This is the last of three volumes of the extensively revised and updated second edition of the Handbook of Superconducti

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*Table of contents : CoverHalf TitleTitle PageCopyright PageContentsForewordPrefaceAcknowledgementsEditors-in-ChiefContributorsPART G: Characterization and Modelling TechniquesG1. Introduction to Section G1: Structure/MicrostructureG1.1. X-Ray Studies: Chemical CrystallographyG1.2. X-Ray Studies: Phase Transformations and Microstructure ChangesG1.3. Transmission Electron MicroscopyG1.4. An Introduction to Digital Image Analysis of SuperconductorsG1.5. Optical MicroscopyG1.6. Neutron Techniques: Flux-Line LatticeG2. Introduction to Section G2: Measurement and Interpretation of Electromagnetic PropertiesG2.1. Electromagnetic Properties of SuperconductorsG2.2. Numerical Models of the Electromagnetic Behavior of SuperconductorsG2.3. DC Transport Critical CurrentsG2.4. Characterisation of the Transport Critical Current Density for Conductor ApplicationsG2.5. Magnetic Measurements of Critical Current Density, Pinning, and Flux CreepG2.6. AC SusceptibilityG2.7. AC Losses in Superconducting Materials, Wires, and TapesG2.8. Characterization of Superconductor Magnetic Properties in Crossed Magnetic FieldsG2.9. Microwave ImpedanceG2.10. Local Probes of Magnetic Field DistributionG2.11. Some Unusual and Systematic Properties of Hole-Doped Cuprates in the Normal and Superconducting StatesG3. Introduction to Section G3: Thermal, Mechanical, and Other PropertiesG3.1. Thermal Properties: Specific HeatG3.2. Thermal Properties: Thermal ConductivityG3.3. Thermal Properties: Thermal ExpansionG3.4. Mechanical PropertiesG3.5. Magneto-Optical Characterization TechniquesPART H: ApplicationsH1. Introduction to Large Scale ApplicationsH1.1. Electromagnet FundamentalsH1.2. Superconducting Magnet DesignH1.3. MRI MagnetsH1.4. High-Temperature Superconducting Current LeadsH1.5. CablesH1.6. AC and DC Power TransmissionH1.7. Fault-Current LimitersH1.8. Energy StorageH1.9. TransformersH1.10. Electrical Machines Using HTS ConductorsH1.11. Electrical Machines Using Bulk HTSH1.12. Homopolar MotorsH1.13. Magnetic SeparationH1.14. Superconducting Radiofrequency CavitiesH2. Introduction to Section H2: High-Frequency DevicesH2.1. Microwave Resonators and FiltersH2.2. Transmission LinesH2.3. AntennaeH3. Introduction to Section H3: Josephson Junction DevicesH3.1. Josephson EffectsH3.2. SQUIDsH3.3. BiomagnetismH3.4. Nondestructive EvaluationH3.5. Digital ElectronicsH3.6. Superconducting Analog-to-Digital ConvertersH3.7. Superconducting QubitsH4. Introduction to Radiation and Particle Detectors that Use SuperconductivityH4.1. Superconducting Tunnel Junction Radiation DetectorsH4.2. Transition-Edge SensorsH4.3. Superconducting Materials for Microwave Kinetic Inductance DetectorsH4.4. Metallic Magnetic CalorimetersH4.5. Optical Detectors and SensorsH4.6. Low-Noise Superconducting Mixers for the Terahertz Frequency RangeH4.7. Applications: MetrologyGlossaryIndex*

Handbook of Superconductivity

Handbook of Superconductivity Characterization and Applications, Volume Three Second Edition

Edited by David A. Cardwell David C. Larbalestier Aleksander I. Braginski

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Second edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2023 Taylor & Francis Group, LLC First edition published by IOP Publishing 2003 CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-4398-1736-0 (hbk) ISBN: 978-0-367-68920-9 (pbk) ISBN: 978-1-003-13963-8 (ebk) DOI: 10.1201/9781003139638 Typeset in Minion Pro by KnowledgeWorks Global Ltd.

Contents Foreword ................................................................................................................................................. ix Preface..................................................................................................................................................... xi Acknowledgements ............................................................................................................................... xiii Editors-in-Chief .....................................................................................................................................xv Contributors ........................................................................................................................................ xvii

PArt G

G1

Characterization and Modelling techniques

Introduction to Section G1: Structure/Microstructure ..........................................................3 Lance D. Cooley

G1.1

X-Ray Studies: Chemical Crystallography ..............................................................................4

G1.2

X-Ray Studies: Phase Transformations and Microstructure Changes .................................22

Lance D. Cooley, Roman Gladyshevskii, and Theo Siegrist Christian Scheuerlein and M. Di Michiel

G1.3

Transmission Electron Microscopy ....................................................................................... 33 Fumitake Kametani

G1.4

An Introduction to Digital Image Analysis of Superconductors..........................................46 Charlie Sanabria and Peter J. Lee

G1.5

Optical Microscopy ................................................................................................................ 71 Pavel Diko

G1.6

Neutron Techniques: Flux-Line Lattice .................................................................................95 Jonathan White

G2

Introduction to Section G2: Measurement and Interpretation of Electromagnetic Properties ............................................................................................. 105 Fedor Gömöry

G2.1

Electromagnetic Properties of Superconductors ................................................................. 106 Archie M. Campbell

G2.2

Numerical Models of the Electromagnetic Behavior of Superconductors ......................... 130 Francesco Grilli

G2.3

DC Transport Critical Currents ........................................................................................... 141 Marc Dhallé

v

vi

G2.4

Contents

Characterisation of the Transport Critical Current Density for Conductor Applications ................................................................................................. 177 Mark J. Raine, Simon A. Keys, and Damian P. Hampshire

G2.5

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep ..............209 Michael Eisterer

G2.6

AC Susceptibility .................................................................................................................. 228 Carles Navau, Nuria Del-Valle, and Alvaro Sanchez

G2.7

AC Losses in Superconducting Materials, Wires, and Tapes ............................................. 238 Michael D. Sumption, Milan Majoros, and Edward W. Collings

G2.8

Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields .................................................................................................................... 251 Philippe Vanderbemden

G2.9

Microwave Impedance .........................................................................................................264 Adrian Porch

G2.10

Local Probes of Magnetic Field Distribution ...................................................................... 278 Alejandro V. Silhanek, Simon Bending, and Steve Lee

G2.11

Some Unusual and Systematic Properties of Hole-Doped Cuprates in the Normal and Superconducting States......................................................................... 301 John R. Cooper

G3

Introduction to Section G3: Thermal, Mechanical, and Other Properties ........................ 322 Antony Carrington

G3.1

Thermal Properties: Specific Heat ...................................................................................... 324 Antony Carrington

G3.2

Thermal Properties: Thermal Conductivity ....................................................................... 333 Kamran Behnia

G3.3

Thermal Properties: Thermal Expansion ............................................................................340 Christoph Meingast

G3.4

Mechanical Properties ......................................................................................................... 352 Wilfried Goldacker

G3.5

Magneto-Optical Characterization Techniques.................................................................. 371 Anatolii A. Polyanskii and David C. Larbalestier

PArt H

H1

Applications

Introduction to Large Scale Applications ........................................................................... 385 John H. Durrell and Mark Ainslie

H1.1

Electromagnet Fundamentals .............................................................................................. 387 Harry Jones

H1.2

Superconducting Magnet Design .........................................................................................400 M’hamed Lakrimi

H1.3

MRI Magnets ........................................................................................................................ 437 Michael Parizh and Wolfgang Stautner

H1.4

High-Temperature Superconducting Current Leads ........................................................... 493 Amalia Ballarino

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Contents

H1.5

Cables ...................................................................................................................................498 Naoyuki Amemiya

H1.6

AC and DC Power Transmission ......................................................................................... 505 Antonio Morandi

H1.7

Fault-Current Limiters ......................................................................................................... 517 Tabea Arndt

H1.8

Energy Storage...................................................................................................................... 523 Ahmet Cansiz

H1.9

Transformers ........................................................................................................................ 535 Nicholas J. Long

H1.10

Electrical Machines Using HTS Conductors.......................................................................542 Mark D. Ainslie

H1.11

Electrical Machines Using Bulk HTS .................................................................................. 553 Mark D. Ainslie

H1.12

Homopolar Motors ............................................................................................................... 562 Arkadiy Matsekh

H1.13

Magnetic Separation ............................................................................................................ 572 James H. P. Watson and Peter A. Beharrell

H1.14

Superconducting Radiofrequency Cavities ......................................................................... 583 Gianluigi Ciovati

H2

Introduction to Section H2: High-Frequency Devices ....................................................... 595 John Gallop and Horst Rogalla

H2.1

Microwave Resonators and Filters ....................................................................................... 596 Daniel E. Oates

H2.2

Transmission Lines ..............................................................................................................609 Orest G. Vendik

H2.3

Antennae .............................................................................................................................. 619 Heinz J. Chaloupka and Victor K. Kornev

H3

Introduction to Section H3: Josephson Junction Devices ................................................. 640 John Gallop and Alex I. Braginski

H3.1

Josephson Effects .................................................................................................................642 Francesco Tafuri

H3.2

SQUIDs ................................................................................................................................. 672 Jaap Flokstra and Paul Seidel

H3.3

Biomagnetism .......................................................................................................................682

H3.4

Nondestructive Evaluation ..................................................................................................690

Tilmann H. Sander Thoemmes

Hans-Joachim Krause, Michael Mück, and Saburo Tanaka

H3.5

Digital Electronics ............................................................................................................... 702 Oleg A. Mukhanov

H3.6

Superconducting Analog-to-Digital Converters ................................................................. 710 Alan M. Kadin and Oleg A. Mukhanov

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H3.7

Contents

Superconducting Qubits ...................................................................................................... 719 Britton Plourde and Frank K. Wilhelm-Mauch

H4

Introduction to Radiation and Particle Detectors that Use Superconductivity .................731 Caroline A. Kilbourne

H4.1

Superconducting Tunnel Junction Radiation Detectors ..................................................... 736 Stephan Friedrich

H4.2

Transition-Edge Sensors ...................................................................................................... 747 Douglas A. Bennett

H4.3

Superconducting Materials for Microwave Kinetic Inductance Detectors ........................ 756 Benjamin A. Mazin

H4.4

Metallic Magnetic Calorimeters .......................................................................................... 766 Andreas Fleischmann, Loredana Gastaldo, Sebastian Kempf, and Christian Enss

H4.5

Optical Detectors and Sensors ............................................................................................. 780 Roman Sobolewski

H4.6

Low-Noise Superconducting Mixers for the Terahertz Frequency Range.......................... 797 Victor Belitsky, Serguei Cherednichenko, and Dag Winkler

H4.7

Applications: Metrology ...................................................................................................... 812 John Gallop, Ling Hao, and Alain Rüfenacht

Glossary ................................................................................................................................................ 821 Index ..................................................................................................................................................... 839

Foreword It is a pleasure to introduce the second edition of the Handbook of Superconductivity, now with the subtitle Theory, Materials, Processing, Characterization and Applications. In combination, the enlarged title expresses the very broad scope of this publication. It is a mark of the ongoing vigour of the field of superconductivity that, in the 15 years or so since the first edition, tremendous progress has been made in theory, materials discovery and applications. Completely new topics have emerged, including topological superconductors, singleatomic-layer superconductivity and twistronics. New superconductors have been predicted and demonstrated, most notably the clathrate superhydride LaH10, which superconducts close to room temperature (though at several megabars). New applied technologies, such as flux pumps, have been demonstrated in motors, generators and MRI. This new edition is therefore timely in presenting the broad vista of our current knowledge in basic and applied superconductivity. Superconductivity is one of the most remarkable physical states yet discovered. More than any other known effect, superconductivity brings quantum mechanics to the scale of the everyday world where a single, coherent quantum state may extend over a distance of metres – or even kilometres – depending on the size of a coil or length of superconducting wire. And perhaps less well known, the underlying physics extends in scale to the very size of the universe, for the Higgs mechanism, by which all matter particles acquire their mass, has the same symmetry-breaking origins as superconductivity. There is something in this mysterious state that never fails to enchant its beholders: researcher, student or lay-person alike. The allure is to be found not only in the demonstrations of infinite conductivity and levitation, not only in the ongoing intellectual puzzle of its root physics, but also in the bold technologies that superconductivity enables. The puzzle of superconductivity resisted explanation for a very long time. The eventual breakthrough with the Bardeen– Cooper–Schrieffer theory, 46 years after their discovery, actually preceded, by about a decade, any significant practical and commercial development of these remarkable materials. Today, half a century later still, the progress of science and technology is greatly accelerated. And yet a theory of the cuprate high temperature superconductors (HTS), discovered 33 years ago, still remains elusive. For these materials

the tables are turned – a range of applications are now nudging their way onto the market even though we do not really understand them. That, of course, overstates the matter. We know that the supercarriers in HTS are Cooper pairs and their symmetry in reciprocal space is predominantly d-wave. We quantitatively understand many properties of HTS materials such as the effect of impurities in suppressing superconductivity and the temperature dependence of the specific heat, thermal expansion and superfluid density. However, this description is primarily based on thermodynamics and the observed d-wave symmetry. What we lack is a clear understanding of the mechanism that binds the pairs in the first place and a relationship between the magnitude of this interaction and the energy scale of the superconductivity set by the maximum d-wave gap amplitude. The difficulty here is the strongly interacting electronic system and HTS cuprates, which, in this sense, are just part of a much wider problem of strongly correlated transition metal oxides that incorporate manganites, ruthenates, cuprates, vanadates and tungstates to name just a few, not to mention hybrid materials, such as the ruthenocuprates, in which magnetism and superconductivity coexist. Such materials not only formally defy a suitable perturbation treatment, but they exhibit many different types of ground-state correlation that compete with each other. Thus, in the HTS cuprates, we currently struggle with the issues of charge ordering, spin ordering, nematicity and superconductivity and the question as to whether these are intimately linked or, in fact, compete. What if we could methodically remove each of these competing states one by one? Would superconductivity remain at all as the ground state? Would it fade away along with its supporters? Or might it even be enhanced? What is the importance of fluctuations or of shortrange versus long-range correlations? A current challenge to the community is to develop the tools to do precisely these kinds of elucidating experiments. The central approach then to these issues is systematic measurement in high-quality materials. With the combined improvement in quality of single crystals as well as resolution in low-energy experimental techniques, much progress has been made in the last several decades leading to many surprising new results. By “systematic” I mean variation in properties ix

x

with carrier concentration, temperature, magnetic field, ion size, pressure and disorder. It is really only recently that the effect of small increments in carefully controlled doping levels have been employed across a variety of spectroscopies and these studies have uncovered abrupt changes in physical properties, such as a ground-state metal insulator transitions and the possibility in many cases of a quantum critical point driving the essential physics and phase behaviour. Materials issues also lie at the heart of commercial application. Superconducting technologies, such as magnets, motors, power cables, transformers, flux pumps, NMR, MRI, telecommunications and computing, all push the present horizons of physical performance and their improvement, not to mention ultimate commercial success, predicated on ongoing materials development. If any relatively uncharted territory were to be identified it might, even today, be high pressure. Not a few pressure-dependent studies have been reported and, somewhat recently, elemental superconductivity has been discovered at high pressure in iron, sulphur and lithium. Nonetheless, high pressures allow one to significantly modify the magnetic exchange interaction in oxides and much more could be done to explore the links between magnetism and other competing correlations through the combined investigation of pressure and doping-dependent systematics. This might have been the situation up until just a few years ago. But beginning in 2015, dramatic developments in high pressure superconductivity have altered the superconductivity landscape forever. Demonstration of transition temperatures as high as 203 K were first reported for sulphur hydride at 155 GPa. Then, following what proved to be a remarkably accurate theoretical prediction, superconductivity was reported in LaH10 at 280 K at pressures around 200 GPa. The result is groundbreaking. It effectively achieves the century-old dream of roomtemperature superconductivity – quantum mechanics is now brought to everyday temperatures. The only remaining barrier, and it is no small challenge, is to reduce these enormous operational pressures to ambient conditions – everyday pressures. Such ongoing achievements continue to enliven the subject. But the point must be emphasized that in all these experiments

Foreword

pressure is used only to enable superconductivity. The challenge to the community is to utilize pressure as a tool to probe the essential physics by tuning spin or charge interactions to investigate their role in pairing. This entails the difficult task of incorporating spectroscopic studies in diamond-anvil experiments. So, while much has been done in this remarkable field, there is much yet to be done. These intellectual and engineering challenges form part of the ongoing puzzle and promise of superconductivity. The present handbook provides a snap-shot of our current knowledge of the field. It is necessarily in introductory form, but its sheer scope illustrates just how broad and multidisciplinary this field is. More than that, the wide range of authors underscores a further critical element of science in any field, namely the organic human element. Those of us who have spent a good few years in the field have collaborated, debated and otherwise interacted with many scientists around the globe and, in the process, formed lasting friendships that otherwise would never have eventuated. Sadly, a number of members of this research community have passed away since the first edition of this handbook. I would note just two, as representative of the many others, Professors John Clem and Koichi Kitazawa who, with me, wrote prefaces to that first edition. Each played a critical role in the development of our subject, each were passionate advocates and each were gentlemen in the finest sense of the word. It is our task to continue the legacy of these and so many others who have built the field to its current highly complex and impactful status. In the present edition, it is especially pleasing to see the names of so many research friends and collaborators as authors of the various sections of this handbook. Collectively, we trust that this volume will not only provide a comprehensive information base for the physics and phenomenology of superconductors, but will also communicate something of the puzzle and promise of superconductivity that continue to fascinate and motivate us as the years roll by. Jeffery L. Tallon

Preface This Handbook of Superconductivity; Theory, Materials, Processing, Characterization and Applications is the second, much expanded edition of the Handbook of Superconducting Materials, edited by David A. Cardwell and David Ginley and published originally by IOP in 2003. That large encyclopedic publication had quite favorable reviews, was rather quickly sold out to major distributors and is no longer readily available in retail. This situation alone would have justified the preparation of the 2nd edition. The second edition also became necessary for the reasons outlined below. The past nearly 20 years have seen rapid and dynamic progress in superconducting materials, with many new compounds and entire isostructural families discovered and eventually synthesized into usable wires, thin film structures and other forms suitable for applications. Of the newly discovered materials, some exhibited the conventional, phonon-mediated Cooper pairing, while others, notably the Fe-based superconductors, still present a formidable challenge to be understood theoretically. The discovery of these new materials is another reason why the 2nd edition is overdue. Progress has been less dynamic in the development of theory and the elucidation of unconventional superconductivity mechanisms, including those of high critical temperature, Tc, cuprates or iron-containing pnictides. As a result, it has been possible to reprint in the 2nd edition a number of rather fundamental chapters published originally in the 1st edition, without or with only minor updates. In a few cases, such reproductions are also in memoriam of 1st and 2nd edition authors and section editors who have passed away over the past two decades. Brian Pippard, E. Helmut Brandt, Heinz Chaloupka, Harry Jones and, the most recently departed, Archie M. Campbell are so honored. The inclusion of reprinted chapters has resulted in some heterogeneity in the Handbook’s style, such as in referencing and citing references within the text.

Viable new applications of superconductivity have emerged in the interim time period of nearly 20 years, and, in some cases, have even became state-of-the art dominant technology, such as, for example, various types of novel highly sensitive radiation and particle detectors, which enabled astronomers to discover many distant galaxies and even distant exoplanets. Both established and novel superconductors have been useful in this particular application. Another example of a currently dominant technology in a completely novel field is that of superconducting qubits and circuits for quantum computing, which most recently reached the milestone of first “quantum supremacy” demonstration. Such new and dominant superconducting technologies have necessitated the addition of several new Handbook chapters and justified changing its title to the present one. Although we have broadened and expanded this Handbook, we do not claim it covers the field of superconductivity in its entirety. Nevertheless, we hope it presents a relatively detailed and up-to-date introduction into that field, suitable for both graduate students and practitioners in experimental physics and multiple engineering disciplines: electronic and electrical, chemical, mechanical, metallurgy and others. Finally, the ultimate phase of our editorial activity coincided with the rapid progression of the Coronavirus pandemic in early 2020, which presented a particular challenge to the timely completion of this work. As a result, the Handbook contains some nonessential defects of presentation, such as occasional use of cgs units system rather than the SI units. We considered it tolerable given the need to publish the work as soon as possible. David A. Cardwell David C. Larbalestier Aleksander I. Braginski

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Acknowledgements The Editors-in-Chief would like to acknowledge the contribution of a number of individuals and institutions who have played a major role in the development of the Handbook of Superconductivity in its evolution from the first edition to its production in a new and substantially revised format. The Handbook would not have been possible without their input. We are particularly grateful to Lara Spieker, the senior editorial assistant at CRC Press/Taylor & Francis Group, for steering the Handbook skilfully from the author submission to the production stage. Her constructive and supportive approach throughout critical phases of the Handbook has been pivotal to its publication. Jeffery L. Tallon, Peter B. Littlewood, Peter J. Lee, and Jianyi Jiang each made a contribution to the Handbook that went way beyond their responsibilities as section editors, invariably at critical times in the publication process, from advising on structure and content to managing the glossary and general format of the Handbook. All four managed to retain their enthusiasm for this exceptionally demanding project over an extended period and provided continuous support to

the Editors-in-Chief. Doug Bennett provided helpful advice on the contents of Volume 3. We acknowledge the contribution of Archie M. Campbell as a section editor and note with great sadness his death in November 2019. Archie made a fundamental contribution to both editions of the Handbook and he will be greatly missed as a friend, colleague and collaborator. We also note the great contributions of Harry Jones to the first edition of the Handbook. He was a section editor of the second edition but passed away in 2015 before being able to complete his efforts for this edition. Finally, we would like to acknowledge the support of our home institutions: the Department of Engineering, University of Cambridge; the National High Field Magnet Laboratory at the University of Florida; and Forschungszentrum Jülich. David A. Cardwell David C. Larbalestier Aleksander I. Braginski

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Editors-in-Chief David A. Cardwell David C. Larbalestier Aleksander I. Braginski

Section Editors and Advisory Board David C. Larbalestier (A1, E3, G3, 2015-2019) Archie M. Campbell (A2, G2, 2015-2019 †) Alexander V. Gurevich (A2, 2017-2020) Alexander A. Golubov (A2, 2019-2020) David A. Cardwell (A3) Peter J. Lee (B) Jeffery L. Tallon (C) Peter B. Littlewood (D) Kazumasa Iida (E1, E2, E5) Jianyi Jiang (E3, 2019-2020) Michael Lorenz (E4, 2015-2018)

François Weiss (E3, E4, 2019-2020) Ray Radebaugh (F1) Lance D. Cooley (E3, G1) Fedor Gömöry (G2, 2015-2019) Antony Carrington (G3, 2019-2020) Harry Jones (H1, † 2015) John H. Durrell (H1, 2016-2020) Mark Ainslie (H1, 2019-2020) Aleksander I. Braginski (E4, H3, 2017-2020) Horst Rogalla (H2, 2019-2020)

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Contributors

Mark D. Ainslie (H1, H1.10, H1.11) Department of Engineering University of Cambridge Cambridge, United Kingdom Naoyuki Amemiya (H1.5) Kyoto University Kyoto, Japan Tabea Arndt (H1.7) Karlsruhe Institute of Technology Institute for Technical Physics Eggenstein-Leopoldshafen and Siemens AG Corporate Technology (until September 2019) Erlangen, Germany Amalia Ballarino (E2.7, H1.4) CERN, TE-MSC Meyrin/Geneva, Switzerland Nobuya Banno (E3.9) Low-Temperature Superconducting Wire Group National Institute for Materials Science Ibaraki, Japan Kees van der Beek (A3.2) Centre National de la Recherche Scientifique Université Paris-Saclay Palaiseau, Paris France Peter A. Beharrell (H1.13) Department of Physics and Astronomy University of Southampton Southampton, United Kingdom

Kamran Behnia (G3.2) Laboratoire Physique et étude de Matériaux (CNRS-UPMC) Ecole Supérieure de Physique et de Chimie Industrielles Paris, France Victor Belitsky (H4.6) Department of Earth and Space Sciences Chalmers University of Technology Gothenburg, Sweden Emilio Bellingeri (C3) Consiglio Nazionale delle Ricerche Genova, Italy Simon Bending (G2.10) Department of Physics University of Bath Claverton Down Bath, United Kingdom Douglas A. Bennett (H4.2) Quantum Sensors Group National Institute of Standards and Technology Boulder, California Luca Bottura (E2.7) CERN, TE-MSC Geneva, Switzerland Aleksander I. Braginski (E4.6, H3) Forschungszentrum Jülich (FZJ), retired Jülich, Germany Marcel ter Brake (F1.4) Faculty of Science and Technology University of Twente NB Enschede, the Netherlands

E. Helmut Brandt (A3.1) † 2011 Max-Plank-Institut für Metallforschung Stuttgart, Germany Archie M. Campbell (A2.1, G2.1) † 2019 Department of Engineering University of Cambridge Cambridge, United Kingdom Ahmet Cansiz (H1.8) Istanbul Technical University Sarıyer/Istanbul, Turkey Haishan Cao (F1.4) Department of Energy and Power Engineering Tsinghua University Beijing, China David A. Cardwell (A1, A3, E2.2) Department of Engineering University of Cambridge Cambridge, United Kingdom Antony Carrington (G3, G3.1) H.H. Wills Laboratory of Physics University of Bristol Bristol, England, United Kingdom Francesco Cerutti (E2.7) CERN, EN-STI Geneva, Switzerland Heinz J. Chaloupka (H2.3) † 2014 Department of High-frequency Technology and Communication University of Wuppertal Wuppertal, Germany xvii

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Contributors

Serguei Cherednichenko (H4.6) Department of Microtechnology and Nanoscience Chalmers University of Technology Gothenburg, Sweden

Liangzi Deng (D4) Department of Physics and Texas Center for Superconductivity University of Houston Houston, Texans

Ching-Wu Chu (D4) Department of Physics and Texas Center for Superconductivity University of Houston Houston, Texans

Marc Dhallé (G2.3) University of Twente NB Enschede, The Netherlands

Gianluigi Ciovati (H1.14) Thomas Jefferson National Accelerator Facility Newport News Virginia Edward W. Collings (G2.7) Center for Superconducting and Magnetic Materials Materials Science Department The Ohio State University Columbus, Ohio Lance D. Cooley (E3.7, G1, G1.1) Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida John R. Cooper (G2.11) Cavendish Laboratory University of Cambridge Cambridge, United Kingdom Timothy Davies (E5.3) Department of Materials University of Oxford Oxford, United Kingdom Nuria Del-Valle (G2.6) Departament de Fisica Universitat Autonoma de Barcelona Bellaterra, Barcelona Catalonia, Spain Gianluca De Marzi (B1) Superconductivity Section Fusion and Technology for Nuclear Safety and Security Department ENEA, Frascati Research Centre Frascati RM, Italy

Pavel Diko (G1.5) Institute of Experimental Physics Slovak Academy of Sciences Košice, Slovak Republic John H. Durrell (H1) Department of Engineering University of Cambridge Cambridge, United Kingdom Michael Eisterer (G2.5) Institute of Atomic and Subatomic Physics Technische Universität Wien Vienna, Austria Jack W. Ekin (E5.1) National Institute of Standards and Technology (NIST), retired Boulder, Colorado Christian Enss (H4.4) Kirchhoff Institute of Physics (KIP) University of Heidelberg INF 227 Heidelberg, Germany Andreas Fleischmann (H4.4) Kirchhoff Institute of Physics (KIP) University of Heidelberg, INF 227 Heidelberg, Germany Jaap Flokstra (H3.2) Faculty of Science and Technology, Emeritus University of Twente NB Enschede, The Netherlands René Flükiger (C3, E2.5, E2.7, E3.6, E3.11) Department of Quantum Matter Physics University of Geneva Geneva, Switzerland

Stephan Friedrich (H4.1) Lawrence Livermore National Laboratory Livermore, California John Gallop (H2, H3, H4.7) National Physical Laboratory, Emeritus Teddington, Middlesex, United Kingdom Loredana Gastaldo (H4.4) Kirchhoff Institute of Physics (KIP) University of Heidelberg, INF 227 Heidelberg, Germany Wilfried Goldacker (G3.4) Institute of Technical Physics Karlsruhe Institute of Technology Eggenstein-Leopoldshafen, Germany Alexander A. Golubov (A2.9) Institute for Nanotechnology University of Twente NB Enschede, The Netherlands Fedor Gömöry (G2) Department of Superconductors Institute of Electrical Engineering Slovak Academy of Sciences Bratislava, Slovakia Colin Gough (A2.7) Department of Physics and Astronomy, retired University of Birmingham Birmingham, United Kingdom Francesco Grilli (G2.2) Karlsruhe Institute of Technology Institute for Technical Physics Hermann-von-Helmholtz-Platz 1 Eggenstein-Leopoldshafen, Germany Chris Grovenor (E5.3) Department of Materials University of Oxford Oxford, United Kingdom Alexander V. Gurevich (A2) Department of Physics Old Dominion University Norfolk, Virginia

xix

Contributors

Damian P. Hampshire (B3, G2.4) Department of Physics Durham University Durham, United Kingdom Ling Hao (H4.7) National Physical Laboratory Teddington, Middlesex, United Kingdom Eric E. Hellstrom (E3.1 and E3.12) Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida Bernhard Holzapfel (E4.1) Institute for Technical Physics Karlsruhe Institute of Technology Eggenstein-Leopoldshafen, Germany Hideo Hosono (C5) Laboratory for Materials and Structures Tokyo Institute of Technology Yokohama, Japan Rudolf P. Huebener (A2.4, A2.5) Experimentalphysik II, retired Universität Tubingen Tubingen, Germany Kazumasa Iida (E1, E2, E5) Department of Materials Physics Nagoya University Nagoya, Japan Yasuo Iijima (E3.9) Low-Temperature Superconducting Wire Group National Institute for Materials Science Ibaraki, Japan

Carmen Jimenez (E4.3) Université Grenoble Alpes, CNRS, Grenoble INP*, LMGP Institute of Engineering Univ. Grenoble Alpes Grenoble, France

Caroline A. Kilbourne (H4) NASA/Goddard Space Flight Center Greenbelt, Maryland

Harry Jones (H1.1) † 2015 Clarendon Laboratory University of Oxford Oxford, United Kingdom

Johannes Kohlmann (E4.5) Department Quantum Electronics Physikalisch-Technische Bundesanstalt (PTB) Braunschweig, Germany

Stephen R. Julian (D1) Department of Physics University of Toronto Ontario, Canada

Victor K. Kornev (H2.3) Department of Physical Electronics Lomonosov Moscow State University Moscow, Russia

Alan M. Kadin (H3.6) Consultant Princeton Junction Princeton, New Jersey

Panagiotis Kotetes (D7) CAS Key Laboratory of Theoretical Physics Institute of Theoretical Physics Chinese Academy of Sciences Beijing, China

Debra L. Kaiser (E2.4) Office of Data and Informatics, MML, NIST Gaithersburg, Maryland Fumitake Kametani (G1.3) Department of Mechanical Engineering FAMU-FSU College of Engineering Florida State University Tallahassee, Florida Sebastian Kempf (H4.4) Kirchhoff Institute of Physics (KIP) University of Heidelberg, INF 227 Heidelberg, Germany Peter H. Kes (A3.2) Kammerlingh-Onnes Laboratory University of Leiden Leiden, The Netherlands

Yoshihiro Iwasa (D3) Department of Applied Physics and Quantum-Phase Electronics Center The University of Tokyo Tokyo, Japan

Simon A. Keys (G2.4) Department of Physics Durham University Durham, United Kingdom

Jianyi Jiang (E3, E3.1) Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida

Akihiro Kikuchi (E3.9) Low-Temperature Superconducting Wire Group National Institute for Materials Science Ibaraki, Japan

Hans-Joachim Krause (H3.4) Research Center Jülich (Forschungszentrum Jülich, FZJ) Jülich, Germany M’hamed Lakrimi (H1.1.2) Siemens HC Ltd Wharf Road Oxford, United Kingdom David C. Larbalestier (A1, E3.7, G3.5) National High Magnetic Field Laboratory Florida State University Tallahassee, Florida Peter J. Lee (B, E3.7, G1.4) The Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida Steve Lee (G2.10) School of Physics and Astronomy, SUPA University of St. Andrews St. Andrews, United Kingdom

xx

Contributors

Anthony J. Leggett (A2.2) Department of Physics, Emeritus University of Illinois at Urbana-Champaign Urbana, Illinois

Christoph Meingast (G3.3) Institute for Quantum Materials and Technologies Karlsruhe Institute of Technology Karlsruhe, Germany

Peter B. Littlewood (D) James Franck Institute and Department of Physics University of Chicago Chicago, Illinois

Brian H. Moeckly (E4.6) Commonwealth Fusion Systems Milpitas, California

Nicholas J. Long (H1.9) Robinson Research Institute Victoria University of Wellington Lower Hutt/Wellington, New Zealand Ralph Longsworth (F1.3) Engineering Department Sumitomo (SHI) Cryogenics of America, Inc. Pennsylvania

Antonio Morandi (H1.6) University of Bologna Bologna, Italy Michael Mück (H3.4) Institut für Angewandte Physik Justus-Liebig-Universität Gießen Gießen, Germany Oleg A. Mukhanov (H3.5, H3.6) Seeqc, Inc. Elmsford, New York

Michael Lorenz (E4) Felix Bloch Institute for Solid State Physics Universität Leipzig Leipzi, Germany

K. Alex Müller (A1.3) Department of Physics University of Zurich Zurich, Switzerland

Bing Lv (D4) Department of Physics and Texas Center for Superconductivity University of Houston Houston, Texans

Luigi Muzzi (B1) Superconductivity Section Fusion and Technology for Nuclear Safety and Security Department ENEA, Frascati Research Centre Frascati RM, Italy

Judith L. MacManus-Driscoll (E3.4) Department of Materials Science and Metallurgy University of Cambridge Cambridge, United Kingdom Milan Majoros (G2.7) Center for Superconducting and Magnetic Materials Materials Science Department The Ohio State University Columbus, Ohio Arkadiy Matsekh (H1.12) Foucault Dynamics Gold Coast, Australia Benjamin A. Mazin (H4.3) Department of Physics University of California Santa Barbara, California

Carles Navau (G2.6) Departament de Fisica Universitat Autonoma de Barcelona Bellaterra, Barcelona Catalonia, Spain Daniel E. Oates (H2.1) Quantum Information and Integrated Nanosystems Group MIT Lincoln Laboratory Massachusetts Institute of Technology Lexington, Massachusetts Terry P. Orlando (A1.2) Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology (MIT) Cambridge, Massachusetts

Michael Parizh (H1.3) GE Research Niskayuna, New York John M. Pfotenhauer (F1.2, F1.5) Department of Mechanical Engineering University of Wisconsin – Madison Madison, Wisconsin Brian Pippard (A1.1) † 2008 Cavendish Laboratory University of Cambridge Cambridge, United Kingdom Britton Plourde (H3.7) Department of Physics Syracuse University Syracuse, New York Anatolii A. Polyanskii (G3.5) Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida Ian Pong (E3.8) Accelerator Technology and Applied Physics Division Lawrence Berkeley National Laboratory Berkeley, California Adrian Porch (A2.6, G2.9) School of Engineering University of Cardiff Cardiff, United Kingdom Kosmas Prassides (D3) Soft Materials Group AIMR Sendai, Japan Ray Radebaugh (F1, F1.1) Applied Chemicals and Materials Division National Institute of Standards and Technology (NIST), retired Boulder, Colorado Mark J. Raine (G2.4) Department of Physics Durham University Durham, United Kingdom

xxi

Contributors

Mark O. Rikel (E2.1) Nexans SuperConductors GmbH Hannover, Germany Horst Rogalla (H2) Electrical, Computer and Energy Engineering Department (ECEE) University of Colorado Boulder, Colorado

Jörg Schmalian (D6) Institute for Theoretical Condensed Matter Physics Karlsruher Institut für Technologie (KIT) Karlsruhe, Germany

Roman Sobolewski (H4.5) Department of Electrical and Computer Engineering University of Rochester Rochester, New York

Lynn F. Schneemeyer (E2.4) Montclair State University Montclair, New Jersey

Susie Speller (E5.3) Department of Materials University of Oxford Oxford, United Kingdom

Alain Rüfenacht (H4.7) Superconducting Electronics Group National Institute of Standards and Technology (NIST) Boulder, Colorado

Thomas Schurig (E4.5) Kryophysik und Spektrometrie Physikalisch-Technische Bundesanstalt (PTB), retired Berlin, Germany

Athena Safa Sefat (E3.3) Materials Science and Technology Division Oak Ridge National Laboratory Oak Ridge, Tennessee

Bernd Seeber (E3.10) scMetrology SARL Geneva, Switzerland

Gunzi Saito (D2) Toyota Physical and Chemical Research Institute Nagakute Aichi, Japan Charlie Sanabria (G1.4) Commonwealth Fusion Systems Cambridge, Massachusetts Alvaro Sanchez (G2.6) Departament de Fisica Universitat Autonoma de Barcelona Bellaterra, Barcelona Catalonia, Spain Tilmann H. Sander Thoemmes (H3.3) Department of Biosignals PTB – The National Metrology Institute of Germany Berlin, Germany Kenichi Sato (E3.2) Cryogenics and Superconductivity Society of Japan Tokyo, Japan Christian Scheuerlein (E5.2, G1.2) TE Department – Magnets, Superconductors and Cryostats (MSC) European Organization for Nuclear Research (CERN) Geneva, Switzerland

Paul Seidel (E4.4, H3.2) Institute for Solid State Physics Faculty of Physics and Astronomy Friedrich Schiller University of Jena Jena, Germany Yunhua Shi (E2.2) Department of Engineering University of Cambridge Cambridge, United Kingdom Jun-ichi Shimoyama (C2, E2.3) Department of Physics and Mathematics Aoyama Gakuin University Tokyo, Japan Theo Siegrist (G1.1) Department of Chemical and Biomedical Engineering FAMU-FSU College of Engineering Florida State University Tallahassee, Florida Alejandro V. Silhanek (G2.10) Experimental Physics of Nanostructured Materials Q-MAT, CESAM Université de Liège Liège, Belgium Enrico Silva (A2.6) Department of Engineering Università Degli Studi Roma Tre Rome, Italy

Tiziana Spina (E2.7) Applied Physics and Superconducting Technology Division Fermi National Accelerator Laboratory Batavia, Illinois Wolfgang Stautner (H1.3) GE Research Niskayuna, New York Michael D. Sumption (G2.7) Center for Superconducting and Magnetic Materials Materials Science Department The Ohio State University Columbus, Ohio Francesco Tafuri (A2.9, H3.1) Department of Physics University of Napoli Federico II Napoli, Italy Takao Takeuchi (E3.9) Human Resources Division National Institute for Materials Science Ibaraki, Japan Jeffery L. Tallon (Foreword, C, C1, C6) Robinson Research Institute Victoria University of Wellington Lower Hutt, New Zealand Saburo Tanaka (H3.4) Toyohashi University of Technology, Toyohashi, Japan Chiara Tarantini (B2) The Applied Superconductivity Center National High Magnetic Field Laboratory Florida State University Tallahassee, Florida

xxii

Edward J. Tarte (A2.8) Department of Electronic, Electrical and Systems Engineering University of Birmingham Birmingham, United Kingdom Sergey K. Tolpygo (E4.5) Lincoln Laboratory Massachusetts Institute of Technology (MIT) Lexington, Massachusetts Volker Tympel (E4.4) Helmholtz Institute Jena Helmholtzweg 5 Jena, Germany Ruggero Vaglio (A2.6) Department of Physics University of Napoli Federico II Napoli, Italy Philippe Vanderbemden (G2.8) Department of Electrical Engineering and Computer Science University of Liège Liège, Belgium Orest G. Vendik (H2.2) Department of Physical Electronics and Technology St. Petersburg Electrotechnical University St. Petersburg, Russia William F. “Joe” Vinen (A1.2), retired Department of Physics and Astronomy University of Birmingham Birmingham, United Kingdom Ming-Jye Wang (D5) Institute of Astronomy and Astrophysics, Academia Sinica Taipei, Taiwan James H. P. Watson (H1.13) Department of Physics and Astronomy University of Southampton Southampton, United Kingdom

Contributors

Harald W. Weber (E2.6) TU Wien – Atominstitut Wien, Austria

Roger Wördenweber (E4.2) Forschungszentrum Jülich (FZJ) Jülich, Germany

François Weiss (E4, E4.3) Université Grenoble Alpes, CNRS, Grenoble INP*, LMGP Institute of Engineering Univ. Grenoble Alpes Grenoble, France

Judy Z. Wu (C4) Department of Physics and astronomy University of Kansas Kansas City Kansas

Jeremy D. Weiss (E3.12) Advanced Conductor Technologies LLC Boulder, Colorado

Maw-Kuen Wu (D5) Institute of Physics, Academia Sinica Taipei, Taiwan

David Welch (A2.3) Brookhaven National Laboratory New York, New York

Phillip M. Wu (D5) Solarcity Inc. San Mateo, California

Frank N. Werfel (E2.1) Adelwitz Technologiezentrum GmbH (ATZ) Torgau, Germany

Mingyao Xu (F1.3) Engineering Department Sumitomo (SHI) Cryogenics of America, Inc. Pennsylvania

Jonathan White (G1.6) Laboratory for Neutron Scattering and Imaging Paul Scherrer Institute Villigen, Switzerland

Akiyasu Yamamoto (E3.11) Department of Applied Physics Tokyo University of Agriculture and Technology Tokyo, Japan

Jörg Wiesmann (E4.1) Incoatec GmbH Geesthacht, Germany

Ayako Yamamoto (E3.5) Shibaura Institute of Technology Graduate School of Engineering and Science Tokyo, Japan

Frank K. Wilhelm-Mauch (H3.7) Institute for Quantum Computing Analytics Research Center Jülich Jülich, Germany Dag Winkler (H4.6) Department of Microtechnology and Nanoscience Chalmers University of Technology Gothenburg, Sweden

Yukihiro Yoshida (D2) Department of Agriculture Meijo University Nagoya, Japan Xiaoqin Zhi (F1.2) College of Energy Engineering Zhejiang University Zhejiang, China

Part G Characterization and Modelling Techniques

1

G1 Introduction to Section G1: Structure/Microstructure Lance D. Cooley This Handbook is in large part a materials science handbook for superconductors. A central tenet of materials science holds that the properties of material are related to the material structure and the fundamental bonding of atoms, whereby property changes can be attained by manipulating the material structure. In this discussion, the context for structural studies spans a wide range of length scales. At the smallest scales, the atomic precision of a material and its defect structure over few nanometers relates to properties connected with the superconducting coherence length, especially in superconductors with high critical temperature and coherence length approaching 1 nm. Somewhat larger structures affect properties related to critical current and flux pinning, including precipitates, strain fields, material grains, boundaries and interfaces, and dispersion of chemical dopants. Synthesis of superconducting phases furthermore depends on microscale diffusional growth at scales of micrometers and higher. Finally, macroscale features of conductors and conductor assemblies in cables, questions related to conductor uniformity, predictability, and reliability span lateral dimensions of millimeters and higher, in lengths of kilometers. This section introduces characterization tools mostly applied at the nano- and microscale, and occasionally extended to the millimeter dimensions of conductors and cables. The section begins with a republication of the original chapter by Roman Gladyshevskii about chemical crystallography, being updated with new material by Theo Siegrist with inclusion of a detailed discussion of iron pnictide materials and new room-temperature superhydride phases. Christian Scheuerlein provides a chapter discussing phase transformations using x-ray tomography techniques, which

complements the chapter provided by Doug Finnemore in the earlier Handbook. Fumitake Kametani provides an updated discussion on transmission electron microscopy to complement the chapter by Hervieu and Raveau in the earlier Handbook. A new chapter about digital microscopy and digital image analysis is provided by Charlie Sanabria and Peter Lee, which describes methods used on conductors across many length scales. Pavel Diko provides a discussion about optical microscopy next, and the section closes with a discussion of neutron techniques applied to the flux-line lattice by Jonathan White, as a complement to the chapter by Lee and Forgan in the earlier Handbook. This section did not intend to cover techniques used broadly in materials science that can also be applied to superconducting materials. Examples include scanning transmission electron microscopy sometimes used in conjunction with electron energy loss spectroscopy, atomic force and magnetic force microscopy, scanning tunneling microscopy, photoelectron spectroscopy, and atom probe microscopy. This section did not include the chapter on Raman and infrared techniques provided by Buckley and Trodahl and the chapter on neutron crystallography by Alan Hewat in the previous Handbook partly for these reasons. The reader is referred to the broader literature for many instances where these techniques have been applied successfully to access important atomic and electronic structure information related to superconducting materials. The reader is also referred to recent literature about the broader class of topological materials, which may be parent materials of superconductors. Their characterization has prompted development of novel techniques and extension of the materials science techniques above.

3

G1.1 X-Ray Studies: Chemical Crystallography Lance D. Cooley, Roman Gladyshevskii, and Theo Siegrist

G1.1.1 Introduction

G1.1.2 Principles of X-Ray Diffraction

Knowledge of crystal structures is essential for understanding physical properties and of great help when defining guidelines for the exploration of new materials. The efforts invested since the discovery of superconductivity during low-temperature studies of mercury have led to the synthesis of more than one thousand different superconducting compounds, among which are several hundreds of high-temperature superconducting oxides. Classical superconductors cover a large spectrum of chemically different substances, ranging from metallic elements and alloys to borides, carbides, chalcogenides and organic compounds, and crystallize with very different structures. High-Tc superconducting oxides contain, in addition to copper and oxygen, which are always present, many other chemical elements, but show common structural features, e.g. a pronounced layer character. A good characterization of all samples for which physical properties are reported is desirable to improve our understanding of the conditions for superconductivity. X-ray diffraction is undoubtedly the experimental technique that has been of great importance in revealing crystal structures. Powder cameras and diffractometers constitute excellent tools for qualitative and quantitative routine phase analysis. Single-crystal diffraction and, to an increasing extent, also diffraction on polycrystalline samples provide valuable information about the detailed constitution of the material, i.e. the interatomic distances, the nearest neighbour atomic environments, element substitutions, thermal motion. X-ray powder devices can, in addition, be used to study samplerelated characteristics that have a great influence on transport properties, such as texture, domain size, internal strain, etc. X-ray diffraction, however, does not allow us to distinguish between atoms having close values of atomic number or to localize light atoms in the presence of heavy atoms. For those cases, neutron diffraction experiments are often more reliable and a combination of x-ray and neutron diffraction data is recommended.

X-rays interact almost exclusively with the electrons of the material, by means of scattering (coherent and incoherent) and absorption. Coherent scattering, which has exactly the same wavelength as the incident radiation, constitutes diffraction. The most general and powerful principle in the diffraction theory for periodic objects such as crystals, is that developed by W L Bragg, which states that the lattice planes reflect radiation like mirrors. Maximum positive interference occurs when the path differences between reflections from successive lattice planes in a family is equal to an integer number of wavelength, i.e.

4

nλ = 2d sin θ,

(G1.1.1)

where n is the order of reflection, λ is the wavelength, d is the lattice plane spacing and θ is the angle of incidence/reflection to the planes. To each family of parallel lattice planes can be associated a point in reciprocal space, which, all together, form a periodic lattice. The geometrical condition for diffraction to occur is that a reciprocal lattice point, defined by the vector d *hkl, d *hkl = 1/ dhkl = 2sin θ / λ ,

(G1.1.2)

lies on the surface of a sphere of radius 1/λ, the Ewald sphere. Only lattice points located inside a sphere of radius 2λ will be able to diffract (see Figure G1.1.1). It follows that the wavelengths used for diffraction experiments must be of the same magnitude as the interplanar distances in crystals (λ ~ 0.5–2.5 Å). The scattering amplitude of an atom is determined by summing the contributions from all of its electrons. The ratio of the scattering amplitude of an atom to that of a single electron defines the atomic scattering factor, f. At a diffraction angle zero, the scattering amplitude is equal to the number of electrons in the atom, however, the amplitude decreases rapidly with increasing diffraction angle. In addition. the atoms in

5

X-Ray Studies: Chemical Crystallography

The scattering amplitude of all atoms in the unit cell is determined by summing the amplitudes from the individual atoms. Its ratio to the amplitude scattered by a single electron, the structure factor, is expressed as a complex number N

Fhkl =

∑ f exp[2πi(hx + ky + lz )], j

j

j

j

(G1.1.4)

j= 1

where 2π(hxj + ky j + lzj) is the phase angle of an atom with fractional coordinates xj, y j, zj and N is the number of atoms in the unit cell. The intensities of the reflected x-ray beams are proportional to the squares of the scattering amplitudes Fhkl, however, for real crystal factors, such as Lorentz polarization, absorption and extinction, modify the intensities. For a non-primitive Bravais lattice, there exists translation symmetry within the unit cell, and some of the potential reflections are systematically absent. Systematic absences occur also for all symmetry elements combining rotation or reflection with a translation, such as screw axes or glide planes, but restricted to smaller groups of reflections (see International Tables for Crystallography [l]). The brief exposure of the principles of x-ray diffraction presented above leads to the following important conclusions. 1. The diffraction angle depends exclusively on the geometry of the lattice, i.e. the unit cell. 2. The intensity of the diffracted beam depends on the square of the number of electrons of the atoms. The diffraction pattern will be dominated by heavy elements such as Pb or Bi, when present, making light elements, such as H, Li, B or C almost invisible. It will also be difficult to distinguish elements following each other in the periodic system.

G1.1.3 Experimental Techniques FIGURE G1.1.1 (a) Diffraction of x-rays by a single crystal. One lattice plane and its corresponding vector d*hkl in reciprocal space are shown. Diffraction takes place because the reciprocal lattice point is located on the surface of the Ewald sphere (radius 1/λ, surface touching the origin of the reciprocal lattice of the crystal). (b) Diffraction of x-rays by a powder. The vector d*hkl for a perfect powder, representing all possible positions, describes a sphere. The intersection with the Ewald sphere (a circle) defines the angular aperture of the cone formed by the diffracted beams.

X-ray diffraction techniques are usually subdivided into singlecrystal and powder (polycrystalline) methods. The former requires the selection of a single crystal not smaller than 50 µm, whereas the latter use a large number of randomly oriented crystallites (mean diameter < 10 µm). Powder methods may also be used for phase identification, quantitative analysis of multiphase samples, determination of texture, domain size and internal strains (see Table G1.1.1).

G1.1.3.1 Single-Crystal Methods a crystal vibrate about their mean positions, due to thermal effects, and the atomic scattering factors are in general corrected by the equation f = f 0exp[− B(sin θ / λ )2 ],

(G1.1.3)

where f0 is the scattering factor at a temperature of absolute zero and B is the Debye–Waller coefficient (also found in the literature as U = B/8π2).

G1.1.3.1.1 Laue Camera There are two ways to experimentally satisfy Bragg’s law: (1) vary the wavelength λ maintaining the angle θ, (2) vary the angle for a fixed wavelength. In the Laue method a stationary crystal is placed between a polychromatic x-ray radiation source and a flat film or area detector. The use of white radiation corresponds to a continuous range of Ewald spheres of different radii. It is possible to determine the orientation of the crystal and the so-called Laue symmetry.

6

Handbook of Superconductivity TABLE G1.1.1

Applications of Different X-Ray Diffraction Methods

Sample

Method/Camera

Characteristics

Applications

Single crystal

Laue

Stationary crystal/polychromatic radiation/ stationary film Rotating crystal/monochromatic radiation/ stationary film Rotating crystal/monochromatic radiation/ moving film Rotating crystal/monochromatic radiation/ moving film

Find the symmetry (Laue group), orient the crystallographic axes, estimate the crystal quality Find the unit cell, detect a possible superstructure

Rotation/oscillation Weissenberg Precession

Powder

Single-crystal diffractometer

Rotating crystal/monochromatic radiation/ counter

Debye–Scherrer

Powder/monochromatic radiation/film

Guinier

Powder/monochromatic radiation/film

Powder diffractometer

Powder/monochromatic radiation/counter

The photographic methods using traditional silver-based films have largely been superseded by modern point and area detector–based diffraction systems, and are not discussed in detail. Modern single-crystal and powder diffractometers, using either laboratory or synchrotron X-ray sources, have revolutionized crystallographic studies. G1.1.3.1.2 Single-Crystal Diffractometer An automatic diffractometer contains an x-ray detector (point detector or area detector) and a goniostat, both computercontrolled. In the commonly used four-circle diffractometers with point detector, three circles regulate the movement of the crystal and the fourth one corresponds to the movement of the detector. During the data collection, the computer will place the circles so that they correspond to the particular angular setting of one potential reflection after the other, measuring the peak intensity and the background on each side. Modern single-crystal diffractometers are usually equipped with area detectors, which are able to record a large number of the reflections that cross the Ewald sphere when the crystal is rotated. Due to the large area of the detector covering a large solid angle range, the data collection time is greatly reduced, with sufficient independent reflections measured within hours. While a large number of diffractometers with CCD-type detectors are in use, hybrid photon counting detectors find increased use in laboratory systems, providing excellent dark counts and large dynamic range. Additionally, modern micro-focus added sources provide increased brilliance, allowing measurements of small crystals with dimensions of less than 100 µm.

G1.1.3.2 Powder Methods An ideal powder contains an infinite number of randomly oriented crystallites. The points of the reciprocal lattice are

Determine a complete structure Find/check the symmetry, find the unit cell, detect a possible superstructure, determine a complete structure Solve a new structure, refine atom coordinates, refine atomic displacement parameters Refine the cell parameters, perform a qualitative phase analysis Perform a qualitative phase analysis, refine the cell parameters Refine the crystal structure(s), perform a quantitative phase analysis, solve a (simple) new structure, study texture, domain size, internal strains

displaced on the surfaces of spheres with radii equal 1/dhkl. The intersection of such a sphere with the Ewald sphere is a circle, and the diffracted reflections corresponding to a particular lattice point form a cone (see Figure G1.1.1). G1.1.3.2.1 Debye–Scherrer Camera The most popular diffraction camera for powders is the Debye–Scherrer camera. A film strip is placed on a cylindrical drum centred on the sample, mounted on a rotating glass needle. The diffraction pattern shows a series of arches resulting from the projection of the diffraction cones. This camera records the entire pattern generated at all possible values of 2θ. The equation of the camera allows the transformation of the values measured in millimetres directly into Bragg angles which can be used to refine the cell parameters. However, the Debye–Scherrer camera produces rather broad diffraction lines, which may be a serious drawback for some applications. A modern implementation of the Debye–Scherrer geometry uses a curved direct recording detector spanning up to 120° in angular range. Diffraction patterns can be recorded in minutes with such a device, providing high throughput. G1.1.3.2.2 Guinier Camera On the contrary, focusing cameras, like the Guinier camera, produce very sharp lines. The sample is ideally an arc of a circle, which is irradiated in transmission by a convergent beam produced by a bent monochromator. Transmission geometry requires very thin samples to avoid absorption effects and line broadening. Guinier line profiles are digitized and computer processed to extract the interplanar distances. It is an excellent method to determine the cell parameters very precisely. A modern Guinier camera may use an image plate detector with the read-out system integrated into the camera, representing a very compact high-resolution diffractometer systems.

7

X-Ray Studies: Chemical Crystallography

G1.1.3.2.3 Powder Diffractometers The powder diffractometer is equipped with a quantum counter detector. The geometry is similar to that of the Debye– Scherrer camera, but the focalization principle is used. In Bragg–Brentano diffractometers, a sample mounted on a plate moves in steps of θ, whereas the detector turns by 2θ. Curved position-sensitive detectors are also devised for use in powder diffraction experiments. Different information may be extracted from the patterns: angular position, intensities and peak shapes. The first two parameters are used to perform phase analyses, solve simple structures, refine known structures and determine texture. The peak shapes give a possibility to study domain sizes and internal strains. Precise information about preferred orientation can be obtained by using a texture goniometer where, for instance, for a chosen reflection the sample rotates around two axes while the detector remains fixed. An area detector is often preferred, since the texture induced intensity variation along a Debye cone can easily be measured with such a device. X-ray reflectometry may also be used to estimate the thickness of thin films. Crystal structure studies of materials in the superconducting state are undoubtedly of great interest and both single-crystal and powder diffractometers can be equipped with low-temperature devices. Working at low temperature decreases the intensity losses due to thermal vibrations and improves the intensity resolution. High-temperature devices and high-pressure (e.g. diamond anvil) cells are also often used in x-ray diffraction experiments to investigate the equation of state of a material. G1.1.3.2.4 Synchrotron Methods Modern synchrotron methods have advanced x-ray diffraction techniques tremendously. The superior collimation and monochromatization of the synchrotron beam give high angular resolution and very sharp profiles due to the extreme brilliance of the synchrotron beam that increases the dynamic range of the measured intensity. Microcrystallography techniques can now handle sample sizes of the order of 10 µm. Powder diffraction patterns with superior resolution and large dynamic intensity range can be routinely obtained by using mail-in services, as well as single-crystal data. Together with ultrafast large area megapixel detectors, full single-crystal data sets can be collected in minutes at dedicated synchrotron beam lines.

G1.1.4 Sample Preparation G1.1.4.1 Single Crystal Synthesis of single crystals of good quality and sufficiently large size is often a matter of great experimental difficulty. On the other hand, absorption is an undesired side effect which increases with the size of the crystal. The effects become particularly difficult to account for when the shape of the crystal

differs significantly from a sphere. The overall absorption coefficient is the sum of the contributions of the constituting elements and depends on the wavelength

∑gµ

µ(λ ) = ρ

i

im

(λ ),

(G1.1.5)

i

where ρ is the density, gi is the mass fraction of element i and µim its mass absorption coefficient (see International Tables for Crystallography [1]). For example, µ, (Mo Kα) = 136.1 and 54.5 mm−1 for elementary Pb and Bi2Sr2CaCu2O8, respectively. Twinning is another common problem, in particular for substances having undergone a phase transition from a structure of higher symmetry (e.g. orthorhombic Ba 2YCu3O7). Crystals of such materials systematically consist of two or more individual crystallites joined together in a well-defined mutual orientation. Relatively recent improvements of software packages make it possible in many cases to study twinned crystals.

G1.1.4.2 Polycrystalline Specimen Serious problems in sample preparation appear also for powder x-ray diffraction methods. A common mistake is to use insufficiently ground samples, which give rise to inaccurate and imprecise intensities. On the other extreme, excessive grinding may cause line broadening, produce small amounts of amorphous surface layers and may even induce phase transitions. Some line broadening usually appears when the crystallites are smaller than ~0.2 µm. Since materials have different hardnesses, an important differentiation of particle size may result in multiphase samples. Wet (acetone or alcohol) grinding helps to produce powders with uniform particle size. Moderate annealing relieves strain and sharpens the diffraction peaks. An anisotropic shape of the particles may induce a preferred orientation, favoring a subset of (hkl)s. This is particularly pronounced for the layer type oxide superconductors, where grinding of the sample will produce anisotropic flakes. The sample must cover the whole cross-section of the incident x-ray beam. It must also be thick enough and packed so that essentially all of the beam interacts with the sample. The minimal sample thickness t can be calculated from the equation (considering that intensity on the back side is 1/1000 of the diffracted intensity on the front side and excluding the spaces between particles): t = 3.45sinθ / µ.

(G1.1.6)

A thickness of > 25 µm is needed to satisfy this equation at diffraction angles between 20 and 60° 2θ for a Tl-based superconducting cuprate sample, using copper radiation. Ideally, the surface of the mounted sample should be flat, any roughness being susceptible to produce systematic deviations in the positions and breadths of the observed peaks. This problem can be solved either by using an internal standard or through

8

analytical processing of the diffraction data. Conventional packing of crystallites with well-defined shape, in particular, plate-like or needle-like crystals, leads to preferred orientation of particles. Common procedures to reduce this significant source of intensity errors include back-packing into a cavity mount, and mixing with an amorphous or viscous material, as well as sample movement during data acquisition.

