Vortex Properties from Resistive Transport Measurements
on Extreme Type-II Superconductors
Andreas Rydh
Stockholm 2001
Doctoral Dissertation
Solid State Physics
Dept. of Physics & Dept. of Electronics
Royal Institute of Technology (KTH)
Typeset in LATEX.
BIBTEX and the natbib package were used for the bibliography.
Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till
offentlig granskning för avläggande av teknisk doktorsexamen fredagen den 25 januari
2002 kl 10.00 i Kollegiesalen, Administrationsbyggnaden, Kungl Tekniska Högskolan,
Valhallavägen 79, Stockholm.
Thesis for the degree of Ph. D. in the subject area of Condensed Matter Physics.
TRITA-FYS-5275
ISSN 0280-316X
ISRN KTH/FYS/FTF/R--01/5275--SE
ISBN 91-7283-175-8
Copyright © Andreas Rydh, 2001
Printed in Sweden by Universitetsservice US AB, Stockholm
Vortex Properties from Resistive Transport Measurements on Extreme Type-II Superconductors
Andreas Rydh, Solid State Physics, Royal Institute of Technology (KTH), Stockholm, Sweden
Abstract
The nature of vortices in extreme type-II superconductors has been an ever growing field of
research all since the discovery of the first high- materials in 1986. This thesis focuses
on the study of vortex phase transitions and vortex-liquid properties through resistive transport measurements on single crystals of mainly YBa Cu OÆ (YBCO). Some important
results are the following:
(i) Location of the vortex glass transition. A study of the second-order vortex glass transition in the magnetic field–temperature ( – ) phase diagram as a function of anisotropy
resulted in a simple description of the glass line, over an extended region. Here is the critical temperature, is a parameter, and
. No evidence was found for any 3D–2D transition in the studied range of field and
anisotropy ( T, ).
(ii) Description of the vortex-liquid resistivity in glassy systems. By introducing a modified vortex-glass correlation length , depending on a mean local pinning energy
and the thermal energy , an explicit description of the vortex-liquid resistivity close to the second-order phase transition was obtained both for varying and .
(iii) Driven vortex dynamics in clean systems. The vortex-solid dynamics below the firstorder melting transition was found to be divided into a flux-creep region with high
critical current density at low temperatures, and an almost Ohmic flux-flow regime with
low .
(iv) Vortex-liquid correlation and multiterminal measurements. Using various multiterminal setups, the vortex liquid in YBCO just above was shown to be rather strongly
correlated along the vortex line also in untwinned crystals. The longitudinal and transversal
resistivities and their mutual relation were investigated.
(v) In-plane anisotropy of YBCO. A new multiterminal configuration allowed measurements of the in-plane anisotropy as a function of temperature and magnetic field direction.
The measurements revealed a possible presence of nearly isotropic fluctuations, corre . This value is clearly lower than that
sponding to a fluctuation anisotropy obtained from the vortex phase diagram or from resistive measurements, .
The thesis starts with a short overview, followed by an introduction to vortices in high-
and subsequent discussions of various vortex properties. The published papers are shortly
summarized and are finally appended.
Keywords: superconductivity, vortex, phase transition, YBa Cu OÆ , (K,Ba)BiO , vortex dynamics, critical
current, single crystal, oxygen deficiency, anisotropy, vortex glass, vortex lattice, melting, vortex liquid, vortex
correlation, disorder, Lorentz force.
ISBN 91-7283-175-8
TRITA-FYS-5275 ISSN 0280-316X ISRN KTH/FYS/FTF/R--01/5275--SE
iii
Vortexegenskaper från resistiva transportmätningar på extrema typ-II-supraledare
Andreas Rydh, Fasta Tillståndets Fysik, Kungliga Tekniska Högskolan, Stockholm
Sammanfattning
Vortexars egenskaper i extrema typ-II-supraledare har varit ett ständigt växande forskningsområde alltsedan upptäckten av de första hög- materialen år 1986. Denna avhandling är fokuserad på studiet av fasomvandlingar i vortexsystemet och på vortexvätskans egenskaper. Den experimentella metod som använts är resistiva transportmätningar
på enkristaller av huvudsakligen YBa Cu OÆ (YBCO). Några viktiga resultat beskrivs
kortfattat:
(i) Läget hos vortexglasövergången. En studie av den andra ordnings fasomvandling
som här tolkas som en vortexglasövergång i fasdiagrammet över magnetfält mot temperatur
( – ), ledde till en enkel beskrivning av glaslinjen, ,
över ett utökat område. Här är anisotropin, den kritiska temperaturen, en
parameter och
. Inga tecken på någon 3D–2D övergång kunde observeras i det
studerade området av fält och anisotropi ( T, ).
(ii) Beskrivning av vortexvätskans resistivitet i glasliknande system. Genom att introducera en modifierad vortexglas-korrelationslängd , bestämd av ett medelvärde av
den lokala pinningenergin och den termiska energin , så erhölls en explicit
beskrivning av vortexvätskans resistivitet både för varierande och i närheten av andra
ordningens fasomvandling.
(iii) Vortexdynamik i rena system vid hög ström. En fasomvandling av första ordningen
inträffar i vortexsystemet vid en smälttemperatur . Dynamiken hos den fasta vortexfas som uppträder under denna fasomvandling befanns vara uppdelad i ett fluxkrypområde
med hög kritisk strömtäthet vid låga temperaturer och ett nästan Ohmskt vortexflödesområde med lågt .
(iv) Vortexvätskekorrelation och multikontaktmätningar. Genom att använda olika konfigurationer med många kontakter så visades att vortexvätskan i YBCO precis ovanför är ganska starkt korrelerad längs med vortexlinjen även i otvinnade kristaller. De longitudinella och transversella resistiviteterna och deras ömsesidiga relation undersöktes.
(v) Anisotropin i planet hos YBCO. En ny kontaktkonfiguration möjliggjorde multikontaktmätningar av anisotropin i planet som funktion av temperatur och magnetfältsriktning.
Mätresultaten indikerade en möjlig närvaro av nästan isotropa fluktuationer, motsvarande
. Detta värde är klart lägre än det som erhålls från
en fluktuationsanisotropi vortexfasdiagrammet eller från resistiva mätningar, .
Avhandlingen inleds med en kort översikt, följd av en introduktion till vortexar i hög och en efterföljande diskussion av olika vortexegenskaper. De publicerade artiklarna
sammanfattas kort och är slutligen bifogade.
Nyckelord: supraledning, vortex, fasomvandling, YBa Cu OÆ , (K,Ba)BiO , vortexdynamik, kritisk strömtäthet, enkristall, syredopning, anisotropi, vortexglas, vortexgitter, smältning, vortexvätska, vortexkorrelation,
oordning, Lorentzkraft.
ISBN 91-7283-175-8
TRITA-FYS-5275 ISSN 0280-316X ISRN KTH/FYS/FTF/R--01/5275--SE
iv
Preface
This thesis is based on my research within the Solid State Physics group at the Royal
Institute of Technology, Stockholm, during the period 1997–2001. It consists of two parts.
The first part of the thesis is intended to assist the reader to become acquainted with
the language of vortices and superconductivity and to describe the context of the present
work in the vast current efforts in vortex physics. After a brief overview I continue with
introducing some basic concepts of superconductivity and the central role of anisotropy,
before discussing various vortex properties and how they can be studied through transport
measurements. The short overview is aimed at readers with a basic knowledge in superconductivity, so the inexperienced reader may prefer to begin the reading with the introduction
to superconductivity (page 4).
The second part consists of the publications, which compose the main part of the thesis.
Some results are also found in the introductory part, while others are presented only in the
papers.
List of publications
The following papers are included in the thesis:
I.
Vortex dynamics in oxygen deficient single crystals of YBa Cu OÆ
B. Lundqvist, A. Rydh, Ö. Rapp, and M. Andersson,
Physica (Amsterdam) 282–287 C, 1959 (1997).
II.
Empirical scaling of the vortex glass line above 1 T for high- superconductors of varying anisotropy
B. Lundqvist, A. Rydh, Yu. Eltsev, Ö. Rapp, and M. Andersson,
Phys. Rev. B 57, R 14 064 (1998).
III.
Consistent description of the vortex glass resistivity in high- superconductors
A. Rydh, Ö. Rapp, and M. Andersson,
Phys. Rev. Lett. 83, 1850 (1999).
IV.
Thermally assisted flux creep of a driven vortex lattice in untwinned
YBa Cu OÆ single crystals
A. Rydh, M. Andersson, and Ö. Rapp,
J. Low Temp. Phys. 117, 1335 (1999).
V.
Magnetic field scaling of the vortex glass resistivity in oxygen deficient
YBa Cu OÆ single crystals
A. Rydh, Ö. Rapp, and M. Andersson,
Physica (Amsterdam) 284–288 B, 707 (2000).
v
vi
List of publications
VI.
VII.
Multiterminal measurements of vortex correlations in the (K,Ba)BiO
system
A. Rydh, T. Klein, I. Joumard, and Ö. Rapp,
Physica (Amsterdam) 341–348 C, 1233 (2000).
Vortex liquid resistivity in disordered YBa Cu OÆ single crystals
M. Andersson, A. Rydh, and Ö. Rapp,
Physica (Amsterdam) 341–348 C, 1239 (2000).
VIII.
Strong vortex liquid correlation from multiterminal measurements on
untwinned YBa Cu OÆ single crystals
A. Rydh and Ö. Rapp,
Phys. Rev. Lett. 86, 1873 (2001).
IX.
Scaling of the vortex-liquid resistivity in optimally doped and oxygendeficient YBa Cu OÆ single crystals
M. Andersson, A. Rydh, and Ö. Rapp,
Phys. Rev. B 63, 184511 (2001).
X.
Multiterminal transport measurements: In-plane anisotropy and vortex
liquid correlation in YBa Cu OÆ
T. Björnängen, A. Rydh, and Ö. Rapp,
Phys. Rev. B 64, 224510 (2001).
XI.
Reply to Comment on “Strong vortex liquid correlation from multiterminal measurements on untwinned YBa Cu OÆ single crystals”
A. Rydh and Ö. Rapp,
Submitted to Phys. Rev. Lett.
The following papers are not part of the thesis and represent group reviews or work with
only minor contributions on my part:
XII.
Different estimates of the anisotropy from resistive measurements in high superconductors
Ö. Rapp, M. Andersson, J. Axnäs, Yu. Eltsev, B. Lundqvist, and A. Rydh,
in Symmetry and Pairing in Superconductors, M. Ausloos and S. Kruchinin
(eds.), Kluwer Academic Publishers, Dordrecht, pp. 289–300 (1999).
XIII.
Vortex liquid properties in optimally doped and oxygen deficient
YBa Cu OÆ single crystals
M. Andersson, Yu. Eltsev, B. Lundqvist, A. Rydh, and Ö. Rapp,
Physica (Amsterdam) 332 C, 86 (2000).
XIV.
In-plane anisotropy and possible chain contribution to magnetoconductivity
in YBa Cu OÆ
T. Björnängen, J. Axnäs, Yu. Eltsev, A. Rydh, and Ö. Rapp,
Phys. Rev. B 63, 224518 (2001).
Comments on my participation
Paper I was written while finishing my diploma work on the oxygen-deficient single crystals of YBa Cu OÆ , a project initiated by Magnus Andersson and Björn Lundqvist.
Sample preparation, measurements, evaluation, and the writing of Paper I and II were carried out in close collaboration with Björn. The evaluations and ideas leading to Paper III
were mine, and this paper and the related Paper V were also mainly written by me. Magnus
did most of the evaluation and writing of Paper VII and IX.
The high current study of the solid state in Paper IV was suggested and carried out by
me. Here I benefitted greatly from the advices of Yuri Eltsev on the subject of making high
quality contacts on small samples.
The collaboration with Thierry Klein and Isabelle Joumard on vortex correlations in the
cubic (K,Ba)BiO system (Paper VI) was initiated through my supervisor, Prof. Östen
Rapp. The Grenoble group provided single crystals for the multiterminal samples which I
prepared and studied.
I initiated the works of Paper VIII and X, and made all samples and measurements of
Paper VIII. Therese Björnängen prepared the sample of Paper X with my guidance. We
otherwise contributed equally to measurements, evaluation and writing.
My supervisor Östen Rapp, finally, has acted as constant source of suggestions, questions, and constructive criticism during all the work. He has especially contributed to the
process of turning manuscripts with no clear structure into the papers presented in this
thesis.
vii
Acknowledgments
First of all I would like to thank my supervisor, Prof. Östen Rapp. I could not have had a
better mixture of guidance and help. Östen always finds some time when you need, and he
really cares about his students.
Magnus Andersson introduced me to low temperature physics and measurements at low
levels. I have benefitted greatly from the many useful discussions we have had.
Björn Lundqvist teached me how to make single crystal samples and we had a close
cooperation on the oxygen-deficient YBa Cu OÆ studies. I have enjoyed our talks about
travelling and research, often over some nice food after a day in the lab.
I would also like to thank Therese Björnängen and Yuri Eltsev. Working with them has
been stimulating and great fun.
Among others at our department I want to mention Johan Axnäs, Shun-Hui Han, Reza
Ghorbani, Markus Rodmar, Pieter Lundqvist, and Mats Ahlgren, who have all contributed
to a nice atmosphere and interesting seminars.
Thierry Klein and Isabelle Joumard sent me many rare (K,Ba)BiO crystals, most of
which I destroyed during the contacting procedure.
A person who has meant a lot to me is Gunnar Benediktsson, who introduced me to solid
state physics and teaching.
Askell Kjerulf has always been willing to help in any practical matter, varying from
harddisk crashes to grounding problems. Also many thanks to Christer Torpling and Hans
Larsson, who have supplied us with liquid helium and nitrogen.
I appreciate useful discussions with Peter H. Kes, Mats Wallin, Peter Svedlindh, and the
group of Petter Minnhagen.
Finally, I would like to thank my parents and brothers. Especially help with numerical
calculations by Markus, and thorough reading, commenting, and discussions on various
manuscripts by David have been valuable.
A travel grant from the Ragnar and Astrid Signeul foundation is acknowledged. Support
for the research work has been obtained from the Swedish Natural Science Research Council (NFR), the Göran Gustafsson Foundation, and the Swedish Superconductivity Consortium (SSF). My time in the Solid State Physics group has been enabled through support
from the Swedish Research Council for Engineering Sciences (TFR), for which I am most
grateful.
Stockholm, December 2001
viii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Comments on my participation . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
Contents
ix
1 Introduction
1.1 Overview . . . . . . . . . . . . . . . . . . .
1.2 Basic concepts of superconductivity . . . . .
1.2.1 Low- and theoretical development .
1.2.2 Type-II superconductors – Vortices .
1.2.3 High- superconductors . . . . . . .
1.3 Anisotropy . . . . . . . . . . . . . . . . . .
1.3.1 Anisotropy in the GL theory . . . . .
1.3.2 Scaling methods . . . . . . . . . . .
1.3.3 Effects of layering . . . . . . . . . .
1.3.4 Anisotropy in high- materials . . .
1.4 Vortex motion . . . . . . . . . . . . . . . . .
1.4.1 Dissipation of a moving vortex . . . .
1.4.2 Elementary forces . . . . . . . . . .
1.4.3 Flux flow . . . . . . . . . . . . . . .
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1
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25
2 Vortex matter and phase transitions
2.1 Pinning in different vortex regimes . . . . .
2.1.1 Flux pinning and creep . . . . . . .
2.1.2 Thermally assisted flux flow . . . .
2.1.3 Depinning and flow . . . . . . . . .
2.1.4 Vortex liquid . . . . . . . . . . . .
2.2 Vortex-lattice melting . . . . . . . . . . . .
2.2.1 Evidence of a first-order transition .
2.2.2 Location of the phase-transition line
2.2.3 Critical points . . . . . . . . . . . .
2.2.4 Peak effect . . . . . . . . . . . . .
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ix
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x
Contents
2.3
Vortex glass . . . . . . . . . . . . . . . .
2.3.1 Glassy vortex systems . . . . . .
2.3.2 Dimensional scaling . . . . . . .
2.3.3 Experimental observations . . . .
2.3.4 Location of the glass transition . .
2.3.5 Critical current density . . . . . .
Vortex correlation . . . . . . . . . . . . .
2.4.1 Flux-transformer measurements .
2.4.2 Local or nonlocal electrodynamics
2.4.3 Phase-coherent liquid . . . . . . .
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42
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51
3 Experimental survey
3.1 Crystal growth and oxygenation . . . . .
3.1.1 Growth process . . . . . . . . . .
3.1.2 Oxygenation . . . . . . . . . . .
3.1.3 Relation between , , and Æ . .
3.2 Sample contact preparation . . . . . . . .
3.2.1 Standard configuration . . . . . .
3.2.2 Multiterminal samples . . . . . .
3.3 Resistive measurements in magnetic field
3.3.1 Temperature control . . . . . . .
3.3.2 Instrumentation . . . . . . . . . .
3.3.3 Low level measurements . . . . .
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4 Summary of results
4.1 Vortex glass line in oxygen-deficient crystals .
4.2 Extended vortex-glass description . . . . . .
4.3 Flux creep at high currents . . . . . . . . . .
4.4 Vortex-liquid correlation . . . . . . . . . . .
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63
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2.4
Bibliography
69
Index
91
Index of citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Papers
105
Chapter 1
Introduction
1.1 Overview
Vortices appear in a large fraction of all superconductors known today. While most superconducting elements are so called type-I superconductors, alloys and compounds belong to a second group, i.e., superconductors forming vortices in strong enough magnetic
fields. A vortex can be seen as a kind of flux tube containing a quantized magnetic flux
, maintained by a vortex current in the superfluid circulating around a normal
conducting core. The physics of vortices has been studied all since their theoretical introduction by Abrikosov (1957) and became a major field of research when the so called
high-temperature superconductors were discovered by Bednorz and Müller in 1986. The
known high- materials are all extreme type-II and possess a rich variety of vortex properties. Vortices are not only very challenging objects to study for the basic scientist. They
also in many ways play a crucial part for the use of high- materials in applications. One
of the problems is that vortices often cause energy dissipation at temperatures even far
below the critical temperature.
The vortices interact through repulsive forces, mediated by the vortex currents, and are
kept together by the magnetic pressure of the external field. Vortex systems can be controlled by both external conditions, such as temperature and magnetic field strength, and
microscopic parameters, such as anisotropy, , coherence length, and penetration depth
(Blatter et al. 1994, Brandt 1995). In addition, vortices can be pinned by crystal defects or
disorder. Due to this high degree of freedom, vortex systems manifest themselves in various configurations and states, as illustrated in Fig. 1.1, where a typical field – temperature
( – ) phase diagram is shown.
Especially in high- superconductors the microscopic parameters act together to form
a very soft vortex system, which allows the low temperature vortex lattice to melt into
a vortex liquid when thermal fluctuations become too strong. This phase transition can
be clearly visualized through the dramatic difference in resistive behavior between the
solid and liquid phases. The vortex solid is more or less strongly pinned and displays a
nonzero critical current, though usually far below the theoretical maximum defined by the
depairing critical current. In the vortex liquid, on the other hand, pinning is effective over
1
2
Chapter 1. Introduction
12
Hc2
2nd order
µ0 H (T)
liquid
8
4
glassy
strong
pinning
CP
weak
pinning
1st order
solid
0
70
75
80
85
90
95
T (K)
Figure 1.1. Typical phase diagram of the vortex system in YBa Cu OÆ single crystals with
a relative low amount of point disorder. At low magnetic fields, a vortex lattice (Bragg glass) is
formed, which melts at a well defined first-order transition. Pinning defects disturb the system
so that the solid state at fields above some critical point (CP) becomes glassy, with properties
depending on the kind of disorder (Fisher et al. 1991, Giamarchi and Le Doussal 1997, Nelson
and Vinokur 1993). Thermal fluctuations allow the vortex lattice to relax so that pinning becomes
relatively strong in a narrow region near the first-order transition. The upper critical field limits the vortex state. Above the normal state is the thermodynamically stable phase.
short distances only, resulting in substantial flow of vortices. This causes dissipation, and
makes the vortex liquid Ohmic over a wide range of currents.
Many interesting subjects and fundamental questions are found within this field of research; – What is the nature of the transitions between the different vortex phases? – Is the
resistivity in the vortex solid truly zero? – How could the glassy state be described?
Some of these questions are addressed in this thesis, where the focus is on vortex dynamics
in the weakly pinned region of the solid state, on vortex correlation in the liquid just above
the first-order transition, and on the vortex glass transition.
Vortex systems can be probed in many different ways. The location of individual vortices can be studied through Bitter decorations (Essmann and Träuble 1967, Brandt and
Essmann 1987, Gammel et al. 1987, Dolan et al. 1989), where smoke from a magnetic
material is evaporated onto the superconductor to form a pattern, like filings around a bar
magnet. Other methods include scanning tunneling microscopy (Hess et al. 1990, MaggioAprile et al. 1995), magnetic force microscopy (Moser et al. 1995), scanning Hall probes
(Oral et al. 1998), and magneto-optical techniques [see, e.g., Brandt (1995) for references].
Information about the structure or crystallinity of the vortex lattice can be obtained through
small-angle neutron diffraction (Cubitt et al. 1993, Johnson et al. 1999) or muon spin rotation (µSR, Lee et al. 1993, Sonier et al. 2000). Vortex phase transitions can be studied
by an equally great number of techniques. With local Hall probes (Zeldov et al. 1995) it is
possible to observe the small change in the local flux density at the melting transition. Measurements of the dc magnetization (Liang et al. 1996, Welp et al. 1996, Nishizaki
et al. 1996a, Deligiannis et al. 1997) or the ac susceptibility (Ishida et al. 1998) provide
1.1. Overview
3
V 23
I14
2
I
1
H
5
2
6
3
7
V 67
3
4
1
4
5
8
8
6
7
Figure 1.2. Multiterminal setups, i.e., configurations with the possibility of a nonuniform current
distribution. Left: The pseudo flux-transformer geometry. Current is fed through the contacts 1
and 4, and voltages and are measured. In the normal state the voltage will be lower
than due to the inhomogeneous current distribution. The ratio
will only depend
on anisotropy and contact geometry. In the vortex state, the resistivity perpendicular and parallel
to the magnetic field will depend on the straightness and properties of the vortices. Lowering
the temperature, the vortices in a clean crystal will align, resulting in an increased ratio for the
shown field direction. Right: By putting two contacts on each side of a plate-like crystal the
symmetry is increased, allowing comparisons between the two in-plane directions for any chosen
orientation of the magnetic field.
information not only on phase transitions, but also on pinning and possible surface barriers. Torque magnetometry (Martínez et al. 1992, Iye et al. 1992, Chien et al. 1994, Ishida
et al. 1997, Willemin et al. 1998) provides a variant of the magnetization measurements
which is powerful for the study of anisotropic properties. By using a mechanical oscillator
(Gammel et al. 1988) one can probe the damping of the vortex system, which peaks at
the vortex-lattice melting. Clear thermodynamic signatures of a first-order transition can
finally be found from calorimetric measurements of the latent heat (Schilling et al. 1996,
1997, Junod et al. 1997).
Resistive transport measurements, upon which this thesis is based to a great extent, are
commonly used to study vortices and vortex phase transitions. Transport measurements
have the advantage of probing the motion of vortices instead of their magnetic signature.
The main drawback is that resistivity is not a thermodynamic property. Strictly speaking
the equilibrium is disturbed by the transport current so that a nonequilibrium condition
is studied. Resistive measurements can anyhow provide much useful information on the
vortex phases and the transitions separating them, since the resistive properties may change
dramatically between different locations in the phase diagram. Resistive measurements
further benefit from the possibility of variation in experimental setup. Besides the normal
four-probe configuration, where a uniform current density is required, one can use various
multiterminal sample configurations, where a deliberately nonuniform current distribution
enables the simultaneous study of properties in different directions. Examples of such
sample configurations that have been used in this work are shown in Fig. 1.2. Multiterminal
measurements are powerful for studying vortex structure and correlation. The coupling
between adjacent superconducting layers along the axis, and questions such as whether
the vortex liquid obeys local or nonlocal electrodynamics can also be tackled.
4
Chapter 1. Introduction
H
Li
He
Be
B
C
N
O
F
Ne
Na Mg
Al
Si
P
S
Cl
Ar
Ga Ge
As
Se
Br
Kr
K
Ca
Sc
Ti
V
Cr
Rb
Sr
Y
Zr
Nb Mo
Cs
Ba
La
Hf
Ta
Fr
Ra
Ac
Unq
...
Ce
Th
Mn
Fe Co
Ni
Cu
Tc
Ru
Rh
Pd
Ag Cd
In
Sn
Sb
Te
I
Xe
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Pr
Nd
Pm
Sm
Eu Gd
Tb
Dy
Ho
Er
Tm Yb
Lu
Pa
U
Np
Pu
Bk
Cf
Es
Fm
Am
Cm
Zn
Md
No
Lr
Figure 1.3. The Periodic Table. Elements with grey background are superconducting at low
temperatures. Those with lighter shades need special conditions, such as higher pressure, to
become superconducting. Neither good metals such as Cu, Ag, and Au or simple metals such as
Na and K have yet been found to superconduct, nor have the elements with magnetic ordering
such as Fe, Co, and Ni (underlined). Unstable elements are printed in smaller letters. Items
marked () are more uncertain or were updated recently. Sources: Ashcroft and Mermin (1976),
Hulm and Matthias (1980), Kittel (1986), Hein (1992), Waldram (1996), and Poole, Jr. (2000).
1.2 Basic concepts of superconductivity
1.2.1 Low- and theoretical development
Superconductivity is a phenomenon found in a variety of materials and alloys at low temperature, including a large number of the elements in the periodic table, as shown in
Fig. 1.3. The highest critical temperature among the elements is found for Nb with
K at ambient pressure. Above the critical temperature the superconducting
state is no longer thermodynamically stable, and the normal-state electronic properties are
recovered. Superconductivity was first observed in Hg ( K) as a sudden disappearance of measurable resistance. The discovery was made by the Kamerlingh Onnes
group (Kamerlingh Onnes 1911a,b,c) when studying the low-temperature behavior of pure
metals.1 In 1913 and 1914, Kamerlingh Onnes, who alone had the capability to produce
liquid helium and thereby access to the low temperatures needed, showed that the new superconducting state was destroyed by too high currents or magnetic fields, in addition to
the effect of temperature. This behavior is illustrated in Fig. 1.4. Usually, materials with
1 See
Gorter (1964) for more history.
