observed dependence is given in Figure 5-8.
The 0.5 resistivity ratio criterion was used in producing this figure. As mentioned
above, the magnitude of the upper critical field appeared to vary for different
samples and different runs. This is most likely due to variation in the phase
separation discussed above.
Critical Magnetic Field Dependence On Temperature
On Temperature
Critical Magnetic Field Dependence
Magnetic Field at Resistivity Ratio 0.5
25-
20-
..
From
----------
resistance vs
---
LL 15
0O
SFrom
---.
5
04.2
resistance vs
temperature data
Data taken 9/17/96
" Substrate 10 Film 2
4.3
4.5
4.4
Temperature (K)
4.6
Figure 5-8 Dependence of the upper critical magnetic field on
temperature for a mixed state indium-bismuth film.
4.7
5.5.3 Critical Current in Various Magnetic Fields
The critical current density in ambient field is simple to measure. Using the basic
four-probe technique, the voltage is measured as the current is increased from zero
to the critical current. When the critical current is reached, the film resistance
jumps from zero to a large value. This is due to thermal runaway, which occurs
easily at these high current densities. As soon as the film resistance exceeds zero,
the power generated by the current is larger than the film cooling rate.
Substrate 11 Film 1 was tested six times over a one month period. The critical
current in zero field ranged from 52 mA to 107 mA. This corresponds to a critical
current density of 5.8x104 A/cm 2 to 1.2x10 5 A/cm 2 . Similar results occurred with
other films.
The critical current can was also measured as a function of applied magnetic
field. Define the critical current as the applied current which produces a one
microvolt voltage drop along the film length. Figure 5-9 presents the measured
critical current density as a function of applied magnetic field.
Critical Current Density vs Magnetic Field
i e+5
E 9e+4
e microvolt
b 8e+4
.
I
criterion
-o- Substate 11 Film 1 - 8/28/96 Data
-4- Substate 10 Film 2 - 9/10/96 Data
7e+4
r 6e+4
i
e: Substrate 10 Film 2
ensions based on estir
5e+4
• 4e+4
3e+4
i
a 2e+4
i
i
Sle+4
.o
i
Oe+O
l
0
2
4
6
8
10
Magnetic Field (mT)
12
14
16
Figure 5-9 Dependence of the critical current density on
magnetic field for two mixed state indium-bismuth films.
5.5.4 Effect of Film Anneal on Temperature Transition and Critical Field
Unless otherwise specified, all experiments in this thesis were performed on asdeposited films stored at room temperature. The storage time was typically weeks
or months between film fabrication and property measurement. The exact storage
100
time for any particular film can be determined by comparing the experiment date
with the fabrication date given in Table 5-4.
This section examines the influence of annealing on the basic film properties.
Substrate 18 Film 1 was deposited on 23 July 1996. It was stored at room
temperature for almost three months. Resistance versus temperature data on
12 October 1996 showed the typical double resistance transition discussed in
section 5.5.1. At 4.7 K a narrow transition occurred from zero resistance to
16.3 ohm. At 5.7 K a second transition occurred from 16.3 ohm to 16.7 ohm.
Resistance versus magnetic field data gave a resistance ratio of one-half at 23 mT.
On 19 November 1996 this film was annealed on a hot plate. The hot plate
temperature was slowly increased over a one hour period, then held at 70-77 OC
for one additional hour. The film was quenched directly into a cryogenic dewar,
cooling from above 70 OC to below -50 OC in 20 seconds. The film was then
cooled further for superconducting property measurement. Only a single narrow
resistance transition was observed. The transition temperature was 4.9±0.3 K. The
low accuracy is due to poor thermal contact between the temperature probe and
the film for this particular run. The normal state resistance just above the critical
transition temperature was 18.4 ohm. Resistance versus magnetic field data gave a
resistance ratio of one-half at 34 mT.
The film was then warmed to room temperature and remained at room
temperature for one hour. The film was cooled again and superconducting
properties were re-examined. A single wide critical temperature transition was
observed, extending from 4.6 K to 5.2 K. This transition is shown in Figure 5-10.
The normal state resistance just above the critical transition temperature was
16.8 ohm. Resistance versus magnetic field data gave a resistance ratio of one-half
at 22 mT.
The film was then warmed a second time to room temperature and remained at
room temperature for one hour. Upon cooling, the low temperature film properties
remained relatively constant. The influence of longer periods at room temperature
were not examined, because the film did not survive the third warming.
The dramatic widening of the resistance versus temperature curve is most likely
due to local compositional variations in the film. The compositional dependence of
the transition temperature is given in Figure 5-1.
The increase upon annealing and subsequent decrease in the normal low
temperature resistance can be accounted for by considering the mean free path.
There are two possible mechanisms for the mean free path to decrease upon
annealing and rapid quenching. First, the larger metastable bismuth solubility
presents more scattering centers and therefore a larger residual resistance. Second,
a very large number of small In2Bi domains may have formed during the quench.
Although the volume fraction would be quite low, these domains could also act as
101
scattering centers. The subsequent room temperature film treatment would result
in a decrease in the bulk solubility of bismuth and in the growth and coalescence of
the In2Bi domains. This would account for the observed decrease in resistance.
Critical Temperature Transition
I1
16
14
oE
•O
12
8
4
2
0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
Temperature (K)
Figure 5-10 Wide critical transition observed in In-Bi film
after temperature treatment
5.6 Lead Films
Type-I superconducting lead films were fabricated using thermal evaporation
onto a quartz substrate through an aluminum mask. These films have a high critical
current density and upper critical field. The measurements below are for a fourprobe film geometry with approximate dimensions of 12.7 mm long by
140 gm wide by 0.58 gm thick. The Pippard coherence length 5o = 83 nm and the
penetration depth XL = '9 nm.83 Using equation (30), K= 0.45. According to
Lasher," the state of this lead film should be (1) singly quantized fluxon lattice for
B < 1 mT, (2) multiply quantized fluxon lattice for 1 mT < B < 3 mT, and (3)
intermediate state for B > 3 mT.
The critical magnetic field vs current was measured using the conductor
proximity effect probe. In ambient field, the critical current was 1.4 A, giving a
critical current density of 1.7x106 A/cm 2. At low current densities, in liquid helium,
the critical field was larger than 45 mT, which was the largest magnetic field that
could be produced with the solenoid. A range of currents were applied to the lead
film. For each current, the magnetic field was increased until the film showed
83 W.
R. Hudson, The Mixed State of Superconductors, (National Aeronautics and Space
Administration, Washington, D. C., 1970) 65.
102
resistance. The transition for superconducting to normal was rapid in all cases.
Once the critical field was reached and resistance appeared the heat generated in
the film caused thermal runaway. The detailed critical field vs current density data
is presented in Figure 5-11.
Figure 5-11 Dependence of the critical magnetic field on
current for a type I lead film.
5.7 YBCO Bulk and Film Samples
Some YBCO thin films were provided by Bill Wilbur and Roland Cadote of the
US Army Research Laboratory (ARL) in Fort Monmouth, New Jersey and by
Dave Rudman of the National Institute of Standards and Technology in Boulder,
Colorado. These films were used to test the pancake coil magnetic critical
temperature measurement system and the four-probe electrical critical temperature
and critical current measurement system. Unfortunately, many of these films had
defects in the materials or patterning or did not survive the shipping to MIT. Some
discussion of the use of these films is presented below. Additionally, bulk YBCO
specimens fabricated by the MIT Specialty Materials Laboratory were used in
some tests. Further tests on YBCO were not possible due the difficulty in attaining
quality patterned films within a reasonable time frame and expense.
5.7.1 Magnetic Critical Temperature Measurements
These experiments were performed with the liquid nitrogen magnetic critical
temperature measurement probe. A 5.0 kilohertz, 500 millivolt rms signal was
applied to the primary with a Philips PM5192 function generator. This corresponds
to an ac magnetic field of approximately one microtesla. The secondary voltage
was monitored with a Stanford SR530 lock-in amplifier referenced to the
frequency of the primary. A General Purpose Interface Bus (GPIB) was used to
103
download the measured lock-in voltage and RTD resistance to a computer,
allowing for continuous data taking as well as easy analysis and plotting.
The sample and the pancake coils were loaded in a sealed copper can. The
copper can was evacuated and backfilled with helium gas just before being
submerged in the liquid nitrogen. Cooling was achieved by slowly lowering the
probe into liquid nitrogen and taking measurements as the temperature changed.
Warming was achieved similarly by raising the probe out of the liquid. The thermal
lag between the sample and the RTD could be reduced by decreasing the warming
and cooling rates, but it could not be eliminated.
Measurements on a bulk YBCO disc were made on 26 Feb 1994. The 0.49 gram
disc has a 9.7 mm diameter and a 1.4 mm thickness. A large hysteresis in seen in
the data, shown in Figure 5-12. This results from the time required to warm or
cool a bulk specimen. The cooling rate at the superconducting transition was about
-5 deg/min, while the warming rate was about +10 deg/min. A magnetic transition
lies somewhere between 85 K and 107 K, bracketing the expected transition
temperature of 90 K.
Secondary Voltage versus Temperature
Bulk YBCO #1
U.1 u
0.115
> 0.110
E
g 0.105
o 0.100
0.095
0.090
75
85
105
115
125
Temperature (K)
Figure 5-12 Magnetic measurement of critical temperature in
bulk YBCO sample #1.
A second bulk YBCO disc was examined 3 Mar 1994. This 1.28 gram disc had a
9.5 mm diameter and a 4.2 mm thickness. In this case the sample was cooled and
warmed three times with successively slower warming rates. The voltage across
the secondary coil is shown in Figure 5-13. As the warming rate decreased from
+6 deg/min to +3 deJ/min to +2 deg/min, the warming curves approached the
cooling curves, which were measured as a rate of -2 deg/min. In this way, it is
104
shown that the hysteresis depends on the warming rate. The measured
superconducting transition temperature is between 85 K and 95 K.