G1.1.5 Choice of Radiation Standard x-ray diffraction experiments use the characteristic Kα lines of metal anticathodes. To obtain a monochromatic radiation, the beam can be filtered through a material that selectively absorbs the Kβ line. For example, a Ni filter strongly absorbs the Cu Kβ peak. Single-crystal monochromators are able to resolve the Kα1 and Kα2 doublet. In addition, modern diffractometer systems may utilize monochromators to eliminate the white radiation background present. Since the wavelength must be of similar magnitude as the interplanar spacings, the choice of characteristic Kα lines is limited to metals such as Cr (2.2910 Å), Fe (1.9374 Å), Co (1.7903 Å), Cu (1.5418 Å), Mo (0.7107 Å) or Ag (0.5609 Å). To refine cell parameters with high accuracy, it is necessary to have large angular dispersion, produced by a radiation with relatively large wavelength, e.g. Cr Kα or Cu Kα. On the contrary, for a complete structure determination, it is important to collect a large number of reflections with good resolution to a relatively large value of sinθ/λ, where a radiation with shorter wavelength is more convenient, typically Mo Kα or Ag Kα. Near an absorption edge the absorption is particularly strong and, for instance, Mo or Fe radiation is more suitable for Fe-containing samples than Cu or Co radiation. If the sample contains heavy elements, it is better to use an anticathode of an element with a large atomic number, in order to decrease intense fluorescence or to use a monochromator. Data collection at a synchrotron source produces higherresolution data and permits single-crystal work with specimens as small as a few µm3.

G1.1.6 Application of the X-Ray Diffraction Techniques G1.1.6.1 Qualitative Phase Analysis The most common use of x-ray powder diffraction data is phase identification. Each crystalline matter is characterized by a specific diffraction pattern, conveniently stored in the form of a list of interplanar distances (d hkl) and relative intensities (I0). The ICDD compilation [2] contains lists of d hkl and I0 reported for hundred thousands of inorganic compounds, and different cross-reference tables. Theoretical x-ray patterns generated by computer programs such as LAZY PULVERIX [3], may also be used for a comparison with the experimental data. The required crystallographic information (space group, cell parameters, atom coordinates) can be obtained

Handbook of Superconductivity

from databases and handbooks such as ICSD [4] (inorganic compounds containing at least one non-metal), TYPIX [5] (data representing structure types found among inorganic compounds not containing oxygen or halogen) and Pearson’s Handbook [6] (inorganic compounds not containing oxygen or halogen). The diffraction pattern of a multiphase sample is the sum of the patterns of all component phases. The phase analysis aims to find a combination of phases that reproduces the experimental spectrum. The reliability of the search/match procedure is mainly determined by the accuracy and precision with which the line positions and intensities have been measured. Difficulties arise when lines of different phases overlap, which is typically the case for samples containing phases with closely related structures. Preferred orientation is a common source of errors during studies of high-Tc superconducting cuprates, which form plate-like crystallites. Note that the limit of phase detection is relatively high, and a sample containing up to ~5% impurities may appear as pure to x-rays.

G1.1.6.2 Quantitative Phase Analysis For the internal standard method, a known amount of a standard material (usually corundum) is homogeneously mixed with the sample. The quantitative analysis is based either on the peak intensities or on the integrated intensities, which are less affected by the peak shape. The analysis is based on the ratio of the intensities of lines from different phases: I iα / I js = KX α / X s ,

(G1.1.7)

where Iiα is the intensity of line i of the phase α, Ijs is the intensity of line j of the internal standard s, K is the slope of the calibration curve, and Xα and Xs, are the weight fractions of the phase α and the standard s, respectively. The method may also be applied to several lines from each phase. In this case, the reference intensity ratio (RIR) and the relative intensities (Ir) are used: I iα / I js = RIR αs ( I riα / I rjs )( X α / X s ).

(G1.1.8)

If all phases in a mixture are identified and each reference intensity value is known, the analysis may be carried out without adding any standard, assuming that the sum of all the weight fractions equals unity N

X α = ( I iα / RIR α I riα )

∑( I

iα

)

/ RIR j I rjα ,

(G1.1.9)

j= 1

where N is the number of phases. Contrary to conventional methods of quantitative analysis, full-profile methods use the complete diffraction pattern. These methods involve fitting of the experimental pattern with a standard pattern, which is, in the case of the widely used

9

X-Ray Studies: Chemical Crystallography TABLE G1.1.2 Cell Parameters and Weight Fractions of the Individual Phases in a Sample of Composition Tl0.5Pb0.5 Sr1.8Ba0.2Ca2Cu3O9 − δ (Rwp = 0.028 for Diffraction Data Containing in Total 517 Reflections and 44 Variable Parameters) [8] Cell Parameters (Å) Phase

Space Group

Tl-1223 (Ca,Sr)2CuO3 Tl-1212 (Ca,Sr)2PbO4

P4/mmm Immm P4/mmm Pbam

a

b

c

Weight Fraction (%)

3.81445(6) 12.288(2) 3.8087(6) 5.875(1)

– 3.807(1) – 9.783(2)

15.2880(3) 3.275(1) 12.079(3) 3.392(1)

88.8(15) 5.4(6) 3.1(2) 2.7(1)

Rietveld refinement, calculated from the structural data. The Rietveld quantitative analysis eliminates the problems connected with extracting intensities and minimizes the effects of preferred orientation. In a typical refinement, individual scale factors, related to the weight percent of each phase, are varied together with profile, background and lattice parameters, atom coordinates and site occupancies. Up to 15 phases may be considered simultaneously by the DBWS program [7] or GSAS-II [8]. The weight fraction of the ith component in a mixture of N phases is obtained from the expression N

Wi = Si ρiVi 2

∑S ρ V , j j

j

2

(G1.1.10)

j =1

where S is the refined scale factor, ρ is the density and V is the unit cell volume. Table G1.1.2 shows the result of a Rietveld quantitative analysis performed on a high-Tc superconducting material of composition Tl0.5Pb0.5Sr1.8Ba0.2Ca 2Cu 3O9 [9]. The sample was found to contain four phases, the weight fraction of the main Tl-1223 phase being 88.8%. The plot of the observed and calculated diffraction patterns is shown in Figure G1.1.2. Preferred orientation (along [001]) was observed for both Tl-1223 and Tl-1212.

G1.1.6.3 Crystal Structure of the Copper Oxide Superconductors Single-crystal data are the most convenient to determine the positions of the atoms in the unit cell, since each reflection can be measured individually. It is, however, not possible to extract the structural information in a unique and automatic way, because only the magnitudes, and not the phase angles, of the structure factors may be directly determined from the experimental data. To solve the phase problem, it is necessary to obtain an initial structure model. This model can then be improved by successive least-squares refinements until an optimal agreement with the experimental data is achieved, minimizing, for instance, the residual R=

∑F

i0

i

− Fic

∑F

i0

,

(G1.1.11)

i

where |Fi0| are the observed moduli obtained from the intensities and |Fic| the corresponding calculated moduli. Typical value of the reliability factor R for an acceptable single-crystal refinement range from 0.03 to 0.07. A low reliability factor is, however, not a sufficient criterion to prove the correctness of a structure and geometrical features such as the interatomic distances and the atomic environments must be carefully

FIGURE G1.1.2 Observed (dotted line), calculated (solid line) and difference (bottom) x-ray diffraction patterns for a sample of composition Tl0.5Pb0.5Sr1.8Ba0.2Ca 2Cu 3O9−δ (Cu Ka radiation). Inset: 2(θ) region where the strongest impurity peaks appear. Bars indicate the peak positions of (1) the Tl-1223 phase, (2) (Ca,Sr)Cu03, (3) the Tl-1212 phase and (4) (Ca,Sr)2PbO4 [9], the residual.

10

Handbook of Superconductivity

analysed. A good structure refinement can be expected from a data collection containing at least 10 unique reflections per refinable parameter. Several program packages provide tools for the structure solution and refinement from single-crystal data, e.g. SHELX [10] and XTAL [11], CRYSTALS [12]. These excellent program packages are often available for free. Two different methods are generally used for extracting structural information from powder diffraction data. One approach, the so-called profile-fitting procedure, is to measure the integrated intensities of individual diffraction lines, convert them to structure factors, and use them as single-crystal data to solve or refine the structure. This technique works well for relatively simple, high-symmetry structures that yield diffraction patterns with little peak overlap. As described above, the Rietveld method employs the entire powder diffraction pattern and each data point is an observation. Structural parameters, profile and background coefficients are varied in a least-squares procedure until the calculated powder profile best matches the observed pattern, i.e. minimization of R=

∑w y i

2

i0

− yic ,

(G1.1.12)

i

where w i is a suitable weight, y i0 is the observed intensity and y ic is the corresponding intensity at ith point calculated from the model. The reliability factor Rwp should not exceed 0.10. The profile-shape functions are typically described by three parameters: the peak height (I(0)), the area (A) and the full width at half maximum intensity (FWHM, 2w). The former two parameters may be combined to give the integral breadth β = A / I (0),

(G1.1.13)

whereas, all three are included in the form factor 'P = 2w / β.

(G1.1.14)

It is widely admitted that the Voight profile-shape function, the analytical convolution product of a Lorentzian and Gaussian

function, is the most powerful. When φ = 0.637, the profile is fully Lorentzian and when φ = 0.940, it is fully Gaussian. However, due to the relative complexity of the Voight function, the majority of the Rietveld refinements use a pseudoVoight function, which is a simple addition of the two basic functions. Depending on the diffractometer, different profile functions, as well as split profile functions to account for peak asymmetries, are often used. Real structures often contain defects. The crystal structure obtained from the diffraction experiment is a statistical model, which expresses the probability that a certain site will be occupied by a particular atom. Different deviations from the original structure model may be observed. Owing to periodic deformations or element substitutions, the unit cell is sometimes multiplied by an integer or rational number, and in the case of an incommensurate superstructure by a real number. The existence of a superstructure is revealed by the presence of weak additional diffraction spots/lines. A small deformation of the unit cell, lowering the symmetry to another crystal system, causes broadening of particular diffraction lines. A stronger deformation will produce splitting into two or several lines, the relative intensities of which are characteristic of the crystal system. The diffraction patterns of incommensurate modulated structures are characterized by satellite reflections, regularly distributed around the main reflections. Crystallographic parameters of the high-Tc superconductors Tl0.53Pb0.50Sr1.43Ba0.20Ca 2.34Cu3O9 (powder) [9] and Bi2.10 Sr1.78Ca1.12Cu2O8 (single crystal) [13], refined on x-ray diffraction data, are given in Tables G1.1.3 and G1.1.4, respectively. Both phases are non-stoichiometric and the refinements represent the average structure. The structure of the latter is complicated, combining displacive and substitutional modulations. An incommensurate modulation was found in the direction of one short cell vector of the average orthorhombic structure; however, the structure can be conveniently described in a monoclinic supercell with ninefold volume (see Figure G1.1.3), an approximation which revealed several interesting structural features. It clearly appears that all layers

TABLE G1.1.3 Fractional Atomic Coordinates, Isotropic Displacement Parameters and Site Occupancies in Tl0.53Pb0.50Sr1.43Ba0.20Ca2.34Cu3O9 (Space Group P4/mmm, a = 3.81445(6), c = 15.2880 Å, RB = 0.055) [8] Coordinates Site Tl Sr Cu(1) Ca Cu(2) O(1) O(2) O(3) O(4)

Wyckoff Position

x

y

z

Biso (Å2)

Occupation

4(l) 2(h) 2(g) 2(h) 1(b) 1(c) 2(g) 4(i) 2(e)

0.0675(10) 1/2 0 1/2 0 1/2 0 0 0

0 1/2 0 1/2 0 1/2 0 1/2 1/2

0 0.1707(1) 0.2865(2) 0.3920(2) 1/2 0 0.1289(7) 0.2930(5) 1/2

1.09(5) 0.67(7) 0.63(7) 0.17(9) 0.63 1.0 1.0 1.0 1.0

Tl0.125Pb0.125 Sr0.715(7)Ca0.187Ba0.098 Ca0.986(2)Tl0.014

11

X-Ray Studies: Chemical Crystallography TABLE G1.1.4 Fractional Atomic Coordinates, Isotropic (O Sites) or Equivalent (Metal Atom Sites) Displacement Parameters and Site Occupancies in the Average Structure of Bi2.10Sr1.78Ca1.12Cu2O8 (Space Group A2aa, a = 5.4161, b = 5.4112, c = 30.873Å, wR = 0.026 for 458 Unique Reflections and 55 Refined Parameters) [11] Coordinates Site

Wyckoff Position

x

y

z

Uiso/eq (Å2)

Occupation

Bi(1) Bi(2) Sr Cu Ca O(1) O(2) O(3) O(4)

8(d) 8(d) 8(d) 8(d) 4(c) 8(d) 8(d) 8(d) 8(d)

− 0.045(2) 0.059(3) 0.494(3) 0.000(3) 0.499(4) 0.690(3) 0.074(3) 0.249(5) 0.246(5)

0.2264(3) 0.2264 0.2504(7) 0.2488(8) 1/4 0.141(7) 0.280(5) 0.506(7) − 0.001(8)

0.0546(2) 0.0491(2) 0.1410(2) 0.1968(2) 1/4 0.0502(12) 0.1196(9) 0.2062(5) 0.1972(7)

0.0307(7) 0.0307 0.045(1) 0.038(1) 0.037(3) 0.073(8) 0.051(7) 0.014(4) 0.026(5)

Bi0.552(9) Bi0.448 Sr0.89(3)Ca0.11

suffer a displacive modulation, with the modulation wavevector along the diagonal [101] of the supercell. All metal atoms show a transverse displacement wave with largest amplitude for the Cu and Ca sites, the Bi and Sr sites being displaced longitudinally. Additional O atoms were found in the BiO layers, so that Bi-rich regions with a distorted rock salttype atom arrangement and Bi-poor regions with chains of

FIGURE G1.1.3 Structure of Bi-2212 (refined composition) Bi 2.09Sr1.90Ca 1.00Cu 2O8.22, space group Cc, a = 37.754, b = 5.4109, c = 41.070 Å, β = 103.58°, wR = 0.057 for 7227 unique reflections and 579 refined parameters) in a projection along [010]. Solid lines delimit the monoclinic supercell, dotted lines the orthorhombic cell of the average structure. The (main) cation of each layer is indicated on the left-hand side of the drawing. Small circles represent O sites [11].

Ca0.90(1)Bi0.10

corner-linked BiO3 ψ-tetrahedra are formed (Figure G1.1.4). The monoclinic symmetry corresponds to a systematic phase shift of the modulation wave.

G1.1.7 Crystal Chemistry The aim of crystal chemistry is to establish relationships between composition, structure and properties of crystalline materials. High-Tc superconductors represent a relatively homogeneous family of compounds with common structural features. Classical superconductors, on the contrary, cover a broad range of compounds and structures and it is difficult to draw general conclusions. The relationships between crystal structure and properties have, however, been well studied within particular structure families, for instance the Chevrel phases. Here, a

FIGURE G1.1.4 Atom arrangement in the BiO layers of Bi-2212 (see figure G1.1.3) in projections perpendicular to the layers and along [010] of the monoclinic cell (central part). The O atoms of the neighbouring SrO layers are included [11].

12

Handbook of Superconductivity

summary of important crystal structures for superconducting materials is presented to augment the discussion about copper oxide superconductors in the previous section.

G1.1.7.1 Low-Temperature (Classical) Superconductors The crystal structures of inorganic compounds are conveniently subdivided into structure types, since it is common that different compounds crystallize with similar geometric arrangements. A structure type is generally named after the compound in which it was first identified [14]. More than one hundred different structure types count among their representatives at least one compound for which a critical temperature above the boiling point of helium, 4.2 K, has been reported [15]. Superconductors range from simple metallic elements, alloys, carbides and borides to chalcogenides and organic charge-transfer salts (see Table G1.1.5).

TABLE G1.1.5

Superconductivity is observed for representatives of each of the three most common structure types adopted by metallic elements, Figure G1.1.5, i.e. the two close-packed structure types (the hexagonal Mg and face-centred cubic Cu types) and the body-centred cubic W type. Elements of comparable atom sizes may form close-packed solid solutions, e.g. Tc0.82Mo0.18. Ordering is, however, often observed around particular compositions, like 1:1 or 3:1, the latter composition corresponding, for instance, to the Cu 3Au type, represented here by La 3In. A series of classical superconducting alloys, among which are Nb3Sn and Nb3Ge, crystallize with the Cr3Si type, often referred to as A15 structure (notation for simple structure types, starting with a capital letter, was introduced at the beginning of the 20th century by the structure compilation Strukturberichte). This structure type belongs to another family of close-packed structures, the so-called tetrahedrally close-packed structures, which contain only tetrahedral

Structural Families of Classical Superconductors and Representative Compounds

Chemical Family

Structural Family

Compound

Tc (K)

Structure Type

Elements

Close-packed structures Other structures Close packed structures Tetrahedrally close-packed and related structures

Tc0.82Mo0.18 Mo0.6Re0.4 MoIr La3In ScTc2 Nb3Ge LaRu3Si2 Nb0.24Tc0.76 Ag0.72Ga0.28 Zr2Rh Y5Os4Ge10 ThIrSi NbB ZrRuP MoC0.5 Ca3Rh4Sn13 Mo6Ga31 NbN0.75C0.25 MoN Zr0.61Rh0.285O0.105 YCI0.75Br0.25 Y1.4Th0.6C3.1 Lu0.75Th0.25Rh4B4 LuNi2B2C

13.7 12.0 8.8 9.7 10.9 23.2 7.6 12.9 7.5 11.3 9.1 6.5 8.3 12.9 5.8 8.7 9.8 18.0 15.1 11.8 11.1 17.0 11.9 16.6

Mg W B′-AuCd Cu3Au MgZn2 Cr3Si LaRu3Si2 Ti5Re24 ξ-(AgZn) θ-CuAl2 Sc5Co4Si10 LaPtSi α-TlI ZrNiAl NiAs Yb3Rh4Sn13 Mo6Ga31 NaCl LT-Nb1−xS W3Fe3C 1s-GdCBr Pu2C3 CeCo4B4 LuNi2B2C

Intermetallic compounds

CsCl type and related structures Structures with atoms in square-antiprismatic coordination Structures with atoms in trigonal prismatic coordination

Interstitial carbides, nitrides, oxides and hydrides Borides, carbides and borocarbides with non-metal polymers

Chalcogenides

Organic compounds

Other structures Structures with close-packed atom layers Other structures Layered carbohalides Structures with non-metal atom dumb-bells Members of the RnT2B2(C/N)n structure series Structures with fullerene molecules Other structures Perovskite, bronze and spinel Layered structures and intercalation compounds Chevrel phases and related structures Other structures Charge-transfer salts

Cs2RbC60

33

K3C60

Mo1.8Rh0.2BC Ba0.6K0.4BiO3 Li0.3Ti1.1S2 Pb0.92Mo6S8 Mo6S6I2

9.0 30 13 15.2 14.0

Mo2BC CaTiO3 4H-Ti1+xS2 HT-Pb0.9Mo6S8 Mo6Se8

NbPS κ-(ET)2Cu[N(CN)2]Cl

12.5 12.5

NbPS BEDT-TTF

13

X-Ray Studies: Chemical Crystallography

FIGURE G1.1.5

fcc, hcp and bcc structures of elemental superconductors Al, Zn, and Nb.

interstices. As described in Figure G1.1.6, the cubic unit cell of the Cr3Si type contains eight atoms and the minority atom site is surrounded by 12 atoms forming a regular icosahedron (polyhedron with 20 triangular faces). The atoms from the majority atom site are arranged in infinite non-intersecting straight chains parallel to each of the cell edges. A relatively high amount of disorder and a certain homogeneity range are generally observed, including anti-site disorder where the majority and minority atoms swap positions. Order and stoichiometry have, in most cases, a positive influence on the superconducting transition temperature, but cannot always be achieved experimentally. The same observation is made for interstitial carbides and nitrides that crystallize with the wellknown rock salt (NaCl) structure. The metal atoms form a close-packed arrangement and the carbon, or nitrogen, atoms occupy the octahedral interstices in the framework.

Other structures adopted by classical superconductors contain well-defined subunits with an important amount of localized bonding, contrasting with the overall metallic character. Many of the superconducting borides and carbides, for instance, contain pairs of covalently bonded non-metal atoms. This is the case for Lu0.75Th0.25Rh4B4 (CeCo4B4 type), Y1.4Th0.6C3.1 (Pu2C3 type) and for YCl0.75Br0.25 (ls-GdCBr type). The structure of LuNi2B2C contains linear B-C-B units, which interconnect relatively rigid Ni2B2 slabs to form a porous framework that accommodates the rare-earth metal atoms. LuNi2B2C and related borocarbides and boronitrides form a homologous structure series of the general formula R nT2B2 (C/N)n. The crystal structures of this family are strongly reminiscent of those of the high-Tc superconducting oxides. The superconducting fullerides contain the characteristic C60 “balls” in a cubic close-packed arrangement. In K 3C60, the

FIGURE G1.1.6 Nb3Sn coordination polyhedron for Sn (top left) is a 12-coordinate Nb icosahedron with Sn-Nb distance 2.951 Å. Coordination polyhedron for Nb (top right) is the 14-coordinate Frank–Cooper polyhedron with 2 Nb at 2.640 Å, 4 Nb at 2.951 Å, and 8 Sn at 3.233 Å. At bottom is the crystal structure of Nb3Sn, where Sn occupies the body-centered cubic sites and Nb atoms form non-intercepting chains along each principal axis direction. Nb atoms are dark spheres, Sn atoms are light gray spheres.

14

FIGURE G1.1.7 Crystal structure of the Chevrel phase Pbx Mo6S8. On the left, a single pseudo-cubic unit cell is shown, with the Mo6 octahedron inscribed in a sulfur cube. The Pb atoms are located in the corners of the unit cell. On the right, the packing of the structure is shown, with the channels formed by the Mo6S8 clusters indicated.

cations occupy both the octahedral and the tetrahedral interstices in the framework. The crystal structures of the well-known Chevrel phases, e.g. Mo6Se8 and Pb0.92Mo6S8, also contain well-defined subunits, in this case, clusters where a Mo6 octahedron is surrounded by eight chalcogen atoms forming a concentric cube. See Figure G1.1.7. The basic structure can host atoms of such different elements as alkaline, alkaline-earth, rare-earth metals or transition metals and metals such as lead or tin in the voids between the clusters. The Mo6 octahedron contracts in agreement with bond-valence rules, when the oxidation state or the concentration of interstitial cations increases. The ideal structure is rhombohedral with a cell angle ranging from 88 to 96°, reflecting the degree of delocalization of the cations.

G1.1.7.2 High-Temperature Copper Oxide Superconductors The crystal structures of the high-Tc superconducting cuprates are conveniently described as a stacking of atom layers and may be grouped into several structure series [16]. Basically, four kinds of atom layer, which have different ‘functions’ in the structure, can be considered: conducting DO2 (a regular square mesh of Cu atoms with O atoms centering the square edges), separating C (a simple regular square mesh of Ca. Y. La or rare-earth metal atoms), bridging BO (a regular square mesh of Ba, Sr or La atoms with O atoms centering the squares) and additional A, AO or AO2 (a regular square mesh of Tl, Pb, Bi, C, Cu or Hg atoms with O atoms in different positions, depending on the cation). The general formula of a structure with AO layers can be written as Ak BlCmDnOk+l+2m+2n. The translation period within the undistorted layers is ~3.85 Å, whereas the periodicity in the third direction depends on the number and kind of atom layers in the stacking unit, the average interlayer distance being ~2.0 Å. Basic structures are commonly referred to by a four-digit code, which is derived from the general formula considering the numbers of cation-containing layers of each kind in the stacking unit: klmn.

Handbook of Superconductivity

The ideal structures are described in one of the following tetragonal space groups: P4/mmm, P4/nmm or I4/mmm. Exceptions are Ba2YCu3O7 and Ba 2YCu4O8, where the undistorted structures are orthorhombic, due to the presence of additional CuO layers with O atoms centering the square edges in only one direction. The lowering from tetragonal symmetry that is often observed for the real structures may be caused by small displacements of the atoms, an ordered arrangement of different cations within layers of the same kind, an ordered arrangement of vacancies or the insertion of extra atoms. These reasons lead, in a majority of cases, to orthorhombic structures with the cell vectors along the diagonals of the base of the tetragonal cell of the ideal structure: a+b, – a+b and c (a ′ = b ′ = 5.4 Å = 21/2 a). The ideal coordination of the cation sites derives from the stacking of the layers, but the real coordination is often distorted to better accommodate a particular cation. Strong covalent bonding is generally observed within the CuO2 layers and within the coordination polyhedra of the A cations. The coordination number of the Cu atoms can be 6, 5 or 4, the surrounding oxygen atoms forming an octahedron, a square pyramid or a square, respectively. Depending on the exact kind of additional layer present, five types of ideal coordination polyhedra are observed for the A sites: two collinear atoms (typical for Hg and Cu), a triangle (C), a square (Cu), a square pyramid (Pb) or an octahedron (Bi, Pb and Tl). When the structure contains more than three consecutive additional layers, the cations of the internal layers have higher coordination numbers, suitable for metal atoms. Only one type of coordination polyhedron, i.e. a tetragonal prism, is observed for the atoms (Ca, Y, La or rare-earth metal) in the separating layers, whereas the coordination number of the B atoms (Ba, Sr or La) can be 12 (cuboctahedron), 10, 9 or 8 (square antiprism). The chemical families of high-Tc superconducting cuprates are usually named after the cation in the additional layers (see Table G1.1.6). The structures of Bi-based cuprates form a series with the formula Bi2B2Cn–1CunO2n+4. The Bi and O atoms in the additional layers are displaced from the ideal positions on the fourfold rotation axes, the Bi sites having ψ-tetrahedral coordination (three O atoms and a lone electron pair). Tl-based cuprates constitute one of the largest chemical families of high-Tc superconductors and form two distinct structure series: TlB2Cn–1CunO2n+3 and Tl2B2Cn–1 CunO2n+4. Their common feature is also the displacement of atoms in the additional layers which results, however, in tetrahedral coordination of the Tl atoms. Structures containing only Pb on the A sites are not known so far, but a large number of compounds where Pb is the majority cation on these sites have been synthesized. In most Pb-based cuprates, Cu atoms are also present in additional layers. Hg-based cuprates form a series with the general formula HgB2Cn–1 CunO2n+2+δ. Carbonbased superconducting cuprates with a single additional layer form the (C,M)B2 Cn–1CunO2n+3 series, carbon being often in part substituted by copper. The O atoms in the additional layer are displaced from the ideal positions on the cell edges

15

X-Ray Studies: Chemical Crystallography TABLE G1.1.6

Structural Families of High-Tc Superconductors and Representative Compounds

Cuprates Rare-earth–alkaline-earth

Structure Series

Four-Digit Code

BCmDnO2m+3

0101 0112 0122

B2Cn2–1CunO2n+2

B2CmCu2O2m+4

CpCuO2p Ba–Y

CukB2CCu2Ok+6

Bi-based

CuB2CmCu2O2m+5 Intergrowth Bi2B2Cn2–1CunO2n+4

Tl-based

Bi2B2CmCu2O2m+6 Intergrowth TlB2Cn2–1CunO2n+3

TlkB2C2Cu2Ok+8

Tl2B2Cn−1CunO2n+4

Pb-based

(Pb,M)kB2C2Cu2Ok+8

(Pb, µ)3B2Cn−1CunO2n+4 Hg-based

HgB2Cn−1CunO2n−2+δ

C-based

(C,M)B2Cn−1CunO2n+3

Others

0132 0201 0212 0223 0234 0222 0232 0011 0021 1212 2212 1222 1212/2212 2201 2212 2223 2222 2201/2212 1201 1212 1223 1234 1222 2222 2201 2212 2223 2234 1222 2222 3222 3201 3212 1201 1212 1223 1234 1245 1201 1212 1223 1234 4212 4222 5212 5222

Compound (Space Group, Tc (K)) LaCuO2.95 (P4/mmm, not supercond.) BaYCuFeO5 (P4/mmm, not supercond.) BaNdCe0.9Cu0.9Fe1.1O7 (I4/mmm, not supercond.) SrY1.65Ce1.35CuFeO9 (P4/mmm, not supercond.) La1.85Sr0.15CuO4 (Bmab, 37.5) Sr2CaCu2O4.6F2.0 (I4/mmm, 99) Sr2Ca2Cu3O6.2F3.2 (I4/mmm, 111) Sr2Ca3Cu4O10 (I4/mmm, 70) La1.8Sr0.4Eu1.8Cu2O8 (P4/nmm, 34) Sr2Ce1.60Y1.40CuFeO10 (I4/mmm, not supercond.) Sr0.9La0.1CuO2 (P4/mmm, 42) Nd1.85Ce0.15CuO3.92 (I4/mmm, 20) Ba2YCu3O7 (Pmmm, 91) Ba2YCu4O8 (Ammm, 80) Ba1.27Nd2.08Ce0.65Cu3O8.91 (I4/mmm, 37) Ba4Y2Cu7O14.94 (Ammm, 91.5) Bi2Sr1.6La0.4CuO6.30 (Amaa, 29.5) Bi1.98Sr1.75Ca0.96Cu2O8 (A2aa, 92) Bi1.72Pb0.28Sr2Ca2Cu3O10.24 (A2aa, 110) Bi2Sr2Gd1.7Ce0.3Cu2O10 (Cmma, 25) Bi4Sr4CaCu3O14 (Pbmm, 84) Tl0.92Ba1.2La0.8CuO4.86 (P4/mmm, 52) Tl0.5Pb0.5Sr2CaCu2O7 (P4/mmm, 90) Tl0.56Pb0.56Sr2Ca1.88Cu3O9 (P4/mmm, 120) Tl1.00Ba2Ca2.96Cu4O11 (P4/mmm, 114) TlBa2Eu1.5Ce0.5Cu2O9 (I4/mmm, 40) Tl2Ba2Eu1.8Ce0.2Cu2O10+δ (P4/nmm, not supercond.) Tl2Ba2CuO6 (I4/mmm, 90) Tl2Ba2CaCu2O8 (I4/mmm, 110) Tl1.94Ba2Ca2.06Cu3O10 (I4/mmm, 125) Tl1.64Ba2Ca3Cu4O12 (I4/mmm, 109) Pb0.5Sr1.75Eu1.75Ce0.5Cu2.5O9 (I4/mmm, 25) Pb0.95Ba0.77Sr1.23PrCeCu3O9 (Cmm2, not supercond.) Pb2Sr2PrCeCu3O10 (Fmmm, not supercond.) Pb2SrLaCu1.84O6 (Pman, 33) Pb2Sr2Y0.73Ca0.27Cu3O7.8 (Pman, 67) HgBa2CuO4.18 (P4/mmm, 95) HgBa2CaCu2O6.26 (P4/mmm, 114) HgBa2Ca2Cu3O8.44 (P4/mmm, 133) Hg0.84Ba2Ca3Cu4O10.4 (P4/mmm, 125) HgBa2Ca4Cu5O12.32 (P4/mmm, 101) C0.9Ba1.1Sr0.9Cu1.1O4.9+δ (P4212, 26) C0.7Sr2Ca0.7Y0.3Cu2.3O7 (Pmmm, 40) C0.5Ba2Ca2Cu3.5O9 (P4/mmm, 120) C0.32Ba2Ca3Cu4.68O11.06 (P4/mmm, 117) Ti2Gd2Ba2CaCu2O12 (I4/mmm, not supercond.) Ti2Nd2CaBa2CeCu2O14 (P4/nmm, not supercond.) Ti3Sm2Ba2CaCu2O14 (P4/mmm, not supercond.) Ti2Sm3.1GaBa2Ce0.9Cu2O16 (I4/mmm, not supercond.)

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Handbook of Superconductivity

such as films, powders, wires, tapes and single crystals are discussed in [19]. Since MgB2 is brittle, wire manufacturing cannot use the MgB2 in traditional wire drawing techniques, but similar techniques that are used for the Nb3Sn and the cuprate superconductors need to be applied.

G1.1.7.4 Iron-Based Superconductors

FIGURE G1.1.8 AlB2 Structure, the parent structure type for the superconductor MgB2. Boron is shown in black, aluminum in grey.

to achieve triangular coordination around the carbon site (carbonate group).

G1.1.7.3 MgB2 The intermetallic phase MgB2 has been known since 1954 [17], but its superconducting properties have not been reported until 2001 [18], with a Tc of 40 K. This is the highest Tc for a binary compound at normal pressure. The structure is the simple AlB2 type (Figure G1.1.8) which arranges boron atoms in a hexagonal honey comb structure (graphite-like layers) separated by magnesium atoms arranged in layers with hexagonal close packing. The light atoms in this intermetallic are expected to support high-frequency phonons that lead to a high Tc within the BCS framework. Even though the structure can be described in terms of layers, the anisotropy in MgB2 is smaller than in the cuprate superconductors. Furthermore, a larger coherence length and transparency of the grain boundaries to the superconducting current makes it a candidate for applications. The Tc of 40 K and its superconducting properties puts MgB2 at the top of the known boride-containing superconductors.

G1.1.7.4.1 The 11 Structure The iron-based superconductors all have two-dimensional layers of FeX (X = pnictide, chalcogenide), with the simplest structure the mackinawite structure of FeS (Figure G1.1.9). While FeS is difficult to obtain, FeSe and Fe(Se,Te) can be grown. This family of compounds is usually referred to as the “11” phases, with FeSe superconducting at Tc = 8 K [22], and the structure is shown in Figure G1.1.9. The 11 structure is tetragonal, symmetry P4/nmm (#129), with lattice constants a = 3.765 Å and c = 5.518 Å, and Fe in the 2a position at (3/4, ¼, 0), and Se in the 2c position at (1/4, 1/4, 0.26). In the 11 structure of FeSe, the tetrahedra are corner connected to form a two-dimensional FeSe layer, with the tetrahedral angles of 105.38° (2×) and 111.56° (2×), indicating a slightly stretched tetrahedral coordination. The height of the tetrahedra is 2.869 Å, with the Se plane above the Fe plane by 1.435Å. Crystallographic Data, 11 Structure of FeSe P4/nmm, #129 Fe Se

Crystallographic Data of MgB2 P6/mmm Mg 1a B 2d

Superconductivity in iron-based compounds has been reported for a few systems, such as Th7Fe3, with a Tc = 1.8 K [20]. More recently, high-temperature superconductivity with Tc about 35 K was observed in fluorine-doped LaOFeAs [21]. This discovery initiated research into finding related systems with common structural features. In addition, moving from the pnictide arsenic to chalcogenides such as Se produced several closely related families of compounds with increasingly complex layers, similar to the families found for the copper oxide–based superconductors. At the time of writing, four of the structures have emerged as important for applications, so these are summarized here.

a = 3.0834(3) Å c = 3.5213(6) Å V = 28.99 Å3 0 0 0 1/3 2/3 1/2

The intermetallic MgB2 is available from commercial suppliers, but laboratory synthesized materials tend to have higher superconducting transition temperatures. The large discrepancy between magnesium and boron volatility presents challenges and opportunities for materials preparation. Different preparation methods for different types of samples,

a = 3.765Å

c = 5.518Å

¾ ¼

¼ ¼

0 0.26

G1.1.7.4.2 The 111 Structure The crystal structure of MFeAs (M = Li, Na), usually referred to as the “111” structure, is based on the PbFCl-type (BiOCltype) structure, with a superconducting transition temperature of Tc = 18 K [23] (Figure G1.1.10). It may be considered a version of lithium/sodium intercalated FeSe-type structure. Similar to the “11” structure, the “111” structure contains tetrahedrally coordinated iron, but instead of a chalcogenide, the pnictide As is present. The symmetry is tetragonal

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X-Ray Studies: Chemical Crystallography

FIGURE G1.1.9 “11” Crystal structure of FeSe: Fe is tetrahedrally coordinated by four Se atoms at a distance of 2.367 Å. The tetrahedra are corner connected to form a two-dimensional FeSe layer. The tetrahedral angles are 105.38° (2×) and 111.56° (4×), indicating slightly flattened tetrahedral coordination. The height of the tetrahedra is 2.869 Å, with the Se plane above the Fe plane by 1.435 Å. The structure is usually referred to as the “11” structure.

primitive, with space group P4/nmm (#129), a = 3.775 Å, c = 6.357 Å. The iron to arsenic distance is 2.529 Å, and the angles in the tetrahedral coordination are 96.45° (2×) and 116.35° (4×), indicating a more stretched tetrahedron as compared to the “11” structure. The tetrahedron height is 3.369 Å, and the height of the arsenic plane from the iron plane is 1.685 Å. As in the 11 structure, the FeAs4 tetrahedra are corner connected to form a two-dimensional FeAs layer in the 111 structure of LiFeAs, while the alkali metals Li and Na are located in between the FeAs layers. Crystallographic Data, 111 Structure of LiFeAs Symmetry P4/nmm, #129 Li Fe As

a = 3.725 Å

c = 6.357Å

¼ ¾ ¼

¼ ¼ ¼

0.657 0 0.26

0

G1.1.7.4.3 The 1111 Structure The crystal structure of the LaOFeAs [24] (Figure G1.1.11) was part of a series include Fe, Ru and Co in place of the transition element, and the larger lanthanides from La to Gd, and is of the ZrCuSi2-type structure, a filled variant of the PbFCl-type. The unit cell is primitive tetragonal, with symmetry P4/nmm (#129), and lattice parameters a = 4.038 Å and c = 8.753 Å. The crystal structure is chemically layered, with LaO and FeAs layers. The Fe atoms are tetrahedrally coordinated, with an Fe-As distance of 2.818 Å. The FeAs4 tetrahedron is stretched in the direction of the c-axis, resulting in two distinct bond angles of 91.53° (2×) and 119.12° (4×). The height of the FeAs layer is 3.931 Å, and the As layer is 1.966 Å above the Fe layer. The LaO layer is sandwiched between the FeAs layers. In the LaO layer, oxygen is tetrahedrally coordinated by four La atoms, with an La-O distance of 2.557 Å. Similarly, the LaO tetrahedra are also stretched, resulting in two distinct

FIGURE G1.1.10 “111” Crystal structure of LiFeAs: Fe is tetrahedrally coordinated by four As atoms at a distance of 2.529 Å. As atoms are shown as large light spheres, Li atoms as dark spheres, and iron atoms as anthracite spheres. The tetrahedra are corner connected to form twodimensional FeAs layers. The tetrahedral angles are 96.45° (2×) and 116.35° (4×), indicating stretched tetrahedral coordination. The height of the tetrahedra is 3.369Å, with the As plane above the Fe plane by 1.685 Å. The structure is usually referred to as the “111” structure.

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Handbook of Superconductivity

FIGURE G1.1.11 “1111” crystal structure of LnOFeAs (Ln = La, Ce, Pr, Nd, Sm, Gd) (“filled” PbFCl-type). The darker spheres represent the LaO layer, the light sphere is the As atom. The FeAs layer consists of corner-connected FeAs4 tetrahedra that are stretched in the c-axis direction. The height of the FeAs layer is 3.931 Å, with the As layer 1.966 Å above the Fe layer. The structure is referred to as the “1111” structure.

angles of 104.28° (2×) and 112.13° (4×). Superconductivity is observed in the fluorine-doped La(O,F)FeAs structure at Tc = 35 K [25]. The structure has a large flexibility and is reported with different size lanthanides, 3d and 4d transition metals (Cr, Mn, Fe, Co, Ni, Zn, Ru, Cd,…), and pnictides phosphorus, arsenic and antimony, as well as fluorine instead of oxygen, with over 600 entries in the Pearson’s Crystal Data. The highest transition temperature of Tc = 56 K is observed in Gd0.8 Th0.2OFeAs [26]. Crystallographic Data, 1111 Structure of LaOFeAs Symmetry P4/nmm (#129) La O Fe As

a = 4.039 Å

c = 8.753 Å

¼ ¾ ¾ ¼

¼ ¼ ¼ ¼

0.6793 ½ 0 0.2246

G1.1.7.4.4 The 122 Structure The structure of (Ba0.6K0.4)Fe2As2 or “122” structure is part of the large family of BaAl4 type structures, or the ThCr2Si2 type structure, with a superconducting transition temperature of Tc = 38 K [27] (Figure G1.1.12). Similar to the 111 structure, two-dimensional FeAs layers sandwich Ba/K atoms. The structure is body-centred tetragonal, with space group I4/mmm (#139), and lattice parameters a = 3.9170 Å and c = 13.2968 Å. The FeAs4 tetrahedra have an Fe–As distance of 2.396 Å, and are almost perfectly regular, with angles 109.64° (2×) and 109.38° (4×). The FeAs4 tetrahedra corner-connect to form the FeAs layer, with Fe–Fe distances of 2.770 Å. The height of the FeAs layer is 2.760 Å, with the As layer 1.380 Å above the Fe layer. In addition, the defect 122 structure of A0.8Fe1.6Se2

with A = K, Rb, Cs, and Tl shows a Tc = 32 K, and may also be found as A2Fe4Se5 [28]. Vacancy ordering is observed, resulting in a √5 × √5 superstructure. Crystallographic Data for (K,Ba)Fe2As2, “122” Structure Symmetry I4/mmm (#139) K, Ba Fe As

a = 3.9170 Å

c = 12.2968 Å

0 0 0

0 ½ 0

0 ¼ 0.3538

Structurally, all these phases share the two-dimensional layers formed by corner-connected FeX4 (X = pnictide, chalcogenide) tetrahedra that either stack directly, or sandwich another structural moiety, the simplest being an alkali metal. This leads to relative highly symmetrical stacks of the FeX layers, based on tetragonal symmetries of P4/nmm and I4/mmm. In principle, homologous series can be envisioned, similar to the large variety of layer stacks based on the LnT2B2C system. The structural features reflect in the physical properties. The superconducting transition temperature peaks for tetrahedral angles close to the ideal tetrahedral angle of 109.54° of the X-Fe-X bonding angles of the FeX4 tetrahedron, similar to the B-Ni-B angles in the NiB4 tetrahedron of the LnT2B2C superconductor system. Furthermore, the superconducting gap symmetry is controlled by the height of the FeX layer, with small heights giving nodal behavior versus large heights giving gapped behavior. The comparably short Fe–Fe distances ensure d-orbital overlap, with the Fe 3d electrons at the Fermi level participating in the superconductivity. Magnetic effects due to the 3d electrons are found in some parts of the phase diagrams of these systems, located close to or co-existing with superconductivity.

19

X-Ray Studies: Chemical Crystallography

FIGURE G1.1.12 The “122” structure of (Ba,K)Fe2As2. The structure is body-centred tetragonal, with space group I4/mmm (#139), and lattice parameters a = 3.9170 Å and c = 13.2968 Å. The FeAs4 tetrahedra have an Fe–As distance of 2.396 Å, and are almost perfectly regular, with angles 109.64° (2×) and 109.38° (4×). The FeAs4 tetrahedra corner-connect to form the FeAs layer, with Fe–Fe distances of 2.770 Å. The height of the FeAs layer is 2.760 Å, with the As layer 1.380 Å above the Fe layer.

The flexibility of the structure is reflected in that both electron or hole-based superconductivity is observed in the 122 and the 1111 parent systems, and within each system correlations between structural features and superconductivity can be found. Furthermore, the tetragonal structure is not fully stable, with structural phase transitions to orthorhombic or lower symmetries observed at low temperatures. Due to the high density of states of Fe 3d electrons, spin density wave transitions occur in some of the systems, with the structural transition temperature and the spin density wave transition temperature not necessarily coinciding [29].

G1.1.7.5 High-Pressure Hydride Superconductors G1.1.7.5.1 H3S Hydrogen-based high-pressure phases have recently been shown to be superconducting at high temperatures. The highpressure phase H3S is reported with a Tc = 203 K at pressures exceeding 180 GPa [30]. The metallic hydride phase H3S is stable above 180 GPa, with proposed cubic symmetry Im-3m and lattice parameter a = 3.089 Å. Figure G1.1.13 shows the proposed structure, with octahedrally coordinated sulfur atoms forming two interpenetrating ReO3 - type networks.

FIGURE G1.1.13 Proposed crystal structure of H3S, cubic symmetry Im-3m (#229). Sulfur atoms are the dark spheres, hydrogen atoms light spheres. The structure forms two interpenetrating ReO3 networks, with sulfur octahedrally coordinated by hydrogen.

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Handbook of Superconductivity

FIGURE G1.1.14 Proposed crystal structure of H3S, cubic symmetry Im-3m (#229). Sulfur atoms are the dark spheres, hydrogen atoms light spheres. The structure forms two interpenetrating ReO3 networks, with sulfur octahedrally coordinated by hydrogen.

An isotope effect is observed for D3S indicating a BCS superconductor. The strong co-valent bonding in metallic H3S is thought to provide the high-frequency phonons necessary for Cooper pair formation. The material is a Type II superconductor, with Hc1 ≈ 30 mT, and estimated Hc2 of the order of 60 to 80 T. The Ginzburg–Landau coherence length is then estimated to by about 2 nm. Since H3S is derived from H2S under high pressure, dissociation of the H2S molecules is required, and an excess of sulfur is expected. An alternate structure builds on the possible dissociation of H2S into (H3S)+ and (SH)−, with a perovskitetype structure [31]. In this model, the (SH)− is at the perovskite A site, and occupies a cavity that is formed by eight sulfur atoms. This structural model does not require the presence of elemental sulfur. G1.1.7.5.2 LaH10 The lanthanide hydride LaH10 has been reported with a Tc of > 250 K at a pressure of about 170 GPa, making this the highest temperature superconductor at the time of this review [32]. This superhydride LaH10 can be considered as an approximation of metallic hydrogen. The superconducting LaH10 at 150 GPa has cubic symmetry Fm-3m (#225), with lattice parameter a = 5.1019 Å. The hydrogen atoms form a sodalite-type cage around the lanthanum atoms (Figure G1.1.14). The structure of LaH10 can also be thought of as a filled up NaCl (rock salt) type structure, four H8 clusters in the anion position, eight hydrogen atoms in the tetrahedral interstitial sites, and four lanthanum atoms in the cation position for La4H40. Due to the complexity of phase formation under high pressures and high temperatures, these high-pressure phases have to be synthesized in situ using diamond anvil cells. The amount of material at present is therefore small, and not necessarily stable at standard pressure and temperature. The fact that systems approximating metallic hydrogen show superconductivity at record high temperatures, make it conceivable

that in the future, systems displaying superconductivity above room temperature are possible.

References [1] 1995/1996 International Tables for Crystallography, Vol A, B and C, eds T Hahn, U Shmueli and A JC Wilson (Dordrecht: Kluwer) [2] X-Ray Powder Diffraction File (Newtown Square, PA 19073: Joint Committee for Powder Diffraction Standards, JCPDS, International Centre for Diffraction Data, ICDD) [3] Yvon K, Jeitschko W and Parthe E 1977 LAZY PULVERIX, a computer program for calculating x-ray and neutron diffraction powder patterns J. Appl. Cryst. 10 73 [4] Kirchhoff A, Pehler A, Warkentin E, Bergerhoff G and Luksch P 1991 ICSD Inorganic Crystal Sll’ucrure Database (release 9712), User Manual (Frankfurt: Gmelin-Jnstitut fiir Anorganische Chemie and Institut fiir Anorganische Chemie der Universitiit Bonn) [5] Cenzual K, Gladyshevskii R and Parthe E 1995 TYPIX 1995 Database of Inorganic Structure Types, User’s Guide (Frankfurt: Gmelin-Institut fiir Anorganische Chemie) [6] Villars P 1997 Pearson’s Handbook (Materials Park, OH 44073: ASM International) [7] Young RA, Larson AC and Paiva-Santos CO 1998 Program DBWS-9807, User’s Guide (Atlanta, GA 30332: School of Physics, Georgia Institute of Technology) [8] B.H. Toby and R.B. Von Dreele 2013 J. Appl. Cryst. 46(2) 544–549 [9] Gladyshevskii RE, Galez Ph, Lebbou K, Allemand J, Abraham R, Couach M, Flükiger R, Jorda J-L and Cohen-Adad M Th 1996 Structural characterization and superconducting properties of (Tl0.5Pb0.5)(Sr2 -xBa x) Ca 2Cu3O9-δ Physica. C 267 93

X-Ray Studies: Chemical Crystallography

[10] Sheldrick G M 1997 Program SHELX-97, Manual (Gottingen: Institut fiir Anorganische Chemie der Universitat Gottingen) [11] Hall SR, Flack H D and Stewart J M 1992 Program XTAL3.2, User’s Guide (Perth: Crystallography Centre, University of Western Australia) [12] P.W. Betteridge, J.R. Carruthers, R.I. Cooper, K. Prout, D.J. Watkin 2003 J. Appl. Cryst. 36 1487, SIR [A. Altomare, G. Cascarano, C. Giacovazzo, A. Guagliardi, M. C. Burla, G. Polidori and M. Camalli 1994 J. Appl. Cryst. 27 435], JANA [M. Dusek, V. Petricek and L. Palatinus 2006 Acta Cryst. A62 s46]. [13] Gladyshevskii RE and Flükiger R 1996 Modulated structure of Bi2Sr2CaCu2O8+δ, a high-Tc superconductor with monoclinic symmetry Acta. Cryst. B52 38 [14] Parthe E, Gelato L, Chabot B, Penzo M, Cenzual K and Gladyshevskii R 1993/1994 TYPIX standardized data and crystal chemical characterization of inorganic structure types Gmelin Handbook of Inorganic and Organometallic Chemist1y (Heidelberg: Springer) [15] Gladyshevskii R and Cenzual K 2000 Crystal structures of classical superconductors Handbook of Superconductivity Ch 6, ed Ch P Poole Jr (San Diego: Academic Press) [16] Gladyshevskii R and Galez Ph 2000 Crystal structures of high-Tc superconducting cuprates Handbook of Superconductivity Ch 8. ed Ch P Poole Jr (San Diego: Academic Press) [17] M.E. Jones and R.E. Marsh 1954 J. Am. Chem. Soc. 76 1434–1436 [18] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y, Zenitami and J. Akimitsu 2001 Nature 410 63

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[19] C. Buzea and T. Yamashita 2001 Supercond. Sci. Technol. 14 R115 [20] B.T. Mathias, V.B. Compton and E. Corenzwit 1961 J. Phys. Chem. Solids 19 130 [21] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya and H. Hosono 2006 J. Amer. Chem. Soc. 128 10012 [22] F.C. Hsu et al. 2008 Proc. Natl. Acad. Sci. USA 105 14262 [23] X.C. Wang, Q.Q. Liu, Y.X. Lu, W.B. Gao, L.X. Yang, R.C. Yu, F.Y. Li and C.Q. Jin 2008 Sol. State Commun. 148 538–540 [24] P. Quebe, L.J. Terbüchte and W. Jeitschko 2000 J. Alloys Comp. 302 70–74 [25] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya and H. Hosono 2006 J. Amer. Chem. Soc. 128 10012 [26] C. Wang et al. 2008 Europhys. Lett. 83 67006 [27] M. Rotter, M. Tegel and D. Johrendt 2008 Phys. Rev. Lett. 101 107006 [28] J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He and X. Chen 2010 Phys. Rev. B 82 180520(R) [29] G.R. Stewart 2011 Rev. Mod. Phys. 83 1589–1641 [30] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov and S. I. Shylin 2015 Nature 525 73–76 [31] E.E. Gordon, K. Xu, H. Xiang, A. Bussmann-Holder, R. K. Kremer, A. Simon, J. Köhler and M.H. Whangbo 2016 Angew. Chem. 55 3682–3684 [32] A. P. Drozdov, P. P. Kong, V. S. Minkov, S. P. Besedin, M. A. Kuzovnikov, S. Mozaffari, L. Balicas, F. F. Balakirev, D.E. Graf, V. B. Prakapenka, E. Greenberg, D. A. Knyazev, M. Tkacz and M. I. Eremets 2019 Nature 569 528–531

G1.2 X-Ray Studies: Phase Transformations and Microstructure Changes Christian Scheuerlein and M. Di Michiel

G1.2.1 Introduction Conventional materials characterisation of superconducting wires or tapes implies the destructive preparation of the samples by cutting, grinding and polishing, for instance, for microscopic studies. A destructive sample preparation is also required for X-ray diffraction (XRD) experiments with laboratory diffractometers, which commonly use Cu Kα radiation (ECu-Kα~8.03 keV) with a penetration depth of some tens of µm in metallic samples. In contrast, both neutrons and high-energy photons can penetrate mm-thick strongly absorbing superconductors, which enables non-destructive diffraction experiments, for instance, with Bi-2223 tapes [1, 2]. The X-ray transmission through typical Ta-alloyed Nb3Sn wires as a function of the X-ray energy is presented in Figure G1.2.1. X-ray energies above 50 keV enable non-destructive experiments in transmission geometry with Nb3Sn wires [3]. Typical acquisition times of neutron diffraction pattern of individual superconducting wires are in the order of hours [4]. State-of-the-art high-energy synchrotron sources can provide very high monochromatic photon flux densities, and diffraction measurements can be performed within seconds when using fast read-out area detectors in Debye–Scherrer transmission geometry. This chapter focuses on high-energy synchrotron radiation in situ studies of entire processes, which can be much faster than conventional materials studies that require a series of samples to be prepared after different processing steps. The non-destructive in situ studies also avoid experimental uncertainties caused by sample quenching, sample inhomogeneity and preparation artefacts. The relatively small scattering angles of high-energy X-rays, facilitate to add auxiliary equipment, for instance, a furnace for heat treatment (HT) studies. For the experiments presented here, two furnaces of the ESRF ID15 beamline have been used. For in situ studies in inert gas or air at ambient pressure, the ID15 diffraction and tomography furnace was 22

used [Figure G1.2.2(a)]. After alignment with respect to the X-ray beam, the position of this furnace remains fixed during the experiment. The sample is mounted onto a ceramic stick that enters the furnace from a bottom hole. For sample alignment and rotation, the ceramic stick is mounted onto a goniometer and the rotation and translation stages. The thin Al foil windows at both sides of the furnace are nearly transparent for high-energy photons. The furnace temperature can be regulated using the temperature reading of a thermocouple that can be inserted into the furnace through the bottom hole. Large sample temperature uncertainties can be avoided when the thermocouple is spot welded onto the sample. For overpressure in situ studies, a set-up consisting of a capillary furnace around a high-pressure cell made of a single crystal sapphire tube has been used [Figure G1.2.2(b)]. This furnace was developed for combined in situ high-energy X-ray diffraction and mass spectrometry investigations during catalysed gas/solid or liquid/solid reactions [5]. The connection of the high-pressure cell to the pressure controller is done with a flexible stainless steel line that allows rotating the highpressure cell up to 360° during the acquisition of diffraction patterns. For the study of superconducting wires, the highpressure cell was modified such that a thermocouple can be spot welded onto the superconductor sample [6]. Other sample environments are possible too, for instance, a cryostat and a tensile rig can be added. This makes it possible to study the superconductor electromechanical behaviour and the damage development by XRD measurements at well-defined uniaxial tensile stress or strain at cryogenic temperatures, for instance, in liquid Helium [3, 7, 8] or in liquid Nitrogen [9]. Lattice distortions, superconducting properties and mechanical properties of high-temperature superconductors can be measured simultaneously [10]. X-ray absorption micro-tomography (µ-CT) can provide three-dimensional images of the superconductor bulk. A spatial resolution of µ-CT in the order of 1 µm is today routinely obtainable with both laboratory and synchrotron sources.

X-Ray Studies: Phase Transformations and Microstructure Changes

23

entire HT cycle, they were continuously moved from the white beam to the monochromatic beam for the alternating XRD and µ-CT measurements. After the installation of the new ID15 insertion device in 2008, the flux density of high-energy monochromatic photons has been further increased such that XRD and fast µ-CT can now be performed both with the same monochromatic photon beam. The goal of this chapter is to illustrate the potential of highenergy synchrotron radiation experiments for in situ studies of the processing of superconductors. We present case studies describing the Nb3Sn wire diffusion HT, the transformation HT of Nb3Al precursor wires and the melt processing HT of Bi-2212 wires.

G1.2.2 Nb3Sn Diffusion HT FIGURE G1.2.1 X-ray transmission through Ta-alloyed Nb3Sn wires of equal composition with different diameters.

Fast µ-CT [10, 11], where the acquisition of about 1000 radiographs needed for the reconstruction of one tomogram lasts not more than 1 minute, is required for in situ studies of microstructural changes and porosity formation during the processing of superconductors with temperature ramp rates in the order of 100°C/h. Different synchrotron techniques, for instance, high-energy XRD and µ-CT, can be combined in one experiment. This combination has been pioneered at the ESRF ID15A beamline for an in situ study of the void growth mechanisms in Nb3Sn wires [12]. This was achieved using two X-ray beams, a highintensity filtered white X-ray beam for µ-CT and a monochromatic X-ray beam (energy 88 keV, energy bandwidth 0.1 keV) for the XRD measurements [13]. The sample and the furnace needed to be aligned in both X-ray beams, and during the

G1.2.2.1 Phase Transformations during the Diffusion HT of Nb3Sn Superconductors The Nb3Sn phase in multifilament wires is produced during a diffusion HT, where the precursor elements Nb and Sn interdiffuse with the Cu matrix, forming various intermetallic phases and finally the superconducting Nb3Sn [14]. The intermediate phase transformations can degrade the microstructural and microchemical homogeneity of the fully reacted superconductor. This is most easily observed in the tubular strand types (powder-in-tube [PIT] [15] and tube type [16]), where typically 25% of the Nb3Sn volume consists of coarse grains that are not well connected and cannot conduct significant supercurrents. High-energy synchrotron X-ray diffraction is an excellent tool to monitor the phase changes in superconducting wires in situ during the processing HT [17, 18]. As an example,

FIGURE G1.2.2 (a) ID15 furnace for in situ XRD and tomography at ambient pressure. (b) Furnace for in situ XRD at pressures up to 200 bar in measurement position and (c) capillary furnace withdrawn from the high-pressure cell.

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FIGURE G1.2.4 Evolution of the integrated intensity of prominent Nb3Sn peaks in the PIT Ø=0.8 mm, PIT Ø=1.25 mm and RRP Ø=0.8 mm wires during identical HT. (Courtesy J. Kadar.)

FIGURE G1.2.3 Summary of the diffraction patterns acquired in situ during the Nb3Sn PIT wire reaction HT. (© IOP Publishing. Reproduced with permission. All rights reserved.)

Sn, Cu6Sn5, NbSn2, Nb6Sn5, Nb3Sn and the ternary phase (Nb0.75Cu0.25)Sn2 [19] can be identified in the diffraction pattern that have been acquired during Nb3Sn PIT wire HT (Figure G1.2.3). The respective temperature intervals where these phases are present are easily revealed in the sequence of diffractograms. In situ XRD measurements during the reaction HT of restacked rod process (RRP) type [20] and tube type [21] Nb3Sn wires revealed a similar phase sequence. In particular, (Nb0.75Cu0.25)Sn2, NbSn2 and Nb6Sn5 are detected in the high Jc strands. In the RRP type wire, the amount of these phases is comparatively small, which presumably explains the relatively small volume fraction of Nb3Sn coarse grains in the fully processed RRP wire. During the processing HT of a low Sn content internal tin wire [22], a markedly different phase sequence is observed, and in particular, the phases (Nb0.75Cu0.25)Sn2, NbSn2 and Nb6Sn5 are not formed [7] (see G1.2.5).