1.2. Basic concepts of superconductivity
5
J
Jc
Hc
Tc
H
T
Hc (T)
Figure 1.4. The superconducting phase only exists at low enough temperature, current and applied magnetic field. Thus, a superconducting magnet operated at high current and producing a
strong magnetic field has to be cooled to a temperature often far below the critical temperature at
zero field.
high have high critical fields2 and indeed Nb has the highest among the elements,
T. Collections of critical values for the superconducting elements are given
by for instance Kittel (1986) and Ashcroft and Mermin (1976).
In the 1920s and 1930s, reports of new superconducting alloys and compounds started to
appear. Interestingly, some of these compounds were formed out of solely nonsuperconducting elements. The critical temperatures remained rather low, however, and when the
high- era began in 1986 the maximum known was around K for Nb Ge (Gavaler
1973). Nevertheless many of these compounds became, and still are, technologically important since they have high critical fields. For example the critical field for Nb Ge
is 38 T and the Chevrel phase material PbMo S discovered in the 1970s has an upper
critical field of 60 T.
A major break-through for the understanding of superconductivity was made by Walther Meissner
effect
Meissner and Robert Ochsenfeld when they showed that magnetic fields in bulk superconducting samples are completely expelled at low applied magnetic fields (Meissner and
Ochsenfeld 1933). This Meissner effect is illustrated in Fig. 1.5. The effect acts as a fingerprint for the superconducting state and indicates its thermodynamic nature. An (imagined)
perfect conductor would tend to keep the field constant inside the sample rather than expelling it, if cooled in magnetic field. The phenomenon was explained by the brothers Fritz
and Heinz London (London and London 1935a,b). They showed that the Meissner state
2 As will be discussed in the next section, things become more complicated for Nb, being a type-II superconductor. The scaling between and , however, works even for type-II superconductors provided that the
thermodynamic critical field (page 8) is used in the comparison.
6
Chapter 1. Introduction
Applied
magnetic field
normal conducting
B=0
superconducting
T < Tc
T > Tc
Figure 1.5. The Meissner effect. A bulk superconductor in low enough magnetic field completely
expels the field, yielding
inside the superconductor. This occurs regardless of the way the
sample is cooled or the field is applied. Thus, if the sample is cooled from a temperature above
, where the field fully penetrates the sample, field expulsion suddenly sets in when entering the
superconducting state.
can be described in terms of supercurrents circulating in a thin surface layer of the superconductor in a way as to oppose the external magnetic field. The thickness of the surface
layer turned out to be a temperature-dependent material parameter, the London penetration
depth . Magnetic field is thus allowed to penetrate a distance of the order of into the suLondon perconductor. The London equations, describing perfect conductivity and flux expulsion,
equations are given by
and
Here
(1.1)
(1.2)
isthe local electrical
field, the supercurrent density, the local induction, and
. The parameters , , and are the mass, concentration,
and charge of the superconducting charge carriers.
The presence of the Meissner effect is fundamental because it shows that the superconducting state is given by equilibrium thermodynamics (London 1950); the lowest energy
state for the system at is found for a superconductor expelling the magnetic field at
low fields, but is given by a normal conducting sample with a uniform field when or at high enough fields.3 Unlike the expected behavior of a simple, perfectly conducting metal, the superconducting transition in magnetic field is reversible. The state of the
superconductor is thus settled if the temperature and applied magnetic field are given.
3 For
a spherical sample, such as the one shown in Fig. 1.5, field penetration starts already at an applied field
due to the self-field of the superconductor. In this case partial field penetration takes place for
fields up to . This intermediate state appears as a system of dendritic lamellae, alternately in the normal
and superconducting state [see for instance Rosenberg (1993) or Livingston and DeSorbo (1969)].
1.2. Basic concepts of superconductivity
7
The London equations describe many effects of the superconducting state well, but they
do not explain why a material becomes superconducting. The temperature dependence
of the London penetration depth, for instance, has to be experimentally determined. It is
commonly observed to follow an expression close to
(1.3)
The penetration depth thus diverges at the critical temperature, indicating that the number
of superconducting charge carriers goes to zero at .
GL
One step further in the theoretical description of superconductivity was taken by Ginzburg
and Landau (1950). Two years earlier Fritz London had expounded his view of the macro- functional
scopic quantum nature of superconductivity (London 1935) by showing how the London
equations could be obtained from a quantum-mechanical treatment of the superfluid (London 1948, 1950). In the quantum description a wave function is introduced, whose
squared modulus describes the local density of superelectrons . Ginzburg and
Landau (GL) argued that the Gibbs free energy should depend on the concentration of superelectrons, and introduced the pseudo-wave function as an order
parameter in the Landau theory of second-order phase transitions (Landau 1937). The GL
Gibbs free-energy functional is given by4
(1.4)
,
is the
where and are coefficients for the expansion in ,
vector potential, is the applied, external field, and is the volume of the superconductor. By minimizing the functional (1.4) with respect to and , two differential equations
are obtained. The first, found from the variation of , is a Schrödinger-like equation with
an extra, nonlinear potential term . The second equation is the so called supercurrent
equation. Each equation has a characteristic length scale, describing the shortest spatial
variations of and (or ). For , the scale is the GL penetration depth , closely
related to the London . The other length scale is the GL coherence length . The parameters and have similar temperature dependence, and the ratio of the two therefore
makes an important number, the Ginzburg–Landau parameter
(1.5)
which is approximately temperature independent.
The GL treatment is useful in describing the superconducting state for cases of spatially
varying , such as when the superconductor is put in contact with a normal conducting
metal, or if a strong enough magnetic field is applied to decrease . The theory also
comprises the London equations, which are obtained from the supercurrent equation in
the limit of constant (Chandrasekhar 1969). While the Ginzburg–Landau theory is
4 See
for instance Blatter et al. (1994) or Werthamer (1969).
8
cond.
energy
Chapter 1. Introduction
powerful in many cases, it also has one important limitation. Since free energy of the
superconducting state is expressed as a series expansion of the order parameter, the result
is only valid close to the point of expansion, i.e., around the critical temperature where
the concentration of superconducting charge carriers goes to zero.
The free-energy difference between the normal and superconducting states is directly
related to (London 1950):
(1.6)
Here and are the Gibbs free-energy densities of the normal and superconducting
material in zero field, respectively. To indicate its thermodynamical nature, is is called
the thermodynamic critical field. The GL result for this field is
! (1.7)
An empirical expression for its temperature dependence is given by
Pippard
(1.8)
In 1953, some years before the GL theory became generally recognized, Pippard introduced another coherence length to generalize the London equations for nonlocal effects.
The Pippard coherence length describes the range over which the supercurrent at a point
is related to the vector potential , . Two contributions to the nonlocality were
suggested. The first arises from the Heisenberg uncertainty principle, giving the superconducting charge carriers a characteristic, shortest length of interaction. Pippard took this
length to be " , where " is the Fermi velocity, i.e., the typical velocity of
the electrons available for interactions in the material. The other length is the mean free
path #, corresponding to the typical distance between scattering events of an electron. The
shortest of these lengths was taken as the coherence length, # .
Despite that both the GL and the Pippard coherence lengths are denoted by , they are
not one and the same. The similarity only goes so far that, under certain conditions, the
GL coherence length at can be found from the Pippard value. Since varies
on the scale of , the nonlocal effects described here are most important in the limit where
² , i.e., for º . A local description is eased by temperatures close enough to ,
where always becomes greater than , and by increased disorder. In the dirty limit,
i.e., for # , it can be shown that will increase by a factor #
, while the
GL will decrease by the same entity (Tinkham 1996). The effect of disorder on is
therefore rather small.
Until this point, a discussion of the nature of the superconducting charge carriers has
been avoided. The explanation was presented by Cooper (1956) and developed in the
subsequent microscopic theory of superconductivity by Bardeen, Cooper, and Schrieffer
(1957a,b). In the electron pairing paper by Cooper, it was shown that electrons in a metal
are able to form at least one bound electron pair if there exists an attractive interaction
BCS
theory
1.2. Basic concepts of superconductivity
9
between the electrons. Furthermore, it was shown that this pairing is feasible for arbitrarily
weak interaction, meaning that if an attractive interaction could be found, the system would
have a lower energy when at least some of the electrons are paired than in the normal,
unpaired state. Since the most stable state should be the one with the lowest free energy,
such a coupling would enable a new phase which could possibly be superconducting.
Two electrons in free space can clearly form no bound pair, but are instead repelled by
each other due to the Coulomb force. Bardeen, Cooper, and Schrieffer however showed
that an attractive coupling between the unbound electrons in a metal may be mediated
through an exchange of virtual phonons, or simply put, through ion motion of the atom
lattice. While this model for the electron-electron interaction has turned out to agree very
well with observations of the conventional superconductors, other mechanisms may be
essential for instance in the high- superconductors. The basics of the BCS theory do
not rely on the explicit coupling, however, but only assume the presence of an attractive
interaction and the creation of so called Cooper pairs. The parameter used earlier
consequently equals the charge of the pair, .
Thus having a state where at least some electrons are paired, and which has an energy
lower than in the normal state, Bardeen, Cooper, and Schrieffer continued to describe the
microscopic system in terms of microscopic quantum mechanics. They showed how to
treat the system when more than one Cooper pair is formed. The electron pair should,
however, not be seen as two electrons moving together through the superconductor. Instead, the most effective pairing is obtained for electrons with opposite spins moving with
the same speed but towards each other.
A rather advanced quantum-mechanical treatment is required to show how electron pairing leads to superconductivity and phenomena such as the Meissner effect. Fortunately,
beginning with the microscopic theory is not always necessary. It may be enough to know
that starting with the BCS theory, it is possible to derive the GL equations and through
them the London equations in the appropriate limits. A strict way to go from BCS to GL
was shown by Gor’kov (1959a,b) and was the break-through for the Ginzburg–Landau theory. More detailed discussions of the BCS theory are provided for instance by Tinkham
(1996), Isihara (1991), or Rickayzen in the extensive collection by Parks (1969).
One of the most important results of the BCS theory is the prediction of the opening of a
gap in the electronic density of states at the Fermi energy $ . The electrons are disallowed
to have energies within around $ , somewhat resembling the semiconducting case of
a band gap. For superconductors, however, the gap is much smaller. In the low temperature
limit, the (weak coupling) BCS result yields , thus typically in the submeV range. (
is the gap at .) The gap is proportional to the order
parameter of the GL theory (Gor’kov 1959a, Van Duzer and Turner 1981). Other BCS
results include a more precise definition of Pippard coherence length, " ! ,
and an expression for the zero-temperature condensation energy, % ,
where % is the density of states at the Fermi level for one spin direction.
Experimentally, effects of the gap were first observed by Giaever (1960a,b) through tunneling studies of electrons over metal/insulator/superconductor junctions. Shortly after,
the tunneling of Cooper pairs through a superconductor/insulator/superconductor barrier
was predicted (Josephson 1962) and confirmed (Anderson and Rowell 1963). This laid
10
Chapter 1. Introduction
the foundation to the superconducting quantum interference device (SQUID) technology,
today of great importance for applications.
1.2.2 Type-II superconductors – Vortices
flux
quantum
In the same year as the BCS theory, the concepts of vortices and type-II superconductivity
were presented by Abrikosov. In his paper (Abrikosov 1957), the results of experiments
that had been carried out on superconducting alloys (Shubnikov et al. 1937) were finally
explained. The magnetic properties of such alloys display a second state at high magnetic
fields, where the magnetic field is only partially expelled. It was known that this violation
of the Meissner effect could not be explained in terms of the partial penetration found close
to for samples of certain shapes. Since really pure metals always seemed to follow
Meissner behavior, the partial penetration had been commonly referred to as arising from
defects and inclusions.
Abrikosov used the
Ginzburg–Landau equations to study what would happen if the GL
parameter . This number had been shown to separate superconductors with
positive surface energy (type-I) from those with negative energy (type-II). It was found
that superconductors belonging to the second group could choose a new way to respond
to applied magnetic fields. At high fields, the external magnetic field could be allowed to
pass through the superconductor, resulting in a smaller amount of magnetic energy to be
overcome and thus a lowered total energy. However, the magnetic field did not penetrate
the superconductor homogeneously. Instead, the calculations indicated that the magnetic
flux should divide itself into as small bundles of flux as possible.
The minimum bundle allowed, the flux quantum, is given by5
vortices
(1.9)
Such quantization had been pointed out already by London (1950), but then in quantums
of . The factor in the denominator comes from the Cooper pair and was settled
only when Deaver, Jr. and Fairbank (1961) and Doll and Näbauer (1961) experimentally
confirmed that flux inside a hollow, superconducting cylinder is quantized in units of .
Abrikosov realized that the description of the type-II superconductors is quite similar to
that of vortex filaments in superfluid helium II. Summarizing the findings, one could say
that along with the flux bundle comes a vortex in the superfluid, with a diverging superfluid
velocity when approaching the singular vortex center. Due to the divergence, the kinetic
energy of the Cooper pairs constituting the fluid eventually increases to a level where the
normal state is energetically favorable. This leads to the presence of a normal conducting
core with radius . (It is not possible to have a smaller radius than this, since corresponds
to the lengthscale of variations of the order parameter.) In the center of this core the order
parameter vanishes, while following a path around it changes the phase & of the order
parameter by ! . The type-II vortex behavior is illustrated in Fig. 1.6.
5 This quantization is a purely quantum-mechanical effect of the superconductor, as can be seen by its vanishing in the classical limit . It is not the flux itself which is quantized, but the flux in a superconductor. To
emphasize this, the term fluxoid quantization is often used.
1.2. Basic concepts of superconductivity
11
Magnetic field
superconducting
normal core
core
Figure 1.6. Type-II superconductor in high magnetic field. The field penetrates the superconductor in small bundles of flux, each corresponding to one flux quantum . The flux is coupled to
vortices in the superfluid, the centers of which are normal conductors.
The superconductor outside the vortex core can be described by the ordinary Meissner
effect. Screening currents flow on a lengthscale around the core and try to expel the
magnetic field from the bulk superconducting material. Since is greater than in the typeII superconductors, the magnetic field does not only penetrate through the core regions,
but over a sometimes much larger area. In the extreme type-II limit ( )
the magnetic field is almost undisturbed throughout the superconductor at high fields. Of
course, the constraint that the area density of singularities in the phase of the order
parameter should correspond to the magnetic flux density remains, .
Thanks to the ability of creating vortices, the type-II materials can be exposed to much
higher magnetic fields before superconductivity disappears. A typical critical field for a
type-I superconductor is T, while the largest field a type-II superconductor
can stand is usually several tesla. Among the elements, Nb, La, Tc, and V are type-II, and
the rest type-I (Rosenberg 1993, Brandt 1995, page 1479). Alloys and other compounds
are mainly type-II. Note that by adding disorder, many type-I elements can be turned into
type-II materials, since decreases with decreasing mean free path as found by the Pippard
relations (page 8).
The vortex state, also called the mixed state or the Shubnikov phase, does not appear
until a high enough magnetic field has been applied. At low magnetic fields, type-II superconductors thus display the usual Meissner effect. This results in two critical fields; a
lower critical field , where vortices enter and an upper critical field , where the
normal state is recovered.
There is a close relation between and . The lower critical field is obtained from the
condition that the free energy is the same without vortices as with the first vortex entered
into the superconductor. It can be shown that the maximum field in the center of a single
vortex will be raised to a field of the order of , due to the vortex currents, if the applied
field is (Tinkham 1996). This field taken over an effective area ! should roughly
correspond to the flux quantum. Correcting with an additional factor 2 in the denominator
critical
fields
12
Chapter 1. Introduction
-M
Type - I
M = -H
Type - II
κ≈1
H
Hc1
Meissner state
Hc
Hc2
Vortex state
Normal state
Figure 1.7. Type-II magnetic response. Up to the type-II superconductor acts as perfect
diamagnet with a susceptibility
. The current-induced magnetization thus
completely opposes the applied field
so that the magnetic flux density
remains zero. The dotted line shows the type-I behavior.
to comply with the GL result, one has
!
(1.10)
This expression is valid up to a factor for high materials (where the core effects are
less significant). In analogy with this, there is a connection between and . The vortex
cores fill the whole superconductor when . Since in this high-field
limit due to the averaging over lengthscales , we directly find which
agrees fairly well with the GL result
! (1.11)
Through equations (1.10) and (1.11) one can find the temperature dependencies of the
critical fields. As expected, both and go to zero at , and follow the same
temperature dependence as . The latter is seen by the useful relation
magnetic
response
(1.12)
Due to the induced supercurrents, the average magnetic field in the superconductor will
not equal the applied magnetic field until is reached. The difference is conveniently
represented through a magnetization corresponding to the screening current. The magnetic
response of the type-II superconductor is illustrated in Fig. 1.7. The area under the '
vs. graph is a measure of the condensation energy. If the curve between the points
' and is approximated by a straight line, the relation is
found, confirming Eq. (1.12). From this it is easy to realize why type-II superconductors
can have large values of the critical field; decreasing with a factor of from ,
1.2. Basic concepts of superconductivity
13
B
Vortex I
Vortex II
x
ns
d
λ
ξ
x
Figure 1.8. Two neighboring vortices in a superconductor. Upper part: Flux density for each
vortex (solid curves) and total flux density (dashed curve). The figure shows the case of close
to since
. Lower part: Variation of the superconducting electron concentration ,
which goes to zero at the center of the vortex cores.
which is by no means an extreme factor, gives a corresponding increase of while the
condensation energy can be kept constant.
In the preceding discussion, it was sometimes implicitly assumed that the vortices some- vortex
how spread out evenly over the available space. This is because the vortices interact with interaction
each other. The interaction is repulsive for two vortices in the same direction,6 and is
caused by the presence of the field or screening current gradient from one vortex in the
neighborhood of the other, see Fig. 1.8. The interaction between different vortices is therefore strong for vortex spacings ( º . This condition is fulfilled for still of the order
of even for moderate values of , since vortices will quickly enter the superconductor until they start to interact. The flux density and superconducting electron concentration
around two vortices are shown in Fig. 1.8. When the applied magnetic field is increased, so
is the average flux density in the superconductor. The field distribution and magnetization
will vary depending on the positions of the vortices, but the variation in is typically
of the order of, or less than, .
One could ask why the repulsive forces between the vortices do not push all vortices Abrikosov
out of the superconductor. The answer is that the vortex collection should arrange itself lattice
into a state of low energy. Since this state is found for a certain ' , the flux density is not
free to change. There are thus two forces competing in the problem, the rather long range
repulsive forces between different vortices and a restoring force keeping the flux density constant up to the small variations allowed in ' . Without other forces the vortices form a
lattice to keep each vortex in equilibrium and to maximize the average vortex spacing. If
there are no preferred directions in the plane perpendicular to the vortices, the two natural
possibilities are the square and the triangular lattices, shown in Fig. 1.9. Comparing the
two lattices, one can see that the triangular lattice has a larger vortex spacing than the
6 There actually exists a temperature
See Brandt (1995) for references.
region where the interaction becomes attractive provided that
.
14
Chapter 1. Introduction
a
a
Figure 1.9. Vortex lattices in two dimensions. The square lattice to the left has a lattice spacing
¾ while the triangular (hexagonal) lattice has a spacing where . The magnetic field is directed out of the paper. The screening currents
circulate counter-clockwise around the vortices as indicated by the arrow.
phase
diagram
pinning
square lattice for fixed . The triangular lattice is therefore the most stable one and is
consequently seen in practise in most cases.
When thermal fluctuations are taken into account, the vortex lattice will no longer remain
perfect. It has thus been found that the vortex lattice can melt into a vortex liquid. It is
customary to illustrate the location of such phase transitions in the field – temperature ( –
) phase diagram. The field here is analogous to the pressure in normal diagrams. A
simple – phase diagram is shown in Fig. 1.10. Also compare with Fig. 1.1.
When an external current is sent through a type-II superconductor in the mixed state,
there will be a force7 acting on the vortices, initiating vortex motion. A moving vortex
could, in turn, cause electrical fields due to the time dependence arising in . The vortices can be pinned in different ways, inhibiting the motion. By introducing defects into
the superconductor the superconductivity is locally depressed. It is advantageous for a
vortex to stick to such pinning centers, since the energy needed to suppress superconductivity in the vortex core does not have to be expended. In practise some disorder is always
present, and there will be a depinning critical current density, usually denoted only as ,
below which the vortices are pinned and “true” superconductivity prevails. Depending on
the magnetic field and temperature the interaction between vortices and pinning centers,
and between different vortices, will change. In addition, the vortices should be regarded
as elastic objects which can bend. This together makes up a very complicated system
discussed in the area of vortex dynamics.
1.2.3 High- superconductors
In 1986, Bednorz and Müller suddenly reported indications of superconductivity around
30 K in the Ba–La–Cu–O system. The discovery was interesting from two points of view.
7 This force is of Lorentz type, i.e., if the magnetic field is along the axis, a current in the direction will give
rise to a force (per volume) in the direction. A more detailed discussion is given in Section 1.4.
1.2. Basic concepts of superconductivity
15
H
Hc2(0)
Hc2(T)
vortex
solid - liquid
transition
Hc1(0)
T
B=0
Meissner state
Hc1(T)
Tc
Figure 1.10.
– phase diagram for a type-II superconductor with . The vortex state
is limited by the upper critical field from above and the lower critical field from below. Above
superconductivity disappears. Interesting transitions take place in the vortex state, such as a
melting of the vortex lattice.
First, the new superconductor was not at all a simple metal or alloy, but belonged to a
totally different class of compounds – the ceramics. Second, there had previously been
discussions about a theoretical upper limit of at 30 K. Bednorz and Müller argued that
there were certain indications that related oxides could have still higher . Investigations
on similar compounds started all over the world, and very soon new compounds with K were found (Wu et al. 1987). In Table 1.1 some of the most studied ones are listed.
With a few exceptions, such as the newly discovered MgB with K (Nagamatsu structure
et al. 2001), the high- superconductors share some characteristic features. They have
a perovskite-like structure8 and are more or less layered. Most of them contain copperoxygen planes. The structures of Ba K BiO , having the cubic perovskite structure,
and YBa Cu OÆ , the first compound found with above the boiling point of nitrogen (77 K), are shown in Fig. 1.11. The high- structures can be built by alternatingly
combining conduction and binding slabs. In YBCO the conduction slab consists of two
CuO conduction layers separated by an Y layer, splitting the perovskite octahedron. The
binding slab, also called charge reservoir, is built up of two BaO bridging layers separated
by the additional CuO-chain layer. The classification in Table 1.1 indicates that YBCO has
one additional layer A (CuO), two bridging layers B (BaO), one separation layer C (Y),
and two conduction layers D (CuO ). Other compounds can be constructed in the same
way, only following some rules how the layers may be ordered (bridging layers, if present,
have to be placed next to conduction layers, etc.). See Poole, Jr. (2000) for further details.
Generally, structures with many conduction layers have higher .
8 The
perovskite, CaTiO , is composed by corner-linked TiO octahedra surrounding the large Ca ion.
16
Chapter 1. Introduction
Table 1.1. High- compounds with approximate maximum critical temperatures. The cuprate
compounds can be classified using a four-letter code usually corresponding to the chemical formula A B C D O . The oxygen deficiency introduces changes in the electronic
structure, which can radically affect both and other properties. Sometimes a compound has to
with .
be stabilized by substitution. Bi-2223 usually has the composition Bi Pb
An extensive overview of various structures is provided by Poole, Jr. (2000).
Æ
Short-name
max (K)
#
(B,K)BiO, BKBO
(La,Sr)CuO
Y-123, YBCO
Y-124
Bi-2212, BSCCO
Bi-2223
Tl-2212
Tl-2223
Hg-1223
30
35
93
80
92
110
110
125
133
perovskite type
0201
1212
2212
2212
2223
2212
2223
1223
Compound
Ba K BiO
La Sr CuOÆ
YBa Cu OÆ
YBa Cu O
Bi Sr CaCu OÆ
Bi Sr Ca Cu OÆ
Tl Ba CaCu OÆ
Tl Ba Ca Cu OÆ
HgBa Ca Cu OÆ
Y
Oxygen
CuO 2 plane
Ba
Bi
c
CuO chain
(K,Ba)
Ba
a
CuO 2 plane
b
Y
a
Figure 1.11. Left: Ba K BiO . The BKBO has an undistorted, cubic perovskite structure
with lattice parameter Å. Right: YBa Cu OÆ . The structure is orthorhombic for
. The oxygen deficiency is mainly taken from the CuO chains. At high , oxygen is
redistributed within the CuO-chain plane and the structure becomes tetragonal. Optimally doped
YBCO with has Å, Å, and Å. Atom sizes are not to
scale.
Æ Æ Æ
1.3. Anisotropy
17
Figure 1.12. Vortices in a traditional type-II superconductor (left) and a strongly layered high- superconductor (right). The superconducting planes in the layered structure are coupled through
Josephson tunneling, which preserves superconductivity between the layers. When the Josephson
coupling is weak, pancake vortices are formed which are only well defined within the conduction
planes.
The high- superconductors are all extreme type-II with typical values . is thus very large while is small. The layered structure of the cuprates also make
them more or less anisotropic, i.e., electrical and other properties are not the same along
different crystal directions. This has a profound influence on and , and through them on anisotropy
most properties of the vortex system. Furthermore, the coherence length along the axis
is very short – typical values of are between 1 and 4 Å. This is less than the -axis
lattice parameter, which may cause superconductivity to prevail only within the conduction
slabs at low temperatures. The most anisotropic materials are therefore often regarded as
stacks of thin, superconducting sheets insulated from each other. Even if the magnitude of
the order parameter does not totally vanish between the conduction layers, is probably
modulated along the axis.
Vortex properties in high- superconductors are much different from those in traditional
type-II materials, mainly because of the high and strong anisotropy but also due to the
relatively high temperatures. In a layered structure where is strongly modulated, the
vortices are only well-defined in the conduction planes, and are therefore sometimes called
pancake vortices (Efetov 1979, Clem 1991). The two cases of a conventional type-II su- pancake
perconductor and a strongly layered high- material are compared in Fig. 1.12. Pancakes vortices
in different layers attract each other through both magnetic coupling and the Josephson
mechanism. The coupling may be strong or weak depending on the distance between the
planes, thermal fluctuations, the size of , and the possible presence of correlated disorder (Brandt 1995).