Secondary Voltage versus Temperature
SML Bulk YBCO #2
Z.LU
2.00
> 1.80
E
M,1.60
0 1.40
1.20
1.00
75
80
85
90
95
100
105
110
Temperature (K)
Figure 5-13 Magnetic measurement of critical temperature in
bulk YBCO sample #2.
ARL provided an unpatterned 500 nm thick YBCO film on a 1 cm x 1 cm MgO
substrate. ARL characterized this film as having a 4-5 K wide superconducting
transition at 78.5 K. A pancake coil characterization on 10 Mar 1994 yielded
Figure 5-14. A transition temperature between 80 K and 85 K was observed. The
transition width appears to be about 2 K. As expected, the susceptibility change
resulting from the superconducting transition in the film is less than the change
from a transition in bulk material.
105
Secondary Voltage versus Temperature
ARL Film #753
Z.UU
1.99
1.98
E 1.97
0 1.96
6 1.95
1.94
1.93
1.92
60
65
70
80
75
85
90
95
Temperature (K)
Figure 5-14 Magnetic measurement of critical temperature in
unpatterned YBCO film.
5.7.2 Electrical Critical Temperature Measurements
Several patterned YBCO films were received from NIST. The mask design is
given by Figure 5-15. Two current contact pads were used to inject a current
through the ten micron line. There were actually four voltage contact pads in this
mask rather than the required two. This provided some flexibility in attaching
voltage leads. Silver was deposited on the pads to enable better electrical contacts.
The most successful electrical contacts to the superconductor were berylliumcopper spring-clip pressure contacts."
L.F. Goodrich, A.N. Srivastava, T.C. Stauffer, A. Roshko, and L.R. Vale, "High Current
Pressure Contacts to Ag Pads on Thin Film Superconductors", IEEE Transactionson Applied
Superconductivity 4, (1994) 61-64.
84
106
Dimensions in millimeters
unless otherwise stated.
1.6
1.6
104
Ill
1.4
1.6
P
Nominal line width
10 microns
/
.i•
I
, " Voltage contact
3.6
Current contact
Current contact
Voltage contact
Voltage contact
Figure 5-15 Typical four-probe resistance measurement
pattern used in NIST films.
The films were specified as being 200 nm thick, with a critical temperature of
90 K. One of the three films in this batch failed due to patterning problems. The
resistance versus temperature was measured for the remaining two films. The
cooling curves are shown in Figure 5-16. Sample 1 had a lower than expected
critical temperature while sample 2 had the predicted critical temperature. The
samples failed at low temperature before the warming curves were measured.
Voltage Versus Temperature Cooling Curve
NIST #L394-433
" 2.5
0
o
6 2.0
* 1.5
.
1.0
cc 0.5
0.0
80
100
110
Temperature (K) (approximate)
Figure 5-16 Transition temperature measurements showing
disappearance of resistance upon cooling for two NIST
patterned YBCO films.
107
--
120
6. Conductor Proximity Effect Experiments
The conductor proximity effect experiment design was discussed in detail in
chapter 4. In this chapter, the experimental results on indium-bismuth films are
presented. The detailed dependence of the flux flow voltage on the gap between
the superconducting film and the normal metal film is examined. The change in the
fluxon velocity due to eddy current damping in the normal metal is viewed in terms
of the eddy current damping coefficient rleddy, which was calculated in detail in
chapter 3.
First, the experimental flux flow data observed for these films is presented. Then,
the experimental relationship between voltage, current, magnetic field, and gap size
is examined in detail. Last, flux flow behavior is modeled and insights into flux
flow for this material are presented.
6.1 Flux Flow Measurements
In this section, flux flow data is presented for indium-bismuth films. Various
models of flux flow are fit to the data. The non-linearity of the experimental data is
contrasted with expected linear behavior of the standard flux flow model.
The term voltage alone always refers to flux flow voltage. Other voltages are
always preceded by descriptors. The terms current and field always refer to applied
current and applied magnetic field, respectively.
6.1.1 Voltage Versus Current at Constant Magnetic Field
As larger magnetic fields are applied to the superconductor, flux flow occurs at
lower current densities and thermal runaway occurs at larger flux flow voltages.
Therefore, more extensive measurements of voltage vs current curves can be made
at larger fields.
For each data point, both positive and negative polarities of the dc current were
applied to the four-probe line. The recorded voltage was the average of the
absolute values of the measured voltages. At a fixed magnetic field, the current
was increased in 0.25 mA or 0.50 mA steps until either (1) the voltage reached
2 mV or (2) thermal runaway occurred. Typical flux flow data is given in
Figure 6-1. The data non-linearity is more clearly demonstrated in a log-log scale,
as shown in Figure 6-2. The slope of these curves is 11 ± 1.The relation
V= Vo(J / Jo)" holds for four orders of magnitude in voltage.
108
Typical Voltage Vs Current Curve At
Various Magnetic Fields
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Data taken 8/25/96
Substrate 11 Film 1
.............
......-.
---· -·-- --1.7
i
0.0-T
4mT
--.
1.7 mT
0
0.0 mT
100
20
Current (mA)
Figure 6-1 Flux flow voltage vs current at various magnetic fields.
Typical Voltage Vs Current Curve At
Various Magnetic Fields
el +1
Data taken 8/25/96
Substrate 11 Film 1
1e+0
>: le-1
E
le-2
0
le-3
.
.
.
. .............
.
.................. .
........... •..... ..............................
.....................
-- - -- -.. ..- -
...... ...
..............
-- - -
. ..........
..-.
. .....
..
. .....
Approximate Noise Floor
......
.....
...
1 4 ...........
le-45
100
10
1000
Current (mA)
Figure 6-2 Log-log scale - flux flow voltage vs current at
various magnetic fields.
6.1.2 Voltage Versus Magnetic Field at Constant Current
The shape of the flux flow voltage versus magnetic field at constant current
curves is very similar to shape of the flux flow voltage versus current at constant
field curves. Again, both polarities of the current are used to remove stray voltage
109
drifts. Typical sets of curves on linear and logarithmic scales are presented in
Figure 6-3 and Figure 6-4, respectively. Again, the log-log plot has a slope of 11
over a wide range of voltages. This gives the relation V = Vo(B / Bo)". The
dependence of the voltage on the current appears to match the dependence of the
voltage on the field. Since the total Lorentz driving force on the fluxons equals the
product of the current and the field, it is reasonable to consider the dependence of
the flux flow voltage on the driving force directly.
Voltage Vs Magnetic Field At Various Currents
2.U
II
1.8
SI= 1 mA *l=5mA
al = 10mA ..I= 20 mA
xl= 30mA xl= 40mA j
1.6
1.4
E 1.2
+1
50mA
..................
I. ..
1=60 mA
S1.0
0.8
0.6
.. ..
........
titrJ
0.4
0.2
0.0
0
2
4
8
10
12
Magnetic Field (mT)
Figure 6-3 Flux flow voltage vs magnetic field at various currents.
110
14
Voltage Vs Magnetic Field At Various Currents
ie+i
1e+O
-le-1
E
* le-2
>
1 e-3
1e-4
1e-5
---
100
0.1
Magnetic Field (mT)
Figure 6-4 Log-log scale - flux flow voltage vs magnetic field at
various currents.
6.1.3 Voltage Versus Driving Force
Flux flow voltage is graphed against Lorentz driving force directly. The log-log
relationships for two films are given in Figure 6-5 and Figure 6-6. The log-linear
relationships are shown in Figure 6-7 and Figure 6-8. The fact that the curves for
various currents and fields overlap demonstrates that the flux flow voltage is
fundamentally a function of the total Lorentz driving force on all fluxons. Note
however that low current and high field data show a higher voltage at a given
driving force, relative to other data.
There are two approaches for describing the empirical voltage vs Lorentz force
data. First, the dependence of flux flow voltage on Lorentz driving force can be
modeled as a power law
V = V.0 JB
(133)
where Vp.o and cx are experimental constants. From the log-log relationships it is
seen that x = 11 for low driving forces and x = 2 for high driving forces. There is a
gradual change in x as the driving force is increased.
The flux flow voltage can also be modeled as exponential in the driving force
with different mechanisms dominating in high and low force regions. Making some
estimates from data in Figure 6-7, it is found that
(134)
V = V,.o exp(BJ / o,)
111
with c, = 2.7x10 4 N/m 3 at low driving forces and cr = 2.9x10' N/m3 at high
driving forces. Using a film thickness of h, = 0.82 p.m, these values correspond to
stresses on the fluxon lattice of 2.2x10 2 Pa and 0.24 Pa, respectively. These
stresses are two to three orders of magnitude less than the shear modulus
calculated in section 1.4.4. Therefore the lattice shears at strains of 10- 2 to 10-3.
Voltage Vs Driving Force At Various Fields
1E+1
Data taken 9/10/96
1E+O
Substrate 10 Film 2
.•
>1E-11E-
- •"- 1E-1
-
B is 2 mT
___
*Bis6mT
E.a
1E-2 S1E-3
B is 11 mT -B is 13 mT
".B
B is 15 mT
Note: Substrate 10 Film 2
...........
1E-4--
B is 4 mT
xBis8mT
-•
dimensions based on estimates
. ..........................
-
.........--------
1E-5
100
10
Field *Current Density (Tesla-Amp/centimeter 2)
Figure 6-5 Log-log scale - flux flow voltage vs Lorentz driving
force at various fields.
112
1000
Voltage Vs Driving Force At Various Currents
It+ 1
1E+1
1E+0
: 1E-1
E
0 1E-2
>
1E-3
1E-4
1E-5
1000
100
10
Field *Current Density (Tesla-Amp/centimrneter 2)
Figure 6-6 Log-log scale - flux flow voltage vs Lorentz driving
force at various currents.
Voltage Vs Driving Force At Various Fields
1 E+0
1E-1
Data taken 9/10/96
Substrate 10 Film 2
Bis4mT
is2mT
is6mT xBis8mT
-E--B
x B is 11 mT - B is 13 mT
Biis 15 mT
S1E-2
S1E-3 ---.........