G1.2.2.2 Nb3Sn Nucleation and Growth Nb3Sn nucleation and growth in multifilament wires is accompanied by changes of the Sn content distribution, and the Nb3Sn grain size distribution. These microstructure and composition changes, which have a strong influence on Jc [23, 24], can be followed in situ by high-energy synchrotron XRD measurements. By monitoring the Nb3Sn diffraction peak area evolution, the Nb3Sn formation kinetics in different wires can be compared. In an internal tin wire with low Sn content, the Nb3Sn phase growth follows a parabolic law [17], indicating that in this wire Nb3Sn growth is diffusion controlled. This is in contrast to the Nb3Sn phase growth in state-of-the-art high Sn content RRP- and PIT-type wires, where Nb3Sn growth is not purely diffusion controlled [17, 20].

Figure G1.2.4 compares the Nb3Sn growth in a RRP wire with 80 µm subelement size with that in PIT wires with 30 µm and 50 µm subelement size. All wires followed the same HT cycle with 100°C/h heating rate and three isothermal steps (4 h-700°C, 1 h-800°C and 1 h-900°C). Usually the processing peak temperature of Nb3Sn superconductors does not exceed 700°C. Here the 800°C and 900°C plateaus were added to explore the full reaction within a duration that allows to perform in situ synchrotron experiments. It is assumed that in the three wires the maximum possible amount of Nb3Sn was formed during the HT. Therefore, the maximum Nb3Sn peak areas measured during the HT cycles can be normalised, and the Nb3Sn growth kinetics in the wires with different elemental composition and architecture can be compared. In the RRP wire with 80 µm subelement size, Nb3Sn is already detected at about 540°C, and about 80% of the maximum possible Nb3Sn volume is formed after the 4 h-700°C plateau. In the PIT wires, Nb3Sn is first detected during the 700°C plateau, and the Nb3Sn formation kinetics and the duration needed to transform Nb6Sn5 entirely into Nb3Sn are significantly influenced by the subelement size. Keeping a small grain size for flux pinning and at the same time have maximum Nb3Sn volume and Sn content are conflicting needs of the HT. In ideally homogeneous wires, the Nb3Sn grain size and Sn content evolution can be monitored simultaneously with the Nb3Sn volume by XRD measurements. The Sn content can be determined from Nb3Sn lattice parameter measurements [25]. Assuming that the Nb3Sn grains nucleate and grow in a nearly stress-free state, the decrease of diffraction peak width after deconvolution of the instrument function is associated with the increase of grain size, and the use of the Scherrer formula allows for a rough calculation of the mean crystallite size [26]. In order to monitor crystallite sizes up to 200 nm, the diffraction experiment needs to be optimised for minimising instrumental peak broadening. Since the RRP wire has a comparatively homogeneous Nb3Sn microstructure, it has been selected for the Nb3Sn

X-Ray Studies: Phase Transformations and Microstructure Changes

25

FIGURE G1.2.5 Average Nb3Sn crystallite size and volume from (a) Nb3Sn (200) reflection and (b) Nb3Sn lattice parameter as a function of temperature. The relative lattice parameter variation induced solely by thermal expansion in the temperature interval 540°C–900°C is shown for comparison. (Reproduced with permission from Appl. Phys. Lett. 99, 122508. Copyright 2011, AIP Publishing LLC.)

nucleation and growth in situ study [27]. Figure G1.2.5 compares the changes of the Nb3Sn volume with the average crystallite size evolution (a) and the average Sn content (b) during 100°C/h HT with the three isothermal steps at 700°C, 800°C and 900°C. The average crystal size is a rough estimate, and only relative changes are meaningful. At the onset of detectability (at 540°C), the mean Nb3Sn crystallite size estimated from the Nb3Sn (200) peak width is about 60 nm. During the 700°C plateau, the average crystallite size increases from 110 nm to about 180 nm. At the same time, the Nb3Sn volume increases by about 35%, and the Nb3Sn lattice parameter increases from 5.3140 Å to 5.3156 Å. This indicates that the average Sn content increases by more than 1%, which in turn corresponds to a strong critical field Bc20 increase of 5 T [23]. At high magnetic field, such a strong Bc2 increase outweighs the reduction of flux pinning force due to the simultaneous Nb3Sn grains growth. Further increase in temperature and HT duration only slightly increases the Nb3Sn volume but has a detrimental influence on the Nb3Sn crystallite size, which results in limited Jc when flux pinning has a dominating influence.

G1.2.2.3 Nb3Sn Texture Formation Static texture analysis of bulk materials can be performed best by neutron diffraction measurements because of the relatively low neutron absorption. Synchrotron XRD in transmission geometry using an area detector can be very fast and is therefore better suited for monitoring texture formation

in situ during processing HT [28]. Diffraction images of different Nb3Sn wires acquired with a Trixell Pixium 4700 two-dimensional flat-panel digital detector are compared in Figure G1.2.6. Preferential crystallite orientation is revealed by intensity fluctuations along the Nb and Nb3Sn diffraction rings [8]. The homogeneous intensity along the Nb3Sn rings of the BR wire shows that the BR process produces randomly oriented Nb3Sn crystallites. In contrast, intensity maxima are seen in the Nb3Sn (200) rings of the RRP- and PIT-type wires, but the intensity maxima are in different positions. Further texture analysis by electron backscatter diffraction (EBSD) revealed an Nb3Sn texture in the PIT-type wire, whilst in the RRP type wire Nb3Sn grows with a texture in the wire axis direction [29]. EBSD also confirmed a strong Nb texture parallel to the wire axis, as it is commonly observed in cold-drawn body-centred cubic (BCC) Nb.

G1.2.2.4 Void Growth Mechanisms in Nb3Sn Superconductors The presence of porosity in superconductors is often unavoidable, and the fabrication route can have a strong influence on the porosity volume and the distribution of voids that remains in the fully processed superconductor. Porosity generally reduces the useful superconductor volume in the composite, and in some cases, it may degrade the irreversible strain limit of brittle superconductors. If the porosity is distributed inside the superconducting phase, it can block the supercurrent.

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FIGURE G1.2.6 Diffraction pattern of a bronze route, PIT and RRP Nb3Sn wire. The dashed arrows indicate the positions of intensity maxima in the Nb3Sn (200) Debye rings. (© IOP Publishing. Reproduced with permission. All rights reserved.)

The visualisation and quantitative description of the distribution of voids in the superconductor can help to better understand the porosity formation and redistribution mechanisms, and how porosity influences the superconducting properties. When the void shape and distribution are irregular, two-dimensional metallographic observations of void formation can be erratic and misleading. In contrast, X-ray µ-CT can provide non-destructively three-dimensional (3D) quantitative information about the porosity and particle size distribution. MAt modern synchrotrons, tomograms can be acquired in less than 1 minute, which enables time-resolved in situ µ-CT studies of entire processes. Figure G1.2.7 shows a sequence of tomograms that was acquired in situ during the processing HT of a low Sn content internal tin Nb3Sn wire [22] with a ramp rate of 60°C/h, using the tomography furnace shown in Figure G1.2.2(a). In order to obtain a 3D view of the porosity

FIGURE G1.2.7 3D view of the porosity inside an internal tin Nb3Sn wire acquired in situ by synchrotron µ-CT at different temperatures. (Reproduced with permission from Appl. Phys. Lett. 90, 132510. Copyright 2007, AIP Publishing LLC.)

inside the wire, the strand materials have been transparently depictured in the image reconstructions. In Figure G1.2.8, the phase evolution during the HT, based on diffraction peak area measurements, is compared with the porosity volume evolution, which is determined from the simultaneously acquired tomograms (Figure G1.2.7). The phase evolution during the HT of the low Sn content internal tin wire differs strongly from that of high Sn content PIT- and RRP-type wires (Figure G1.2.3). In particular, the Nb-containing phases NbSn2, (Nb0.75Cu0.25)Sn2 and Nb6Sn5 are not formed in the low Sn content wire. The analysis of the simultaneously acquired µ-CT and XRD results allows to distinguish between different void formation mechanisms. The growth of the globular voids up to a temperature of about 200°C is driven by a gain in free energy through a reduction of the total void surface area when smaller voids

FIGURE G1.2.8 Evolution of prominent diffraction peak areas of all Sn-containing phases, apart from α-bronze, that exists in the IT Nb3Sn strand during the reaction HT up to 540°C. Diffraction peak areas have been scaled such that the values correspond with the relative phase volume in the wire. The liquid Sn evolution is estimated from the amount of the detected phases. The total void volume is shown for comparison. (Reproduced with permission from Appl. Phys. Lett. 90, 132510. Copyright 2007, AIP Publishing LLC.)

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27

present in the as-drawn wire agglomerate to larger globular voids. At 200°C, the maximum ratio of void volume to void surface area is obtained. At this temperature the total void volume corresponds to 2.5% of the pure Sn volume in the asdrawn wire. The correlation between void volume and Cu3Sn content, which is obvious in Figure G1.2.8, is due to the 4% higher density of Cu3Sn with respect to the Cu and Sn in their stoichiometric quantities.

G1.2.3 Transformation HT of Rapidly Quenched Nb3Al Precursor The Nb3Al phase in superconducting wires is produced during a Rapid heating, quenching and transformation (RHQT) process [30]. During a HT at roughly 1900°C, a Nb(Al)SS solid solution is obtained, which can be retained during rapid quenching to ambient temperature. The rapid heating and quenching (RHQ) stages are followed by a transformation HT with a peak temperature of typically 800°C, during which fine-grained Nb3Al with high Al content is formed from the Nb(Al)SS solid solution. The phase evolution during this transformation HT can be studied in situ by high-energy synchrotron X-ray diffraction [31]. The two-dimensional diffraction pattern acquired in transmission geometry with an area detector can be caked into sectors, in order to distinguish between reflections from the crystalline planes oriented both perpendicular and parallel to the wire drawing axis, which are in the following referred to as the axial and transverse directions, respectively. The pattern presented in [31] have been caked into 128 sectors. The 222 filament Nb3Al precursor wire without Cu stabiliser that was studied has a partial interfilamentary Ta matrix. Since the strain-free Ta and Nb lattice parameters at RT differ by about 0.03% only, these phases could not be distinguished by their lattice spacing. Therefore, in the following, Nb diffraction peak refers to both overlapping peaks of Nb and Ta. The axial and transverse Nb (110) diffraction peaks of the RHQ wire are presented in Figure G1.2.9. The axial Nb (110) peak is about eight times more intense than the transverse peak, which shows that the Nb (and/or Ta) texture, which is developed during the cold drawing of BCC metals, is partially retained during the RHQ process. The axial and transverse Nb peaks exhibit two maxima, which are characteristic for pure Nb and Ta (larger d-spacing) and Nb(Al) ss supersaturated solid solution (with roughly 1% smaller d-spacing). The evolution of the Nb (110), Nb3Al (200) and Nb3Al (211) diffraction peak shape and intensity during the RHQ Nb3Al precursor wire transformation HT with a ramp rate of 800°C/h and a final 800°C plateau lasting 30 minutes can be seen in Figure 5 of Reference [31]. The Nb (110) peak

FIGURE G1.2.9 Axial and transverse Nb(110) diffraction peak, consisting of two components characteristic for pure Nb and for Nb(Al)SS.

shape change is caused by the vanishing of the Nb(Al)SS peak component, upon formation of Nb3Al. When heating with a ramp rate of 800°C/h, Nb3Al (200) and Nb3Al (211) peaks are detected at about 780°C. When heating with a ramp rate of 160°C/h, the transformation from a Nb(Al)SS supersaturated solid solution into Nb3Al occurs at roughly 60°C lower temperature than during the 800°C/h HT [31].

G1.2.4 Bi-2212 Wire Melt Processing G1.2.4.1 Phase Evolution during Bi-2212 Wire Melt Processing In order to form well-connected and textured Bi-2212 filaments, the Bi-2212 precursor particles in the as-drawn Bi-2212 PIT wire need to be melted when the wire is at its final size and shape [32]. During the melt processing HT, an external oxygen supply through the oxygen-permeable Ag wire matrix is needed in order to re-form Bi-2212 out of the melt. The phase evolution during the melt processing HT can be studied in situ by high-energy synchrotron XRD measurements. Oxygen can be supplied conveniently in a flow of air at ambient pressure, using the X-ray transparent furnace shown in Figure G1.2.2(a). The sequence of diffraction pattern acquired during the melt processing of a state-of-the-art Bi-2212 PIT wire in air (oxygen partial pressure pO2 =0.21 bar) is presented in Figure G1.2.10 [33]. An initial Bi-2212 diffraction peak growth with increasing temperature is observed, which is attributed to crystallisation of Bi-2212 that was amorphised during the wire drawing process. The main impurity phase Bi-2201 is first detected when the temperature exceeds approximately 200°C, and a maximum amount of Bi-2201 is detected at about 500°C.

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bar, the Bi-2212 precursor particles in the state-of-the art Bi-2212 multifilament wire decomposes completely in the solid state.

G1.2.4.2 Void Formation and Redistribution during Bi-2212 Wire Melt Processing

FIGURE G1.2.10 Sequence of XRD patterns acquired during Bi-2212 wire HT in ambient air. The diffraction peaks which are labelled with arrows have been tentatively identified as the Cu-free phase Bi 2(Sr4-yCay)O7). (© IOP Publishing. Reproduced with permission. All rights reserved.)

Bi-2201 decomposes completely at 850°C, and reforms again upon cooling at approximately 850°C. The diffraction peaks that occur upon Bi-2212 melting around 880°C in a pO2 =0.21 bar process gas have been tentatively identified as Cu-free phase Bi2(Sr4-yCay)O7. Overpressure (OP) processing at pressures of up to 100 bar is a key for achieving homogeneous high critical currents in long lengths of Bi-2212 wires [34]. OP processing also enables varying the oxygen partial pressure in a wide range, and it is of interest to verify how pO2 influences the phase sequence and the Bi-2212 precursor melting and recrystallisation behaviours. In order to study the influence of pO2 on the Bi-2212 phase stability inside the Bi-2212/Ag wire by in situ highenergy synchrotron XRD measurements, the high-pressure cell and capillary furnace shown in Figure G1.2.2(b, c) have been used. This furnace allows to explore pO2 above ambient pressure, with total process gas pressures up to 200 bar. Another advantage of this furnace is that 5-cm-long wire samples with closed ends identical to the samples typically used for Bi-2212 critical current measurements can be studied. The diffraction pattern acquired in situ during HTs at different pO2 shows that increasing pO2 reduces the Bi-2212 stability [6]. At pO2 =1.5 bar, Bi-2212 decomposes partly prior to melting, and the precursor decomposition temperature is about 20°C lower than it is at pO2 =1.05 bar. At pO2 =5

Porosity and second phase particles formed during melt processing are considered to be the main current limiting defects in Bi-2212 wires. It is therefore of great interest to visualise and to quantify the porosity and second phase distribution during the different processing steps. The potential of µ-CT to visualise these features inside a superconducting wire depends equally on the spatial and density resolution of the µ-CT experiment. The calculated linear absorption coefficients of 70 keV photons in the main wire constituents Ag and Bi-2212, and the main impurity phase Bi-2201, are µAg=40 cm−1, µBi−2212=15 cm−1 and µBi−2201= 18 cm−1, respectively. Because of the different X-ray attenuation in Ag, Bi-2212 and porosity, high-energy synchrotron µ-CT is particularly well suited to monitor Bi-2212 microstructure changes and the porosity formation and redistribution inside Bi-2212/Ag wires [33]. On the other hand, because of the relatively small difference of the X-ray attenuation in Bi-2212 and Bi-2201, these phases cannot be distinguished in the X-ray absorption tomograms. The void redistribution during the melt processing of a 37 × 7 filament Bi-2212 wire at ambient pressure can be followed in the longitudinal µ-CT cross-sections shown in Figure G1.2.11, which have been acquired in situ at different temperatures. In order to show a more detailed view of the voids, the images have been cropped from the longitudinal cross-sections showing the entire wire cross-section. The black areas represent voids, the bright grey areas the strongly absorbing Ag matrix, and the dark grey areas are Bi-2212 with a small amount of Bi-2201. Filament microstructure changes can be first observed at about 850°C when the Bi-2201 impurity phase decomposes (as seen in the simultaneously acquired XRD pattern). At this temperature, the finely divided porosity, which is in the asdrawn wire uniformly distributed between the precursor particles, coalesces into lens-shaped defects. On Bi-2212 melting, the lens-shaped voids grow to bubbles of a filament diameter. Upon cooling, nucleation of Bi-2212 is first observed in the tomogram acquired at 877°C at the filament periphery. The Bi-2212 formed upon cooling partly bridges the void space, but bubbles remain and cause an obstacle to the current flow in the Bi-2212 wires that are melt processed at ambient pressure [35]. The importance of complete Bi-2212 precursor melting is obvious when comparing the longitudinal µ-CT cross-section acquired at the end of a processing HT to Tmax=915°C, during which Bi-2212 was completely melted [Figure G.1.1.2.12(a)],

X-Ray Studies: Phase Transformations and Microstructure Changes

29

FIGURE G1.2.11 Detailed view of Bi-2212 wire longitudinal tomographic cross-sections acquired in situ at different temperatures during HT to Tmax=915°C in air. A time lapse movie showing the changes occurring over the whole heating and cooling cycle is available (https:// edms.cern.ch/document/1153082/1). (© IOP Publishing. Reproduced with permission. All rights reserved.)

FIGURE G1.2.12 Tomographic cross-sections of the Bi-2212 wire acquired after in situ HT to (a) Tmax=875°C and (b) Tmax=915°C. 3D reconstructed images of selected filaments are shown in the insets. (© IOP Publishing. Reproduced with permission. All rights reserved.)

30

and to Tmax=875°C, in which only a fraction of the Bi-2212 powder was melted [G.1.1.2.12(b)]. The tomograms show clearly that after the Tmax=875°C HT, the filaments remain interrupted by a regular array of lens-shaped voids, and that filament connectivity is only achieved after the porosity rearrangement that occurs during complete Bi-2212 melting and recrystallisation. The overall porosity volume in Bi-2212 wires that are short enough to allow relief of internal pressure through the open ends does not strongly change, because the Bi-2212 processing does not involve phase transformations associated with important density variations, as it is, for instance, the case in Nb3Sn conductors [7]. In long Bi-2212 wires and in wires with closed ends, additional porosity is formed during processing at ambient pressure when internal gas pressure leads to creep of the Ag matrix [36]. OP pressing strongly reduces the porosity volume that is present in the as-drawn wire [34].

G1.2.5 Outlook Today the time-resolved combined XRD and µ-CT experiments for in situ studies of superconductors that are described above can be routinely performed at advanced high-energy synchrotron beamlines. Thanks to the fast data acquisition at modern synchrotron beamlines, XRD and u-CT results can be compared with those obtained with identical ramp rates by other in situ techniques like differential scanning calorimetry (DSC) to achieve an even deeper understanding of the endothermic and exothermic phase transformations [37]. The continuously improving brilliance of synchrotron sources and new efficient X-ray focusing optics makes it possible to use nanometer scale X-ray beams, enabling new nondestructive in situ experiments on length scales that so far were only accessible to destructive techniques [38]. Grain size and grain orientation have a dominant influence on the performance of most superconductors, and studies of the thermal growth of grains and the grain orientation evolution are examples where future superconductor research can profit from new synchrotron experiments with X-ray nanobeams. Such studies can be performed in two dimensions, averaging over the sample depth that is penetrated by the X-ray beam. When applying tomographic methods (e.g. XRD-tomography [39]) spatially resolved in situ studies of phase composition, crystallite size distribution and texture become possible.

Acknowledgments All XRD and µ-CT experiments presented here have been performed at the ESRF ID15 beamline. We are grateful to Julian Kadar for the Nb3Sn diffraction peak analysis of Figure G1.2.4.

Handbook of Superconductivity

References 1. T.R. Thurston, P. Haldar, Y.L. Wang, M. Suenaga, N.M. Jisraw, U. Wildgruber, “In situ measurements of texture and phase development in (Bi, Pb)2Sr2Ca 2Cu3O10–Ag tapes”, J. Mater. Res., 12(4), (1997) 2. D.K. Finnemore, “X-ray studies: phase transformation and texture”, in Handbook of Superconducting Materials 1st edition, edited David A. Cardwell, David S. Ginley, (2003) 3. C. Scheuerlein, M. Di Michiel, F. Buta, “Synchrotron radiation techniques for the characterisation of Nb3Sn superconductors”, IEEE Trans. Appl. Supercond., 19(3), (2009), 2653–2656 4. C. Scheuerlein, U. Stuhr, L. Thilly, “In-situ neutron diffraction under tensile loading of powder-in- tube Cu/ Nb3Sn composite wires: effect of reaction heat treatment on texture, internal stress state and load transfer”, Appl. Phys. Lett., 91(4), (2007), 042503 5. J. Andrieux, C. Chabert, A. Mauro, H. Vitoux, B. Gorges, T. Buslaps, V. Honkimäki, “A high pressure and high temperature gas loading system for the study of conventional to real industrial sized samples in catalyzed gas/solid and liquid/solid reactions”, J. Appl. Cryst., 47, doi:10.1107/S1600576713030197, (2014) 6. C. Scheuerlein, J. Andrieux, M.O. Rikel, J. Kadar, C. Doerrer, M. Di Michiel, A. Ballarino, L. Bottura, J. Jiang, T. Kametani, E.E. Hellstrom, D.C. Larbalestier, “Influence of the oxygen partial pressure on the phase evolution during Bi-2212 wire melt processing”, IEEE Trans. Appl. Supercond., submitted 7. L. Muzzi et al, “Direct observation of Nb3Sn lattice deformation by high energy xray diffraction in internal tin wires subject to mechanical loads at 4.2 K”, Supercond. Sci. Technol., 25, (2012), 054006 8. C. Scheuerlein, M. Di Michiel, F. Buta, B. Seeber, C. Senatore, R. Flükiger, T. Siegrist, T. Besara, J. Kadar, B. Bordini, A. Ballarino, L. Bottura, “Stress distribution and lattice distortions in Nb3Sn/Cu multifilament wires under uniaxial tensile loading at 4.2 K”, Supercond. Sci. Technol., 27, (2014), 044021, 9. R. Bjoerstad, C. Scheuerlein, M. Rikel, A. Ballarino, L. Bottura, J. Jiang, M. Matras, M. Sugano, J. Hudspeth, M. Di Michiel, “Strain induced irreversible critical current degradation in highly dense Bi-2212 round wire”, Supercond. Sci. Technol., 28, (2015), 062002 10. M. Di Michiel, J. M. Merino, D. Fernandez-Carreiras, T. Buslaps, V. Honkimäki, P. Falus, T. Martins, O. Svensson, “Fast microtomography using high energy synchrotron radiation”, Rev. Sci. Instrum., 76, (2005), 043702 11. J. Y. Buffiere, E. Maire, J. Adrien, J. P. Masse, E. Boller, “In situ experiments with X ray tomography: an attractive tool for experimental mechanics”, Experimental Mechanics, 50, (2010), 289–305,

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12. C. Scheuerlein, M. Di Michiel, A. Haibel, “On the formation of voids in Nb3Sn superconductors”, Appl. Phys. Lett., 90, (2007), 132510 13. “Void formation in Nb3Sn Superconductors”, http:// www.esrf.eu/Apache_files/Highlights/HL2007.pdf, (2007), 27–28 14. M.T. Naus, P.J. Lee, D. C. Larbalestier, “The interdiffusion of Cu and Sn in internal Sn Nb3Sn superconductors”, IEEE Trans. Appl. Supercond., 10(1), (2000) 983–987 15. H. Veringa, E. M. Hornsveld, and P. Hoogendam, Adv. Cryo. Eng., vol. 30, (1984), 813–821 16. E. Gregory, X. Peng, M. Tomsic, M. D. Sumption, A. Ghosh, “Nb3Sn superconductors made by an economical tubular process”, IEEE Trans. Appl. Supercond., 19(3), (2009), 2602–2605 17. M. Di Michiel, C. Scheuerlein, “Phase transformations during the reaction heat treatment of powder-in-tube Nb3Sn superconductors”, Supercond. Sci. Technol., 20, (2007), L55–L58 18. A.B. Abrahamsen, J.C. Grivel, N.H. Andersen, M. Herrmann, W. Häßler, K. Saksl, “In-situ synchrotron x-ray study of MgB2 formation when doped by SiC”, J. Phys. Conf. Ser., 97 (2008), 012315 19. S. Martin, A. Walnsch, G. Nolze, A. Leineweber, F. Léaux, C. Scheuerlein, “The crystal structure of (Nb0.75Cu0.25) Sn2 in the Cu-Nb-Sn system”, Intermetallics 80, (2017), 16–21 20. C. Scheuerlein, M. Di Michiel, G. Arnau, F. Buta, “Phase transformations during the reaction heat treatment of Internal Tin Nb3Sn strands with high Sn content”, IEEE Trans. Appl. Supercond., 18(4), (2008), 1754–1760 21. C. Scheuerlein, M. Di Michiel, L. Thilly, F. Buta, X. Peng, E. Gregory, J.A. Parrell, I. Pong, B. Bordini, M. Cantoni, “Phase transformations during the reaction heat treatment of Nb3Sn superconductors”, J. Phys. Conf. Ser., 234, (2010), 022032 22. M. Durante, P. Bredy, A. Devred, R. Otmani, M. Reytier, T. Schild, F. Trillaud, “Development of a Nb3Sn multifilamentary wire for accelerator magnet applications“, Physica C, 354, (2001), 449–453 23. R. Flükiger, D. Uglietti, C. Senatore, F. Buta, “Microstructure, composition and critical current density of superconducting Nb3Sn wires”, Cryogenics, 48, (2008), 293–307 24. P.J. Lee, D.C. Larbalestier, “Microstructural factors important for the development of high critical current density Nb3Sn strand”, Cryogenics, 48, (2008), 283–292 25. H. Devantay, J. L. Jorda, M. Decroux, J. Müller, R. Flükiger, J. Mater. Sci., 16, (1981), 2145 26. J.I. Langford, D. Louër, P. Scardi, “Effect of a crystallite size distribution on X-ray diffraction line profiles

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and whole-powder-pattern fitting”, J. Appl. Cryst., 33, (2000), 964–974 L. Thilly, M. Di Michiel, C. Scheuerlein, B. Bordini, “Nb3Sn nucleation and growth in multifilament superconducting strands monitored by high resolution synchrotron diffraction during in- situ reaction”, Appl. Phys. Lett., 99, (2011), 122508 H.R. Wenk, S. Grigull, “Synchrotron texture analysis with area detectors”, J. Appl. Cryst., 36, (2003), 1040–1049 C. Scheuerlein, G. Arnau, P. Alknes, N. Jimenez, B. Bordini, A. Ballarino, M. Di Michiel, L. Thilly, T. Besara, T. Siegrist, “Texture in state-of-the-art Nb3Sn multifilamentary superconducting wires“, Supercond. Sci. Technol., 27 (2014) 025013 T. Takeuchi, “Nb3Al Conductors –Rapid Heating, Quenching and Transformation Process”, IEEE. Trans. Appl. Supercond., 10(1), (2000), 1016–1021 C. Scheuerlein, A. Ballarino, M. Di Michiel, X. Jin, T. Takeuchi, A. Kikuchi, K. Tsuchiya, K. Nakagawa, T. Nakamoto, “Transformation heat treatment of rapidly quenched Nb3Al precursor monitored in situ by high energy synchrotron diffraction“, IEEE Trans. Appl. Supercond., 23(3), 6000604, (2013) K. Heine, J. Tenbrink M. Thoner, “High field critical current densities in Bi2Sr2CaCu2O8-/Ag wires”, Appl. Phys. Lett. 55, (1989), 2441–2443 C. Scheuerlein, M. Di Michiel, M. Scheel, J. Jiang, F. Kametani, A. Malagoli, E.E. Hellstrom, D.C. Larbalestier, “Void and phase evolution during the processing of Bi-2212 superconducting wires monitored by combined fast synchrotron micro-tomography and X-ray diffraction”, Supercond. Sci. Technol., 24 (2011), 115004 D.C. Larbalestier, J. Jiang, U.P. Trociewitz, F. Kametani, C. Scheuerlein, M. Dalban-Canassy, M. Matras, P. Chen, N.C. Craig, P.J. Lee, E.E. Hellstrom, “Isotropic round wire multifilament cuprate superconductor for generation of magnetic fields above 30 T”, Nature Mater., 13, (2014), 375–381 F. Kametani, T. Shen, J. Jiang, A. Malagoli, C. Scheuerlein, M. Di Michiel, Y. Huang, H. Miao, J.A. Parrell, E.E. Hellstrom, D.C. Larbalestier, Supercond. Sci. Technol., 24, (2011), 075009 A. Malagoli, P.J. Lee, A. Ghosh, C. Scheuerlein, M. Di Michiel, J. Jiang, U.P. Trociewitz, E.E. Hellstrom, D.C. Larbalestier, “Evidence of length-dependent wire expansion, filament dedensification and consequent degradation of critical current density in Ag-alloy sheathed Bi-2212 wires”, Supercond. Sci. Technol., 26, (2013), 055018 C. Scheuerlein, J. Andrieux, M. Michels, F. Lackner, C. Meyer, R. Chiriac, F. Toche, M. Hagner, D. Michiel, “Effect of the fabrication route on the phase and volume

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changes during the reaction heat treatment of Nb3Sn superconductors”, Supercond. Sci. Technol. 33, (2020), 034004 38. G.E. Ice, J.D. Budai, J.W.L. Pang, “The race to X-ray microbeam and nanobeam science”, Science, 334, (2011), 1234–1239

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39. S.D.M. Jacques, M. Di Michiel, A.M. Beale, T. Sochi, M.G. O’Brien, L. Espinosa-Alonso, B.M. Weckhuysen, P. Barnes, “Dynamic X-ray diffraction computed tomography reveals real- time insight into catalyst active phase evolution”, Angewandte Chemie International Edition, 50(43), 2011, 10148–10152

G1.3 Transmission Electron Microscopy Fumitake Kametani

G1.3.1 Introduction Important properties of ‘real’ materials (catalytic, physical, electrical, mechanical…) are often derived from their micro-, nano- and/or atomic structures. We can hardly gain full understanding in materials’ properties only with the information of exteriors that the scanning electron microscope (SEM) or optical microscope (OM) provides, or of averaged structures for which X-ray diffraction techniques, for example, are used. We must explore their very local structures which often determine all the mechanisms liable to disturb the local atomic arrangements. Among the experimental techniques that explore the structure of materials, transmission electron microscopy (TEM) has the strong advantage of providing information in reciprocal space (working in electron diffraction mode, ED), real space (imaging at very different scales ranging from the micrometer to the ångstrom) and stoichiometry. High-resolution electron microscopy (HREM) and atomic resolution scanning electron microscope (STEM) are the only technique that can directly access the structure, density and distribution of the defects at the nano- or atomic scale. Now that the spherical aberration (Cs) can be corrected by the Cs corrector, the spatial resolution in imaging is subångstrom. Also, the strong interaction between electrons and materials provides the information regarding the local stoichiometry and electron state, making the electron microscopies and related techniques essential in materials science. Understanding nanostructural correlations to physical properties is so crucial to investigate new functional materials. TEM is a key tool in this task. Such nanostructural correlations is very important for superconductivity as well. Especially practical applications of superconductors strongly demand high critical current density Jc at temperature and/or magnetic field where the devices actually operate. The strong vortex pinning and intergrain superconducting connectivity are the primary factors for attaining high Jc, prompting the structural investigation and evaluation of these at the nano and atomic scale. Thus, the role of TEM has become more and more important for research and deployment of superconductors.

This chapter gives a brief introduction to TEM and related techniques as a tool for researches of superconducting materials, and provides a few examples how to use these techniques for micro-, nano- and atomic structural characterizations. In particular, the studies of vortex pinning nanostructures, stoichiometry and intergrain connectivity are shown here since Jc of conductors is determined primarily by these factors for which nanostructural information is crucial to control them.

G1.3.2 Transmission Electron Microscopy Electron microscopy provides a unique capability to obtain, by direct observation, information which can only be obtained in a statistical form by x-ray and neutron crystallography. The microscopy concept implies the acquisition and interpretation of magnified images. As a direct consequence, a discussion of TEM involves the necessity to specify the scale of the observation and the resolution of the microscope. ‘Atomic’ resolution observations of materials are offered by two different techniques: high-resolution transmission electron microscopy (HRTEM) and scanning transmission electron microscopy (STEM). First, we discuss about the former. In fact, numerous different effects result from the strong and complex interactions between an electron beam and materials, most of them being exploited by electron microscopy techniques. A detailed course in transmission electron microscopy is given in specialized books [1–4]. Here only a brief presentation of the technique is given. The working principle of the imaging system in transmission electron microscopy (TEM) is sketched in Figure G1.3.1. Assuming we work with a phase object and a coherently accelerated electron beam, the wave function f(x,y) at the exit plane of the object is a projection of the crystal potential in a material. Passing through the crystal, the electron beam undergoes diffraction following Bragg’s law. The diffracted beams are focused onto the back focal plane of the objective (electromagnetic) lens. Their amplitudes are given by the Fourier transform of the object function: F (u, v ) = F [ f ( x , y )] . This 33

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FIGURE G1.3.1 Schematic illustration of forming an image in a TEM. The incident electron beam is split into the direct and diffracted beams that focus onto the back focal plane where the spatial information (locations A and B) is not seen. Then the direct and diffracted beams form the magnified image on the image plane that shows the locations A and B again.

back focal plane acts as a set of Huyghens sources of spherical waves which interfere in the image plane: F ( x , y ) = F [ f (u, v )] ; i.e. the inverse Fourier transform restores the object function. This means that crystallographic information from the diffracted beams is obtained at the back focal plane of the objective lens, namely in the electron diffraction mode, and that an ideal microscope would provide a magnified image of the projected potential of the object, i.e. of the structure of the sample, so-called the image mode, on the image plane.

G1.3.2.1 Electron Diffraction In the electron diffraction mode, the spots appear in addition to the direct beam spot at the center, making a particular pattern based on a crystal structure and its crystallographic orientation. Such a pattern is called as the electron diffraction pattern (DP). The geometrical arrangement of the diffracted electrons is seen at the back focal plane of the objective lens. Figure G1.3.2 shows the selected-area electron diffraction pattern (SADP) from an orthorhombic YBa 2Cu3O7-δ single crystal grain. In a classical kinematical approach, the electron diffraction pattern is formed when the Ewald sphere cut through the reciprocal lattice points for which Bragg’s condition are satisfied. For a perfect single crystal, the SADP pattern exhibits sharp diffraction spots that correspond to the crystallographic orientation of specimen. The relative positions of the spots on the zero-order Laüe zone (ZOLZ) give information about the cell parameters, the existence of symmetry elements and the orientation of the crystal. Information on the three-dimensional crystallography of the sample can

be known by tilting around selected crystallographic axes (rotation method), or by working with high-order Laüe zones (HOLZ). Alternatively, convergent beam electron diffraction (CBED) and microdiffraction [4, 5] allow the crystal point group to be obtained. Electron diffraction is also very sensitive to local deviation from the perfect structure. Any crystal defects can locally displace the lattice, changing the reflection angle at which the Bragg condition is satisfied. Electron diffraction is thus a very efficient tool for determining the degree of long-range order (LRO) as well as short-range order (SRO). Other features in the diffraction pattern, such as the shape of the individual spots, give information concerning the shape or perfection of the crystal.

G1.3.2.2 TEM Imaging To obtain images of the object, it is necessary to work out the intensity distribution on the lower surface of the crystal, which is given by the lenses at the level of the image plane (Figure G1.3.1). Diffraction contrast is produced by placing an aperture (Figure G1.3.1) in the microscope optic. Selecting the direct beam or diffracted beam provides the bright field (BF) or dark field (DF) image, respectively. Collecting the direct beam and one diffracted beam in the two-beam condition produces lattice fringe images which are sometimes sufficient to elucidate certain reaction mechanisms and to distinguish polytypes or different phases. An example of DP with the zone axis and corresponding BF image of twinning domains of YBa 2Cu3O7 is shown in Figure G1.3.2.

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FIGURE G1.3.2 (a) (b) Diffraction pattern and bright field TEM image of an orthorhombic YBa 2Cu 3O7-δ single grain (crystal) from the [001] projection, respectively. The diffraction pattern taken with using the selected-area aperture is called selected-area diffraction (SAED).

For detailed understanding of nano- or atomic structure of various defects, or of interface between constituent phases in complex state-of-art materials, high-resolution electron microscopy (HREM) imaging is very useful. But we need further consideration how HREM image forms and how to interpret it [4]. It is no longer sufficient to consider kinematic scattering and an “ideal electron microscope, and we must pay attention particularly to lens aberrations (spherical and chromatic). It is the microscope “point resolution”, which limits the precision with which we can achieve structure determination. The actual scattered amplitude on the imaging plane is expressed as ψ ( x , y ) = F [F (u, v )]exp(iχ)] , where exp (iχ) is the contrast transfer function (CTF) of the microscope and

accounts for the phase shifts, spherical aberration and focus values. An important aspect of HREM imaging is the fact that, by changing the objective lens defocus or specimen thickness, the phase shift can easily occur, inverting the image contrast. Interpreting an HREM image is not always straightforward. Generally, the positions with high electron density can be either bright or dark. Only under the specific condition (specific defocus values (the so-called Scherzer value), for thin crystals (a few nanometers thick) with relatively short periodicities along the viewing direction), we can directly correlate the phase contrast to the atomic potential. This is illustrated in Figure G1.3.3 (left and right, respectively) by two HREM images recorded for the same ‘1223’-type CaBa2Ca2Cu3O9 crystal [5].

FIGURE G1.3.3 HREM images of the 1223-type CaBa 2Ca 2Cu 3O9 cuprate. The cation positions are imaged as (left) bright dots or (right) dark dots, depending on the focus value.

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G1.3.2.3 Scanning TEM Imaging and Analytical Capabilities For TEM imaging, we presume the parallel beam condition where a certain area of the specimen is illuminated by spreading the coherent electron beam. Instead we can also converge the electron beam to make a fine probe. Scanning TEM (STEM) images are formed by scanning the specimen area with such a fine probe (Figure G1.3.4). Combining the field emission electron gun and spherical aberration (Cs) corrector, the signal-to-noise ratio and spatial resolution of STEM imaging have been dramatically improved over last 15 years [6–10]. Unlike TEM or HREM imaging where the diffraction and phase shift dominantly influence the contrast, we can record only Z-contrast if the high-angle annular dark field (HAADF) detector is employed. Thus interpretation of atomic columns in HAADF-STEM images is more straightforward compared with HREM. Also the sub-angstrom size, high luminosity of Cscorrected beam probe is beneficial for various spectroscopies, particularly energy dispersive X-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS). Both of them

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provides quantitative information on local chemical composition in the specimen. EELS also gives information on the light atom content (such as C, N or O) and on the element bonding, allowing, for example, the oxidation states of metallic atoms to be studied. Diffraction patterns and diffraction contrast images can be recorded over a range of temperatures (commonly from 1000°C to 90 K with liquid N2 cooling and sometimes down to 10 K with liquid He cooling) when the cold or heating sample stage is used. Combining the fast imaging camera, in situ TEM imaging is also possible.

G1.3.2.4 Sample Preparation The very strong interaction between the specimen and the electron beam is an unquestionable advantage – we can observe even very small deviations from the perfect crystal structure and to obtain images from very small volumes of material. The latter is essential for studying some types of specimens, such as nanoparticles. However, we must keep in mind that the TEM specimens have to be very thin; a few hundred nanometers for conventional microscopy and 10–50 nm maximum for atomic resolution imaging and spectroscopy. Depending on the chemical nature of material, we usually use one of four specimen preparation methods: a. Powder method: Simply crush the specimen into small flakes/powders in a suspension such as alcohol, and then deposit them on a holey carbon film. b. Ion milling: Mechanically grind and polish the specimen down to ~20 µm, and then do ion milling with using Ar ions at a very low glancing angle. c. Chemical polishing: Mechanically grind the specimen, and then chemically etch with applying the voltage in a certain echant relevant to the specimen materials. d. Focused ion beam (FIB): The TEM lamellae are cut out in the FIB instrument by utilizing a well-controlled beam of Ga ion. This method is particularly beneficial to investigate non-uniform samples for which you need to observe pin-point locations. It should be noted that the electron beam itself can also cause irradiation damage (heating, structural changes, introduction of lattice defects, amorphisation) in the material, especially in organic compounds. In metal and ionic compounds, such as the cuprates, this kind of damage has been rarely reported for electron beams of 200 kV energy or less.

FIGURE G1.3.4 Schematic showing images available in the STEM as a probe is scanned across a crystal. A Z-contrast image is formed by mapping high-angle scattered electrons as the probe is scanned, and, for a sufficiently small probe, will show atomic resolution with strong atomic number (Z) contrast. The image resolves the YBa 2Cu 3O7-δ lattice along the a-axis. The intensity of each atomic column is dependent on Z.

G1.3.3 Nanoparticle Precipitations in REBa2Cu3Ox Thin Films Thanks to its high critical temperature Tc, high critical current density Jc, high irreversibility field Hirr, and moderate anisotropy parameter γ, REBa 2Cu3Ox (REBCO, where RE = rare

Transmission Electron Microscopy

earth) thin films grown on flexible and mechanically strong substrates can exceed the temperature and field application limits of the Nb-based low-temperature superconductors, and enable superconducting applications in a broad temperature and magnetic field regime that now exceeds 35 T at 4 K [11, 12]. For practical applications in such a wide range of temperature and field, Jc and Hirr enhancement and anisotropy reduction have been pursued by embedding nano-sized 2nd phases in the REBCO matrix [13–20], for which S/TEM imaging has played a crucial role to correlate their shape, size, spacing and density to the superconducting properties. Figure G1.3.5 shows a nanostructure of the prototype REBCO coated conductor (CC) manufactured by MOCVD. The bright field image of diffraction contrast [Figure G1.3.5(a)] was taken under the two-beam condition for which g = (004) of REBCO matrix was preferentially excited so that any nanostructural defects such as precipitates, dislocations, etc. appear in dark contrast. A high density of RE2O3 precipitates are found to form ‘planes’ tilted at ∼7° to the IBAD template, as shown in Figure G1.3.5(a). Also seen are numerous c-axis oriented threading dislocations, which are strong pinning centers in small applied fields [21, 22]. Figure G1.3.5(b) represents a typical HREM image of RE2O3 nanoprecipitate. Strain contrast on the top and bottom of precipitate due to the semi-coherent interface between YBCO and RE2O3. As shown in the inset, the diffraction pattern derived from the fast Fourier transform (FFT) shows the crystal orientation relation between RE2O3 and REBCO. The [110] direction on the (001) plane of RE2O3 is semi-coherent to (001)[100] of REBCO. Figure G1.3.5(c) shows that the RE2O3 nanoparticle planes lie ∼5° away from the ab planes of YBCO. The total misalignment of ∼7° results from this ∼5° tilt and a 2° tilt between the IBAD MgO and the ab planes of the YBCO, as shown in the inset of Figure G1.3.5(c). From Figure G1.3.5(b), the effective size of the precipitates, including their top and bottom strain field,

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is ∼10–15 nm, which is ∼5 times the vortex core diameters 2ξ, where ξ is the coherence length of YBCO (∼1.6 nm at 4.2 K). So the precipitates are larger than optimized for 4.2 K use [11, 23]. Nevertheless, these RE2O3 nanoprecipitates form very effective tilted correlated pinning centers, while the high-field measurements of this particular specimen suggested loss of correlation at high vortex densities. From Figure 1.3.5(c), the average c-axis spacing between precipitate planes is ∼20–30 nm, which corresponds to a matching field BΦ = Φ0/a 2 of ∼3–5 T for H//ab, where Φ0 = 2.07 × 10 −15 Wb is the flux quantum, and a is the separation of vortex pinning precipitate planes. Even with the same 2nd phase of RE2O3, its volumetric shape significantly influences the flux pinning properties. Figure G1.3.6 show TEM images of YBCO thin film with a high-density RE2O3 nanoprecipitates. As shown in the low magnification images, the distribution of precipitates is rather uniform compared with the CC sample we discussed in the previous section. However, although its diameter is 10–15 nm, which is similar to those in Figure G1.3.5, the effective thickness of RE2O3 is 3–5 nm, so they are a thin disk rather than a particle as suggested in the HREM image [(Figure G1.3.6(c)]. The theoretical analysis suggests that for precipitates of similar diameters, the elementary pinning force increases linearly with increasing precipitate thickness [24]. Comparing these two specimens with RE2O3 of different thicknesses, the elementary pinning force f p is ~3.3 times higher for the CC specimen (Figure G1.3.5) than the specimen of Figure G1.3.6. The greatly enhanced pinning strength of the MOCVD CC increases its resistance to thermal fluctuations, especially at high temperatures and fields. On the other hand, such high density, uniform arrays of thin disk RE2O3 significantly increases f p along the ab-plane direction, because Jc is also strongly influenced by the extension of pinning precipitates along the field direction, which directly affects the resistance to thermal fluctuations.

FIGURE G1.3.5 (a) A diffraction contrast bright field (BF) TEM image shows tilted plane of RE2O3 precipitates and c-axis threading dislocations. (b) High-resolution TEM (HREM) image of a typical RE2O3 precipitate. (c) A higher magnification diffraction contrast BF image shows that the precipitate planes tilt at ~5° to the ab planes of REBCO. The inset is the HREM image showing a 2° tilt between the IBAD buffer and the REBCO ab planes.

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FIGURE G1.3.6 (a) Low magnification cross-sectional TEM image shows a uniform distribution of the precipitates through thickness. (b) At a higher magnification, it is observed that the precipitates are formed in planes separated by ~10 nm, as was originally designed. The YBCO/SrTiO3 interface area has a higher precipitate concentration than the matrix. (c) High-resolution TEM image shows a typical precipitate of ~10 nm wide and ~3 nm thick. The diffraction pattern derived from FFT shows the structural relationship between Y2O3 and YBCO.

G1.3.4 Nanorod Formation in REBa2Cu3Ox Thin Films G1.3.4.1 Nanostructural Features As the nanostructural analyses indicate, the pinning force should be significantly enhanced if the pinning precipitates with the diameter of ~2ξ continuously elongate along the field direction. Considering the geometrical anisotropy derived from the thin-film form and the fact that the crystallographic direction of REBCO normal to the tape plane is the c-axis along which Hc2 or Hirr is at least 5–6 times lower than along the ab plane, enhancement of pinning along such a direction is highly beneficial for practical uses of REBCO conductors in magnetic fields [25–27]. Indeed, the vortex pinning along the c-axis in REBCO CCs has been improved further by adding the BaZrO3 (BZO) nanorods [13, 16, 19, 26, 28]. By utilizing well-controlled deposition processes with BZO addition, BZO grows continuously along the REBCO c-axis so as to minimize the free energy associated with lattice mismatch against the REBCO matrix. Figure G1.3.7 represents the cross-sectional and planar view of the REBCO thin-film matrix with the high density of BZO nanorods. Technically the diffraction contrasts in TEM bright field or dark field images are derived from Bragg diffraction of lattice planes in the crystalline specimen whose local contrasts are significantly influenced by fluctuation of lattice plane tilt caused by local strain and/or 2nd phases with different crystal structures. Thus diffraction contrast imaging is highly beneficial to visualize the vertically aligned nanostructure such as REBCO with BZO nanorods. Compared with the sample that contains only RE2O3, strain field produced by BZO causes more complicated diffraction contrasts especially in the cross-section as seen in Figure G1.3.7(a) and (b). The BZO nanorods are quite

continuous (several hundred nm long and spaced by 20–30 nm) and provide correlated pinning centers along the normal to the film plane, thus along the c-axis. Interestingly, BZO addition improves the pinning not only along the c-axis, but also in the whole angular range at 4.2 K, indicating that BZO nanorods increase the density of random pinning [14, 29].

G1.3.4.2 Important Aspects of Atomic Structure The vertically aligned nanorod structure in the layer-by-layer perovskite matrix such as REBCO creates a unique atomic structure at the interface since the crystal structure is isotropic along the ab plane, whereas anisotropic along the c-axis. The atomic structure of BZO and the REBCO matrix is shown in Figure G1.3.8 and G1.3.9 of high-angle annular dark field STEM images taken in a Cs-corrected STEM. With spherical aberration corrected and the highly coherent field emission electron gun, the probe size of electron beam is now sub-angstrom. When the specimen is on the zone axis, we can acquire the images of each atomic column whose contrast is proportional to its Z2. They were taken under such a condition. Since BZO is semi-coherent to REBCO, the BZOREBCO interface always accompany with misfit dislocations. As denoted as red marks in the planar view and cross-section, there are two types of misfit dislocations at the BZO-REBCO interface. The one is a threading dislocation which runs along the BZO column as seen in the plan view HAADF-STEM of Figure G1.3.8 (left). The other is a loop dislocation which lies on the ab plane of REBCO as seen in Figure G1.3.9(a). These misfit dislocations produce complicated strain fields along both the ab plane and c-axis of REBCO. Figure G1.3.8 (left) and (center) and G1.3.9(b) show the map of local strain field along the ab-plane and c-axis direction of REBCO. The

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FIGURE G1.3.7 Low and high magnification TEM images of REBa 2Cu 3O7-δ thin film with BaZrO3 nanorods. (a) (b) and (c) (d) represent cross-sectional and planar views, respectively. Continuous growth of BZO along the REBCO c-axis results in the nanorods formation. The uniform distribution of BZO nanorods is achieved by the well-controlled deposition process.

FIGURE G1.3.8 (Left) Planar view high-angle annular dark field STEM (HAADF-STEM) image showing a BZO nanorod and the REBCO matrix. The locations of misfit dislocations are denoted as a red mark at the interface between BZO and REBCO. (Center and Right) Elastic strain εxx and εyy map of (Left), respectively.

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FIGURE G1.3.9 Cross-sectional HAADF-STEM image showing a BZO nanorod and the REBCO matrix. (a) The locations of misfit dislocations are indicated at two points on the dashed line marking the interface. (b) Elastic strain along the c-axis direction (εzz) is mapped based on the image (a). The white dotted line represents the REBCO-BZO interface.

geometric phase analysis (GPA) was used to calculate the strain field [30]. Three regions are defined based on the strain distribution: the YBCO matrix, the interface region and the BZO nanorod. The difference in εxx, εyy and εzz indicates the large elastic strain in the BZO nanorod, as observed in the FEM and XRD, results. The interface region contained large strain and misfit dislocations, as observed in Figure G1.3.3. A typical pattern of tensile and compressive strain along the c-axis was observed around a dislocation core in the εzz mapping. Basically the BZO is under tensile strain due to the lattice mismatch with REBCO. However, interestingly as is seen in Figure G1.3.9(b), the regions under tensile strain locally extend to the REBCO region near the interface. According to the first principle calculations, the REBCO becomes oxygen-deficient when it is under tensile strain [31]. It is considered that these complicated strain fields around the BZO nanorods create the local fluctuation of oxygen stoichiometry in REBCO that affects superconductivity such as Tc and produces the point defects which become dominant in flux pinning at the lowtemperature regime [29, 31].

G1.3.5 Flux Pinning Nanostructure in Fe-Based Superconductors The newly discovered Fe-based superconductors (FBS) exhibit intriguing physical properties despite of co-existing magnetism, such as high Tc, large upper critical field Hc2 and lower anisotropy compared with HTS cuprates [32–34]. These FBS’s complexity and superconducting mechanisms are of great interest not only in physics, but also in practical applications as a conductor. Similar to HTS cuprates, establishing the methodology of enhancing flux pinning is one of critical issues to make a viable conductor. FBS have already shown high intragrain Jc, irreversibility field Hirr close to Hc2 [35–38]. Especially

the studies of Co-doped BaFe2As2 (Co-Ba122) thin films have shown great possibility of improving Jc by introducing effective pinning centers [39–42]. It was demonstrated that vertically aligned, self-assembled BaFeO2 (BFO) nanorods could be introduced without suppressing Tc [42]. These nanorods act as strong c-axis correlated pins which enhance Jc (H//c) above Jc (H//ab), inverting the intrinsic material anisotropy because of the very high density of nanorod whose diameter is comparable to 2ξ (ξ being the superconducting coherence length). Ba122 appears to be unique compared with other FBS and HTS cuprates as it can accept a much higher density of 2nd phase as a pinning center than other HTS, for example, REBCO. Figure G1.3.10 represents such a flux pinning nanostructure in a Co-Ba122 thin film with 2nd phase nanorods and nanoparticles. The cross-sectional TEM image of Figure G1.3.10(a) shows the nanorods extended along the Ba122 c-axis as well as additional arrays of nanoparticle arranged along the ab planes with spacing of about 18 nm along the direction normal to the film plane. The higher magnification image [Figure G1.3.10(b)] shows that both nanoparticles and nanorods have a 4–5 nm diameter. The nanorods, which in other studies were continuous from the buffer layer to the top surface [49, 50] here show some discontinuity but still maintain long lengths. Their lattice is coherent to the Ba122 matrix in the planar view, so appear as a coffee bean-like contrast in the two-beam condition of TEM imaging. From the planar view of Figure G1.3.10(c) and (d), an average spacing is estimated as ∼12.5 nm for the randomly distributed thin nanorods, corresponding to a matching field Bφ of ∼13.2 T. The formation of nanoparticles is likely related to pre-existing oxygen that contributes to the formation of BFO secondary phase. The target material used in the thin-film deposition often includes pre-existing oxygen, and the volume fraction

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FIGURE G1.3.10 TEM images of Co-doped BaFe2As2 thin film with 2nd phase nanorods and nanoparticles. (a) (b) Cross-section images showing the nanorods aligned along the c-axis and the nanoparticles arranged along the ab plane of Ba122. (c) (d) Planar view reveals a high density of nanorods, corresponding to a matching field Bφ = 13.2 T.

of these 2nd phase nanorods and nanoparticles appeared to be related to how much oxygen is incorporated [43]. It has also been reported that superconducting properties such as Tc or Jc of thin-film FBS such as FeSe1-xTex, SmFeAs(O,F), NdFeAs(O,F) or Ba122 are generally improved when they are grown on CaF2 substrates, presumably because of FBS’s high sensitivity to strain [44–48]. Such Tc or Jc improvements are considered to relate to a compressive strain along the ab plane in the FBS matrix. In principle, there are two possible mechanisms by which the substrate produces such strain. The one is the strain correlated to a lattice mismatch between the substrate and film that, in case of Ba122 and CaF2, is ~2.7% [48]. The other possibility is thermal compression caused by the substrate during cooling from the high deposition temperature as a result of very different thermal expansion coefficient between the two materials [49]. Nevertheless, it is reported Tc of the Co-doped Ba122 films on CaF2 is enhanced by about 3 K with respect to similarly grown films on LSAT, and that, in the

multilayered films with more 2nd phase precipitation, Jc and the pinning force Fp increase above what is expected for the simple Tc enhancement, reaching a Fp maximum of 84 GN/m3 at 22.5 T and 4.2 K, and the pinning enhancement by precipitation is 2–3 times more effective than multilayer deposition on LSAT/STO [50]. Diffraction contrast TEM imaging is useful to investigate the pinning nanostructures that produced such a large pinning enhancement. The cross-sectional TEM image of Figure G1.3.11(a) reveals that the single-layer film on CaF2 has a remarkably clean Co-Ba122 layer: only threading dislocations that extend through the whole thickness and a thin transition layer at the interface between the Co-Ba122 layer and CaF 2 substrate are observed. On the other hand, as shown in Figure G1.3.11(b) and (c), the multilayered Co-Ba122 film on CaF 2 has short nanorods along the c-axis, nanoparticles some of which align along the ab-plane direction. The distribution is clearly not uniform through the thickness. In

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FIGURE G1.3.11 TEM cross-sections of Co-Ba122 thin films on CaF2. (a) Single-layer Co-Ba122 film with a very clean microstructure with only threading dislocations and few stacking faults. (b) Multilayer Co-Ba122 films with various pinning nanoprecipitates such as short c-axis nanorods, round nanoparticles some of which align along the Ba122 ab plane. (c) HREM image of some of these precipitates present in the multilayer film shown in (b).

fact, a notable decrease in the nanoparticle density is seen near the substrate [Figure G1.3.11(b)]. Another feature of the multilayered film is a thicker reaction layer at the Ba122/substrate interface compared with the single-layered film. It should be noted that the density of threading dislocation is larger in the films on CaF2 than those on other oxide substrates, indicating that these films of Figure G1.3.11 have larger compressive strain compared with those in Figure G1.3.10. Combined with EELS with inserting a slit to collect a certain range of energy, the intensity of EELS signals can be seen as a contrast in TEM imaging, which is called as energy-filtered TEM (EFTEM). For mapping the compositional fluctuation and distribution in the films, EFTEM was performed on the multilayered Co-Ba122 thin film of Figure G1.3.11, as shown in Figure G1.3.12. The Ca map of Figure G1.3.12 confirms that the Co-Ba122 layer is Ca-free, except for a transition layer. The Fe and As distributions are uniform from the substrate interface (represented as the white dotted line) to the top of the Co-Ba122 layer. Ba is also uniform in the Co-Ba122 layer, but there is a clear depletion at the reaction layer just above the interface. Figure G1.3.12 reveals the presence of oxygen in the reaction layer and the top surface of substrate; moreover bright spots of oxygen in the Co-Ba122 layer correspond to the location of precipitates, confirming that they contain oxygen. The elemental mapping suggests that the transition layer is mainly Fe-As-O with some slight inter-diffusion of Ba and Ca, and that the top part of the CaF2 substrate actually transformed into calcium oxide during deposition, as judged by clear segregation of oxygen on both sides of the interface.

G1.3.6 Grain Structures and Grain Boundaries Superconducting magnets require high Jc in long-length wire forms, which are inevitably polycrystalline [11]. In this case, the Jc across the grain boundary (GB) network, the so-called

intergrain Jc, determines the overall Jc of any long-length conductor, whereas the intragrain Jc defines the ultimate Jc that the conductor can develop. LTS materials such as Nb-Ti and Nb3Sn do not have GB Jc degradation because of their high carrier density and insensitivity to structural distortions at the GB. In fact, the small variation of superconducting properties that occurs at their GBs enables strong vortex pinning [51–53]. But all HTS, including FBS, have lower carrier densities that are controlled by the dopants and with their shorter coherence lengths ξ, superconductivity is much more easily degraded at GBs. Thus, careful evaluation and engineering of GBs are needed to avoid GB blocking of supercurrent. TEM and STEM have played the crucial role to investigate the GB structure of HTS materials. FBS have some important similarities to HTS cuprates. The layer-by-layer structure has FeAs layers sandwiched between carrier doping layers that yield an almost two-dimensional electronic structure of superconducting FeAs layers into which superconductivity is induced by the Ba/Sr/dopant layer, as does the Bi-O layers in Bi-2212 and Bi-2223 cuprates. However, there are still many more unknowns regarding the difference of GB properties between HTS cuprates and FBS, for example Ba122. Wetting FeAs impurity phase, impurity oxygen segregation that disrupts the local Fe-As layer at the GB, the complex multi-band superconductivity, the more beneficial smaller anisotropy of FBS all potentially play significant roles. After the careful synthesis studies [54], the FeAs wetting GB phase could be minimized, leading to an unexpected result in which untextured random polycrystalline K-Ba122 bulks and short round wires with high GB density show no electromagnetic granularity and have transport Jc well over 0.1 MA/cm 2 at 4.2 K and self-field, more than 10 times higher than that of any other random polycrystalline FBS [54]. Figure G1.3.13 shows a TEM image of the polycrystalline K-doped Ba122 bulk. The diffraction contrast in Figure G1.3.13(a) clearly shows that the average grain size

Transmission Electron Microscopy

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FIGURE G1.3.12 Elemental distribution map of Ca, As, O, Fe and Ba in the Co-doped Ba122 thin film of Figure G1.3.10. Elemental mapping was performed by using energy-filtered TEM (EFTEM) that utilizes electron energy loss spectroscopy (EELS).

FIGURE G1.3.13 Microstructures of K-doped Ba122 bulk investigated by TEM imaging. (a) TEM image of polycrystalline bulk K-doped Ba122, showing equiaxed grains with average grain diameter of 100–200 nm. (Inset) A selected-area electron diffraction pattern of (a) that indicates that the Ba122 grains are randomly oriented with many high-angle GBs. (b) HRTEM image of a typical K-doped Ba122 GB.