1.3 Anisotropy
Electrical anisotropy has a profound influence on the characteristics of superconductors.
A short discussion of its main effects is therefore of interest. Anisotropy can be taken into
18
Chapter 1. Introduction
account in an ad hoc way through the anisotropic Ginzburg–Landau theory. Only in case
of strongly layered structures it may be necessary to go beyond this treatment to use the
Lawrence–Doniach theory (Lawrence and Doniach 1971) which takes the modulation of
the order parameter perpendicular to the superconducting layers into account.
effective
mass
1.3.1 Anisotropy in the GL theory
In the anisotropic GL theory the superconductor is assumed to be homogeneous but anisotropic. The anisotropy is assumed to come from the effective mass , which becomes
an effective mass tensor in the anisotropic case. The mass tensor is assumed diagonal
when considering the principal axes:
(1.13)
and is suitably normalized as . The anisotropy in the mass tensor
directly enters into in the London equations, Eqs. (1.1) and (1.2), making it a diagonal
tensor too, . This leads to three possibly different penetration depths
where ), *, or and . Usually the anisotropy in
high- compounds is more or less unidirectional, with a much larger effective mass along the direction than within the )* planes. In this case it is common to define the
anisotropy parameter as:
(1.14)
where . This gives with defined in line with . The
high- superconductors thus have . Note that , for instance, gives the screening length in the * direction for a magnetic field directed along the axis. The reason is
that the length scales are connected to , not directly to the magnetic field. The modification of the coherence length by anisotropy is found in a similar way. The definition of
in the GL theory is , where is the scalar parameter used in
the expansion of , see Eq. (1.4). Thus , where . The
coherence length in the )* plane is therefore a factor larger than .
Perhaps the most important result of the anisotropy is that the upper and lower critical
, we have (using the GL
fields change with the direction of the applied field. For
equations):
!
(1.15)
while the upper critical field along the )* plane becomes a factor larger:
! !
(1.16)
The opposite is true for , disregarding the complication of the factor (Orlando and
Delin 1991). One has to be careful when is found in an equation, since the ratios 1.3. Anisotropy
19
depend on the anisotropy. From and , it is seen that the isotropic [e.g. in Eq. (1.5)] scales as so that
(1.17)
in the case of unidirectional anisotropy. It is common to refer to this as the GL parameter
of anisotropic materials (Brandt 1995). The anisotropy of the high- superconductors
is varied by changing the metallicity of the charge reservoirs, which does not necessarily
leave unchanged. Therefore, modifying the layer coupling may cause changes in both
and .
Unlike the upper and lower critical fields, the thermodynamic critical field remains unchanged by modifications of :
! (1.18)
where indicates any direction. The anisotropic relations between , , and are
found directly from Eq. (1.15), (1.16), and (1.18). Expressed in the parameter as defined
in Eq. (1.17), the isotropic expressions are recovered:
(1.19)
where logarithmic factors are omitted as before. The convenient relation is
thus still valid. Equation (1.19) indicates that, for unchanged and , a high anisotropy
favors the maximum upper critical field, while it disfavors the minimum upper critical field
(along the axis), since .
If the magnetic field is applied in an arbitrary direction to the principal axes, the London angular
and GL equations become hard to solve in general. The vortices could for instance have a dependence
direction other than that of the applied field (Kogan 1981, Blatter et al. 1994). In extreme
type-II materials such effects can usually be neglected at intermediate to high magnetic
fields ( ). Let + denote the angle between the applied magnetic field and the
)* plane (leaving , ! + for the angle to the axis). With this notation, the angular
dependence of the upper critical field is given by
+ !- (1.20)
where - is defined as
- - + + +
(1.21)
When the magnetic field is aligned with the planes we have - and Eq. (1.20)
is simplified to Eq. (1.16). For the field along the axis we obtain -! in agreement with Eq. (1.15). The angular dependence of the lower critical field is more complex
(Klemm and Clem 1980).
20
Chapter 1. Introduction
Table 1.2. The BGL scaling approach. Quantities in the scaled, isotropic system are denoted .
Replace all quantities according to the table to introduce anisotropy into an isotropic expression.
Isotropic quantity
Anisotropic replacement
Description
, - , - $
., /
0
, magnetic field
energy
fluctuation temperature
volume
in-plane lengths
out-of-plane length
GL lengths
GL parameter
$
., /
0
, 1.3.2 Scaling methods
In many cases the isotropic expression for a quantity is known and the anisotropic counterpart is wanted. In the previous section the anisotropy was introduced into the solution of
the GL equations, as obtained for the isotropic case. In general the anisotropic equations
also have to be solved. An easier way is to look at the starting equations and scale them
as much as possible before considering each specific problem. It is thus not necessary to
minimize the GL functional [Eq. (1.4)] to find out how anisotropy changes the isotropic
results. By suitably rescaling the parameters, the way anisotropy enters becomes more
Klemm & clear. One path to do this was provided by Klemm and Clem (1980). They showed that, in
Clem
the mean field limit ( and ) the gradient and vector potential terms can be
transformed into their isotropic forms by introducing anisotropy into the magnetic energy
term. Alternatively, the anisotropy can be mixed into the GL parameter . The free energy
can then be minimized, yielding the same, isotropic equations, but in a transformed frame
of reference, where the isotropic is replaced by
BGL
scaling
-
(1.22)
for the unidirectional anisotropy case. It is customary to define the directional dependent
GL parameters in analogy with this relation, i.e., (Clem 1998).
A similar approach to the scaling of the anisotropic GL free-energy functional was made
by Blatter, Geshkenbein, and Larkin (BGL) (Blatter et al. 1992). Instead of mixing the
anisotropy into , they kept the magnetic term anisotropic. In the mean field limit the
magnetic field energy part of can then be minimized separately. This gives a scaling
rule which is equivalent to Eq. (1.22), but in a form that allows problems with thermal
fluctuations and disorder to be handled. According to the BGL scaling approach isotropic
in an equation should be replaced by the anisotropic counterparts 1 following
quantities 1
the rules of Table 1.2.
1.3. Anisotropy
21
1.3.3 Effects of layering
The layered structure of the cuprate high- compounds will in some cases demand a
description beyond the anisotropic GL approach. This is especially true when studying
vortices, where the size of the vortex core may be comparable to the layer spacing 2.
Anisotropy and layering are closely related. If a system of superconducting layers with
thickness ( are stacked with a periodicity 2 along the axis, the in-plane penetration depth
will be modified to become 2(
, where is the bulk penetration depth
for the assumed isotropic superconducting sheets (Clem 1989). The decay length for
screening currents along the axis will, on the other hand, be controlled by Josephson
tunneling. With a Josephson critical current density 3 , the penetration depth is given by
23 in the limit of weak coupling. This results in an effective anisotropy
. Close to where is much larger than 2, the system can similarly
be described by effective coherence lengths and the anisotropic GL theory is
recovered.
The exact expression for 3 depends on the model assumed for the binding slabs and
the corresponding . The limit of 3 is discussed by Clem (1991). In this case
the order parameter vanishes between the superconducting planes, and vortices are formed
only as true pancakes in the planes. These pancakes interact only through magnetic coupling. Effects of a nonzero 3 have to be considered when 2 is no longer much larger
than (Clem 1991). This is always the case at high temperatures, and in most materials at all temperatures. A similar condition, ² 2, describes when can be
considered uniform so that layering is no longer important9 (Lawrence and Doniach 1971,
Kes et al. 1990).
Possibly more realistic descriptions of the layered structure include taking into account
a nonzero order parameter in the binding slabs as a proximity effect (where electron pairs
diffuse into the normal metal), or as arising from nonequivalent layers (CuO, CuO for
YBCO) with different or effective masses. The proximity effect should lead to both
field- and temperature-dependent effective anisotropy (Feinberg et al. 1994), while the
nonequivalent layers should have an effect on the temperature dependence of the upper critical field (Bulaevskii and Vagner 1991). Many models use the formalism of the
Lawrence–Doniach theory (Lawrence and Doniach 1971), which discretizes the order parameter to two-dimensional, superconducting sheets stacked with a distance 2. Daemen
et al. (1993) considered the effect of thermal fluctuations on the vortex pancakes and found
that pancakes may thermally decouple above a certain field-dependent transition temperature. Misalignment of the pancakes causes a phase difference between the layers, which
locally depresses the -axis critical current.
When a field is applied close to the )* plane, the shape and direction of the vortices will
be strongly affected by anisotropy and layering. Vortices prefer to align with the planes,
since the vortex cores then can be situated between the layers, where the order parameter is
lowered. This is, however, disadvantageous to the screening currents, which prefer to flow
in the planes and therefore try to locally turn the vortex directions. In between the planes
9 A small
does not by itself imply layering. It is, however, a condition for a layered structure to appear
layered.
22
Chapter 1. Introduction
Josephson vortices are formed in case of weak coupling. These vortices do not have a core
with vanishing order parameter since the Josephson currents never diverge.
Even if the magnetic field is both high and aligned to the axis, there could be cases
where the anisotropic GL description is less good. When the vortex-lattice spacing )
approaches the plane distance 2 scaled by a factor , the interaction between different
vortices will become stronger than the interaction between the pancakes in each vortex.
This leads to a sort of decoupling or quasi two-dimensional behavior (Feigel’man et al.
1990, Fisher et al. 1991, Glazman and Koshelev 1991). The corresponding field is of the
order of
3D–2D 2
(1.23)
1.3.4 Anisotropy in high- materials
The anisotropy for optimally doped YBa Cu OÆ is one of the lowest among the layered
high- materials. The variation of anisotropy with the oxygen deficiency Æ has been
studied by Janossy et al. (1991) and Chien et al. (1994). Measurements at oxygen content
Æ yield an anisotropy (Janossy et al.), a value which increases up to between
30 (Janossy et al.) and 70 (Chien et al.) for Æ , where K. Optimally doped
YBCO (Æ ) has in the range 7 – 9 according to these and most other measurements.
There is also an in-plane anisotropy in YBa Cu OÆ due to its orthorhombic structure
and CuO chains. This anisotropy is sometimes masked by twinning, i.e., the presence
of a planar defect structure where the ) and * directions are spatially mixed, but can be
studied in untwinned single crystals. According to Ishida et al. (1996), the )* anisotropy
is .
Both Bi-2212 and Tl-2212 have higher anisotropy than YBCO. Values often mentioned
are between 55 and 200 for Bi-2212 and around 90 for Tl-2212 (Brandt 1995, Martínez
et al. 1992). Further references to measurements are given by Blatter et al. (1994). Changing the oxygen content seems to have a similar effect on the anisotropy in these materials
as in YBCO.
The usual way to determine the anisotropy is to perform torque measurements, where
the magnetic torque 4 is measured as a function of magnetic field direction.
These measurements thus use the fact that and are no longer parallel in anisotropic
materials. An expression for the torque has been given by Kogan (1988), derived from the
3D anisotropic GL / London model. Results quoted above all used torque measurements.
The angular dependence predicted by Eq. (1.22) or the BGL scaling, in general describe
experiments well except in a narrow range close to the )* plane. The agreement between
various models and experiments is discussed by Kadowaki et al. (1994) for the case of
resistivity and by Trajanovic et al. (1997) for the case of critical current density.
1.4 Vortex motion
Perfect conductivity is a property of superconductors only in static conditions. Losses
could be expected when the supercurrent density or the amplitude of the order parameter
1.4. Vortex motion
23
(i.e., ) varies with time, since such variation gives rise to electrical fields, as seen from
the first London equation, Eq. (1.1). A major implication of this is that the mixed state
becomes dissipative when vortices start to move.
1.4.1 Dissipation of a moving vortex
The relation between vortex motion and dissipation is complex. A complete description
would require using the time-dependent GL theory and a nonlocal description of the vortex
core. In the limit of high the vortex can, however, be reasonably viewed as consisting of
a normal core of radius (Caroli et al. 1964) surrounded by supercurrents and described
locally (Bardeen and Stephen 1965).
Two main effects are suggested to contribute to the dissipation. First, when a vortex
moves, a field distribution is created around the vortex due to the change in relative
to the stationary superconductor. Inside the vortex core an electrical field is found which
matches the surrounding field. Since the core can be viewed as normal conducting, there
will be a core current creating dissipation. Second, the electrical field in a transition region outside the vortex core is suggested to give rise to dissipation of a similar order of
magnitude (Bardeen and Stephen 1965, Tinkham 1996).
The first contribution can be estimated using the first London equation. The supercurrent
of a vortex expressed in the velocity of the Cooper pairs is given by and thus
. Assuming that the vortex moves with a velocity , the time derivative
in Eq. (1.1) can be rewritten as
(1.24)
which follows from the chain rule of differentiation when the total derivative is zero. The
microscopic electrical field around a moving vortex thus is
!
(1.25)
where the relation was used to describe the supercurrent around a stationary vortex centered at . Equation (1.25) is the field of a dipole in the plane
perpendicular to the vortex line. From the boundary conditions at , the field inside
the vortex core can be found. This field is given by
!
(1.26)
where is a vector with magnitude directed along the vortex line. With an assumed
normal-state resistivity 5 of the core, and assuming that current can flow in the direction
of the electrical field, the dissipated power per unit length of the vortex will be 6 ! $ 5 " ! 5 (Stephen and Bardeen 1965). If the vortex velocity is zero but
a supercurrent external to the vortex is applied, no energy will be dissipated. In this case
the external supercurrent flows around the vortex core.
24
Chapter 1. Introduction
A third contribution to dissipation, suggested as an alternative view of the previous picture, has been given by Tinkham (1964). When the vortex moves, the order parameter is
temporarily depressed and thus has to be restored behind the vortex. Tinkham suggested
that there should be a relaxation time for this process given by 4 .
The voltage created between two probing contacts when a vortex passes by is most easily
considered as an induced Faraday voltage . There are some problems with
such an interpretation, however, and the effect is best referred to the motion of vortices in
a way as if it were due to induction (Kim and Stephen 1969). For a superconductor in a
magnetic field , the corresponding average electrical field is directed as the field
in the vortex core, Eq. (1.26), and is thus perpendicular to both the vortex direction and the
velocity of the vortex. It is given by (Josephson 1965)
(1.27)
1.4.2 Elementary forces
A main force acting on a vortex is the Lorentz-like force which is caused by the magnetic
pressure gradient (Gorter 1962a,b) or the attempt to straighten magnetic flux lines (Tinkham 1964), in general arising when an external current is applied. The supercurrent is
thus reduced by the external current on one side of the vortex and enhanced on the other,
causing a force on the vortex in the direction where the screening current density is increased. The same force acts between different vortices, causing a repulsive interaction
when the vortices are aligned in the same direction. For a single vortex the force per unit
length is given by
(1.28)
An ensemble of vortices characterized by the mean flux density thus feels the force
per unit volume, where is the applied current density.
The Lorentz force of Eq. (1.28) is valid when the vortex is at rest or moving slowly
compared to the driving current. An additional Magnus contribution arises when the vortex
is moving, since it is only the relative motion between the supercurrent and the vortex
which gives rise to a force. The applied current density can be written 7 where
the charge density of the condensate is 7 . Including the Magnus contribution, the
Lorentz force (1.28) is replaced by
7 (1.29)
This force, first suggested by de Gennes and Matricon (1964), is of the same origin as the
force on a vortex in a classical uncharged fluid and tries to drag the vortex along with the
current flow (Vinen 1969). The validity of Eq. (1.28) or Eq. (1.29) has been a subject of
intense debate. It has been argued that the charged background should cancel the effect of
the relative motion (Bardeen 1964). Detailed calculations however showed that Eq. (1.29)
is correct (Nozières and Vinen 1966) and recent studies confirm this view [see Brandt
(1995) for references].
1.4. Vortex motion
25
y
ff
vs x Φ0
x
θH
(η/ρ s)vv
-vv x Φ0
fM
j
vs
vv
Figure 1.13. Forces acting on a moving vortex in the flux-flow state when no pinning is present.
The direction of the driving force at steady-state is the same as the direction of the vortex
velocity. It is most easily seen as composed by a Lorentz force perpendicular to the driving
current and a Magnus contribution perpendicular to the vortex velocity. The Hall angle is a
measure of the vortex motion in the direction of the driving current .
Since the force (1.29) is the only external force in a clean enough system, the equilibrium
vortex flow in such a system is expected to be found when the vortices are moving with
velocity 7 along with the supercurrent fluid (Blatter et al. 1994). This limit has not been
found in experiments, however. Due to the dissipation in the vortex core of a moving vortex
there should be a friction force acting to restore the motion. The result of such a force is
an equilibrium flow state where the is not necessarily equal to . The friction force
corresponds to a viscous damping (with the coefficient 8 characterizing the superfluid) and
is commonly assumed to be directed opposite to the vortex velocity,
8 (1.30)
with defined per unit length of the vortex. The size of the coefficient 8 depends on the
model of electron scattering in the vortex core. A microscopic treatment of this problem
has been given by Kopnin and Kravtsov (1976a,b).
Except for the driving and friction forces, real materials contain defects and disorder
giving rise to an additional pinning force , which tries to prevent the vortex motion. The
effect of such pinning is discussed in Section 2.1.
1.4.3 Flux flow
In clean enough materials or when the driving force far exceeds the pinning force, the vortex flow gives rise to a flux flow with Ohmic resistance (Kim et al. 1963, 1964, Anderson
and Kim 1964). This steady-state flow corresponds to the dynamical equilibrium where
the friction force has the same magnitude as the Magnus force; 8 7 .
The direction of is given by the Hall angle as defined in Fig. 1.13. Equating velocities
26
Chapter 1. Introduction
along and perpendicular to the vortex velocity then gives
"
,
" ,
(1.31)
7 (1.32)
8
where " 7 . It is seen that the dissipation is Ohmic since the electrical field $ . It can be shown that the current flowing in the core during flux flow is such
that it corresponds to the transport current (Bardeen and Stephen 1965). Only the field
component parallel to the current will give rise to resistance. The flux-flow resistance
therefore is
"
5
$
" , , 8
(1.33)
The dissipated energy during a time is given by the force acting over the covered distance
9 " 8" . A comparison of this with the calculated power of Section 1.4.1, reduced by , due to the angle between and , gives
8 , ! 5 where the dissipation was taken to be two times that of the core.
Inserting this coefficient into Eq. (1.33), the resistivity can be expressed as
" . The power is thus given by 6
5
5
(1.34)
with ! . This is the traditional result for flux flow, first formulated in this
way by Strnad et al. (1964). Experimentally, the observed is usually best taken as the
zero-temperature value (Kim et al. 1965).
As presented here, the Hall angle can be found from the relation , , 7 5 . The right hand side of this equation can be written as : 4 by
using the relations 7 , 5 4 , and : , where .
Discussions and other expressions for the flux-flow Hall angle can be found in the works
by Bardeen and Stephen (1965), Nozières and Vinen (1966), Kim and Stephen (1969),
S̆imánek (1994), and Blatter et al. (1994).
Chapter 2
Vortex matter and phase transitions
2.1 Pinning in different vortex regimes
As long as lattice distortions are small, the flux-line lattice (FLL) can be described with
an elastic continuum approximation (Brandt 1995). The FLL interacts with pinning in a
collective manner, described by the pinning strength and concentration, the presence of
thermal fluctuations, and the elastic properties of the lattice. In the high- superconductors, more exotic effects are found when thermal fluctuations become large.
2.1.1 Flux pinning and creep
There is an additional pinning force trying to prevent vortices from moving when defects
are present in a superconductor. The pinning will give rise to a depinning critical current
density , below which there will be no vortex motion at low temperatures (where thermal
fluctuations are small). For a single vortex the maximum pinning force is given by
(2.1)
When looking at an ensemble of vortices the interaction between the vortices has to be
taken into consideration, and therefore one cannot simply sum the pinning of all vortices
to get the total pinning force. The relation between and pinning force can still be written
on the same, simple form, as Eq. (2.1) provided that is allowed to have additional field
and temperature dependencies arising from vortex interactions, i.e.,
(2.2)
where is the pinning force per unit volume and is the average flux density. Equation (2.2) does not show how to calculate on an ensemble of vortices, but transfers the
problem to a question of determining .
The effective pinning depends on the ability of vortices to adopt to pinning centers. In
addition to the properties of the FLL, factors such as size, strength, density, and distribution
of pinning centers could all be important. The basics of the problem of collective pinning
has been given by Larkin and Ovchinnikov (1979). In their description, a distorted vortex
27
28
Chapter 2. Vortex matter and phase transitions
F
FL = jB
Fp
jcB
coherent
volume V
V c (B,T,…)
Figure 2.1.
and the pinning force
Comparison between the driving Lorentz force as a function of coherent volume . At low current densities, only large coherent
volumes can be depinned. Large enough volumes would actually always be depinned. Such
volumes are not necessarily available, however, since the FLL prefers to respond to the pinning
only within coherent bundles of size .
lattice is assumed to be in a landscape of equally strong, randomly distributed pinning
centers, which are characterized by their number density . Looking at a volume , the
total pinning is suggested to be given by the variance of the sum of random numbers,
since the pinning forces acting on the lattice do not all add in the same direction. The total
pinning then scales as , which gives a pinning force density
; (2.3)
The driving force density [see Eq. (1.28)] is, of course, independent of .
From a comparison of the two forces it is therefore seen that small enough volumes will be
pinned, while large volumes cannot withstand the Lorentz force, as illustrated in Fig. 2.1.
A condition for this argument to be valid is that the vortex system acts coherently. By
splitting up a large FLL into smaller, coherent volumes acting individually, each volume
can stand a higher driving force, and the total pinning is increased. The division into
smaller volumes is dependent on the properties of the FLL. The actual coherent volume is given by a comparison of the pinning energy with the cost in (elastic) energy to deform
the FLL between the coherent volumes. Strong or dense pinning tend to decrease while
low magnetic fields or high elastic moduli increase .
thermal
When the temperature is increased, the role of thermal fluctuations becomes increasingly
activation important. It is shown (Anderson 1962, Anderson and Kim 1964) that vortices will be able
to move, or creep, due to thermal activation of the vortices, even when the driving current
alone would not suffice to depin the system. The flux creep can usually be expressed in
terms of an activation energy for flux motion, . In conventional type-II superconductors the creep is experimentally observed as a decay of the persistent current in a hollow
cylinder containing trapped flux (Kim et al. 1962, 1963). Due to the flux gradient in such
a setup, there will be a driving force trying to push vortices out through the cylinder wall.
2.1. Pinning in different vortex regimes
29
The vortex motion will be unhindered as long as the screening current density is higher
than the depinning , and therefore the trapped flux will rapidly decay until . But
as soon as º the pinning becomes effective and no further decay should occur, were
it not for the thermal fluctuations.
In the basic flux-creep model vortex bundles are assumed to be thermally excited over
barriers separating one place of strong pinning from another. The rate of successful jumps
then is proportional to a Boltzmann factor !" , where is given by the pinning
energy of a typical bundle of volume , but modified by the driving force times
the jumping distance Æ . For a jump along the driving force, the effective activation energy
then is
Æ (2.4)
where is the critical current density in absence of flux creep, defined to give a disappearing at . Assuming that Æ stays the same for all jumping directions, the electrical
field becomes proportional to the difference in rate of jumps along and opposite to the
Lorentz force (Tinkham 1996, Dew-Hughes 1988), so that
$ ¼ #
(2.5)
This is a strongly nonlinear response, $ !" for , and
corresponds to a logarithmic decay of screening current with time, in general agreement
with experiments on low- materials (Beasley et al. 1969). For and ,
jumps in the forward direction are dominating. Equation (2.5) can then be simplified to
$ !" , with given by Eq. (2.4). The prefactor of Eq. (2.5) is
usually taken from experiments, and can be both field and temperature dependent, but does
not depend on in the simple description of Eq. (2.4).
In low- materials, flux creep can only be studied at , where Eq. (2.4) is
a good approximation. The creep rate in high- materials is, however, much larger. To
describe the case, the theory of collective flux creep was developed by Feigel’man
et al. (1989). In this theory the relation comes from an underlying variation of bundle
size and hopping distance. The main result is an activation energy that diverges at ,
(2.6)
where takes different values depending on the type of creep. For the important case
where is independent of at low temperatures, corresponding to creep of collectively
pinned single vortices, the theory predicts . The two current regimes of Eqs. (2.4)
and (2.6) are often combined into a single relation (Geshkenbein
et al. 1989).
Measurements interpreted within this modernized theory of collective creep have been
presented by for instance Krusin-Elbaum et al. (1992) and Küpfer et al. (1994). Flux creep
in YBCO over long periods of time have been studied by Thompson et al. (1991) and
support the picture of barriers that increase with time and, thus, decreasing current density.
30
Chapter 2. Vortex matter and phase transitions
The collective creep theory is valid for cases of elastic deformations. When the barriers
associated with such creep becomes large enough, other mechanisms may start to dominate. One such mechanism could be plastic deformations of the FLL at high temperatures
in the low current limit (Geshkenbein et al. 1989, Vinokur et al. 1990a,b).
TAFF
2.1.2 Thermally assisted flux flow
Even at low currents , some type-II superconductors in magnetic field display a
linear (Ohmic) resistance well below . This is especially the case for high- materials.
An explanation for this phenomena, known as thermally assisted flux flow (TAFF, Kes
et al. 1989), was suggested by Dew-Hughes (1988). The TAFF concept is closely related
to flux creep, and is found in the limit where the argument of the #. factor in Eq. (2.5)
is small, i.e., where # . .. This gives an Ohmic, resistivity
5 5 ¼ (2.7)
with appearing as a characteristic activation energy. In low- materials, the ratio
is large over the major part of the vortex phase diagram, which makes the TAFF
regime normally inaccessible.1 In thin films, however, the pinning is reduced and an Ohmic
resistivity appears even at low currents when a magnetic field is applied (Masker et al.