....
Note: Substrate 10 Film 2
dimensions based on estimates
1E-5
0
20
40
60
80
100
120
Field * Current Density (Tesla-Amp/centimeter 2)
Figure 6-7 Log-linear scale - flux flow voltage vs Lorentz
driving force at various fields.
113
140
Voltage Vs Driving Force At Various Currents
1E+1
1E-1S:EData
taken 8/28/96
1E-2
>
..
."............
-"•........
Substrate 11 Film 1
.......
* I= 1 mA
l = 10 mA
xl=30 mA
1E-3
1 E-4 ------.-. . . . . . . . . . . .... . . . . . ....+
50 mA
* I=5mA
I = 20 mA
i.
xl=40mA
=60 mA
1E-5
0
20
40
60
80
100
120
140
Field * Current Density (Tesla-Amp/centi meter2 )
Figure 6-8 Log-linear scale - flux flow voltage vs Lorentz
driving force at various currents.
6.1.4 Analysis of Flux Flow Data
Observed flux flow voltage will be analyzed in terms of a linear fractional motion
model. For a given applied current and magnetic field, a fraction 3 of fluxons move
with a velocity proportional to applied current. The rest of the fluxons remain
stationary, due to pinning. Under this model, the flux flow voltage is
V = 3Bl,v = 3Bl ,o(J
- Jc)/ r71o. The fraction of moving fluxons 3 is expected to be
5
an increasing function of driving force. As the driving force increases, more
fluxons break free of pinning sites, the lattice shears more easily, etc.
The stress applied to the fluxon lattice is the total driving force per unit fluxon
length BJ times the film thickness h,. Figure 6-9 shows how P depends on fluxon
lattice stress. This graph uses the intrinsic viscous coefficient calculated in section
5.1. The fraction of moving fluxons increases markedly as stress increases from
0.3 Pa to 0.6 Pa.
At higher currents, 3 is a function of stress, independent of current. At lower
currents (higher fields), there is a larger fraction of mobile fluxons for a given
driving force. This is qualitatively consistent with equation (46), in which the
fluxon lattice shear modulus decreases with increasing magnetic field. To see this
relationship more explicitly, the fraction of mobile fluxons is plotted against the
lattice strain in Figure 6-10. The strain is defined here as the applied stress divided
by the magnetic field dependent shear modulus. The curves for different applied
currents overlap more when graphed in this manner. The fraction of mobile fluxons
114
increases suddenly at 4 to 6 % strain. This is consistent with the ideal strength of a
crystal lattice, which is about 7 % of Young's modulus.8s
Fraction of Mobile Fluxons vs Applied Lattice Stress
1e+0
----------
i...
aken 8/28/96.
ate 11 Film 1
1e-1
1e-2
.......... ..
.
.....
. .......
.--....... ..
.
.. .......-
.
.................-
~x5IP
r
1e-3
1e-4
X
i)&:
•. :
1e-5
-
~`~'-~~
`~
0.2
0.1
0.0
'
'
.
~
.
~
'
,
,
'
0.3
'
'
'
"
'
5 mA
= 10
20 mA
40 mA
60 mA
= 30
= 50
~
n__
0.5
0.4
0.7
0.6
Fluxon Lattice Stress (Pascal)
Figure 6-9 Fraction of mobile fluxons as a function of applied stress.
Fraction of Mobile Fluxons vs Lattice Strain
le+0
Data taken 8/28/96
g
~
Substrate 11Film 1
.
":S
o
Assume Hc2= 35 mT
le-2
m
10mAl
m x l= 3
i :l= mA2
. ..
. .......
... ..........
........
"0.1le-3
- .... .
X=20mA *I=30mA
+1 =40"mA -l=50mA
. l =6 00mA
o
.m.
."=
le-5+
0%
I = 60 mA-
A
2%
4%
6%
8%
10%
12%
14%
Lattice Strain
Figure 6-10 Fraction of mobile fluxons as a function of lattice strain.
as M. F. Asbhby and D. R. H. Jones, EngineeringMaterials:An Introduction to their Properties
and Applications,(Pergamon Press, Oxford, 1980) 87.
115
6.1.5 Temperature Stability
To maintain temperature stability, most measurements are performed with the
substrate and film submerged in the helium bath. Since the voltage-current relation
is a strong function of temperature, it is important to consider if the observed flux
flow behavior is a result of current dependent local temperature changes. The two
primary heat sources are flux flow resistance and contact resistance. The flux flow
power generated is the flux flow voltage multiplied by the applied current. The
contact resistance power generated equals the contact resistance multiplied by the
square of the applied current.
Cooling occurs by direct contact of the superconducting film to the helium
liquid, or in some cases, to helium gas in the probe can. The film is also thermally
well connected to the quartz substrate, which is effectively cooled by the helium
bath. Contact lead wires provide additional cooling at the contacts. The thermal
boundary conductance of liquid helium is 0.1 W/cm 2'K. The thermal conductivity
of quartz at this temperature is 1 mW/cm-K. The thermal conductivity of copper
magnet wire is 4 W/cm-K.86
Assume that the observed sudden voltage increases are due to thermal runaway.
Note that if the contact resistance was the sole cause of thermal runaway, then
runaway would always occur at a constant current, independent of magnetic field.
If the flux flow power is important, the thermal runaway current would be field
dependent. Field dependent sudden voltage increases are observed experimentally.
Heating at the current leads can only effect the four-probe flux flow voltage
measurement if it changes the four probe line temperature. Since that part of the
film is far from the current contacts and the substrate cooling rate is large, the
influence is small. However, heat generated at current contacts may cause the
entire film to go normal.
The measured contact resistance is 0.1 ohm for the current lead pair, or
0.05 ohm per lead. A large applied current of 100 mA therefore generates 0.5 mW
at a single lead. Assume all this power is generated at the superconducting film
side and calculate the local temperature increase. Since there are many cooling
mechanisms operating in parallel it is difficult to determine the exact cooling rate.
Consider the heat transfer to the bath from a small section of the superconductor
near the contact. The line width at this part of the film is 1.6 mm. Take this as the
diameter of a small heat dissipation region. The area of this region is 0.02 cm2 and
therefore this small region would dissipate 2 mW/K. Next consider the contact
lead, a 5 mil diameter copper wire 1 cm long. The cross-section is 1.3x10 4 cm2. If
one end is attached to an ideal heat sink, the wire dissipates 0.5 mW/K. The
86
R. B.Scott, Cryogenic Engineering,(D.Van Nostrand Company, Inc., Princeton, New Jersey,
1959) 344-345. A. C. Rose-Innes, Low Temperature Techniques: The Use of Liquid Helium in
the Laboratory, (Tbe English Universities Press Ltd., London, 1964) 82.
116
cooling path through the substrate is more difficult to calculate, due to the
geometry involved. Since the substrate has a large top surface area of 7.4 cm2, it
has a total possible dissipation of 700 mW/K. The substrate will therefore be
treated as a heat sink to the bath temperature. The film is in intimate contact with
the substrate, therefore the heat dissipation rate into the substrate will be
comparable to the dissipation rate into the helium. These rough estimates add up
to a heat dissipation rate of 4.5 mW/K, or a temperature increase at the contacts of
approximately 0.1 K. This gives an approximation of the temperature change at
large applied currents.
Next consider flux flow power dissipation. Consider high values of flux flow
power just below thermal runaway. For example, choose a 10 mA current
producing a 1.6 mV voltage for a total power of 16 gW. This is much less power
than at the current leads, but at a more critical area of the film. The surface area of
a typical superconducting line pattern is 0.02 cm2 within the four-probe area. The
four-probe area can therefore dissipate 2 mW/K to the helium bath. Without even
considering heat dissipation into the substrate, the temperature change of this part
of the film must be less than 8 mK.
These calculations indicate that the measured flux flow voltage curves are taken
at nearly constant temperature, until thermal runaway occurs. Thermal runaway
may be caused by local heating in the film which cascades to drive the entire film
normal. The sample was submerged in helium liquid for some runs and helium gas
for others. If liquid helium cooling had a large influence on thermal runaway, then
the data would reflect this trend. The data does not indicate any dependence of the
voltage versus current relationship on cooling method.
Figure 6-11 shows the detailed data. Voltage versus current curves at constant
same magnetic field were measured on different days for a single film. The
variables from run to run include storage time, thermal cycling, and cooling
method. The legend He gas indicates that the probe was sealed, evacuated, and
backfilled with He gas. The legend He liquid/gas indicates that the probe bottom
was open but the top was pressure tight. He liquid may have covered the film, but
a pressure build-up in the probe may have prevented the He liquid from covering
the film. The legend He liquid indicates that the probe bottom and top were open
such that liquid covered the film. No relationship is observed between the curve
shifts and the cooling method.
117
Voltage vs Current at Fixed Field - Multiple Days
2.0
1.8 - o 8-14-96 He gas
r8-17-96 He gas
1.6
1.4
>
1.8-20-96
i
He liquid/gas
:x 8-25-96 He liquid
.2x 8-28-96 He liquid
_
I
"
x
n
S0.8
Field = 2.6 mT
> 0.6
Substrate 11 Film 1
0.4
"
0o
0.2
0
5
10
20
15
25
30
35
40
Current (mA)
Figure 6-11 Voltage versus current curves over multiple days on the same
film - various cooling methods.
6.2 Aluminum Film Conductivity
Aluminum films were deposited using the thermal evaporator in the MIT
MicroLab Facility. This was convenient, since this evaporator was dedicated solely
for use with aluminum and was calibrated to deposit films of the desired thickness.
The operation of this evaporator is similar to the Specialty Materials Laboratory
evaporator described in section 5.2. It is important to know the electrical
conductivity of aluminum in liquid helium, since the eddy current damping is
proportional to the normal metal conductivity. The conductivity of 99.996% pure
aluminum at room temperature is 3.77x107 -•'m-1. 87
An aluminum film was deposited using a four-probe pattern mask with the
MicroLab thermal evaporator. A Dektak 3 profilometer characterized the width
and thickness of the film. The length was measured from the mask itself. Due to
mask geometry (lead pattern has same width as line pattern), line length was
known only to within a factor of line width. The expected bulk room temperature
resistance is calculated for this geometry.
s7 CRC Handbook of Chemistry and Physics, 63" edition, edited by R. C. Weast, (CRC Press,
Inc., Boca Raton, Florida, 1982), page F-133.