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is approximately 100–200 nm. The electron diffraction pattern from this area (inset of Figure G1.3.13) indicates a randomly oriented polycrystalline structure containing many high-angle GBs. The HREM observation confirms clean and well-connected GBs in a randomly oriented polycrystalline bulk, as shown by a typical GB in Figure G1.3.13(b). When Figure G1.3.13(b) was acquired, the TEM specimen was tilted so the electron beam was almost parallel to the GB plane so as to minimize the contour effect of the GB lying across the beam path. The lattice fringes of upper and bottom grains meet at the GB without an amorphous contrast, indicating that the GB is clean without a wetting impurity phase. However, TEM images do reveal some porosity and secondary phase that obstruct some current flow, indicating room for further process optimization. Nevertheless, the very fine grain size which is comparable to or smaller than the penetration depth, and the low Hc2 anisotropy that K-doped Ba122 possesses, provides a basis for high vortex stiffness. The enhanced phase purity at the GBs and low anisotropy seem to enable high Jc across GBs of randomly polycrystalline K-doped Ba122 bulks.

References 1. Gevers G, van Landuyt J and Amelinckx S 1978 Diffraction and imaging techniques in materials sciences (Amsterdam: North-Holland) 2. Spence J C H 1980 Experimental high resolution electron microscopy (Oxford Science Publications) 3. Cowley J M 1993 Electron diffraction techniques (Oxford: Oxford University Press) 4. Williams D B and Carter C B 2009 Transmission electron microscopy (Springer) 5. Morniroli J P and Steeds J W 1992 Microdiffraction as a tool for a crystal structure identification and determination Ultramicroscopy 45 21 6. Batson P E, Dellby N and Krivanek O L 2002 Subångstrom resolution using aberration corrected electron optics Nature 418 617 7. Krivanek O L, Dellby N and Lupini A R 1999 Towards sub-Å electron beams Ultramicroscopy 78 1 8. Krivanek O L et al. 2010 Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy Nature 464 571 9. Okunishi E, Sawada H, Kondo Y and Kersker M 2006 Atomic resolution elemental map of EELS and a Cs corrected STEM Micros. Microanal. 12.S02 1150 10. Freitag B, Kujawa S, Mul P M, Ringnalda J and Tiemeijer P C 2005 Breaking the spherical and chromatic aberration barrier in transmission electron microscopy Ultramicroscopy 102 209 11. Larbalestier D, Gurevich A, Feldmann D M and Polyanskii A 2001 High-Tc superconducting materials for electric power applications Nature 414 368

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12. Trociewitz U P et al. 2011 35.4 T field generated using a layer-wound superconducting coil made of (RE) Ba 2Cu3O7−x (RE = rare earth) coated conductor Appl. Phys. Lett. 99 202506 13. MacManus-Driscoll J L et al. 2008 Strain control and spontaneous phase ordering in vertical nanocomposite heteroepitaxial thin films Nat. Mater. 7 314 14. Wimbush S C et al. 2009 Interfacial strain-induced oxygen disorder as the cause of enhanced critical current density in superconducting thin films Adv. Funct. Mater. 19 835 15. Llordés A et al. 2012 Nanoscale strain-induced pair suppression as a vortex-pinning mechanisms in high temperature superconductors Nat. Mater. 11 329 16. Xu A, Braccini V, Jaroszynski J, Xin Y and Larbalestier D C 2012 Role of weak uncorrelated pinning introduced by BaZrO3 nanorods at low-temperature in (Y,Gd) Ba 2Cu3Ox thin films Phys. Rev. B 86 115416 17. Civale L et al. 2004 Understanding high critical currents in YBa 2Cu3O7 thin films and coated conductors J. Low Temp. Phys. 135 87 18. Puig T et al. 2008 Vortex pinning in chemical solution nanostructured YBCO films Supercond. Sci. Technol. 21 034008 19. Maiorov B et al. 2009 Synergetic combination of different types of defect to optimize pinning landscape using BaZrO3-doped YBa 2Cu3O7 Nat. Mater. 8 398 20. Miura M et al. Mixed pinning landscape in nanoparticle-introduced YGdBa 2Cu3Oy films grown by metal organic deposition Phys. Rev. B 83 184519 21. Chen Z, Kametani F, Kim S I, Larbalestier, Jang H W, Choi K J and Eom C B 2007 Influence of growth temperature on the vortex pinning properties of pulsed laser deposited YBa 2Cu3O7-x thin films J. Appl. Phys. 103 043913 22. Chen Z, Kametani F, Chen Y, Xie Y, Selvamanickam V and Larbalestier D C 2009 A high critical current density MOCVD coated conductor with strong vortex pinning centers suitable for very high field use Supercond. Sci. Technol. 22 055013 23. Haugan T, Barnes P N, Wheeler R, Meisenkothen F and Sumption M 2004 Addition of nanoparticle dispersions to enhance flux pinning of the YBa 2Cu3O7-x superconductor Nature 430 867 24. Chen Z, Kametani F, Gurevich A and Larbalestier D 2008 Pinning, thermally activated depinning and their importance for turning nanoprecipitate size and density in high Jc YBa 2Cu3O7-x films Physica C 469 2021 25. Kang S et al. 2006 High-performance high-Tc superconducting wires Science 311 5769 26. Goyal A et al. 2005 Irradiation-free, columnar defects comprised of self-assembled nanodots and nanorods resulting in strongly enhanced flux-pinning in YBa 2Cu3O7−δ films Supercond. Sci. Technol. 18 1533

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27. Yamada Y et al. 2005 Epitaxial nanostructure and defects effective for pinning in Y(RE)Ba 2Cu3O7−x coated conductors Appl. Phys. Lett. 87 132502 28. Xu A et al. 2014 Strongly enhanced vortex pinning from 4 to 77 K in magnetic fields up to 31 T in 15 mol.% Zr-added (Gd, Y)-Ba-Cu-O superconducting tapes APL Mater. 2 046111 29. Xu A et al. 2010 Angular dependence of Jc for YBCO coated conductors at low temperature and very high magnetic fields Supercond. Sci. Technol. 23 014003 30. Hytch M J, Putaux J L and Penisson J M 2003 Measurement of the displacement field around dislocations to 0.03A by electron microscopy Nature 423 270 31. Horide T, Kametani F, Yoshioka S, Kitamura T and Matsumoto K 2017 Structural evolution induced by interfacial lattice mismatch in self-organized YBa2Cu3O7−δ nanocomposite film ASC Nano 11 1780 32. Kamihara Y, Watanabe T, Hirano M and Hosono H 2008 Iron-based layered superconductor La[O1-x Fx] FeAs (x = 0.05−0.12) with Tc = 26 K J. Am. Chem. Soc. 130 3296 33. Chen X H, Wu T, Wu G, Liu R H, Chen H and Fang D F 2008 Superconductivity at 43 K in SmFeAsO1-xFx Nature 453 761 34. Tarantini C et al. 2011 Significant enhancement of upper critical fields by doping and strain in iron-based superconductors Phys. Rev. B 84 184522 35. Yuan H Q et al. 2009 Nearly isotropic superconductivity in (Ba,K)Fe2As2 Nature 457 565 36. Yamamoto A et al. 2009 Small anisotropy, weak thermal fluctuations, and high field superconductivity in Co-doped iron pnictide Ba(Fe1-xCox)2As2 Appl. Phys. Lett. 94 062511 37. Yamamoto A et al. 2008 Evidence for electromagnetic granularity in the polycrystalline iron-based superconductor LaO0.89F0.11FeAs Appl. Phys. Lett. 92 252501 38. Yamamoto A et al. 2008 Evidence for two distinct scales of current flow in polycrystalline Sm and Nd iron oxypnictides Supercond. Sci. Technol. 21 095008 39. Lee S et al. 2010 Template engineering of Co-doped BaFe2As2 single-crystal thin films Nat. Mater. 9 397 40. Miura M et al. 2013 Strongly enhanced flux pinning in one-step deposition of BaFe2(As0.66P0.33)2 superconductor

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51. 52.

53.

54.

films with uniformly dispersed BaZrO3 nanoparticles Nat. Commun. 4 2499 Iida K et al. 2009 Strong Tc dependence for strained epitaxial Ba(Fe1-xCox)2As2 thin films Appl. Phys. Lett. 95 192501 Tarantini C et al. 2012 Artificial and self-assembled vortex-pinning centers in superconducting Ba(Fe1−xCox)2As2 thin films as a route to obtaining very high critical-current densities Phys. Rev. B 86 214504 Zhang Y et al. 2011 Self-assembled oxide nanopillars in epitaxial BaFe2As2 thin films for vortex pinning Appl. Phys. Lett. 98 042509 Hanawa M et al. 2012 Empirical selection rule of substrate materials for iron chalcogenide superconducting thin films. Jpn. J. Appl. Phys. 51 010104 Ueda S et al. 2011 High-Tc and high-Jc SmFeAs(O,F) films on fluoride substrates grown by molecular beam epitaxy Appl. Phys. Lett. 99 232505 Iida K et al. 2013 Oxypnictide SmFeAs(O,F) superconductor: a candidate for high–field magnet applications Sci. Rep. 3 2139 Uemura H et al. 2012 Substrate dependence of the superconducting properties of NdFeAs(O,F) thin films Solid State Commun. 152 735 Kurth F et al. 2013 Versatile fluoride substrates for Fe-based superconducting thin films Appl. Phys. Lett. 102 142601 Lei Q Y et al. 2014 Structural and transport properties of epitaxial Ba(Fe1-xCox)2As2 thin films on various substrates Supercond. Sci. Technol. 27 115010 Tarantini C et al. 2014 Development of very high Jc in Ba(Fe1-xCox)2As2 thin films grown on CaF2 Sci. Rep. 4 7305 Scanlan R M et al. 1975 Flux pinning centers in superconducting Nb3Sn J. Appl. Phys. 64 2244 Suenaga M and Jansen W 1983 Chemical composition at and near the grain boundaries in bronzeprocessed superconducting Nb3Sn Appl. Phys. Lett. 43 791 Dew-Hughes D 1987 The role of grain boundaries in determining Jc in high-field high-current superconductors Phil. Mag. B 55 459 Weiss J D et al. 2012 High intergrain critical current density in fine-grain (Ba0.6K0.4)Fe2As2 wires and bulks Nat. Mat. 11 682

G1.4 An Introduction to Digital Image Analysis of Superconductors Charlie Sanabria and Peter J. Lee

G1.4.1 Introduction: The Link between Superconductivity and Microscopy Because the dimensions that are critical to superconducting properties range from the atomic-scale to the macro-scale, being able to evaluate structures quantitatively across this wide range is essential for the understanding of superconductors. This quantification allows us to understand fundamental material properties and provides insight into how they can be improved. Technological advances in digital image acquisition, resolution, and analysis are continually making these microscopy techniques more powerful and accessible. In this chapter, we will demonstrate a wide range of these techniques by showing examples of their application to practical superconducting materials.

G1.4.1.1 The Power of Observation In superconductivity, just like in almost every branch of science, observation is essential to the understanding of the principles that govern phenomena. Many advanced techniques used nowadays for the characterization of superconductors— such as magnetization measurements [1], x-ray diffraction [2], and heat capacity measurements [3]—have very little impact without the visual support of micrographs taken from the features behind the signal obtained. Furthermore, aside from this qualitative visual support, the quantification of these images through image analysis (IA) can provide measurements that are necessary for an accurate interpretation of the results obtained from other characterization techniques.

available software—replacing earlier techniques such as planimetry and cut-and-weigha. There are many software packages available today. Many of these software are native to the microscope manufacturers with proprietary aspects (as well as financial costs), so for the purposes of this chapter we will use examples from one of the most widely used IA software: ImageJ, which is both multi-platform and open-source. ImageJ was originally developed by Wayne Rasband [4] for the National Institutes of Health and therefore used to carry the name of NIH Image. This tool has evolved into the powerful program that is known as ImageJ or Fijib today. Rasband was a visionary; he knew that as long as NIH Image had a robust (yet simple) base code and interface, he could have it preform complex tasks through additional features in the form of plugins and scripts (macros) which the users could implement and develop themselves. Today, the ever-expanding ImageJ/Fiji [5] package is maintained by numerous contributors around the world and has a vast online community of users constantly helping each other achieve better results. Although most of this work is in the biological fields, where images can be very difficult to quantify routinely, the advanced user-written plugins, scripts, and macros are just as useful for material science and metallography.

G1.4.1.3 Sample Preparation Is Essential Images of superconductors are most commonly taken of polished cross-sections. Which implies that a metallographic samplec must be prepared before acquiring the images. However, this sample preparation can affect the outcome in many ways: it can slightly alter shapes, it can introduce a

G1.4.1.2 Image Analysis Software The qualitative information in an image (taken by a microscope, a telescope, or any image-capturing device) can now be easily transformed into quantitative digital data using widely 46

b

c

Where photographs were literally cut into areas for analysis and weighed. Fiji is a standardized packaging of ImageJ that includes many useful add-ons that can be automatically updated. A metallographic sample is used for light microscopy, atomic force microscopy, or scanning electron microscopy, while an electron-transparent foil is needed for transmission electron microscopy.

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FIGURE G1.4.1 Light microscope images of the same region of an unreacted Internal-Tin Nb3Sn strand showing Nb rods inside a Cu matrix after the application of good (a) and bad (b) polishing procedures. The sample in (b) was polished using excessive force. (c) and (d) show the apparent isolated rod cross-sections from (a) and (b), respectively; the dimensional difference quantified in (d) is an artificial effect obtained from a poor metallographic preparation.

foreign particles to the sample, or it can hide features within the sample. A few examples of features that can be missed (or misinterpreted) if the sample preparation is not optimal are the presence and size of voids, boundary phases, and secondary phases, among others. An example of this is shown in Figure G1.4.1, where the same features are shown under two different sample preparation techniques. A poor polishing technique in the right-hand image reveals what appear as larger features that might lead to erroneous conclusions. Aside from dimensional artefacts, sample preparation can introduce many artificial features that may or may not affect the IA or the interpretation of the image. Given that the samples are normally prepared through a series of material removal steps using abrasive media (sand paper, diamond solutions, silica solutions, etc.), minor scratches are the most common artefacts in metallographically polished samples. These scratches are often times purely aesthetic, and may be ignored if they do not influence the analysis—as is the case of Figure G1.4.2. However, in some cases they can drastically alter the interpretation of the results. An example of a misleading micrograph can be seen in Figure G1.4.3 where a straight line of cold-worked copper grains is seen across the sample. This scratch was most likely produced by a large abrasive particle carried over from a coarse grinding step, but its damage can still be seen thanks to the micro-misorientations picked up by the backscattered electron detector (more on BSE images later). It should be noted that such a scratch might have

been overlooked if a light micrograph had been taken (since dislocations are not detected by light microscopy without specialized etching techniques), and therefore the false positive filament fractures observed would have led to a very different conclusion. Soft components in particular can lead to potentially incorrect conclusions since they are easily smeared across other

FIGURE G1.4.2 An image of sub-elements in a fully reacted RRP® Nb3Sn strand with a minor scratch produced during the final polishing stages or sample cleaning.

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FIGURE G1.4.3 (a) A longitudinal cross-section of an Nb3Sn ITER strand (after cyclic electromagnetic testing in the SULTAN facility) imaged using a backscattered electron detector in a Field Emission Scanning Electron Microscope (FESEM). Cold work introduced by a hard particle during polishing is evident from the strain contrast. These samples are used to assess the damage caused by cyclic strains. Without the evidence of the scratch, false positive filament fractures shown in (b) could be misinterpreted as being due to electromagnetic testing. The light micrograph in (c) shows the same region where the buried scratch is not detected.

(harder) components during sample preparation. Such is the case of Bi-2212 strands, which are very difficult to prepare for metallographic examination as the interaction with water must be avoided and the Ag matrix of these composites is very soft. Figure G1.4.4 shows the same Bi-2212 strand polished by two different techniques. Notice that both images are virtually scratch-free, and appear to be of high quality, but the amount of bubbles (voids, in black) is drastically different. Another way of imaging superconducting materials, without metallographic preparation, is what has come to be known as fractography. This technique is most effective on cross-sections of brittle materials, and has proven to be a useful tool to analyze microstructures in Nb3Sn. However, it should be noted that the fracture surfaces may not be planar. In Figure G1.4.5 we compare a fracture surface of a filament of an Nb3Sn strand (left) with a 3D rendering of that filament (right). From the 3D rendering it is clear that the fracture cannot be correctly interpreted as a planar cross-section. The rendering was made possible by taking an additional “stereo-pair” image of the same area at a tilt (typically ~7°). Such stereoscopic images are recommended for all cases of fractured surfaces. An additional feature of stereoscopic images and their analysis is that the geometric

shifts—that occur because of height differences when a sample is tilted—can be converted to quantitative topographic information using a variety of commercial programs that can obtain good accuracy [6].

G1.4.2 Image Acquisition Since the most common and accessible microscopy techniques are light microscopy (LM) and scanning electron microscopy (SEM), these two techniques will be the primary focus of this section. Nonetheless, most of the techniques shown here can be applied to images acquired by other widely used microscopes such as atomic force microscopy (AFM) and transmission electron microscopy (TEM). However, it should be mentioned that for TEM (being a projection through a foil of finite thickness), quantification is made more complex by the contrast gradients produced by foil thickness variations, diffraction contrast from microstructural defects, and deconvolution from the volumetric nature of TEM images. Nevertheless, quantification of TEM images has been crucial to the development of our understanding of superconductor properties [7–9]. In the case of AFM, since the images are produced using quantitative information from the force response of the instrument [10], there is no need to re-quantify

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FIGURE G1.4.4 Images of the same Bi-2212 strand (fabricated by OST, Now Bruker) showing significantly different contents of voids (or bubbles) depending on the polishing technique. The image in (a) is a properly prepared sample, and the image in (b) is a sample prepared using excessive force and short polishing times.

FIGURE G1.4.5 A fractograph (in-lens secondary electron image) of a filament from an Nb3Sn strand (left) compared with a 3D rendering of the same filament showing the non-planar nature of the fracture.

the images, and it is preferred to use other methods of data manipulation to process the information [11, 12].

G1.4.2.1 Light Microscopy Some advantages of light microscopy are low cost, color contrast, ease of use, and availability. However, the wavelength of visible lightd limits the spatial resolution for non-fluorescing materials to ~200 nm with a good quality objective lens d

Between 400 nm and 700 nm for violet and red, respectively.

(~120 nm for a Scanning Laser Confocal Microscope) compared to Field Emission Scanning Electron Microscopes (FESEMs), which routinely achieve spatial resolutions of ~1 nm. Below, we will expand on both the limitations and advantages of light microscopy. G1.4.2.1.1 Depth of Field Considering only conventional light microscopy, another challenge that arises is the shallow depth of field (DOF) of light microscopes. The DOF represents the range of topographic heights over which the aperture and lens combination

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FIGURE G1.4.6 In this focus stack example, 14 images of a grain boundary in a chemically polished Nb sample have been combined to produce both a fully focused output image and 3D rendering of the surface.

can produce a fully focused image—and this depth of field decreases as the magnification increases. Reducing the objective aperture diameter increases the depth of field in any optical instrument, but in light microscopy this is often insufficient to overcome specimen height variations—especially for large specimens and samples with significant topology. However, digital image processing provides techniques that can expand the in-focus range. G1.4.2.1.2 Focus Stacking and Scanning Laser Confocal Microscopy Unfortunately, during metallographic preparation, different phases and components in superconducting samples are often polished at different rates, producing a topography which can make certain features fall outside the DOF. Furthermore, unpolished samples such as fracture surfaces often have uneven topography (see Figure G1.4.5), and polished crosssections of large samples will most likely have height variations across them. All these circumstances, when uncorrected, result in poorly focused regions that will introduce errors to the analysis. One software solution to the inherent shallow DOF in light microscopes that can be applied to any light microscope is “focus stacking”. Focus stacking uses a series of images of the same area at a range of working distances (the distance from the objective lens to the sample surface) which covers the entire range of sample topography. It then combines the images using software algorithms that identify the focused areas from each stacked image providing a relatively sharp image across the entire sample. A bonus is that if the objective focus height is known, the height at which each pixel is in optimum focus is also known, and thus a three-dimensional image map can be obtained. Thus, the inherent depth of field disadvantage of light microscopy can be exploited to produce height [13], surface roughness [14], and volumetric

measurements. An example of this is shown in Figure G1.4.6, using an extended depth of field plug-in for ImageJ [15] and rendered using Gwyddion [16] (a free and open-source modular software package designed for 3D surfaces). The accuracy of this technique is subject to the ability of the software to reliably identify the best-focused planes and noise and artefacts are typical. A more accurate method is laser scanning confocal microscopy (LSCM). In this case, the infocus height is accurately identified by the strongest reflected laser beam intensity passed through the confocal optics to the detector. The surface height measurements obtained by this technique are sufficiently accurate to provide a depth resolution down to ~10 nm—and when combined with a digitally controlled stage, it can assemble fully focused composite images that span large samples regardless of sample topography—such as the large micrographs taken of ITER CICCs in [13]. Unfortunately, metallurgical LSCMs typically cost 5 to 10 times that of conventional light microscopes. Figure G1.4.7 shows examples with poorly focused areas of samples due to uneven sample surface height. Figures G1.4.8, G1.4.9, and G1.4.10 show focus improvements using different methods. Note that because the LSCM image is monochromatic, it may miss useful color contrast. This is particularly evident comparing Figure G1.4.9(c) (white LED illumination) with Figure G1.4.10(c) (obtained with a 408-nm wavelength violet laser). The availability, ease of use, and low cost of light microscopes is the main reason why this technique is still widely used today. In addition, there are many effects and filterse one can use to make features more discernable, making light microscopy very powerful even in metallography. However, most IA of metallographic samples benefit greatly from SEM e

Among these, we have, light polarization, oblique illumination, differential interference contrast (DIC), etc.

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FIGURE G1.4.7 Images with poorly focused areas due to (a) the slanted orientation of a penny, (b) etched off silver on a Bi-2212 strand (produced by OST, now Bruker), and (c) the slanted sample of Nb rods inside a Cu matrix (precursors of RRP® strands).

imaging, due to much greater available spatial resolution, depth of field, and responsiveness to IA algorithms. There are, however, two situations in addition to cost where light microscopy has an advantage over SEM for digital image

analysis: Firstly, as it will be discussed below, an SEM image can have a certain degree of drift and distortion, which can make their measurements inaccurate if not properly controlled. This is where the fixed optics, constant illumination

FIGURE G1.4.8 Images of the same area as those in Figure G1.4.7 corrected for focusing using an open-source ‘focus stack’ algorithm (a) the slanted orientation of a penny, (b) etched off silver on a Bi-2212 strand (produced by OST, now Bruker), and (c) the slanted sample of Nb rods inside a Cu matrix (precursors of RRP strands).

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FIGURE G1.4.9 Images of the same area as those in Figure G1.4.7 corrected for focusing using OLYMPUS Stream Motion® (a) the slanted orientation of a penny, (b) etched off silver on a Bi-2212 strand (produced by OST, now Bruker), and (c) the slanted sample of Nb rods inside a Cu matrix (precursors of RRP strands).

FIGURE G1.4.10 Images of the same area as those in Figure G1.4.7 corrected for focusing using an Olympus Laser Scanning Confocal Microscope. Note that for (c) the monochromatic blue/violet laser image does not provide as good contrast for the multifilamentary superconductor as the full spectrum (white LED) light microscope [(Figure G1.4.9(c)].

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FIGURE G1.4.11 (a) A backscatter electron image of an artificial pinning center in a Nb–Ti strand at an intermediate stage in wire drawing. This image is compressed and distorted (by upwards and lateral movements) during acquisition. (b) A drift corrected version of the same image using a fast-scan image underlay (for reference) shows that the pin is slightly less aspected than suggested by the original image.

intensity, and low optical distortion of modern light microscopes makes it robust and cost-effective solution for accurate measurements of very large samples by combining multiple images into mosaics [17]. Secondly, as noted above, color contrast can be useful to detect different phases and compositions, for instance, the color of the Cu–Sn α phase is very sensitive to Sn composition and can be used to identify local Sn compositions at spatial resolutions beyond EDS [18].

G1.4.2.2 Scanning Electron Microscopy Scanning electron microscopy (SEM) has played a crucial role in the development of superconductors given the broad range of the magnifications that it can cover and the wide variety of detectors providing topographic, chemical, and crystallographic data. The principles of the technique will not be discussed here, but we will touch on some of the most relevant considerations when acquiring images of metals, in particular superconducting strands. G1.4.2.2.1 Noise, Drift, and Distortion Electron beam deflection is a well-known issue for SEM imaging [19–21], it happens because electrons coming from the irradiating beam can locally charge the sample surface if the sample surface has low conductivity. Such negatively charged areas can deflect the beam itself and introduce noise, drift, or (in some cases) abrupt distortions of the sample. Therefore, it is crucial for the sample surface to have a low-resistance path to ground in order to reduce these effects. Effective grounding can be achieved with the use of conducting mounting materials, conductive tapes or paints providing a path to ground, and/or sample coating (typically with thin layers of carbon or gold). G1.4.2.2.1.1 Noise and Poor Signal There are many ways to increase signal and reduce image noise. The most obvious ones are inherent to the microscope

itself such as electron source, image acquisition settings, beam settings, vacuum state, and overall cleanliness of the vacuum chamber and parts. However, one must keep in mind that the sample preparation can introduce noise as well. Thin native oxide surface layers (usually undetectable by LM or SEM) can increase the noise of the image by micro-charging. For this reason, a freshly polished or ion-etched sample is recommended for optimal signal. This is particularly important for the Kikuchi band diffraction images used for crystallographic orientation mapping. G1.4.2.2.1.2 Drift Low-noise SEM images suitable for image analysis are typically acquired by averaging multiple frames (frame-averaging) or lines (line-average), or by increasing the raster time. The latter (increasing raster time) can be obtained by increasing the dwell time and/or by averaging pixels. However, all these techniques increase acquisition time, and significant sample movement may occur during acquisition. At high magnification, this can result in blurring of the image (for frame-averaging) or distortion of the image (line- or pixel-averaging) because each serially acquired frame, pixel, or line is sequentially offset by the drift that occurs while each frame, pixe,l or line is obtained. The resulting imaged area from a rectangular scan is a parallelogramf despite the final image appearing to be a rectangle. Figure G1.4.11 shows an image where drift has compressed and distorted the shape of an artificial Nb pinning center (APC) in a Nb–Ti strand sampled during processing to final size [22]. If any detailed measurement is to be extracted from images with detectable drift, a drift correction is recommended. This can be achieved by acquiring a fast-scan (sufficiently fast for insignificant drift to occur) of the same area with enough detail to discern the most representative features. The long-scan (desired image) can then be corrected for drift by aligning it f

A parallelogram for constant drift, or a trapezoid for changing drift.

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G1.4.2.2.2 Secondary Electrons Images

FIGURE G1.4.12 SEM image showing a sudden electron discharge affecting the image scan. Smaller discharges can be harder to identify unless compared with a fast-scan frame.

with the fast-scan image. For best results, the fast-scan may be taken at a lower pixel lower resolution than the long-scan and scaled up in size prior to image matching—since this does not affect the drift correction, but it does make the fast-scan even faster. An alternative technique is to acquire multiple images of the same area using short-scan times to minimize drift and then align and average each frame externally. An advantage of averaging aligned frames is that the short dwell times result in less micro-charging. Drift correction may also be included in the microscope acquisition software. G1.4.2.2.1.3 Electron Discharge Distortion If there is a sufficient build-up of trapped electrons, a sudden discharge can occur when a new grounding path is suddenly found. This discharge will typically offset the scan momentarily resulting in an abrupt distortion. Figure G1.4.12 shows how electron discharge can severely affect the continuity of an image. Other more subtle scan discontinuities due to charging can also be detected using the fast-scan overlay technique (explained above for drift correction) given that the charge build-up should be different for the fast-scan. Sample charging needs to be avoided if accurate image analysis is to be performed.

FIGURE G1.4.13

Modern scanning electron microscopes often have more than one detector providing complementary information. Of these detectors, the secondary electron detector has been the most commonly used historically. Because secondary electrons have very low energies, they only escape from the top atomic layers, and thus the resolution of secondary electron images is less degraded by internal scattering within the sample. However, this also means that the images are sensitive to surface contamination. Such surface contamination may not be obvious in backscattered electron images (BSE); therefore, it is very desirable to acquire SE images even if the relevant information will be extracted from the backscattered electron image (see Section G1.4.4.2). Secondary electron (SE) images are also highly sensitive to surface topography, which makes them very useful for qualitative evaluation of features that are sensitive to polishing and etching conditions—or for fractography. G1.4.2.2.3 Backscattered Electron Images Backscattered electrons do not lose significant energy from the incident beam and thus are emitted from a much greater depth range than the low-energy electrons acquired for SE images. They are also much less sensitive to surface topology and surface contamination. However, what makes backscattered electron images most desirable is the fact that that their intensity is proportional to the average atomic number (Z) of the volume under the electron probe. This Z-contrast, combined with the reduced topographic sensitivity makes it typically much easier to obtain quantitative data from backscattered electron images than SE images as will be shown in Section G1.4.4.2. Figure G1.4.13 shows SE and BSE images of the same sample. In the SE image, one can appreciate the different rates of material removal during polishing—where the hardest materials, Ta and Nb3Sn, seem to form higher plateaus while softer materials like Cu and bronze appear like valleys in between. The BSE image, on the other hand, has no apparent topography,

Transverse cross-sectional images of an Internal Tin Nb3Sn strand designed for ITER. (a) SE image and (b) BSE image.

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FIGURE G1.4.14 (a) and (b) Backscattered electron images of the filamentary area of an Nb3Sn ITER strand before reaction taken at a working distance of (a) 9 mm with good phase contrast and another one with a working distance of (b) 5 mm showing higher grain contrast. Similar cases in (c) and (d) of an Nb3Sn layer of an RRP® strand.

but an outstanding contrast for the different atomic numbers ranging (from light to dark) between Ta and Cu. Voids are obviously depicted as absence of signal (i.e. black). Both images provide complimentary information and both should be acquired, particularly as the surface-sensitive SE images clearly show regions of surface contamination that can introduce errors in image analysis of the BSE. G1.4.2.2.3.1 Crystallographic Orientation Contrast versus Phase Contrast Because BSE detectors are typically annular detectors (sometimes divided into quadrants) with the initial beam passing through the hole in the center of the detector, the distance between the sample and the detector determines the range of scattering angles accepted by the detector. This phenomenon can be exploited to obtain better grain contrast by adjusting the working distanceg (WD). A large WD allows only those

electrons whose backscattering direction is almost parallel to the incident beam, while a shorter WD allows for higherangle electrons to reach the detector. High-angle scattering favors crystallographic orientation backscatter, while low-angle backscattering is mostly Z-contrast related, thus two distinct contrast origins can be favored by the selection of the WD using the same detector. Some backscatter detectors use two concentric annular rings so that images from both large and small angles of scatter can be collected simultaneously. As shown in the examples in Figure G1.4.14, using a long WD can produce an image that is dominated by Z-contrast, where a short WD provides better contrast of grains with different orientations. Short working distance BSE images can also be used to identify areas of local cold-work, which introduces micro-misorientations within grains as shown in Figure G1.4.3.

G1.4.3 Image Properties g

WD is the distance between the objective lens and the sample. Since the distance between the BSE detector and the objective lens is fixed, reducing the WD shortens the distance between the sample and the detector.

Before analyzing the image acquired, we must understand how an image works, its components, and properties. A digital image is discretized in small squares or rectangles called

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FIGURE G1.4.15

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(a) A 221-pixel image showing some gray values and (b) the histogram of the image in 8-bit space (256 levels of gray).

picture elements or pixelsh. Somewhat confusingly, a pixel can describe a length as well as an area, depending on the context. We will assume here that a pixel has an area of one square pixel. Therefore, image resolution can be described in two ways, either by the total number of horizontal and vertical pixels, or by the number of total square pixels. That is, an image of 3072 × 2304 pixels has a total of 7 megapixels or 7 × 106 pixels2—or, because of the above-mentioned definition of pixels, 7 × 106 pixels.

G1.4.3.1 Intensity Values and Histograms Aside from having a dimension of one pixel by one pixel, a pixel has an intensity value, either a single value for B&W (grayscale) images, or combination of three values (blue, red, and green) in the case of color images. The most important properties (regarding digital images) we need to keep in mind for the rest of this chapter are (1) that pixels have an intensity value and (2) that pixels can be arranged in a histogram. These two properties are described below. G1.4.3.1.1 Intensity Values and Bit Depth When talking about digital images one often talks about bit depth, that is, the range of intensity values which a single pixel can have using a binary system of information. In 8-bit space, each pixel has a value between black (with a value of zero) and white (with a value of 255). In other words, each pixel can have one of 256 (that is 2 8, hence 8-bit) levels of gray (intensity). In an RGB color image, each pixel has three values (red, green, and blue), therefore the image is said to be 24-bit (8 times 3). However, in a 16-bit image, a pixel has the option of having any of 216 levels of gray; that is, an astonishing, 65,536 options where 0 is black and 65,535 is white. Although 256 intensity levels are sufficient for the human eye’s contrast sensitivity, digital analysis is not limited to this range, and modern electron microscope h

In 3-D images such as MIR scans the most basic unit is called voxel after “volume element”.

detectors can distinguish intensity level variations well beyond 256 levels. G1.4.3.1.2 Histograms A histogram is another way to describe an image in terms of the number of pixels and intensity values. In its most basic definition, a histogram is a frequency chart where the vertical axis represents the number of pixels and the horizontal axis represents the measured values. The histogram of a very small picture (17 by 13 pixels) is shown in Figure G1.4.15, notice that there are only 256 available intensity values for this 8-bit image, also that the picture is dominated by lighter pixels and therefore the histogram is more weighted to the right-hand side (higher tone values). Figure G1.4.16 shows a much larger (and richer) image in 16-bit space and its histogram. One can intuitively see that this backscattered electron image can be separated into three different components noted by the three peaks in the histogram (or four, if the zero values are taken into account).

G1.4.4 Image Analysis Having understood the properties of an image, we can now move on to the analysis of such images using open-source algorithms.

G1.4.4.1 Feature Recognition We will mostly concentrate on digital image analysis based on algorithms which solely use the histogram shape and pixel position—without manipulating or moving the pixelsi—to isolate features and separate them into slices. These slices are then transformed into binary images in which the individual components are represented by black pixels with everything else being white. Figure G1.4.17 shows an image that has been separated into binary slices of each component according to their Z-contrast as seen in the histogram. i

Algorithms that manipulate pixel position and values are more associated with photography rather than IA

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FIGURE G1.4.16 (a) A backscattered electron image of the filamentary area of Nb3Sn ITER strand taken in 16-bit space showing (b) the histogram of the image.

G1.4.4.2 Quantitative Analysis by Image Intensity As described in Section G1.4.3, a pixel is a unit of area. This means that an image (i.e. an array of pixels) represents an area as well. And just like any two-dimensional plane, one can perform simple measurements such as the distance between two points or the area of a particular section of such plane. Other common measurements are shape description parameters such as perimeter, circularity, aspect ratio, center of mass, skewness, and many more. Obtaining these measurements is the essence of IA. However, before doing so, the different

features of the image have to be separated (thresholded) adequately from the original image into binary slices or layers like those in Figure G1.4.17. Additionally, non-invasive algorithms are often applied to the binary images to best extract the measurements. Let us explore these processes in detail. G1.4.4.2.1 Thresholding A threshold by definition is a point at which a quantity stops (or starts) meeting a criterion. In IA we want to set these criteria so that they include a particular feature that is typically associated with a Gaussian distribution of intensities around

FIGURE G1.4.17 (a) SEM image of an Nb3Sn ITER strand and its (b) histogram showing different regions of gray values which can be easily separated into black objects representing (c) voids, (d) bronze and Cu, (e) Niobium, and (f) Nb3Sn.

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FIGURE G1.4.18

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Histograms of (a) an SE image and (b) a BSE image of the same Nb3Sn strand cross-sectional area.

a peak in the histogram. In other words, one uses the histogram to select the pixels that belong to a particular component (peak) as it was demonstrated in Figure G1.4.17. However, histograms can be complicated, and the valleys between peaks are often not very well defined. Fortunately, there are many different ways to separate these peaks, for instance, there are 16 different thresholding algorithms in Fiji which use sophisticated statistical principles based on the histogram shape, relative heights, entropy, position, etc. [23]. Selecting the right algorithm can result in a very accurate and reproducible selection of the threshold pointj. However, depending on the type and quality of the image, the use of threshold algorithms can be inaccurate or misleading. Therefore, it is important to start with an image (or image area) that facilitates a reproducible and sensible selection. For better IA, the digital image should have a histogram with easily discernable peaks. For example, Figure G1.4.18 shows histograms of two images from Figure G1.4.13 (an SE image and a BSE image), notice that identifying the different components is much easier for the BSE image while the Nb and Cu peaks are indiscernible in the SE image. For this reason, BSE images preferred over SE images for IA of superconducting materials. However, not all BSE imaging conditions are optimal; technical superconductors are typically composites of different materials often containing multiple phases with a variety of Z-contrast ranges which may overlap. In addition, these peaks can broaden and result in overlap if a short working distance is chosen (to improve grain orientation contrast in BSE imaging as discussed earlier in Section G1.4.2.2.3.1), or if the image is acquired with insufficient spatial resolution, poorly focused, or with too much noise—affecting our ability to separate them as illustrated in the histograms of Figure G1.4.19. Despite there being plenty of algorithms that work in different situations, the easiest and most reproducible thresholding will always be that in which only two peaks are involved. To get to this advantageous situation, one can make a separate image containing only the boundary between the two j

The word reproducible in this sentence is essential.

relevant components, and perform the threshold on that particular section of the image—as done in the example in Figure G1.4.20. Notice that the threshold on the original image cuts through the Nb peak, resulting in some incomplete features at the center of the filaments, while the thresholds on the specific selections have identified the valleys adequately, and show solid features. It is important to note that in order to obtain the full area of the components of Figure G1.4.20 we must fill-in the features since the threshold applies only to the boundary. There are commonly used (automated) algorithms that can perform simple processes like filling-in holes inside the features. This and other non-invasive processes are discussed below. G1.4.4.2.2 Non-Invasive Algorithms Always keeping in mind the consequences of altering pixel count and position, one can perform a wide variety of algorithms to produce more sensible and (in many cases) more representative results. We will call these processes “non-invasive” algorithms. G1.4.4.2.2.1 Fill-Holes As it was shown in Section G1.4.4.2.1, in order to most accurately determine the phase boundary it is often desired to perform a threshold including only those pixels surrounding the phase boundary to produce two well-separated Gaussian peaks (see Figure G1.4.20). However, under this constraint, the resulting binary image is often a ring—having the true boundary on the outside, and an arbitrarily chosen region of the feature on the inside. In such cases, a “fill-holes” algorithm is recommended to solidify the features and turn this ring into the true feature being analyzed. A fill-holes algorithm identifies white pixels that are completely surrounded by black pixels and “fills” them in with black to complete the feature. This of course requires certain judgement from the user, but is very beneficial in most composite material cases where very large arrays of identical features are common—and filling them manually would be very tedious. An example using a fully reacted Internal Tin (RRP®) Nb3Sn

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FIGURE G1.4.19 (a) Histogram of a BSE image of an Nb3Sn strand cross-sectional area taken with a working distance of 5 mm, enhancing the grain contrast and broadening each peak in the histogram. (b) A histogram of a BSE image of the same Nb3Sn strand cross-sectional area as (a) but taken with a larger working distance (9 mm). Notice the histogram in (b) has enhanced contrast and each peak is much narrower.

FIGURE G1.4.20 Fiji’s default threshold applied to an image as a whole and to different selections specific to the phase boundaries. Such selections are more convenient and accurate as shown by the histogram and the resulting binary images.

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FIGURE G1.4.21 (a) A transverse cross-section of a fully reacted RRP® strand to which an automated threshold on the full image has been applied, producing an acceptable guess between the two main regions of the (b) histogram. This threshold resulted in a (c) binary image from which a fill-holes algorithm produces a (d) binary image of the sub-element features in their entirety.

strand is shown in Figure G1.4.21, where the fill-holes algorithm produced a binary image having the sub-elements in their entirety (including the cores, not selected by the threshold). Notice that in this case the thresholding algorithm has been applied to the entire image—which contains more than two peaks—but the threshold is acceptable since it falls well within the valley of the two main components of interest, that is, the Nb + Nb3Sn peaks, and the Cu/Bronze peak. Such thresholding procedures involving more than two peaks must be used with caution as it can often fall on a minor peak (see Figure G1.4.21) producing irregular binary images.

G1.4.4.2.2.2 Watershed Segmentation Watershed segmentation algorithms are very useful when features are joined together by small bridges of pixels (such as the Nb3Sn filaments in Figure G1.4.22), but their individual properties are required. The watershed name derives from the geographical concept of dividing the image into ridges and values so that each feature is separated by watersheds simulated by progressive flooding of the valleys. Although there are multiple variants of the watershed technique, the end result is a pixel-thick line segmenting the image into individual features. As shown in Figure G1.4.22, this method can effectively separate superconducting filaments with very satisfactory results.

FIGURE G1.4.22 (a) Nb3Sn filaments of an ITER strand slightly bonded during reaction. The filaments were isolated into binary images (not shown) which were then measured and (b) shaded by area for the filaments as thresholded (left) and as separated by watershed segmentation (right). In (c), the shading was reapplied across the corrected range, showing a much more valuable piece of information, that the filaments closer to the center show the widest variation in area. Notice that the color scales of (b) and (c) are different.

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FIGURE G1.4.23 (a) Nb3Sn filaments of an ITER strand. The filaments were isolated into binary images and their diameters measured in (b) as thresholded and in (c) after rejecting features smaller than 100 pixels as well as those features on the image edges. The objects are colorcoded according to their area-equivalent diameters. Once again, the corrected range shows more valuable information as the variation is more easily seen. Notice that the color scales of (b) and (c) are different.

Notice that this pixel-thick line represents an area that could belong to either of the connected features and therefore the error of this manipulation should be taken in consideration. G1.4.4.2.2.3 Filtering by Feature Values

G1.4.4.2.3 Measurements Once an adequate binary image is produced, a wide variety of geometrical measurements can be made. A detailed description of these can be found in the ImageJ user guide [25]k. Some measurements of particular importance in this chapter are:

It is very often the case that a threshold produces features that are not intended to be measured. This can happen with dust, voids, scratches, and other sample features that may or may not be introduced by the polishing procedure. Either way, the removal of such features (often very small) is desirable. Such features can be removed by filtering the analyzed objects by their size (e.g. dust) or shape (e.g. cracks). Except for analyzing total volumes, it is also important to remove features that intersect with the edge of the image—since they are most likely incomplete. Figure G1.4.23 shows an example of an Nb3Sn ITER strand where a very small piece of filament was trapped inside a void. This small piece was removed using a reject features algorithm as well as those filaments touching the edge of the image. Special care must be taken when dealing with a large number of features since it may result in the inadvertent removal of relevant features. The standard particle analysis tool in ImageJ/Fiji can filter by area and circularity, and this can be extended to filtering by any measurements using the BioVoxxel Toolbox plug-in [24] or by customized scripts. Such value filters may be used to separate out image components according to their shape (e.g. elongated features can be extracted and analyzed separately using an aspect ratio filter).

• • • •

Area Object (x, y) coordinates Perimeter Feret diameter (or maximum caliper diameter) – The furthest distance between any two points in the object outline • Minimum Feret – the narrowest caliper dimension (imagine rotating the object between two calipers to find the minimum separation)

G1.4.4.2.2.4 Outlines Outline algorithms use an already thresholded binary image and reject any pixel that is not at the edge of such feature. This could be useful since every one of the remaining black pixels has a position (x and y coordinates) and therefore can be used for more complex measurements. Figure G1.4.24 shows the Ta barrier of an ITER strand that has been outlined.

FIGURE G1.4.24 (a) the Ta barrier of an ITER strand. (b) The barrier isolated into a binary image. (c) Outlines (pixel-thick lines) produced by rejecting all black pixels that are not at the edge of the thresholded barrier.

k

The analyses are output in pixel units, or scaled units if the magnification of the image has been previously calibrated.

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• Feret angle – Angle of the Feret diameter • Shape descriptors: • Aspect ratio: In ImageJ this is the ratio of the major and minor axes of a fitted ellipse and can be quite different from the ratio of the Feret diameter to the minimum Feret. The inverse of the aspect ratio is called the roundness 4 π⋅Area • Circularity: Perimeter 2 ; has a maximum of 1 for a perfect circle. The value is decreased by both deviation of circularity and by increasing roughness of the object surface • Area equivalent diameter: 4⋅Area π ; this measurement of diameter has been historically useful fluxpinning studies because it is inversely proportional to the grain boundary density • Regions of interest (ROIs): ROIs are certain regions (or selections) in a particular image that can be saved in order to perform more advanced measurements on specific regions of the image when the rest of the image is not relevant.

FIGURE G1.4.25 manufacturers.

Cu to Non-Cu ratios of all eight ITER strand

superconductor properties such as secondary phase fractions, void fractions, unreacted fractions, etc. In all these cases, the measurement significance and appropriate level of sampling should always be considered. G1.4.5.1.2 Filament Uniformity (Sausaging)

G1.4.5 Capabilities of Image Analysis As we have seen throughout this chapter, IA requires a series of steps where various artefacts can affect the validity of the observations. However, even when all steps are done correctly, the most important aspect to consider before performing IA is the relevance (and significance) of the measurement itself. This can only be achieved through an adequate knowledge of the sample studied, and good experiment design. Below we will discuss several measurements commonly performed on superconducting strands and their relevance.

G1.4.5.1 Measurements and Their Relevance

Given that technical superconductors are made using multiple consecutive processing steps, there is a high chance that the smallest components (i.e. filaments or layer) have variations along their length. For multifilamentary wires this inhomogeneity is produced by the inevitable instabilities that occur in the filament packs during the extensive wire drawing process, an example of which is sausaging (represented by the sketch in Figure G1.4.26). Sausaging is quite common in Nb–Ti and Nb3Sn superconducting composites [26, 27]. As multi-filamentary strands typically have hundreds and sometimes thousands of filaments in a single transverse cross-section, sausaging (despite being a longitudinal effect) can be estimated from individual transverse cross-sectional imagesl. Using this approach, a single strand cross-section provides us with the equivalent of multiple measurements

Once the binary image has been obtained, there are several IA measurements that are useful for explaining tape, strand, or cable quality, and performance. G1.4.5.1.1 Area Fractions One of the most important specifications for superconductors is the stabilizer-to-superconductor ratio, a measurement that can be easily obtained using IA. Figure G1.4.25 shows the Cu to Non-Cu ratio for all eight ITER strand manufacturers, obtained using a single cross-section and dividing the Cu area by the Non-Cu area (measured from the binary images produced of each). For demonstration purposes, this comparison uses a single cross-section per measurement, but a proper analysis would require an average of multiple cross-sections. Such practice provides a distribution with a mean value and a standard deviation that can describe the homogeneity of the superconductor along the length. Area fraction measurements are also performed for many other features relevant to

FIGURE G1.4.26 Sketch of a wire length depicting very dramatic filament sausaging.

l

That is if we assume that each neighboring filament in the strand cross section can be regarded as the same filament but a couple of millimeters further down the length, which is a very common and fair assumption.

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FIGURE G1.4.27 Binary images of Nb3Sn filaments of two ITER manufacturers (more than 1000 filaments each). The wire on the right shows locally pinched filaments that contribute to larger standard deviation.

“along the length”m. Such measurements can provide us with the degree of sausaging, an estimate that has shown to be representative of strand performance [26]. In Figure G1.4.27 we show how the filament area of two similar ITER strands varies, where one of the manufacturing processes has more outliers observed to be only adjacent to specific local defects in the sub-element cores. G1.4.5.1.3 Circularity and Aspect Ratio Another measure of instabilities during wire drawing is a low circularity value for the filaments. The circularity or roundness of filaments is affected during wire processing by a variety of factors such as ductility of the material, texture, work-hardening, component relative hardness, etc. Variations in the circularity, or in more extreme cases the aspect ratio, can be very useful to track the shape effects in superconductors.

G1.4.5.2 Examples of More Complex Measurements As mentioned in Section G1.4.1.2, Fiji’s open-source philosophy allows users to build upon its features and create macros enabling more complex analyses. Below we explore ImageJ/ m

It should be noted that such measurement would not be possible from a longitudinal cross section, as the location of a longitudinal cross section with regards to the filament dimeters is ambiguous.

Fiji’s capabilities a little further than the basic IA shape measurements. G1.4.5.2.1 Using Non-Binary Images IA enables measurements of percent area fractions if we add a third color to binary images (other than black and white). This is particularly useful when measuring percentages of a particular phase inside wire components. Such measurement is enabled by taking a normal binary image—where the feature has an intensity value of 0 (black)—and coloring the phase of interest with a value of 100. Doing so, one can measure the average gray valuen of the component using Fiji’s “regions of interest” (ROI) manager. Such mean gray value of the feature should be the percentage of non-black pixels inside it—therefore the percentage of the phase of interest. This is very helpful when calculating phase fractions such as those shown in Figure G1.4.28. The SEM image in Figure G1.4.28(a) shows partially reacted RRP® sub-elements having three different phases inside the cores—if the voids are considered a third “phase”. The binary images of the cores (which includes everything inside the light ring of Nausite, but excludes the ring itself) are used to establish the ROIs in order to produce three different slices. These slices are obtained by filling the different phases separately with a gray value of 100 (while leaving the rest as black). The mean values of the ROIs are obtained n

This measurement is labeled “mean” in Fiji.

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FIGURE G1.4.28 (a) An SEM image of partially reacted RRP® sub-elements having three different phases inside the cores (namely the Cu–Sn phases η and ε, as well as voids). (b) A binary image of the cores used to establish the regions of interest and to measure the areas of each. The different phases are shown in (c), (d), and (e) colored using a gray value of 100 and leaving the rest black, these images are used to calculate the percentage of each phase in the cores. (e) A composite image of all phases colored for emphasis. (f) Residual phases not analysed within the cores.

A = getResult("Area",i); // T h e area is retrieved from the results table DA = 2*(sqrt(A/PI)); //A r e aequivalent "Heywood diameter" is calculated setResult("Da _ equiv", i, DA); // T h i s new measurement is added to the results table FT = A/((0.5*P)-(2*(A/P))); //F i b e r thickness is estimated from Area and Perimeter setResult("FbrTh", i, FT); // A d d s the new thickness estimate to the table }

and shown in (c), (d), and (e) for each phase and each feature (core). This procedure can be automated as will be described in the next section. G1.4.5.2.2 Extending Analysis Capabilities by the Use of Macros ImageJ can record a complex sequence of actions into a script, which can be edited to produce a “macro” that can then run the same process automatically. Macros created in this way have the advantage of reproducibility and speed. Furthermore, the ImageJ macro language, which is similar to JavaScript, has additional capabilities such as looping statements and array manipulation. Such macros can extend the measurement options, by adding geometrical descriptions not found on the standard analysis output. They can also perform complex and repetitive tasks such as the phase analysis shown in the previous section. In the simple example below, we show how we can add values for the area-equivalent diameter and an estimate fiber width for long fibers [28] to the strand ImageJ results table (the action of each line is explained by the comments placed after “//”): for (i=0; i α211, whereas in the a/b-plane α123 < α211 [24, 39, 40]. The higher thermal expansion coefficient in the c-direction, αc123/211, in the c-GS than in the a-GS induces tensile stresses, σcc, in the c-GS. On the other hand, lower αa/b123/211 in the c-GS than in a-GS causes tensile tangential stress in the a/b-plane, σta/b, in the a-GS. Tangential stresses are also induced in the a-GS due to the observed gradual increase of 211 concentration with distance from the growth sector boundary. An approximate profile of the residual thermal macrostresses in two different sections of a single-grain sample is depicted in Figure G1.5.17. The value of the tensile stress σcc in the half height of the typical MG Y123 sample was estimated to be 150 MPa. This value is much higher than the measured tensile fracture stress in the c-direction, σFRc, (σFRc ≈ 10 MPa for MG Y123 [41]) and is apparently responsible for macrocracking of

grains. The macrocracks in the grains are usually parallel to the a/b-plane [Figure G1.5.15(a)] but can sometimes take random orientation [Figure G1.5.15(b)] [34]. At a collision of the 211 particle with a macrocrack, it can be cross-cracked or the 211/123 interface delaminates [Figure G1.5.15(a), (b)]. Another reason for macrocracking is the difference in thermal expansion of 123 phase and larger secondary phase islands [35]. G1.5.3.4.2 Macrocracks Induced by Macroscopic Inhomogeneity of the 211 Particle Concentration The solidification process itself introduces macroscopic inhomogeneity. In the TSMG process the growth starts from a seed and 211 particles can be pushed by the growth front, depending on their size and the speed and crystal orientation of the advancing growth front [30, 36]. Consequently

FIGURE G1.5.16 Simplified diagrams for the calculation of residual thermal dilatation macrostresses in the single-grain melt-grown 123/211 bulk superconductor. The differences in Young’s modulus, E, are neglected. σa/b is the tangential (hoop) stress in the a-GS, σcc is the normal stressin the c-GS, ν is Poisson’s ratio.

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FIGURE G1.5.19 Long c-growth subsector (c-GSS) in the a-growth sector with lower Y211 concentration.

FIGURE G1.5.17 The approximate profile of the residual thermal macrostresses in two different cross-sections of a single-grain sample.

the c-GS (Figure G1.5.18). The estimated maximum value of the tangential stress at the sample surface reaches 50 MPa. The fracture stress in the a/b-plane, σFr,a/b, is higher than that in the c-direction (σFra/b ≈ 25 MPa for MG Y123 [41]), but still much lower than the value estimated. The tangential stresses in the

a-GSs of real samples are apparently lower than estimated. They can be relaxed during cooling at higher temperatures by plastic deformation and by microcracking at lower temperatures. The growth process also introduces some microinhomogeneity of 211concentration into the growth sectors [42]. Microregions of smaller and higher 211 concentration are produced by the combination of two growth directions at the steps on the growth front and are called growth subsectors (Figure G1.5.19) [42]. Other microinhomogeneity appears as 211 concentration oscillation parallel to the growth front and is caused by oscillation of the growth rate [38]. The third kind of 211concentration microinhomogeneity is the occurrence of 211 low-concentration regions (211LCR). They develop when large 123 particles or an agglomeration of fine 123 powder particles are present in the starting powder [43, 44] and by filling of large pores with melt [23]. All these microinhomogeneities induce dilatation stresses during cooling from the processing temperature and can therefore contribute to cracking.

G1.5.4 Crystal Defects Influenced by Volume Fraction and Size of 211 Particles G1.5.4.1 Microcracks in a/b-Planes (a/b-Microcracks) It was shown in previous studies [35, 45–48] that a/b-microcracks (microcracks in the a/b-planes) are formed in meltgrown (MG) bulks (Figure G1.5.20) due to microstresses induced by 211 particles. G1.5.4.1.1 Stresses Around 211 Particles

FIGURE G1.5.18 a/b-macrocracks in the c-growth sector with lower 211 concentration.

Because the expansion coefficients of the two phases are generally different, thermal stresses are set up within and around the particle as the body cools down from the fabrication temperature. An exact theoretical calculation of stresses

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TABLE G1.5.1 Values of Estimated Radial Stress σ MR and Tangential Stresses σ MΘ, σ MΦ in the a/b-Plane and the c-Direction in the 123 Matrix (Orthorhombic, Fully Oxygenated) at the 123/211 Interface c-Direction −2.0

σ MR [GPa] σ MΘ [GPa]

1.0 No component

σ MΦ [GPa]

FIGURE G1.5.20 light).

a/b-microcracks visualised by etching (normal

exists for an isotropic particle in an isotropic matrix. Selsing [49] described the radial stresses, σMR(r), and the tangential stresses σMΘ(r) = σMΦ(r), in a matrix Rp < r < ∞ in terms of the hydrostatic stress P0 in a spherical particle embedded in the infinite matrix, as follows −2σMΘ(r ) = −2σMφ(r ) = σMR(r ) = P0 (R p /r )3 P0 =

( α P − α M ) ΔT

(G1.5.4) (G1.5.5)

1 + νM 1 − 2 νP +2 2 EM EP

where the subscripts P and M signify the embedded particle or the surrounding matrix, the subscripts R, Θ and Φ indicate components in spherical coordinates (R, Θ, Φ), r is the distance from the center of the embedded particle. The condition αP > αM is placed in the thermal expansion coefficients for convenience. The signs of +σ and −σ signify tensile or compressive stress; R P is the radius of the embedded particle; ν is Poisson’s ratio; E is Young modulus; ΔT= TFZ − TRM (TFZ: freezing temperature, TRM: room temperature) is the range of temperature in the cooling stage within which the stresses are not released by atomic diffusion. The stresses at the P–M interface

TABLE G1.5.2 Constant E [GPa] ν α [K−1] KIC [MPa m1/2] a

G1.5.4.1.2 The Criterion for the a/b-Microcracking The production of an a/b-microcrack, in the case where the particle has a lower thermal expansion than that of the matrix, requires presence of some flaw at the 123/211 interface and a supply of energy for the flaw to grow. Energy for a growing flaw is provided by the elastic energy stored in the particle and in the surrounding matrix [51–53]. The elastic energy available is proportional to R P3 whereas the energy absorbed in cracking is proportional to R P2. Therefore, critical particle size exists below which spontaneous microcracking does not occure [51]. The formula for the critical Y211 particle size was derived [47] for usual 211 interparticle distance MFD211 = 2R P as follows Rc211 ≥

18γ S + ν 1 1 − 2 νP M 2 +2 POc E EP Mc

YBa2Cu3O7 a/b-Plane

Y2BaCuO5

143 0.255

182 0.255

213 0.25a

3.2 × 10−5b 0.8

0.86 × 10−5b 0.32

1.24 × 10−5 -

Ref. 31 47 39,47 47

typical value for oxides. includes both thermal expansion and oxygen uptake. E, Young’s modulus; ν, Poisson’s ratio; α, thermal expansion coefficient; KIC, fracture toughness. b

0.43 −0.215 −0.215

do not depend on the particle size. They decrease with (R/r)3 and are at 12.5% of the maximal value at r = 2R. Despite the high anisotropy of the 123 phase, this theory was applied to estimate the stress around 211 particles [50]. The calculation is possible for the Y123/Y211 because the elastic constants have been measured in this system. The results summarised in Table G1.5.1 were obtained when the values of elastic constants and thermal expansion coefficients in the c-direction from Table G1.5.2 were used and TFZ = 925°C was taken. The stress field around the spherical 211 particle can be qualitatively depicted. A schematic presentation of the radial and tangential stresses in the 123 matrix at the 211/123 interface is presented in Figure G1.5.21. The stress field is symmetric with the c-axis and the a/b-plane symmetry.

Elastic Constants, Thermal Expansion and Fracture Toughness Values YBa2Cu3O7 c-Axis

a/b-Plane

(G1.5.6)

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FIGURE G1.5.22 211 particles smaller than critical size do not nucleate a/b-microcracks. Etched.