1969, Horn and Parks 1971, Graybeal and Beasley 1986, Garland and Lee 1987). Support for an interpretation of these observations in terms of TAFF (Palstra et al. 1990) is
found in the activated behavior, the low flow resistivity 55 , and the appearance of nonlinear response at increased current densities. In high- materials, many
factors such as short coherence lengths and high temperatures act together to decrease the
magnitude of the exponent in Eq. (2.7). This gives rise to pronounced TAFF behavior and
relatively low values of .
Figure 2.2 shows a typical resistive transition of disordered YBa Cu OÆ , measured
at low current density with Ohmic response. Let us first focus on the resistivity at low
levels, 55 . At such levels, thermally assisted flux flow is prevailing. The TAFF is
characterized by an activation energy which is field and slightly temperature dependent.
In addition to the temperature dependence deduced from the slope of the curves, a factor
is probably present in but cannot be directly probed, since such a factor
leaves the slope unchanged. Taking this factor into account, the activation energy is of
the order of 20 for all magnetic fields if determined at a constant, low level of 55 .
Measurements on Bi and Tl compounds usually show good agreement with a TAFF model
described by an energy
(2.8)
with possibly field dependent (Palstra et al. 1988, 1990) but typically around 0.5 (Nikolo
et al. 1992, Yamasaki et al. 1993, Wagner et al. 1995, Attanasio et al. 1995, Miu et al.
1998). The temperature dependence for YBCO with relatively strong pinning is usually
slightly more complex and the exponent somewhat higher [ (Fendrich et al.
1 Note
that the crossover from TAFF to flux creep at decreases with increasing
.
2.1. Pinning in different vortex regimes
31
10 0
YBa 2Cu3O7-δ
Tc
ρ/ρ n
10 -1
10 -2
10 -3
10 -4
1.1
1.2
1.3
1.4
1.5
100 / T ( K -1)
Figure 2.2. Resistive transition of a disordered YBa Cu OÆ single crystal in magnetic fields
, , , , and
T (from left to right). The Arrhenius graph emphasizes the activated
behavior and resistivities at low levels of dissipation. The activation energy can be found from
the slope . Two different dissipative regimes are clearly seen, separated by a
crossover at
–
. Thermally assisted flux flow is found at low resistivities, while
flux flow prevails at higher temperatures. The zero field critical temperature is indicated by .
! "! "" 1995, López et al. 1998), (Wang 1999)]. The experimental differences between
various high- materials do not seem to be of fundamental character, however, since a
temperature dependence similar to that of YBCO can be found also in for instance Bi2212 (Kucera et al. 1992), Bi-2223 (Wagner et al. 1995), and Tl-2212 (Lundqvist et al.
1998a).
Plastic deformations of a FLL are thought to be due to thermally activated motion of
dislocations over Peierls barriers (Geshkenbein et al. 1989, Vinokur et al. 1990a), corresponding to the creation of double-kinks. The energy for such kinks has been estimated
to ) . The predicted behavior of plastic TAFF is thus in fairly good agreement
with Eq. (2.8) and experimental observations.
As expected from Eq. (2.5), dissipation eventually becomes nonlinear when the current
is increased in the TAFF regime. The current dependence does not strictly follow that of
Eq. (2.4), however. Zeldov et al. (1989) using the simple, activated expression of Eq. (2.7)
reported a logarithmic current dependence
(2.9)
for larger than some number of the order of . [At lower currents the
activation energy remains constant and true TAFF response prevails, as can be seen in
measurements by for instance Wang et al. (1997).] The logarithmic variation at high currents corresponds to a power-law2 in $ , instead of the exponential variation expected
from Eq. (2.5).
2 The
exponent of the power-law should be given by in this description.
32
Chapter 2. Vortex matter and phase transitions
The most direct explanation for the logarithmic current dependence is that it could be
caused by the spatial shape of the pinning potential (Zeldov et al. 1989, 1990). The differences between Eq. (2.9) and Eq. (2.6) are not very pronounced, though. Many mechanisms
may therefore be able to describe the current dependence. Magnetic relaxation measurements have been found to be consistent with Eq. (2.9) down to low temperatures (Maley
et al. 1990) and in various high- materials (Hergt et al. 1993, Gjølmesli et al. 1995).
The main difference between low and high temperatures is that at low temperatures no
Ohmic regime is found, but the activation energy seems to diverge with decreasing current.
The divergence is sometimes even stronger than a power-law in $ (Koch et al. 1989,
Sengupta et al. 1993), indicating a possibly truly superconducting state for .
2.1.3 Depinning and flow
As seen in Fig. 2.2, there seems to be a crossover at 55 – between the TAFF
regime at low resistivities and a highly dissipative regime. Above this crossover the resistivity is Ohmic at all current levels (Zeldov et al. 1989), suggesting that the flux-flow
mechanism is limiting the dissipation. The crossover can therefore be thought of as a
depinning transition, above which the barrier is no longer important for the vortex motion (Vinokur et al. 1990a).3 This does not necessarily mean that becomes zero at the
crossover. With a large prefactor 5 5 of Eq. (2.7) flux flow could start when is
still large compared to .
Even the flux-flow regime in Fig. 2.2 appears to be thermally activated. Likely, however, this is an artifact of an analysis limited to the small range over which flow is found.
Gordeev et al. (1999) showed that the resistivity can be rather well described by a simple
flux-flow relation, 5 5 . The flow resistivity can be studied over a wider
range in temperature by driving the FLL into the flux-flow state by an applied current, as
shown in Fig. 2.3. When is increased into the range , the
flux creep increases rapidly until the (nonlinear) creep resistivity exceeds 5 . When this
happens, a crossover from creep to flux flow takes place, corresponding to current-induced
depinning.
Pinning in the flow regime is not totally ineffective. For the pinning gives rise to
a friction-like force ; , which is approximately independent of flow velocity and
driving current. Taking this force into consideration will change the flux-flow expression
so that is effectively replaced by (Strnad et al. 1964, Kim et al. 1965). In this
case the true flux-flow resistivity, expected at , can be found from the slope $ .
As seen in Fig. 2.3, the flux-flow resistivity obtained in this way agrees well with the flow
resistivity at high temperatures. The slight curvature of the resistivity in the Arrhenius
graph is consistent with Bardeen–Stephen flux flow. Note that the dip in 5 appearing
at the temperature in Fig. 2.3 is also present in the differential resistivity and in the
apparent flux-flow resistivity. The dip is related to a melting of the FLL at and is
further discussed in Section 2.2.4.
3 The onset of TAFF could, similarly, be regarded as due to thermal depinning. Brandt (1995) uses depinning
to denote both the onset of Ohmic TAFF and flux flow.
2.1. Pinning in different vortex regimes
33
10 2
4T
ρ (µΩcm)
10 1
Tm
YBa2Cu3O7-δ
C
2T
10 0
A
B
7 A/cm2
1500
A/cm2
0.7 T
10 -1
0T
10 -2
10 -3
1.10
1.15
1.20
1.25
1.30
100/T (K-1)
Figure 2.3. Resistive transition of a relatively clean YBCO single crystal. Magnetic fields are as
indicated. The change from TAFF to flux flow is illustrated by the three curves A – C. Curve A
corresponds to Ohmic TAFF. Curve B displays nonlinear flux creep at low levels of , crossing
over to almost Ohmic flux flow at higher levels. Curve C shows the flux flow at , as
obtained from the differential resistivity
. The close connection between the flux flow
above and below is illustrated by the dashed curve. The dip in at is related to a melting
of the FLL. See also Paper IV (Rydh et al. 1999a).
!#!
"
" 2.1.4 Vortex liquid
General considerations of the elastic properties of the FLL show that the shear modulus,
i.e., the stiffness to shear forces acting between different vortices, should vanish when
the upper critical field is approached. This should lead to diverging vortex vibrational
amplitudes. The FLL could consequently be expected to undergo a transition to a liquid
state at high enough temperatures or fields (Labusch 1969).
Early experimental signs of a vortex solid-to-liquid phase transition in the mixed state of
high- superconductors were the observation of a finite, low-current resistivity well below
(Palstra et al. 1988, 1990, Tinkham 1988, Koch et al. 1989) and the occurrence
of a corresponding irreversibility line , above which reversible magnetization is
found (Müller et al. 1987, Yeshurun and Malozemoff 1988). The irreversibility line reflects a boundary between a pinned regime, where relaxation of the vortex system is slow
compared to measuring times, and an almost unpinned regime with quick relaxation. Further support of a transition were obtained by mechanical oscillator measurements (Gammel
et al. 1988) and the current – voltage (< – ) characteristics around a transition line separating reversible, Ohmic response from a superconducting phase with nonzero critical
current density (Koch et al. 1989).
The intuitive picture of a liquid as a state of highly mobile vortices is far from obvious,
since a soft vortex system could adopt to the pinning more easily, which could lead to
increased pinning in the liquid. As the number of experimental investigations increased,
it turned out that the amount and type of disorder is important for the behavior of the
transition. In very weakly disordered materials at low and intermediate magnetic fields,
34
Chapter 2. Vortex matter and phase transitions
signs of a first-order melting transition are found, while more strongly disordered materials
show a behavior in better agreement with a glassy, second-order transition. In both cases
the liquid close to is Ohmic at all current levels with a resistivity of the order of the
Bardeen–Stephen flux flow.
The main differences in dissipative behavior of the vortex liquid between the clean and
disordered cases are found in the neighborhood of the solid-to-liquid transition. In moderately clean materials, the transition from a weakly pinned solid to a flowing liquid takes
place in a narrow temperature range, K, corresponding to curve A in Fig. 2.3.
Uncorrelated defects increase the width of the TAFF regime by displacing the onset of
dissipation to lower temperatures, but also by raising the temperature of the crossover to
the flux-flow regime, see Paper I (Lundqvist et al. 1997) or Petrean et al. (2000). The
qualitative difference between weakly and strongly disordered systems can be understood
by considering the correlated volume. In systems with weak disorder the volume changes
rapidly at the melting temperature from a vortex lattice with vortex–vortex correlation over
long distances, to a system of independently moving vortices. Point disorder thus disturbs
the vortex lattice structure so that the transition into a liquid becomes more gradual. These
observations show that the liquid state is favored by point disorder (Fendrich et al. 1995).
This is in accordance with for instance the usual lowering of the freezing temperature when
salt is dissolved into water.
It is clear from the reasoning in Section 2.1.1 that dense, strong disorder should pin
independent flux lines of a vortex liquid more effectively than a FLL could be pinned.
Such strong liquid pinning may be found in some low- materials, as indictated by a peak
effect close to (Berlincourt et al. 1961, DeSorbo 1964, Pippard 1969). In disordered
high- materials, however, the rapid increase of the resistivity when the vortex liquid is
entered indicates that disorder is either (a) dense, but weak in the liquid, or (b) strong,
but not too dense. In the first case one has to explain how a change from strong to weak
disorder is related to a melting of the FLL. In case b, the question why vortices that are
unpinned do not simply flow around the pinned ones has to be answered.
Many models and theories have been suggested to describe the vortex liquid. It was
early noted that the expected dependence of the resistivity on the angle between current
and magnetic field was not found. The contribution from the Lorentz force seemed
to constitute only a minor fraction, if any, of the total dissipation. This observation, and
the general broadening of the resistive transition, have frequently been ascribed to a thermally fluctuating system of stacked Josephson junctions or weak links (Kim et al. 1990,
Briceño et al. 1991, Gray and Kim 1993, Blackstead and Kapustin 1994). A model for the
resistivity of such a system was provided by Tinkham (1988), who based it on a junction
description by Ambegaokar and Halperin (1969). Suppression of the macroscopic Lorentz
force is weaker in less disordered samples, however [Kwok et al. 1990, Rydh and Rapp
2001 (Paper VIII), Björnängen et al. 2001 (Paper X)].
Vinokur et al. (1990a) explained the activated TAFF behavior in the liquid by large barriers for plastic vortex motion. Such barriers have been suggested to arise from a very
viscous liquid due to vortex entanglement (Nelson 1988, Nelson and Seung 1989) with
possible cutting and reconnecting of the vortex lines (Vinokur et al. 1990a). It has been
argued that, due to the layered structure, the energy of cutting vortices is much lowered in
plastic
motion
2.2. Vortex-lattice melting
35
high- materials compared to conventional type-II superconductors. Plastic creep, possibly governed the motion of dislocations in the FLL (Abulafia et al. 1996), or deformations
through entanglement (Schönenberger et al. 1996) have been suggested to also explain the
dynamics of the vortex solid in some cases.
A different approach to the vortex-liquid dissipation is the nonlocal, hydrodynamic the- hydroory by Marchetti and Nelson (1990, 2000), where the liquid behavior is controlled by a dynamics
set of friction and viscosity coefficients. The nonlocality in this theory arises from vortex interactions and entanglement, and becomes important if the viscous terms are large
compared to frictional drag terms from pinning and the Bardeen–Stephen mechanism.4 It
is argued that the nonlocal vortex flow, caused by the viscous forces, are accompanied by
nonlocal electrodynamics, where $ depends on the current density , (Huse
and Majumdar 1993).
2.2 Vortex-lattice melting
2.2.1 Evidence of a first-order transition
A first-order transition (FOT) is characterized by a discontinuity in the derivative of the
free energy . The entropies of two phases at a FOT are thus not the same.
Second- and higher-order transitions have their first discontinuity in the th derivative
of , where is the order of the transition. Observation of latent heat is
therefore a clear indication of a FOT. Transitions without latent heat are denoted continuous
transitions. A transition between phases of different spatial symmetry, such as melting, is
theoretically expected to be of first order (Landau 1937).
Indications of a first-order transition in the vortex system have been found by a number of experimental methods. The most direct evidence of a FOT in YBCO is given by
latent
calorimetric measurements of the latent heat (Schilling et al. 1996). Measurements of the
heat
specific heat reveal a narrow peak at the transition, from which and
can be found by integrating the peak area of (Junod et al. 1997, Schilling et al.
1997).
The vortex density of the two phases at the FOT are slightly different. This is reflected
by a jump ' in the magnetization (Liang et al. 1996, Welp et al. 1996, Nishizaki et al.
local
1996a,b) or in the local magnetic induction ' (Zeldov et al. 1995) when
passing the FOT. Magnetization measurements of Bi-2212 are complicated by the fact that induction
the FOT is found at relatively low magnetic fields only, T. Platelet samples
at such fields may give a flux density which is nonuniform, i.e, with slightly increasing
towards the center of the sample. This effect of surface barriers smears the result of global
magnetization measurements (Zeldov et al. 1995) and induces irreversible response (Majer
et al. 1995). While such magnetization measurements therefore show wide transitions and
low values of (Pastoriza et al. 1994), the jump in from local induction measurements
takes place in a very narrow field and temperature range, mT, mK (Zeldov et al. 1995).
4 The vortex viscosities here should not be confused with the “viscous damping” coefficient
which is a friction force arising from the motion of a vortex in the supercurrent fluid.
$ of Eq. (1.30),
36
Chapter 2. Vortex matter and phase transitions
The jump at the FOT follows from the equality of of the two phases at the transition. It is related to the slope of the transition line and to the jump in entropy, as described
by the Clausius–Clapeyron equation
(2.10)
Here is the volume and is the phase-transition line. As indicated by Figs. 1.1 and
1.10, the slope is usually negative so that , as well as , is higher in the liquid,
i.e., the vortex liquid is denser than the solid. The entropy change at the FOT is usually
specified for a volume 2 , where 2 is the layer spacing, and is found to be around
0.5 per vortex and superconducting layer for both Bi-2212 and YBCO ( ). The
first-order transition in YBCO is observed for fields typically in the range 0.5 – 8 T. Due
to this very different scale of as compared to Bi-2212, the detection of a jump in
YBCO by local induction measurements will require a much improved resolution .
A more direct evidence of a first-order melting transition is given by the differential
magneto-optical (MO) technique developed by Soibel et al. (2000). With this technique
the melting process of Bi-2212 can be visualized directly. Soibel et al. found that the width
of the melting transition comes from a combination of disorder and the small, intrinsic
gradient in the internal magnetic field. Importantly, droplets of vortex liquid coexist with
the solid, as expected for a first-order melting, and expand throughout the melting process.
Studies of the vortex-lattice decomposition can also be made by for instance local scanning
Hall probe measurements (Oral et al. 1998), small-angle neutron diffraction (Cubitt et al.
1993, Forgan et al. 1996), line-shape analysis of O NMR spectra (Reyes et al. 1997),
and muon spin rotation measurements (Lee et al. 1993).
vortex
As previously noted, transport measurements do not directly probe thermodynamic quantransport tities. Their sensitivity to the motion of vortices in the mixed state, however, enables the
study of various vortex phase transitions. The presence of some disorder is here a prerequisite, since measurements of a freely flowing system would reveal few details of the
vortex structure. As was seen in Section 2.1.4, low current resistivity measurements on
clean YBCO single crystals display a sharp step at the onset of resistivity (Charalambous
et al. 1992, Safar et al. 1992b, Kwok et al. 1992). The step disappears if the current is
directed along the applied magnetic field so that the driving Lorentz force becomes zero,
as can be concluded from the measurements by Kwok et al. (1990). See also Paper VIII
(Rydh and Rapp 2001) and Paper X (Björnängen et al. 2001). A corresponding step occurs
in the resistivity of Bi-2212, but only up to a lower level of 55 (Pastoriza and Kes 1995,
Fuchs et al. 1996, 1997, Mirković et al. 1999). Step-like features resembling the case of
YBCO have also been seen in some other materials (Mun et al. 1996, Hussey et al. 1999).
The location of the resistive step has been shown to agree well with the FOT determined
from magnetic measurements (Welp et al. 1996, Fuchs et al. 1996).
Transport studies of the Seebeck coefficient (Ghamlouch et al. 1996) and the Hall effect
(D’Anna et al. 1998) are in agreement with resistivity measurements, and show similar
steps at the FOT. The Hall effect in the mixed state of many type-II superconductors displays an anomalous negative Hall angle, in disagreement with simple descriptions of vortex
2.2. Vortex-lattice melting
37
flow (Section 1.4.3). The influence of the melting transition, however, clearly shows the
close relation to vortex dynamics (D’Anna et al. 1998).
Hysteresis in the resistive FOT can be observed if the measuring current density is low hysteresis
enough (Safar et al. 1992b, 1993). This hysteresis is not directly connected to the presence
of latent heat since no time dependent relaxation is found, nor any subloops if the transition
direction is reversed in the middle of the transition (Jiang et al. 1995). Charalambous
et al. (1993) showed that superheating dominated the effect in their measurements. The
direct imaging of the melting process by Soibel et al. (2000) and observations of hysteretic
behavior in the resistive transition by Crabtree et al. (1996) give evidence for supercooling,
on the other hand. Kwok et al. (1994a) emphasized the importance of disorder for the
observation of a certain hysteretic behavior. The experiments thus indicate that the resistive
hysteresis is rather due to a nucleation barrier than being of the thermodynamic nature
found from simulations (Domínguez et al. 1995), and expected for a transition without
any pinning under adiabatic conditions. Regardless of the mechanism, the presence of
hysteresis supports the first-order nature of the transition.
2.2.2 Location of the phase-transition line
Melting of the FLL in high- superconductors is facilitated by the high GL parameter
and the strong anisotropy, which enhance the already large thermal fluctuations. This
gives a melting transition far below . The location of the melting line can
be estimated by studying the mean-square displacement = of the flux lines. The so
called Lindemann criterion, saying that melting usually takes place when the vibrational
amplitudes reach some fraction of the (vortex) lattice spacing, can then be used.
A detailed calculation of = by Houghton et al. (1989) gives an expression
>
(2.11)
for the melting transition close to at intermediate magnetic fields, .
Here (Blatter and Ivlev 1994b), , and > is the Ginzburg number, which
is defined from a comparison of and the condensation energy at (Brandt
1995),
> ! (2.12)
By inserting Eq. (2.12) into Eq. (2.11) and rewriting the critical fields in terms of and ,
the melting transition can be written as (Tinkham 1996)
?
-
(2.13)
where ? is a numerical constant. This equation shows that the main temperature dependence comes from , assumed to vary as in Eq. (2.11). This
temperature dependence is probably somewhat altered by quantum fluctuations, as shown
by Blatter and Ivlev (1993, 1994a,b). A more important contribution is the suppression of
ÀÌ 38
Chapter 2. Vortex matter and phase transitions
12
Tm (K)
µ0H (T)
8
88
4
γ ca = 7.0
γ cb = 8.9
4T
86
84
γ ab = 1.27
H || a
90
b—c
H || b
0
30
60
ϑ (°)
90
H || c
0
84
86
88
90
92
T (K)
Figure 2.4. Phase diagram of the melting transition for an untwinned YBCO single crystal annealed at 400ÆC. The magnetic field is directed along the axis and Æ away from the and
. Inset: Meltaxes. Solid curves are power-law fits to the expression ing temperature as a function of magnetic-field direction for T. The angle is defined
from the axis and lies within the plane. [From Paper X (Björnängen et al. 2001)].
%
the order parameter with magnetic field (Blatter and Ivlev 1994b). Such suppression results
in a change of the penetration depth *, where * .
Various expressions for the temperature dependence of have been compared by Blatter and Ivlev (1994b). The experimental location of the melting line in YBCO is usually
well described by a simple power-law with in the range 1.35 –
1.45. Eq. (2.11) gives a too steep rise of , but is improved if the assumed field dependence of is introduced. For Bi-2212, the temperature dependence is closer to that of
Eq. (2.11). This is reasonable, since the high anisotropy suppresses to a location
far from . It is also in agreement with the expected result of a decreased importance of
quantum fluctuations with increasing anisotropy (Blatter and Ivlev 1994b).
If the position of the melting transition is compared to the location of the upper critical
anisotropy field, it is seen from Eq. (2.13) that anisotropy reduces with a factor . While
the size of cannot directly be obtained from this factor, due to the uncertainty in and , the anisotropy can be found from - , defined in Eq. (1.21). The angular
dependence of due to anisotropy is the same as that expected for . In Fig. 2.4,
the temperature and angular dependence of are shown for an untwinned YBCO single
crystal. As seen in the figure, there is a small in-plane anisotropy due to the
orthorhombic structure with CuO chains along the * axis. The anisotropy between the axis and the plane is found from , where . The result is
for this sample.
2.2.3 Critical points
The signs of a first-order melting transition are usually limited to intermediate magnetic
fields below a kind of critical point (Safar et al. 1993). Above this critical point ,
2.2. Vortex-lattice melting
39
the sharp drop in resistivity disappears and is replaced by a continuous transition, resembling the behavior of more disordered samples. In YBCO the critical point is normally
found at – T while in Bi-2212 it is limited to much lower fields, – mT
(Zeldov et al. 1995).
Small-angle neutron diffraction by Cubitt et al. (1993) on Bi-2212 has indicated that the
vortex system becomes strongly disordered or decoupled at fields above at low temperatures. Further investigations showed that at least some correlation between the vortex
pancakes remains even at these fields, since a re-emergence of the diffracted intensity was
observed at higher temperatures but still above (Forgan et al. 1996). Thus, assuming
that the field corresponds to a transition or crossover in the vortex solid, the transition is more likely induced by disorder than being the field-induced 3D–2D transition
mentioned in Section 1.3.3.
In both YBCO and Bi-2212, a second peak sometimes appears in the magnetization
' , in addition to the usual maximum at . This anomalous peak is named the fishtail effect in YBCO and the arrow-head in Bi-2212. The reason for it still remains somewhat unclear. Some suggested explanations are effects from surface barriers, a disorderinduced transition, or a field-induced 3D to (quasi) 2D transition. The magnetic field where
the second peak in magnetization appears is only weakly temperature dependent and more
or less coincides with . This has led to the thought that this magnetization peak is related to a possible transition in the vortex solid (Yang et al. 1994, Khaykovich et al. 1996,
1997, Deligiannis et al. 1997, Fuchs et al. 1998, Vinokur et al. 1998). Especially the onset of the otherwise rather broad second magnetization peak is pronounced (Khaykovich
et al. 1996, Nishizaki and Kobayashi 2000, Nishizaki et al. 1998, 2000). Experiments
thus indicate at least three different phases in weakly disordered vortex systems; the rather
well-ordered vortex solid or Bragg glass,5 the disordered solid or vortex glass, and a vortex
liquid, possibly turning into a pancake gas. Other phases may be present as well (Fuchs
et al. 1998, Bouquet et al. 2001).
The size of is sensitive to sample preparation. Especially the oxygen content and
structure are important for YBCO and Bi-2212. Thus, while is usually lower than
T in YBCO, the measurements by Nishizaki et al. (1996b) display signs in resistivity
of a first-order melting up to their highest available magnetic fields ( ) of T.
Junod et al. (1997) showed that the specific-heat peaks in fully oxygenated YBa Cu OÆ
(i.e., Æ ) remain up to at least T, and that the second magnetization peak in this
case is suppressed. They also noted that it is possible to obtain optimally doped samples
without the fishtail effect. This indicates that the location of the critical point is influenced
by oxygen vacancies and ordering (see Section 3.1 for a discussion of oxygen structure).
Annealing at high partial pressure and temperature seems to give a higher critical point for
approximately the same oxygen content (Däumling et al. 1996). Oxygen cluster formation
therefore appears to lower .
Twin boundaries in YBCO act as correlated defects along the axis. The presence of
twin structure appears to extend the first-order character of the melting transition to high
5 The main difference between a Bragg glass (Giamarchi and Le Doussal 1995, 1997) and a vortex lattice is
the presence of some pinning. The Bragg glass is still rather well ordered and shows Bragg peaks of characteristic
shape in diffraction (Klein et al. 2001).
second
peak
40
Chapter 2. Vortex matter and phase transitions
Jc (A/cm2)
300
A
200
B
100
YBa 2Cu3O7-δ
0
80
µ0H = 2 T
82
84
T (K)
86 Tm
88
Figure 2.5. Critical current density below the first-order transition for an untwinned YBCO crystal at T, . Two different methods were used to determine . Curve A was
found by applying the relation
at high current densities, thus describing a flowing
vortex system. Curve B was obtained through the criterion
µV/cm, corresponding to the
onset of flux creep. Solid curves are guides for the eye. (A. Rydh, unpublished.)
LCP
# " #
magnetic fields, as can be concluded from calorimetric measurements on twinned YBCO
(Junod et al. 1997, Roulin et al. 1998).6 Kwok et al. (2000) clearly showed that columnar
defects, introduced by heavy ion irradiation, increase . Point defects and correlated
disorder thus have opposite effects on .