118
I= 20 mm
w = 740 gpm
h = 0.88 plm
R
a 20 C wh
(135)
815 mf
The actual room temperature measurement gave a resistance of 911 mR,which is
10% larger than the bulk value.
The change in resistance upon cooling to 4.2 K was measured for another film
deposited in the same thermal evaporator. Conductivity in liquid helium was found
to be a factor of twenty-one greater than the conductivity at room temperature for
these aluminum films. Therefore the aluminum film conductivity at 4.2 K is taken
to be 21x3.77x107 -'mn - ' = 7.9x10's -m-I.
6.3 Gap Dependence of Flux Flow Data
This section examines the direct experimental confirmation of the conductor
proximity effect. Each part of this section addresses a different aspect of the flux
flow dependence on the superconductor to metal gap size. The gap size is
determined by measuring the capacitance between the beam and substrate. Due to
large stray capacitance, it is difficult to determine the absolute magnitude of the
gap size. As the gap decreases, however, the capacitance rises sharply. The
position uncertainty is therefore quite substantial for large gaps, but much less for
small gaps:
6.3.1 Voltage Versus Field Curves at Two Gaps
The voltage versus magnetic field at constant current curves are measured for
two gaps. The variation of these curves is a repeatable function of the gap size.
The two different gap values chosen correspond to zero beam force and finite
beam force. If the gap is assumed to be approximately 1V2 microns when there is
no beam force, then the measured capacitance change indicates a gap of
approximately 0.8 pm with a beam force.
To ensure that the change observed with gap was real, the voltage vs field curves
were measured twice with no beam force, twice with beam force, and then twice
again with no beam force. The data was repeatable. The average voltage vs field
for large and small gaps is plotted in Figure 6-12. The imbedded plot shows that
the voltage difference is close to linear in V, which is the expected result for the
flux flow models in Chapter 3. A closer examination of AV versus V'on a log-log
scale reveals a dependence slightly less than linear. This is an indication that the
voltage change is reduced at larger magnetic fields.
119
Voltage Vs Field Curve at Two Gap Spacings
,,
1.d
1.6
1.4
1.2
-
1.0
> 0.8
0.6
o
0.4
0.2
0.0
1.50
1.75
2.00
2.25
2.50
2.75
3.00
Magnetic Field (mT)
Figure 6-12 Gap dependence of voltage versus field data.
6.3.2 Voltage Versus Current Curves at Two Gaps
Voltage versus current at constant magnetic field curves were measured at two
different gaps and several field values. As in the above section, the large gap and
small gap data are slightly different, in a repeatable and consistent manner. The
observed voltage for large gaps is always larger than the observed voltage for
small gaps (ignoring small noise and drift voltages). Figure 6-13 shows a set of six
runs for a fixed magnetic field. Two runs are taken at a large gap, two at a small
gap, and then two more are repeated at a large gap. The four large gap runs form a
single curve and the two small gap runs form a single curve. The small gap curve is
very close to the large gap curve, but it is distinct.
The voltage difference AV= VLARGE - VsAxU. is plotted against current for
various magnetic fields in Figure 6-14. VLARGE and Vs.~ý.. are defined as averages
over large gap and small gap curves respectively. These curves all have a similar
shape. They are offset from each other since the critical current varies with
magnetic field.
Analogous to Figure 6-12, voltage difference is plotted against flux flow voltage
squared in Figure 6-15. For each value of magnetic field the curve is close to
linear. A closer examination shows that AV= VX with 2.5 < x < 3.5. Equations
(116) and (118) motivate plotting BxAV versus V2 . This relationship is shown in
Figure 6-16. The curves are much closer to overlapping, however, the observed
voltage difference is still reduced for larger magnetic fields.
120
A qualitative comparison of theory and experiment may be performed using data
in Figure 6-16. Taking the slope of the data to be 0.06 T/V and using equation
(118), the observed eddy current damping coefficient is found to be 7x10-' 2 kg/ms.
The capacitance change for each of these curves is +9.4 pF. The small gap spacing
is therefore between 0.6 4lm and 0.9 gm, depending on the assumed large gap
spacing. At these spacings, the expected damping from Figure 3-10 is between
2x10 - 2 kg/ms and 6x10 - 12 kg/ms. Accounting for the finite penetration depth in
this material, the actual expected damping is about one half of this value.
Agreement between theory and experiment is therefore within an order of
magnitude, with the experimental result larger than the predicted value.
Voltage Vs Current Curve at Two Gap Spacings
1.2
Six runs: four at large gap
1.0
0.8
and two at small gap
-
Field = 0.9 mT
=Larg e
...
Data taken 11/21/96
E
Substrate 19 Film 1 .......
g0.6 .....
........
............
..
....
..
...
~....
.......
.~
o
> 0.4
.......
..
0.2
0.0
.......... ...........
..
1 i~
I
30
Current (mA)
Figure 6-13 Gap dependence of voltage versus current data.
121
Gap Dependent Voltage Difference Vs Current
0.14
0.12
-
0.10
-
--e-0.9 mT Data taken 11/21/96
1.
... 1.7mT
S....
7............1.7
mT Substrate 19 Film
0.08 .
-
0.06
-
0.04
-
-.......
- 2.6 mT...
.
IN1
0.02
2
0.00
"
~kk~aa
~2~-~--
25
20
15
10
5
0
30
35
40
Current (mA)
Figure 6-14 Gap dependent voltage change versus current at
various magnetic fields.
Voltage Difference Vs Voltage Squared
0.14
-
S-e-0.9 mT Data taken 11/21/96
1.7 mT Substrate 19 Film 1
........
0.12
-~- 2.6 mT
.. .......................
0.10...........
-- - -
>' 0.08 -
.............
.............
..
.-......
....
.. ............
. .....
--..........
>" 0.08 ............
0.02
1 . ..
0.00
0.0
0.5
1.0
.
.
2.0
1.5
...
2.5
3.0
Voltage (mV) 2
2
Figure 6-15 Gap dependent voltage change versus voltage
squared at various magnetic fields.
122
3.5
Voltage Difference * Field Versus Voltage Squared
~ ::: ""-.-.D-:·-· ·-~-:~:-~-~-. _.--..""".t-....-....-__-._-..._-_.:-(--...-....-...-....-.....,.T-....-...-....-....-....-........--....-...-....-...-....
-:"j-._-_..-....-...-........,
...
~
>E
~3.4mT
a.15
_
_
_
: : . x
.....:
.,-"~
.~,.."r
:
"'C
~
<l
",'"
0.1 0
····1:··:,..·.<:·::············
i-.••...•••._.__•.:.•...+..............
.----.·.·.·······-··················.·.---·-::,·.~;;:~.7""::::::L.················ --············
:
r
'
o. 05 -t----i-----::i~ ~~----;..------;-____i Data taken 11/21/96
Substrate 19 Film 1
I
0.0
0.5
1.5
1.0
2.0
2.5
3.0
3.5
Voltage 2 (mV)2
Figure 6-16 Voltage change times magnetic field versus voltage
squared at various magnetic fields.
6.3.3 Voltage Versus Current Curves at Many Gaps
In the prior section it was demonstrated that flux flow voltage for a small gap is
less than the flux flow voltage for a large gap. The data presented here will show
that as the gap is changed, flux flow voltage changes in a regular and consistent
manner. As the beam force increases, the gap between the superconductor and the
normal conductor decreases, increasing the measured capacitance. Figure 6-17
shows a series of D. V versus I curves at a constant magnetic field. As the
capacitance change increases, the voltage change increases as well. The legend
indicates the capacitance increase D.C for each data set, relative to the zero beam
force capacitance Co. Using equation (III. 1), gap size is given by
h1
w.!s
= CoCo+D.C
(136)
where Ws and Is are the width and length respectively of the superconducting line
pattern under the beam. The zero force gap size is approximately 1Y2 to 2 microns.
123
Voltage Difference Vs Current at Various Gaps
U.14
0.12
0.10
>" 0.08
E
>
0.06
0.04
0.02
0.00
20
Current
(mA)
40
Current (mA)
Figure 6-17 Gap dependent voltage change versus current at various gaps.
6.3.4 Voltage Versus Gap at Constant Current and Field
To determine the dependence of the conductor proximity effect on gap size, the
capacitance and flux flow voltage were measured at constant current and field as
the beam force was varied. The data is shown in Figure 6-18. Here the force is
represented by the angle of the shaft on the probe head. As the angle increases, the
string length decreases, and the spring applies a larger force to the beam. The
capacitance measures the size of the gap, increasing as the gap decreases. The flux
flow voltage decreases as the gap decreases, as predicted by the conductor
proximity effect. To avoid false signals from slow time variations of stray voltage,
measurements were made during loading and unloading.
124
Capacitance and Flux Flow Voltage Versus Force
125
Data taken 8/14/96
Substrate 11 Film 1
120
1.54
1.52
1.50 '
E
1.48
-
LL
-
I AAM
C
o
~
--*--Capacitance
-
I =
1_I5
mA
- -
I. -rv %s
- 1.44>
I
t
-
1A
1.40
..
1.38 L.
1.36
1.34
105
100
-90
0
180
90
270
360
Angle (degrees)
Figure 6-18 Capacitance and flux flow voltage variation with beam force.
When the flux flow voltage is plotted against the capacitance directly, the
loading and unloading parts of the curve fall on top of each other, as expected.
Capacitance isconverted to a gap size by assuming a gap of 2 gm when the beam
force iszero. The flux flow voltage is plotted versus gap size inFigure 6-19. This
plot shows that the eddy current damping viscous coefficient increases
approximately linearly with decreasing gap. This dependence is consistent with the
finite penetration depth extension to the eddy current viscous coefficient
calculation.