FIGURE G1.5.21 Schematic presentation of the radial and tangential stresses in the 123 matrix at the 123/211 interface.

where γs is the effective surface energy of the matrix. γs can be derived from the fracture toughness of the 123 phase in the a/b-plane, K ICa/b, and Young’s modulus in the c-direction, Ec, as γsa/b = (K ICa/b)2/Ec. For a Y123/Y211 composite, substituting calculated P0c and material constants from Table G1.5.2 one obtains Rc211 = 0.24 μm for the orthorhombic YBa 2Cu3O7 which is in good agreement with the value estimated experimentally [47]. The subcritical 211 particles, which are not able to nucleate any a/b-microcrack, can be seen in Figure G1.5.22. The existence of the critical 211 particle size can explain deviation from the linear dependence of the a/b-microcrack spacing, λa/b-MIC, on the 211 particle size (Figure G1.5.23) for small 211 particle sizes [17]. The stress field around a 211 particle found in the analysis is also confirmed by the observed microcrack arrangement pattern around 211 particles (Figure G1.5.24). A high radial compressive stress at the poles of the 211 particle causes it to be difficult for an a/b-microcrack to propagate close to the poles. That is why a/b-microcracks shun the 211 poles and turn to the radial direction with tangential tensile stresses (compare Figure G1.5.24. with the schematic representation of the stresses in G1.5.21). More precise analytical

FIGURE. G1.5.23 Dependence between the a/b-microcrack spacing, λa/b-MIC, and the Y211 particle size, d 211, (corrected by the V123/V211 ratio). The solid curve represents supposed suppression of cracking below the critical 211 particle size.

FIGURE G1.5.24 Microcrack pattern around 211 particles. a/bmicrocracks shun the 211 poles and turn to the radial direction. Lower twin spacing in the region with higher 211 density. Twin matching along a/b-microcracks.

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that this critical 211 particle size is higher for the orthorhombic than tetragonal 123 phase. The question is: where does misinterpretation of a/b-MIC come from? The answer can be found in the microstructure of 123/211 bulks. The size of the 211 particles in the melt-grown 123/211 bulks produced in the early stages of this technology was much higher (more than 10 μm) than now, and it is therefore not surprising that a/b-MIC were present also in as-grown samples. This fact could lead to incorrect interpretation of these defects and to the development of some not proved mechanisms of their formation during solidification.

FIGURE G1.5.25 Dense a/b-microcracks in the orthorhombic Nd123 phase and no a/b-microcracks in the tetragonal Nd123 phase. OCMG Nd123 with small 422 particles. Polarised light, etched.

G1.5.4.2 Twin Structure

expression of stresses around a 211 particle in a 123 matrix supports the a/b-microcrack formation as described above [54, 55]. Earlier it was supposed that the a/b-microcracks in the melt-grown 123/211 bulks are so-called growth-related a/bplanar defects produced by solidification [56]. Only later was it shown that these defects do not exist in as-grown samples [35, 47]. In Figure G1.5.25, the microstructure of a deeply etched Nd 123/211 melt-processed sample partially transformed into orthorhombic phase is presented. The appearance of a/b-planar defects practically only in the orthorhombic part is evidence that a/b-planar defects do not originate from solidification. Examination of an etched sample with much larger Nd211 particles showed that a/b-microcracks also develop in the tetragonal parts of the sample (Figure G1.5.26). The crack spacing in the transformed orthorhombic Nd123 is much smaller than in the tetragonal part. This finding confirms that, for both the orthorhombic and tetragonal 123 phases, there is a critical 211 particle size below which a/b-microcracking associated with 211 particles does not occur, and

The twinning of 123 phase, with the (110) and (−110) twinning planes, occurs at its transformation from the tetragonal to orthorhombic state [57, 58]. Twins are well visible in polarised light [Figures G1.5.15(b) and G1.5.18] but also in normal light after etching [17, 59]. The traces of (110) and (−110) twin boundaries and traces of a/b-microcracks can be used in determination 123 crystal orientation according the calculation developed by Verhoeven and Gibson [58]. The declination angle αc is then defined as the angle between the normal to the polished surface and [001] crystal direction. The angle αc lays in the plane perpendicular to the a/b-microcrack traces on the polished surface. A lamellar assembles of twins (twin complexes) can be visualised under slightly uncrossed polarisers when polariser and analyser are nearly parallel or perpendicular to the “a” and “b” directions (Figure G1.5.27) [59]. Contrast between twin complexes is highest in the a/b-plane. Twin complexes are visualised because of the higher optical conductivity for light impinging with the light polarisation vector parallel to the twin walls than for the light polarisation vector perpendicular to the walls [59]. The mean linear size of the twin complexes dtc is essentially depressed by lowering mean free

FIGURE G1.5.26 Higher a/b-microcrack density in the orthorhombic than in the tetragonal Nd123 phase. OCMG Nd123 with large 422 particles. Polarised light, etched.

FIGURE G1.5.27 Twin complexes visualised in polarised light. The darkest are 211 particles. Bright and gray are twin complexes. αc = 25°.

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TABLE G1.5.3 Measured Values of 211 Volume Fraction V211, Mean Free Distance between 211 Particles MFD211 and Mean Twin Complex Size dtc [17] Sample

V211

MFD211 [mm]

dtc [mm]

1 2 3

0.36 0.32 0.29

50 12 7.3

25 8 4.5

distance between 211 particles (MFD211) (Table G1.5.3), and linear dependence between the twin complex size dtc and MFD211 may be suggested. Such a refinement of the twin complexes is typical for regions of the sample which are little influenced by the stresses arising during cooling (due to the anisotropy in the thermal expansion of 123 grains). Macrostresses cause the predominance of one twin complex type in each grain [60]. The twin spacing lt depends on the local microstructure. Generally, lt is larger in the regions with lower concentration of the 211 particles (Figures G1.5.15 and G1.5.18). The relationship between lt and the crystal dimension is expressed [61] as: lt = ( γ ⋅ g / CMΦ 2 )1/2 ,

(G1.5.7)

where Φ = 2(b − a)/(a + b) is the orthorhombicity, and g and M are the grain size and the shear modulus, respectively. C is a constant nearly equal to 1. γ is the twin boundary energy. In the case of 123-211 composite the mean free distance between 211 particles MFD211 is a characteristic dimension that determines the twin spacing lt [17], therefore MFD211 has to be used instead of the grain size, g, in the Equation (G1.5.6). Twin structure is significantly influenced by the stress field around 211 particles. The stresses evoked by 211 (or 422) particles cause detwinning or predominance of one twin variant in the directions by the twin boundary motion (Figure G1.5.28) [17, 62, 63]. The motion of twin boundaries

FIGURE G1.5.28 Detwinned areas and areas with predominant one twin domain variant extended in the directions around 422 particles in OCMG Nd123. Polarised light and Laws–Ernst compensator. αc = 0°.

FIGURE G1.5.29 Plastic strain in the and directions associated with the detwinning process.

indicates that residual stresses exist in the a/b-plane at a level that is necessary for the detwining process. The stress necessary for the twin boundary motion has been theoretically estimated [64] to be in the range of 130–400 MPa which is in agreement with experimental determinations [65–67] carried out at 350°C (50–100 Mpa). The estimated stresses necessary for twin boundary motion are in good agreement with estimated stresses around 211 particle in the a/b-plane directions. Closer analysis shows that simple motion of twin boundaries is not sufficient to cause the relaxation of stresses around a 211 particle in all directions lying in the a/b-plane [60]. The illustrations in Figure G1.5.29 confirm the dimension changes due to diminishing of one twin type. Clearly, normal stresses perpendicular and parallel to the twin boundaries do not relax by the twin boundary motion, so in the directions the twinning structure does not change by thermal stresses around 211. Microstresses evoked by 211 particles can also cause splitting of the twins at the 211/123 interface (Figure G1.5.30). The twin boundaries and twin complex boundaries are places of higher distortion of 123 lattice and therefore they can also contribute to the pinning of flux lines. Their contribution to the critical current density can be treated similarly as in the work of [60].

FIGURE G1.5.30 Twin splitting caused by stresses around 211 particle. Polarised light.

Optical Microscopy

G1.5.5 Crystal Defects Related to Oxygenation of Melt-Grown REBa2Cu3O7−x

85

It was supposed that macrocracks in 123/211 bulks are formed due to macroscopic stresses, which are induced mainly by macroscopic inhomogeneity of the 211 particle concentration in the samples. Later it was shown that the formation of macrocracks starts on the sample surface at the initial stage of oxygenation [68]. A study was performed on TSMG singlegrain bulks, which were annealed at 900°C in oxygen for 1 h and cooled down in argon with a cooling rate 600°C h−1 to have starting tetragonal samples with low and defined oxygen content. For further treatment, bars of 2 × 2 mm2 cross-section and with bar walls parallel to a/c-planes were cut and oxygenated at different temperatures in a tubular furnace. Heating

to and cooling from the oxygenation temperature was carried out in Ar atmosphere. Samples were oxygenated in a flow of oxygen. At first, macrocracking on sample surfaces polished before oxygenation was examined. Here, a parallel array of the macrocracks could clearly be seen (Figure G1.5.31). Then the oxygenation process of the bars was studied on the crosssections of the oxygenated layers perpendicular to the plane. At lower oxygenation temperatures the oxygenation was very in homogeneous [Figure G1.5.32(a), (b)]. With higher oxygenation temperature the oxygenated layer becomes more regular, and at 700°C and higher temperatures the oxygenation process was nearly homogeneous [Figure G1.5.32(c)]. As the oxygen diffusion at higher temperatures is fast, a layer of lower oxygen content developed at the sample surface during cooling in argon atmosphere from the oxygenation temperature [Figure G1.5.32(c)]. We counted the spacing of the macrocracks (Table G1.5.4, Figure G1.5.33) and found out that with increasing temperature the crack spacing increases to infinity at 800°C, where no cracking was observed. As cracks also appear in the tetragonal phase (at 700 and 750°C), this cracking is not associated with the tetragonal–orthorhombic

FIGURE G1.5.31 a/b-macrocracks seen on the surface of samples oxygenated at 500°C/15 min (a) and 600°C/6 min (b).

FIGURE G1.5.32 Oxygenated surface layer seen under polarized light. Oxygenation at lower temperatures is highly inhomogeneous due to cracking along a/b-planes and oxygen transport along the cracks. Oxygenation at (a) 400°C/5 h. (b) 600°C/1 h. (c) 700°C/9 min. The oxygen depleted surface layer developed during cooling down in argon from the oxygenation temperature 700°C.

G1.5.5.1 Macrocracks Introduced by the Oxygenation Process G1.5.5.1.1 a/b-Macrocracks

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TABLE G1.5.4 Spacing of the a/b-Cracks, λa/b, Measured on Samples Oxygenated at Different Oxygenation Temperatures. The Equilibrium Strain in the Oxygenated Layer, εc = Δc/c Means the Strain Induced by the Room Temperature c-Parameter Difference between the Sample Oxygenated at 900°C, c900, and the Sample Oxygenated at Temperature T, ct (Δc = c900 − cT). The Critical Crack Spacing, λabcr, the Critical Thickness of Oxygenated Layer, dabcr and the Critical Stress in the Layer, σabcr, Were Calculated Using Thoules’ Model Oxygenation Temperature [°C]

400

500

600

700

750

800

850

Time [min] λa/b [μm] εa/b10−2 λabcr [μm] dcr [μm] σabcr [MPa]

24 60 1.05 4 0.4 1660

18 100 0.88 5.6 0.6 1350

6 200 0.52 18 1.6 830

10 350 0.18 106.5 13.6 280

5 426 0.14 175 22.5 220

5 ∞ 0.09 427 54.3 140

5 ∞ 0.055 1142 146 87

region and the energy necessary to create the cracks, i.e. the new surface energy. The crack spacing λ will be adjusted in order to minimize the total energy of the system to λ = 5.6 K IC 2d /( Eε)2

{

1/2

}

(G1.5.8)

with the mode I fracture toughness K IC, thickness d of the films, Young’s modulus E and strain ε. This results in minimum crack spacing and is consistent with the requirement that the energy release by cracking should be higher than the energy that is necessary to create the cracks. This means that below a critical thickness FIGURE G1.5.33 a/b-macrocrack spacing measured on samples oxygenated at different oxygenation temperatures.

transformation of the123 phase. The reason for cracking is a change of the c-lattice parameter of the 123 phase with the oxygen content (Figure G1.5.34). The oxygenated layer has lower c-lattice parameter than the matrix and therefore it is under tensile stresses perpendicular to the plane of the 123 lattice. To describe macrocrack formation during the oxygenation process, we try to apply a model for cracking of brittle films on elastic substrates developed by Thoules [69] (Figure G1.5.35). According to the model suggested in [69], the total energy is the sum of the strain energy in the cracked

FIGURE G1.5.34 The room temperature c-lattice parameter of the Y123 phase corresponding to the oxygen concentration in the Y123 phase. The oxygen concentration is expressed by annealing temperature in a pure oxygen atmosphere [49, 71].

dcr = 0.5 K IC 2 /( Eε)2

(G1.5.9)

no cracks will appear. During isothermal oxygenation the first cracks can form in the surface oxygenated layer only when the thickness of the layer is critical. The tensile stress (σa/bcr = Eε) in the oxygenated layer of critical thickness, da/bcr, is given in Table G1.5.4. Further fast growth of these macrocracks is provided by the stress generated at the tip of the crack due to oxygenation of the matrix ahead of the crack tip by oxygen flowing from the sample surface along the macrocrack.

FIGURE G1.5.35 Development of parallel cracks with spacing λcr in a brittle ﬁlm with thickness d under uniaxial tensile stress σ.

Optical Microscopy

At lower temperature of oxygenation, the measured crack spacing values, λa/b, (Table G1.5.4, Figure G1.5.33) are much higher than the critical spacing, λa/bcr, calculated for critical layer thickness, da/bcr, (Table G1.5.4). The difference can be caused by the fact that not all cracks created at the critical layer thickness propagate into the sample. The final macrocrack spacing in the TSMG Y123/211 bulks oxygenated with standard conditions (at 400°C) was found in the range of 200–300 μm (Figure G1.5.36), and is obviously higher for the a-GS with higher 211 particle concentration (consequently higher fracture toughness) and with compressive stresses in the c-direction. However, at higher oxygenation temperatures the λa/b and λa/bcr values are closer. At higher temperatures the rate of oxygen diffusion and the velocity of the MAC propagation are not very different, and therefore the cracking process is closer to the conditions of Thoules’ model.

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G1.5.5.1.2 a/c-Macrocracks

FIGURE G1.5.36 A later stage of oxygenation shows that the oxygenated orthorhombic lamellae propagating along the a/b-MAC into the tetragonal part of the sample have higher spacing than at the beginning of oxygenation.

The cracks parallel to the plane (macrocracks [68]) which formed in the first stages of oxygenation provided a passage for oxygen delivery to the sample interiors. The oxygenated regions along the macrocracks appear very uniform after shorter times of oxygenation even at the oxygenation

temperature 400°C [Figure G1.5.37(a)]. In fact, the tip angle of these oxygenated lamellae expresses the ratio of the effective oxygen diffusion rates along the plane over the c-direction. At longer times of oxygenation, the lamellae become irregular

FIGURE G1.5.37 (a) Regular orthorhombic lamellae along a/b-macrocracks. Oxygenation at 400°C for 20 min. (b) Irregular orthorhombic lamellae developed at longer oxygenation time. Oxygenation at 400°C for 5 h. (c) The c-macrocracks perpendicular to the a/b-macrocracks are formed in the orthorhombic layer. (d) The tip of the orthorhombic lamellae along the a/b-MAC. Oxygenation in the c-direction is significantly enhanced by the c-MAC formation.

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FIGURE G1.5.38 (a) Regular oxygenated surface layer seen in polarized light with a/b-macrocracks. (b) a/b-macrocracks in the oxygenated layer visualized in normal light. (c) No c-macrocracks can be seen in the oxygenated layer even at higher magnification (polarized light).

[Figure G1.5.37(b)], reflecting some events which locally increase the diffusion rate in the c-direction. Observation at higher magnifications revealed that some cracks perpendicular to the marginal macrocrack have been formed in the oxygenated lamellae [Figure G1.5.37(c), (d)]. Obviously, oxygen can flow much faster along these cracks (c-macrocracks) than the bulk oxygen diffusion rate in the c-direction would allow. In this way the oxygenation process of the TSMG bulk samples is significantly enhanced. With higher oxygenation temperature the oxygenated layer became more regular. The formation of c-cracks in the oxygenated lamellae was not observed when the oxygenation temperature was increased to higher than 680°C (Figure G1.5.38). The oxygenation process at this temperature was nearly homogeneous besides a few sharp thin oxygenated lamellae along the macrocracks; see Figure G1.5.38(a). The micrographs taken in normal light [Figures G1.5.38(b) and G1.5.19(b)] and at higher magnification [Figure G1.5.38(c)] prove that no c-macrocracks have been formed in the sample. As the oxygen diffusion at higher temperatures is fast, a layer of lower oxygen content, however, has

been developed at the sample surface and along the surfaces of the macrocracks during cooling in an argon atmosphere from the oxygenation temperature [Figure G1.5.38(a)]. The Thoules model was considered to be appropriate also for c-crack formation [70]. Seemingly there is one obstacle which must be defeated. It is generally supposed that stresses in the plane can be relaxed by twin boundary motion. Actually it was shown that the uniaxial stress in the plane parallel to the a- or b-directions ([100] directions) can be relaxed by motion of twin boundaries (schematically illustrated in Figure G1.5.29), while the uniaxial stress parallel to the [110] directions cannot be relaxed by twin boundary motion [17]. This phenomenon is well known and is used in the detwinning process of 123 single crystals [62]. However, the situation is significantly different in the case of the oxygenated layer along the macrocrack and at the a/b sample surface. Here, the lattice parameters shorten with oxygen content in both the a- and b-directions [49, 71], which induces biaxial tensile stress on the oxygenated layer. In this case the stress cannot be relaxed through plastic deformation provided by twin boundary motion (Figure G1.5.29). The

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FIGURE G1.5.39 The room temperature (a + b)/2 lattice parameters of the Y123 phase corresponding to the oxygen concentration in the Y123 phase. The oxygen concentration is expressed by the annealing temperature in a pure oxygen atmosphere [49, 71].

room temperature dependence of the a-lattice parameter (for the tetragonal 123 phase) and the average (a + b)/2-lattice parameter (for the orthorhombic 123 phase) on the oxygen concentration (the oxygen concentration is expressed by the annealing temperature in oxygen) is shown in Figure G1.5.39. It is clear from this dependence that tensile stresses are induced on the oxygenated layer; nevertheless, they are much lower than in the case of the oxygenated a/c-surface layer (Table G1.5.5) as this dependence is much weaker than for the c-lattice parameter [49, 71]. For the same reason the calculated critical layer thickness, dcr, and critical crack spacing, λcr, values are higher for the c-MAC than for the a/b MAC (Table G1.5.5). The Young modulus (E = 188 GPa in the a-direction) and fracture toughness (K IC =1.48 MPa m1/2 for the a/c-plane) values for Y123/Y211 composite, which were taken from published data [49], and strain, ε, induced on the oxygenation layer due to the difference of the average (a + b)/2-lattice parameters corresponding to equilibrium oxygen contents at 680°C and the temperature of oxygenation were used for these calculations. The lattice parameters of the YBa2Cu3Ox compounds for different x were taken from [49, 71], and the differences in thermal expansion for different x were neglected in the calculation. The calculated critical crack spacing value λccr =75 μm

is not very far from the c-macrocrack spacing λc = 34.2 ± 17 μm measured in the samples oxygenated at 380°C [72]. During isothermal oxygenation the first c-cracks can appear only when the thickness of the layer along the macrocrack is critical. Further growth of c-macrocracks is provided by the stress generated at the tip of the crack due to the oxygenation of the matrix ahead of thecrack tip by the oxygen flowing from the surface through the and c-macrocracks. Obviously, it is necessary to take into account that macrocracking during oxygenation is influenced by mechanical stresses present in the sample. The final macrocrack pattern (local macrocrack spacing) in the TSMG bulks is therefore determined by the parameters of the oxygenation process and by the dilatation stresses induced due to 211 concentration inhomogeneity of the sample. The starting microstructure of the samples also influences the crack formation at the oxygenation process [73]. The c- and a/b-MAC start to form at first in the 211 lowconcentration regions (211LCR) due to its lower fracture toughness (Figure G1.5.40). The formation of a dense structure of a/b- and c-macrocracks causes faster and more homogeneous oxygenation of the sample. Opening of created macrocracks is

FIGURE G1.5.40 Optical micrograph showing the macrocrack pattern in a 123/211 skeletonand 211LCRs. Note the higher a/b- and c-macrocrack density in 211 low concentration regions (211LCRs) than in 123/211skeleton.

TABLE G1.5.5 Equilibrium Strain in the Oxygenated Layer, ε = Δab/ab, Means the Strain Induced by the Average (a + b)/2 Lattice Parameter Difference between the Sample Oxygenated at 900°C, ab900, and the Sample Oxygenated at Temperature T, abT (Δab=ab900 − abT). The Critical Crack Spacing, λccr, the Critical Thickness of Oxygenated Layer, dccr and the Critical Stress in the Layer, σccr, Were Calculated Using Thoules’ Model Oxygenation Temperature [°C]

380

400

500

600

εab λccr [μm] dccr [μm] σccr [MPa]

1.8 75 10 316

1.55 105 12.5 283

1.04 240 28 188

0.55 950 110 95

90

FIGURE G1.5.41 Orthorhombic twinned Nd123 formed around a/b- microcracks in furnace cooled sample. Polarised light.

the way that the macroscopic stresses induced by macroscopic 211 particle inhomogeneity are released. This can be very important, because it prevents the formation of fatal c-macrocracks, which divide the sample into more domains [73, 74, 75], during cooling from oxygenation temperature or during sample performance.

G1.5.5.2 Oxygen Concentration Inhomogeneity It is easy to recognise tetragonal and orthorhombic phase in polarised light due to changes in optical anisotropy. This leads to different contrast and colour of orthorhombic and tetragonal phases. Besides that twinning is characteristic for orthorhombic 123 phase (Figure G1.5.41). As it was shown the oxygenation of MG bulks is highly inhomogeneous [68, 70, 72, 75] and is assisted by formation of oxygenation a/b- and c-macrocracks. The macrostresses induced by inhomegeneity of 211 particle concentration developed during growth by pushing effect influences formation of oxygenation macrocracks and consequently oxygenation. The compressive stresses in the a-growth sectors in the c-direction (Figure G1.5.17) are the highest close to the sample top surface, and consequently, the formation of a/b-macrocracks is suppressed there. This leads often to observation of a layer of tetragonal phase in these regions (Figure G1.5.42), which can be detected under polarised light.

FIGURE G1.5.42 Tetragonal 123 phase in the a-growth sectors with higher compressive stresses (compare with figure G1.5.17).

Handbook of Superconductivity

FIGURE G1.5.43 Aligned secondary phases in the Bi-2223 matrix seen in normal light. Polished.

G1.5.6 BiSrCaCuO, TlBaCaCuO, HgBaCaCuO Superconductors The microstructure micrographs of BiSrCaCuO, TlBaCaCuO, HgBaCaCuO superconductors obtained under polarised light are also full of information. As example, the microstructures of polycristalline (BiPb)2Sr2Ca 2Cu3Ox (Bi-2223), TlBa 2Ca 2Cu3Ox (Tl-1223) and Hg xPbyBa 2Ca 2Cu3Ox (Hg-1223) ceramics will be presented. Alignment of Bi-2223 plates as well as secondary phases after hot forging can be seen in Figures G1.5.43–G1.5.46. Combination of different observation conditions provides more information about perfection of Bi-2223 particle alignment. The micrograph obtained in normal light after polishing (Figure G1.5.43) clearly indicates alignment of secondary phases of different composition and little information about Bi-2223 matrix. Similar, but more colourful information, is obtained in polarised light when polarised light vector is 45 degrees to the forging direction (Figure G1.5.44). The Bi-2223 platelet alignment is better pronounced when the platelet

FIGURE G1.5.44 Aligned secondary phases in the Bi-2223 matrix seen in polarised light when the vector of polarised light is 45 degrees to the forging axis. Polished.

Optical Microscopy

FIGURE G1.5.45 The Bi-2223 platelet alignment is better pronounced when the platelet boundaries are visualized by etching in the solution of 1 wt. % of Br in ethanol. Polarised light.

boundaries are visualized by etching (Figure G1.5.45). The declination of Bi-2223 platelets from the preferential orientation is very clearly observable when the vector of polarised light is parallel to the forging axis (Figure G1.5.46). The inner structure of the Bi-2223 platelets can be studied when the sample surface parallel to the forging direction is polished. In this case so-called rotation (001) twins are well pronounced (Figure G1.5.47). An example of microstructure of Tl-1223 polycristalline ceramic after synthesis is presented in Figure G1.5.48. In polarised light different Tl-1223 crystal orientations are pronounced by different colour (contrast in black and white). Cracks, formed due to the anisotropy of Tl-1223 phase thermal expansion, at the grain boundaries can be seen in normal light (Figure G1.5.49). Similar microstructure exhibits Hg-1223 ceramic after synthesis (Figure G1.5.50). Platelet Hg-1223 grains contain small amount of secondary phases, which induce microcracks into 1223 grains obviously due to differences in the thermal expansions of present phases.

FIGURE G1.5.46 The declination of Bi-2223 platelets from the preferential orientation is very clearly observable when the vector of polarised light is parallel to the forging. Polished.

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FIGURE G1.5.47 Rotation (001) twins are well pronounced in polarised light when the sample surface parallel to the forging direction is polished.

FIGURE G1.5.48 In polarised light different Tl-1223 crystal orientations are pronounced by different colour (contrast in black and white).

FIGURE G1.5.49 Cracks, formed due to the anisotropy of Tl-1223 phase thermal expansion, at the grain boundaries can be seen in normal light.

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FIGURE G1.5.50 Platelet Hg-1223 grains contain small amount of secondary phases, which induce microcracks into 1223 grains due to differences in the thermal expansions of present phases.

G1.5.7 Conclusions It is shown that polarised light microscopy is very effective technique for microstructure analysis of high-temperature superconductors of RE (rare earth), Bi, Tl and Hg types. After surface polishing, visualisation of structure features in polarised light is allowed due to high optical anisotropy of layered cuprate crystals. Another advantage is that observation of objects in the dimension scale from 2 mm to 0.2 μm is possible in a very simple way. Under observation in polarised light secondary phases can be identified. This is particularly important for single-grain melt-grown REBCO consisting of RE Ba 2Cu3 O7 (123) single crystal in which RE2BaCuO5 (211) particles are distributed. In this way, the microscopic as well as macroscopic inhomogeneity of 211 concentration in the bulk can be effectively studied. This knowledge is essential for optimisation of superconducting and mechanical properties of these bulks. In the case of REBCO 123 bulks, it is possible to determine crystal orientation from observed twin pattern. Visualisation of twin structure gives possibility to study interaction of twins with stress fields generated by 211 particles and in this way to describe indirectly the dilatation residual micro-stresses around 211 particles. Additional etching of the samples reveals also those microcracks which were not visible after polishing (filled with sample powder produced during polishing), and quantitative information about sample cracking can be obtained. Changes in the optical anisotropy of 123 phase with oxygen concentration causes different colour or contrast in 123 phase and, therefore, homogeneity of oxygen concentration in the sample can be studied.

References [1] Samuels L E 1971 Metallographic Polishing by Mechanical Methods (Melbourne: Pitman) [2] Bradbury S 1991 An Introduction to the Optical Microscope (Oxford: Oxford University Press)

Handbook of Superconductivity

[3] Phillips V A 1971 Modern Metallographic Techniques and Their Applications (New York: Wiley) [4] McCrone W C, McCrone L B and Delly J G 1978 Polarised Light Microscopy (Ann Arbor: Ann Arbor Science) [5] Schumann H 1990 Metallographie (Leipzig: Deutscher Verlag fur Grundstoffindustrie) [6] Diko P, Fox S, Moore J C, Grovenor C R M and Goringe M J 1997 Microstructural studies of Ag and Ag alloy sheathed Tl-1223 tapes J. Mater. Chem. 7 947 [7] Konig W, Gritzner G, Diko P, Kovac J and Timko M 1995 Relationship between the microstructure and properties of PbOdoped BiSrCCO 2223 superconductors synthesized from malic acid gels and from spray-dried nitrates J. Mater Chem. 5 879 [8] Diko P, Zezula I, Timko M, Kavecansky V, Csach K, Miskuf J and Zentko A 1990 The microstructure changes in Tl2Ba 2Ca 2Cu3Ox samples during annealing J. Mater. Sci. Lett. 9 391 [9] Sargankova I, Diko P, Kovac J and Timko M 1997 Microstructure and superconducting properties of Bi1.7Pb0.3Ca 2.4Sr1.6Cu3Oy textured by uniaxial pressing Superlattices Microstruct. 21 95 [10] Sargankova I, Konig W, Mair M, Gritzner G, Diko P, Kavecansky V, Kovac J and Longauer S 1997 Influence of the variation of Hg and Pb stoichiometry on the microstructure and Tc of HgxPbyBa 2Cu3O8+d Superlattices Microstruct. 21 367 [11] König W and Gritzner G 1998 Microstructure and properties of textured bulk Tl(Bi)–1223 superconductors Physica C 294 225 [12] Heddrich R, Schuster T h, Kuhn H, Geerk J, Linker G and Murakami M 1995 Critical current limiting structural defects in melt-textured YBa 2Cu3O7−x Appl. Phys. Lett. 66 3212 [13] Diko P, Takebayashi S and Murakami M 1998 Origin of subgrain formation in melt-grown Y–Ba–Cu–O Bulks Physica C 297 216 [14] Klapper H 1980 Defects in non-metal materials crystals Proc. NATO Advanced Study Institute on Characterization of Crystal Growth Defects by X-Ray Methods, eds B K Tanner and D K Bowen (Durham, UK: Plenum) p 133 [15] Jackson K A, Solidification, American Society for Metals 1971, p 121 [16] Rutter J W and Chalmers B 1953 A prismatic substructure formed during solidification of metals J. Phys. 31 15 [17] Diko P, Gawalek W, Habisreuther T, Klupsch Th, Gornert P 1995 Influence of Y2BaCuO5 particles on the microstructure of YBa 2Cu 3O7-x (123) – Y2BaCuO5 (211) melt-textured superconductors Phys. Rev.B 52 13658 [18] Sudhakar Reddy E and Rajasekharm T 1997 Origin of subgrain structure within a domain in melt-processed YBa 2Cu3O72x Phys. Rev. B 55 14160

Optical Microscopy

[19] Fujimoto H, Murakami M, Nakamura N, Gotoh S, Kondoh A, Koshizuka N and Tanaka S 1991 Advances in Superconductivity IV, Proc. 4th Int. Symp. on Superconductivity (ISS’91), (October, Tokyo), eds H Hayakawa and N Koshizuka (Berlin: Springer) p 339 [20] Ogava N, Hirabayashi I and Tanaka S 1991 Preparation of a high-Jc YBaCuO bulk superconductor by the platinum doped melt growth method Physica C 177 101 [21] Kim C J, Lai S H and McGinn P J 1994 Morphology of Y2BaCuO5 and segregation of second phase particles in melt-textured Y–Ba–Cu–O oxides with/without BaCeO3 addition Mater. Lett. 19 185 [22] Diko P, Wende Ch, Litzkendorf D, Klupsch Th and Gawalek W 1998 The influence of starting Y123 particle size and Pt/Ce addition on the microstructure of Y123– 211 melt processed bulks Supercond. Sci. Technol. 11 49 [23] Kim C, Lee H G, Kim K B and Hong G W 1995 Nonuniform distribution of second phase particles in melt-textured Y–Ba–Cu–O oxide with metal oxide (CeO2, SnO2, and ZrO2) addition J. Mater. Res. 10 1605 [24] Diko P, Todt V R, Miller D J and Goretta K C 1997 Subgrain formation, 211 particle segregation and high-angle 90° boundaries in melt-grown YBaCuO Physica C 278 192 [25] Uhlman D R, Chalmers B and Jackson K A 1964 Interaction between particles and a solid-liquid interface J. Appl. Phys. 35 2986 [26] Endo A and Shiohara Y 1996 Macrosegregation of Y2BaCuO5 particles in YBa 2Cu3O72x crystal grown by an undercooling method J. Mater. Res. 11 801 [27] Utech H P and Flemings M C 1966 Elimination of solute banding in indium antimonide crystal by growth in a magnetic field J. Appl. Phys. 37 2021 [28] Diko P and Goretta K C 1998 Macroscopic shape change of melt-processed YBaCuO/YBaCuO bulk superconductors Physica C 297 211 [29] Varanashi C, Black M A and McGinn P J 1996 The demonstration of Y2BaCuO5 particle segregation in meltprocessed YBa2Cu3O72x through computer visualisation model J. Mater. Res. 11 565 [30] Honjo S, Cima M J, Flemings M C, Ohkuma T, Shen H, Rigby K and Sung T H 1997 Seeding crystal growth of YBa 2Cu3O6.5 in semisolid melts J. Mater. Res. 12 880 [31] Goyal A, Funkenbusch P D, Kroeger D M and Burns S 1991 Fabrication of highly aligned YBa 2Cu3O72x–Ag melt-textured composites Physica C 182 203 [32] Evans G 1978 Microfracture from thermal expansion anisotropy—I. Single phase systems Acta Metall. 26 1845 [33] Clarke D R, Shaw T M and Dimos D 1989 Issues in the processing of cuprate ceramic superconductors J. Am. Ceram. Soc. 72 1103 [34] Diko P, Gawalek W, Habisreuther T, Klupsch T h and Gornert P 1996 Macro- and microcracking, subgrains, twins and thermal stresses in YBa2Cu3O72xð123Þ– Y2BaCuO5 (211) melt textured superconductors studied by means of polarized light microscopy J. Microsc. 184 46

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[35] Diko P, Pelerin N and Odier P 1995 Microstructure analysis of melt-textured YBa 2Cu3O72x ceramics by polarized light microscopy Physica C 247 169 [36] Diko P, Todt V R, Miller D J and Goretta K C 1997 Subgrain formation, 211 particle segregation and highangle 90° boundaries in melt-grown YBaCuO Physica C 278 192 [37] Diko P and Goretta K C 2000 Macrocracking in meltgrown 123 bulk superconductors Inst. Phys. Conf. Ser. 167 67 [38] Diko P 2000 Growth-related microstructure of meltgrown REBa 2Cu3Oy bulk superconductors Supercond. Sci. Technol.13 1202 [39] Dubrovina I N, Zakharov R G, Kostsin E G, Antonov A V, Balakirev F V and Valotin N A 1990 Supercond. Phys. Chem. Technol. 3 3 [40] Goyal A, Oliver W C, Funkenbusch P D, Kroeger D M and Burns S J 1991 Mechanical properties of highly aligned YBa 2Cu3O72x Effect of Y2BaCuO5 particles Physica C 183 221 [41] Sakai N, Seo S J, Inoue K, Miyamoto T and Murakami M 1998 Adv. Supercond.XI 1 685 [42] Diko P, Zmorayova K, Granados X, Sandiumenge F and Obradors X 2003 Growth related Y2BaCuO5 particle concentration micro-inhomogeneity in the growth sectors of TSMG YBa 2Cu3O7/Y2BaCuO5 bulk superconductor Physica C 384 125 [43] Diko P, Kracunovska S, Ceniga L, Bierlich J, Zeisberger M, Vasalek W 2005 Microstructure of top seeded melt-grown YBCO bulks with holes Supercond. Sci. Technol. 18 1 [44] Kracunovska S, Diko P, Litzkendorf D, Habisreuther T and Gawalek W 2003 The influence of the starting YBa 2Cu3Ox powder on the microastructure of melt-textured YBa 2Cu3O7-x/Y2BaCuO5 bulks Physica C 397 123 [45] Diko P, Kojo H and Murakami M 1997 Microstructure of oxygen controlled melt-grown Nd-Ba-Cu-O superconductors Physica C 276 185 [46] Sakai N, Munakata F, Diko P, Takebayashi S, Yoo S I and Murakami M 1997 Adv. Supercond.X 1 645 [47] Diko P 1998 Cracking in melt-processed RE-Ba-Cu-O Supercond. Sci. Technol. 11 68 [48] Tancret F, Monot I and Osterstock F 2001 Toughness and thermal shock resistance of YBa 2Cu3O7−x composite superconductors containing Y2BaCuO5 or Ag particles Mater. Sci. Eng. A 298 268 [49] Casalta H et al 1966 Neutron-scattering determination of the structural parameters versus oxygen content of YBa 2Cu3O6+x single crystals Physica C 258 321 [50] Diko P, Fuchs G and Krabbes G 2001 Influence of silver addition on cracking in melt-grown YBCO Physica C: Superconductivity 363 60 [51] Davidge R W and Green T J 1968 The strength of twophase ceramic/glass materials J. Mater. Sci. 3 629 [52] Weyl D 1959 Ber. Dtsch. Keram. Ges. 36 319

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[53] Selsing J 1961 Internal stresses in ceramics (page 419) Am. Ceram. Soc. 44 419 [54] Ceniga L and Kovac F 2001 Cracking of isotropic particle–matrix system: globular shape particle Mater. Sci. Eng. B 86 178 [55] Ceniga L and Diko P 2003 Matrix crack formation in Y-Ba-Cu-O superconductor Physica C 385 329 [56] McGinn P, Chen W, Zhu N, Lanagan M and Balachandran U 1990 Microstructure and critical current density of zone melt textured YBa2Cu3O6+x Appl. Phys. Lett. 57 1455 [57] Zandbergen H W, van Tendeloo G, Okabe T and Amelinckx S 1987 Electron diffraction and electron microscopy of the superconducting compound Ba 2YCu3O72x Phys. Stat. Solidi 103 45 [58] Verhoeven J D and Gibson E D 1988 Determination of crystallographic orientation of YBa2Cu3Ox grains from their optical twin patterns Appl. Phys. Lett. 52 1190 [59] Rabe H, Rivera J P, Schmid H, Chaminade J P and Nganga L 1990 Reflected polarized light microscopy of the ferroelastic domain structure of YBa 2Cu3O72x Mater. Sci. Engn. B 5 243 [60] Diko P, Gawalek W, Habisreuther T, Klupsch Th, Gornert P 1996 Macro- and microcraking, subgrains, twins and thermal stresses in YBa 2Cu3O7-x (123) – Y2BaCuO5 (211) melt textured superconductors studied by means of polarised light microscopy J. Microsc. 184 46 [61] Streiffer S K, Zielinsky E M, Lairson B M and Bravman J C 1991 Thickness dependence of the twin density in YBa 2Cu3O7-x thin films sputtered onto MgO substrate Appl. Phys. Lett. 58 2171 [62] Favrot D, Dechamps M and Revcolevschi A 1991 De-twinning of YBa 2Cu3O72x: elastic interpretation mechanism Phys. Mag. Lett. 64 147 [63] Schmidt H, Burkhard E, Sun B N and Rivera J P 1989 Uniaxial stresses induced ferroelastic detwinning of YBa 2Cu3O72x Physica C 157 555 [64] Kaiser D L, Gayle F W, Roth R S and Swartzendruber L J 1989 Thermomechanical detwinning of superconducting YBa 2Cu3O72x single crystals J. Mater. Res. 4 745

Handbook of Superconductivity

[65] Hatanaka T and Savada A 1989 Ferroelastic domain switching in YBa 2Cu3Ox single crystals by external stress Japan. J. Appl. Phys. 28 L794 [66] Kes P H, Pruymboom A, van der Berg J and Mydosh J A 1989 Pinning force of twin planes or grain boundaries in high Tc superconductors Cryogenics 29 228 [67] Miletich R, Murakami M, Preisinger A and Weber H W 1993 Microstructural characteristics of melt-powdermelt-grown YBa 2Cu3O72x crystals Physica C 209 415 [68] Diko P and Krabbes G 2003 Macro-cracking in meltgrown YBaCuO superconductor induced by surface oxygenation Supercond. Sci. Technol. 16 90 [69] Thoules M D 1990 Crack spacing in brittle films on elastic substrates J.Am. Ceram. Soc.73 2144 [70] Diko P and Krabbes G 2003 Formation of c-macrocracks during oxygenation of TSMG YBa 2Cu3O7/Y2BaCuO5 single-grain superconductors Physica C 399 151 [71] Jorgensen J D et al 1987 Oxygen ordering and the orthorhombic-to-tetragonal phase transition in YBa 2Cu3O7−x Phys. Rev. B 36 3608 [72] Diko P, Fuchs G and Krabbes G 2001 Influence of silver addition on cracking in melt-grown YBCO Physica C 363 60 [73] Kracunovska S, Diko P, Litzkendorf D, Habisreuther T, Bierlich J and Gawalek W 2005 Oxygenation and cracking in melt-textured YBCO bulk superconductors Supercond. Sci. Tecnol. 18 S142 [74] Eisterer M, Haindl S, Wojcik T and Weber H W 2003 ‘Magnetoscan’: a modified Hall probe scanning technique for the detection of inhomogeneities in bulk high temperature superconductors Supercond. Sci. Technol. 16 1282 [75] Diko P 2004 Cracking in melt-grown RE-Ba-Cu-O single-grain bulk superconductors Topical Reviev in Supercond. Sci. Technol. 17 R45 Eisterer M, Haindl S, Wojcik T and Weber H W 2003 ‘Magnetoscan’: a modified Hall probe scanning technique for the detection of inhomogeneities in bulk high temperature superconductors Supercond. Sci. Technol. 16 1282

G1.6 Neutron Techniques: Flux-Line Lattice Jonathan White

G1.6.1 Introduction The emergence of the flux-line lattice (FLL) in the mixed state of Type II superconductors is an entirely quantum phenomenon. Due to the phase-locked property of the Cooper pair wavefunction, the magnetic flux that penetrates the bulk of a Type II superconductor must be a multiple of the magnetic flux quantum Φ0 = h / 2e . Correspondingly, Φ0 is the actual flux carried by individual f lux lines, themselves often called ‘vortices’ since a supercurrent of finite vorticity circulates the flux-line axis. Due to magnetic flux quantisation, the flux-line density depends directly on the average internal induction inside the material, and is hence easily tunable by the applied magnetic field. In the idealised case of a perfectly isotropic superconductor, the f lux lines form a two-dimensional crystal that is triangularly coordinated in the plane perpendicular to the applied field. Of course, the FLLs in real systems display far richer behaviour, since their properties are strongly inf luenced by material-dependent factors such as pinning due to defects, superconducting gap and/or crystal anisotropy, electron spin susceptibility and thermal f luctuations. Experimental insight into all of these phenomena and more are readily obtained from neutron scattering studies. Not only is the neutron a penetrating bulk probe of a material, its intrinsic magnetic moment interacts with and diffracts from the periodic variation of the internal field due to the f lux lines. Indeed, after Abrikosov’s seminal prediction for the emergence of the FLL in Type II superconductors in 1957 (Abrikosov, 1957), the first neutron diffraction measurement from the FLL in elemental Nb was reported just 7 years later in 1964 (Cribier et al., 1964). Since this pioneering work, the neutron techniques used to study FLLs have matured and continue to diversify. Here we will introduce the recent advances, including both time-resolved and imaging techniques. To set the stage for these developments we begin by introducing the small-angle neutron scattering (SANS) technique, which in its standard

form continues to be the technique most commonly used for FLL studies.

G1.6.2 Small-Angle Neutron Scattering G1.6.2.1 Measurement Technique A cursory inspection of the flux-line density, and hence fluxline plane spacing, confirms the FLL to crystallise with a mesoscopic length scale. Since each two-dimensional unit cell of the FLL contains a single flux quantum, the internal induction is simply B = Φ0 /d 2 / sin β, where d is the planar spacing, and β is the angle between the two basis vectors that define the FLL primitive cell. Angle β can be chosen to adopt any value that preserves the two-dimensional Bravais lattice, with the cases β = 60° and β = 90° defining perfectly triangular and square FLLs, respectively. For the triangular lattice, the plane spacing is thus d = 3Φ0 /2 B , which at B = 1 T corresponds to 40 nm, indeed a mesoscopic distance that extends over many atomic unit cells. The study of magnetic structures displaying such length scales falls naturally within the nano- to near micro-metre regime covered by small-angle neutron scattering (SANS). The fundamental relation that describes the elastic scattering of neutrons by the FLL is Bragg’s law, nλ n = 2d sin θ, where θ is the scattering angle, and λ n is a neutron wavelength typically in the range of ‘cold’ neutrons (~5-20 Å). For λ n = 10 Å, the scattering angle from planes of the triangular FLL at 1 T is only ~0.7°. By inference the small scattering angles necessitate large sample-to-detector distances and the use of instruments at large-scale facilities. Figure G1.6.1(a) shows a typical SANS instrument used for FLL studies. Neutron beams are produced by either fission or spallation processes, and moderated to the cold wavelengths suitable for SANS. At pulsed neutron sources, the incident neutron wavelengths are determined by measuring the length

(

)

95

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FIGURE G1.6.1 (a) Schematic of a typical layout for a SANS instrument at a continuous neutron source. See the text for the details. (b) and (c) show sketches of the two possible experimental geometries for applied magnetic field with respect to the incoming beam, namely the parallel and perpendicular field geometries, respectively. (d) depicts the diffraction condition in reciprocal space from a FLL in the parallel field geometry like that illustrated in (b). The shaded grey part is a section of the Ewald sphere, the radius of which is defined by ki which spans the origin of diffraction (OD ) to the origin of reciprocal space (OR ). To satisfy the Bragg condition, the two-dimensional reciprocal FLL is rotated by a small angle ω about the vertical axis so that a reciprocal lattice vector q forms a chord of the Ewald sphere. The final wavevector k f connects OD with the end of q, and the angle between ki and kf is the Bragg 2θ. The Bragg peaks are depicted as spheres to indicate their finite size in reciprocal space. Their spatial extent depends on both the instrumental resolution and the properties of the FLL. Correspondingly, the spot lengths indicated in the inset can be estimated from the angular widths of the Bragg peaks measured at the detector.

of time taken to travel from the source to the detector, the so-called time-of-flight (TOF) technique. At continuous neutron sources, a helical tilt-slot velocity selector can be used to extract a narrow portion of the spectrum that is centred on the chosen wavelength, which has a full width at half maximum spread of typically ~10%–20%. The beam subsequently passes through a monitor; this provides a measure of the incident flux that can be used for an absolute normalization of the scattered intensity. After the monitor, the neutron beam profile is optimised using pinhole slits and a variable length collimation section before impinging on to the sample. Behind the sample, the scattered neutrons are detected by a rail-mounted two-dimensional multidetector. Since cold neutrons are

readily deflected due to air scattering, the detector is placed inside an evacuated flight tube to improve the measurement sensitivity. Likewise, the collimation section is also evacuated. Finally, to prevent both damage and overload of the detector, a neutron-absorbing beamstop is placed in front of the detector to protect it from the intense unscattered beam. In a standard setup, the collimation and sample-to-detector distances are chosen to be identical; this classic ‘pinhole’ geometry defines an optimal trade-off between incident flux and resolution. The superconductor of interest is typically installed inside a cryomagnet that provides both cryogenic sample temperatures and a nominally uniform horizontal magnetic field profile at the sample position. The dynamic range afforded by

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SANS is suitable for FLL studies over a wide field range, from a few mT up to the presently available maximum field of 17 T (Holmes et al., 2012). As indicated in Figure G1.6.1(a), using a superconducting split-coil (or a Helmholtz pair) to apply the magnetic field provides flexibility to conduct SANS studies in two standard geometries; with magnetic field applied approximately parallel [Figure G1.6.1(b)] or perpendicular [Figure G1.6.1(c)] to ki of the incident neutron beam. The parallel field geometry [Figure G1.6.1(b)] is chosen most often for experiments aimed at studying the FLL to learn about the fundamental properties of the Type II superconductor. In this geometry, any FLL plane can be brought to the Bragg condition by small rotation angles – or ‘rocking’ angles of the cryomagnet sample ensemble (by angles φ, ω in Figure G1.6.1). Bespoke motorised goniometer tables installed at the beamline enable the necessary tilting and rotation. Figure G1.6.1(d) shows a schematic illustration of diffraction from a triangular FLL. From SANS measurements in this geometry, information concerning the FLL coordination and alignment, and also the strength of the scattered intensity at each Bragg peak, can be used to learn about both the anisotropy and characteristic length scales of the superconducting state (see Section G1.6.2.2). As indicated in the inset of Figure G1.6.1(d), the Bragg peaks have a finite size in reciprocal space; they are broadened by both instrumental factors and FLL properties, such as its perfection and mosaicity. Therefore, collection of the full scattered intensity for a Bragg peak requires measurements over a range of rocking angles that cover the entire Bragg scattering volume. From these measurements, insight may be obtained concerning the FLL perfection. To leading-order the volume of the Bragg spot can be described by the three indicated lengths Wl, Wq and W⊥ , which give rise to the respective angular widths τ l, τ q and τ ⊥ measured experimentally. In the parallel field geometry, τ q is the width parallel to q in the detector plane and is dominated by the resolution function. Width τ ⊥ also lies in the detector plane but is orthogonal to τ q , and can give a measure of the FLL orientational order around the field axis. Width τ l is the so-called ‘rocking curve width’ measured orthogonal to the detector plane (perpendicular to the Ewald sphere), and is sensitive to the flux-line correlations along the applied field. The associated correlation length is finite if the flux lines deviate from the applied field direction due to pinning, or ‘bending’ over large distances. Although these two effects cannot be distinguished, a lower estimate for the longitudinal correlation length can nonetheless be estimated as σ L ~ 2π / τ l qh,k , where τ l is the angular spread (in radians) of the peak at wavevector qh,k . Since width τ l is also dependent on instrumental factors, the accurate determination of σ L requires the deconvolution of the instrumental and sample contributions to τ l. A more complete discussion concerning these effects is given by (Cubitt et al., 1992). In general, the contribution to the observed spot widths due to the SANS resolution function is smaller along ki by a factor of ~ k /q compared with directions perpendicular to ki

Esklidsen (1998b). Therefore, measurements done in the parallel field geometry are most sensitive to flux-line correlations along the field, as discussed above. In the perpendicular field geometry [Figure G1.6.1(c)], the instrument is instead most sensitive to the correlation lengths associated with the flux line orientational and positional order (these being less easily extractable in the parallel field geometry). Consequently, the perpendicular field geometry may be chosen for studies where the FLL plays the role of a model crystal on which to test theories of crystallisation. (Laver et al., 2008) provides an example study where a combination of both SANS in the perpendicular field geometry and a reverse Monte Carlo analysis provide insight concerning the structural correlations in the Bragg glass phase of the FLL in an impure Nb sample. For further discussion on the application of SANS for the study of vortex matter, the reader is referred to the review of Mühlbauer et al. (2019).

G1.6.2.2 Physical Properties of the Superconducting State For studies aimed at elucidating fundamental properties of the host superconductor, SANS experiments in the parallel field geometry are preferred. In this geometry, measurement of the FLL coordination and alignment, and also the Bragg scattering intensity can be done both simultaneously and efficiently. Measuring these properties as functions of magnetic field and sample temperature can constitute the content of a typical SANS experiment. In general, FLLs are observed to display a diverse range of field- and temperature-dependent coordinations and alignments across a broad range of superconductors, so much so that observations of simple triangular FLLs are comparatively rare. As an example, Figure G1.6.2 summarises a variety of FLL coordinations observed at different applied fields in single crystals of High-Tc YBa 2Cu3O7 (White et al., 2009). For low temperatures and fields that are moderate fractions of H c2 , square and square-like FLL coordinations are observed in many materials such as YBa 2Cu3O7 [Figure G1.6.2(e)], borocarbide LuNi2B2C (Eskildsen et al., 1997) and heavy-fermion CeCoIn5 (Bianchi et al., 2008). In these cases, the physical mechanism that stabilises these high-field coordinations can be generally assigned to ‘nonlocal’ effects that originate from a superconducting anisotropy in the system. Candidate anisotropies include the Fermi surface or a nodal superconducting gap, since each is expected to become increasingly influential on the FLL coordination at higher fields when the flux lines interact more strongly. Even in the conventional s-wave superconductor Nb, for certain directions of applied field with respect to the crystal a rich array of low-symmetry FLL structures has been observed, thus evidencing the complexity of the flux-line interactions in simple elemental materials (Laver et al., 2006; Mühlbauer et al., 2009). Due to the generally tiny differences in free energy between different FLL coordinations, it remains a significant challenge to

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FIGURE G1.6.2 FLL diffraction patterns obtained in the parallel field SANS geometry from YBa 2Cu 3O7 at T = 2 K and in applied fields, B||c of (a) 1.5 T after a field-cooling (FC) protocol, (b) 1.5 T after an oscillation field-cooling (OFC) protocol, (c) 4 T after FC, (d) 6.5 T after FC and (e) 10.8 T after FC. Each image is constructed by summing together the multidetector measurements done over a range of rocking angles. This allows the observation of the reciprocal FLL coordination in a single image. Due to the two-dimensional structure of the reciprocal FLL, the real-space FLL coordination is obtained by a 90° rotation of the reciprocal space image around the magnetic field axis, and the addition of a single flux line at the origin. [Reprinted with permission from White et al., (2009). Copyright © 2009, American Physical Society.]

obtain unambiguous and precise theoretical descriptions of the mechanisms that stabilise certain FLL phases. In contrast, measurements of FLL diffraction intensities prove more effective at providing important information concerning the microscopic superconducting state. For a structurally well-ordered FLL, the local field variation inside the superconductor B ( r ) is readily expressed by the sum over the spatial Fourier components at the various reciprocal space wavevectors, qh,k of the two-dimensional reciprocal FLL B(r ) =

∑F

q h ,k

e (iq h ,k ⋅r )

(G1.6.1)

h,k

In principle, each Fourier component can give rise to diffracted intensity when the associated wavevector is brought to the diffraction condition [Figure G1.6.1(d)]. Then the scattered neutron intensity measured on the detector at qh,k depends on Fq h ,k according to Christen et al., (1977): 2

−1 I q h ,k = 2πφ ( γ / 4 ) Vλ 2n Φ0−2qh,k Fq h ,k

2

(G1.6.2)

where φ is the intensity of the incident neutron beam, γ is the neutron magnetic moment in nuclear magnetons, V is the sample volume and λ n is the neutron wavelength. To determine the quantity Fq h ,k accurately, measurements of the scattered intensity are required over the range of rocking angles that covers the entire Bragg volume. Plotting the diffracted intensity as a function of rocking angle yields the so-called rocking curve, and the quantity I q h ,k is obtained by integrating the area under a suitable lineshape fit of the data. From such measurements, the absolute magnitude of Fq h ,k can be obtained. According to Equation (G1.6.1), measurements of Fq h ,k can then be used for a reconstruction of B ( r ) from experimental data, and compared with model expectations for the microscopic superconducting state. By means of example, Figure G1.6.3 summarises results from a study of the FLL in LuNi2B2C (Densmore et al., 2009). The SANS pattern shown in Figure G1.6.3(a) was obtained by summing many multidetector measurements taken over a range of rocking angles around the vertical axis. Numerous orders of diffraction from the FLL were observed,

FIGURE G1.6.3 Data from a SANS study of the FLL in LuNi 2B2C. Panel (a) shows the SANS diffraction pattern from a square FLL at H = 0.5 T and T = 2 K. The image shows a sum of detector measurements recorded over a sufficiently broad range of sample rotation angles such that the intensities for many orders of diffraction are simultaneously observable. Panel (b) shows the rocking curves for q10 and q32 Bragg peaks each fitted by an appropriate lineshape to obtain values of I q h ,k . From the resulting values of Fq h ,k the real-space reconstruction of B ( r ) is computed and shown in panel (c). [Reprinted with permission from Densmore et al. (2009). Copyright © 2009, American Physical Society.]

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and the various values of Fq h ,k at each qh,k were obtained from rocking curves, some examples of which are shown in Figure G1.6.3(b). From these measurements, the amplitudes of the various Fourier components were obtained, though it is not possible to determine their relative signs. To help resolve this problem in general, one can turn to theoretical expectations for the sign scheme, and/or determine them experimentally using the µSR technique. Using these approaches the authors could use the SANS data to perform the real-space reconstruction of B ( r ) as shown in Figure G1.6.3(c). A similar study was also done earlier on the candidate spin-triplet superconductor Sr2RuO4 (Kealey et al., 2000). The study outlined above done on LuNi2B2C ultimately amounted to a model-free real-space reconstruction of B ( r ) in the mixed state. In many superconductors, however, it turns out to be only experimentally feasible to measure Fq h ,k for the lowest q Bragg spots. Therefore, a model is required for the data analysis. For strongly type II materials with Ginzburg–Landau parameter κ = λ / ξ 1/ 2 , a modified London model has proved useful for this purpose (Yaouanc et al., 1997): Fq h ,k

B (− cq 2 ξ2 ) e h ,k = 2 2 1 + qh,k λ

(

)

(G1.6.3)

where B is the average internal field, λ is the London penetration depth, ξ is the Ginzburg–Landau coherence length, and c is a constant expected to lie between ¼ and 2. Various theoretical and experimental studies indicate a value for c close to ½ to be usually appropriate. Equation (G1.6.3) amounts to the usual London model modified by a Gaussian cut-off term. The inclusion of the cutoff is justified on physical grounds; being a purely local model, the London model is obtained in the limit λ / ξ → ∞ which describes infinitesimally narrow flux-line cores. According to this model, the internal field diverges logarithmically at the flux-line axis. To avoid this unphysical situation, the cut-off is included to affect a suppression of the field divergence over a length scale of order the flux-line core size ξ. A different form for the cut-off derived by Clem (1975) has a qualitatively similar effect as the gaussian term. Both models are conveniently algebraic and enable a simple analysis of Fq h ,k data, though they are nonetheless approximations, especially for the lowtemperature T Tc , low-field B Bc2 , regime where SANS FLL studies are typically carried out. Nonetheless, more rigorous calculations confirm the approximations as reasonable (Brandt, 2003; Suzuki et al., 2010). According to Equation (G1.6.3), the expectation is that at constant temperature, Fq h ,k falls exponentially with increasing field at a rate determined by the cut-off term. In the lowfield limit where the cut-off → 1, the size of Fq h ,k depends on the absolute size of λ. Quantitative estimates for both λ and ξ length scales in the zero-field limit are usually obtained by using Equation (G1.6.3) to fit Fq h ,k data measured as a

FIGURE G1.6.4 The field dependence of Fq h ,k (F) for first-order Bragg spots in KFe2As2 at T = 50 mK and 1.5 K as reported by KawanoFurukawa et al. (2011). The solid lines are theoretical fits to Equation (G1.6.3) as described in the text. [Reprinted with permission from Kawano-Furukawa et al. (2011). Copyright © 2011, American Physical Society.]

function of field, at constant temperature. By means of example, Figure G1.6.4 shows such measurements of Fq1,0 taken on the iron–pnictide superconductor KFe2As2 with Tc ( H = 0 )=3.6 K (Kawano-Furukawa et al., 2011). Datasets obtained at temperatures of 50 mK and 1.5 K are each well-described by fits to Equation (G1.6.3). In this instance, the values of ξ were fixed according to measurements of Bc2 by magnetometry; and the c parameter fitted instead. The best fit at 50 mK gave λ=203 nm and c = 0.52 with ξ=13.5 nm (from Bc2=1.8 T), while at 1.5 K the best fit gave λ=240 nm and c = 0.55 with ξ=15.9 nm (from Bc2=1.3 T). The monotonic decrease of Fq1,0 with increasing field observed in KFe2As2 is displayed by virtually all Type II superconductors. Notable exceptions include TmNi2B2C (DeBeer-Schmitt et al., 2007) and CeCoIn5 (Bianchi et al., 2008) where Pauli paramagnetic effects are important. The data in Figure G1.6.4 also evidence the variation of Fq h ,k with temperature at constant field, thus demonstrating experimental access to the temperature-dependent superfluid density ns (T ) (since λ (T ) ∝ 1/ ns (T ) ). Therefore, temperature-dependent measurements of Fq h ,k are sensitive directly to thermal excitations of quasiparticles above the superconducting gap, and are performed to obtain direct insight into both the anisotropy of the superfluid density (White et al., 2014), and details of the superconducting gap structure such as the existence of nodes (Kawano-Furukawa et al., 2011; Gannon et al., 2015). This is particularly the case at very low field where the cut-off term in Equation (G1.6.3) → 1. In this

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situation, the more ambiguous core effects can be neglected, and the neutron intensity is essentially directly proportional to the superfluid density. By combining Equations (G1.6.2) and (G1.6.3), we can point out some notable aspects of the modified London model in the context of conventional SANS experiments. Firstly, we see that I q h ,k ∝ λ −4 , which makes SANS experiments on many unconventional superconductors with long penetration depths, e.g. heavy-fermion systems, significantly more challenging than on conventional Type II superconductors like Nb. Secondly, −5 we find that I q h ,k ∝ qh,k so that higher-order Bragg peaks beyond the fundamental I q1,0 become progressively difficult to observe with increasing {h, k}. Related to this, since q ∝ B , the cut-off term in Equation (G1.6.3) leads to a reduction in I q h ,k with increasing field due to the finite extent of the fluxline cores, and their increasing overlap as the flux lines move closer together. Careful consideration of these factors in advance often provide reliable judgement for the feasibility of a SANS study.