At low fields, experiments indicate the existence of a lower critical point (LCP), below
which the transition appears to be of second order (Welp et al. 1996, Schilling et al. 1997,
Roulin et al. 1998). The effect of correlated disorder is to increase the corresponding ,
which can be as large as T in heavily twinned or columnar defected YBCO
(Roulin et al. 1998, Kwok et al. 2000). The solid-to-liquid phase boundary below is shifted upward in temperature (Konczykowski et al. 1991, Civale et al. 1991, Samoilov
et al. 1996, Kwok et al. 2000). In combination with current – voltage characteristics, this
has led to the identification of the phase below the LCP as a Bose glass, described theoretically by Nelson and Vinokur (1992, 1993) and Larkin and Vinokur (1995).
2.2.4 Peak effect
A local maximum in the critial current density appears just below the first-order transition
of YBCO (Kwok et al. 1994b, 1997). This peak effect in is accompanied by a dip
in the high-current resistivity, as was described in Fig. 2.3. In low- materials the peak is
located just below , possibly with the maximum in the vortex liquid (DeSorbo 1964,
Bhattacharya and Higgins 1993, Henderson et al. 1998). The prevailing explanation for
this peak is a softening of the shear modulus of the FLL, falling to zero faster than the
pinning force from defects (Pippard 1969, Larkin and Ovchinnikov 1979).
Figure 2.5 shows at T for the same untwinned sample as in Fig. 2.3. Two curves
6 In resistivity measurements, however, the vortex flow around
may be suppressed by the presence of just
a few twin boundaries (Kwok et al. 1992, 1994c, Fleshler et al. 1993).
2.2. Vortex-lattice melting
41
8
88 K
E / j (µΩcm)
6
86 K
84 K
90 K
82 K
80 K
4
79 K
2
78 K
0
0
2
4
µ0 H (T)
6
8
#
Figure 2.6. Measurement of the nonlinear resistivity
as a function of magnetic field for a
constant current density A/cm . The upper critical point is located close to (79 K, 7 T).
The minimum resistivity of the peak effect follows a dependence close to that of the
K
isotherm. The curves at low temperatures are affected by relaxation, possibly connected to surface
barriers. See discussion in Paper IV (Rydh et al. 1999a). (A. Rydh, unpublished.)
for are shown. Curve A describes for a flowing vortex system, i.e., as found
from the relation $ 5 at , while Curve B corresponds to the onset of
a measureable flux creep. It is apparent that the peak effect is mainly found when the
dynamics is slow. This indicates that the flowing FLL does not fully adopt to the pinning.
The difference is most pronounced just at the peak maximum, where the pinning force
is reduced by a third when going from the static to the dynamic case. Access to higher
driving currents may reduce the of Curve A in this region even further.
A softening of the FLL prior to melting cannot alone explain why the high-current differential resistivity of Fig. 2.3 around is reduced below the estimated free flux-flow
resistivity. The observation is consistent with an apparent increase of the viscouse damping, as found at MHz frequencies (Wu et al. 1997). An alternative explanation is that some
parts of the defective FLL are pinned, while unpinned vortices or small vortex bundles flow
around the occupied pinning sites (Jiang et al. 1997). It has been found that just a few correlated defects may give a sharp peak effect (Kwok et al. 1994b). Sample surfaces could
possibly have a similar pinning effect, see Paper IV (Rydh et al. 1999a). A defective FLL
with different flow velocities is also consistent with current – voltage characteristics. In the
region between the peak maximum and , measurements indicate both a low current and
a high current Ohmic regime (Berghuis et al. 1990, Kwok et al. 1994c, Rydh et al. 1999a).
Many phenomena related to the peak effect remain to be explained. One example is
the coincidence of the high-current resistivity minima of the peak effect with the flow
resistivity of an isotherm close to the upper critical point, see Fig. 2.6. Other features
include the frequency dependence and noise response below (D’Anna et al. 1995, Safar
et al. 1995, Kwok et al. 1997), and the importance of the shape of the probing current in
ac measurements (Gordeev et al. 1997, 1998, Henderson et al. 1998).
42
Chapter 2. Vortex matter and phase transitions
2.3 Vortex glass
While the vortex system in a clean enough material undergoes a first-order melting if the
magnetic field is not too high, strong disorder causes a continuous, glassy transition of
presumed second order. Experimentally, the resistivity of disordered single crystals and
not-too-thin films is found to vanish smoothly without any sharp features, when the irreversibility line is approached from the vortex-liquid side. If this is interpreted as a crossover from thermally assisted flux flow to flux creep, no distinct phase boundary would
be expected between the high-temperature and low-temperature states. It has been argued, however, that a thermodynamic, sharp second-order phase boundary exists in the
– phase diagram of disordered systems in three dimensions (Fisher 1989). The lowtemperature vortex glass below this boundary is predicted to be a thermodynamic phase
with a truly zero resistivity in the limit.
2.3.1 Glassy vortex systems
Disorder can be introduced in various ways. Columnar defects can be obtained by heavyion irradiation (Thompson et al. 1992, van der Beek et al. 1995, Doyle et al. 1996), while
proton or electron irradiation is expected to give point-like disorder (Civale et al. 1990,
Fendrich et al. 1995). An additional source of pinning is the intrinsic oxygen vacancies
found in almost all high- compounds. Since generally depends strongly on the oxygen
content, the local , and hence also the condensation energy and pinning landscape, will
vary with the oxygen content and structure. In YBCO, twin boundaries act as correlated
disorder, particularly when the magnetic field is applied in the planes of the boundaries.
Correlated, intrinsic pinning is also provided by the layered structure of the high- materials.
Various models and theories have been suggested to describe the glassy systems with
different disorder structure. In the case of columnar defects, the theory describes a so
called Bose glass (Nelson and Vinokur 1992, 1993, Larkin and Vinokur 1995). The Bose
glass resembles the vortex glass, i.e., the glass with isotropic disorder structure, but in the
Bose glass the vortex lines are localized on the correlated defects, giving a pronounced
angular dependence and anisotropy. Columnar defects can also be introduced in a splayed
fashion, forcing the vortex system to become entangled (Hwa et al. 1993, López et al.
1997). Other alternatives to the vortex glass are, except for collective flux-creep theories
(Feigel’man et al. 1989), the polymer- and window-like glasses (Nelson 1988, Reichhardt
et al. 2000), with growing time scales when the glass temperature is approached, but with
no true thermodynamic equilibrium. In the following, the discussion will be restricted to
the kind of vortex glass described by Fisher (1989) and Fisher, Fisher, and Huse (1991).
2.3.2 Dimensional scaling
The vortex glass theory and critical scaling have been discussed in more detail by for
instance Fisher et al. (1991) and Blatter et al. (1994). In the vortex glass, infinite vortexmotion barriers are developed, locking the vortices into specific states (Blatter et al. 1994).
The glass state is characterized by a nonvanishing Edwards–Anderson order parameter
, where is the (gauge-invariant) order parameter, and indicates an
2.3. Vortex glass
43
average over thermal fluctuations or disorder (i.e., the latter is an average over points ).
Long-range phase coherence is described by a correlation function
> (2.14)
which is expected to assume a finite value in the vortex glass phase even for .
The behavior of > at long distances is monitored by the glass correlation length . In
the vortex fluid, > should fall off exponentially with .
The vortex glass theory is based on dimensional scaling arguments that should apply
rather generally to any second-order phase transition. It is assumed that near such a transition, physical quantities could be expressed as functions of the correlation length and
a characteristic relaxation time 4 . Here 0 is the dynamic critical exponent, which
should be a constant within each universality class or symmetry of the glass phase. Approaching the glass transition temperature , the correlation length diverges, with the
divergence described by
! (2.15)
where @ is the static critical exponent. Applying dimensional scaling, it is argued that the
vector potential scales as an inverse length [see Eq. (1.4)] and hence the electrical field
$ should scale as 4 . Similarly, the current density should scale
as the derivative of the free-energy density with respect to , and is thus found to scale
as " " , where ( is the dimension. Combining the two scaling forms, one
arrives at a scaling ansatz
$ ; " (2.16)
Here ; is a scaling function describing the behavior above (+) and below (–) the transition. ; . is only partially known, but at small . and for it is given by
; . .. This results in a linear resistivity
5 $ "
! "
(2.17)
at low currents, i.e., for " . To investigate Eq. (2.17) experimentally, one
usually calculates the inverted derivative 5 , which should be proportional to
with a slope @ 0 (. Scaling expressions for other properties, such as the
frequency dependence of the complex conductivity, can be obtained in a similar way as
above.
It is not settled whether the vortex glass is isotropic or anisotropic. In systems with
strong disorder and with an entangled vortex liquid the glass could be isotropic, but an
anisotropic description may be required for the high- materials (Blatter et al. 1994).
A possible anisotropy would not need to be directly related to the effective mass tensor,
but could arise from correlated disorder or from the applied magnetic field, breaking the
symmetry. Anisotropy can be introduced through an anisotropy exponent A , relating the
#
(Nelson and Vinokur
correlation lengths along and perpendicular to the field, 1992, 1993, Marchetti and Nelson 2000). The expected exponent for the Bose glass is
A .
44
Chapter 2. Vortex matter and phase transitions
2.3.3 Experimental observations
The first experimental indications of a vortex glass were presented by Koch et al. (1989),
studying epitaxial films of YBCO. They confirmed general agreement of their $ measurements with the prediction of Eq. (2.16), and found critical exponents 0 and @ – , resulting in the combined exponent 2 @ 0 ( for
( . Further experiments on single crystals of both YBCO and Bi-2212 were found to
agree with the vortex-glass picture (Gammel et al. 1991, Safar et al. 1992a). Other materials which have been reported to display glass-like behavior include Bi-2223 (Wagner
et al. 1995), Tl-2212 (Deak et al. 1995b), and oxygen-deficient YBCO (Hou et al. 1994,
1997, Deak et al. 1995a). See also Paper I and II (Lundqvist et al. 1997, 1998b). Signatures of a second-order transition can furthermore be found for low- materials such as
Nd$ Cu$ CuOÆ (Yeh et al. 1992, Herrmann et al. 1996), Ba K BiO (Klein et al.
1998), and disordered Al (Fujiki et al. 1998). Frequency response in agreement with the
observation of a vortex glass transition in YBCO has been reported by Deak et al. (1993)
and in more detail by Kötzler et al. (1994a,b).
Comparisons of critical exponents between different samples and systems can be found
in many papers (Kötzler et al. 1994a, Roberts et al. 1994, Hou et al. 1997, Cohen and
Jensen 1997). For thin films, exponents @ and 0 are usually found
(Kötzler et al. 1994a). It seems, however, that the critical exponents of single crystals are
more scattered and possibly lower than those of the films. Some measurements also yield
exponents that appear to be field dependent, thus questioning the validity of the vortex glass
theory (Cohen and Jensen 1997). Other reports support the universality of the glass scaling
(Friesen et al. 1996, Moloni et al. 1997). The variation of exponents between different
studies has been suggested to come from systematic errors introduced when determining
(Moloni et al. 1997).
Differences in the disorder structure of samples complicate the comparison and evaluation of measurements. Twinned YBCO crystals without additional, artificial point disorder have been described in terms of a Bose glass (Grigera et al. 1998). Measurements
on untwinned, proton irradiated YBCO, on the other hand, favor a vortex glass (Petrean
et al. 2000). A certain, minimum amount of point disorder may be required to stabilize
this vortex-glass state (Petrean et al. 2000). The possibility of partial first-order melting
in twinned crystals and other relatively clean samples complicates the evaluation of such
systems. Some studies indicate that the first-order transition can be followed by a glassy
transition at lower temperature, the intermediate state named vortex slush (Worthington
et al. 1992, Safar et al. 1993, Reyes et al. 1997). Thus, experimental conditions are far
from ideal. Additional problems with the interpretation of experiments in terms of a vortex
glass are provided by Reichhardt et al. (2000). Nevertheless, the glass theory has proved
powerful in describing all main features of the transition from a vortex liquid to a pinned,
glassy phase.
2.3.4 Location of the glass transition
The first-order transition can be completely suppressed in strongly disordered samples. For
such samples, the glassy, second-order transition has been found to extend over the major
2.3. Vortex glass
54
3
2
101
1
Sample µ0H0 (T)
1
2
3
4
5
100
10
H/H0
µ0 H (T)
15
45
10-1
5
36.9
16.2
7.5
3.7
2.3
10-2
0
0
20
40
60
T (K)
80
100
10-3
10-2
10-1
100
1–T/Tc
Figure 2.7. Left: Glass transition for a series of oxygen-deficient YBCO crystals. The
anisotropy increases from for sample 1 to for sample 5. [See Paper II (Lundqvist
et al. 1998b) for more details.] Dashed curves are fits to the relation with
around
. Deviations from this expression are seen for the most anisotropic samples at high
magnetic fields. Solid curves are fits to the modified relation with
. Right: The glass transitions of the left figure can be scaled fairly well onto
one single curve by plotting as a function of on a logarithmic scale. Deviations
from the simple power-law (dashed line,
) are seen for
, but can be described
by the modified expression (solid curve,
).
& ' & ' range of accessible fields. Close to , the temperature dependence of the glass transition
usually follows the relation
(2.18)
with – . This expression was first used to describe the irreversibility line
(Müller et al. 1987, Yeshurun and Malozemoff 1988). It is also similar to that of the firstorder melting, mainly differing in the prefactor.
The glass transition is displaced to lower temperatures and fields when the anisotropy
is increased, as shown in the left graph of Fig. 2.7 for a series of oxygen-deficient YBCO
crystals. Deviations from Eq. (2.18) are found at low temperatures and high magnetic
fields for the most anisotropic samples. As can be seen in Fig. 2.7, the glass transition
in this high-field region becomes less temperature dependent. The deviations start at a
field of the order of 2 , and have therefore been suggested to indicate the presence
of a field-induced 3D–2D crossover (Safar et al. 1992a, Deak et al. 1995a, Almasan and
Maple 1996). It was, however, found that the glass transitions for different samples could
be scaled to one common curve, including both high and low temperatures (Seidler et al.
1991, Hou et al. 1994, 1997, Deak et al. 1995a). A dimensional crossover should also
result in a change of the critical exponent 2 @ 0 ( for the vortex glass, going from
( to ( , while Hou et al. (1997) found no systematic variations of the exponent.
An alternative explanation for the observed temperature dependence of was presented in Paper II (Lundqvist et al. 1998b). By slightly modifying Eq. (2.18), a single
46
Chapter 2. Vortex matter and phase transitions
expression
(2.19)
was found to describe the glass transitions of various samples and systems, including
fields both below and above the purported crossover. This is illustrated in Fig. 2.7. Since
Eq. (2.18) and Eq. (2.19) can be seen to have the same critical behavior close to , there
are no theoretical arguments in favor of Eq. (2.18). Therefore, it seems reasonable to assume that the deviations from Eq. (2.18) are due to a too simplified expression for far from .
In Paper III (Rydh et al. 1999b), it was suggested that the temperature dependence of
Eq. (2.19) could come from a competition between thermal energy and some local,
average pinning energy , following similar arguments by Vinokur et al. (1998). By
introducing a modified vortex-glass correlation length,
!
(2.20)
it was shown in Paper III that both Eq. (2.19) and the vortex-liquid resistivity close to
could be consistently described. From the requirement that is given by the
equation , the pinning energy corresponding to Eq. (2.19) is found
to vary as , where . The expression for is
thus similar to the activation energy found to describe the thermally assisted flux flow, see
Eq. (2.8). It is seen that increases with decreasing temperature, consistent with the
expected behavior of a generalized condensation energy.
2.3.5 Critical current density
It is known that the pinning force density ; of disordered materials can be
scaled onto a single curve ; ; versus for a wide range of temperatures in the
vortex solid (Yamasaki et al. 1993, Mittag et al. 1994, and references therein). Interpreting
the irreversibility line in these materials as a glass transition, it is tempting to apply
a vortex-glass description to explain this scaling and the appearance of instead of in the scaling relation.
The critical current density of the vortex glass can be found from a study of the current
density scale " . For currents , relatively large distances are probed, and the
response is that of a glass, with vanishing resistivity in the limit (Blatter et al. 1994).
By increasing the current density above , the critical behavior at short distances is probed.
In the vortex liquid, high currents again probe critical behavior while low currents probe the
Ohmic, liquid response. Extending this argument to describe in the vortex glass even at
low temperatures may be well beyond the applicability of the theory. It nevertheless gives
2.4. Vortex correlation
47
1.0
Fp /Fp,max
0.8
β = 1.0
0.6
β = 0.7
0.4
0.2
0.0
0.0
0.2
0.4
0.6
H/Hg
0.8
Figure 2.8. Pinning force as a function of reduced magnetic field
relation with
.
( 1.0
, as given by the
a remarkably good description of the observed behavior. The field dependence of could
!
either be assumed to follow , or be described by Eq. (2.20). Writing
; ;
where ;
,
!
(2.21)
one arrives at
; ; *
*
!
(2.22)
Here * . This functional dependence is described in Fig. 2.8 for an assumed
value @ and some reasonable values of .
2.4 Vortex correlation
The nature of vortices in the vortex liquid has so far been only briefly discussed. At a melting transition, the vortex solid should transform into a line liquid. In layered superconductors, however, the vortices may not always be intact, but could break-up into pancakes or
short vortex segments. Thus, if the coupling between different layers is weak, other transitions could occur, such as a further evaporation of the line liquid into a gas of pancakes or a
sublimation transition directly from the vortex solid to the pancake gas. Moderately anisotropic superconductors should be 3D in character close to , where the coherence lengths
are much larger than any interplanar distance. At high temperatures, however, fluctuations
become increasingly important. This promotes vortex meandering, so that a vortex could
become entangled and loose its correlation.
The discussion of vortex correlation reaches over a wide variety of problems. A main
question is when the coupling along a vortex line becomes so weak that the system could
48
Chapter 2. Vortex matter and phase transitions
be viewed as consisting of decoupled pancakes. Another question is if vortex loops appear spontaneously, or only as vortex line bending. The electrodynamics of the vortex
system could finally be local or nonlocal, with major implications for the interpretation of
experiments.
Descriptions of the vortex liquid, where loss of vortex integrity is most likely to take
place, include pictures of hydrodynamic, viscous flow (Marchetti and Nelson 1990, 2000,
Huse and Majumdar 1993), the appearance of closed vortex loops (Nguyen and Sudbø
1998, 1999), and models of entanglement. An entangled liquid could in turn be characterized by its entanglement length (Nelson 1988, Nelson and Seung 1989, Crabtree and
Nelson 1997) or by some correlation length (Righi et al. 1997b, Olsson and Teitel 1999).
PFT
2.4.1 Flux-transformer measurements
A large fraction of the experiments that study vortex structure and correlation are based on
the pseudo flux transformer. The original flux transformer was invented by Giaever (1965,
1966) to investigate the magnetic coupling between two superconducting films separated
by a thin, insulating layer, and with a magnetic field applied perpendicular to the film
surfaces. Giaever showed that there was an induced voltage in the secondary film when a
current was sent through the primary film. Furthermore, it was found that a current in the
secondary film could assist or neutralize the driving Lorentz force on the vortices in the
primary film. The effect was attributed to the magnetic coupling between the vortices in
the two films. The experiment was later investigated in great detail both theoretically and
experimentally, with excellent agreement between the two (Clem 1974, 1975, Ekin et al.
1974, Ekin and Clem 1975).
The pseudo flux-transformer (PFT) setup, shown in the left part of Fig. 1.2, was first
used by Busch et al. (1992) and Safar et al. (1992c) to study the vortex behavior in Bi2212. It has a design which is both similar to, but also very different from the Giaever
flux transformer. In the PFT, the insulating section is replaced by the intrinsically layered
structure of the studied superconductor. Current is sent through contacts < on one side of
the sample and the voltage is measured on this side ( ) and on the opposite side
( ), see Fig. 1.2. The magnetic field is directed along the axis, connecting top
and bottom surfaces. Two limiting cases can be found. If there is no coupling between the
pancakes in different layers, the system could be viewed as a gas of fluctuating pancakes.
Such a system would not preserve any phase coherence, and should therefore be dissipative
in all directions, possibly with resistive properties similar to the normal state. On the other
hand, if the vortices act like straight, intact rods, the vortex flow is the same at top and
bottom contacts, so that could be expected.
In the normal state, the (two-dimensional) potential distribution . / is given by the
Laplace equation,
B 5 .
5% / (2.23)
Here
and B is the conductivity tensor. With suitable boundary conditions
on , this equation can be solved by some finite element or finite difference method.
2.4. Vortex correlation
49
2
H
2
3
1
a
4
5
3
1
4
5
8
6
7
b
Figure 2.9. Left: Corbino disk. Current flows from the center to the perimeter of the sample.
Voltages are probed at different distances from the center. Middle: A uniform current flows
between contacts 1 and 4. Voltages and are probed. Right: Symmetric pseudo flux
transformer with calculated equipotential curves for current and an anisotropy .
)
)
& Equation (2.23) shows that the potential distribution is a function only of the anisotropy
% 5% 5 and boundary conditions. For a known sample geometry, the voltage
ratio is therefore directly related to the anisotropy, and should not depend
on the absolute size of the resistivities.
Some variants of the PFT configuration have been invented. The Corbino disk (Fig. 2.9,
left) is a circular disk where current is fed from the center to the full perimeter (López et al.
1999). This gives a current falling off as the inverse of the distance from the center. The
magnetic field is directed perpendicular to the disk surface, so that the Lorentz force will
try to rotate the vortices around the disk center. Therefore, if the vortex system acts as a
rigid body, the vortex velocity will increase with the distance from the center. On the other
hand, if the vortices only feel the local Lorentz force from the applied current, the velocity
will be highest at the center of the disk. In the geometry of Pautrat et al. (1998, 1999), the
current is flowing uniformly through a rectangular sample, with the magnetic field directed
at a certain angle to the current (Fig. 2.9, middle). The generated electrical field is probed
by contact pairs along and perpendicular to the current. In the normal state, no voltage will
appear between contacts 3 and 5, as defined in the figure. In a liquid of intact vortices,
due to flux flow (assuming
however, the electrical field should have an angle Æ to
that ). A third sample configuration is the symmetric PFT (Fig. 2.9, right), which
can be used to to study in-plane anisotropy and vortex correlation for both in-plane and
out-of-plane magnetic fields, see Paper X (Björnängen et al. 2001).
The results from various transformer experiments reveal many interesting properties of
the vortex liquid. Busch et al. (1992) and Safar et al. (1992c) found no indications of
correlated vortices in Bi-2212. Subsequent measurements on twinned YBCO, however,
showed a range with at the onset of the resistive transition (Safar et al. 1994,
Eltsev et al. 1994, López et al. 1994a,b). A similar measurement is shown in Fig. 2.10.
In a central paper by López et al. (1996a), it was shown that correlation with ratio was lost exactly at the melting transition for untwinned YBCO. This result is discussed in
Paper VIII (Rydh and Rapp 2001). While López et al. interpreted their measurements as a
results
50
Chapter 2. Vortex matter and phase transitions
V top , Vbot (V)
10-6
V top
I14 = 0.3 mA
V bot
10-7
10-8
10-9
75
80
85
T (K)
90
95
Figure 2.10. Pseudo flux-transformer measurements of a twinned YBCO crystal. Magnetic fields
are (from left to right) 12, 9, 6, 4, 2, 1 and 0.1 T. At the onset of resistivity, a range is found
where , indicating correlation over the sample thickness. (A. Rydh, unpublished.)
liquid with no correlation, it was later pointed out that the change in ratio 5 5 indicated
a finite vortex correlation of a few micrometers (Righi et al. 1997b). In Paper VIII it is
argued that the vortex liquid correlation close to can be at least of the order of 50 µm,
i.e., beyond the thickness of thin samples.
The effect of disorder on the correlation strength depends on the nature of the disorder.
An increased correlation is found for columnar defects (Doyle et al. 1996, Righi et al.
1997a) and a decreased correlation for point-like disorder (López et al. 1998). Doyle et al.
(1996) thus showed that the introduction of columnar defects into Bi-2212 resulted in a
ratio increasing from less than above to close to 1 at the onset of dissipation.7
The measurements by Pautrat et al. (1999) with the geometry described in the middle
part of Fig. 2.9 can be interpreted as a Lorentz force contribution to the resistivity that
decreases continuously from 100% at to zero at . This is in line with the observations
of Paper VIII and X (Rydh and Rapp 2001, Björnängen et al. 2001).
2.4.2 Local or nonlocal electrodynamics
The question of local or nonlocal electrodynamics has been a subject of discussion all since
the first reports of correlations in YBCO from pseudo flux-transformer measurements. Safar et al. (1994) interpreted their measurements of correlated behavior as due to a nonlocal
conductivity. Motivated by these results, Huse and Majumdar (1993) proposed a model for
nonlocal resistivity in the vortex liquid, based on the hydrodynamic theory of Marchetti
and Nelson (1990). Eltsev and Rapp (1995) and Doyle et al. (1996), however, found that
their measurements were consistent with local conductivity.
In the nonlocal picture, the current is flowing close to the top surface. The Lorentz force
at this surface causes the vortices to move, but the intact vortices or viscous vortex system
7 Measurements by Fuchs et al. (1997) on Bi-2212 without columnar defects indicated that even such crystals
have a ratio that is higher at than in the normal state. However, these measurements were interpreted as a
sublimation transition at .
2.4. Vortex correlation
51
will have the same velocity both at the top and the bottom surfaces. In the local picture, it
is argued that the resistivity along the magnetic field decreases when the vortices become
less fluctuating. For a perfectly straight vortex the resistivity should be zero (Brandt 1995).
Equation (2.23) shows that the current in this case will distribute evenly over the sample
thickness, resulting in . The resistive anisotropy will thus depend on the direction of
the magnetic field. This was verified in Paper X (Björnängen et al. 2001). The best way
to determine if the electrodynamics is nonlocal, is to study the resistivities for different
paths of the current. If nonlocal, such paths will give contradictory results. The results of
Paper VIII and X (Rydh and Rapp 2001, Björnängen et al. 2001) were in both cases found
to be consistent with local behavior.