The measured eddy current damping coefficient may be calculated from this data.
Use equation (118) and reasonable values for critical current (Ic = 4 mA) and the
non-linearity factor (x = 11). The measured damping coefficient is 9x10T" kg/ms
at closest approach. At the closest approach distance of 0.6 gm, the predicted
damping coefficient is 6x10 -' 2 kg/ms. Accounting for the finite penetration depth
in this material, the actual predicted damping is about one half of this value. The
measured damping therefore exceeds the predicted damping by a factor of thirty.
The agreement is not as good as was found in section 6.3.2.
Curves such as the one shown here were successfully repeated for this film
during three different runs over the course of two weeks of experimentation.
Qualitatively similar results were seen on a second film. Additionally, the
experiment was repeated with an uncoated beam. In the absence of the metal
conductor, no variation in voltage was seen as the beam force was varied.
125
r
Flux Flow Voltage vs Gap
rn
I.OZ
1.50
> 1.48
E
m 1.46
0 1.44
> 1.42
o
I- 1.40
x
= 1.38
LL
1.36
1.34
0.0
0.5
(microns)
Gap
2.0
Gap (microns)
Figure 6-19 Flux flow voltage variation as a function of normal
conductor to superconductor gap size.
6.3.5 Voltage Versus Capacitance Data at Various Magnetic Fields
Table 6-1 and Table 6-2 show the results compiled from flux flow voltage versus
capacitance data taken at different fields on different days (such as Figure 6-18).
During these runs, there were various degrees of success achieving close spacings
(high capacitances). Data was selected in which the change in capacitance achieved
is approximately equal, in order to allow for a comparison of the voltage change.
In drawing conclusions from this data, it is important to remember that data was
taken over several days under varying conditions. Detailed quantitative comparison
is probably not completely reliable. Qualitative comparisons of trends is possible,
however.
The tables contain the date of the experiment and the data file number. The
minimum capacitance corresponds to the capacitance circuit output signal when no
beam force is applied. A large part of this value is due to stray capacitance in the
system. The maximum capacitance corresponds to closest spacing achieved or
used in this comparison. When larger beam forces are applied, a non-zero
conductivity between the normal film and the superconductor would appear. This
indicates contact or shorting at some point on the film. The capacitance circuit is
highly sensitive to conductivity, giving substantial output signals for resistance less
than one megaohm. The table specifies flux flow voltages at the maximum and
minimum gaps. The constant applied currents and magnetic fields used during the
experiment are also listed.
126
The first observation to note is the order of magnitude difference in voltage
percentage change for the two films. Qualitatively, Substrate 17 Film 1 shows a
smaller voltage percentage change. Note, however, that the capacitance change
achieved for this film is about half what was achieved for the other film. Therefore
the minimum gap achieved for Substrate 17 Film 1 is much larger than the
minimum gap for Substrate 11 Film 1.It is therefore not surprising that the two
films have different voltage percentage changes.
It is interesting to examine the magnetic field dependence of the observed
voltage percentage change. For both films, as the magnetic field is increased, the
percentage change in voltage decreases. The dependence of the percentage change
in flux flow voltage versus the applied magnetic field in shown in Figure 6-20.
The magnetic field determines the fluxon lattice parameter. Larger magnetic
fields cause the fluxons, on average, to be spaced closer together. As shown in
Figure 3-3, the overlapping magnetic fields of closely spaced fluxons reduce the
magnetic field gradient. Therefore, as the magnetic field increases, the eddy current
damping coefficient is expected to decrease. The effect is best observed when the
fluxon lattice parameter is comparable to the film thicknesses and gap size. The
fluxon lattice parameter is given by equation (45). As the magnetic field varies
from 1.7 mT to 4.3 mT, the lattice parameter decreases from 1.16 pm to 0.73 gm.
Table 6-1 Flux Flow Voltage Change Observed For Various
Magnetic Fields - Substrate 11 Film 1
8-17-96 8-25-96 8-14-96 8-17-96 8-25-96
Date
22
21
14
20
22
File #
90
89
90
90
88
Capacitance(min) (pF)
111
104
107
109
Capacitance(max) (pF) 111
0.353
1.27
0.485
1.52
Voltage (at min C) (mV) 0.38
0.334
1.1
0.395
1.34
0.3
Voltage (at max C) (mV)
18
20
21
16
(pF) 21
Change in C
19%
12%
13%
5.4%
- Change in V
(%) 21%
Current
(mA)
Field
(mT)
34
2.6
37
1.7
127
16.5
3.4
21.5
3.4
20
4.3
Table 6-2 Flux Flow Voltage Change Observed For Various
Magnetic Fields - Substrate 17 Film 1
Date
10-2-96
10-2-96
10-2-96
10-2-96
File #
17
14
16
19
(pF)
95
95
Capacitance(min)
95
95
(pF) 105
Capacitance(max)
106
106
105
(mV) 0.501
Voltage (at min C)
0.562
0.553
0.382
(mV)
0.493
0.557
Voltage (at max C)
0.3805
0.5475
(pF)
10
11
Change in C
11
10
1.6%
0.9%
1.0%
- Change in V
0.4%
(%)
24
(mA)
17
Current
17
9
Field
1.7
(mT)
2.6
2.6
4.3
Change in Flux Flow Voltage vs Applied Field
Z370
20%
1.6%
0o
> 15%
m10%
o
1.2%
.. ............
- ----------...
-----------
0.8%
Substrate 11 Film 1
-0-% Substrate 17 Film 1
-I+-
C
5%0
0.4%
0%
0
0.0%
1
2
3
Magnetic Field (mT)
4
5
Figure 6-20 Change in flux flow voltage as a function of magnetic field.
128
7. Computer Model of Lattice Behavior
This chapter discusses a computer model of fluxon motion which the author
developed and implemented. The model examines circular flux flow in a disc
superconductor resulting from a radially applied current distribution. The purpose
of the model was to evaluate methods of simulating individual fluxon motion. The
results of the simulations proved to be quite interesting.
Consider a cylindrical superconducting disc in the mixed state in a magnetic field
parallel to its axis. While the magnetostatic energy, demagnetization factor, and
fluxon formation energy determine the density of fluxons penetrating the cylinder,
the distribution of the fluxons depends on their mutual repulsive interaction. Flux
flow as a radial current is applied to the disk has been studied experimentally." 89
Here, a computer model of this system is described. Each computer simulation
begins with a fixed number of rigid fluxons in specified initial positions. The
software then calculates the fluxon motion through an iterative process. It takes
into account fluxon-fluxon interaction forces, the Lorentz force due to the
externally applied radial current, viscous forces, and boundary conditions.
The presence of pinning centers affects the positions of one or more fluxons
locally. Other fluxons are affected indirectly through the fluxon-fluxon repulsive
force. Differences are to be expected between the behavior of a lattice based on an
attractive force, such as a typical solid, and one based on a repulsive force.
Fluxons are considered to have zero inertia for the purposes of this model. This
assumption is justified by the very short relaxation time of typical fluxons, which is
less than 10-12 seconds. 90 The model limits the maximum velocity to ensure
simulation stability. There has been no introduction of thermal motion into the
system, but the limited damping inherent in the model results in some randomness
in the stable lattice behavior, even under no external forces.
The software was designed and implemented entirely by the author in the C++
language. It runs in the Microsoft Windows 3.1 and above operating system
environment.
88 M. P. Shaw and P. R. Solomon, "Flux-Flow in a Superconducting Disk", PhysicalReview 164,
535-537 (1967).
89 J. B. McKinnon and A. C. Rose-Innes, "Measurements on Corbino Discs of Type-II
Superconductors", Physics Letters 26A, 92-93 (1967).
90 H. Suhl, "Inertial Mass of a Moving Fluxoid", Physical Review Letters 14, 226-229 (1965).
129
7.1 Model Parameters and Terminology
The geometry, as shown in Figure 7-1, is a radius R
section of a homogeneous disc. The N fluxons are
confined to approximately the region R. Each fluxon is
a singly quantized straight line of length h, and
magnetic flux 0o, parallel to the axis of the disc.
Each simulation consists of iterations or steps. For
(vertical)
e rh iteratinn the. velocitv of All thp. flnxnns it
calculated, a time step for that iteration is determined,
and positions of the fluxons are updated. A frame is
defined as a preset number of iterations and simulation
data is saved for each frame for later display and
analysis.
Figure 7-1 Geometry
of superconductor for
computer model.
7.1.1 Forces
Fluxon-fluxon interaction force is the mutual repulsion between fluxon pairs. The
interaction force is the gradient of the free energy. The interaction force is
F"" h,
e-PA
4AUL
o
2 7r
pi
(137)
p
for large fluxon separations. In the simulation an attempt is made to only consider
those pairs of fluxons with a significant interaction, to speed execution time.
The Lorentz force is a result of the interaction of the electric current I with the
magnetic field of the fluxon. The Lorentz force per unit length is given by equation
(50). For an applied current I, the total Lorentz force on each fluxon is
F6"",=
2Io
(138)
where pi is the radial distance of the fluxon from the disc center.
The viscous damping force per unit length is given by equation (51). The total
viscous damping is Fis " s = hfi"s s.
7.1.2 Boundary Conditions
The fluxons are confined to a finite region by one of a number of possible
r
boundary conditions. A power law boundary condition FiBc Pow"
is an inward radial
force proportional to a power of the fluxon's distance outside the disc. Fluxons
will tend to expand outside the disc, but remain confined to a local region. In an
~g' ,each fluxon is repelled by a single image fluxon of the
Bc
image boundary Fi
same orientation on the opposite side of the boundary. In a lattice boundary
condition FiBc-ai• " , a triangular lattice of immobile fluxons several lattice spacings
130
thick is located just outside of disc and the repulsion by these fluxons keeps the
fluxons in the disc confined.