G1.6.3 Developments beyond Usual Small-Angle Neutron Scattering G1.6.3.1 Polarised Small-Angle Neutron Scattering The standard application of the SANS technique for FLL studies neither depends on nor requires any knowledge of the neutron polarisation state. In general, however, since neutrons are spin-1 2 quantum objects, they can display two spin eigenstates with respect to a quantisation axis, and be made to reorient, or flip, between these two states due to certain magnetic scattering processes. Therefore, knowledge of the neutron polarisation states before and after the scattering process can provide deeper insight concerning the local field distribution in the bulk of a material than otherwise achievable using only unpolarised beams. A common method of polarisation analysis used in SANS experiments is longitudinal analysis, which describes the analysis of the neutron polarisation state before and after the sample along one direction, usually parallel to ki. In a simple modern setup, a polarising supermirror is placed after the velocity selector to produce a polarised neutron beam, and a 3He cell placed behind the sample to analyse the polarisation of the scattered beam. Using this setup allows distinction between non-spin-flip (NSF) and spin-flip (SF) crosssections, + + and + − where the first (second) symbol describes the incoming (outgoing) neutron polarisation state; so-called half polarisation analysis. Inclusion of an additional spin flipper before the sample allows access to the two further crosssections − + and − − in a full polarisation analysis setup. In the context of the FLL, polarised neutron techniques have found use in so-called depolarisation studies (Roest and Rekveldt, 1993), and small-angle neutron spin echo studies of moving FLLs (Forgan et al., 2000). Here we describe briefly how polarised SANS is a useful probe of superconducting anisotropy.

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In a purely isotropic superconductor, flux-line screening supercurrents flow in the plane perpendicular to the direction of the average field, and so the local fields everywhere are parallel, or ‘longitudinal’ to this direction, i.e. Fq h ,k contains components only along the field. In the parallel field geometry, scattering from local longitudinal fields will not flip the spin of a longitudinally polarised neutron beam, with incident polarisation either parallel or anti-parallel to the applied field. In anisotropic superconductors such as High-Tc cuprates, screening supercurrents may lie preferentially within an easy plane, even when the field is applied at an angle far from perpendicular to this plane. In this situation, while the flux lines still follow the average field direction, Fq h ,k has both longitudinal and perpendicular (or ‘transverse’) field components with respect to the average field direction. The transverse field components lead to SF scattering of a longitudinally polarised neutron beam (Böni and Furrer, 1998), and hence a polarisation analysis of the neutron spin state allows distinction between scattering due to the longitudinal and transverse local fields. Furthermore, the relative weight of NSF/SF scattering can be compared quantitatively with model estimates obtained using an extended form of the London theory that caters for the effective mass anisotropy of the supercarriers (Thiemann et al., 1988). These calculations also show that the longitudinal and transverse field components can be expected to be of the same order in size, but that the latter vanish for a field applied perfectly along a main crystal axis. The study of transverse field components by polarised SANS was first done on the High-Tc cuprate YBa 2Cu3O7-δ (Kealey et al., 2001). An observable consequence of the SF scattering due to perpendicular field components is shown in Figure G1.6.5. A neutron SF in a magnetic field leads to a change of the neutron potential energy by an amount equivalent to the Zeeman

FIGURE G1.6.5 The rocking curves for NSF and SF scattering from a FLL Bragg peak in YBa 2Cu 3O7-δ and with and applied field of 0.5T at 60° to the c-axis. The two curves for SF scattering denote the two different initial neutron spin states. The splitting of the peak positions of the SF curves arise from the Zeeman splitting of the neutron potential energy due to the applied field at the sample. In each instance this leads to kf ≠ ki and a small shift in the Bragg angle as indicated in the inset. [Reprinted with permission from Kealey et al., (2001). Copyright © 2001, American Physical Society.]

Neutron Techniques: Flux-Line Lattice

splitting for the neutron. The scattering is still elastic, yet due to energy conservation, there is necessarily a small change in the neutron kinetic energy. The result is that kf ≠ ki and the rocking angle at which the Bragg condition is satisfied becomes shifted relative to the usual angle expected for NSF scattering. Experimental observation of this effect in the study of YBa 2Cu3O7-δ is shown in Figure G1.6.5, where the SF rocking curves are shifted relative to the NSF curve to higher (lower) in angle for a SF process involving a loss (gain) in neutron kinetic energy. As also seen in Figure G1.6.5, without polarisation analysis the NSF and SF scattered intensities would overlap, and thus be indistinguishable. In a different study of the FLL in superconducting Sr2RuO4 (Rastovski et al., 2013), the extreme anisotropy of this system leads to such large transverse field components for fields applied nearly in the basal plane that only SF scattering can be observed.

G1.6.3.2 Time-Resolved Small-Angle Neutron Scattering The elastic moduli of the FLL are measured typically by bulk transport techniques on macroscopic samples, or by surfacesensitive probes on thin samples. Interpretation of the results can be complicated by parasitic pinning and/or geometric affects, and furthermore these studies can provide no direct insight into the momentum-resolved bulk FLL elasticity. Scientists at FRM-II in Münich, Germany have recently developed a time-resolved SANS technique that enables the study of slow FLL dynamics with ms time resolution (Mühlbauer et al., 2011). Using this technique, the bulk FLL elastic moduli can be studied directly, and a quantitative estimate of the FLL tilt modulus in a bulk Nb sample was obtained. The time-resolved SANS method amounts to the usual SANS technique extended by a time-resolved multidetector. The general experimental approach is to firstly excite the

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sample using an external control parameter, and then to measure the time-dependent response/relaxation of the system in response to this excitation. To improve statistics the measurement is done stroboscopically; i.e. repeated cyclically with the data from all cycles summed coherently, and with the repetition cycles of the excitation and the time-resolved detector phase-locked. For the stroboscopic SANS study of the FLL in Nb (Mühlbauer et al., 2011), the magnetic field direction, which defines the longitudinal alignment of the FLL, is chosen as the control parameter. The sample is placed at the heart of a bespoke arrangement of two orthogonal pairs of Helmholtz coils as shown in Figure G1.6.6(a). The outer pair generates the usual large static field H stat along the Y-axis approximately parallel to ki. A smaller orthogonal field H osc along the X-axis is generated by the inner pair. H osc is time-varying and driven by a rectangular pulse waveform generator. For a suitable nonzero value of H osc , the field at the sample position is rotated with respect to H stat by an angle ε = arctan ( H osc /H stat ). This yields two equilibrium field orientations for H osc = 0 and ≠ 0 , and hence two equilibrium FLL orientations with associated Bragg angles ε apart. In the experiment, the relaxation of the FLL on moving between the two equilibrium conditions is studied as H osc is varied. This is done by measuring the time-dependent FLL scattered intensity at a certain rocking angle, and repeating the measurement for a range of rocking angles to obtain a time-resolved mapping as shown in Figure G1.6.6(b). As seen from the map, when H osc is altered between its zero and non-zero values, the Bragg angle changes and the relaxation of the FLL between the two conditions can be followed by tracking the intensity as a function of time. Using the observed ~100 ms timescales associated with the relaxation processes, (Mühlbauer et al. 2011) obtained quantitative estimates for the FLL tilt modulus in a Nb sample, as functions of both main field H stat and temperature.

FIGURE G1.6.6 (a) Sketch of the double Helmholtz coil setup for the time-resolved stroboscopic SANS experiment. The sample is located at the centre of two coils that generate orthogonal fields, H stat and H osc . Combining the large static field H stat with the smaller time-varying field H osc allows a rotation of the net field within the X–Y plane. The authors chose values of H stat and H osc so that the field was rotated by ε ∼ 2° between zero and non-zero H osc . This corresponds to the two equilibrium FLL orientations with rocking angles 2° apart as shown in the timeresolved map in panel (b). Changing H osc leads to a change in equilibrium FLL position, and the time-dependent relaxation of the FLL on switching between the two positions is thus monitored experimentally. [Reprinted with permission from Mühlbauer et al. (2011). Copyright © 2011, American Physical Society.]

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FIGURE G1.6.7 (a) Schematic of the general nGI setup showing a source grating G0, the phase grating G1 and an analyser grating G2. At the sample position, a cryomagnetic environment for studying the IMS can be installed. (b) Sketch showing the expected variation of the neutron intensity in a pixel of the detector as one of the gratings is moved along direction x which is transverse to the incoming beam. [Reprinted with permission from Grünzweig et al. (2008). Copyright © 2008, AIP Publishing LLC.]

The ms time resolution of the stroboscopic SANS technique is largely determined by time-smearing arising from the finite collimation and wavelength spread of the incoming beam. In principle, this smearing can be suppressed and the resolution reduced towards the µs regime using the TISANE technique, which makes use of a chopper installed upstream from the sample (Kipping et al., 2008).

G1.6.3.3 Neutron Grating Interferometry For low κ Type II superconductors like Nb, where κ lies close to border between Type I and Type II superconductivity, attractive interactions between flux lines can exist in addition to the usual repulsive interactions that lead to the formation of the Abrikosov FLL. Consequently, at very low flux-line densities just above H c1, and in the presence of a non-zero demagnetization coefficient, an intermediate mixed state (IMS) can form. In this state, ~10-µm-sized FLL domains nucleate that are embedded within a field-free Meissner region. For this reason, the IMS has long attracted interest as a model system for the general study of domain nucleation and morphology. However, the inherent length scale of the IMS domains lies beyond the realm of conventional SANS which can only directly probe the FLL within the domains. Consequently, most studies of the IMS domain structures have been done using surfacesensitive probes (Sarma 1968; Krägeloh 1970). Recently, scientists at both the Paul Scherrer Institut and FRM-II have applied the technique of neutron grating interferometry (nGI) to conduct the first direct-space imaging study of the IMS domain formation in the bulk of a high purity, single crystal Nb sample (Grünzweig et al., 2008; Grünzweig et al., 2015). The general setup for the nGI experiment, shown schematically in Figure G1.6.7(a), makes use of three diffraction gratings (Grünzweig et al., 2008). The source grating G0 creates a periodic array of coherent line neutron sources that allows the use a relatively large beam of ~cm-sized cross-section, but which does not compromise the coherence requirements of the interferometer. Phase grating G1 behind the sample imprints a periodic phase modulation onto the incoming wave field. Due to the Talbot effect, this phase modulation results

in an intensity modulation in the plane of the multidetector. This intensity modulation is typically smaller than the pixel size of the detector, but it can nonetheless be measured using an analyser grating G2 placed directly in front of the detector. By scanning one of the gratings transversally along x, the intensity observed in each detector pixel oscillates as shown in Figure G1.6.7(b), and can be written as a Fourier series: I ( x ) = ∑ i aicos[ikx + φi ] ≈ a0 + a1cos[ikx + φ1 ]. Here, ai are amplitude coefficients, φi the corresponding phase coefficients, and k = 2pπ2 , where p2 is the grating period of G2. Extraction of the zeroth-order Fourier coefficient a0 for each detector pixel yields the transmission image (TI), as would be obtained using standard neutron radiography. Already from the TI, information concerning the FLL morphology can be inferred because the TI is sensitive to a similar range of length scales as SANS. The spatially resolved information concerning the IMS domain structure is obtained from a1 , the amplitude of the intensity oscillation. Due to ultra-SANS from IMS domains of length scale 10 µm, the neutron beam coherence behind G1 becomes degraded, leading to a reduction in the value of a1 expected for a fully coherent neutron beam. The coherence degradation is the origin of a contrast that is visualised in realspace in the form of ‘dark-field images’ (DFIs). Therefore, it is the DFIs that yield information on the long length scales associated with the IMS domains that lie out of reach for conventional SANS. Most recently, DFI analysis has provided new insight into the nucleation and morphology of the IMS domains in a high purity Nb sample (Grünzweig et al., 2015).

G1.6.4 Summary In this entry, we have showcased the neutron techniques presently being used to study the flux-line lattice (FLL) in Type II superconductors. Due to the crystallisation of the FLL with a mesoscopic length scale, the workhorse technique of small-angle neutron scattering (SANS) continues to be the choice probe for studying FLLs and their properties in many superconducting classes, ranging from conventional superconductors such as elemental Nb (Laver et al., 2006), (Mühlbauer et al., 2009), to more exotic systems such

Neutron Techniques: Flux-Line Lattice

as borocarbides (Yaron et al., 1996), (Eskildsen et al., 1998a), High-Tc cuprates (Cubitt et al., 1993), (Chang et al., 2012), pnictides (Eskildsen et al., 2011) and heavy-fermion materials (Bianchi et al., 2008), (Gannon et al., 2015). As an experimental probe of FLLs, SANS has a special status; in contrast to surface-sensitive techniques, SANS is a non-perturbative bulk probe of the FLL that allows a direct imaging of the FLL coordination and alignment. In favourable situations, information concerning the superconducting gap function of the material can be obtained. In addition, SANS studies can also be conducted for a wide range of applied magnetic fields ranging from typical values for H c1 up to 17 T. This field range overlaps with, and even expands beyond those accessible by other probes of the FLL, enabling SANS to provide unique and complementary information concerning the microscopic FLL structure and perfection both in the bulk, and under extreme conditions. We have focused on three developments that in various ways broaden the parameter space afforded by conventional SANS. By using polarised neutron techniques, the deeper interrogation of the local field distribution due to the FLL in anisotropic superconductors becomes possible. Time-resolved detection as used in the stroboscopic SANS technique permits the direct study of slow FLL dynamics, which in turn enables quantitative estimates of the FLL elastic moduli to be obtained. By using neutron grating interferometry, ultraSANS scattering from >10-µm -sized FLL domains in the intermediate mixed state (IMS) generates a visible real-space contrast in the form of dark-field images. This has allowed the first study of bulk FLL domain nucleation and morphology in the IMS of a high purity Nb crystal. Promisingly, there remains plenty of scope for further broadening the accessible spatial and temporal regimes relevant for the FLL, and also with improved resolution (Rekveldt et al., 2001), (Pautrat et al., 2012). This promises many future opportunities for new and novel neutron scattering experiments.

References Abrikosov AA (1957) On the magnetic properties of superconductors of the second group Sov. Phys. JETP 5:1174−1182 Bianchi AD et al. (2008) Superconducting vortices in CeCoIn5: toward the Pauli-limiting field Science 319:177−180 Böni P, Furrer A (1998) Neutron Scattering in Layered Copper Oxide Superconductors (Furrer A, ed) Dordrecht: Kluwer Brandt EH (2003) Properties of the ideal Ginzburg-Landau vortex lattice Phys. Rev. B 68:054506 Chang J et al. (2012) Spin density wave induced disordering of the vortex lattice in superconducting La 2−xSr xCuO4 Phys. Rev. B 85:134520 Christen DK, Tasset F, Spooner S, Mook HA (1977) Study of the intermediate mixed state of niobium by small-angle neutron scattering Phys. Rev. B 15:4506−4509 Clem JR (1975) Simple model for the vortex core in a type II superconductor J. Low Temp. Phys. 18:427−434

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Cribier D, Jacrot B, Madhav Rao L, Farnoux B (1964) Mise en evidence par diffraction de neutrons d’une structure periodique du champ magnetique dans le niobium supraconducteur Phys. Lett. 9:106−107 Cubitt R et al. (1992) Neutron diffraction by the flux lattice in high-Tc superconductors Physica B, 180 & 181:377−379 Cubitt R et al. (1993) Direct observation of magnetic flux lattice melting and decomposition in the high-Tc superconductor Bi2.15Sr1.95CaCu2O8+x Nature 365:407−411 DeBeer-Schmitt L et al (2007) Pauli paramagnetic effects on vortices in superconducting TmNi2B2C Phys. Rev. Lett. 99:167001 Densmore JM et al. (2009) Small-angle neutron scattering study of the vortex lattice in superconducting LuNi2B2C Phys. Rev. B 79:174522 Eskildsen MR et al. (1997) Structural stability of the square flux line lattice in YNi2B2C and LuNi2B2C studied with small angle neutron scattering Phys. Rev. Lett. 79:487−490 Eskildsen MR et al. (1998a) Intertwined symmetry of the magnetic modulation and the flux-line lattice in the superconducting state of TmNi2B2C Nature 393:242−245 Eskildsen MR et al. (1998b) Small angle neutron scattering studies of the flux line lattices in the borocarbide superconductors PhD Thesis: Risø National Laboratory, Denmark. Eskildsen MR, Forgan EM, Kawano-Furukawa H (2011) Vortex structures, penetration depth and pairing in iron-based superconductors studied by small-angle neutron scattering Rep. Prog. Phys. 74:124504 Forgan EM et al. (2000) Measurement of vortex motion in a type-II Superconductor: a novel use of the neutron spinecho technique Phys. Rev. Lett. 85:3488−3491 Gannon WJ et al. (2015) Nodal gap structure and order parameter symmetry of the unconventional superconductor UPt3 New J. Phys. 17:023041 Grünzweig C et al. (2008) Bulk magnetic domain structures visualized by neutron dark-field imaging App. Phys. Lett. 93:112504 Grünzweig C et al. (2015) Visualizing the morphology of vortex lattice domains in a bulk type-II superconductor submitted Holmes AT, Walsh GR, Blackburn E, Forgan EM, SaveyBennett M (2012) A 17 T horizontal field cryomagnet with rapid sample change designed for beamline use Rev. Sci. Instrum. 83:023904 Kawano-Furukawa H et al. (2011) Gap in KFe2As2 studied by small-angle neutron scattering observations of the magnetic vortex lattice Phys. Rev. B 84:024507 Kealey PG et al. (2000) Reconstruction from small-angle neutron scattering measurements of the real space magnetic field distribution in the mixed state of Sr2RuO4 Phys. Rev. Lett. 84:6094−6097 Kealey PG et al. (2001) Transverse-field components of the flux-line lattice in the anisotropic superconductor YBa 2Cu3O7−δ Phys. Rev. B 64:174501 Kipping D, Gähler R, Habicht K (2008) Small angle neutron scattering at very high time resolution: Principle and simulations of ‘TISANE’ Phys. Lett. A 372:1541−1546

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Krägeloh U (1970) Der Zwischenzustand bei Supraleitern zweiter Art Phys. Stat. Sol. (b) 42:559−576 Laver M et al. (2006) Spontaneous symmetry-breaking vortex lattice transitions in pure niobium Phys. Rev. Lett. 96:167002 Laver M et al. (2008) Uncovering flux line correlations in superconductors by reverse Monte Carlo refinement of neutron scattering data Phys. Rev. Lett. 100:107001 Mühlbauer S et al. (2009) Morphology of the superconducting vortex lattice in ultrapure niobium Phys. Rev. Lett. 102:136408 Mühlbauer S et al. (2011) Time-resolved stroboscopic neutron scattering of vortex lattice dynamics in superconducting niobium Phys. Rev. B 83:184502 Mühlbauer S et al. (2019) Magnetic small-angle neutron scattering Rev. Mod. Phys. 91:015004 Pautrat A, Brulet A, Simon C, Mathieu P (2012) Flux-lines lattice order and critical current studied by time-of-flight small-angle neutron scattering Phys. Rev. B 85:184504 Rastovski C et al. (2013) Anisotropy of the superconducting state in Sr2RuO4 Phys. Rev. Lett. 111:087003 Rekveldt MT, Keller T, Golub R (2001) Larmor precession, a technique for high-sensitivity neutron diffraction Europhys. Lett. 54:342−346

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Roest W and Rekveldt MT (1993) Three-dimensional neutrondepolarization analysis of the magnetic flux distribution in YBa2Cu3O7−δ Phys. Rev. B 48:6420−6425 Sarma NV (1968) Transition from the flux lattice to the intermediate state structures in a lead-indium alloy Philos. Mag. 18:171−176 Suzuki KM, Inoue K, Miranović P, Ichioka M, Machida K (2010) Generic first-order orientation transition of vortex lattices in type II superconductors J. Phys. Soc. Jpn. 79:013702 Thiemann SL, Radović Z, Kogan VG (1989) Field structure of vortex lattices in uniaxial superconductors Phys. Rev. B 39:11406−11412 White JS et al. (2009) Fermi surface and order parameter driven vortex lattice structure transitions in twin-free YBa 2Cu3O7 Phys. Rev. Lett. 102:097001 White JS et al. (2014) Magnetic field dependence of the basalplane superconducting anisotropy in YBa 2Cu4O8 from small-angle neutron scattering measurements of the vortex lattice Phys. Rev. B 89:024501 Yaouanc A, Dalmas de Réotier P, Brandt EH (1997) Effect of the vortex core on the magnetic field in hard superconductors Phys. Rev. B 55:11107−11110 Yaron U et al. (1996) Microscopic coexistence of magnetism and superconductivity in ErNi2B2C Nature 382:236−238

G2 Introduction to Section G2: Measurement and Interpretation of Electromagnetic Properties Fedor Gömöry Superconductor is not merely an ordinary conductor whose electrical resistivity has become vanishingly small; indeed, almost invariably, the dissipation is finite, but not ohmic, with the electric field E being a rapidly increasing function of the current density J. In this context, the crucial difference between normal conductors and Type II superconductors (all the HTS compounds are extreme type II materials) is that relevant voltages and electric fields are generated via electromagnetic induction from magnetic fields changing within the sample. It is therefore essential to include in the problem the magnetic fields generated by the supercurrents themselves; the dissipation may then be viewed as arising from magnetic hysteresis as flux moves in and out of the sample. Furthermore, it is often much more convenient to apply contactless magnetic methods to measure the relationship between E and J, particularly when current densities often exceed 1010Am−2, and electric fields of interest may drop below 10 −9Vm−1. Chapter G2.1 sets out the relevant electromagnetic theory in a form suitable for description of superconductors. Laws of general validity in physics are applied on superconductors providing formulas of invaluable use. Then, Chapter G2.2 summarizes recent rapid progress of computer technology and numerical modelling that allowed to improve significantly the correspondence with real applications. Superconducting wires are usually best characterized by transport measurements (Chapter G2.3) akin to those used for conventional conductors. For construction of large magnets, e.g. for particle accelerators, an accurate and detailed technical specification of the high current conductors is essential (Chapter G2.4). The magnetic techniques to obtain E and J are particularly useful for other kinds (bulks, thin films) of

superconducting artefacts. Experiments using slowly ramping magnetic fields usually exceeding several Tesla are described in the Chapter G2.5. Basic information along the same lines can also be obtained when exposing superconductors to AC fields with millitesla amplitudes (Chapter G2.6). Normal conductors and superconductors behave very differently when AC currents are involved, even at frequencies as low as 50 or 60 Hz, and AC loss measurement is not a trivial task (Chapter G2.7). Several applications foresee that the superconductors already magnetized are later exposed to magnetic fields with a direction different from the original one. Unusual development of current and field distributions is expected in crossed magnetic fields (Chapter G2.8). Because of their much larger energy gap, the HTS materials have found a role at microwave frequencies that was inaccessible to the LTS superconductors. Their low losses at GHz frequencies make them attractive as filters and other components (Chapter G2.9). A major difficulty with most electromagnetic characterizations is that they average the response over a macroscopic scale, millimeters or more. However, the HTS materials are often inhomogeneous on a much finer scale as was discussed in Chapter C1; this has given the incentive to develop a number of techniques that can examine the electromagnetic response on a scale of microns or less (Chapter G2.10). Unsurprisingly, the main interest in experimental investigation of superconductor materials focuses on the superconducting state at temperatures below the critical one. However, understanding of superconductivity requires also the knowledge of normal-state properties. This topic is of particular interest for cuprate superconductors (Chapter G2.11).

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G2.1 Electromagnetic Properties of Superconductors Archie M. Campbell

G2.1.1 Introduction This article concentrates on the physics of electromagnetic fields in superconductors. For many practical purposes it is possible to assume that it is a material with permeability μo, so that B = H and M = 0. This is generally the case in high fields in the mixed state when the critical state model can be used. This regime has more in common with eddy currents in copper than a magnetic material. There may be a magnetic moment, but you cannot usefully define a local M or effective permeability. However, the incorporation of superconductors into Maxwell’s equations provides a fascinating example of how general these equations are, although Maxwell never envisaged superconductors. It is commonly assumed that the description of electromagnetic fields in materials is well established and that superconductors can easily be accommodated. However, the standard treatment of magnetic fields was developed before superconductors were discovered, and it is necessary to go back to first principles to avoid confusion in understanding the electromagnetic properties of these materials. It is also necessary to distinguish between approaches which use the external field and a measurable macroscopic quantity, such as the magnetic moment, and those which attempt to describe internal, or local, fields. With the first approach the sample is treated as a ‘black box’ and all properties are related to the external sources of energy such as a solenoid. Work and heat are defined in terms of inputs and outputs crossing the boundaries between the external sources and the system, and the state is described purely by external measurements. This approach is appropriate to complex materials, or samples with large demagnetising factors as is usually the case with high-Tc superconductors. The second, local field, description is appropriate for understanding the physics of the material on a molecular level. Both approaches are required, but the terms used must be defined carefully since the same letters are often used with different meanings in the two pictures. This article will start with conventional electromagnetism in a form most suitable for the description of superconductors. 106

The equations of fields in free space are straightforward for the purposes of this article. In this case, there is only one magnetic field, or flux density, which is defined by the force between currents, but also produces a voltage when it is changed. This field is called H or B, with B = μοH, according to whether we want to work in A/m or Tesla, with the latter giving more convenient numbers. For this reason, it is increasingly the practice when superconductors are involved to measure all fields, including magnetisation, in Tesla. Superconducting materials present a number of complications in comparison with ferromagnetic materials. In ferromagnetic materials there is a local magnetisation which is determined by the local dipole density, and all other currents are dissipative transport currents. In superconductors the distinction is less clear since transport currents give a DC magnetisation, and these currents can flow on a scale comparable with the sample size. It is also necessary to consider samples with large demagnetising factors. Although the algebra is simple, some of the concepts addressed in this article are difficult and have been the source of much confusion for researchers in the field. The main aims of this article, therefore, are twofold: firstly to highlight these potential sources of confusion, and secondly to describe the basic differences in the analysis of magnetic fields in superconductors compared with conventional magnetic materials. The main points made in this article are summarised below to help the reader follow the structure of the article.

G2.1.1.1 Summary of Main Points 1. Fields in materials must be defined as averages over a specified length scale. 2. To define H and M we need a different approach for superconductors compared with conventional magnetic materials. 3. Demagnetising factors are useful for certain situations, but the shape effects in the critical state cannot be described in terms of a demagnetising factor, and

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4.

5.

6.

7.

samples small compared with the penetration depth require different treatment. The velocity of a flux line and its displacement are uniquely measurable from the electric field. For small displacements the response is linear. Displacements large compared with the vortex size are needed to build up the critical state. For practical purposes, the thermodynamics can be most conveniently expressed in terms of the external field and the total moment, rather than local values in the sample. The most general equilibrium condition is that the change in free energy is equal to the work done by an external source, i.e. the availability is minimised. Minimising the Gibbs function is restrictive and can lead to errors. For the majority of practical applications, the Bean model with B = μoH is entirely satisfactory. However, the magnetisation must be defined carefully.

G2.1.2 Fields in Materials G2.1.2.1 The Magnetisation The field of a current loop at large distances compared with its size depends on the product of the area, δS, and current, j. This product is defined to be the magnetic moment, m = iδS as the field at large distances is a dipole filed. Atomic dipoles are the source of magnetic moments in ferromagnetic and paramagnetic materials, so the dipole approximation is extremely accurate at distances greater than a few nanometers. Also, since the fields due to atoms are additive, it is possible to define a local magnetisation, M, as the magnetic moment per unit volume, averaged over a small volume δV containing many atoms, i.e.; M=

∑ iδS δV

(G2.1.1)

The scale over which this average is taken should be small compared with the sample size but large compared with the microstructure. For a dipole array the total magnetic moment Mo is found by integrating M in Equation (G2.1.1) over the volume of the sample. (Mo will always be used for the total moment in this article, even if the sample has unit volume). Provided no current flows in or out of a sample, we can also define a magnetic moment due to currents flowing in loops on a scale comparable with the sample dimensions. This occurs in superconductors in DC fields and conventional conductors in AC fields, which induce eddy currents. The most general expression is Mo =

1 r × j.dV 2

∫

(G2.1.2)

FIGURE G2.1.1 A loop of current can be replaced by a mesh of magnetic dipoles.

where r is the radius vector from an arbitrary origin, j is the local current density, including any atomic currents and the integral is over the whole sample. [1]. In ferromagnetic materials, Mo is the sum of the moment due to macroscopic currents and the integral of M over the sample volume. (The magnetic moment of a sample with current flowing across its boundaries cannot be defined since in this case Equation (G2.1.2) is not independent of the origin of r). It is common to divide the total moment by the volume and call the result the ‘magnetisation’, but it must be recognised that if the currents flow on the scale of the sample, this has no physical meaning on a local scale within the sample. In this case, the magnetisation of a sample carrying eddy currents or supercurrents will depend on its radius, which is in contrast to a conventional magnet where the magnetisation tells us about the density of dipoles on an atomic scale, and the moment is the magnetisation multiplied by the volume. Nevertheless, this magnetisation due to macroscopic currents can be useful. In Section G2.1.2.4.1 an array of dipoles is converted to an equivalent surface current. In Figure G2.1.1 the reverse operation is demonstrated where a current stream line is spanned by a mesh of arbitrarily small current loops whose currents cancel within the loop. They therefore have the same effect as the loop. However, the currents can also be regarded as an array of dipoles with a uniform magnetisation M across the loop. There are many different ways of constructing such a dipole array, any surface spanning the loop can be used, so M is not unique. For a bulk current density, a series of dipole layers is needed. However, all such surfaces, whatever the value of M, have two properties. The first is that within the sample curl M = J, where J is the current density. This is why M is not uniquely defined by J. The second property is that integrating M over the volume gives the total magnetic moment of the sample. Therefore, if the reversible magnetisation [defined in Section G2.1.2.3 Equation (G2.1.9)] can be neglected, we can find the total magnetic moment of a superconducting sample

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FIGURE G2.1.2

Handbook of Superconductivity

(a) A disc or cylinder. (b) The field if the cylinder is long. (c) A local magnetization with a uniform current density.

by integrating the moment of the macroscopic current loops. This is what determines the field at large distances compared with the largest dimension of the sample, and is what is measured in most magnetometers. For example, Figure G2.1.2(a) shows a cylinder or disc carrying a uniform circular current density J. The total moment

∫ πr ( Jdr ) = JLπa / 3 and the moment/vol a

is given by M o = L

0

2

3

by M / V = Ja / 3. This result is true whatever may be the length of the cylinder. However, the field distribution shown in Figure G2.1.2(b) is only true for an infinitely long cylinder. Only in this geometry is the magnetisation the mean difference between external flux density and the mean flux in the cylinder. It is possible to use an M as the vector potential of the current. The main application of this is when currents are confined to the xy plane so that M has only a component perpendicular to the plane. It then becomes a scalar and the stream function of the currents. If a problem involving the current density is solved in terms of the scalar magnetisation, most numerical packages will plot the current streamlines easily by contouring the magnetisation. For the cylinder this M = J(a − r). (M must be continuous and be zero in the space outside the cylinder). This is shown in Figure G2.1.2(c). However, this local M has no physical significance. In particular, it cannot be used in the expression B = μ ο (H + M). A common misconception is that the magnetisation of a sample is the difference between the internal and external field. Firstly, the term ‘internal field’ has many possible interpretations and should only be used if it is the deliberate intention to be vague. However, what can be shown is that if the sample is a long cylinder, of any cross-section, with the axis parallel to the external field, then the magnetisation is the difference between the external field and the average field across the sample. The statement is true only for this geometry, and not for other shapes. Although the geometry was common in low-Tc materials, it is rare in high-Tc superconductors, which often have large demagnetising factors [see Equation. (G2.1.26)].

G2.1.2.2 Measurement of Magnetic Moment The following derivation is modeled on that in Pippard’s text “Classical Thermodynamics” [2]. The results are slightly extended and introduced here since the expressions will be used throughout this article. Consider the magnetic moment, m, of a current loop defined by m = idS. If the current i in the loop is changing, the voltage induced in a search coil is given by: V = M ind

di dt

(G2.1.3)

where Mind is the mutual inductance between the loop and the search coil. Since the mutual inductance is independent of which coil carries the current, it is most easily found by putting a test current I into the search coil and finding the voltage in the dipole loop. The simplest case is when the sample is in the centre of a long search coil of n turns per metre. The flux through the loop is then μοnIδS. In this case, Mind = μοnδS and: di dt

(G2.1.4)

dm dt

(G2.1.5)

V = µ onδS so V = µ on

Adding the effect of all the dipoles which change their moments, we see that the voltage is directly proportional to the rate of change in total magnetic moment of the sample. The constant of proportionality depends on the field at the current loop due to a current in the search coil, i.e. for a long search coil with n turns per meter: o V = µ onM

(G2.1.6)

(Strictly the component of the moment in the direction of the field due to a current in the search coil is what is measured since magnetisation is a vector).

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Electromagnetic Properties of Superconductors

FIGURE G2.1.3 Two search coil geometries. (a) The voltage is proportional to Mo. (b)The voltage is proportional to M.

For the voltage to be simply related to the total moment, the only condition is that the field from the coil must be uniform over the sample. For example, the sample can be outside the coil as illustrated in Figure G2.1.3(a). This is how SQUID and vibrating sample magnetometers measure the moment. They have large coils (such as a Helmholtz pair) well away from the sample, which is moved to generate a voltage. (The movement can be regarded as creating a magnetisation at one end of the sample and removing it at the other). Then, instead of n in Equation (G2.1.6), the constant of proportionality is the ratio of the external field at the sample to the current in the search coil that would produce it. This is a very general result and as true for eddy currents as local dipoles and supercurrents Another simple geometry is illustrated in Figure G2.1.3(b) where a coil is wound round the centre of a cylindrical sample. In this case, the signal is proportional to the change in flux in the sample, or average flux density across the section at the search coil, i.e. the magnetisation rather than the total moment. However, for the best signal-to-noise ratio, the search coil should be similar in size to the sample. In this case, the voltage due to induced currents will depend on where they are flowing, and some assumptions must be made about the mutual inductances between the current loops and the search coil before the experiments can be interpreted. There is then no simple relation between the voltage measured and either the magnetic moment or the magnetisation, even if details of the current distribution are known.

similar to Maxwell’s equations in free space. However, the way this is done depends on the nature of the material. Following Landau and Lifshitz [1], B in a material is always defined as the average of the microscopic field, b, over a convenient volume determined by the material or particular application. In most magnetic materials, for example, this average is taken over a large number of atoms. It may be over a similar scale in superconductors, but can also be over a much larger volume containing many flux lines, or even over many grains (typically ≈ μm in size). In electrical machines the average can be taken over the teeth of the rotor (≈ several cm) and in principle fields may be averaged over the length scales of a solar system or galaxy although at present this does not seem to be very useful. Nevertheless, the length scale must be defined in any derivation and used consistently throughout the analysis. Consistency in the length scale over which averages are made is particularly important when applying the Ginzburg– Landau equations. Here the scale is much larger than an atom, but much smaller than either the coherence length or the penetration depth. (see Chapter A2.1 Phenomenological Theories). As discussed above [Equation (G2.1.1)], a local magnetisation M can be defined easily in conventional magnetic materials as the sum of local moments, in which case it is convenient to define a vector H by: This is more usually written as: B = µ o (H + M)

(This was done because early workers thought the magnetic field was due to poles and wanted the magnetic equations to look like the electrostatic ones). With this definition of H, Maxwell’s equations in a conventional material can be written in a similar form to those in free space (see Appendix VI). It is important to distinguish between H, which is defined within the material, and the external, or applied field, Ho. The latter is the field in free space that would be present if the sample were to be removed and all other currents, including magnetisation currents, kept fixed. (It must be done in this order as a magnet near a permeable material is in an applied field due to its image in the material). If we want to measure the applied field in Tesla, we define an applied Bo = μoHo. (This can be done because the external field is nearly always in free space, fields in magnetic fluids are rare and require more detailed analysis). Equation (G2.1.7) is a local relation between average fields at any point, and M is a local magnetisation not due to macroscopic currents. On the other hand the equation: B = µ o (H o + M o )

G2.1.2.3 Fields in Materials The field on an atomic scale inside a material, which we define as b, is far too complex to be useful, so it is necessary to work with average fields. We will see that these can be formulated in such a way that the field equations in a material are very

(G2.1.7)

(G2.1.8)

which is in terms of the total moment and the applied field, can be used whatever the source of the magnetic moment, including macroscopic currents, but only if the sample is a long cylinder parallel to a uniform applied field, and B is an average over the whole cross-section.

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In a superconductor there are no obvious localised magnetic moments to define M, and alternative definitions are required in order to derive the standard equations. In the mixed state B is defined as the average field over many flux lines, and H is defined as the external field which would produce the same value of B in a reversible sample in the absence of a demagnetising factor [3]. (This is also the field in a hole parallel to the field). In a reversible sample this is equivalent to the definition of Josephson [4] who defined H(B) as the derivative of the free energy with respect to the flux density. H(B) in superconductors can be calculated from the flux lattice theory of Abrikosov [5]. In this case, the magnetisation M defined is by: M = B/µ o − H

(G2.1.9)

Thus, in a material with localised moments we define B then M then H. In the superconductor we define B then H and then M. M is commonly called ‘the reversible magnetisation’ and was first calculated by Abrikosov. Since the energy needed to insert unit length of flux line is Hφ o, where φ o is the flux quantum, it follows that H can be regarded as the chemical potential of a flux line. Because these are lines, we need to integrate the energy along a short length, and this leads to a driving force proportional to curl H rather than the gradient of the potential which occurs for particles. These fields can be used in Maxwell’s equations in any geometry, (see Appendix I) and also in the Onsager theory of irreversible thermodynamics, which deals with linear thermal effects such as the Seebeck and Peltier effects. There is a full treatment in references [3] and [4]. Having defined the B-H curve, we can then use the same techniques for calculating fields in superconductors as for conventional materials. Two particularly important results from this approach are for the transport current density: J = curl H

(G2.1.10)

and the driving or Lorentz force on flux lines; FL = J × B

(G2.1.11)

This is the force which must be borne by pinning centres. (Those who come to the subject from fluid dynamics may refer to the latter as the Magnus force, although numerically equal to the Lorentz force, on a microscopic scale it can be attributed to the different electron velocities on opposite sides of a vortex). As in ferromagnets, there can be contributions to the total magnetic moment in superconductors from two types of currents, the magnetisation and transport currents. The total magnetic moment will be the sum of the reversible magnetisation M defined above [Equation (G2.1.9)] and that due to the transport current density J. In the case of superconductors

these are only distinguishable by thermodynamic arguments as both are supported by a change in the vortex configuration and velocities of the same electrons. In practice, the reversible magnetisation in Type II superconductors is very small over most applicable ranges of field, and we can usually assume H = B/μo (i.e. M = 0). In this case, the driving force on flux lines becomes, FL = B × curl (B/µ o )

(G2.1.12)

This has the simple interpretation of the imbalance in repulsive force due to a density gradient of vortices, plus a line tension if the flux lines are curved. Note that H and M are defined only on a scale larger than the flux lines in the above analysis, on any smaller scale, currents should be treated as flowing in free space with B = μοH. The magnetisation cannot be defined on a smaller scale since it arises from the arrangement of flux lines (unless there is a separate atomic scale magnetisation due to the presence of paramagnetic atoms). Field definitions on this basis and some alternatives are discussed further in Appendix I.

G2.1.2.4 Shape Effects Usually, the effects of sample shape on magnetic field distribution are addressed only briefly in text books of physics or electrical engineering. Most high-Tc materials, however, are made with large aspect ratios in which case shape effects are non-trivial. There are three different types of sample we need to consider here: I. Conventional magnets, macroscopic reversible Type II superconductors and large simply connected superconductors in the Meissner state. These materials are the only samples in which a conventional demagnetising factor can be used, although the concept may remain qualitatively valid in other samples. In this case, we can use the B-H curve, as measured on a long thin sample, in Maxwell’s equations. II. Samples obeying the London equation in which J = A/(μoλ 2), where A is the vector potential and λ the penetration depth (see Section G2.1.5). These include not only samples in the Meissner state which are small compared with λ, but also small oscillations in a pinned vortex lattice. The mathematics is identical to that of eddy currents in a normal conductor where: J = σE − iωσA

(G2.1.13)

Here, σ is the conductivity, and ω is the frequency. The solutions of the eddy current equation for many shapes can be found in electrical engineering text books. To obtain the solution to the London equations we replace ωσ by –i/μολ2.

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Electromagnetic Properties of Superconductors

III. Irreversible superconductors which obey the critical state model and for which there are analytic solutions for relatively few shapes [11–13]. These should not be discussed in terms of demagnetising effects. If, as is common, the sample is saturated with the critical current density, the fields can be calculated directly from the Biot–Savart law. In general, samples intermediate between these limits are difficult to analyse. G2.1.2.4.1 Demagnetising Fields We show first how demagnetising fields can be derived for a conventional magnetic material. This involves calculating the field due to a dipole array using a construction of equivalent surface currents. Since the dipoles are small, the field outside the sample does not depend on the exact arrangement of dipoles, only on their density. It is less obvious, but can be shown to be true, that the average field inside the dipole array, B, is also independent of how the dipoles are placed within the sample (see Appendix VI). As a result, the most convenient dipole array can be used. This consists of closely packed squares placed on top of each other to form an array of rectangular solenoids. Details of this calculation are as follows: If each loop carries a current i, is of width a and separated from its neighbouring loop by a distance d in all three directions as illustrated in Figure G2.1.4(a), then the magnetisation is given by, M = ia 2 /d 3

(G2.1.14)

The field in each solenoid is μοi/d, in which case the average field across the sample is given by, B = µ oia 2 /d 3 = µ o M

(G2.1.15)

Each dipole is now enlarged whilst keeping its moment constant by reducing the current in proportion to the increase in area, until the array exactly fills the interior of the sample as illustrated in Figure G2.1.4(b). In this case, the currents cancel, leaving only a surface current ia2/d2 at each layer, or ia 2/d3 A/m as shown in Figure G2.1.4(c). The surface current will

FIGURE G2.1.4

vary as the cosine of the angle if the surface is at an angle to the magnetization. Hence, a magnetisation, M, can be replaced by an equivalent surface current, flowing in free space, of magnitude equal to the component of M parallel to the surface. Here, equivalent means that the surface current produces the same average field B inside the sample and generates an identical field in the region outside. H, therefore, will be different inside the sample since the current in the equivalent system is in free space. A uniform magnetisation has been assumed in this analysis. If the magnetisation is non-uniform, the internal currents will not cancel as illustrated in Figure G2.1.4(b), and an equivalent current density, j = curl M, will remain within the bulk of the sample (see Appendix VI). Note that lower case j is used to distinguish between local currents, which include atomic orbitals and electron spins as well as any transport current, and the transport current density, J, which is given by curl H. These results will now be applied to spheres and cylinders in which a uniform magnetisation M in the θ = 0 direction leads to an equivalent surface current M sin θ. It can be shown from the Biot–Savart law that the B field at the centre of a sphere due to the surface current is (2/3) μοM. The corresponding value for a cylinder with M perpendicular to the axis is (1/2) μοM. It can be shown further that in an ellipsoid of uniform magnetisation the B produced by the surface currents is uniform, so these central values apply throughout the sample. Any external field must then be added to that due to the magnetisation to give the total field in the sample. The above formulation is the logical way to proceed. However, magnetostatics were developed initially in the nineteenth century when it was thought that magnetism might be due to magnetic poles, in direct analogy with electrostatic charges. Since the average electric field E obeys similar field equations to H, magnetostatics were developed using H as the primary field rather than B. For example, in a sphere B = Bo + (2/3) μοM is not the starting expression, as suggested by the analysis above, but an equivalent statement which is the direct analogy of an electric polarisation, H = Ho – (1/3) M. The two can be shown to be equivalent by substituting B = μο (H + M)

(a) A dipole array. (b) Larger dipoles with the same magnetisation. (c) The equivalent surface current.

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and Bo = μοHo. In general, in any magnetic ellipsoid in a uniform external field Ho applied along a principal axis: H = Ho − nM

(G2.1.16)

The factor n is defined as the demagnetising factor for that axis and superposition of components can be used for off-axis fields. (The difference between the electric and magnetic cases is due to the different geometry of the dipoles responsible for the fields). Given that H is a difficult vector to define it might seem that Equation (G2.1.16) is a purely formal statement for which the only advantage is that the magnetostatic field equations can be formulated by direct analogy with electrostatics. However, there is a little more significance to H than is evident at first sight, which is implied by the term demagnetising field. The surface current produces an average field in the same direction as the dipoles producing it, so it is difficult to see why they should wish to demagnetise (i.e. turn around to align oppose the direction of the average field in the region). However, what is important to an individual dipole is not the average field, which includes its own field, but the field applied to it by the other dipoles. If a dipole is removed from a material, the field in the space left behind can be in the opposite direction to M and B but in the same direction as H (depending on the demagnetising factor of the sample). For a magnetic material, therefore, there is some reason to regard H as a field tending to demagnetise dipoles. However, the exact connection between the field on an atomic dipole and the fields B and H depends on the crystal structure. For simple structures it is given by the Lorentz theory of dielectrics [6]. (This theory applied to composite materials has also proved useful on a larger scale in relating the susceptibility of a powder composite to the susceptibility of individual particles, as discussed further in Appendix II). The use of demagnetising factors simplifies the calculation of fields in ellipsoids, which would otherwise require a solution of Maxwell’s equations with appropriate boundary conditions. The arguments above based on local dipoles cannot be used in superconductors. However, with the definitions of M and H given in Equation (G2.1.9), superconductors in equilibrium, (i.e. no pinning), do obey the same differential equations as conventional materials, so that the same equations can be used with appropriate demagnetising factors to determine B, H and M in a superconducting ellipsoid as if the magnetisation was due to local dipoles. In this case, there is no local field which can be identified as the demagnetising field, and the physical meaning of H is the chemical potential of the flux lines [3]. It is misleading to call this ‘an internal field’ since no magnetic field of this size exists in the superconductor. A comprehensive discussion of demagnetising factors with a number of references is given in [7]. It can be shown that the demagnetising factors along the three Cartesian axes sum to unity so from symmetry considerations, n = 1/3 for a sphere

and n = 1/2 for a cylinder with its axis oriented transverse to the magnetic field. For spheroids, n can be expressed in terms of the aspect ratio γ = b/a, where b is along the axis of rotation and the field direction. Other expressions for n are as follows: For a prolate spheroid (i.e. cigar shaped); γ > 1 and; n=

1 γ arccosh( γ ) − 1 γ − 1 ( γ 2 − 1) 2

(G2.1.17)

For an oblate spheroid (i.e. plate transverse to the field); γ < 1 and;

n=

1 γ arccos( γ ) 1 − 1− γ 2 (1 − γ 2 )

(G2.1.18)

For a thin disc or strip perpendicular to an applied field an approximate expression is: n =1−

πb 2a

(G2.1.19)

where a/b is the aspect ratio of the sample which is large. This can be seen physically from the argument in Section G2.1.2.4.2. G2.1.2.4.2 Susceptibilities In a linear material, the susceptibility, χ, and relative permeability, μ, of a material are defined by, M = χH

(G2.1.20)

µ =1+ χ

(G2.1.21)

These are local relations in which χ and μ are material properties. These equations can be used to derive the fields for an ellipsoid in an external field Ho applied along a principal axis. Inside an ellipsoid sample in a field parallel to an axis, H = Ho – nM and M = χH. Hence: M=

χHo 1 + nχ

(G2.1.22)

H=

Ho 1 + nχ

(G2.1.23)

µ oHo (1 + χ ) 1 + nχ

(G2.1.24)

B=

These equations are also valid for a non-linear material since the field is uniform, but in this case χ must be consistent with the value of H (not Ho). Equations (G2.1.22) to (G2.1.24) apply only to a local magnetisation and not to macroscopic currents. However, an ‘effective’ susceptibility χeff can also be defined for a sample of

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unit volume by Mo = χeff Ho even if there are macroscopic currents flowing. If no macroscopic currents flow, then, χeff =

χ 1 + nχ

(G2.1.25)

If there are macroscopic eddy currents or pinned supercurrents in the sample, the effective susceptibility will depend on both the size and shape of the sample and cannot be related to a local material property such as χ. It is not useful in the numerical solution of the field equations. From Equation (G2.1.23) we deduce that the field just outside the edges of a plate oriented perpendicular to the applied field changes by a factor (1 + nχ)−1, since the parallel component of H must be continuous across the sample surface. If χ is –1, as for a superconductor in the Meissner state, this tends to infinity for very thin plates normal to the applied field (for which configuration the demagnetising factor tends to unity). Hence, thin films are penetrated by vortices at very low external fields since vortices are generated when the field at the edge reaches Hc1. The field is concentrated at the edges by a factor approximately equal to the aspect ratio. Combining Equations (G2.1.7) and (G2.1.16) gives: B − µ oHo = µ o M(1 − n)

(G2.1.26)

which demonstrates that, unless n = 0, the magnetisation is not the difference between B and the external magnetic field. This can be appreciated physically for thin flat crystals oriented perpendicular to the field for which n ≈ 1. In this case, B = μοHo [from Equation (G2.1.26)], which ensures that the normal component of B is continuous. As the sample becomes thinner, the difference between B and the external field tends to zero while the magnetisation tends to a constant value χHo/(1+χ). A corollary of this is that flat superconducting samples oriented perpendicular to the applied field do not usually show a Meissner effect. In this case, there is not enough energy in the sample to push the flux to the edge, since this would also require the field to be excluded from the free space round the sample (see Figure G2.1.5). This is the origin of the so-called ‘geometric barrier’ [8]. Figure G2.1.5(a) illustrates how this geometric barrier arises. A field is applied to three different shapes of Type II superconductor. The top shape is a ring with a hollow centre. This will exclude flux until the surface field reaches Hc1 and will trap a field of a similar value when the external field is reduced to zero. The internal field cannot escape to achieve thermodynamic equilibrium because this can only be done by nucleating a vortex which can migrate to the outside. Once the field in the centre is less than Hc1, no vortex can be nucleated. It follows that an ideal reversible Type II superconductor in this geometry will have an irreversible magnetisation curve at fields close to Hc1 due entirely to its shape. Figure G2.1.5(c) is an ellipsoid, and if the material has no pinning centres, the magnetisation will follow standard

FIGURE G2.1.5 The geometric barrier. A field is applied to three shapes of reversible superconductor: (a) a hollow ring, (b) a rectangular sample, (c) an elliptical sample.

electrodynamics and be reversible. The difference is because although the flux lines want to go to the centre as in the top shape, there is sufficient material for the Meissner effect to balance this by trying to expel them. Intermediate is the shape of most high-Tc samples which is a thin rectangle as in Figure G2.1.5(b). Clearly there is less material in the centre to expel flux lines, so this too should have an irreversible magnetisation curve even if the material is ideally reversible. This may seem a rather esoteric point but it is of considerable importance in showing that magnetic measurements of the irreversibility line, one of the most important parameters in high-Tc superconductors, may be dominated by geometric effects rather than telling us about the pinning strength. The physical reason for the increased moment per unit volume of a flat sample is illustrated in Figure G2.1.6. A long flat diamagnetic sample oriented perpendicular to the applied field will change the field in a volume equal to a

FIGURE G2.1.6 Lines of force bent round a diamagnetic ellipsoid with major axis a, minor axis b (vertical). Field is screened from the circumscribing cylinder.

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circumscribed cylinder (i.e. of diameter comparable to the sample dimension). Figure G2.1.6 shows a superconducting sample, semi-axes a and b, with b parallel to the applied field. It can be seen that magnetic flux is excluded from a region which is roughly the circumscribed cylinder, as if there was diamagnetic material in this region. The magnetic moment per unit length is approximately –πa 2H0, where H0 is the external, distant field. Dividing by the volume of the sample (πab per unit length) gives a magnetisation of ≈ a/b H0, so, according to Equation (G2.1.22), n must be approximately 1−b/a. This result, which also applies to spheroids, is consistent with Equation (G2.1.19). Finally, χ = −1 in the Meissner state so that the field at the edges of a flat plate is increased by a factor approximately equal to the aspect ratio. G2.1.2.4.3 The Mixed and Intermediate States The fields in ellipsoids obey Equations (G2.1.16) and (G2.1.8), [H = Ho – nM and B = μο(H + M)]. In the Meissner state B = 0 so H = −M, and therefore M = Ho/(n – 1). In a Type I superconductor, the critical field Hc is reached at the edges when Ho = Hc(1 – n). However, were the sample to be driven normal, the field in the sample at this point would revert to the applied field Ho and the sample would become superconducting again. The solution to this paradox is that the sample splits into normal regions containing a field Hc, (the critical field) and superconducting regions, which contain zero field. This is called the intermediate state. The boundaries between the superconducting and non-superconducting regions of the sample must be parallel to the field and so H must be the same in both and equal to Hc. Hence, the average of H in the sample is Hc. Since H = Ho – nM, it follows that M = (Ho – Hc)/n. As a result, the magnetisation decreases linearly to zero at Hc, and the magnetisation curve remains triangular with an area ½μοHc2. The ideal curves for a sample with demagnetising factors of zero and finite n are shown in Figure G2.1.7.

Handbook of Superconductivity

If n is not zero, the thermodynamic transition in a field becomes second order since it takes place over a range of fields with no discontinuity in M. Such a spreading out of the first-order transition, so that it becomes second order, is not confined to Type I superconductors but will occur for any first-order magnetic transition, such as flux lattice melting, in a sample with a finite demagnetising factor. In a Type II superconductor, the magnetisation starts off as for a Type I superconductor, but lines of flux quanta enter when the surface field reaches a lower critical field Hc1 to produce regions of superconducting and non-superconducting material. This is called the mixed state. The subsequent behaviour of a reversible ellipsoidal Type II material in the mixed state can be found by defining the susceptibility χ by M = χH as calculated from the Abrikosov theory [5]. In this case, demagnetising effects are only important at fields very close to Hc1 since χ decreases very quickly with field. However, the magnetisation is usually dominated by pinning effects which are treated using the Bean model. If after reaching a field above Hc1 in such a material, the field is decreased, the initial slope is close to that of the Meissner state. This has been used to determine the scale on which currents flow in granular materials by making use of the fact that the initial slope of minor hysteresis loops is proportional to the aspect ratio [9]. The scale of the subdivision of the normal and superconducting phases in the intermediate state of a Type I material depends on the surface energy between normal and superconducting states, but decreases with thickness. The minimum subdivision is set by the size of the flux quantum so that a thin film of a Type I superconductor in a perpendicular field is almost indistinguishable from a Type II material. Originally, these boundaries were thought of as planes, but clearly a twodimensional division is more favourable, so that the idea of a negative surface energy leads naturally to a lattice of normal and superconducting regions which is another way of looking at the vortex lattice. G2.1.2.4.4 Small Samples Anybody which excludes field, such as that shown in Figure G2.1.6, will lead to a field concentration at its extremities so that flux penetrates further than it would in a sample with no demagnetising factor. Consider first a superconductor in the Meissner state. This can be approached by analogy with the behaviour of a normal metal in an oscillating field. At low frequencies, B can be assumed to be equal to the value of the applied field, Bo. For a slab of half thickness a oriented parallel to a field in the y direction, the (complex) electric field Ez and (complex) current density J are given by, E z = iωBo x

(G2.1.27)

J = iσωBo x

(G2.1.28)

and FIGURE G2.1.7 Magnetisation curves for a Type I superconductor with zero and finite demagnetising factors.

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Electromagnetic Properties of Superconductors

In this case, the magnitude of the central field, Bc, due to induced currents flowing in the surface of the conductor is given by: 1 Bc = µ o σωa 2 .Bo 2

(G2.1.29)

The regime of flux penetration changes to flux exclusion when the field due to the induced surface currents is approximately equal to the applied field. This cross-over occurs when μoσω = 2a 2 or δ = a, where δ is the skin depth. A superconductor obeys the same equations with: λ 2 = (µ o σω )−1

(G2.1.30)

where λ is the penetration depth of the superconductor. Therefore, if we use the same criterion, flux exclusion begins when a = √2λ, which is around the expected value. The same calculation can be performed for a sheet of half length b in the direction of the field and a perpendicular to it with b ≪ a. In this case, the current density is the same but the field at the centre, Bc, becomes: B c = µ oσωab.B o /(2π )

(G2.1.31)

Flux exclusion occurs when λ2 = 2ab (i.e. when the penetration depth is between the width and the thickness, as expected). For samples with non-zero demagnetising factors in general, flux exclusion takes place when the cross-section area is about λ2, i.e. λ is intermediate between the two dimensions. It is not necessary to have a thickness comparable with λ for nearly complete screening. This is also the effective penetration depth for vortices in thin films. G2.1.2.4.5 The Critical State A similar shape effect to that described in Section G2.1.2.4.4 occurs for Type II superconductors in the critical state. Again, consider a long strip of dimensions 2a in the x direction and 2b in the y direction, which is the direction of the field. We assume it obeys the Bean model with constant Jc, .(i.e. the slab carries uniform current density when fully penetrated with field). We can find the external field which just penetrates to the axis and the field on the axis when saturated with current from the Biot–Savart law. We find the penetration field for a field by calculating the field at the centre when the slab is saturated with currents plus and minus Jc. This is the maximum field that can be excluded. If b ≫ a, then the full penetration field for slabs and cylinders oriented parallel to the field is μoJca and the surface field is exactly half this. The following are the expressions for a general size.

Strip Penetration B =

Strip Surface B =

µo Jc π

2µ o J c b a2 −1 b a tan + ln 2 π a 2 b a2 −1 2b a tan + b ln 2 a 4b

Disc Penetration B = µ o J cb sinh −1 ( a / b ) Disc Surface B = µ o J cb sinh −1 ( a / 2b )

(G2.1.32)

A particular application is the use of cylindrical YBCO pucks to generate high magnetic fields from the trapped flux. To saturate a sample fully at low temperature, we need to apply twice the penetration field. This means that we need four times the trapped surface field for a long cylinder which is quite a problem for practical applications. The surface field is only reduced by 4% if the length is reduced to equal the radius, but we still need to apply 3.7 times the surface field to trap the maximum. Calculations of the critical state in thin samples can be found in references [10–14].