The measurements by López et al. (1999) with the Corbino disk geometry showed that
inter-vortex correlations in the vortex liquid are negligible. In the vortex solid, on the
other hand, the vortex system could be made to rotate as a rigid body, indicating a nonlocal
vortex velocity. Since the electrical field is proportional to the vortex flow, it may be argued
that this measurement also shows nonlocal electrodynamics. However, the vortex solid is
non-Ohmic and the nonlinear resistivity could therefore increase with the distance ) from
the center of the disk, i.e., $ ), ), and thus 5 ) .
Summarizing, to interpret measurements by local electrodynamics, the possibility of an
anisotropy depending on the magnetic field cannot be excluded. The resistivity in nonOhmic regimes also has to depend on both vortex velocity (electrical field) and current.
The presence of nonlocal electrodynamics on a scale longer than the coherence length or
vortex lattice spacing seems not yet to be evident from experiments.
2.4.3 Phase-coherent liquid
An open question is if correlated disorder can stabilize a phase-coherent vortex liquid in
some region above . In clean materials, it is clear from both theoretical simulations
(Koshelev 1997, Chin et al. 1999, Nguyen and Sudbø 1999) and experiments (López et al.
1996a, Rydh and Rapp 2001) that phase coherence is lost along the field direction when the
vortex lattice melts. The liquid can thus not support any supercurrent in the field direction.
Measurements in the PFT configuration and of the -axis resistivity on twinned YBCO, on
the other hand, indicate a temperature range with perfect vortex velocity correlation and
finite -axis critical current (López et al. 1994a, de la Cruz et al. 1994).
The highest temperature with is found to be thickness dependent, decreasing
with increasing sample thickness (López et al. 1996b, Righi et al. 1997a). Such thicknessdependence could come from nonlocal electrodynamics, with the onset of nonzero -axis
resistivity corresponding to the temperature where the correlation length drops below the
sample thickness. An alternative explanation could be an inhomogeneous system with
some vortices in a Bose glass state and others forming an incoherent vortex liquid. Pinning
effects would then explain the nonlinear response.
52
Chapter 3
Experimental survey
3.1 Crystal growth and oxygenation
The discussion of crystal growth and oxygenation in this section is limited to the case
of YBa Cu OÆ . For details on Ba K BiO crystals and other materials, see for instance Baumert (1995), Rosamilia et al. (1991), or references in the handbook by Poole,
Jr. (2000).
3.1.1 Growth process
The YBCO crystals in this work were grown using a flux growth method, where crystals
form from a melt with an excess of BaO and CuO (Kaiser et al. 1987). The main reason
for using this method is that the melting temperature of Y O is very high, ÆC.
If the temperature is increased too much, the crucible will dissolve into the melt and the
amount of impurities in the final crystals would be significant. By dissolving the yttrium
oxide into an eutectic melt of 28% BaO + 72% CuO, the so called flux, the temperature
can be kept low.
The growth process has been reviewed in greater detail by Lundqvist (1998, 2000). Dry
powders of Y O , BaCO , and CuO with the highest possible purity are carefully mixed
and ground ( min) in an agar mortar to obtain a grain size suitable for dissolving
the Y O . The reason to use BaCO instead of BaO is the higher purity of the former
compound. To obtain crystals displaying the first-order melting transition of the flux-line
lattice, it is recommended to use at least 99.999% CuO, 99.999% BaCO , and 99.99%
Y O . A suitable batch size for a typical crucible with an inner diameter of 19 mm and a
height of 29 mm, as used here, is around 6 g. The relative amount of YBCO in the melt
is not critical. Good results are usually obtained for Y O corresponding to 10 – 15% YBCO. A typical batch aiming at 15% YBCO can be obtained by mixing 0.154 g Y O ,
3.346 g BaCO , and 3.241 g CuO.
The process is developed for a tube furnace. Phase diagrams for the BaO – Y O – CuO
system can be found in Poole, Jr. (2000). The first process step is to decalcinate the BaCO
to BaO by keeping the crucible at 900ÆC during 10 h. The temperature is then raised to
1 000ÆC, to let the flux melt and the yttrium oxide dissolve. After 10 additional hours,
53
flux
54
overflow
crucible
material
Chapter 3. Experimental survey
the crucible is displaced in the furnace during a couple of minutes to create a temperature
gradient of 1ÆC/mm over the melt. After displacement the temperature in the middle of the
crucible is around 965ÆC. In the following growth step, the temperature is slowly decreased
with a rate of 2ÆC/h until the temperature in the middle of the crucible is approximately
900ÆC, close to the melting temperature of the flux. At this point the flux starts to withdraw
to the cold side of the crucible, leaving clean, free-standing crystals behind. If the temperature is right, the flux even creeps up along the crucible wall and out of the crucible. This
overflow can be taken as a sign that the process has finished. After holding the temperature
3 – 5 h at the overflow temperature, the process is stopped and the crystals are allowed to
cool to room temperature. A successful process should give between 20 and 100 nicely
shaped crystals with shiny surfaces. Typical dimensions are mm , with
the short distance along the axis.
A good choice of crucible material is important to maintain a high purity of the final
crystals. Two main crucible materials are commonly found in the literature. The yttria
stabilized zirconia (YSZ) crucibles used in this work consist of mainly ZrO . The solubility
of this material in the melt is very low, due to its high melting temperature. The smallness
and charge of the Zr ion makes it furthermore unlikely to enter into the YBCO structure
(Liang et al. 1992). Dissolved yttrium could slightly modify the composition of the melt,
but should have no influence on the final crystal quality. The barium zirconate (BaZrO )
crucible introduced by Erb et al. (1995) has been suggested to give an equally good result,
and may be better when growing other 123 compounds such as PrBa Cu OÆ . Gold can
be used as crucible material, but has the disadvantage of replacing copper to a rather large
extent. Its use has therefore been diminished.
3.1.2 Oxygenation
The as-grown YBa Cu OÆ crystals are oxygen deficient and have an inhomogeneous
oxygen distribution. They therefore have broad superconducting transitions with low ’s.
To improve the properties, the crystals need to be annealed in oxygen atmosphere at 400 –
700ÆC. Both the annealing temperature and oxygen partial pressure affect the oxygen content of the crystals. The oxygen transport is governed by diffusion, where the final oxygen
content is found from a temperature-dependent equilibrium with the oxygen partial pressure.
diffusion
The diffusion rate of oxygen in YBCO is rather high, with the average oxygen content
typically reached within 24 h at 400ÆC and within one hour at 700ÆC. The diffusion is
much slower along the axis than within the )* plane (Rothman et al. 1991), a fact which
should be considered when annealing thick crystals. It is not enough to merely anneal
during the time needed for an average equilibrium oxygen content to become established.
Concentration gradients need to be diminished, which in practise means annealing times
much longer than what can be inferred from weight-change measurements or from the
Einstein law, . C
, where . is the diffusion distance, C the temperature-dependent
diffusion coefficient, and the annealing time.
Single crystals of size mm require annealing during 5 – 10 days at ÆC. Shorter
times may be necessary for higher temperatures, since the crystals degrade easily with time
3.1. Crystal growth and oxygenation
55
p(O2) = 1 atm.
600
80
O–T
p(O2) = 0.21 atm.
500
Tc (K)
annealing temperature (°C)
100
700
60
40
20
400
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.0
oxygen deficiency δ
0.2
0.4
0.6
0.8
oxygen deficiency δ
Æ
Æ
Figure 3.1. Left: Relation between annealing temperature and oxygen deficiency
in
YBa Cu OÆ for oxygen partial pressures corresponding to pure oxygen and air. Data are
taken from Lindemer et al. (1989) () and Meuffels et al. (1989) (). At high values of , an
orthorhombic-to-tetragonal transition takes place, as indicated by the thick, dashed line. The
tetragonal phase is stable at room temperature for
. Right: Critical temperature as a
function of . Data with error bars are from Claus et al. (1990) (- - -) and Jorgensen et al. (1990)
( ). Results on our single crystals (—) indicate that, for these crystals, the oxygen deficiency
may be underestimated if taken from the left graph at low annealing temperatures.
Æ
Æ by reaction with impurities in the atmosphere, which may cause insulating surface layers
to start forming. Annealing temperatures below ÆC are not likely to sustain equilibrium
conditions. The oxygen diffusion process in YBa Cu OÆ has been further discussed by
for instance Rothman et al. (1991), LaGraff and Payne (1993), and Erb et al. (1996).
The oxygen deficiency Æ can be controlled through the annealing temperature and oxygen annealing
partial pressure (Lindemer et al. 1989, Meuffels et al. 1989, Jorgensen et al. 1990). In
the left part of Fig. 3.1, the annealing conditions for different values of Æ are shown for
annealing in oxygen or air. The benefit of using air is the lower and very stable oxygen
partial pressure, which allows moderate annealing temperatures. A drawback is that carbon
dioxide in the air may eventually degrade the crystals (Cooper et al. 1991).
When the oxygen content is reduced, oxygen is mainly removed from the CuO chains.
Some oxygen is, however, also redistributed to usually empty sites in the chain plane.
This leads to an orthorhombic-to-tetragonal phase transition at a transition line (O – T)
dependent on both temperature and deficiency Æ , as indicated in Fig. 3.1.
By varying the annealing conditions, the superconducting properties can be readily modified. The critical temperature of YBCO is sensitive to the oxygen structure and content.
The relation between and Æ has been studied by for instance Jorgensen et al. (1990) and
Claus et al. (1990). has a maximum at Æ in the Lindemer scale (Rykov et al.
1999), but generally decreases with Æ and goes to zero at Æ , corresponding to the
O – T transition. Usually, the Æ curve shows plateau-like behavior both for Æ ( – K) and for Æ – ( – K). The second plateau may be
Chapter 3. Experimental survey
temperature (arb. units)
56
normal metal
strange
metal
a.f.
s.c.
hole concentration (arb. units)
Figure 3.2. Schematic picture of a phase diagram for typical hole-doped high- superconductors. A reduction of the oxygen content corresponds to a decreased hole concentration, which
reduces . At low hole concentrations superconductivity (s.c.) is replaced by an insulating, antiferromagnetic phase (a.f.). The optimum for YBa Cu OÆ is at . The so called
overdoped region (corresponding to low ) can be only partially accessed through oxygen doping.
Æ
Æ related to an ordering of the oxygen vacancies in alternatingly full and empty chains [see
for instance Cava et al. (1990)], as discussed further below.
In the right graph of Fig. 3.1, the Æ relation obtained for single crystals annealed in
oxygen at 450ÆC () and in air at various temperatures between 500ÆC and 700ÆC (Æ) are
shown and compared with literature data. For the single crystals the oxygen deficiency
was estimated from the annealing conditions, using the left graph of the figure. While the
agreement at high deficiencies is good, the seems to drop off faster than the literature
data. This may indicate that the oxygen deficiency is higher than estimated. A possible
reason is diffusion barriers in the crystals due to clustering of the oxygen defects. Such
an explanation is supported by a relatively large -axis lattice parameter, as observed by
x-ray analysis, see Paper I (Lundqvist et al. 1997). Part of the deviations may also be
connected to the problem of determining the absolute oxygen deficiency. A small error
in the reference measurement, which is typically a fully oxygenated sample with assumed
Æ , may displace the scale of Æ by a corresponding amount.
The strong influence of Æ on is related to the electronic phase diagram. The superconducting planes are doped through charge exchange with the chains, which act as charge
reservoirs. Modifying the chain oxygen content thus changes the doping state of the superconductor. YBCO, as most high- materials, is hole doped. The (electronic) phase
diagram is fairly complex and still not fully explored. Figure 3.2 shows a schematic picture of some main features.
3.1.3 Relation between , , and Æ
In addition to the doping effect, the removal of oxygen from the CuO chains reduces the
metallicity of the binding slabs, which causes the coupling between the (double) CuO
conduction layers to be decreased. This, in effect, increases the anisotropy . The relation
3.2. Sample contact preparation
57
between and Æ for YBCO has been investigated by for instance Janossy et al. (1991) and
Chien et al. (1994). The size of for YBCO is found to grow approximately as the square
of Æ . Since the suppression of varies in a similar way, the relation between and is approximately linear.1
A complication when studying changes in and with Æ is the effect of short-range
ordering (SRO) of the oxygen vacancies in oxygen-deficient YBCO (Claus et al. 1990).
The vacancies have a tendency to order in alternatingly full and empty chains. The superconducting for an ordered sample is possibly different from that of a sample with totally
disordered chain structure. does thus depend not only on the average oxygen content,
but also on the local oxygen arrangement. By quenching a sample after annealing, the
oxygen defect structure may be varied. Claus et al. (1990) showed that a rapid quenching
can decrease with more than 20 K for large oxygen deficiencies, where the vacancy
ordering is as most important. The large error bars in the Æ graph of Fig. 3.1 reflect
this possible variation of .
The vacancy ordering also leads to another problem, namely a possible broadening of
the zero-field transition width (Claus et al. 1990). Slow cooling of a crystal after annealing
gives a well-ordered chain-vacancy structure, but risks to alter the average oxygen content.
Higher quench rate, on the other hand, could lead to a gradient in the degree of order,
going from total disorder at the surfaces, where quenching is fast, to a somewhat ordered
structure in the center of the crystal. A high enough quench rate gives sharp transitions,
but the sample may age with time at room temperature. A possible solution to this problem
of finding a suitable quench rate could be to make a rapid quench to room temperature,
followed by short annealing at low temperature, ÆC, where the oxygen atoms are
allowed to relax without too much diffusion. Some attempts were made in this direction,
but with inconclusive results.
3.2 Sample contact preparation
Resistivity measurements and measurements in multiterminal configurations rely on the
possibility of attaching contacts to the material under investigation. The small size of
available crystals of high enough quality makes many standard methods, such as sputtering,
unfeasible for normal use. Contacting is therefore mostly performed by using conducting
silver paste, which is applied by hand and then cured in a short annealing step to obtain
stable, Ohmic contacts with low contact resistances.
3.2.1 Standard configuration
In the standard, four-contact configuration current should flow uniformly through a preferably rod-like crystal with a constant cross section. Two potential contacts are placed on the
same side of the crystal. The contacts should define a probing distance in the current direction, and should be placed far enough away from the current contacts to ensure a uniform
current.
&
1 For the same suppression of
, it is found that chain doping gives a strong increase of , while doping
directly into the CuO planes (for instance by Zn) leaves the anisotropy almost unchanged (Rapp et al. 1999).
SRO
58
Chapter 3. Experimental survey
A typical contacting procedure is outlined below. The current contacts are made with
silver paint and the potential contacts with silver paint and gold wires. The main working
tool is a sharpened needle and a microscope. A small heater is also required. The
choice of silver paste is critical. DuPont no. 5504 or silver epoxy (EPO-TEK H20E, Epoxy
Technology, Inc.) have been found to be good for YBCO, DuPont no. 7713 works rather
well for Ba K BiO .
I. Prepare a mm sapphire glass to use as a substrate. The thickness is not critical, but
1.5 mm is suitable. The sapphire glass is used to obtain a good heat conduction.
II. Place a small piece of paraffin on the middle of the substrate, and heat the substrate at
150ÆC for one or two seconds to melt the paraffin. The droplet should be approximately
mm in size.
III. Cut out a nicely shaped sample from a suitable crystal with a razor blade. Carefully check
the locations of possible twin boundaries by a polarized microscope with high resolution.
Cutting is easier if the crystal is placed on a thin sheet of soft plastic before cutting. Place
the cut piece on top of the paraffin droplet and remove any surplus of paraffin which is not
covered by the crystal. Heat for one second to let the paraffin melt and the crystal become
attached to the substrate.
IV. Carefully remove the paraffin from one of the short edges by a sharpened needle or
wooden stick. Place a fresh drop of paint in a corner of the substrate. Clean the needle, take a small amount of paint from the drop and apply it quickly to the exposed crystal
edge. Paint a connection line between the drop of paint in the substrate corner and the
crystal. Repeat this step for the other crystal edge.
V. Heat the substrate for 6 – 10 s (at 150ÆC) to dry the paint. Avoid heavy boiling. The
crystal should have been fixed to the paint and the paraffin should have evaporated. Make
sure not to dislodge the crystal after heating, since this will destroy the possibility of good
contacts.
VI. Carefully clean the top surface from possible paraffin. Prepare pieces of gold wire in
lengths of 1 – 1.5 mm. The diameter of the wire depends on the crystal size; usually
25 µm is suitable for large crystals ( mm) and 10 – 15 µm for smaller crystals.
VII. Put a fresh drop of paint in a free corner. Paint a connection to a place approximately
0.7 mm from the crystal. Clean the needle. To make point-like contacts, take a small
amount of new paint onto the needle and grab a gold wire in one end with it. Dip the other
end of the gold wire into the paint on the substrate to get a tiny drop of paint on the wire.
Place the wire between the painted lead on the substrate and the crystal (fix the crystal
end of the wire first). To make a contact strip of the same width as the crystal, paint the
strip directly onto the crystal. A second strip of paint can be applied on top of the gold
wire when it is in place. Repeat with the other potential contact.
VIII. Heat at 150ÆC for a few seconds and let cool again. Check that all leads and connections
look good.
IX. Cure the contacts in conditions as similar as possible to the annealing. Good contacts are
usually obtained after 30 min at 400 ÆC. This step is the most critical, due to the possibility of disturbances in oxygen content or distribution. The importance of quenching, as
discussed in Section 3.1.3, should be considered.
3.3. Resistive measurements in magnetic field
59
X. Check that the contact resistances are all good (0.1 – 1.5 ). Finally place a small piece
of paraffin close to the crystal and heat for a second at 150ÆC to cover crystal and contact
leads with paraffin. This will protect the contacts from degrading.
3.2.2 Multiterminal samples
The procedure to prepare multiterminal samples is somewhat different from the method
described above, since contacts have to be applied on both top and bottom surfaces of the
crystal. The first three steps are the same.
IV. Remove any excess paraffin around the crystal on the substrate without touching it. Place
a drop of paint on some free space of the substrate. This paint will be used as a source of
paint and should be kept fresh.
V. Paint a strip on the crystal where the contact should be. Then quickly pick up a gold wire
with the needle and attach it to the crystal. A sharp needle makes the wire easier to handle.
Do not use silver paint to grab the wire, since this will make the wire stay on the needle.
Let the second end of the gold wire reside in the air. Cover the wire on the crystal with
more paint. Repeat for all contacts on the surface.
VI. Remove remaining paint on the substrate. Then heat the sample at 150ÆC for 5 – 15 s to
allow the paint to dry. When cooling, move the crystal around on the substrate, to get rid
of remaining paraffin.
VII. Flip the crystal with the needle and attach the gold wires to painted contacting pads on
the substrate. Heat for 10 s. Avoid to bend the wires, since this will easily destroy the
contacts to the crystal.
VIII. Fix the crystal to the substrate with some paraffin. Any movement of the crystal will
destroy the bottom contacts. Paint a pad for the next contact and prepare leads to a suitable
place for the gold wire to be attached. Repeat step V. Finally, proceed from VIII of the
standard configuration.
The use of slightly bent gold wires is recommended, since the wires will be strained at
low temperatures. A modification of the procedure above is to only paint contacts on the
crystal before the curing. The wires are then attached afterwards and heated at 150ÆC.
This gives physically strong contacts with good contact resistances. The drawback is the
required short heating in conditions different from the annealing.
3.3 Resistive measurements in magnetic field
3.3.1 Temperature control
All measurements in this work were made in a flowing He gas cryostat from Oxford
Instruments, described in more detail by Andersson (1992). Its main features are shown in
Fig. 3.3. Cold He gas is pumped through the sample cell, where the sample holder (and an
extra shield) is placed. Through a suitable heating on the sample holder, the cooling power
can be equilibrated at any temperature between 4.2 and 300 K. For temperatures between
1.5 and 4.2 K the sample cell is filled by liquid He, which is then pumped on.
The cryostat is equipped with a 12 T NbTi/Nb Sn superconducting magnet, which can
be operated in persistent mode, i.e., with the external current source switched off. By
60
Chapter 3. Experimental survey
Needle valve
to He recovery
system
to pump
Cu thermal link
Radiation shields
N2(l)
He (l)
Outer vacuum chamber
Inner vacuum chamber
Radiation shield
Superconducting magnet
Sample cell
He inlet tube
Figure 3.3. Schematics of the flowing gas cryostat. The cryostat is equipped with a 12 T superconducting magnet (13.5 T if operated at the lambda point). By pumping on the sample cell,
helium is vaporized at the needle valve and cold gas will flow through. Radiation shields, liquid
nitrogen, and vacuum chambers reduce the helium boil off. Redrawn from Andersson (1992) and
Lundqvist (1998) with permission.
pumping on the main He bath through a needle valve with a consequent cooling loop (not
shown in the figure), the temperature of the He bath can be decreased to the lambda point
& K of He. This allows the magnet to be operated at a maximum field of T.
The standard sample holder is shown in Fig. 3.4. To obtain a good temperature stability,
the incoming He gas can be preheated. The inner vacuum shield provides an extra insulation of the sample from the streaming gas. Various thermometers are used for temperature
control, regulation, and measurement. The carbon-glass thermometer is well suited for low
temperatures, especially below 30 K, where platinum thermometers cannot be used.
To compensate for magnetoresistance in the thermometers, the thermometers need to
be field calibrated. Carbon-glass thermometers are less sensitive to magnetic fields, with
a maximum temperature error of about 0.2 K at low temperatures and 12 T. The error
increases with temperature, however, and can be above 0.5 K at 90 K. The best temperature
3.3. Resistive measurements in magnetic field
61
Sample holder
Heater winding
Carbon thermometer
Pt thermometer
Sample space
Heater winding
Pt thermometer
Plug-in
Sample position
Carbon-glass thermometer
Sample cell
Inner vacuum shield
Heater winding
(for incoming gas)
He inlet tube
Figure 3.4. Sample holder with shield in the sample cell. The sample with substrate is mounted
on a plug-in sample holder for easy handling. The carbon-glass thermometer is the main thermometer for low to intermediate temperatures (
K), while the platinum thermometer has
a somewhat better sensitivity at the highest temperatures. (Courtesy of B. Lundqvist.)
control in varying magnetic fields is found by using a capacitive sensor or by stabilizing the
temperature with a thermometer located in zero field. Zero field is created by cancellation
coils 20 cm above the sample position. (This requires a different sample holder than the
one shown in Fig. 3.4).
3.3.2 Instrumentation
Various instruments are used in a typical measurement. All instruments are connected
through GPIB to a computer based control and acquisition system (HP-VEE). The temperature is controlled or stabilized by a temperature regulator (Lake Shore 340), which
measures the thermometer resistance and compares it to a known calibration curve. A constant current source (Keithley 220, 2400) provides currents in the range 10 nA – 1 A with
4 – 5 significant figures. Voltage is measured through a DC picovoltmeter (EM Electronics, Model P13), acting as a preamplifier to a standard voltmeter (Hewlett Packard 3458,
Schlumberger 7081). The resolution of the picovoltmeter is limited by source resistance
and variation of the thermal power in the wires. In normal cases a resolution of 0.1 – 0.3 nV
62
Chapter 3. Experimental survey
10-5
µ0 H = 6 T
∆T = 10 mK
V (V)
10-6
10-7
∆T = 20 mK
T = 80.6 K
10-8
10-9
10-10
10-6
10-5
10-4
10-3
10-2
10-1
I (A)
Figure 3.5. Current – voltage characteristics of an untwinned YBCO single crystal at an applied
field T. The temperature ranges from 79.9 K to 80.6 K (from right to left), which
corresponds to the vicinity of the melting transition. The temperature step
between curves is
as indicated, except for the lowest lying isotherms, which are spaced by
mK.
is obtained. The direction of the magnetic field can, finally, be controlled by a rotatable
sample holder with an angular resolution of 0.01Æ (Andersson 1999).
3.3.3 Low level measurements
Measurements at low resistance levels need special care to yield good results. At low
measurement frequencies, all current and voltage pairs should be twisted, to reduce induced
voltages from magnetic fields, and noise from thermal power. To further decrease the
effects from thermal power, the current direction should be switched and the average taken.
The best is to sample data with the current switched twice, ++, which also compensates
for linear drifts in thermal-power background. With these actions the noise level can be
suppressed to below or around nV. For the model P13 picovoltmeter, such low levels
require a total source resistance not larger than 3 – 4 $. Good contact resistances are
therefore imperative.
Long stabilization times for the picovoltmeter do not necessarily increase resolution,
since the sampling time constant has to be much shorter than the time constant of variations
in the drifting thermal power. Better is to average multiple datapoints, which have been
individually compensated for thermal power noise by current reversal. An example of a
current – voltage (< – ) measurement for an untwinned crystal at 6 T, close to the upper
critical point, is shown in Fig. 3.5.
Chapter 4
Summary of results
4.1 Vortex glass line in oxygen-deficient crystals
In Paper I, we initiated a study of the glass transition and its dependence on anisotropy in
optimally doped and oxygen-deficient, twinned single crystals of YBa Cu OÆ (YBCO).
The anisotropy in this system can be rather easily changed over a wide
range by varying the oxygen deficiency Æ (Chien et al. 1994). The desired oxygen content
can, in turn, be obtained through a suitable annealing procedure (Meuffels et al. 1989,
Jorgensen et al. 1990). Annealing in oxygen at a rather low temperature ÆC provides
close-to-optimally doped crystals. Higher temperatures or annealing in air decreases the
oxygen content.
A series of single crystals with varying anisotropy provides a useful system for studies
of superconducting properties as a function of . By varying oxygenation conditions, such
a system was obtained with monitored in the range – . The investigated crystals all
showed signatures of a transition from a dissipative vortex liquid to a glassy vortex solid.
No signs of the first-order melting transition were observed at any magnetic fields, possibly
due to the heavy twinning, but also due to point defects suppressing the upper critical point.
Such point defects were introduced at the crystal growth, by using starting powders not of
highest available purity. In addition, oxygen disorder increases with the oxygen deficiency,
thus adding to the pinning.
The glass line of the disordered sample with close to optimal doping was found to be
located well below the first-order melting transition for clean crystals with comparable
oxygen deficiency. This effect of disorder can be understood as an increased vortex bending and meandering, promoted by point disorder. Several recent studies investigate the
effect of point disorder on the location of the solid-to-liquid transition (Khaykovich et al.