F
=Ic-••a
h 0
F4CPow
(2(R -p,))
- 2e 2-
(pr R
-h
F4rcA-owr
),
)
~A
(139)
(140)
7.1.3 Statistical Analysis
Various statistics of the fluxon positions and velocities can be calculated in order
to best characterize different aspects of the fluxon behavior.
The degree of lattice order can be measured by examining its radial distribution
function. To calculate the radial distribution function, the distance r between each
pair of fluxons is measured. A box size dr = 2R / N is chosen, where R is the radius
of the disc and N is the number of fluxons. The radial distribution function (RDF)
used here s(r)dr is the number of pairs with separation between r and r + dr,
normalized to be independent of the size of dr. For a random distribution of
fluxons in a circular disc of radius R, the radial distribution function looks like
Figure 7-2. The shape is a result of the confined geometry of the disc, so this shape
should be kept in mind as other RDF curves are examined. Short range order
would be indicated by the presence of peaks and valleys for small separations,
while long range order would be indicated by the presence of peaks and valleys for
larger separations.
Mean Radial Distribution Function: Complete Disorder
Separation Distance from 0 to 2R
Figure 7-2 Mean radial distribution function for complete
disorder in a disc of radius R.
Since the various boundary conditions allow the fluxon lattice to compress and
expand, it is useful to know the actual area filled by the lattice. This information
can be used to calculate the optimum lattice spacing I of a perfect triangular lattice
131
filling this area. The area is determined using the radius of the fluxon farthest from
the center of the disc in any given fluxon distribution. For a simulation involving
multiple frames of distributions, the maximum from each frame is taken and these
maximums are averaged. This statistic is called the radial magnitude versus frame.
The radial voltage due to the dissipation by fluxon motion is proportional to the
average tangential velocity of the fluxons. The voltage-current curve is an
important property of a superconducting system and one which is frequently
measured. The average, maximum, and minimum tangential fluxon velocities in any
given frame can be plotted for each frame in a run. This statistic is called the
tangential velocity versus frame (VFT).
7.2 Predictions and Results: No External Forces, No Pinning
Theory predicts and imaging experiments 12.13 show that fluxons will form a
triangular lattice. The first test of the simulation software was to put a number of
fluxons randomly into a confined area and allow them to move only under their
mutual repulsion. The resulting stable configuration, a multi-grain triangular
lattice, is shown in Figure 7-3.
Figure 7-3 Triangular lattice resulting after 5000 iterations
from random initial fluxon positions. Distinct boundaries can
be seen between the different grains that have formed.
The radial distribution function is shown in Figure 7-4. The width of the
separation axis is 2R. The minor tick marks are one lattice spacing apart. Lattice
order extends to about one radial distance or about fifteen lattice spacings. The
peak locations and relative heights correspond to a triangular lattice. This can be
seen by comparing this figure with Table 1-1.
132
Radial Distribution Function
Separation (m)
I--
No current
I
Figure 7-4 Radial distribution function for the fluxon lattice
shown in Figure 7-3.
7.3 Predictions: ExternallyApplied Radial Current, No Pinning
A radial electric current causes a tangential Lorentz force. The current I and
thickness of the superconductor h, are constant Local current density and
therefore the Lorentz force vary inversely with the distance from the center of the
disc.
If the lattice were completely rigid, all fluxons would rotate together at the same
angular velocity. The fluxons at the outer edge of the disc would have to move at a
much larger linear speed than those near the center of the disc. The fluxons that
would have to move fastest are those with the smallest Lorentz force applied to
them. The lattice would have to support very large shear stresses to maintain this
type of motion.
The fluxon lattice is much stronger in tension and
compression than in shear.23 9 For low current
densities, then, the lattice will tend to slide along the
easiest shear line that is most parallel to the local
direction of the Lorentz force. This will allow the
fluxons near the center of the disc to move at a larger
angular velocity than those at the edge. Since the lattice
has six-fold symmetry, the resulting motion will be in a
hexagonal pattern as shown in Figure 7-5.
I
#.TY
dl
I
Figure 7-5
Hexagonal fluxon
flow pattern due to
lattice shear.
91 R. Labusch, "Elastische Konstanten des FluBfadengitters in Supraleitern zweiter Art", Physica
Status Solid 19, 715-719 (1967).
133
For high current densities, the lattice isunable to maintain the cohesion of the
local lattice structure against the tendency of the Lorentz force to produce circular
motion. Long rangre lattice order disappears. The
motion will hp in qcircula;r pa;ttern of the- tynpe
n-eitluser
'H
shown in Figure 7-6.
For any given applied current, there is a high current
density very close to the center of the disc and low
current density far from the center. Both of the above
types of motion should occur therefore, depending on
the relative strengths of the local Lorentz and lattice
forces.
I
1
Figure 7-6 Circular
flux flow pattern for
large currents.
Take the fluxon pair interaction at one lattice spacing,
equation (137) with pij equal to 1,as a measure of the
local lattice strength and compare it to the local Lorentz force, equation (138).
Setting magnitudes of these two forces equal gives an equation approximating the
radial distance pcni, where the transition from low to high current density will
occur.
h, 0
2r pctl
4p 1 Vi2cla
e-1'
(141)
A more detailed calculation would use the actual lattice stress and strain
coefficients. This simple calculation gives rather good results, however.
7.4 Results: Externally Applied Radial Current, No Pinning
Below the results of the most complete series of basic simulations are presented.
In these simulations, the number of fluxons is fixed at N = 750. The penetration
depth is taken to be X = 100 nm, the thickness of the disc is hs = 100 gjm, and the
radius considered is R = 10 jim. The boundary condition allows the fluxon lattice
to spread past the 10 gjm outer radius by about 15%. All simulation parameters
were kept constant except for the magnitude of the applied current, which was
varied on a logarithmic scale from 0.0001 A to 0.75 A. For the above number of
fluxons and effective radius, the equivalent magnetic flux is 3.6x10 "3 Tesla.
The fluxons were initially distributed in a jumbled triangular lattice of radius
11.5 gjm, with a lattice parameter I of about 800 nm. By jumbled lattice it is meant
that fluxons were arranged in a perfect triangular lattice and then each fluxon was
randomly moved by up to 21% of the lattice spacing in a random direction. The
lattice was then allowed to settle for 900 iterations with only the mutual fluxon
interaction considered and no current applied. This was then taken at the starting
fluxon distribution for 26 runs.
134
1
Each run consists of thirty frames, with fifteen iterations per frame, for a total of
450 iterations per run. For each frame, the two dimensional vectors for all fluxon
positions and velocities at the end of the fifteen iterations of that frame were saved
and are used in the following analysis.
7.4.1 Visual Character
Table 7-1 shows the run number, the applied radial current, the lattice spacing I
calculated from RFM data as described in Section 7.1.3, and the predicted critical
distance p,,inai as given by equation (141). When the predicted critical distance is
much less than the 11.5 gm effective radius hexagonal behavior is expected, but
when the critical distance exceeds 11.5 gm circular motion is expected.
The table shows a visual evaluation of the character of the fluxon motion. The
visual evaluation is made by overlaying the fluxon positions from all thirty frames
for a particular run and examining the appearance of the resulting figure. An
example for each type of observed motion is shown in Figure 7-7, where the circle
indicates the base radius R of the disc and the fluxons appear both inside and
outside this radius. For very low currents, as shown in Figure 7-7a there is
insufficient fluxon motion over the thirty frames to evaluate the character of the
motion. Each fluxon has moved only a fraction of a lattice spacing over the course
of the entire simulation run. For low currents, as shown in Figure 7-7b, a
hexagonal pattern of motion is revealed by a general hexagonal shape and by
six-fold symmetry in the spiral arms. The hexagon is actually tilted slightly from
the original lattice configuration revealing an added complexity to the motion. For
intermediate currents, as shown in Figure 7-7c, the lines of motion appear to be
circular, but the six-fold symmetry in the spiral arms is still visible. This state is
considered mixed hexagonal-circular. For high currents, as shown in Figure 7-7d, a
circular ring pattern has clearly developed and the six-fold symmetry has
disappeared.
The theoretical critical radius corresponds well with the visually observed
transition. The motion visually appears hexagonal for predicted critical distances
less than one third of the disc size. For 4.6 rmn< pt, < 7.7 gm, the motion
appears mixed. Part of the motion is circular and part is hexagonal. No sharp
transition is seen, since this is not a sharp phase change. The fluxon motion
changes character slowly. For predicted critical distances greater than the radius of
the simulated disc, the motion appears circular.
135
Run
Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
Run 7
Run 8
Run 9
Run 10
Run 11
Run 12
Run 13
Run 14
Run 15
Run 16
Run 17
Run 18
Run 19
Run 20
Run 21
Run 22
Run 23
Run 24
Run 25
Run 26
Table 7-1 Parameters Of Computer Simulation Runs
Visual
Current (A)
Lattice (pm)
Critical Distance (,urnm)
0.0001
0.0003
0.0010
0.0013
0.0018
0.0024
0.0032
0.0042
0.0056
0.0075
0.01 00
0.0130
0.0180
0.0240
0.0320
0.0420
0.0560
0.0750
0.1000
0.1300
0.1800
0.2400
0.3200
0.4200
0.5600
0.7500
0.80
0.80
8.87E-03
2.84E-02
0.80
8.87E-02
0.80
0.12
0.16
0.21
0.28
0.37
0.50
0.62
0.89
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.79
0.80
0.79
0.79
0.79
0.79
0.80
0.79
1.2
1.6
2.0
2.6
3.5
4.6
6.2
7.7
10.7
13.8
18.4
24.5
32.2
46.1
57.4
136
-see text-see text-see text-see text-see text-see text-see texthex
hex
hex
hex
hex
hex
hex
hex
hex
mixed
mixed
mixed
circle
circle
circle
circle
circle
circle
circle
(d)
(c)
Figure 7-7 Overlays of 30 frames of fluxon positions in each of
runs 6 (7-7a), 12 (7-7b), 18 (7-7c), and 24 (7-7d).