G2.1.2.5 Internal Electric Fields The electric field in a material, E, is defined as the average of the microscopic electric field, just as B was defined as the average of the magnetic field. There are two mechanisms in homogeneous superconductors which lead to an electric field. The first is the inertia of the electrons, which allows an electric field to exist as the superconducting electrons accelerate to achieve a constant speed. This effect is described by the following equation: mv = eE

(G2.1.33)

where m is the mass of an electron and e its charge. During this time the normal electrons are scattered and dissipate energy determined by the normal-state resistivity. This is the only loss mechanism in the Meissner state, apart from surface hysteresis, and it is difficult to detect below GHz frequencies (see Chapter H2.1 Microwave Resonators and Filters). The second source of electric field is the motion of flux lines. This follows from the fact that flux moves under the action of the Lorentz force and so must dissipate energy (i.e. work done = force × distance moved), which can only occur if the moving flux generates an electric field. In a sense this follows directly from Faraday’s law which can be illustrated as follows. The e.m.f. generated if flux lines leak out of a thin-walled hollow cylinder of radius r is given simply by the rate of change of flux. If the flux density B moves with velocity v, then the flux crossing the cylinder wall per second is 2πrBv. The electric field in this case is therefore Bv, to give the correct e.m.f. The vector relation between the electric field and B and v can be written: E = B× v

(G2.1.34)

since a velocity parallel to the flux line can have no physical significance. This is the same as the classical expression for the electric field induced when a conductor cuts magnetic flux. Initially, Equation (G2.1.34) caused some confusion when applied to a DC resistive measurement since it appears to describe an inductive voltage, and there is no flux change in a DC measurement. The situation was clarified by Josephson [15] who argued that the conditions at the vortex level are defined uniquely by local conditions, and therefore if

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Equation (G2.1.34) applies to the simple geometry of flux leaking out of a hollow tube, it must be universally true. It is true even if there is a charge density present and is a stronger statement than can be made in conventional conductors where we may need to add the field due to electrostatic charges, i.e. E = B × v + ∇φ

(G2.1.35)

In a superconductor with v the vortex velocity, the electrostatic potential ∇ϕ is zero even in the presence of electrostatic charges. The simplest example is the case of a uniform field moving past a conducting wire. There will be charges induced at the ends which cancel the induced field in a normal conductor and we use Equation (G2.1.35) with v the velocity of the field. We can do the same for a superconductor, but alternatively we can also use Equation (G2.1.34) where v is the velocity of the vortices. This is zero whether the vortices are pinned or not so E = 0 without putting in a ∇ϕ term. A more complex example of charge densities in superconductors is in Appendix IV. The mechanism by which a moving flux line generates a voltage is subtle. The moving vortex creates a gradient in the chemical potential of the normal electrons, which is measured as an electric field in a transport measurement. At the microscopic level the electric field can be described as being due to currents in the normal core, or as the difference in equilibrium electron density at the leading and trailing edges of the vortex because of time lags. Either description leads to the Bardeen–Stephen expression for the flux flow resistivity, ρff [16]: ρff = ρn B / Bc2

(G2.1.36)

where ρn is the normal-state resistivity.Although approximate, this result is physically reasonable in that the resistivity rises in proportion to the amount of normal core material present, reaching the normal-state value at Bc2. In a Type II material with pinning centres, this is the detailed mechanism by which the unstable depinning of flux lines dissipates energy on a local scale. However, as in the case of hysteresis in ferromagnets, where domain walls behave in a similar way, it is not necessary to analyse the details of the motion. The loss, and hence the electric field, can be calculated from the critical current density using the Bean model (i.e. from a macroscopic parameter). The generation of electric fields by moving flux lines is discussed in more detail in Section G2.1.5.2.

G2.1.2.6 The Voltage Measured along a Wire In conventional materials the voltage drop along a wire at low frequencies depends only on the resistivity. However, the flux flow resistivity of a superconductor is far lower than the resistivity of copper, and inductive effects must be taken into account even at low frequencies. The measurement of the

FIGURE G2.1.8 (a) Faraday’s law applied to a wire carrying a current. (b) More flux is enclosed by the edge contacts so the voltage is greater.

voltage drop along a wire depends on the nature of the current and the geometry of the voltage taps [17]. A DC electric field can be measured directly with voltage taps, whereas an AC field introduces complications due to the voltage induced by the varying self-magnetic field. As an aid to understanding what a voltmeter measures, it is useful to think of using an electrostatic voltmeter attached to potential taps positioned unit length apart on a superconducting wire, as shown in Figure G2.1.8(a). (The reading is of course independent of the nature of the voltmeter). Faraday’s law is then applied to a loop defined by the centre line of the wire, lines from there along the potential leads and the gap between the plates of the electrostatic voltmeter. The field along the centre of the wire, Ec, is zero if the critical current has not been exceeded, as is the field in the leads since no current is drawn. Hence, the integral of the field between the plates Ev, which is the voltmeter reading, is equal to the rate of change of flux in the loop: V = φ1 + φ2

(G2.1.37)

It can be seen from Figure G2.1.8(b) that, if the wire is of a tape geometry, the voltage from taps on the top surface will be much less than that from taps on the side because there is less flux between the contact and the centre. In this situation, the true loss is obtained by extending the voltage loop to at least three tape widths away from the sample before twisting the wires together and measuring the out-of-phase voltage [17]. Notice that although the increased size of the loop includes only free space, the voltage measured will include both inductive and loss components since the field

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in the loop is not sinusoidal. If the field along the centre of the wire is not zero, then the measured voltage will be given by: V = φ1 + φ2 + Ec

(G2.1.38)

where Ec is the field along the centre of the wire. The voltage can also be used to measure the surface field since by the same argument the voltage is given by: V = φ1 + Es

(G2.1.39)

where Es is the surface electric field. In a normal sample, the dominant term in the integral is the resistive electric field along the axis of the wire, and inductive voltages are usually negligible. An alternative arrangement which gives the true loss voltage is to wind the potential contact wires into a one turn helix along the sample [18]. The above analysis will now be applied to the measurement of the voltage along a wire which obeys the London equations (i.e. A=μ ο λ 2J, where A is the vector potential). We assume that the wire is thin compared with the penetration depth and that it carries a sinusoidal current. This is the opposite extreme from a typical Type II wire. In the notation of Figure G2.1.8, the voltage measured from taps d apart is, V = iω(φ1 + φ2 + dEc )

(G2.1.40)

If the wire is small, the current density J, and the electric field E are uniform. Hence, E c = − A = −iωµ o λ 2 J

(G2.1.41)

It can be seen the voltage measured is inductive and consists of the flux change in the voltage contact loop plus a term which is proportional to the current density. This latter term is called the kinetic inductance since the energy associated with it goes into the kinetic energy of the superelectrons rather than the magnetic field. (see Chapter A2.1 Phenomenological Theories). A similar effect is seen in a pinned vortex lattice, in samples small compared with the effective penetration depth, and in Josephson Junctions small compared with the Josephson penetration depth, which also obey the London equation (see Section A2.1.). The kinetic inductance can also be interpreted as the voltage due to reversible movement of vortices in their potential wells. The voltage along a wire is commonly used to measure the loss in a superconductor. This is valid if the only source of power is driving the current in the wire. However, if the wire is also in an external AC field, there is a loss due to this field, and a voltage measurement is insufficient. You can get the total loss in a coil from the voltage at the terminals, but not the loss in individual turns from voltage taps.

G2.1.3 Thermodynamics G2.1.3.1 Magnetic Energies The subject of magnetic energies is a minefield of confusing and apparently inconsistent expressions. It is worth making explicit why this is so by summarising the situation in conventional magnets. In most other circumstances, the energy, (for example, elastic energy), and also entropy, of a material is determined by local conditions on the scale of a number of atoms. A local energy density can then be defined based on these conditions and integrated over the sample volume to obtain the total energy, entropy or any other thermodynamic quantity. This approach is not possible with magnetic energies since the effect of any magnetic phase change in the material generally extends out in free space to distances comparable with the sample size. The effect of magnetic fields is not limited to the immediate environment of the atoms concerned, and it is therefore necessary to return to first principles to describe phase changes and energy densities in magnetic materials. The safest route through this conceptual quagmire, although not always the most elegant, is to use thought experiments to calculate the work done. The work done is an unambiguous quantity, while the question of where the energy resides is complex and should not be asked unless we are prepared to consider the details of micromagnetics. Even then the results are rather arbitrary. For example, we can put all the energy into the system magnetically from a solenoid, but it may be found locally as elastic energy or electrostatic energy, as well as in the space outside the sample. The most straightforward expression for the work done on a magnetic system is found in Pippard [2]. For a sample in a solenoid of n turns per metre, Equation (G2.1.6) shows that when the magnetic moment, and hence magnetisation, o. changes there is an induced voltage given by V = µ onM The power P needed from the power supply is VI so that the rate of doing work is, o P = µ onI M

(G2.1.42)

Here Mo is the component of the total moment in the direction of the applied field, which has a magnitude nI. Hence: o P = µ oH o .M

(G2.1.43)

As a result, the work done for any small change in magnetisation of the sample is given by, δw = µ oH o .δM o

(G2.1.44)

This is a very general expression of great utility which can be used even if the magnetisation is hysteretic or due to eddy currents. Note that the variables in Equation (G2.1.44) are the external field Ho and the total moment Mo, and not the local

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values H and M. It avoids integration of fields inside and outside the sample but gives no information on how the energy is distributed within the sample. Equation (G2.1.44), therefore, may be a very general expression for the work done on a sample, but it yields little information about local energy densities. In addition, a term μοHo.δHo integrated over all space should be added to Equation (G2.1.44) since the applied field Ho may be changed as well as the magnetisation. This is simply the work done by the solenoid on the free space inside it at constant magnetisation. Although in principle this term should appear in all expressions for the work done, it can usually be omitted since we are only interested in energy differences of the sample (although this is not the case within the framework of the Ginzburg–Landau theory, see Chapter A2.1 on Phenomenological Theories). If Ho is non-uniform, μοHo.δM can be integrated over the sample volume, using the artifice of equivalent dipole current sheets to give a local magnetisation equivalent to the macroscopic currents. (This is just a formal magnetisation which is useful because it can be integrated over the volume to give the correct total magnetic moment, see Section G2.1.2.1). As an example we show that the area under the reversible magnetisation curve of a superconductor is equal to the condensation energy. The most appropriate energy to use is the Helmholtz free energy, F, defined by: F = U − TS

(G2.1.45)

Here U is the internal energy of the system, T is the temperature and S is the entropy. The Helmholtz free energy is used because the change in F is equal to the work done for any reversible change at constant temperature, and this is the type of experiment performed most frequently. Under these conditions an unknown amount of heat can enter or leave the sample without affecting the argument. A solenoid is used to take a Type II superconductor of unit volume from zero field to just above Hc2, where it is normal. From Equation (G2.1.44), the work done in this case is μο∫Ho.dMo. Hence,

∫

Fn ( H c2 ) − Fs (0) = µ o H o . dM o

(G2.1.46)

where Fn and Fs are the Helmholtz free energies in the normal and superconducting states. The analysis is now repeated starting with the normal state. (Although this is not the equilibrium state, we can imagine a nucleation barrier which allows us to take the sample below Tc while remaining in a metastable normal state and hence apply equilibrium thermodynamics). In this case: Fn ( H c2 ) − Fn (0) = 0

(G2.1.47)

Subtracting Equation (G2.1.47) from Equation (G2.1.46) gives,

∫

Fn (0) − Fs (0) = µ o H o . dM o

(G2.1.48)

Given also that, δ ( H o M o ) = H oδMo + M oδH o

(G2.1.49)

and HoMo is zero at both limits of the integration then:

∫

∫

µ o H o . dM o = −µ o M o . dH o

(G2.1.50)

for these limits of integration. This can also be seen geometrically from the graph of the magnetisation curve. Hence:

∫

Fn (0) − Fs (0) = −µ o M o . dH o

(G2.1.51)

(The signs in this equation are correct since M is negative so the free energy of the superconducting state in zero field is lower than the normal state, as expected). The above analysis shows that the area under the reversible magnetisation curve is the difference in free energy between the normal and superconducting states in zero field, i.e. the condensation energy. This result is independent of the shape or size of the superconductor. Hence, by using the external field and total moment, we have shown that the result holds for samples small compared with the penetration depth, and for samples of any shape. The analysis becomes less general if local values of H and M are used because, if there is a demagnetising factor, the work done cannot be written as μοH.δM integrated over the sample volume. This expression is only valid for cylinders oriented parallel to the external field and large compared with the penetration depth. The correct expression for the energy using local values is H.δB integrated over all space, which is algebraically intractable unless the demagnetising factor is zero. A proof of the equivalence of the two expressions and a detailed discussion of magnetic energies can be found in Stratton [19] and the paper by Heine [20]. An alternative expression for the change in energy is δU = −μοMo.δΗο. To understand how this quite different expression can lead to the same results, it is important to visualise the sort of experiment where it applies. The force on a magnet depends on the gradient of the applied field and can be written as: F = µ o (M o .∇)H o

(G2.1.52)

The term μοMo.δHo is the mechanical work done when a magnet of moment Mo is moved from a point where the field is Ho to one where it is Ho + δHo. In principle, a superconductor could be taken through its magnetisation loop by bringing it up to a permanent magnet, instead of keeping it stationary and applying the field with a solenoid. In the former case the work done is that by the force on the superconductor and it is not necessary to consider changes in the energy of the permanent magnet since it is outside the system.

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Electromagnetic Properties of Superconductors

Now F = U − TS so at constant T: δF = −µ oM o .δH o

Equation (G2.1.55) with the first law of thermodynamics, δU = δq + δw, gives: (G2.1.53)

where δF is the change in Helmholtz free energy (note that with this experiment the Gibbs and Helmholtz free energies are interchanged in comparison with the previous expressions). Writing the work done in this form often significantly reduces the algebra. For example, it leads immediately to the equivalence of the area under the magnetisation curve and the condensation energy. The work done in bringing the sample up to the permanent magnet from infinity gives:

∫

Fn ( H c2 ) − Fn (0) = −µ o H o . dM o

(G2.1.54)

Since in the normal state M = 0; it follow that Fn(Hc2) = Fn(0) Hence:

∫

Fn (0) − Fs (0) = −µ o H o . dM o

(G2.1.55)

However, since experiments are not usually performed in this way, energies based on this form of the work done are less easy to understand physically. As a result, few authors use them, despite the brevity of the derivations which can result. Furthermore, Equation (G2.1.53) is incompatible with the standard Ginzburg–Landau expression for the free energy of a superconductor, since no work is done in bringing a normal sample into the field, while the G–L free energy includes a term b2/2μο in the normal state. However, Equation (G2.1.53) is often used by authors who wish to relate the magnetic properties to processes on an atomic scale and the energy levels of electrons in atoms. This requires a very different analysis from that presented here, although both can start at the same point, long samples parallel to the field. To show this, it is only necessary to point out that Equation (G2.1.44) shows that when we apply a field to a paramagnetic sample, we do work so its energy increases. However, the field splits the energy levels of the electrons into a high-energy antiparallel level, and a low-level parallel level into which the electrons go, reducing their energy. We will not attempt to reconcile these approaches in this article.

G2.1.3.2 Equilibrium Conditions Again following Pippard [2] we derive the equilibrium condition from the Clausius which is: δq ≤ ToδS

(G2.1.56)

where To is the temperature of the source of heat δq, and δS is the change in entropy of the sample. Combining inequality

δU − δw ≤ ToδS.

(G2.1.57)

From the definition of the Helmholtz free energy, F = U − TS, and given that the temperature of the sample T is usually equal to To, inequality Equation (G2.1.55) can be written as: δF ≤ δw

(G2.1.58)

In equilibrium at constant temperature, the work done is equal to the change in F, while the difference between δF and δU is the heat exchange. This leads to the following general and very useful expression of the condition for equilibrium with uniform T: δF = δw

(G2.1.59)

δF = µ oH o .δM o

(G2.1.60)

For a magnetic system:

If the external world can do no magnetic work on the sample (i.e. δMo = 0) but the temperature remains constant, so that heat goes in or out, from Equation (G2.1.58) the equilibrium state is when δF = 0, i.e. a minimum in F. Analytically, F is minimised at constant T and Mo. (In minimising thermodynamic functions it is essential to state explicitly what is kept constant). δw can be set to zero in a thought experiment by surrounding the sample with Type I superconductor which allows no change in total flux. This is usually the simplest procedure since most results of the G–L theory are not dependent on the external field and can therefore be derived from F at a given fixed B.

G2.1.3.3 The Availability and Magnetic Gibbs Functions The next step often taken in the thermodynamic analysis of the superconducting state is to write δw in terms of magnetic fields and to minimise the Availability or Gibbs function. This requires much more care and can be restrictive. For example, if a current is supplied from a battery, the work term is Vδq, where V is the voltage, and q is the charge drawn. Therefore, it is always a good idea to refer back to Equation (G2.1.59) (δF = δw) if in doubt. Many problems only involve the internal structure, so we can fix a value for B and minimise F at constant B and T. It is only necessary to introduce the work done and Gibbs functions if we want to determine the external field in equilibrium with a particular flux structure. (In effect this is minimising the free energy of the combined system consisting of the solenoid and sample, assuming that the solenoid behaves reversibly).

Handbook of Superconductivity

120

Writing the work done on the sample as δw = μοHo.δMo gives the following condition on δF: δF ≤ µ oH o .δM o

(G2.1.61)

or δF − µ oH o .δM o ≤ 0

(G2.1.62)

The Availability, A, is defined by, A = U − ToS − µ oH o .M o

(G2.1.63)

= F − µ oH o .M o δA = − SδT0 − µ 0M 0δH 0

(G2.1.64)

Usually T = T0. Hence, if T and Ho are constant, δA ≤ 0

Ginzburg–Landau only use F per unit volume and so avoid the complications of external fields and H.

G2.1.3.5 Summary of the Thermodynamics The following summarises the main thermodynamic arguments presented in this section. The most general and the most useful expressions for the work done by a solenoid on a Type II superconductor are:

(1)

∫

δw = H o .δMo + µ o H o δH o dV

where Mo is the total magnetic moment of the body, Ho is the applied field and the integral is over all space, V. In many cases, the integral term can be omitted.

∫

(2) δw = H.δ B dV (G2.1.65)

The equilibrium condition is then a minimum of A, at constant T and Ho. The Availability includes external fields and is not, therefore, a thermodynamic function of the state of the material such as the Helmholtz free energy or Gibbs function. Some authors refer to A as the Gibbs function, but this is a misleading terminology. Only if there is local magnetisation M, and H = Ho at all points in the sample does the Gibbs function, G = U – TS – μοH.M, become equal to the Availability. This is a fairly severe restriction. An example of the use of the Availability to derive the equivalent of the Clausius–Clapyron equation to relate the flux lattice melting temperature [21] to the entropy of the transition is given in Appendix III.

G2.1.3.4 Long Thin Samples The various expressions for the Availability and the Gibbs function converge if the analysis is limited to long thin samples oriented parallel to the external field. The differences between expressions can then be traced to various combinations of the following: 1. The free space energy ½μοHo2 is omitted. 2. H = Ho is assumed at all points in the sample, either on the scale of several flux lines, or on a scale of less than the vortex spacing. The lack of a distinction between the external field Ho and H, or a statement of the length scale over which fields are defined, is a major defect in many treatments. The treatment of superconductors via the Availability or the Gibbs function becomes equivalent in these conditions. Note that it is important to distinguish between local fields in the G–L equations, usually called b, the external field Bo = μο Ho and average flux densities over the whole sample B.

(G2.1.66)

(G2.1.67)

over all space. This is algebraically intractable if there are demagnetising factors. Equations (G2.1.66) and (G2.1.67) can be shown to be identical [19, 20]. (3) Equally general is the mechanical work done on a sample where the field at the sample is changed by moving the sample up to a permanent magnet. The work is done by the force on the sample, and is similar to the pressure on the gas times its area, commonly used in thermodynamic texts. This work is given by, δw = −µ oM o .δH o

(G2.1.68)

Equation (G2.1.68) is less easy to relate to real experiments, and is incompatible with the Ginzburg–Landau free energy, but simplifies some important derivations. (4) The condition for equilibrium is δF = δw. (5) A high proportion of the problems we need to solve do not involve the value of the external field. We can assume a fixed total flux, δw = 0 and we minimise F at constant T and flux. However, if we want to know the equilibrium external field, we must minimize the availability, A. (6) Since transport currents are not in thermodynamic equilibrium, the application of equilibrium thermodynamics to them is of doubtful validity, although it can give useful results. (A similar situation arises with thermoelectric effects).

G2.1.4 Flux Displacement In classical electromagnetism, the movement of magnetic flux is only considered to have meaning if the magnet producing it moves. However, in Type II superconductors the flux is quantised in vortices, and any change in field produces an observable displacement of the flux lines. The scale of this displacement has considerable physical significance. If this distance is much less than the size of the vortex core, then the vortices do not move far in their potential wells and remain

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Electromagnetic Properties of Superconductors

pinned. This produces a linear and reversible response to the applied field. If the flux is displaced by a distance of several vortex spacings, on the other hand, then vortices unpin, and we can use the critical state model. Slab geometry simplifies the algebra significantly and provides a useful example of the motion of flux. We begin with a large uniform field oriented parallel to the axis of a nonsuperconducting slab of width 2d and increase the flux density by small amount b. The increase in flux per unit length inside the slab is 2bd as shown in Figure G2.1.9(a). From the point of view of a flux lattice entering the slab, if the external flux density is B, and it moves a distance y,

then the change in flux per unit length in the sample is 2By. Figure G2.1.9(b) shows this as viewed from the top of the slab. Equating these two expressions, it follows that the distance moved by the flux is y = bd/B. If b is varying with the distance x (parallel to the slab thickness), then a similar argument shows that at any point dy/dx = −b/B. Consider, for example, a field of 1 T in a sample of thickness 10 μm. If the field is oscillated by 10 mT, then the amplitude of oscillation is 50 nm. This displacement will decrease if there is pinning in the slab material, and the response is extremely dependent on whether this distance is large or small compared with the flux spacing or coherence length. A more general approach for three-dimensional problems is to use the electric field. Assuming Equation (G2.1.34) (i.e. E = B × v, where v is the flux line velocity see Section G2.1.2.5), then for a harmonic oscillation: E = iω B × y

FIGURE G2.1.9 The increase in flux per unit length in a slab shown through a cross-section of the slab (a) perpendicular and (b) parallel to the direction of the applied field.

(G2.1.69)

where y is the amplitude of oscillation, and ω is the angular frequency. For more general motion, integration of the velocity will give the total distance moved. Hence, a measurement of the local electric field gives a direct measurement of how far the flux lines are moving, independent of any model. Typically, the response is linear if this distance is less than about one-tenth of the vortex spacing, although it is much smaller in high-Tc superconductors. Figure G2.1.10 shows E-J curves for an YBCO single crystal in a field of 4 T and a frequency of 200 Hz. BJ gives the force and E/ωB the displacement. In Figure G2.1.10(a) the field is at an angle to the ab planes, and it can be seen that the curve becomes non-linear at a displacement of about 2 nm, which is of the order of the coherence length. In Figure G2.1.10(b) the field was aligned accurately with the ab planes, and steps can be seen as the amplitude coincides with the spacing of the atomic planes. This is the most convincing demonstration of intrinsic pinning by atomic planes. Although this is a simple calculation, the flux line displacement involves a number of parameters. Small displacements usually occur at high-field amplitudes, or if the sample is small, as in multifilamentary superconductors. All high-frequency measurements tend to be in the linear regime since the field amplitudes are small, as is the case with oscillating reed experiments. Experiments in zero applied field, on the other hand, are usually large in amplitude since the vortices at the flux front must move macroscopic distances. The displacement can be closely related to the vector potential, and is measureable directly from the electric field., as discussed further in Section G2.1.5. The idea of flux movement in superconductors can be extended to normal metals and free space, since there is no discontinuity in behaviour between a material just below and just above Bc2. This concept can be useful in predicting the fields and currents in conventional materials. However, there are occasions where it can be misleading. For example, flux

122

FIGURE G2.1.10

Handbook of Superconductivity

(a), (b) Force-displacement for YBCO for B perpendicular to a single crystal at 4 T showing steps at lattice spacings.

can be pinned in a superconductor, which can then be moved through a uniform external field without experiencing a force (although rotation will produce a torque). Such difficulties can always be traced to the fact that flux lines can cut each other without difficulty in a normal material or free space, and at fields very close to Bc2 in a superconductor. Flux cutting normally occurs when the local current is not perpendicular to the local field, and in these circumstances, the concept of flux lines moving as a continuous line will break down since a unique velocity for the line cannot be defined where two vortices cross.

which is the flux enclosed in the loop. Also,

The vector potential A appears in many different physical situations and is defined by: ∇×A = B

(G2.1.70)

(The letter A is used both for the vector potential and the Availability, but the context should make clear which is meant). This definition of A, however, is not unique in that we can add the gradient of any scalar to A and still obtain the same value of B. Changing A in this way is a gauge transformation and must not change any observable quantity. A unique solution for A can be obtained by defining ∇.A and by specifying the boundary conditions. However, just as for the electrostatic potential, there are a number of different boundary conditions which can lead to different definitions. In computational electrodynamics, A is frequently useful because it reduces the number of dimensions involved in the calculation. For example, a two-dimensional vector field with zero divergence can be defined by a one-dimensional vector potential. Contours of A are then the lines of force or streamlines. If the vector is the current density J, then the vector potential for J is the magnetisation due to these currents (see Section G2.1.2.1) From the definition of A, it follows that for a closed loop,

∫ A.dl = ∫ ∇ × A.dS = ∫ B.dS

(G2.1.71)

(G2.1.72)

∇ × (A + E ) = 0

(G2.1.73)

E = − A + ∇φ

(G2.1.74)

Hence,

So,

G2.1.5 The Vector Potential G2.1.5.1 Classical Treatment

∇ × A = B = −∇ × E

where ϕ is any scalar function of position and time. If we add the condition that ∇.A = 0, then ∇ 2 φ = ∇.E = ρ / µ o .

(G2.1.75)

In this gauge ϕ is no longer a mathematical abstraction but is the negative of the electrostatic potential derived from the charges present.A can also be related to the movement of magnetic flux (see Section G2.1.5.2). In all potential problems there are many different sets of boundary conditions which lead to a unique solution, and this is also true of the vector potential. It is therefore quite difficult to define a gauge with a unique vector potential from the divergence of A, without adding a wide range of sufficient boundary conditions. A unique value can be defined as follows. From Equation (G2.1.71), ∇ × (∇ × A ) = ∇ × B

(G2.1.76)

∇ 2 A = −µ o J

(G2.1.77)

So, in free space:

(The vector potential is normally relevant on the scale of a few atoms, so in a superconductor we only need to consider a permeability μ if there are local atomic moments which cause a permeability. This is an unnecessary complication at this stage, but magnetic superconductors are an active research area).

123

Electromagnetic Properties of Superconductors

This vector equation is three scalar equations of the form familiar from electrostatics where the potential and charge density are related by Poisson’s equation. The solution of this equation is the potential due to the individual charges in the charge density so it follows that an expression for a vector potential with div (A) = 0 due to free space currents is:

∫

A = µ o J.dV / (4πr)

(G2.1.78)

(Note that to use this equation it must be in Cartesian co-ordinates, not polar or spherical). Ways of obtaining a vector potential for a known magnetic field are to calculate the flux from an arbitrary starting point, or the electric field from a sinusoidal oscillation of the field. Another is the use of a mesh of small loops, as in Section G2.1.2.1.

− A = E = B × v = B × s

The vector potential has much more physical significance in quantum mechanics since it appears in the classical Hamiltonian of a charged particle in a magnetic field. A is therefore very important in superconductors, where the electrons behave as if they were in a single state with a single wave function. A number of results describing the superconducting state involving A are summarised in this section without proof, which will be found in more detail elsewhere (see Phenomenological Theories A3.1). The analysis begins with the Meissner state and the London equations. The London equation: (G2.1.79)

(G2.1.82)

Thus, the vector potential is directly related to the flux movement by, δA = δs × B

G2.1.5.2 The Vector Potential in Superconductors

curl (µ o λ 2 J ) = − B

sufficient to define A. In article A3.1 the difference is illustrated for two simple examples. The vector potential can also be related to the movement of flux lines. We generalise the one-dimensional situation of Figure G2.1.9 to where small changes are made to the field in a pinned vortex lattice. Although apparently very different, this also leads to a London equation provided it can be assumed that the field is large compared with any changes and that it is oriented perpendicular to the current. (Fields with a component oriented parallel to the current lead to force free configurations and flux cutting, which are dealt with elsewhere). In the flux lattice, if the flux moves a distance s,

(G2.1.83)

Also E = −A. with no ∇ϕ term. In this gauge, however, ∇.A is not necessarily zero as can be seen from Appendix IV where ∇.A. is used to calculate the charge density. It is apparent from Equation (G2.1.83) that the change in vector potential is directly related to how far the flux moves. Now suppose that the movement is resisted by an elastic force due to pinning centres which is proportional to the displacement, s i.e., B × J = −αs

(G2.1.84)

where α is the curvature of the potential well. Then, for small changes from the original state, A = − B × s = α −1B × (B × J ) = −α −1B2 J

(G2.1.85)

can be written; curl (J + ΛA ) = 0

(G2.1.80)

If we also define div A = 0, since Div (J) = 0, it follows that: J = −µ o λ 2 A .

(G2.1.81)

The physical interpretation of this is that the current at any point is proportional to the flux that has crossed that point. London also added the boundary condition that An = − μο λ2Jn, where n refers to the normal component at the surface. In this gauge (the London gauge), the phase of the order parameter is constant and can be taken to be zero so the order parameter is real. Strictly this can only be used in simply connected samples (see Chapter A3.1 Phenomenological Theories). Notice that although the London and Coulomb gauges both define ∇.A = 0, they usually give different results because of the different boundary conditions, so that defining ∇.A is not

This is the same as the first London equation which yields an effective penetration depth λʹ where λ ′ = B (1/ αµ o )

(G2.1.86)

This penetration depth is determined by the curvature of the pinning wells [22]. Given this parameter, the electrodynamics is the same as that of a London superconductor, an AC signal decays exponentially from the surface within an effective penetration depth. The Josephson equations lead to essentially identical electrodynamics. A Josephson junction behaves as a single pinning centre for a Josephson vortex, and arrays of junctions pinning Josephson vortices are almost indistinguishable from arrays of bulk pinning centres populated with Abrikosov vortices. In a wire small compared with λ’, the voltage will be inductive and is known as the kinetic inductance (see Chapter A3.1). The vector potential also appears in microscopic and phenomenological theories. In the Hamiltonian equations of

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Handbook of Superconductivity

classical mechanics, forces on charged particles in magnetic fields can be included easily only if they are expressed in terms of A. For example, the canonical momentum p of a body with mass m and charge q is given by: p = mv + qA

(G2.1.87)

The appearance of A in the Hamiltonian of classical charged particles shows that magnetic fields should be included in wave mechanics by defining the wave vector k as k = p/ħ since for any wave the change in phase on moving a small distance δl is δθ = k.δl. Hence, p = k = ∇θ

(G2.1.88)

For example, the vector potential can be used to show how the BCS pairing of electrons in states with opposite k vectors is consistent with the phenomenon of resistance less current flow. Consider electrons in states with wave vectors k and –k and velocities v1 and v2, respectively. Then, if −e is the electronic charge, mv 1 = k − eA

FIGURE G2.1.11 SQUID ring.

current and stops, generating a voltage known as the kinetic inductance while it is moving. Finally, if the critical current is exceeded, a series of vortices enters the ring, each generating a voltage pulse and bringing in a flux ϕo. If the frequency is f vortices per second, then the voltage V across the junction is:

(G2.1.89)

and

A Josephson vortex approaches a weak link in a

V = fφo = fh/2e

(G2.1.93)

f = 2eV/h

(G2.1.94)

Hence, mv 2 = − k − eA

(G2.1.90)

Adding Equations (G2.1.89) and (G2.1.90) gives the average electron velocity as v = eA/m. Hence, putting J = −nev gives the London equation: µoλ 2J = −A

(G2.1.91)

λ = (m / (µ one 2 ))

(G2.1.92)

with

This is the same as found from the free electron picture. The supercurrent arises because, if the momentum of a pair is zero, the presence of a magnetic field means that the velocities do not sum to zero. Note that a hidden assumption in this analysis is that the wave function is rigid with respect to small changes in field, this is a result of the BCS theory, (see Chapter A2.2). The link between these quantities is illustrated qualitatively in Figure G2.1.11. Here an increasing external current causes a ‘virtual’ vortex to approach a weak link in a SQUID ring. When the vortex enters the ring, it becomes a Josephson vortex. The distance it moves is proportional to δA, and this is related to the phase difference across the junction θ, which will change by 2π if the vortex gets into the ring. If the current driving the vortex is below the critical value, the vortex moves a distance proportional to the

where f is the Josephson frequency. This result was obtained originally by Josephson from tunneling theory [23]. If an external source of field coincides with this frequency, it will synchronise with the vortices as they enter and give a step in the V-I characteristic. It is clear that A has a great deal of physical significance in superconductors.

Acknowledgements I have benefited from the advice of a number of people in writing this article, but I would particularly like to thank the late Sir Brian Pippard who pointed out to me a large number of errors and sections of muddled thinking which I have tried to put right.

References [1] L. D. Landau and E. M. Lifshitz, The Electrodynamics of Continuous Media, Pergamon, Reading, (1960). [2] A. B. Pippard, The Elements of Classical Thermodynamics, Cambridge University Press, Cambridge (1957). [3] J.E. Evetts and A.M. Campbell, Proceedings of the 10th Conference on Low Temperature Physics, Moscow, 1966, Vol IIB, pp. 33-37. A. M. Campbell and J. E. Evetts, Rev Mod Phys. (1971).

Electromagnetic Properties of Superconductors

[4] B.D. Josephson, ‘Macroscopic Field Equations for Metals in Equilibrium’, Phys. Rev. A, 152, 211–217 (1966). [5] A. A. Abrikosov, JETP, 5, 1174 (1957). [6] C. Kittel, Introduction to Solid State Physics, Chapman and Hall, London. [7] R. Goldfarb, Encyclopaedia of Magnetic and superconducting Materials, J. E. Evetts editor. Pergamon, Oxford (1992). [8] E. Zeldov, A.I.Larkin, M. Konczykowski, B. Khaykovich, D. Majer,. B. Geshkenbein, and V.M.Vinokur, ‘Geometrical Barriers in Type-II Superconductors’, Physica C, 235, 2761–2762 (1994). [9] A.D. Caplin, M.A. Angadi, J.R. Laverty, and A.L. de Olivera, Supercond. Sci. Tech., 5, 161–164 (1992). [10] W.T. Norris, J.Phys. D, 3, 489 (1970). [11] E. H. Brandt, Phys. Rev. B, 50, 4034–4050 (1994). [12] E. H. Brandt and M Indenbom, Phys. Rev. B, 48, 12893– 12906 (1993). [13] E. H. Brandt, Phys. Rev. Lett, 67, 2219–2222 (1991) [14] M. Daeumling and D. C. Larbalestier, Phys. Rev.B, 40, 9350 (1989). [15] B. D. Josephson, Phys.Lett., 16, 242–243 (1965). [16] J. Bardeen and M. J. Stephen, Phys. Rev. A, 140, 1197– 1207 (1965). [17] A. M. Campbell, IEEE Trans. Magn. Appl. Supercond., 5, 682–687 (1995). [18] S.Fukui, Y.Kitoh, T.Numata, O.Tsukamoto, J.Fujikami, and K Hayashi, Adv. Cryog. Eng., 44, 723 (1998). [19] J.A.Stratton, Electromagnetic Theory, McGraw Hill, New York (1941). [20] V. Heine, Proc. Cambridge Philos. Soc., 52, 546–562 1956 (1961). [21] E. Zeldov, D. Majer, M. Konczykowski, V. B.Geshkenbein, V. M.Vinokur, and H.S. Shtrikman, Nature, 375, 373–376 (1995). [22] R A Doyle, A M Campbell and R E Somekh, Phys. Rev. Letts., 71, 4241–4244 (1993). [23] B.D. Josephson, Phys. Lett., 1, 251–253 (1962).

APPENDICES G2.1.A1 Appendix I G2.1.A1.1 A Discussion of the Magnetic Field H Maxwell’s equations in free space are possibly the most fundamental laws of physics, and can be written in terms of two fields E and B, and the fundamental constants μo and ε0. One of either μo and εo can be chosen arbitrarily to define electrical units. If we accept the validity of special relativity, the equations are relatively uncontroversial in free space. The problems arise in applying these equations to materials and they can be traced to two sources. One was the desire to base the

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equations on macroscopic experiments rather than attempt to use our knowledge of the atomic structure of real materials. Most of these experiments were impractical and remained ‘thought’ experiments. The other was to regard free space as a particular example of a more general class of material, i.e. linear materials, and describe other materials by means of a non-linear and hysteretic permeability. This would be similar to attempting to describe plastic deformation and fracture mechanics as merely examples of non-linear hysteretic elastic moduli. This may allow solutions of simple problems, but it completely obscures the physics. The correct procedure using average fields is not new, it was originated by Lorentz and will be found in Landau and Lifshitz [2]. However, it is only in the last twenty years that it has become standard in physics courses, and is only now beginning to appear in electrical engineering books. Although Maxwell’s equations must inevitably describe materials on an atomic scale, the results are far too complex to be useful and this is why it is necessary to apply them to average fields over macroscopic dimensions. We then divide any currents flowing into atomic scale and large scale currents. This is usually achieved in conventional magnetic materials by writing the equations in terms of two fields, B and H. Historically, however, these fields, have been defined in many different ways. Before the structure of materials was understood, the fields were defined in macroscopic terms from the forces on bodies in fluids, and extrapolated to solids, although the forces could not be measured in solids. This was extremely unsatisfactory since forces in liquids include the effects of hydrostatic pressure, while in a solid the stress is a tensor, which depends not only on the fields but also the rate of change of permeability with strain. It is not possible to express the force on a boundary solely in terms of the magnetic fields on either side of the boundary [1]. To take a simple example, the force between two capacitor plates with a fixed charge decreases if a dielectric fluid is poured into the capacitor, but is unaltered if a solid dielectric is inserted between the plates. If we define B at a point in a solid as the force on a current element j at that point, we conclude that the force on a wire is BI, which will be wrong by a factor of up to μ, depending on the shape of the wire. Furthermore, experiments were confined to linear reversible materials, and so the definitions could not be used for hysteretic ferromagnetic materials. Forces on currents in materials are a very complex subject and can play no part in the development of electromagnetism in materials. This approach also led to physical significance being attached to the units in which fields were measured, which can be very misleading. The SI system of units has removed this confusion. Units are discussed further in Appendix V. A second, more mathematical, approach has been to define the fields involved as vectors which obey Maxwell’s equations. However, this requires a relation between B and H to be determined experimentally which, in turn, requires an explicit method of how such a measurement should be performed. Such a prescription requires an understanding of the materials being measured; a method devised for ferromagnetic

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materials will not work for superconductors, so this approach is only useful for simple well-understood materials. A common definition is in terms of the fields in slots in a material. This has something to recommend it since it gives the right answer in virtually all situations. It still needs some knowledge of the material since we need to know what size and shape of slot is appropriate. We must not cut slots which divert a supercurrent, and the size must be appropriate to the microstructure. However, the main disadvantage of this approach is that it gives little physical understanding of what is happening inside the material. Knowledge of the microscopic nature of materials, therefore, must be used to define fields in such a way that Maxwell’s equations in their conventional form in free space can be applied inside a material. The flux density B is always defined as the average field. The definition of H, however, presents more problems. In conventional magnetic materials with atomic or electronic magnetic dipoles we can define the magnetisation, M, as the dipole moment density, and then H as: H = B/µ o − M

(G2.1.A1)

Appendix VI gives further details of how to derive Maxwell’s equation in a magnetised material. This definition works well both for reversible and hysteretic ferromagnetic materials, but not for superconductors in which the magnetisation cannot be traced to local dipoles. Josephson [4] defines H as the derivative of the Helmholtz free energy per unit volume, f, with respect to B, i.e for each component H = (∂f/ ∂B)T

(G2.1.A2)

δf = H.δB

(G2.1.A3)

or

This is equivalent to the definition in reference [3] as the external field in equilibrium with a flux density B. With this latter picture H can be regarded as the chemical potential of a flux line. It leads to conventional electrodynamics in normal materials, in Type II superconductors in thermodynamic equilibrium, and also in hysteretic superconductors. However, it cannot easily be applied to hysteretic ferromagnets. The Meissner state raises particular difficulties in interpreting the field H. The simplest case is that of a simply connected sample in thermodynamic equilibrium, of dimensions much larger than the penetration depth. In this case, Josephson’s treatment of H is applicable. The material is treated as a conventional magnetic material with μ = 0. The field in a long cylinder of material oriented parallel to an external field Ho is then given by: H = Ho

(G2.1.A4)

However, this implies that the field in a hole drilled along the axis of the cylinder will also be Ho, which is not necessarily the case. It is only true if a field Ho is trapped in the hole, which is the equilibrium state of lowest energy, but there are many metastable states with arbitrary amounts of trapped flux. This means there is no entirely satisfactory definition of H in the Meissner state. Fortunately, the above arguments are mainly of academic interest, and it is usually unnecessary to use H in a superconductor. The main point to be made here is that H can be a useful quantity, but must be defined carefully according to the nature of material. It is nearly always possible to find a definition of H which allows Maxwell’s equations to be used in a material. It is necessary, however, to understand the material before we can decide what the definition should be, and it cannot usually be closely related to any of the internal fields in the material on a microscopic scale. Having defined the fields so that Maxwell’s equations are obeyed, conventional magnetostatics can be used to calculate the fields in samples of any shape. The local relation between B and H, which can be measured in long thin samples, takes care of the magnetisation currents which in ferromagnets are provided by local dipoles. The fact that the flux exists in the form of lines in superconductors, rather than atomic dipoles, makes no difference to the equations or energies involved so that the results obtained from the magnetic case of local dipoles can be used to determine fields in superconductors. Currents carried by pinning centres must be treated as transport currents, J. This is illustrated in Figure G2.1.A1. Figure G2.1.A1(a) shows a material with no pinning but a gradient in a material property such as κ. Since J = 0, it follows that H is uniform, so there is a gradient in B and in the vortex density. This causes a force on the vortices, but this force is taken up by the lattice, and there is no net force on the vortices. The force on the bulk is balanced by differing surface pressures due to the different surface currents, and there is no force on the body in a uniform field. The bulk current due to the changing vortex density is balanced by the different surface currents and there is no net current. The body is in thermodynamic equilibrium, and the bulk current is a magnetisation current, not a transport current.

FIGURE G2.1.A1 rial with pinning.

(a) No pinning, gradient in κ. (b) Uniform mate-

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Figure G2.1.A1(b) shows a material with uniform superconducting properties but with pinning centres. J is the critical current density, and there is a gradient in μoH which is equal to the gradient in B. The surface currents are equal, there is a force on the body and a net current. This shows how it is the gradient in H, not B, which determines whether a material is in thermodynamic equilibrium and the transport current. This is a one-dimensional picture. For three-dimensional fields it is shown in refs [3] and [4] that it is the curl of H, not the gradient, which determines the transport current and Lorentz force. This allows the line tension of the flux lines to be included. To summarise, Maxwell’s equations in a conventional material are derived first using the concept of local dipoles, which are then used to derive the free energies of a material in terms of the fields. In the case of superconductors, there are no local dipoles so arguments based on the energy of flux lines have to be used to derive equivalent equations. The equations are derived for both classes of material in equilibrium in reference [4] using purely thermodynamic arguments without reference to either dipoles or flux lines. However, hysteretic effects in the two types of material require separate treatment.

G2.1.A2 Appendix II G2.1.A2.1 The Susceptibility of Powders Superconducting samples frequently consist of powders in a non-magnetic medium. In this appendix, the effective susceptibility of a single particle χeff is related to that of the whole sample using the Lorentz theory of dielectrics. The apparent magnetisation, Mp, of a single particle (i.e. total moment/volume) is given by: M p = χeff h1

(G2.1.A5)

where hl is the local field applied to the particle, assumed to be uniform over the particle. [The relation of χeff to the susceptibility of the material of which the particle is composed is discussed in Section G2.1.2.4.2. Equation (G2.1.A5) gives the relationship for a linear material]. According to the Lorentz theory, the contribution of dipoles in a sphere around a particle sums to zero if the dipoles are randomly placed. Therefore, local dipoles over a spherical region centred on the dipole can be removed from the analysis, leaving the individual dipole of interest isolated in a spherical cavity [Figure G2.1.A2(a)]. If the sample has magnetisation M, demagnetising factor n and is in an external field Ho then, h1 = H o − nM + M/3

(G2.1.A6)

This follows directly from the electrostatic analogue in which a polarisation can be replaced by surface charges which give the same E, as is shown in Figure G2.1.A2(b).

FIGURE G2.1.A2 (a) The field on a dipole due to the magnetisation is divided into that from a sphere round the dipole, which is zero, and that from the remainder of the sample. (b) The electrostatic analogy and its equivalent charge distribution.

Now M = fMp where f is the volume fraction of the particles so, M = fχeff H o /(1 + nfχeff − fχeff /3)

(G2.1.A7)

This correction to the dilute expression (M = fχeff Ho) fails as f approaches unity since it is no longer valid to consider the particles as being in a uniform local applied field. If the particles are spherical and composed of a material of susceptibility χ, then Equation (G2.1.A7) shows that: χeff =

χ 1+ χ / 3

(G2.1.A8)

The susceptibility of the dispersed powder χp is then: χp =

fχ 1 + χ (1-f ) / 3

(G2.1.A9)

G2.1.A3 Appendix III G2.1.A3.1 The Clausius–Clapyron Equation As an example, the results derived in Section G2.1.3.3 are used to find the equivalent of the Clausius−Clapyron equation when applied to the flux melting transition in high-Tc crystals. (Note that a transition of this nature must become second order in

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samples of flat geometry although this is not too important if the range of the transition is small). In either case, the analysis refers to the whole sample, and not to local values. At the transition, the material can be changed reversibly between liquid and solid states, i.e., FL − FS = Δw = µ oH o (M oL − M oS )

(G2.1.A10)

where subscripts L and S refer to liquid and solid states, respectively; Mo is the total moment, and F is the Helmholtz free energy. In other words, the availability AL = AS. Now, if the external field and temperature are changed then in general, δA = δ(U − TS − µ oH o M o ) = −SdT − µ oM o dH o

(G2.1.A11)

If this change is made in such a way as to remain on the melting line, the condition AL = AS must still apply so δAL = δAS. Hence, −SLdT − µ oM oL dH o = − SSdT − µ oM oSdH o

(G2.1.A12)

Rearranging and assuming Ho is parallel to Mo, µ odHo /dT = ( SS − SL ) / ( M oL − M oS ) (G2.1.A13) Note that this derivation cannot be done using B, unless we are prepared to integrate the magnetic energy over all space. In the experiments showing flux lattice melting [21], it is the flux density that is measured with miniature Hall probes, not the magnetisation. To find the magnetisation (Mo divided by the volume), the change in B measured by this technique must be multiplied by 1 − n, where n is the demagnetising factor. 1 − n is approximately equal to the aspect ratio. The change in the local B-H curve is independent of shape, and this is what is calculated by theoreticians. If melting produces a change in susceptibility Δχ, Equation (G2.1.A7) shows that in principle the changes in B and M will both be dependent on the aspect ratio. However, since χ is small, we can assume ΔM = ΔχΗο, and ΔB is μοΔχΗο (b/a), where (b/a) is the aspect ratio. The entropy of a liquid is greater than that of a solid, and the melting field, and like all other critical fields in a superconductor, decreases with temperature. As a result, Equation (G2.1.A12) shows that the magnetisation must increase on melting so that the density of the vortex liquid is greater than that of the vortex solid.

G2.1.A4 Appendix IV G2.1.A4.1 Charge Densities in a Superconductor This is an example of a situation in which an electrostatic charge exists in a superconductor.

FIGURE G2.1.A3 An increasing magnetic field applied parallel to the axis of a square cylinder produces a separation of charge.

A large uniform field B is applied a rate b. parallel to the axis of a cylinder of square section of a Type II superconductor obeying the Bean model (i.e. with constant Jc), as shown in Figure G2.1.A3. The critical state leads to a uniform current density flowing in the material in a square pattern. The electric field must be parallel to the current lines, corresponding to flux flowing parallel to the y axis in the top and bottom quadrants and parallel to the x axis in the left and right quadrants. The velocity of the flux is proportional to the distance from its position of zero velocity, at its intersection with the diagonal. For x and y positive (here y is one of the Cartesian components, not the flux line displacement), and y > x, the flux displacement d is given by, d = b(y − x)/B

(G2.1.A14)

E x = b (y − x)

(G2.1.A15)

The velocity is d. and so,

There is then an electrostatic charge density given by, ρ = ε o∇.E = −ε o b

(G2.1.A16)

A similar charge density appears in the other quadrants. Although at first sight the existence of an electrostatic charge in a superconductor may appear strange, there is no paradox. Such charges are inevitable in conductors if either the conductivity changes with position or the current density is nonuniform in a non-linear material, which is the case here. Note that E = B × v still applies despite the presence of electrostatic charges. In addition, a vector potential proportional

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to the flux displacement may be used so that E = −A., there is no ∇ϕ term.

G2.1.A5 Appendix V G2.1.A5.1 Units The old cgs system of units created endless confusion as it had three different units for each quantity. The situation was made worse by the idea that there was some physical significance in our choice of units. For example, there was a major controversy as to whether we should measure magnetisation in Gauss or Oersted. The SI system, although not perfect, has removed nearly all of this confusion. B is called the flux density and measured in Tesla, H is called the field and measured in amps/metre, and M is the magnetisation in amps/metre. Some modifications have crept into this system, which are convenient and fairly harmless. For example, we normally quote the magnetic field of a magnet in Tesla, i.e. we quote μoH. To insist on calling it a flux density rather than a field because we measure it in Tesla seems a little pedantic. Similarly, we quote μoM and call it the magnetisation in Tesla. This gives convenient numbers and allows the three fields to be compared in magnitude. If necessary, any ambiguity can be avoided by calling the field the H field or B field as appropriate. (It would have been better if the SI system had defined the units of B, H, and M as Tesla, but it is too late to change this now). Chemists are often concerned with the moments of atoms and quote magnetisation as moment per unit mass, as this will not depend on the density of the sample. This magnetisation cannot be used in the field equations which are in terms of the moment per unit volume. Theoreticians, particularly those working with relativity, like a system in which the fundamental constants are εo, and the speed of light, but this is not very convenient for practical purposes. The SI system of units is therefore recommended as standard. It is important to separate the definition of a quantity from the definition of the unit in which it is measured. The former requires an understanding of the physics. The latter will depend on the most accurate experiment which can be performed, and will change as technology changes.

G2.1.A6 Appendix VI G2.1.A6.1 Maxwell’s Equation in a Material (Neglecting the Displacement Current) On a local scale Ampères theorem states that ∫b.dl = μο I and on averaging ∫B.dl = μοII-.

FIGURE G2.1.A4 A random array of dipoles projected onto the xy plane. The dot is the projection of the line element δl.

I- is the mean current density, which for the moment we assume to be due to local dipoles. Hence, we need to work out the average current intersected by a line in a material containing small current loops. Figure G2.1.A4 shows the projection of the loops of the z component of the magnetisation onto the xy plane for a length δl in the z direction. The line δl is then a point in this plane and the probability of it lying within a particular loop of area δS is δS/A, where A is the area of the array. If the loop is intersected, there will be a contribution to the line integral of B of I, where i is the current in the loop. Otherwise, there will be no contribution. Therefore, the mean contribution will be the current multiplied by the probability, iδS/A or m/A, where m is the moment of the loop. The total current for the volume A x δl is Σm/A for this volume. Now, the magnetisation M is defined as Σm/(Aδl), so I- = M δl. Since I- is the current due to localised dipoles, we must add any macroscopic current I to this mean current to get the total field. If we divide the currents in this way between atomic dipoles and macroscopic transport currents I, and include components of M in other directions, then Ampères theorem becomes:

∫ B.dl = µ I − µ ∫ M.dl o

o

(G2.1.A17)

We now define Η = B/μο − M so that Equation (G2.1.A15) becomes:

∫ H.dl = I

(G2.1.A18)

or ∇ × H = J.

(G2.1.A19)

This shows that by taking averages, we can write Maxwell’s equation in materials in the conventional form.

G2.2 Numerical Models of the Electromagnetic Behavior of Superconductors Francesco Grilli

G2.2.1 Introduction Numerical modelling of superconductors is a vast field because different aspects of superconductors can be modeled, including electromagnetic behavior, thermal effects, mechanical forces, fluid dynamics (for the coolants), and network behavior in an electric grid. This chapter focuses on the modelling of the electromagnetic behavior of superconductors (with a section dedicated to the coupling of electromagnetic and thermal effects), because it is what makes superconductors different from conventional materials. Consider, for example, the current penetration in a conductor carrying an AC transport current: in normal metals, this gives rise to the so-called skin effect, which can be investigated with stationary models in the frequency domain. In superconductors, the highly non-linear current-voltage relationship gives rise to a more complex electrodynamic behavior, which usually requires time-dependent simulations able to describe the movement of the magnetic flux during an AC cycle. In addition, the non-linear resistivity of superconductors also depends on the magnetic field amplitude and orientation (sometimes in a quite complex fashion), on the temperature, and possibly also on the position (due to non-uniformities created by the manufacturing process). These complexities require the development of specific numerical models. This chapter aims to give readers an overview of such models, also providing references where more details can be found. Most of the examples shown here deal with hightemperature superconductors (HTS) and coated conductor tapes in particular, because of the increased interest in developing applications based on these materials (Eisterer, Moon, and Freyhardt 2016). However, the same modelling techniques can be used also for low-temperature superconductors. The latter are actually simpler objects to simulate, both in terms of physical and geometric properties: their current-voltage characteristic is sharp so that the Bean model (and the analytical formulas based on it) can often be

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used. Their geometry is characterized by conductors with round cross-sections (or assemblies of such conductors), which are generally easier to handle than the high aspect ratio geometry typical of HTS coated conductors.

G2.2.2 Analytical Models for Electromagnetic Behavior Several analytical models for superconductors have been developed in the past decades based on the critical state model (CSM), originally developed by Bean in the 1960s (Bean 1962; 1964). An example is displayed in Figure G2.2.1, which shows the evolution of the current density and magnetic field profiles in a round superconducting wire carrying AC transport current, according to Bean’s critical state. The current and field patterns depend on the ‘history’ of the sample: note for example the differences between the instants ωt = π and ωt = 2π. They both correspond to a zero total current, but they are reached from different magnetization paths: in particular, after the current has gone through its maximum (ωt = π/2) and minimum (ωt = 3π/2) values, respectively. Under the critical state assumptions, the current and field distributions do not depend on the speed of such history: exactly the same patterns are obtained if the frequency of the sinusoidal current is changed. The example shows two important features of the critical state model: it is intrinsically frequency-independent and it cannot account for overcritical currents, since the maximum amplitude of current density is Jc. The main advantage of analytical models is that they provide concise expressions for the current density and magnetic field profiles in the superconductor in a variety of working conditions. The most commonly used expressions are those for AC losses, which can be used to check whether a superconducting sample behaves according to the theory.

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Numerical Models of the Electromagnetic Behavior of Superconductors

FIGURE G2.2.1 Temporal evolution of the current density in a round superconducting wire carrying AC transport current (sinusoid), according to the Bean model. Significant instants of the first AC cycle are selected. The black lines represent the magnetic field profile inside the wire. Image courtesy: Philipp Krüger, Karlsruhe Institute of Technology, Germany.

Norris provided formulas for the losses caused by an AC transport current in a conductor with elliptical or infinitely thin rectangular cross-section (Norris 1970). i ellipse I c2µ 0 (1 − i )ln(1 − i ) + (2 − i ) 2 (Jm −1 ), QT = × π (1 − i )ln(1 − i ) + (1 + i )ln(1 + i ) − i 2 thin strip (G2.2.1)

where i = Ia /Ic and Ia is the amplitude of the transport current. For both shapes the losses do not depend on the actual dimensions of the conductor, but just on its critical current Ic and on the fraction of it that is transported. Brandt and Indenbom (1993) and Zeldov et al. (1994) calculated the expressions for the current and field penetration in an infinitely thin rectangular conductor (strip) and the corresponding losses for the cases of transport current only, external field only (perpendicular to the conductor’s wide face), and combinations of the above. These thin strip models differ from the original Bean model in a number of ways. One remarkable difference is that, when the flux has partially penetrated the strip and a critical state is established near the edges, the currents flow over the entire width of the strip in order to shield the flux-free region. In the original Bean model, the flux-free region is also current-free, as shown in Figure G2.2.1. More details are discussed in Brandt and Indenbom (1993) and Zeldov et al. (1994).

A frequently used formula is that for the magnetization losses of a thin strip of width 2a and sheet current density jc in a field of amplitude Ha perpendicular to the strip’s surface, originally derived by Halse (1970):1

QM = 4µ 0a 2 jc H a g ( H a / H c ) (Jm −1 ),

(G2.2.2)

where g(x) = (2/x) ln cosh(x) − tanh(x) and Hc = jc/π. The curve of the magnetization losses of a thin strip presents a change of slope: from QM ∼ H a4 at low fields to QM ~ Ha at high fields. A similar change of slope, related to the full penetration of the field in the superconductor, is observed in other geometries, too. The formula also reveals a quadratic dependence of the losses on the strip’s width 2a, which is why a common way to reduce the dissipation of coated conductors is by introducing narrow filaments.2 Another frequently used model is that for the magnetization of a slab in parallel field (Wilson 1983), which Iwasa used and extended for analyzing a variety of situations relevant 1

2

Due to differences in notation and approach, the expression for the losses has a different form in the papers Brandt and Indenbom (1993); Halse (1970) and (Zeldov et al. 1994), but they are in fact equivalent. Here we adopt the one from Brandt and Indenbom (1993). When the magnetic field fully penetrates the superconductor, under the assumption of absence of electrical coupling between the filaments and of negligible width of the separation between the filaments, the losses of a superconducting strip striated into N filaments are lower than those of the original (non-striated) strip by a factor N × 1/N2 = 1/N.

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for superconducting magnet applications, with the simultaneous presence of transport current and external field (Iwasa 2009). Analytical models exist also for infinite assemblies of thin tapes like horizontal arrays and vertical stacks, as well as for thin tapes with magnetic substrate (of infinite permeability). An exhaustive review of the available analytical models is given in Mikitik et al. (2013). Although analytical models are very attractive because of their ease of use, they have important limitations. Most analytical models are based on the critical state model and on a constant Jc (although some models with field-dependent Jc have been developed – see dedicated section in (ibid.)). If superconducting wires have magnetic materials, the magnetic permeability of those materials is assumed to be constant or infinite, whereas in reality it varies very nonlinearly. These limitations mean that in several instances analytical models cannot be used to make quantitatively accurate predictions, e.g. of AC losses. Another limitation is the fact that the considered current and field excitations usually have simple temporal variations (ramps, sinusoids, or their combination) and spatial uniformity. These limitations can be overcome by numerical models, which are the subject of the next section.

G2.2.3 Numerical Models for Electromagnetic Behavior A common way of studying the electromagnetic response of superconductors is by using finite elements to solve a differential form of time-dependent Maxwell’s equations, which can be written in different formulations, i.e. using different electromagnetic quantities as state variables. Independently of the chosen formulation, the superconductor’s electrical behavior is usually described by means of a non-linear resistivity (or conductivity).3 For example a 3

The superconductor’s magnetic constitutive relation is usually assumed to be µ = µ0, because the lower critical field Bc1 is very small.