1997, Nishizaki and Kobayashi 2000, Paulius et al. 2000).
The temperature dependence of the glass transition in strongly disordered samples is
commonly observed to follow a simple power-law dependence, , at
temperatures close to . Here . However, deviations from such a behavior are
63
Paper I
64
Paper II
Chapter 4. Summary of results
found for a combination of high anisotropy and strong magnetic fields. Some reports suggested that these deviations could correspond to a dimensional crossover from 3D to quasi2D behavior, consistent with a steeper slope of the glass transition line above the crossover
field (Safar et al. 1992a, Hou et al. 1994, 1997, Deak et al. 1995a). For the irreversibility
line, which, in principle, describes the glass transition, a universal scaling relation with the
above power-law was reported by Almasan et al. (1992), who noted that the crossover to a
steeper temperature dependence takes place at for all magnetic fields.
In Paper II, we showed that the vortex glass transition in oxygen-deficient YBCO follows
a temperature dependence close to the empirical expression
(4.1)
valid at all temperatures and accessible fields T. The difference between this
equation and the simple power-law expression may seem insignificant. The successful description of the full temperature range, however, suggests that the observed deviations from
the simple power-law expression can be explained without invoking any 3D–2D transition
or field-induced crossover. This does not contradict the general idea of a field-induced
transition, which may be located at even higher magnetic fields.
By applying the proposed temperature dependence to data from the literature on thin
films of YBCO, Tl-2212, and Bi-2223, we further found that all these systems can be described by the same equation, provided that the exponent is allowed to vary between
different systems. The role of anisotropy is to change the prefactor . We showed that
Eq. (4.1) is consistent with the expected variation for a solid-to-liquid transition with
. The same anisotropy dependence is, however, expected also for the
3D–2D transition, see Eq. (1.23).
4.2 Extended vortex-glass description
Paper III
In Paper III, the resistivity in the vortex liquid close to the glass line was discussed. The
resistivity at fixed magnetic field should, according to the vortex glass theory, follow a
relation 5 with 2 @ 0 being a constant exponent. It has furthermore
been suggested that the position of the glass line (irreversibility line) may be found from a
comparison between thermal energy and pinning energy (Cohen and Jensen 1997, Vinokur
et al. 1998). To unify these two points of view, we proposed a simple model with two
main assumptions. (a) The glass line was assumed to be given by a competition between
a mean pinning energy and the thermal energy , i.e., .
(b) The vortex-glass correlation length was assumed to depend on the relative difference
between thermal energy and this pinning energy.
The pinning energy referred to in point (a) above deserves a comment. If the resistivity
is written as a thermally activated process, 5 !" , the barrier will diverge
in the vortex glass below for . This is current dependent, and is given by
where and is current independent (Blatter et al. 1994).
Our corresponds roughly to in this equation, and does not diverge in
4.2. Extended vortex-glass description
65
Energy
x
U0
kBT
Figure 4.1. Schematic pinning potential of a small vortex segment. The segment is localized at
low temperatures, since no other states are available with as low energy. If , thermal
fluctuations are able to overcome the pinning, even in the zero-current limit.
the vortex glass or when approaching the glass line. A plausible view is that corresponds
to the condensation energy of a short vortex segment in a typical, local pinning defect.
The size of the jumping volume diverges in the low-current limit of the vortex-glass
state. Therefore, access to new metastable states on the local scale cannot be provided by
the thermal fluctuations. If the pinning energy of a segment becomes lower than , on
the other hand, the location of the segment is no longer restricted to the pinning center, and
thermally excited vortex motion could set in. Figure 4.1 illustrates this simple model. At
low temperatures is large and thermal fluctuations are negligible. Close to the
condensation energy should vanish. There will thus be a point where . It seems
reasonable that the vortex-glass correlation length should scale with the relative excess
(deficit) energy as the main parameter. The results can be summarized in
the modified expression for the vortex-glass correlation length,
!
(4.2)
where @ is the static critical exponent, and the prefactor was argued not to be strongly field
or temperature dependent. The usual expression recovered if is replaced by its value at
the glass line, i.e., by .
Paper V provided some supplementary measurements of the vortex-liquid resistivity in
the vicinity of the glass transition as a function of magnetic field. This enabled the investigation of the field dependence of the proposed scaling in a more direct way.
Paper V
The quantitative description of the vortex-liquid resistivity close to the glass transition by
Eq. (4.2) was emphasized in Paper VII and IX. We experimentally verified that the energy
varies with magnetic field in the same way as the activation energy in a thermally activated description, provided that the activation energy is evaluated at a constant resistivity
level 55 . In Paper IX, the field and temperature dependence of were also visualized
by inverting the suggested expression for the vortex-liquid resistivity close to the glass line.
Papers
VII, IX
66
Chapter 4. Summary of results
4.3 Flux creep at high currents
Paper IV
In Paper IV, we discussed the vortex-solid dynamics of untwinned, clean single crystals
of YBCO. At low currents, the resistive transitions of these crystals showed the sharp
step associated with the first-order melting transition (Charalambous et al. 1992, Zeldov
et al. 1995). The main focus of the work was to investigate the region of weakly pinned
vortex solid at temperatures below the melting, see Fig. 1.1. Dynamics in this part of the
phase diagram is rich on exotic features. The main observation is a peak effect just below
melting (Kwok et al. 1994b, Jiang et al. 1997). Other effects include a large vortex noise
(Safar et al. 1995, D’Anna et al. 1995), oscillatory dynamics at frequences much lower
than the driving frequency (Gordeev et al. 1997, 1998, Kwok et al. 1997), and relaxation
processes (Fendrich et al. 1996). Not much is known, though, about the boundary to the
more strongly pinned solid at low temperatures. It is also of interest to further investigate
the conditions for a flowing lattice.
The main result in Paper IV was the observation of a pronounced crossover with temperature, from flux creep to flux flow, of the driven vortex lattice. We also confirmed that the
vortex-lattice flux flow gives a resistivity consistent with the vortex-liquid resistivity above
the melting temperature . The flux flow was characterized by a linear differential resistivity $ , while the flux creep was found to be irregular with a high noise level. The
onset of linear, Ohmic resistivity close to was finally noted to agree with the location
of maximum critical current at the peak effect.
4.4 Vortex-liquid correlation
Paper
VIII
The vortex state above the first-order melting transition has been, and still is, a question of
intense debate. In Paper VIII, we studied the vortex-liquid correlation of clean, untwinned
YBCO by using multiterminal transport measurements in two different contact geometries.
The original flux transformer by Giaever (1965, 1966) was devised to study the magnetic
coupling between two superconducting films separated by a thin, insulating layer. Multiterminal measurements in the pseudo flux-transformer (PFT) geometry, shown in Fig. 1.2,
were designed to study the coupling strength between the pancakes of a vortex line in
the strongly anisotropic high- superconductors (Busch et al. 1992, Safar et al. 1992c).
These first studies on Bi-2212 revealed no signs of correlation. In the less anisotropic
YBCO, however, the vortices appeared to be correlated across the sample thickness at low
temperatures (Safar et al. 1994, Eltsev et al. 1994).
In the PFT geometry, current is sent through contacts on one side of the crystal, and
the voltage from vortex flow is probed both on that side (“top”) and the opposite one
(“bottom”). The magnetic field is usually directed along the axis, connecting top and
bottom surfaces. The correlation manifests itself as an increased ratio between bottom and
top voltages, ultimately reaching the value when approaching the vortex
solid. Interpretations of the vortex correlation experiments are divided into two main points
of view; local or nonlocal electrodynamics. In the nonlocal description, the injected current
is suggested to flow very close to the top surface. The vortices, if intact, are dragged by the
4.4. Vortex-liquid correlation
Tm
V/I (mΩ)
t , lc (µm)
60
40
67
40
20
t = 10 µm
4T
Tth 0 T
0
85
20
0
84
88
91
94
T (K)
lc
86
88
90
92
Tth , T (K)
* Figure 4.2. Left: Temperature dependence of the correlation length at T, as calculated from resistivities and and a condition of 99% bottom-to-top voltage ratio. Symbols
mark the location of the temperature , defined by the 99% ratio, for three samples of different
thickness. The inset shows top and bottom voltages (divided by current
mA) at T
for the thin sample. Right: Relatively straight vortices. Current in the field direction cause no
macroscopic Lorentz force on the vortex system. Locally, however, thermal fluctuations bend the
vortices on a scale of . The vortices feel the current and longitudinal resistivity may appear.
"
*
"
) "
Lorentz force, creating a vortex flow also at the bottom surface (Safar et al. 1994, López
et al. 1994b). In the local picture, the resistivity in the field direction becomes zero if the
vortices are absolutely straight lines. The current is then distributed homogeneously over
the sample, also resulting in (Eltsev and Rapp 1995, Doyle et al. 1996).
The project of Paper VIII started out from the results of López et al. (1996a), where
vortex-lattice melting and loss of vortex correlation were reported to coincide for untwinned YBCO. In our paper, we presented measurements indicating much stronger correlation than suggested by López et al. (1996a) and subsequent studies (López et al. 1996b,
Righi et al. 1997b). The results were shown to be described well by local conductivity, so
that the resistivities perpendicular to (5 ), and along (5 ) the magnetic field could be obtained. From the resistivities, we estimated the temperature at which could be expected for typical samples of different thickness. This rough method defines a
vortex velocity correlation length, which is reproduced in Fig. 4.2.
It should be noted that the temperature range between and was found to be
Ohmic in all directions. While the resistivity 5 could not be resolved at temperatures close
to , the temperature dependence of # indicated that 5 could become finite already at
. The temperature should thus not be interpreted as a transition line, below which
superconducting phase coherence is preserved in the field direction.
A question which is still unresolved is the mechanism causing the loss of correlation
with increasing temperature. In Paper VI we studied vortex correlations in the cubic Paper VI
Ba K BiO (BKBO) system. The BKBO superconductor is extreme type-II with – , but the coherence length – Å is much larger than in the high-
cuprates (Baumert 1995). The vortices in this system are consequently expected to be
line-like, and effects of layering should not be found. Two main results can be pointed
68
Chapter 4. Summary of results
out. First, we observed correlation over the sample thickness which was gradually lost
with increasing temperature. This indicates that the similar loss of correlation in YBCO
may be explained by a nonlayered model. Second, we noted that some correlation effects
remained up to temperatures where the resistivity is close to the normal-state resistivity.
The location of the upper critical field in BKBO is rather diffuse and the midpoint of the
resistive transition does not follow the expected curve for (Samuely et al. 1997).
The point where the bottom-to-top voltage ratio becomes different from the normal-state
ratio may be a better estimate of the lower limit of from transport measurements.
Paper X
Paper X introduced a modified PFT geometry, where two contacts are placed on each
edge of a square-shaped, untwinned YBCO crystal. This allowed a comparative study of
)- and *-axis properties with the magnetic field applied along any of the three crystal directions. The main result of the paper is the strong effect of the Lorentz force, verifying
Paper VIII. It was settled that the resistivity 5 exhibits a smooth, continuous increase,
while the step at is a cause of the Lorentz force, which suddenly overcomes pinning.
When the field was aligned exactly within the )* plane all signs of the Lorentz force disappeared. The remaining, featureless transition of 5 rather well agreed with the longitudinal
resistivity.
For magnetic fields along the axis, the ratio % of the in-plane resistivities, % 5 5 , was found to be fairly constant. In a narrow range close to , however, the
than for within the plane. This was interpreted as an
size of % was higher for
effect of almost isotropic fluctuations.
Paper XI
In Paper XI, finally, we explained our interpretation of Paper VIII. In a Comment to
Paper VIII, it was argued by Maiorov et al. (submitted) that our measurements on the untwinned crystals should be interpreted as observations of an “uncorrelated” vortex system.
To support this point of view, they presented PFT measurements on twinned and untwinned
YBCO crystals, indicating qualitatively different current – voltage characteristics in the region above where the voltage ratio is close to . While the twinned crystal had a range
above with nonlinear response, the untwinned crystal was Ohmic in all directions. In
our Reply, we noted that the measurement by Maiorov et al. on the untwinned crystal confirms our results in Paper VIII. We further made clear that our measurements were Ohmic
in the vortex liquid, indicating strong, but not perfect correlation. Thus, we argued that
a perfectly correlated system should be phase coherent in the field direction and have a
bottom-to-top voltage ratio exactly equal to , while the ratio of an uncorrelated system
should equal the normal-state value. The correlations are strong when the ratio is far from
the uncorrelated case.
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Index of citations
Abrikosov (1957), 1, 10
Abulafia et al. (1996), 35
Almasan and Maple (1996), 45
Almasan et al. (1992), 64
Ambegaokar and Halperin (1969), 34
Anderson (1962), 28
Anderson and Kim (1964), 25, 28
Anderson and Rowell (1963), 9
Andersson (1992), 59, 60
Andersson (1999), 62
Andersson et al. (2000), vii, 65
Andersson et al. (2001), vii, 65
Ashcroft and Mermin (1976), 4, 5
Attanasio et al. (1995), 30
Bardeen (1964), 24
Bardeen and Stephen (1965), 23, 26
Bardeen et al. (1957a), 8, 9
Bardeen et al. (1957b), 8
Baumert (1995), 53, 67
Beasley et al. (1969), 29
Bednorz and Müller (1986), 1, 14, 15
Berghuis et al. (1990), 41
Berlincourt et al. (1961), 34
Bhattacharya and Higgins (1993), 40
Björnängen et al. (2001), vii, 34, 36, 38,
49–51, 68
Blackstead and Kapustin (1994), 34
Blatter and Ivlev (1993), 37
Blatter and Ivlev (1994a), 37
Blatter and Ivlev (1994b), 37, 38
Blatter et al. (1992), 20
Blatter et al. (1994), 1, 7, 19, 22, 25, 26,
42, 43, 46, 64
Bouquet et al. (2001), 39
Brandt (1995), 1, 2, 11, 13, 17, 19, 22,
24, 27, 32, 37, 51
Brandt and Essmann (1987), 2
Briceño et al. (1991), 34
Bulaevskii and Vagner (1991), 21
Busch et al. (1992), 48, 49, 66
Caroli et al. (1964), 23
Cava et al. (1990), 56
Chandrasekhar (1969), 7
Charalambous et al. (1992), 36, 66
Charalambous et al. (1993), 37
Chien et al. (1994), 3, 22, 57, 63
Chin et al. (1999), 51
Civale et al. (1990), 42
Civale et al. (1991), 40
Claus et al. (1990), 55, 57
Clem (1974), 48
Clem (1975), 48
Clem (1989), 21
Clem (1991), 17, 21
Clem (1998), 20
Cohen and Jensen (1997), 44, 64
Cooper et al. (1991), 55
Cooper (1956), 8
Crabtree and Nelson (1997), 48
Crabtree et al. (1996), 37
Cubitt et al. (1993), 2, 36, 39
Daemen et al. (1993), 21
D’Anna et al. (1995), 41, 66
D’Anna et al. (1998), 36, 37
Däumling et al. (1996), 39
de Gennes and Matricon (1964), 24
de la Cruz et al. (1994), 51
Deak et al. (1993), 44
Deak et al. (1995a), 44, 45, 64
91
92
Deak et al. (1995b), 44
Deaver, Jr. and Fairbank (1961), 10
Deligiannis et al. (1997), 2, 39
DeSorbo (1964), 34, 40
Dew-Hughes (1988), 29, 30
Dolan et al. (1989), 2
Doll and Näbauer (1961), 10
Domínguez et al. (1995), 37
Doyle et al. (1996), 42, 50, 67
Efetov (1979), 17
Ekin and Clem (1975), 48
Ekin et al. (1974), 48
Eltsev and Rapp (1995), 50, 67
Eltsev et al. (1994), 49, 66
Erb et al. (1995), 54
Erb et al. (1996), 55
Essmann and Träuble (1967), 2
Feigel’man et al. (1989), 29, 42
Feigel’man et al. (1990), 22
Feinberg et al. (1994), 21
Fendrich et al. (1995), 30, 34, 42
Fendrich et al. (1996), 66
Fisher et al. (1991), 2, 22, 42
Fisher (1989), 42
Fleshler et al. (1993), 40
Forgan et al. (1996), 36, 39
Friesen et al. (1996), 44
Fuchs et al. (1996), 36
Fuchs et al. (1997), 36, 50
Fuchs et al. (1998), 39
Fujiki et al. (1998), 44
Gammel et al. (1987), 2
Gammel et al. (1988), 3, 33
Gammel et al. (1991), 44
Garland and Lee (1987), 30
Gavaler (1973), 5
Geshkenbein et al. (1989), 29–31
Ghamlouch et al. (1996), 36
Giaever (1960a), 9
Giaever (1960b), 9
Giaever (1965), 48, 66
Giaever (1966), 48, 66
Giamarchi and Le Doussal (1995), 39
Giamarchi and Le Doussal (1997), 2, 39
Index of citations
Ginzburg and Landau (1950), 7
Gjølmesli et al. (1995), 32
Glazman and Koshelev (1991), 22
Gordeev et al. (1997), 41, 66
Gordeev et al. (1998), 41, 66
Gordeev et al. (1999), 32
Gor’kov (1959a), 9
Gor’kov (1959b), 9
Gorter (1962a), 24
Gorter (1962b), 24
Gorter (1964), 4
Gray and Kim (1993), 34
Graybeal and Beasley (1986), 30
Grigera et al. (1998), 44
Hein (1992), 4
Henderson et al. (1998), 40, 41
Hergt et al. (1993), 32
Herrmann et al. (1996), 44
Hess et al. (1990), 2
Horn and Parks (1971), 30
Hou et al. (1994), 44, 45, 64
Hou et al. (1997), 44, 45, 64
Houghton et al. (1989), 37
Hulm and Matthias (1980), 4
Huse and Majumdar (1993), 35, 48, 50
Hussey et al. (1999), 36
Hwa et al. (1993), 42
Ishida et al. (1996), 22
Ishida et al. (1997), 3
Ishida et al. (1998), 2
Isihara (1991), 9
Iye et al. (1992), 3
Janossy et al. (1991), 22, 57
Jiang et al. (1995), 37
Jiang et al. (1997), 41, 66
Johnson et al. (1999), 2
Jorgensen et al. (1990), 55, 63
Josephson (1962), 9
Josephson (1965), 24
Junod et al. (1997), 3, 35, 39, 40
Kadowaki et al. (1994), 22
Kaiser et al. (1987), 53
Kamerlingh Onnes (1911a), 4
Kamerlingh Onnes (1911b), 4
Index of citations
Kamerlingh Onnes (1911c), 4
Kes et al. (1989), 30
Kes et al. (1990), 21
Khaykovich et al. (1996), 39
Khaykovich et al. (1997), 39, 63
Kim et al. (1990), 34
Kim and Stephen (1969), 24, 26
Kim et al. (1962), 28
Kim et al. (1963), 25, 28
Kim et al. (1964), 25
Kim et al. (1965), 26, 32
Kittel (1986), 4, 5
Klein et al. (1998), 44
Klein et al. (2001), 39
Klemm and Clem (1980), 19, 20
Koch et al. (1989), 32, 33, 44
Kogan (1981), 19
Kogan (1988), 22
Konczykowski et al. (1991), 40
Kopnin and Kravtsov (1976a), 25
Kopnin and Kravtsov (1976b), 25
Koshelev (1997), 51
Kötzler et al. (1994a), 44
Kötzler et al. (1994b), 44
Krusin-Elbaum et al. (1992), 29
Kucera et al. (1992), 31
Küpfer et al. (1994), 29
Kwok et al. (1990), 34, 36
Kwok et al. (1992), 36, 40
Kwok et al. (1994a), 37
Kwok et al. (1994b), 40, 41, 66
Kwok et al. (1994c), 40, 41
Kwok et al. (1997), 40, 41, 66
Kwok et al. (2000), 40
Labusch (1969), 33
LaGraff and Payne (1993), 55
Landau (1937), 7, 35
Larkin and Ovchinnikov (1979), 27, 40
Larkin and Vinokur (1995), 40, 42
Lawrence and Doniach (1971), 18, 21
Lee et al. (1993), 2, 36
Liang et al. (1992), 54
Liang et al. (1996), 2, 35