7.4.2 Voltage-Current Relation
The radial voltage produced by fluxon motion is proportional to the average
fluxon tangential velocity. Fluxon tangential velocity is a result of the balance
between total Lorentz force and total viscous force, since fluxon-fluxon interaction
forces produce no net tangential force on the fluxons. Lorentz force on each
fluxon depends on the radial position of the fluxon. Total Lorentz force therefore
depends only on the radial distribution of the fluxons. Total Lorentz force is not
affected by the transition.
137
Average Tangential Velocity versus Radial Current
I U..- i
le+1
le+O
Oo
Z
le-1
le-2
1e-3
1e-4
1 e-3
1e-2
le-1
1e+O
Current (A)
Figure 7-8 Flux Flow voltage vs current relation on log-log scale.
The total viscous force is simply the viscous coefficient multiplied by the number
of fluxons times the average fluxon velocity. The number of fluxons and the
viscous coefficient are constant. Even if the fluxon velocity distribution changes at
the transition, the average velocity, and therefore the voltage, are not affected by
the transition.
The VFT statistic is calculated and the resulting velocity versus current relation
is given in Figure 7-8. From the calculations above and Table 7-1 a transition, if it
exists, should appear around I = 0.1Amp. No change is seen in the V-I curve,
agreeing with the prediction.
7.4.3 Lattice Order
The lattice order can be measured in terms of the radial distribution function
described in Section 7.1.3. The RDF is plotted for various applied current levels in
Figure 7-9. For a perfect triangular lattice, the first few RDF peaks should be given
by Table 1-1. The grid lines on the separation axis are one lattice spacing apart.
For run 3, the lowest current level, long and short range order can be seen in the
detailed structure of the curve. This order can be seen to decrease in magnitude
and the peaks can be seen to merge together as the applied current is successively
increased in runs 7, 11, 15, 19, and 23.
138
Radial Distribution Function
Figure
Separation
Figure 7-9 Radial distribution functions showing decreasing
order range and intensity for increasing applied current.
139
8. Conclusions
8.1 Flux Flow in Indium-Bismuth Films
Flux flow voltage V was measured as a function of magnetic field H and current
density J in as-deposited Ino 95Bio o5 films. The observed voltage was a function of
the total Lorentz driving force BJ (Figure 6-5 and Figure 6-6). For high magnetic
fields and low current densities, the measured voltage is slightly larger than the
expected voltage (section 6.1.3).
This flux flow voltage is not consistent with the standard linear model of rigid
fluxon lattice translation. To account for the observed behavior, lattice shear or
dislocation motion must considered. The observed flux flow voltage is explained
by assuming some fluxons move with a velocity linear in applied current. The
remaining fluxons are stationary. The fraction of moving fluxons is graphed versus
Lorentz force induced lattice strain (Figure 6-10). The fraction of mobile fluxons
increases dramatically at approximately 5% strain. Fluxon lattice shear is therefore
a function of lattice strain.
8.2 Conductor Proximity Effect
Fluxon motion produces eddy currents in a nearby normal metal film. This
damping mechanism causes drag on the fluxons. The flux flow voltage, which is
proportional to average fluxon velocity, is therefore reduced.
The damping coefficient is only appreciable when the normal metal is very close
to the superconductor. For separation gaps larger than the fluxon spacing,
magnetic fields of adjacent fluxons overlap (Figure 3-3). As the magnetic field at
the normal conductor becomes more uniform, eddy currents become negligible.
The conductor proximity effect therefore only occurs at low magnetic fields. It is
expected to decrease with increasing field and increasing gap, especially when the
fluxon spacing is comparable to the gap size.
Detailed calculation of the eddy current damping coefficient leddy for individual
fluxons was presented in chapter 3. For the geometry and materials of interest
here, the damping coefficient is approximately inversely linear in separation gap
(section 3.5.2).
Experimental measurements observe a flux flow voltage change when the gap
size between the normal metal and the superconductor is varied. As predicted, flux
flow voltage decreases when the normal metal is brought closer to the
superconductor (Figure 6-18). This voltage change is inversely linear in gap size
(Figure 6-19), consistent with theoretical predictions. As expected, the fractional
voltage change decreases as magnetic field increases (Figure 6-20).
140
Determination of the magnitude of the expected eddy current damping
coefficient is limited by uncertainty in the normal metal to superconductor gap
size. The gap size uncertainty is limited by stray capacitance in the position sensing
circuit as well as substrate and beam curvature. Qualitative comparison between
theory and experiment depends on the data chosen and the assumed fluxon motion
model. Using plots of AV-B versus V and assuming a fraction of the fluxons move
with a velocity non-linear in applied Lorentz force, agreement is very good
(section 6.3.2). Using plots of voltage versus gap gives agreement within a factor
of thirty (section 6.3.4). In each case, the measured damping coefficient is larger
than the predicted value.
There are several factors which may account for the larger than expected
magnitude of 1eday. First, fluxon bundling would cause an increase in e,ddy
proportional to the number of fluxons per bundle (section 3.6).
Second, long range fluxon interaction at the film surface could significantly
increase the fluxon dipole moment mi. Since film thickness is on the order of X(T),
perhaps the entire film should be treated as a surface region. Considering that
,eddyc mn2, this correction factor could be rather large.
Third, non-uniformities in the superconductor or normal metal film surfaces
would enable some regions of the two films to be much closer than the average
spacing. At spacings much smaller than the film thickness and fluxon lattice
parameter, the eddy current damping coefficient has a much sharper distance
dependence than capacitance. Therefore, if the two films were very close over a
small area, overall capacitance would not increase significantly, but eddy current
damping would increase.
A smaller than predicted value may result from the overlap of the magnetic fields
of adjacent fluxons. A decrease in eddy current damping is expected as the
magnetic field uniformity increases (Figure 3-3). This effect has been discussed
qualitatively. It had not been included quantitatively in the theoretical model,
however, which considers only isolated fluxons.
8.3 Computer Model of Lattice Flow
To study lattice shear under flux flow, the author developed a computer model.
Fluxon motion was simulated under the influence of the Lorentz driving force, a
linear viscous damping force, and the fluxon interaction forces. The geometry
considered was a circular disk subjected to a radial applied current. Since fluxons
move tangentially, creation and annihilation of fluxons did not need to be
considered.
The character of fluxon motion in a circular disc subject to radial current changes
with applied current magnitude. For low current densities, motion is hexagonal
with the fluxon lattice shearing along the local easy direction most aligned with the
141
tangential direction. Long range and short range lattice order is maintained. For
high current densities, motion is circular. Only short range lattice order survives.
For a given total current, there is a high current density for small radii and a low
current density for large radii. Therefore, the character of fluxon motion varies
with radius. The smooth transition from circular to hexagonal motion can be
determined by comparing the fluxon nearest neighbor interaction force with the
local Lorentz force. A basic theoretical approach yields equation (141), which
describes this transition point with reasonable accuracy.
In the absence of pinning, this transition does not affect the basic voltage vs
current characteristics of the material to first order. The transition is therefore not
seen in simple electrical measurements. Imaging technology should permit visual
observation of this change in character of the motion.
8.4 Recommendations for Future Work
Theoretical prediction of the conductor proximity effect is presently limited by
two factors. The first is computing power. Since there are several characteristic
lengths which are all the same order, a detailed analysis will require numerical
integration to determine magnetic fields, electrical fields, and power losses. A
more detailed numerical calculation would account for the actual current
distribution around a fluxon and for the interactions of neighboring fluxons. All of
the formulas used to date have been solvable in closed form, however, which
vastly simplifies numerical calculations.
The second theoretical aspect is the proper determination of the fluxon
supercurrent distribution. The real current distribution in a superconducting film
lies somewhere between the bulk specimen results and the pure surface results.
Thus the true current distribution would optimally be calculated directly from
Ginzburg-Landau theory, under appropriate boundary conditions.
There are two primary ways in which the experimental work can be enhanced to
improve measurement of the conductor proximity effect and extract more
information about fluxon motion. First, an improved position sensor or controller
would allow more accurate measurement of the gap dependence of the effect
Second, a different superconducting material which conforms better to the
standard Kim linear flux flow model would eliminate some unknowns in flux flow
behavior.
The position sensor could be improved by providing a means of eliminating or
more accurately measuring stray capacitance. If stray capacitance were known,
then gap size could be measured much more accurately. Alternately, the beam
could be moved by means of a piezoelectric device, rather than by mechanical
means, as was done here. Then the applied force could be calibrated and ac
techniques at a variety of frequencies could be used. The piezoelectric would
142
introduce an electric field near the superconductor. However, the probe could be
designed to maximize the distance between the piezo and the superconductor,
thereby minimizing the impact of this electric field.
The choice of a superconducting material is not simple. The conductor proximity
effect measurement is best performed at a constant temperature (unless an ac
measurement could be performed with a shorter time constant than the
temperature change). Thus it is preferable to operate in a liquid helium bath for
low temperature superconductors or a liquid nitrogen bath for high temperature
superconductors. The optimum material would be superconducting above 4.2 K,
have a relatively low critical current density, and have a linear voltage-current
relationship over a wide range of applied fields and currents. In-Bi alloys are good
choices, but the composition could be chosen to avoid precipitate formation at
room temperature. An anneal and quench procedure could also be employed to
increase the bismuth solubility in indium.
It would be interesting to perform this measurement on high transition
temperature oxide superconductors. The much higher operating temperature
would simplify mechanical design. Piezoelectrics have better operating
characteristics at higher temperatures. Thermal contraction and substrate warping
become less significant issues.
143
Appendix I:Constants and Major Symbols
Vectors are given in bold face type (Roman) or with vector symbols (Greek). Hats
are used to indicate unit vectors. Scalars are given in italics (Roman).
Constants
e
h
h
kB
m
So
Lto
0o
1.6x10" 9 C
6.626x10-34 J.s
1.05x10-34 J.s
1.38x102"3 J/K
9.1x10-3'kg
8.854x10-'2 C2/N.m 2
4xx107" Ns2/C2
2x10 "'• T-m2
electron charge
Planck's constant
reduced Planck's constant h / 2nr
Boltzman's constant
electron mass
permittivity of free space
permeability of free space
flux quantum
Symbols
Coordinates
spherical coordinate position vector
r
x, y, i rectangular coordinates unit vectors
two-dimensional delta function in x and y
8XY
b,,e2
cylindrical coordinates unit vectors
Experimental Parameters
ho
thickness of normal conductor
hbwb,lb thickness, width, and length of beam
h,,w,,l, thickness, width, and length of superconducting line pattern under beam
hi
gap thickness between normal conductor and superconductor
s
separation distance between fluxon magnetic dipole and normal conductor
Young's modulus of beam
Y
a
conductivity of normal metal film
General Symbols
B
magnetic induction
E
electric field
total force on a fluxon
F
f
force per unit fluxon length
free energy
f
H
magnetic field
I
applied current
applied current density
J
V
flux flow voltage
v
fluxon velocity
o0 singly quantized flux line with a given orientation
144
Superconductor Properties
Hc
thermodynamic critical magnetic field
Hce
lower critical magnetic field
Hc2
upper critical magnetic field
critical current density
Jc
I
fluxon lattice spacing
mean free path of electrons
1MP
superconducting electron density
n,
N(O) zero temperature density of states at Fermi surface
Tc
superconducting critical transition temperature
average electron Fermi velocity
VF
Ao
energy gap
total viscous damping coefficient
T"
intrinsic viscous damping coefficient
rio
normal conductor eddy current damping viscous coefficient
leddy
K
London kappa, K X(T) / 4(T)
X(T) temperature dependent penetration depth
London penetration depth XL = X(O)
XL
(T7) temperature dependent coherence length
Pippard coherence length
o0
position vector of flux line in xy plane
ij
displacement vector between two flux lines in xy plane
normal conductivity of superconductor
Ginzburg-Landau complex order parameter
145
Appendix II: Bessel Functions and Related Functions
The Bessel functions and related functions used below and throughout this work
follow the conventions of Gradshtein and Ryzhik.92 Order v Bessel functions of the
first and second kinds of order are written as J,(z) and Nv(z), respectively, where z
is a real number.
The Bessel functions of the third kind are also called Hankel functions H,()(z)
and are defined in terms of Bessel functions of the first and second kinds.
H(1 (z)= J (z)+iN(z)
(II.1)
Hc ) (z) = J (z) - i N, (z)
(1.2)
The most relevant Bessel functions here are the Hankel functions of imaginary
argument Kv(z), of order v and real z.
(II.3)
K, (z) = ~2 ie 'n H t(it) = nrie- v"x' H ' (iz)
Some useful Bessel function identities are 93
Z(z) = (-1)V ZV(z)
K_v (z) = Kv (z)
(114)
where the generic term Z, is used to mean any of J,, N,, Hv,( or HV,
The limits of the Hankel functions are
lim Ko (z) -4 0.12 -In z
),
but not K,.
0(1.5)
lim K, (z) - 1/z
for small z and
lim KV (z)
"
(II.6)
for large z. The derivatives of Bessel functions are also Bessel functions.
d Ko(z)
dz - K,(z)
dz
One useful integral involving Bessel functions is"
u`Kv(au)du = 2
i'a-'1'
2l
(II.7)
)1
+
(11.8)
0
for Re(L.+l+v) > 0 and Re(a) > 0.
92 I. S. Gradshtein and I. M. Ryzhik, Table of Integrals,Series, and Products: Correctedand
EnlargedEdition, prepared by Alan Jeffrey, (Academic Press, Inc., New York, 1980).
93 Ibid., page 952 equation 8.407 and page 968 equation 8.472.
94
Ibid., page 684, equation 6.561 #16.
146
Appendix II: Capacitance Position Sensing Circuit
Capacitive distance sensing is used to measure the spacing between the
superconductor on a flat substrate and the normal conductor on a bent beam. The
capacitance between any two conductors is a function of their geometry.
Approximate the normal conductor and superconductor combination as a parallel
plate capacitor. The area of the superconducting line pattern under the beam is
used for the area of the capacitor plates A. Measurement of the capacitance C
determines the separation gap hi between the plates.
C=
A
hi
(.1)
Figure HI-1, Figure II1-2, and Figure HI-3 show the electrical circuit used for
capacitance sensing. The "capacitor" is formed by connecting the beam film to
CAP 1 and the substrate film to CAP 0. A high frequency signal (typically 10
kilohertz, 20 millivolt rms) is applied to CAP 0, causing an ac charging current on
CAP 1. An operational amplifier converts this ac current to a corresponding ac
voltage. Active rectification, amplification, and low-pass filtering produce a dc
voltage proportional to the capacitance between CAP 0 and CAP 1.
The input buffer takes an ac signal from a Phillips 5192 signal generator and
generates a corresponding ac output signal. The input is ac coupled so that the
output dc level remains close to signal ground. For low current levels, the Phillips
5192 tends to have an undesired dc offset. The CAP 0 output charges one side of
the capacitor.
The capacitance measurement takes the form of a current-to-voltage converter.
Thus the charging current of the capacitor is measured directly and converted to a
signal voltage level for further processing. A passive notch filter (not shown in a
figure) is used to eliminate an external interfering 35 kilohertz noise signal. The
signal is then buffered for output. The LED lights when the two capacitor plates
are shorted. This is a useful indicator of the beam position. The ac output is
measured directly with a Stanford SR530 lock-in amplifier at 10 kilohertz. An
oscilloscope is used to check the ac output signal for noise interference. Once all
the phase offsets are taken into account, the real part of the ac output corresponds
to conductance while the imaginary part corresponds to capacitance.
147
The ac-to-dc converter consists of a rectifier and a low pass filter. The dc output
is used when low frequency variations in the capacitance need to be measured. For
example, to measure a 10 hertz variation of superconductor to normal conductor
spacing, the dc output is connected to a lock-in amplifier triggered to 10 hertz.
The capacitance circuit is battery powered to eliminate line noise interference.
Furthermore, capacitors are used to prevent noise spikes on the power rails.
Input Buffer
CAP 0
Output
5
V. (excite)
in
I
.
.. .. .. .. ... .... ....
.
.° .
.
° .. .
.
.
.
.
.
.
.
.
.
.
.
.
.
°
°
°
CAP 1
Input
Vout(ac)
LED
Figure M-1 Capacitance distance sensing circuit with ac
outputs.
148
AC to DC Converte
10 kM 1%
V
(d
I!
------------------:---------------------------------------
Power Supply
9 volts
9 volts
--- l---It---
100 par
Electrolytic Electrolytic
10oo0
LF
0.03 IF
.1.
ss
0.03 LF
-V
u
ss
Figure III-3 Capacitance circuit power supply.
The capacitance circuit was calibrated by measuring the ac and dc output signals
when four known capacitors were attached to the CAP 0 and CAP 1 terminals.
Calibration capacitors ranged from about 40 pF to 225 pF. The resulting
calibration was
C(pF) = 28 pF*- (
C(pF) = 59.7 pF * (V.e
R(Q) =0.55 Mf *
+ 1.9 pF
)
+ 6.4 pF
(V1•d)I
for operation with a 20 mV,, 20 kHz input voltage excitation.
150
(111.2)
Appendix IV: Motor Timing Circuit
A Hurst brand model AB synchronous motor is used for ac conductor proximity
effect measurements. The motor applies an ac force to the back of the beam, as
discussed in section 4.2.2. This appendix describes the motor timing circuitry.
The motor rotates 600 per line current cycle for a speed of 600 revolutions per
minute or 10 Hz. A lock-in amplifier reference signal at this frequency is required
to make ac measurements corresponding to the motor speed. This reference signal
is produced by dividing the line current frequency by six and providing the lock-in
amplifier with a 10 Hz digital logic signal. The battery operated circuit shown in
Figure IV-1 takes 60 Hz line power as input and generates the required lock-in
reference 10 Hz output. The motor is driven from the same input source.
The line voltage step down rectifier converts the 120 Volt rms line voltage to a
signal range of zero to 72 Volts. After the switching and safety circuit, a 10:1
transformer provides voltage reduction and ground isolation. A single diode
performs half-wave rectification. A simple voltage divider is used to generate the
desired output level.
A CMOS Schmitt trigger inverter performs the analog to digital conversion. The
Schmitt trigger prevents multiple clock signals from being produced by noise on
the ac line. Due to the high input impedance and fast speed of the CMOS, a
parasitic capacitor is used to stabilize the output
The CMOS digital logic consists of a frequency divide-by-three circuit followed
by a frequency divide-by-two circuit. Both circuits utilize the properties of JK flipflops. The JK flip-flop is a simple memory device. The output Q. changes during a
clock pulse (HI) as a function of the inputs J and K during the clock pulse and the
previous state of the device Q,t. The specific logic is given in Table IV-1.
151
The divide-by-three circuit is achieved by having three logic states among two
flip-flops. The clock pulses generate transitions from each state to the next. The
detailed inputs and outputs in each state are presented in Table IV-2. The output
Q2 is used as the clock pulse for the divide-by-two circuit. This circuit's flip-flop
alternates between two states. The logic table is given in Table IV-3. The output
Q3 is used to drive the lock-in amplifier.
Table IV-1 JK Flip-Flop Logic Table
Qo
J
K
O (LOW)
1 (HI)
0 (LOW)
0 (LOW)
0 (LOW)
1 (HI)
Q_~- (previous)
1 (HI)
0 (LOW)
1 (HI)
1 (HI)
not Q,_-1 (toggle)
Table IV-2 State Table For Frequency Divide-By-Three Circuit
State # Q,
1
0
2
0
3
1
1
0
Q2
J1
0
1
0
0
J2
K,
0
1
0
0
0
0
1
0
1
1
0
1
K2
0
1
0
0
Table IV-3 State Table For Frequency Divide-By-Two Circuit
State #
1
2
1
Q3
0
1
0
J3
1
1
1
153
K3
1
1
1