Lindemer et al. (1989), 55
93
Livingston and DeSorbo (1969), 6
London (1935), 7
London (1948), 7
London (1950), 6–8, 10
London and London (1935a), 5
London and London (1935b), 5
López et al. (1994a), 49, 51
López et al. (1994b), 49, 67
López et al. (1996a), 49, 51, 67
López et al. (1996b), 51, 67
López et al. (1997), 42
López et al. (1998), 31, 50
López et al. (1999), 49, 51
Lundqvist (1998), 53, 60
Lundqvist (2000), 53
Lundqvist et al. (1997), vii, 34, 44, 56,
63
Lundqvist et al. (1998a), 31
Lundqvist et al. (1998b), vii, 44, 45, 64
Maggio-Aprile et al. (1995), 2
Maiorov et al. (submitted), 68
Majer et al. (1995), 35
Maley et al. (1990), 32
Marchetti and Nelson (1990), 35, 48, 50
Marchetti and Nelson (2000), 35, 43, 48
Martínez et al. (1992), 3, 22
Masker et al. (1969), 30
Meissner and Ochsenfeld (1933), 5
Meuffels et al. (1989), 55, 63
Mirković et al. (1999), 36
Mittag et al. (1994), 46
Miu et al. (1998), 30
Moloni et al. (1997), 44
Moser et al. (1995), 2
Müller et al. (1987), 33, 45
Mun et al. (1996), 36
Nagamatsu et al. (2001), 15
Nelson (1988), 34, 42, 48
Nelson and Seung (1989), 34, 48
Nelson and Vinokur (1992), 40, 42, 43
Nelson and Vinokur (1993), 2, 40, 42, 43
Nguyen and Sudbø (1998), 48
Nguyen and Sudbø (1999), 48, 51
Nikolo et al. (1992), 30
94
Nishizaki and Kobayashi (2000), 39, 63
Nishizaki et al. (1996a), 2, 35
Nishizaki et al. (1996b), 35, 39
Nishizaki et al. (1998), 39
Nishizaki et al. (2000), 39
Nozières and Vinen (1966), 24, 26
Olsson and Teitel (1999), 48
Oral et al. (1998), 2, 36
Orlando and Delin (1991), 18
Palstra et al. (1988), 30, 33
Palstra et al. (1990), 30, 33
Parks (1969), 9
Pastoriza and Kes (1995), 36
Pastoriza et al. (1994), 35
Paulius et al. (2000), 63
Pautrat et al. (1998), 49
Pautrat et al. (1999), 49, 50
Petrean et al. (2000), 34, 44
Pippard (1953), 8, 11
Pippard (1969), 34, 40
Poole, Jr. (2000), 4, 15, 16, 53
Rapp et al. (1999), 57
Reichhardt et al. (2000), 42, 44
Reyes et al. (1997), 36, 44
Rickayzen (1969), 9
Righi et al. (1997a), 50, 51
Righi et al. (1997b), 48, 50, 67
Roberts et al. (1994), 44
Rosamilia et al. (1991), 53
Rosenberg (1993), 6, 11
Rothman et al. (1991), 54, 55
Roulin et al. (1998), 40
Rydh and Rapp (2001), vii, 34, 36, 49–
51, 66–68
Rydh and Rapp (submitted), 68
Rydh et al. (1999a), vii, 33, 41, 66, 68
Rydh et al. (1999b), vii, 46, 64
Rydh et al. (2000a), vii, 67
Rydh et al. (2000b), vii, 65
Rykov et al. (1999), 55
Safar et al. (1992a), 44, 45, 64
Safar et al. (1992b), 36, 37
Safar et al. (1992c), 48, 49, 66
Safar et al. (1993), 37, 38, 44
Index of citations
Safar et al. (1994), 49, 50, 66, 67
Safar et al. (1995), 41, 66
Samoilov et al. (1996), 40
Samuely et al. (1997), 68
Schilling et al. (1996), 3, 35
Schilling et al. (1997), 3, 35, 40
Schönenberger et al. (1996), 35
Seidler et al. (1991), 45
Sengupta et al. (1993), 32
Shubnikov et al. (1937), 10
S̆imánek (1994), 26
Soibel et al. (2000), 36, 37
Sonier et al. (2000), 2
Stephen and Bardeen (1965), 23
Strnad et al. (1964), 26, 32
Thompson et al. (1991), 29
Thompson et al. (1992), 42
Tinkham (1964), 24
Tinkham (1988), 33, 34
Tinkham (1996), 8, 9, 11, 23, 29, 37
Trajanovic et al. (1997), 22
van der Beek et al. (1995), 42
Van Duzer and Turner (1981), 9
Vinen (1969), 24
Vinokur et al. (1990a), 30–32, 34
Vinokur et al. (1990b), 30
Vinokur et al. (1998), 39, 46, 64
Wagner et al. (1995), 30, 31, 44
Waldram (1996), 4
Wang (1999), 31
Wang et al. (1997), 31
Welp et al. (1996), 2, 35, 36, 40
Werthamer (1969), 7
Willemin et al. (1998), 3
Worthington et al. (1992), 44
Wu et al. (1997), 41
Wu et al. (1987), 15
Yamasaki et al. (1993), 30, 46
Yang et al. (1994), 39
Yeh et al. (1992), 44
Yeshurun and Malozemoff (1988), 33, 45
Zeldov et al. (1989), 31, 32
Zeldov et al. (1990), 32
Zeldov et al. (1995), 2, 35, 39, 66
Author index
Baratoff, A., 2
Bardeen, J., 8, 9, 23, 24, 26
Barnard, J. C., 2, 36
Batlogg, B., 30, 33, 53, 56
Baumann, M., 44
Baumert, B. A., 53, 67
Beasley, M. R., 29, 30
Bednorz, J. G., 1, 14, 15, 33, 45
Behr, R., 44
Bending, S. J., 2, 36
Berghuis, P., 41
Berlincourt, T. G., 34
Berseth, V., 36, 37
Bhattacharya, S., 40
Bishop, A. R., 37
Bishop, D. J., 2, 3, 33, 36–38, 41, 44, 45,
48–50, 64, 66, 67, 73
Björnängen, T., vii, 34, 36, 38, 49–51, 68
Blackstead, H. A., 34
Blanchard, S., 39
Blatter, G., 1, 7, 19, 20, 22, 25, 26, 35,
37, 38, 42, 43, 46, 64, 76
Bonn, D. A., 2, 35, 54
Bouquet, F., 39
Brabers, V. A. M., 46
Brandt, E. H., 1, 2, 11, 13, 17, 19, 22, 24,
27, 32, 37, 44, 51
Brewer, J. H., 2
Briceño, G., 34
Brongersma, S. H., 3, 22
Brown, B., 44
Brynestad, J., 55
Bulaevskii, L. N., 21
Burlachkov, L., 35
Burns, C. A., 63
A
Aarts, J., 21, 30
Abell, J. S., 39
Abrikosov, A. A., 1, 10
Abulafia, Y., 35
Adachi, S., 36
Adrian, H., 30, 31, 44
Aegerter, C. M., 2
Ager, C., 2
Akimitsu, J., 15
Alers, G. B., 41, 66
Almasan, C. C., 30, 44, 45, 64
Ambegaokar, V., 34
Amer, N. M., 31, 32
Anderson, P. W., 9, 25, 28
Andersson, M., vii, 31, 33, 34, 41, 44–
46, 56, 57, 59, 60, 62–66, 68
Andrei, E. Y., 40, 41
Aomine, T., 44
Arribére, A., 35
Asaoka, H., 2, 3, 22, 35
Ashburn, J. R., 15
Ashcroft, N. W., 4, 5
Assmus, W., 44
Attanasio, C., 30
Aubin, M., 36
Ausloos, M., 57
Axnäs, J., 57
B
Baar, D. J., 54
Bachman, H. N., 36, 44
Baker, J. E., 54, 55
Balachandran, U., 55
Balseiro, C. A., 44
95
96
Busch, R., 48, 49, 66
C
Calemczuk, R., 39
Campbell, A. M., 42, 50, 67, 81
Canfield, P. C., 36
Cao, X. W., 31
Carolan, J. F., 54
Caroli, C., 23
Cava, R. J., 53, 56
Chakhalian, J., 2
Chakoumakos, B. C., 42
Chandrasekhar, B. S., 7
Charalambous, M., 36, 37, 66
Chaud, X., 49
Chaudhari, P., 2
Chaussy, J., 36, 37, 66
Chen, J. L., 31
Chien, T. R., 3, 22, 57, 63
Chin, S-K., 51
Chisholm, M. F., 53
Cho, B. K., 36
Christen, D. K., 42
Chu, C. W., 15
Civale, L., 29, 40, 42, 50, 51
Claus, H., 55, 57, 63
Clem, J. R., 17, 19–21, 40, 48, 70
Coccorese, C., 30
Cohen, L. F., 44, 64
Conde-Gallardo, A., 44
Cooper, E. A., 55
Cooper, L. N., 8, 9
Coulter, J. Y., 21
Crabtree, G. W., 2, 3, 31, 34–37, 39–42,
44, 48–51, 55, 63, 66, 67
Crommie, M. F., 34
Cubitt, R., 2, 36, 39
D
Daemen, L. L., 21
Dalichaouch, Y., 64
D’Anna, G., 36, 37, 41, 66
Dasgupta, D., 3, 35
Datars, W. R., 3, 22, 57, 63
Author index
Däumling, M., 39
de Andrade, M. C., 44, 64
de Gennes, P. G., 24
de Groot, D. G., 3, 22
de Groot, P. A. J., 2, 32, 39, 41, 66
de la Cruz, F., 31, 35, 44, 48–51, 66–68,
73
Deak, J., 44, 45, 64
Deaver, Jr., B. S., 10
Deligiannis, K., 2, 39
Delin, K. A., 18
DeSorbo, W., 6, 34, 40
Dew-Hughes, D., 29, 30
Dickey, R. P., 44
Dinger, T. R., 2
Dolan, G. J., 2
Doll, R., 10
Domínguez, D., 37
Doniach, S., 18, 21
Dosanjh, P., 54
Downey, J. W., 36, 37, 40, 41, 45, 55, 57
Doyle, R. A., 36, 39, 42, 46, 50, 63, 64,
67, 81
Drost, R. J., 36, 50
Duan, H. M., 30
Dunsiger, S. R., 2
Dye, R., 44
E
Eckstein, J. N., 31
Efetov, K. B., 17
Ekin, J. W., 48
Eltsev, Yu., vii, 44, 45, 49, 50, 57, 64, 66,
67, 86
Emmen, J. H. P. M., 46
Endo, K., 30, 46
Erb, A., 2, 3, 35–37, 39, 40, 54, 55
Escribe-Filippini, C., 44, 68
Essmann, U., 2
Ettouhami, A. M., 21
Evetts, J., 4
F
Face, D. W., 44
Author index
97
Fairbank, W. M., 10
Fang, J., 31
Feigel’man, M. V., 22, 29–32, 34, 40, 42
Feild, C., 2, 31, 42, 50
Feinberg, D., 21
Fendrich, J. A., 2, 31, 34–37, 40–42, 44,
51, 63, 66, 67
Ferguson, S., 63
Fischer, Ø., 2
Fisher, D. S., 2, 22, 42
Fisher, M. P. A., 2, 22, 32, 33, 42, 44
Fisher, R. A., 3, 35, 39, 40
Fleshler, S., 31, 34, 36, 37, 40–42
Flükiger, R., 39, 54
Foglietti, V., 32, 33, 44
Foltyn, S., 44
Forgan, E. M., 2, 36, 39, 55
Forró, L., 36, 37
Fossheim, K., 32
Frey, U., 30, 31, 44
Friesen, M., 44
Fritz, O., 2
Fruchter, L., 22, 57
Fuchs, D. T., 36, 39, 50
Fujiki, H., 44
Giamarchi, T., 2, 39
Giapintzakis, J., 31, 34, 37, 38, 41, 42,
44, 66
Ginsberg, D. M., 36–38, 41, 44, 66
Ginzburg, V. L., 7
Gjølmesli, S., 32
Glarum, S. H., 53
Glazman, L. I., 22
Goerke, F., 44
Goffman, M. F., 35
Gordeev, S. N., 29, 32, 41, 66
Gor’kov, L. P., 9
Gorter, C. J., 4, 24
Gough, C. E., 39
Gray, K. E., 34
Graybeal, J. M., 30
Greb, B., 55
Greene, R. L., 44
Griessen, R. P., 3, 22
Grigera, S. A., 31, 44, 48, 50, 51, 67
Grønbech-Jensen, N., 37
Gu, C., 3, 22, 57, 63
Guertin, R. P., 64
Güntherodt, H.-J., 2
Gupta, A., 31–33, 44
G
Gagnon, R., 2, 32, 36, 39, 41, 66
Gallagher, P. K., 55
Gallagher, W. J., 32, 33, 44
Gambino, R. J., 31, 32
Gammel, P. L., 2, 3, 33, 36–38, 41, 44,
45, 48–50, 64, 66, 67, 73
Gangopadhyay, A. K., 55
Gao, L., 15
Garland, J. C., 30, 64, 76
Gates, J. E., 55
Gavaler, J. R., 5
Genoud, J.-Y., 3, 35, 39, 40
Geshkenbein, V. B., 1, 2, 7, 19, 20, 22,
25, 26, 29–32, 34–37, 39, 42,
43, 46, 64, 66, 76
Ghamlouch, H., 36
Giaever, I., 9, 48, 66
H
Haibach, P., 30
Hake, R. R., 34
Halperin, B. I., 34
Halperin, W. P., 36, 44
Hammel, P. C., 36, 44
Hao, Z., 70
Hardy, W. N., 2, 35, 54
Harris, D. C., 76
Hebboul, S. E., 76
Heeb, E., 35
Hein, R. A., 4
Heinz, D. L., 45
Hempstead, C. F., 25, 26, 28, 32
Henderson, W., 40, 41
Henini, M., 2, 36
Hergt, R., 32
Hermann, A. M., 30
98
Hermann, B. A., 44
Herrmann, J., 44
Herting, A., 31
Hess, H. F., 2
Hewat, A. W., 56
Hewat, E. A., 56
Hiergeist, R., 32
Higgins, M. J., 40, 41
Hillmer, F., 30, 31, 44
Hofer, J., 3
Hofman, D. J., 40
Holm, W., 49, 66
Holstein, W. L., 44
Holtzberg, F., 37, 40, 41, 44, 66
Holtzberg, F. H., 42
Holzberg, F., 2, 29, 53
Hor, P. H., 15
Horn, P. M., 30
Horovitz, B., 88
Hou, L., 44, 45, 64
Houghton, A., 37
Hu, D., 46
Huang, Z. J., 15
Hubbard, C. R., 55
Hug, H. J., 2
Hulm, J. K., 4
Hulscher, R., 34, 36
Hunley, J. F., 55
Huse, D. A., 2, 22, 35–38, 41, 42, 44,
48–50, 66, 67, 73
Hussey, N. E., 36
Hwa, T., 42
I
Inoue, K., 22
Ishida, T., 2, 3, 22
Isihara, A., 9
Ivanov, Z. G., 31
Ivlev, B. I., 3, 22, 37, 38
Iwase, A., 39
Iye, Y., 3, 36
J
Jahn, W., 29
Author index
Jakob, G., 30
Janossy, B., 22, 57
Jansen, A. G. M., 44, 68
Jensen, H. J., 44, 49, 64, 67
Jiang, W., 37, 41, 44, 66
Jiang, Wu, 44
Jiang, Y., 3, 22, 57, 63
Johansson, L.-G., 31
Johnson, S. T., 2
Johnston, D. C., 36
Jorgensen, J. D., 55, 63
Josephson, B. D., 9, 24
Joumard, I., vii, 39, 67
Junod, A., 3, 35, 39, 40
K
Kadowaki, K., 22, 36, 42, 50, 67, 81
Kaiser, D. L., 53
Kajimura, K., 30, 46
Kamerlingh Onnes, H., 4
Kampwirth, R. T., 34
Kanda, Eizo, 18, 21
Kapitulnik, A., 44, 45, 64
Kapustin, G. A., 34
Karapetrov, G., 40
Kaufmann, M., 44
Kawae, T., 44
Kaya, I. I., 2, 36
Kazumata, Y., 3, 22
Kealey, P. G., 2
Keller, H., 2, 3, 36
Kerchner, H. R., 42
Kes, P. H., 2, 3, 21, 22, 30, 36, 39, 41,
42, 46, 50, 63, 64
Khaykovich, B., 39, 46, 63, 64
Khotkevich, W. J., 10
Kiefl, R. F., 2
Kiehl, W., 30
Kim, D. H., 34
Kim, Y. B., 24–26, 28, 32
Kimura, K., 36
Kirk, M. A., 40, 42
Kitazawa, K., 3, 22
Kittel, Ch., 4, 5
Author index
Klein, T., vii, 39, 44, 67, 68
Klemm, R. A., 19, 20
Kobayashi, N., 2, 35, 39, 63
Koch, R. H., 32, 33, 44
Kogan, V. G., 19, 22
Konczykowski, M., 2, 35–37, 39, 40, 42,
46, 50, 63, 64, 66
Kopnin, N. B., 25
Koren, G., 31–33, 44
Kosaka, S., 30, 46
Koshelev, A. E., 3, 22, 30, 51, 66
Kostic, P., 3, 22, 57, 63
Kötzler, J., 44
Krajewski, J. J., 56
Kravtsov, V. E., 25
Kreiselmeyer, G., 48, 49, 66
Kresse, R., 29
Kriplani, U., 37
Kruchinin, S., 57
Krusin-Elbaum, L., 31, 40, 42, 50
Kucera, J. T., 31
Küpfer, H., 29
Kuric, M. V., 64
Kushnir, V. N., 30
Kusmartsev, F. V., 55
Kussmaul, A., 44
Kwo, J. R., 2
Kwok, W. K., 2, 3, 31, 34–37, 39–42,
49–51, 55, 63, 66, 67
Kwon, C., 22
L
Labusch, R., 29, 33
LaGraff, J. R., 55
Landau, L. D., 7, 35
Langan, R. M., 2, 32, 39, 41, 66
Larkin, A. I., 1, 7, 19, 20, 22, 25–27, 29–
32, 34, 35, 40, 42, 43, 46, 64,
76
Larsson, J., 31
Lawrence, W. E., 18, 21
Le Doussal, P., 2, 39, 42
Lee, D., 29
Lee, H. J., 30
99
Lee, S. L., 2, 36, 39
Lee, Sung-Ik, 36
Lee, W. C., 36–38, 41, 44, 66
Lejay, P., 36, 37, 40, 66
Leslie, D. H., 34
Lessure, H., 32
Levin, G. A., 73
Levy, P., 50, 51
Li, K. B., 31
Li, S., 44
Li, T. W., 2, 36, 39, 42, 63
Liang, R., 2, 35, 54
Lindemer, T. B., 55
Liu, J. Z., 34, 36
Livingston, J. D., 6
Lloyd, S. H., 2, 36, 39
Lobb, C. J., 22
London, F., 5–8, 10
London, H., 5
López, D., 31, 42, 49–51, 67
Lundqvist, B., vii, 31, 34, 44, 45, 53, 56,
57, 60, 63, 64
M
MacFarlane, W. A., 2
Maggio-Aprile, I., 2
Maiorov, B., 68
Majer, D., 2, 35, 36, 39, 50, 63, 66
Majumdar, S. N., 35, 48–50, 66, 67, 73
Maley, M. P., 21, 32
Malozemoff, A. P., 33, 42, 45
Mao, S. N., 44
Maple, M. B., 44, 45, 64
Marčelja, S., 30
Marcenat, C., 39
Marchetti, M. C., 35, 43, 48, 50
Marcus, J., 39, 44, 68
Marezio, M., 56
Maritato, L., 30
Martindale, J. A., 36, 44
Martínez, J. C., 3, 22
Marwick, A. D., 40, 42
Masker, W. E., 30
Mason, T. O., 55
100
Matricon, J., 23, 24
Matthias, B. T., 4
McElfresh, M. W., 31, 32, 42, 44, 45, 64
McGinn, P. J., 32
McHenry, M. E., 32
McKay, D. M., 34
Meier-Hirmer, R., 29
Meissner, W., 5
Meng, R. L., 15
Menovsky, A. A., 2, 3, 22, 36, 39
Mermin, N. D., 4, 5
Metcalf, P., 44, 45, 64
Meuffels, P., 55, 63
Miller, B., 53
Miller, M. M., 36, 40
Miller, R. I., 2
Mirković, J., 36
Mittag, M., 46
Mitzi, D. B., 44, 45, 64
Miu, L., 30
Mochiku, T., 42, 50, 67, 81
Moloni, K., 44
Mook, H. A., 2, 36, 39
Mori, N., 36
Morré, E., 44
Mortensen, K., 2, 36, 39
Moser, A., 2
Motohira, M., 3
Moulton, W. G., 40
Muenchausen, R., 44
Müller, K. A., 33, 45
Müller-Vogt, G., 55
Mun, Mi-Ock, 36
Muranaka, T., 15
Murray, C. A., 2
Myasoedov, Y., 36, 37
Mydosh, J. A., 30
N
Näbauer, M., 10
Naeven, R., 55, 63
Nagamatsu, J., 15
Naito, T., 35, 39
Nakagawa, N., 15
Author index
Nakielski, G., 44
Nelson, D. R., 2, 34, 35, 40, 42, 43, 48,
50
Neumeier, J. J., 64
Neumueller, H. W., 32
Nguyen, A. K., 48, 51
Nieva, G., 44, 48–51, 66, 67, 73
Nikolo, M., 30
Nishizaki, T., 2, 35, 39, 63
Noakes, D. R., 2
Noda, K., 3, 22
Nowicki, L. J., 55, 63
Nozières, P., 24, 26
O
Ochsenfeld, R., 5
Oguro, I, 3
O’Kane, D., 49, 67
Okayasu, S., 39
Okuda, K., 2, 3, 22
Olsson, P., 48
Olsson, R. J., 3, 40, 49, 51, 63
Ong, N. P., 41
Onodera, Y., 2, 35, 39
Ooi, S., 2, 36, 37, 39, 50
Oral, A., 2, 36
Orlando, T. P., 18, 31
Osquiguil, E., 44, 68
Oussena, M., 2, 39, 41, 66
Ovchinnikov, Yu. N., 27, 40
P
Palstra, T. T. M., 30, 33
Parashikov, I., 2
Parks, R. D., 6, 7, 9, 24, 26, 30
Pasquini, G., 50, 51
Pastoriza, H., 35, 36
Paul, D. McK., 2, 36, 39
Paulikas, A. P., 3, 22, 45, 55, 57, 63
Paulius, L. M., 31, 34, 40, 41, 44, 49–51,
63, 66, 67
Pautrat, A., 49, 50
Payne, D. A., 55
Peck Jr., W. F., 56
Author index
Pekker, S., 22, 57
Pelcovits, R. A., 37
Peligrad, D., 46
Petrean, A. M., 34, 44, 49, 51, 63
Phillips, N. E., 3, 35, 39, 40
Pinfold, S., 2, 39, 41, 66
Pippard, A. B., 8, 11, 34, 40
Poole, Jr., C. P., 4, 15, 16, 53
Preosti, G., 44, 45, 64
Prischepa, S. L., 30
Prost, D., 22, 57
Prozorov, R., 35
R
Rabe, K. M., 56
Rajeswari, M., 22
Rapp, Ö., vii, 31, 33, 34, 36, 38, 41, 44–
46, 49–51, 56, 57, 63–68, 82,
86
Rappaport, M. L., 36, 37, 39
Rassau, A. P., 32, 41, 66
Reed, D. S., 37, 44
Reichhardt, C., 42, 44
Renner, Ch., 2
Revaz, B., 3, 35, 39, 40
Reyes, A. P., 36, 44
Riabinin, J. N., 10
Rice, A. P., 41, 66
Rice, J. P., 36, 37
Rickayzen, G., 9
Ries, G., 32, 48, 49, 66
Righi, E. F., 31, 48–51, 67
Riseman, T. M., 2
Roberts, J. M., 44
Robinson, R. B., 2
Rodriguez, E., 48, 49, 66
Rosamilia, J. M., 53
Rosenbaum, T. F., 45
Rosenberg, H. M., 6, 11
Rosenberg, M., 46
Rossel, C., 3
Rothman, S. J., 54, 55
Roulin, M., 3, 35, 39, 40
Routbort, J. L., 54, 55
101
Rowell, J. M., 9
Rullier-Albenque, F., 40
Rupp Jr., L. W., 56
Rydh, A., vii, 33, 34, 36, 38, 41, 44–46,
49–51, 56, 57, 63–68, 82, 86
Rykov, A. I., 2, 49, 50, 55
S
Saemann-Ischenko, G., 48, 49, 66
Safar, H., 31, 36–38, 41, 42, 44, 45, 48–
51, 64, 66, 67, 73
Salama, K., 29
Sales, B. C., 42
Salvato, M., 30
Samoilov, A. V., 40
Samuely, P., 44, 68
Sanfilippo, S., 49
Savić, I. M., 2, 36
Schauwecker, R., 2, 36
Schilling, A., 3, 35, 39, 40
Schneemeyer, L. F., 2, 3, 30, 33, 44, 48–
50, 66, 67, 73
Schönenberger, A., 35
Schrieffer, J. R., 8, 9
Schwartz, J., 32
Seaman, C. L., 64
Seidler, G. T., 45
Sengupta, S., 32
Seow, W. S., 36, 42, 50, 67, 81
Serin, B., 48
Seung, H. S., 34, 48
Shaulov, A., 35, 40
Shepelev, J. D., 10
Shi, D., 32
Shinozaki, B., 44
Shtrikman, H., 2, 35, 39, 66
Shubnikov, L. W., 10
Silhanek, A. V., 68
S̆imánek, E., 26
Simon, C., 49, 50, 55
Smith, M. E., 32
Smith, M. K., 40
Soibel, A., 36, 37
Songliu, Y., 22
102
Sonier, J. E., 2
Souw, V., 44
Steep, E., 39
Steifel, B., 2
Stephen, M. J., 23, 24, 26
Strnad, A. R., 25, 26, 28, 32
Stronach, C. E., 2
Sudbø, A., 37, 48, 51
Sun, J. Z., 42
Sun, Y., 40
Sun, Y. R., 29, 32, 42
Sutton, Jr., A. L., 55
Szabó, P., 44, 68
T
Taillefer, L., 2, 32, 36, 39, 41, 66
Tajima, S., 2, 49, 50, 55
Takagi, H., 36
Takashige, M., 33, 45
Takeda, K., 44
Takei, H., 2, 3, 22, 35
Takeshita, N., 36
Takeuchi, I., 22
Tamegai, T., 2, 3, 36, 37, 39, 50
Tanabe, K., 36
Tanaka, S., 44
Tang, X. P., 36, 44
Tarnawski, Z., 2, 3, 22, 36, 39
Tate, J., 44
Taubert, J., 32
Teitel, S., 48
ter Haar, D., 80
Terasaki, I., 2
Theodorakis, S., 21
Thomas, H., 2
Thompson, J. R., 29, 40, 42
Thomson, J. O., 42
Tinkham, M., 8, 9, 11, 23, 24, 29, 33, 34,
37
Tobos, V., 63
Tombrello, T. A., 41, 66
Torng, C. J., 15
Trajanovic, Z., 22
Träuble, H., 2
Author index
Turner, C. W., 9
U
Umeda, M., 30, 46
V
Vagner, I. D., 21
van den Berg, J., 30
van der Beek, C. J., 42
van der Slot, A. L. F., 41
van Dover, R. B., 30, 33
Van Duzer, T., 9
van Otterlo, A., 42, 44
Vandervoort, K. G., 34, 36
Veal, B. W., 2, 3, 22, 35–37, 40, 45, 55,
57, 63, 66
Venkatesan, T., 22
Vinen, W. F., 24, 26
Vinokur, V. M., 1, 2, 7, 19, 21, 22, 25,
26, 29–32, 34–37, 39, 40, 42,
43, 46, 64, 66
Virshup, G., 31
Viswanathan, H. K., 31, 34, 42
Volkozub, A. V., 41, 66
W
Wagner, P., 30, 31, 44
Waldram, J. R., 4
Walker, E., 2, 3, 35–37, 39, 40, 54
Wan, Y. M., 76
Wang, Y. Q., 15
Wang, Z., 32
Wang, Z. H., 31
Warden, M., 2, 36
Warmont, F., 49
Waszczak, J. V., 2, 3, 30, 33
Webb, W. W., 29
Welp, U., 2, 3, 31, 34–37, 39–42, 54, 55,
66
Wenzl, H., 55, 63
Wernhardt, R., 46
Werthamer, N. R., 7
Werthner, H., 48, 49, 66
Willemin, M., 3
Author index
Willis, J. O., 32
Wirth, G., 42, 50, 67, 81
Wolf, T., 29
Wolfus, Y., 35
Worthington, T. K., 40, 42, 44, 53
Wu, H., 41
Wu, M. K., 15
Wühl, H., 35
Wyder, P., 68
Wylie, M. T., 36, 39
Y
Yacoby, E. R., 40
Yamasaki, H., 30, 46
Yan, Y., 42, 50, 67, 81
Yang, G., 2, 36, 39
Yang, S., 55, 57
Yeh, N.-C., 37, 41, 44, 66
Yeshurun, Y., 33, 35, 40, 45
Yethiraj, M., 2, 36, 39
Yoshida, S., 30, 46
Z
Zech, D., 2, 36
Zeldov, E., 2, 31, 32, 35–37, 39, 46, 50,
63, 64, 66
Zenitani, Y., 15
Zettl, A., 34
Zhukov, A. A., 29
Zimányi, G. T., 42, 44
Zimmermann, P., 2, 36
103
104
Papers
I
Paper I
Vortex dynamics in oxygen deficient single crystals of YBa Cu OÆ
B. Lundqvist, A. Rydh, Ö. Rapp, and M. Andersson,
Physica (Amsterdam) 282–287 C, 1959 (1997).
II
Paper II
Empirical scaling of the vortex glass line above 1 T for high- superconductors
of varying anisotropy
B. Lundqvist, A. Rydh, Yu. Eltsev, Ö. Rapp, and M. Andersson,
Phys. Rev. B 57, R 14 064 (1998).
III
Paper III
Consistent description of the vortex glass resistivity in high- superconductors
A. Rydh, Ö. Rapp, and M. Andersson,
Phys. Rev. Lett. 83, 1850 (1999).
Paper IV
Thermally assisted flux creep of a driven vortex lattice in untwinned
YBa Cu OÆ single crystals
A. Rydh, M. Andersson, and Ö. Rapp,
J. Low Temp. Phys. 117, 1335 (1999).
IV
Paper V
Magnetic field scaling of the vortex glass resistivity in oxygen deficient YBa Cu OÆ
single crystals
A. Rydh, Ö. Rapp, and M. Andersson,
Physica (Amsterdam) 284–288 B, 707 (2000).
V
Paper VI
Multiterminal measurements of vortex correlations in the (K,Ba)BiO system
A. Rydh, T. Klein, I. Joumard, and Ö. Rapp,
Physica (Amsterdam) 341–348 C, 1233 (2000).
VI
Paper VII
Vortex liquid resistivity in disordered YBa Cu OÆ single crystals
M. Andersson, A. Rydh, and Ö. Rapp,
Physica (Amsterdam) 341–348 C, 1239 (2000).
VII
Paper VIII
Strong vortex liquid correlation from multiterminal measurements on untwinned
YBa Cu OÆ single crystals
A. Rydh and Ö. Rapp,
Phys. Rev. Lett. 86, 1873 (2001).
VIII
Paper IX
Scaling of the vortex liquid resistivity in optimally doped and oxygen-deficient
YBa Cu OÆ single crystals
M. Andersson, A. Rydh, and Ö. Rapp,
Phys. Rev. B 63, 184511 (2001).
IX
Paper X
Multiterminal transport measurements:
In-plane anisotropy and vortex liquid correlation in YBa Cu OÆ
T. Björnängen, A. Rydh, and Ö. Rapp,
Phys. Rev. B 64, 224510 (2001).
X
Paper XI
Reply to Comment on “Strong vortex liquid correlation from multiterminal measurements
on untwinned YBa Cu OÆ single crystals”
A. Rydh and Ö. Rapp,
Submitted to Phys. Rev. Lett.
XI