Handbook of Superconductivity

popular choice is a power-law resistivity depending on the magnitude of the current density J

ρ( J ) =

Ec J Jc Jc

n −1

,

(G2.2.3)

where Jc is the critical current density, Ec is the critical electrical field at which Jc is determined, and n is an exponent describing the flux-creep regime. Both Jc and n may depend on the amplitude and orientation of the magnetic field, as well as on the position inside the tape as a result of non-uniformities caused by the manufacturing process. The power-law model converges toward the critical state model for n → ∞ . Using a continuous (although highly non-linear) resistivity allows considering over-critical currents and introduces a dependence of the results on the frequency. For example, Figure G2.2.2 displays the current density distribution in a round wire at the peak of a sinusoidal cycle, obtained with a transport current equal to 80% of Ic at different frequencies. At very low frequencies, there is enough time for the current density to ‘relax’ toward a uniform distribution. In that case the wire’s cross-section is uniformly filled with a current density equal to 80% of Jc. In contrast, at high frequencies, the current density does not reach the center of the wire and is distributed at the periphery of the wire, where it assumes a value close to Jc. This situation resembles that of the critical state (compare to Figure G2.2.1), the difference being that with the CSM J assumes only the value Jc, whereas with the power-law model J locally reaches over-critical values. The power-law (G2.2.3) is a good choice for describing the behavior of the superconductor in the vicinity of Jc. However, alternative E − J relationships are preferable in other cases, e.g. to simulate the superconductor in situations when knowing its behavior at electrical fields substantially lower or higher than Ec is important. (See for example Li et al. 2018; Sirois, Grilli, and Morandi 2019; Roy, Dutoit, et al. 2008). Methods different from finite-element analysis, such as integral and variational methods, have been developed to

FIGURE G2.2.2 Distribution of the current density (normalized to the critical value Jc) at the peak of an AC cycle of different frequencies, obtained with a power-law resistivity and n = 25. The amplitude of the transport current is 80% of Ic.

Numerical Models of the Electromagnetic Behavior of Superconductors

simulate the electrodynamic behavior of HTS. A comprehensive open-access review can be found in Grilli et al. (2014). Numerical models of superconductors have greatly improved over the past few years and most of them are now able to simulate complex problems (Grilli 2016). In the following, we consider several examples of transient problems involving superconducting assemblies, such as stacks of tapes, cables and coils. When tapes are brought close together, e.g. in stacks or coils, they interact electromagnetically. This has a great influence, for example, on AC losses, whose level is a key factor for the practical realization of many superconducting applications. If AC losses are prohibitively high, the associated cryogenic cost can make the applications not attractive or even unrealistic. In the case of the magnetization losses caused by an external magnetic field applied perpendicularly to the wide face of the tapes, the effect of stacking tapes is positive: the AC loss per tape decreases with increasing number of tapes. This is because of two reasons: first, the stack of tapes can be seen as a conductor with a width-to-thickness ratio much lower than that of an individual tape, because of the extra thickness provided by the space between the tapes and by the non-superconducting materials (in a typical HTS coated conductor, for example, the superconductor is only about 1 µm thick, but the metallic substrate is 50–100 times thicker). Conductors with lower aspect ratio have generally lower magnetization losses, due to a reduced demagnetization effect (Campbell 1982). Second, the tapes situated in the interior of the stack are ‘protected’ by the ones situated at the top and bottom of the stack, and experience a lower field. The magnetization losses for stacks made of different numbers of tapes are shown in Figure G2.2.3 for HTS coated conductors

FIGURE G2.2.3 Magnetization AC losses of a stack of HTS coated conductors with different numbers of tapes: experimental data (symbols) and simulation results (continuous lines). The loss prediction for an infinite stack is also shown, calculated both by means of an analytical formula and finite-element method (FEM) simulations.

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FIGURE G2.2.4 Magnetic flux density distribution in a stack composed of six tapes for an external field of 50 mT. The image refers to the peak instant.

1 cm wide and put in close contact. The agreement between experimental data and simulation is excellent. The two effects (described above) leading to lower losses in stacked tapes are visible in Figure G2.2.4, which shows the magnetic flux density distribution in the stack composed of six tapes for an external field of 50 mT. It is worth emphasizing that in this kind of comparison there is no fit of the experimental data: the only inputs for the simulations are the superconductor’s properties (in this case, Ic = 150 A and n = 19), which are measured independently of the AC losses, and the geometric layout of the stack. It is also worth noting that the AC loss values span over five orders of magnitude and that the agreement is consistent over the whole range. Another reason why numerical models are useful is that they can simulate situations that cannot be considered with analytical formulas. In this specific tape arrangement, for example, it is possible to analytically calculate the magnetization losses of a single tape4 and of a stack composed of an infinite number of tapes (Mawatari 1996). However, these are two extreme situations, which can only provide the upper and lower limits for the losses of a realistic stack of tapes. These limits are situated orders of magnitude apart, so any accurate estimation of the AC losses of a realistic stack (with a finite number of tapes) can only be made with the help of numerical models. Numerical models are nearly indispensable for the detailed analysis of superconducting wires made of different materials. Consider, for example, a multi-filamentary MgB2 wire as shown in Figure G2.2.5. In addition to being composed of several filaments of superconductor material, it is also characterized by the presence of materials (such as nickel and Monel in this case, and sometimes also iron) whose magnetic properties strongly influence the penetration of the 4

The magnetization losses of a single tape can be calculated by means of Equation (G2.2.2). The results are very close to those calculated with the finite-element model, and are not shown in Figure G2.2.3 for clarity.

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FIGURE G2.2.5 A finite-element model of a MgB2 wire, with realistic filament shape. The filaments are embedded in a nickel matrix, which is surrounded by Monel alloy. Both metals present ferromagnetic properties. Image courtesy: Columbus Superconductors.

magnetic field in the wire, and hence the wire’s performance. The main issue of modelling this material is their non-linear and hysteretic magnetic permeability, which requires a specific formulation of the magnetic constitutive equation. Implementing magnetic hysteresis cycles in time-dependent electromagnetic problems is not an easy task (Mayergoyz 2003). If no simultaneous electromagnetic and thermal coupling is required, the AC losses in the ferromagnetic materials can be calculated a posteriori by considering only the maximum field experienced by the ferromagnetic material and by using an empirical relation linking it to the losses, as done in Nguyen et al. (2010). Real applications such as cables, coils, and magnets consist of a large number of tapes or turns, and they can usually be simulated in 2D by considering them to be infinitely long (cables) or in an axisymmetric description (coils). This allows simulating relatively complex devices in great detail, while keeping the size of the problem and the computation time at acceptable levels. Figure G2.2.6 shows the 2D axisymmetric modelling scheme for a pancake coil made of HTS coated conductors without and with flux diverters on top and bottom. The latter are used to reroute the magnetic flux away from the superconductors, hence reducing the AC losses. Since the focus of this kind of simulations is the magnetic field penetration in the superconductor, all the constituents of the HTS tapes (substrate, buffer, silver cap, stabilizer) that are magnetically neutral can be neglected in the model. The copper stabilizer is sometimes taken into account because it can contribute significantly to dissipation in the form of eddy current loss. The magnetic flux results of Figure G2.2.6 clearly show the beneficial effect of flux diverters to redirect the magnetic field away from the superconductor (Ainslie, Di, et al. 2015). One can also note, however, that the magnetic flux gets concentrated in the diverter, which is made of magnetic material and exhibits its own losses. One therefore needs to evaluate if these extra losses make the use of diverters advantageous with respect to a non-shielded coil. Numerical simulations are the perfect tool for this purpose and one can quickly determine the best diverter configuration

FIGURE G2.2.6 Magnetic flux field lines in and around a pancake coil made of HTS coated conductors, without and with magnetic flux diverters. A 2D axisymmetric model is used, so that only the transversal cross-section of the coil is simulated.

before conducting costly and time-consuming experiments (Song et al. 2016; Liu et al. 2017). Numerical models can be useful to handle complexities originating not only from the materials’ properties but also from the geometry. An example is represented by Roebel cables assembled from HTS coated conductors (Figure G2.2.7) (Goldacker et al. 2014). This kind of cables is particularly interesting because it has a large engineering current density (thanks to the tightly packed strands), while having fully transposed strands (i.e. all strands are electromagnetically equal since they occupy all the possible positions along the cable’s length). The full transposition of the strands is a very important feature to reduce the losses caused by an external varying magnetic field. These cables are often simulated in 2D by considering a transverse cross-section. This approach is justified by the fact that the cross-over region is much shorter than the straight region of one transposition length. Full 3D simulations are rather complex for this type of geometry and an example is shown in Figure G2.2.8 (Zermeno, Grilli, and

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FIGURE G2.2.7 2D numerical model of a Roebel cable (shown on top): (a) The cable is modeled as two stacks of superconducting layers separated by vacuum (drawing not to scale). (b) Self-field magnetic flux density distribution in the cross-section. Image courtesy: Christian Barth and Anna Kario, Karlsruhe Institute of Technology, Germany.

Sirois 2013). The figure does not correspond to the actual geometry of the cable, but to a ‘unit cell’: the conductors in the figure represent the different segments of a strand along a transposition length. In this example, only the superconductor layer is simulated. Despite the use of a relatively coarse mesh, the unit cell has a number of mesh elements in the hundreds of thousands. The simulation of one AC cycle can take up to several days on powerful desktop workstations. In general, 3D full simulations of time-dependent problems with superconductors are very demanding and so they are rarely performed, at least for routine use. If the simulation of 3D objects is necessary, simplifying assumptions

FIGURE G2.2.8 Full 3D simulation of a Roebel cable carrying an AC transport current: simulation of a periodic cell (top) and results of current density distribution in the reconstructed strand (bottom). Reprinted with permission from Zermeno, Grilli, and Sirois 2013.

can be made, e.g. by reducing the dimension of the problem while still considering 3D geometries and 3D effects. For example, in the case of the Roebel cable, one can neglect the variation of electromagnetic quantities across the thickness of the superconductor and make an infinitely thin tape approximation. Since the superconductor is typically 4–12 mm wide and only about 1 µm thick, the assumption is justified. This allows treating the problem in a 2D environment (Nii, Amemiya, and Nakamura 2012; H. Zhang, M. Zhang, and Yuan 2017). This kind of approximation has been also used to study the quench propagation (Chan and Schwartz 2012) or the coupling between filaments (Kasai and Amemiya 2005) in HTS coated conductors. The T – A formulation introduced in H. Zhang, M. Zhang, and Yuan (2017) follows this infinitely thin tape approximation: it is gaining popularity because of its relatively easy implementation in the commercial software package Comsol Multiphysics, and it has been applied to simulate the 3D electromagnetic behavior of Roebel (Yan et al. 2019) and CORC (Wang et al. 2019) cables. Other types of approximations to treat 3D problems in a 2D environment include the change of coordinates (Stenvall, Grilli, and Lyly 2013) and the use of anisotropic conductivity (Honjo et al. 2003). The former uses the idea of changing the coordinate system of helicoidally wound conductors, so that the conductors can be represented as straight objects and therefore analyzed in 2D. The latter uses an anisotropic conductivity to account for the fact that, in HTS power cables, the current does not flow straight along the cable’s axis, but at an angle, because HTS tapes are helicoidally wound on the cable’s former. These approximations can be very effective for reducing the complexity of the calculations, but they are also application-specific. Even if one stays confined to 2D, simulations can quickly become very demanding for certain devices for which a large

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number of superconductors needs to be simulated. Typical examples are large superconducting coils, e.g. for magnet applications, for which one needs to simulate hundreds or thousands of tapes. The availability of relatively cheap computing power has made the task easier. Although timedependent simulations with hundreds or even thousands of individual tapes have been reported [see, for example, references in Grilli (2016) and in particular Pardo (2016)], alternative models that do not require the simultaneous simulations of all tapes have proved to be much faster without loss of accuracy. The so-called homogenization and multi-scale methods have been developed specifically for HTS coated conductors (Quéval, Zermeño, and Grilli 2016). The idea behind the homogenization method is the following: the cross-section of a large stack of tapes can be substituted by a homogeneous bulk conductor, where the superconductor’s critical current density Jc is ‘diluted’ to take into account the distance between the layers of superconducting material in the actual stack. This distance includes the non-superconducting materials (substrate, buffer layers, stabilizer) as well as the separation between the tapes (e.g. electrical insulation). If the original object of simulation is a coil, each tape in the simulated cross-section represents a turn of the coil, and all the simulated tapes must carry the same current. This means that in the homogenized bulk the current cannot be let free to distribute in the whole crosssection, but must be constrained to mirror the real situation. One can, for example, use current constraints to impose that the integral of the current density in the direction of the tape’s width is always the same: this corresponds to the fact that each tape (turn) carries the same current. Transport current simulation results for ten 200-tape stacks are shown in Figure G2.2.9, both for the original (reference) and homogenized geometry. Due to the symmetry of the problem, only five 100-tape stacks are simulated. In the homogenized geometry the details of the tapes are ‘washed out’, but the overall current and magnetic flux density distributions are very similar to the original ones. The AC losses calculated with the homogenized model are within 1% of those calculated with the reference model. The calculation times are 30 minutes and 50 hours, respectively, which demonstrate the great advantage of using the homogenization method.5 The homogenized problem is much simpler to solve because one can use a relatively coarse mesh (resulting in a much lower number of degrees of freedom) and less current constraints (whose number influences the speed of the solving process). The main drawback is that this method cannot take into account magnetization currents caused by a magnetic field parallel to the wide face of the tapes. 5

The advantage of the homogenization becomes evident for the simulation of large systems, For example, in Pardo (2016) the homogenization technique is used to calculate the screening currents in a coated conductor magnet composed of 40,000 turns.

Another approach for modelling large numbers of interacting tapes is the multi-scale method (Queval and Ohsaki 2013): the idea is to simulate one conductor at a time using a boundary condition for the field it is subjected to as the result of the environment (for example, the field created by the other conductors). This magnetic field can be calculated in a relatively simple way, with magnetostatic models. If one is, for example, interested in calculating the AC losses of the whole device, the position of the tape of interest can then be moved and a ‘map’ of the AC loss distribution in the device created. The advantage is that the simulations involving one tape are very fast and they can be truly parallelized. In addition, it is not necessary to consider all the positions of the tape in the device, but only some key ones. The method applied to the same geometry of Figure G2.2.9 is discussed in Quéval, Zermeño, and Grilli (2016), where 25 equally distributed positions are used to produce a sufficiently accurate AC loss map. In the case considered in (ibid.), it is found that the current distribution used for the magnetostatic calculation has an important influence on the final AC loss value and hence on the accuracy of the results: for example, starting with a uniform current distribution in the superconductors is too rough an assumption and produces errors in the losses as large as 20%. The homogenization and multi-scale methods have been recently applied to the T – A formulation (Juarez et al. 2019): the improved speed allows for real-time simulation of superconducting magnets made of HTS coated conductors.

G2.2.4 Coupling of Electromagnetic and Thermal Effects When the critical current density is reached and exceeded, thermal effects play an important role in the behavior of the tape. Firstly one needs to take into account that the critical current density depends on the temperature as well and that the E – J relation is generally more complex due to the different physical states characterizing the superconductor (e.g. flux creep, flux flow, normal state). For example, a simplified description of the superconductor’s resistivity can take the following form (Roy, Dutoit, et al. 2008)6 ρpl ( J ,T ) ρ( J ,T ) = ρn 6

,T < Tc ,T ≥ Tc ρpl ( J ,T ) ≥ ρn

(G2.2.4)

Formulas (G2.2.4)–(G2.2.6) do not fit any particular real sample, but they are inserted here to show that the resistivity can assume a fairly complex form, including the combination of power-law dependencies and piece-wise function definitions. The precise characterization of the superconductor’s resistivity as a function of the current density, magnetic field and temperature is a complicated task, especially because the superconductor material is often embedded in a composite structure, so that it is difficult to extract its intrinsic properties. Interested readers can, for example, find a more precise characterization and modelling of coated conductors in Therasse et al. 2008; Roy, Therasse, et al. 2009; Riva et al. 2019.

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FIGURE G2.2.9 Simulation of a matrix of 10 × 200 tapes, representative e.g. of the turns in the cross-section of a high-field magnet. Due to the symmetry of the problem, only 5 × 100 tapes are simulated. The top figures represent the current density and magnetic flux density distributions obtained by simulating all tapes with the time-dependent H-formulation of Maxwell’s equations (Brambilla, Grilli, and Martini 2007). The bottom figures represent the same quantities obtained with an homogeneous bulk approximation: the results are very similar, but the computation is about 100 times faster. Reprinted with permission from Quéval, Zermeño, and Grilli 2016.

ρpl ( J ,T ) =

E0 J J c (T ) J c (T )

20 , J < J c n= . 10 , J ≥ J c

n −1

(G2.2.5)

the temperature variation in a superconductor, one has to solve the heat equation ρm cp

(G2.2.6)

Secondly, one has to remember that electromagnetic and thermal phenomena are coupled. For example, in order to study

∂T − ∇ ⋅ (− k∇T ) = Q , ∂t

(G2.2.7)

where ρm is the mass density, cp the specific heat capacity, k the thermal conductivity, and Q is the power density dissipated in the superconductor, which is given by the joule dissipation J ⋅ E that one can compute by solving the electromagnetic problem.

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FIGURE G2.2.10 Schematic illustration of the coupling between the electromagnetic and thermal models and of their exchange of variables.

The coupling between the electromagnetic and thermal models and the exchange of calculated quantities can be schematically illustrated as in the diagram of Figure G2.2.10: the temperature is calculated by the thermal model and passed to the electromagnetic model, which solves Maxwell’s equations with temperature-dependent parameters. In turn, the joule dissipation calculated by the electromagnetic model is passed to the thermal model, where it plays the role of source term in the heat equation. The coupling of time-dependent electromagnetic and thermal models is computationally challenging and usually time-consuming. The investigation of magnetization in stacks of HTS coated conductors to be used as permanent magnets is a significant example, which is further complicated by the presence of numerous layers of different materials with a very large aspect ratio. In this case, the use of a structured mesh is of the utmost importance (Page et al. 2015). Superconducting bulks have a simpler geometry to simulate, and 3D electromagnetic-thermal calculations have already

FIGURE G2.2.11 Comparison of the temperature distributions generated in a REBCO bulk at different instants of a magnetizing pulse. The results are obtained with inhomogeneous (left) and homogeneous (right) models for the critical current density Jc. Image courtesy: Mark Ainslie, University of Cambridge, UK.) (See Ainslie, Fujishiro, et al. 2014; M. Ainslie and Fujishiro 2019 for details.

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been performed, also taking into account non-uniformities of Jc (Ainslie and Fujishiro, et al. 2014; Ainslie and Fujishiro 2015); Ainslie and Fujishiro 2019). An example of these 3D calculations is shown in Figure G2.2.11, which compares the temperature distributions generated in a REBCO bulk at different instants of a magnetizing pulse. The results are obtained with inhomogeneous and homogeneous models for the critical current density Jc. If one is interested in modelling quench and normal zone propagation effects, for example, in tapes for fault current limiter applications, simplified models assuming uniform current density distribution in the superconducting layers are usually sufficient. This assumption is justified by the very different time constants of the electromagnetic and thermal phenomena (Roy, Therasse, et al. 2009). These electrothermal models neglect the magnetic flux dynamics inside the superconductor and are capable of handling 3D geometries (Chan and Schwartz 2012; Colangelo and Dutoit 2014). The quench propagation in coated conductors can also be studied from a macroscopic point of view with network model approaches (Colangelo and Dutoit 2014; Ruiz, Zhong, and Coombs 2015).

References Ainslie, M. and Fujishiro, H. (2019). Numerical Modelling of Bulk Superconductor Magnetisation. 2053–2563. IOP Publishing. ISBN: 978-0-7503-1332-2. Ainslie, M. D., Di, H., et al. (2015). “Simulating the in-field AC and DC performance of high-temperature superconducting coils”. In: IEEE Transactions on Applied Superconductivity 25.3, pp. 1–5. ISSN: 1051–8223. Ainslie, M. D. and Fujishiro, H. (2015). “Modelling of bulk superconductor magnetization”. In: Superconductor Science and Technology 28.5, p. 053002. Ainslie, M. D., Fujishiro, H., et al. (2014). “Modelling and comparison of trapped fields in (RE)BCO bulk superconductors for activation using pulsed field magnetization”. In: Superconductor Science and Technology 27.6, p. 065008. Bean, C. P. (1962). “Magnetization of hard superconductors”. In: Physical Review Letters 8.6, pp. 250–252. Bean, C. P. (1964). “Magnetization of high-field superconductors”. In: Reviews of Modern Physics 36, pp. 31–39. Brambilla, R., Grilli, F., and Martini, L. (2007). “Development of an edge-element model for AC loss computation of high-temperature superconductors”. In: Superconductor Science and Technology 20.1, pp. 16–24. Brandt, E. H. and Indenbom, M. (1993). “Type-IIsuperconductor strip with current in a perpendicular magnetic field”. In: Physical Review B 48.17, pp. 12893–12906. Campbell, A. M. (1982). “A general treatment of losses in multifilamentary superconductors”. In: Cryogenics 22.1, pp. 3–16.

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Chan, W. K. and Schwartz, J. (2012). “A hierarchical threedimensional multiscale electro-magneto-thermal model of quenching in coated-conductor-based coils”. In: IEEE Transactions on Applied Superconductivity 22.5, p. 4706010. Colangelo, D. and Dutoit, B. (2014). “Analysis of the influence of the normal zone propagation velocity on the design of resistive fault current limiters”. In: Superconductor Science and Technology 27.12, p. 124005. Eisterer, M, Moon, S H, and Freyhardt, H C (2016). “Current developments in HTSC coated conductors for applications”. In: Superconductor Science and Technology 29.6, p. 060301. Goldacker, W. et al. (2014). “Roebel cables from REBCO coated conductors: a one-century-old concept for the superconductivity of the future”. In: Superconductor Science and Technology 27.9, p. 093001. Grilli, F. (2016). “Numerical modeling of HTS applications”. In: IEEE Transactions on Applied Superconductivity 26.3, p. 0500408. Grilli, F. et al. (2014). “Computation of losses in HTS under the action of varying magnetic fields and currents”. In: IEEE Transactions on Applied Superconductivity 24.1, p. 8200433. Halse, M. R. (1970). “AC face field losses in a type II superconductor”. In: Journal of Physics D: Applied Physics 3, pp. 717–720. Honjo, S. et al. (2003). “Efficient finite element analysis of electromagnetic properties in multi-layer superconducting power cables”. In: IEEE Transactions on Applied Superconductivity 13.2, pp. 1894–1897. Iwasa, Y. (2009). Case Studies in Superconducting Magnets: Design and Operational Issues. 2. ed. New York: Springer. ISBN: 978-0-387-09799-2. Juarez, E. Berrospe et al. (2019). “Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using T – A formulation”. In: Superconductor Science and Technology (in press). Kasai, S. and Amemiya, N. (2005). “Numerical analysis of magnetization loss in finite-length multifilamentary YBCO coated conductors”. In: IEEE Transactions on Applied Superconductivity 15.2, pp. 2855–2858. Li, Y. et al. (2018). “Influence of E–J characteristics of coated conductors and field ramp-up rates on shieldingcurrent-induced fields of magnet”. In: IEEE Transactions on Applied Superconductivity 28.3. Liu, G. et al. (2017). “Numerical study on AC loss reduction of stacked HTS tapes by optimal design of flux diverter”. In: Superconductor Science and Technology 30.12, p. 125014. Mawatari, Y. (1996). “Critical state of periodically arranged superconducting-strip lines in perpendicular fields”. In: Physical Review B 54.18, pp. 13215–13221. Mayergoyz, I. D. (2003). Mathematical Models of Hysteresis and Their Applications. 1. ed. Elsevier series in electromagnetism. Amsterdam: Elsevier Academic Press.

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Mikitik, G. P. et al. (2013). “Analytical methods and formulas for modeling high temperature superconductors”. In: IEEE Transactions on Applied Superconductivity 23.2, p. 8001920. Nguyen, D. N. et al. (2010). “A new finite-element method simulation model for computing AC loss in roll assisted biaxially textured substrate YBCO tapes”. In: Superconductor Science and Technology 23, p. 025001. Nii, M., Amemiya, N., and Nakamura, T. (2012). “Threedimensional model for numerical electromagnetic field analyses of coated superconductors and its application to Roebel cables”. In: Superconductor Science and Technology 25.9, p. 095011. Norris, W. T. (1970). “Calculation of hysteresis losses in hard superconductors carrying ac: isolated conductors and edges of thin sheets”. In: Journal of Physics D: Applied Physics 3, pp. 489–507. Page, A. G. et al. (2015). “The effect of stabilizer on the trapped field of stacks of superconducting tape magnetized by a pulsed field”. In: Superconductor Science and Technology 28.8, p. 085009. Pardo, E (2016). “Modeling of screening currents in coated conductor magnets containing up to 40000 turns”. In: Superconductor Science and Technology 29.8, p. 085004. Queval, L. and Ohsaki, H. (2013). “AC losses of a grid-connected superconducting wind turbine generator”. In: IEEE Transactions on Applied Superconductivity 23.3, p. 5201905. ISSN: 1051–8223. Quéval, L., Zermeño, V. M. R., and Grilli, F. (2016). “Numerical models for ac loss calculation in large-scale applications of HTS coated conductors”. In: Superconductor Science and Technology 29.2, p. 024007. Riva, N. et al. (2019). “Over-critical current resistivity of YBCO coated conductors through combination of pulsed current measurements and finite element analysis”. In: IEEE Transactions on Applied Superconductivity 29.5, p. 6601705. Roy, F., Dutoit, B., et al. (2008). “Magneto-thermal modeling of second-generation HTS for resistive fault current limiter design purposes”. In: IEEE Transactions on Applied Superconductivity 18.1, pp. 29–35. Roy, F., Therasse, M., et al. (2009). “Numerical studies of the quench propagation in coated conductors for fault current limiters”. In: IEEE Transactions on Applied Superconductivity 19.3, pp. 2496–2499. Ruiz, H., Zhong, Z., and Coombs, T. (2015). “Resistive type superconducting fault current limiters: concepts, materials, and numerical modelling”. In: IEEE Transactions on Applied Superconductivity 25.3, p. 5601405. Sirois, F., Grilli, F., and Morandi, A. (2019). “Comparison of constitutive laws for modeling high-temperature superconductors”. In: IEEE Transactions on Applied Superconductivity 29.1, p. 8000110. Song, S. et al. (2016). “Analysis of the notch effect on flux diverters for high-temperature superconducting magnets”. In:

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IEEE Transactions on Applied Superconductivity 26.4, p. 4601604. Stenvall, A., Grilli, F., and Lyly, M. (2013). “Current-penetration patterns in twisted superconductors in self-field”. In: IEEE Transactions on Applied Superconductivity 23.3, p. 8200105. Therasse, M. et al. (2008). “Electrical characteristics of DyBCO coated conductors at high current densities for fault current limiter application”. In: Physica C 468.21, pp. 2191–2196. Wang, Y. et al. (2019). “Study of the magnetization loss of CORC cables using 3D T-A formulation”. In: Superconductor Science and Technology 32.2, p. 025003. Wilson, M. N. (1983). Superconducting Magnets. Clarendon Press Oxford.

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Yan, Y. et al. (2019). “Experimental and numerical study on the magnetization process of Roebel cable segments”. In: IEEE Transactions on Applied Superconductivity 29.8201005. Zeldov, E. et al. (1994). “Magnetization and transport currents in thin superconducting films”. In: Physical Review B 49.14, pp. 9802–9822. Zermeno, V., Grilli, F., and Sirois, F. (2013). “A full 3-D timedependent electromagnetic model for Roebel cables”. In: Superconductor Science and Technology 26.5, p. 052001. Zhang, H., Zhang, M., and Yuan, W. (2017). “An efficient 3D finite element method model based on the T–A formulation for superconducting coated conductors”. In: Superconductor Science and Technology 30.2, p. 024005.

G2.3 DC Transport Critical Currents Marc Dhallé Superconductors have many exceptional properties, but arguably the most eye-catching of these is their ability to carry an electrical current without ohmic loss. It is this property which enables their use in devices such as compact high quality filters, high power transmission lines, high field magnets, to name just a few of the many established or emerging applications of these materials. The term ‘high’ in the previous sentence is used loosely, comparing superconducting devices to their normal-state counterparts. If we want to quantify just how much higher these quality factors, amounts of transmitted power or magnetic fields might be, we need to explore the limits of this loss-less charge transport, i.e. we need to consider the ‘critical current’ of a superconductor. When doing so, one finds that the experiments used to explore these limits not only give the answer needed for practical applications, but often also shed light on the fundamental properties of the superconducting state itself. This Section G2.3 introduces the issue of DC transport critical currents. It is laid out as follows. Subsection G2.3.1 discusses critical currents in general terms. It gives a working definition of the concept ‘transport critical current’, introduces the underlying current–voltage behaviour of superconducting materials and makes a classification of the different dissipation mechanisms which lie at the basis of this behaviour. Subsection G2.3.2 puts the emphasis on the experimental methods used to determine the transport critical current. It discusses the strategy underlying such measurements and shows how they can be compared to the many other experiments used to investigate superconducting materials. Then it introduces the nuts and bolts of critical current measurements, discussing the necessary instruments and the experimental details. It finishes with some points of caution about things that can go wrong. Subsections G2.3.3 to G2.3.6 make up for the bulk of this section. They consist of a selection of some of the critical current experiments discussed in literature. The range of phenomena related to the topic and the extent of the literature treating transport experiments in superconductors is too large for this to be more than an introduction to the more common concepts. Some of the examples were chosen on grounds

of their historical importance, others for their experimental elegance or for the way they illustrate general measuring techniques. They are roughly classified as to the underlying dissipation mechanism which is being explored. Because of experimental similarities with critical current measurements, also resistive measurements of the superconducting transition are briefly discussed.

G2.3.1 Introduction G2.3.1.1 Definitions and E(J) Curves What is the ‘transport critical current’ of a superconductor? The term ‘transport’ refers to a current that is externally injected at some point and extracted at another. It is included to make the distinction with magnetically induced currents. A ‘critical current’ can be defined as the maximum electrical current that a superconductor can carry without dissipation. It is usually given the symbol Ic. When the current is lower than Ic, charge is transported without interaction with the ionic lattice and the resistance of the material is zero. If one imposes a current higher than Ic, superconductivity ‘disappears’ in the sense that the charge flow is no longer loss-less. Energy is dissipated, a voltage develops across the material and the resistance is no longer zero. Note that this does not necessarily mean that the paired charge carriers break up and that the superconducting state is lost, but rather that the motion of charge is no longer possible without some kind of dissipation process taking place. This definition provides a clear picture, but should be used with caution. It is based on an idealisation which in many real systems is only true as a first approximation. In order to describe an actual superconductor more accurately, it is necessary to consider the current–voltage curve V(I) or the current density–electric field relation E(J) (the latter is more general because it is independent of sample size and geometry). The approximate definition of the critical current Ic or the critical current density Jc given above idealises the E(J) relationship for superconducting materials: superconductors are often modeled in such a way that for current densities 141

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FIGURE G2.3.1 Schematic electric field–current density curves for an ohmic and a superconducting material plotted on (a) a linear or (b) a double logarithmic scale.

below Jc the electric field E is strictly zero, as for the dashed curve in Figure G2.3.1(a). Once the current density J exceeds Jc, E abruptly and discontinuously rises to its normal-state equivalent. In reality, this is often not true. Depending on the material and the external conditions (such as temperature and magnetic field), several dissipation mechanisms can occur in superconductors and all give rise to different types of E(J) behaviour. Nevertheless, a common feature of all superconducting materials is the occurrence of a J-range where the E(J) relation is strongly non-linear. The solid curve in Figure G2.3.1(a) is a schematic example of such non-linear but not infinitely steep behaviour. For sufficiently low current densities, the electric field is so small that in an experiment it will be non-detectable. As the current increases, the electric field rises faster than linear, a rise which eventually slows down again as E approaches its value corresponding to the normal-state resistivity. In order to bring out this nonlinear behaviour more clearly, E(J) curves of superconductors are often plotted on a double-logarithmic scale such as in the Figure G2.3.1(b). We will treat different types of E(J) behaviour in more detail later. If the E(J) curve is not extremely steep, some criteria are needed in order to have an unambiguous definition of Jc or Ic. Usually, one takes an electric field criterion, for instance, by defining ‘the’ critical current density Jc as the current density which gives rise to an electric field E of 10 −4 V/m (this popular criterion coincides with the typical voltage detection limit in laboratory scale experiments). Also, a given resistivity level ρc or a specified level of dissipated power per unit volume pc are sometimes used as criteria, e.g. ρc = 10 −12 Ωm or pc = 104 W/m3. Apart from Jc, there is another parameter that is used to describe such non-linear E(J) curves: the n-factor. As indicated in Figure G2.3.1(b), it is defined as the logarithmic derivative of the E(J) curve at Jc, n ≡ dln(E)/dln(J)|Jc, and has both

fundamental and practical importance. From a fundamental point of view, the temperature- and field dependence of n can provide insight into the nature of the dissipation process that determines the shape of the E(J) curve. From a practical viewpoint, the n-factor describes how fast induced or injected supercurrents decay. An ohmic conductor has an n value of 1, whereas a superconductor whose E(J) curve approaches the idealised behaviour in Figure G2.3.1 will be characterised by an n-factor approaching infinity. A word of caution regarding the critical current density Jc. In transport experiments, this quantity is calculated dividing the critical current Ic of a given sample by its cross-sectional area A perpendicular to the current direction. Inherent to this calculation is the assumption that the current density is uniform throughout the sample. It should be born in mind, however, that this assumption can break down. Two general causes are, on the one hand, ‘current crowding’ at the current terminals or at geometrical sample irregularities (Gurevich and McDonald, 1998) and, on the other hand, the possible occurrence of different intrinsic properties in different parts of non-homogeneous samples. Furthermore, due to the intrinsic electromagnetic properties of the superconducting state (see Section C2.1), currents well below Ic tend to flow near the sample surface (Brandt and Indenbom, 1993; Zeldov et al., 1994). A case apart from non-homogeneous materials is composite superconductors, such as multifilamentary wires or multistrand cables. Authors interested in the electrotechnical behaviour of such conductors often refer to the overall- or engineering critical current density JE. It is the critical current Ic divided by the total conductor cross-section, including nonsuperconducting parts such as metallic matrix, reinforcement, isolation, etc. The symbol JE is used to make a clear distinction with Jc, which refers to the cross-section of the superconducting material only.

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DC Transport Critical Currents TABLE G2.3.1 Possible Current-Limiting Mechanisms in Superconductors, with an Indicative Range of the Resulting Critical Current Density; the Responsible Dissipation Mechanism; the Underlying Structure that Controls the Dissipation and the Section in Which They Are Discussed Mechanism

Current Density Jc

Limited By

Controlled By

Section

Depairing Depinning Decoupling

10 – 10 A/m 108 – 1010 A/m2 106 – 108 A/m2

Kinetic energy Magnetic flux motion Josephson junctions

Pairing mechanism Nano-structure Microstructure

C2.2.4 C2.2.5 C2.2.6

10

12

2

G2.3.1.2 Dissipation in Superconductors A dissipating superconductor doesn’t seem to be true to its name. But as we stated above, the presence of dissipation doesn’t necessarily mean that the superconducting state is lost. If the superconducting state is lost, we will term the currents that induce such transition to the normal-state thermodynamic critical currents (discussed in Section G2.3.4). They can be treated in terms of the energy difference between the current-carrying superconducting- and normal state, which is described in part A2 of this handbook. Often these currents are sharply defined, and the corresponding E(J) curve resembles the idealised curve in Figure G2.3.1. Examples of such thermodynamic critical currents are the depairing current; the Silsbee Ic of a wire in which the transport current induces too large a field at the surface; and the surface currents that can flow in a thin outer sheet between the critical fields Hc2 and Hc3 of a Type II superconductor. A second and large class of dissipation processes in superconducting materials involve the motion of magnetic flux (treated here in subsection G2.3.5, but also in parts A4, G2.1 and G2.6 of this volume). In Type II superconductors subjected to a field larger than Hc1, magnetic flux enters in the form of discreet flux lines or vortices. These flux lines interact with a transport current and can be driven to move through the material. Whether or not this occurs depends on the socalled pinning properties of the material. If they do move, they will dissipate energy. The resulting E(J) curves will depend strongly on the amplitude of the externally applied magnetic field and are often not extremely steep, but rather like the schematic solid curve in Figure G2.3.1. The third form of dissipation processes of which we will treat examples in G2.3.6 occurs at superconducting junctions. Here the superconducting order parameter is depressed in the region between two superconducting electrodes, giving rise to locally modified properties and a reduced Jc. The characteristics of single junctions are treated in parts A2.7 and E4.1 of the book. In this part, we will discuss how such junctions may determine the overall critical current of bulk superconducting materials containing extended inhomogeneities. If a transport current is forced to pass such inhomogeneous regions, the macroscopic Jc may reflect their ‘averaged’ junction behaviour. Due to their relatively low coherence length, this can easily happen in high-Tc materials. Mathiessen’s rule in principle applies also to superconductors: all dissipation processes act in an cumulative fashion.

Which of the three dissipation mechanisms discussed above dominates and therefore determines the overall E(J) behaviour in general depends on the nature of the superconducting material, and also on externally imposed conditions such as temperature, applied magnetic field and injected current density. In practice, however, typically Jc,decoupling ≪ Jc,depinning ≪ Jc,depairing so that – unless they are ‘disabled’ by proper materials engineering – decoupling or depinning dominate well before depairing kicks in. Table G2.3.1 gives an overview of the typical critical current density ranges and the strategy that needs to be followed to optimise them.

G2.3.2 Experimental Techniques G2.3.2.1 Measurement Strategies and Experimental Windows Generally speaking, two different experimental strategies can be used to carry out transport experiments on superconducting samples. In the first, the temperature T and the external magnetic field H are kept fixed and one measures the voltage drop across the sample whilst ramping up the current (either continuously or in steps). In other words, the E(J) curve is measured directly. Repeating the measurement at various temperatures and/or magnetic fields and applying some Jc criterion then allows to determine the Jc(H,T) behaviour. The second strategy is to fix the measuring current and to vary either temperature or magnetic field, again whilst recording the voltage drop. These types of experiments straightforwardly yield the temperature and/or field dependence of the resistivity ρ(H,T) and are mostly used close to the superconducting transition temperature. Sometimes they are repeated at different current levels. All transport measurements have in common that the current density is imposed and the electric field is measured. This is in contrast with inductive characterisation techniques, which are discussed in parts G2.5, G2.6 and G2.9 of this book. There it is the rate of change of the external magnetic field that is controlled, and the sample’s magnetisation that is measured. It’s important to realise that these two on first sight very different characterisation techniques, i.e. transport and inductive measurements, essentially probe the same thing: the relation between electric field E and current density J. (In inductive measurements, E can typically be deduced from the rate of change of the induction B through Faraday’s law, whilst J can be related to the magnetisation M.) The main differences

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between transport and inductive experiments concern the experimentally accessible window and the geometry of the current path. As discussed below, there are practical limits to the transport current amplitude which can be injected into a sample without too many experimental difficulties. Generally speaking, transport experiments are more suited to probe lower current densities and higher electric fields, whereas inductive techniques usually involve higher current densities and lower electric fields. Details depend of course strongly on the measuring equipment and sample size, but the typical windows which are accessible to respectively transport and inductive experiments are roughly as indicated by the two hatched areas in Figure G2.3.2(a). As an illustration, Figure G2.3.2(b) shows an example of E(J) data obtained with both types of measurements on the same sample (Dhallé et al., 1997). The second important difference between transport and magnetisation experiments concerns the current geometry. In transport measurements, the current has to flow from one current contact to the other, whereas magnetically induced currents form closed loops which are confined to the sample. In a homogeneous sample, both currents will be governed by the same E(J) curve and therefore, all other conditions being equal (H, T and E), both experiments will yield the same critical current density (Caplin et al., 1994; Zhukov, 1992). If, however, the sample is inhomogeneous, it is possible that induced current loops that are confined to localised high Jc parts of the sample dominate the overall magnetisation signal, whilst the transport current is forced to pass also through lower Jc regions. In this case, the measured transport Jc will seem to be lower than the one derived from an inductive experiment. Because of these two differences, transport and inductive experiments should be seen as complementary to one another. Whereas magnetisation experiments allow access to a wider experimental window and enable one to measure without too

Handbook of Superconductivity

many problems high J values or low E fields, transport experiments necessarily involve current flow on a macroscopic scale and therefore may give a better view of the overall behaviour of the material under investigation. Which of both is preferred depends on the precise goal of the experiment and on the availability of the experimental instrumentation.

G2.3.2.2 Instrumentation A typical set-up for low-temperature transport measurements consists of several parts, as schematically shown in Figure G2.3.3. The sample is mounted in a cryostat which may be equipped with a magnet. Temperature T and magnetic field H are controlled and monitored by some kind of device, usually a personal computer. There exists a large range of cryostats (bath cryostats, gas flow cryostats, closed cycle cryocoolers,…) and magnets (Cu-based electromagnets, superconducting solenoids, permanent magnets,…). Which one is best suited for a given experiment depends on a number of factors such as desired versatility, precision, stability, homogeneity, degree of automation, etc. A discussion concerning this part of the experimental design falls outside the scope of this text and readers requiring more information are referred to the specialised literature (White, 1979; Ekin, 2006). Once the desired temperature and magnetic field are reached, a current is fed into the sample and the corresponding voltage drop is measured. This requires essentially two pieces of equipment: a suitable current source and a voltmeter. The most commonly used sources and detectors are discussed below. G2.3.2.2.1 Current Sources The choice of current source depends on the required range and precision. As an illustration, let’s consider the case of a typical YBa 2Cu3O7 thick film and a typical (Bi,Pb)2Sr2Ca 2Cu3O10/Ag

FIGURE G2.3.2 (a) The experimental electric field and current density windows which typically can be investigated with transport and inductive experiments. As an illustration, (b) shows E(J) data measured both with a transport – (open symbols) and magnetisation experiment (closed symbols) on the same (Bi,Pb)2Sr2Ca 2Cu 3O10/Ag tape in various magnetic fields. (After Dhallé et al., 1997.)

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DC Transport Critical Currents

FIGURE G2.3.3 Schematic representation of a typical transport measurement set-up.

tape. Their Jc in low magnetic field might be expected to lie in the range 109 A/m2 < Jc < 1010 A/m2 at T = 77 K for the film and T = 4.2 K for the tape. Typical cross-sectional areas of the film and the tape might be S = 10 −10 m2 and S = 10 −7 m2, respectively. This straightforwardly gives us 0.1 A < Ic < 1 A for the film and 100 A < Ic < 1000 A for the tape. Clearly, a different instrument is needed for the measurement of both. If one is interested in the 1 mA to 10 A current range, there is a large number of manufacturers offering low-cost, highprecision current sources which are externally controllable. Any electronics or lab equipment catalogue will have several of them. In the 0.1 A to 1 kA range, things get more expensive (roughly proportional to the maximum current required) and the number of manufacturers is lower. However, there is still ample choice from a range of quality instruments. Magnet manufacturers or companies specialised in low-temperature experimental equipment are a good place to look. Apart from the desired current range, the compliance voltage of a current source must be taking into consideration. This is the maximum voltage that the apparatus can source. Although the sample under investigation will usually have quite a low resistance, the current leads leading from the source to the sample and the contacts between these leads and the sample also have to be taken into account. Because of considerations of thermal stability and available cooling power, the current leads going into the cryostat cannot be made to have an arbitrarily large cross-section, but need to be optimised balancing thermal conductance against electrical resistance. The typical residual resistance in the current circuit of a transport experiment designed to work in the I = 1 mA to 10 A range will be around R residual = 0.1 – 1 Ω, whilst an experiment working in the I = 0.1 to 1 kA window will typically have R residual = 0.01 – 0.1 Ω. This means that the voltage drop along the current circuit can be as high as several tens of volts. If one wants to use the full current range of a source, the apparatus must be able to provide this voltage.

Although specifications can vary a lot, most commercial current sources have a precision of around 1 part in 104 to 105, i.e. it will be possible to set and read-out, for instance, a current source with maximal rating of 10 A with a precision of 0.1 to 1 mA. For many transport experiments such as straightforward E(J) measurements, this is largely sufficient and the current can usually be read back directly from the source. If a more precise read-out is required, as may be the case in detailed ρ(H,T) measurements, a reference resistor can be added in series with the sample to read out the current independently of the source (as in Figure G2.3.3). When opting for this solution, it should be carefully verified that ohmic heating does not compromise the linearity of this resistor over the desired current range. In high-current measurements, a shunted reference or contactless current probe can be used. For the vast majority of experimental requirements, it’s these days hard to beat the cost-effectiveness of commercial current sources. Nevertheless, sometimes it might be necessary to resort to home-made solutions. A typical example might be the need to find a floating source (i.e. with both terminals at an arbitrary voltage with respect to ground) in a specified current range. Although high-precision-, high-stability sources need expert knowledge to design, one can get quite a far way with rechargeable accumulators, some operational amplifiers and a good transistor. An excellent introductory text treating such DIY solutions can be found in Horowitz and Hill (1989). Finally, some words on pulsed current sources. Some cryostat systems might not have enough cooling power to cope with the heat generated in the current leads and sample contacts during a protracted high-amplitude DC current experiment. In that case, a pulsed current source might provide a solution. The idea here is to apply the measuring current only during a limited time (typically a few ms) to record the voltage during this time window and then to let the temperature stabilise during a relatively long period (e.g. for 1 s) before giving the next pulse with a slightly higher amplitude. The pulse shape should be carefully chosen to avoid measuring transient effects, inductive coupling between the current and voltage circuit should be minimised and the design of current source and voltage detection system will require some careful thinking, but with some trial and error good results can be obtained (Meisels et al., 1989). G2.3.2.2.2 Voltmeters The voltage drop over the sample is most commonly measured using a digital voltmeter. A large number of manufacturers offers excellent and reasonably priced instruments. Things to consider when buying one are the required precision (number of digits), accuracy, versatility (number of measuring ranges), interfacing possibility and noise rejection Oliver and Cage (1971). When measuring superconducting samples, an accuracy of 1 μV or better is a must. Most manufacturers will offer this. If one wants to extend the voltage window downwards, the choice gets smaller and prices go up, but commercial instruments providing a 1 nV accuracy can still readily

146

be found. The high-precision preamplifiers which are used in these nanovoltmeters should be treated with some respect: they are relatively vulnerable to overloads (e.g. when a sample ‘burns out’ and causes an open circuit) and repairs can be fairly expensive. When measuring in this nV range, extra care also has to be taken to avoid parasitic voltages which may arise from thermal effects, electromagnetic pick-up or electronic instability. Some possible remedies to these common plagues are discussed below. Another instrument which provides a 1 nV accuracy without too many precautions is a lock-in amplifier or phasesensitive detector. Because it generally uses a low-amplitude AC measuring current, it’s not suited for direct E(J) type measurements, but is commonly used to measure ρ(T,H). Essentially, it consists of a preamplifier combined with a reference signal mixer and a low-pass integrator (Bozic et al., 1975; Diefenderfer, 1979). This combination allows to select a very narrow-frequency (and phase) window: typically, only the voltage with the same frequency (and in phase with) the measuring current is detected whilst parasitic components are efficiently filtered out. Again, there are several manufacturers. It’s worthwhile to shop around a bit, since prices can vary considerably. A special application of lock-in detection are noise measurements. The instrument’s ability to single out a sharply defined frequency window makes it very suitable to measure the noise spectral density which, as discussed in Section C.2.2.5, can sometimes be used to probe the dissipation mechanism in more detail. The most accurate means of voltage detection at present is without doubt the SQUID voltmeter (Gallop, 1991). This instrument essentially feeds the sample voltage into a superconducting coil, the flux change in which is then measured by a SQUID magnetometer (see Section E4.2). If the voltage source has a sufficiently low impedance, as can be made to be the case when measuring superconducting samples, the voltage sensitivity may in principle be as low as 10 −17 V. In practice, parasitic noise will limit the sensitivity of these measurements. Although the large extension of the experimental voltage window gives access to valuable information (Gordeev et al., 1997), SQUID-based voltage measurements remain a specialised domain and – to the author’s knowledge at least – such instruments are not commercially available.

Handbook of Superconductivity

propagate into the sample and may drive it normal prematurely, well below Ic and long before the cryostats thermometers signal a problem. Heating problems in superconducting transport measurements often have a ‘thermal runaway’ character (Wilson, 1997), with the voltage continuing to rise whilst the current is stable. Therefore, a good test to verify that there is no such problem is to check the voltage stability whilst maintaining a relatively high and constant current amplitude. Although less obvious and to some extent less crucial, the low contact resistance principle applies also to the voltage contacts: a low-resistance voltage circuit is much less susceptible to noise, be it thermal or electromagnetic in nature. The second general principle, avoiding sample damage, may seem trivial but in practice cannot be forgotten with impunity. Sample damage or modification can occur in subtle ways, such as chemical contamination by solvents whilst cleaning the surface prior to applying the contacts or microcracking due to local thermal expansion whilst soldering contacts. Such damage is often hard to spot and the only remedies are trial and error on the one hand and a healthy suspicion of totally unexpected experimental results on the other. Apart from the nature of the electrical contacts, also their position should be carefully considered. Electrical measurements on different types of material pose different geometry requirements (Wieder, 1979) and for superconductors it’s essential that measurements be made in a so-called fourterminal layout [Figure G2.3.4(a)]. Since the voltage drop across the contact resistance of the current leads is generally much larger than the voltage across the sample itself, the latter must be measured with a separate pair of terminals. If one wants to check explicitly whether the electric field is uniform throughout the sample, more than two voltage terminals can be used. We will discuss examples of such multiple-terminal measurements in subsections G2.3.5 (flux transformers) and in G2.3.6 (homogeneity tests). As mentioned above, heating of the current contacts may cause the temperature to rise locally, which can drive the sample normal prematurely. If the material under consideration is easily shaped, as may be the case, for instance, with

G2.3.2.3 Current & Voltage Terminals The art of making good electrical contacts to superconducting samples may have its subtleties and depends to a very large extent on the nature of the superconductor in question. Different techniques are discussed in detail in Section B5 of this book. We limit ourselves here to the two crucial general principles: keep contact resistances as low as possible and avoid damaging the sample. The first principle has to do with thermal stability and noise. If the current contacts are too resistive, the ohmic heat generated in them will cause important thermal gradients which

FIGURE G2.3.4 Different possible contact geometries: (a) standard four-terminal, (b) four-terminal with ‘necking down’ of the measured section, (c) Van der Pauw’s method and (d) Corbino’s disk. Thick lines represent the current leads, thinner ones the voltage probes.

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DC Transport Critical Currents

patterned thin films or some bulk superconductors, this might be avoided by necking down the measurement area as in Figure G2.3.4(b). The idea is to keep the current density close to the current terminals lower than in the section where the voltage is measured, so that even when these terminal regions heat up slightly, their critical current density is not reached, they do not turn normal and ‘thermal avalanches’ are avoided. If on the contrary, the material which is being characterised is not readily available in a well-defined geometry, as might be the case for instance with single crystals, one runs into the problem of how to calculate the electric field E, current density J and resistivity ρ from the voltage V, current I and resistance R. This problem might be partially solved using the Van der Pauw configuration (Van der Pauw, 1958) shown in Figure G2.3.4(c). Four taps are placed on the edge of the sample. Two successive resistance measurements are made, one using terminals AB as current taps and CD as voltage probes (resistance RCD) and one using BC as current and DA as voltage leads (resistance R DA). If the sample is homogeneous, has a uniform thickness t and has no holes through it, the resistivity ρ can be calculated from the relation: πt πt exp RCD + exp RDA = 1. ρ ρ

(G2.3.1)

Note that implicit in Van der Pauw’s derivation of this expression the assumption is made that the E(J) curve is linear; the method is useful for determining ρ(H,T) in the ohmic regime, but cannot be used to construct a non-linear E(J) curve. A final contact geometry discussed here is the more exotic Corbino configuration shown in Figure G2.3.4(d). The current is injected in the middle of the sample and extracted with a second ring-shaped contact placed symmetrically around the first one. Voltage taps are aligned radially. This configuration offers two distinguishing features: (1) the current density decreases as one moves away from the centre (J ∝ 1/r, with r the distance to the central tap) and (2) in a magnetic field perpendicular to the plane of the disk, there is no sample boundary parallel to both current and field. These two features have allowed for some elegant experiments which will be discussed in Sections G2.3.4 and G2.3.5.

G2.3.2.4 Some Common Problems and How to Avoid Them Most problems in measurements of the DC transport critical current involve random voltage noise or parasitic voltages proportional to the current. Possible sources of such disturbances can be divided into two categories: the measuring instruments or the experimental environment. G2.3.2.4.1 Instrumental Noise and Parasitic Components A simple first test of a new experimental set-up is to shortcircuit the voltmeter’s inputs and to record the voltage during

some time. Random noise generated inside the voltage detector can usually only be avoided by using a better instrument. Sometimes, however, the problem is sitting in the mains, especially when close by things such as furnaces or electrical motors are using the same grid. Electronic mains filters come in several forms and prices and can offer some relief. Alternatively, some voltmeters have an option to power their preamplifier from a rechargeable accumulator, which improves things drastically. As a last resort, one might want to postpone the most sensitive measurements to the evening or weekend, when the offending machinery is switched off. It’s also a good idea to move the instruments around a bit in order to find an optimal position, since mains interference may have an inductive origin. Transformers in high-power current sources (e.g. magnet power supplies) are notorious in this respect. Generally speaking, keep the voltage detection instruments as close as possible to the cryostat, and move away current sources and power supplies. Apart from the noise floor of a voltage detector, its offset and drift need to be considered. The offset is the error on the null-detection (i.e. the small DC voltage measured when the inputs are short-circuited). This error can cause awkward distortion of the low-voltage part of the E(J) curves and therefore needs to be eliminated. In principle, this can be done after the measurement, by subtracting the zero-current limiting value of the detected voltage. Often, however, the offset has a tendency to drift slowly, which makes this correction procedure more difficult. A simple remedy is to switch the equipment on about half an hour before the experiment, since offset drift typically originates from thermal drifts inside the voltmeter. Once the instrument is ‘warm’, offsets tend to be more stable. More precise measurements usually require longer acquisition times and are therefore more prone to this problem. In this case, it might be best to get rid of the offset altogether by measuring each data point with two opposing current polarities and subtracting both measurements from each other: V + ( I ) = Voffset + I ⋅ R( I ) V − ( I ) = Voffset − I ⋅ R( I ) V + (I ) − V − (I ) V (I ) = = I ⋅ R( I ) 2

(G2.3.2)

When the current source itself does not offer the possibility to reverse polarity, the switching solution shown in Figure G2.3.5 can be used. If the required current is not too high, power relays to switch it on or off can readily be found. Two of these relays (or two groups for higher amplitudes) allow to change the polarity smoothly in a ‘make-beforebreak’ switching sequence, which avoids large inductive voltages when opening a relay. When measuring at high current amplitudes, a commonmode voltage might plague the experiment. This problem manifests itself as a voltage which grows linear with the current, sometimes with a polarity opposite to the ‘true’ voltage

148

FIGURE G2.3.5 sequence.

Handbook of Superconductivity

The use of external power relays to switch the polarity of the measuring current in a three-step make-before-break

drop Vs across the sample. A good diagnostic test is to connect one single sample terminal to both inputs of the voltmeter and to check for the occurrence of a linear I(V) curve. The origin of common-mode voltages is shown in Figure G2.3.6. Nearly all voltmeters use some kind of differential amplifier as a first instrumental stage. If the sample as a whole sits on a voltage VCM (the common-mode voltage) with respect to the supply voltage Vo of the amplifier, the output of the amplifier will not only be the amplified difference between its two input voltages, αVs, but will also have a small component βVCM of the common-mode voltage mixed in. How much off a problem this poses can be worked out from the common-mode rejection ratio (CMRR) of the instrument, which is defined as: α CMRR ≡ 20log β

(G2.3.3)

As an example, assume a voltmeter has a CMRR = 140 dB. If the parasitic resistance in the current circuit (due to leads and contacts) is Rp = 1 Ω, a 10 A measuring current will generate a common-mode voltage of VCM = 10 V. This will show up as a 1 μV component to the measured signal, which might well be unacceptable. Three possible solutions to common-mode problems are reducing the parasitic resistance Rp, grounding the sample or

‘booth-strapping’ the amplifier. The first one is trivial, a lower Rp means a lower VCM. In the second solution, an extra terminal is attached to the sample and connected to earth potential [Figure G2.3.6(b)]. This indeed very straightforwardly gets rid of the common-mode voltage, but should be done only with the utmost care! If there is a second earth in the measuring circuit (typically an internal ground in the current source), this solution will lead to ‘earth loops’ which, as discussed below, cause even greater problems. The third solution also consists of adding a terminal to the sample, this time connecting it to the common of the preamplifiers power supply, so that the amplifier ‘floats’ on the same voltage as the sample. This works quite well, but is, in practice, hard to achieve with commercial instruments. The last but all too common instrumental problem discussed here is the occurrence of so-called earth loops. If the measuring circuit is grounded in several places, part of the current will not flow in the desired path, but will instead take a circuitous route bypassing the parasitic resistance (Figure G2.3.7). Not only will this ‘lost’ current component falsify the read-out of the sample current, but it can also give rise to awkwardly high voltages appearing in unexpected places and possibly even damage the measuring equipment. Tell-tale signs are large ohmic components to the measured signal, unexpected overloads of the voltmeter and excessive

FIGURE G2.3.6 (a) Schematic representation of the common-mode problem and two possible solutions; (b) grounding the sample (careful!, see text) and (c) ‘booth-strapping’ the preamplifier.

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FIGURE G2.3.7

Schematic representation of a possible earth loop.

noise (in the case of ‘unstable’ earth loops). Earth loops are often quite subtle (for instance, poor electrical insulation between the voltage circuit and the metal body of a grounded cryostat) and therefore hard to track down. Measuring the voltage with respect to ground at different points of the circuit and checking for unexpectedly large voltages might do the trick. If this fails, the experiment should be taken apart and redesigned. G2.3.2.4.2 Sources of Error in the Experimental Environment Problems not involving the measuring instruments but rather due to the experimental environment (i.e. inside the cryostat) fall into two categories: electromagnetic pick-up and thermal instabilities. Electromagnetic pick-up manifests itself as a noise level in excess of the instrumental noise floor measured with the voltage inputs short-circuited. It’s inductive in nature, stemming from changes in the magnetic flux treading the loop made by the voltage leads between the sample and the voltmeter. The obvious solution is therefore to minimise the area of this loop and to stabilise the magnetic flux through using some simple precautions. Twisting the voltage leads together minimises the loops area. Keeping them as much as possible separated from the current leads and other current-carrying wiring (resistive thermometer leads, heaters, Hall probes, etc.) minimises the flux through them. Fixing them to the sample holder reduces their motion and thus flux changes due to vibration (which can be a surprisingly important noise source when measuring in an TABLE G2.3.2

applied magnetic field). In extreme cases, twisted wires within a grounded shield might reduce high-frequency pick-up noise. Nevertheless, sometimes all these measures aren’t enough, for instance, when the applied magnetic field itself is noisy, and this noise is picked up by the unavoidable opening in the voltage loop at the sample. In that case, the integration time of the voltmeter will have to be increased. Random noise diminishes as the inverse of the square root of this integration time, e.g. if with a 100 ms integration the noise level is 100 nV, it will go down to 10 nV by increasing the integration time to 10 s. This noise reduction will have to be balanced against the disadvantages of slower experiments and a higher sensitivity to instrumental drifts. Thermal gradients and instability can also lead to parasitic voltages. Thermal runaways due to heating of the current contacts has been discussed above. It can be prevented by careful thermal anchoring of the sample and by necking down the measuring section. Another source of thermal voltage is the thermo-power developed at the voltage contacts on the sample, at solder joints on the sample holder or feed-troughs into the cryostat. This type of problem usually manifests itself as a slowly drifting offset of a few μV. Although it can be remedied by measuring data points bipolar, as discussed when treating instrumental drift, it’s good practice to ensure that at all interconnections in the voltage circuit both wires are at the same temperature. This can be achieved by making sure that such connections are plunged in the cryogenic bath or by wrapping the twisted wires several times around a large thermal mass in the immediate vicinity of interconnections. Table G2.3.2 summarises the experimental difficulties and their solutions discussed above.

G2.3.3 The Resistive Transition G2.3.3.1 Introduction The resistive transition is only partly related to the transport critical current. Nevertheless, from an experimental point of view as well as from the point of view of the dissipation processes involved, there exist sufficient similarities between the two to merit a brief discussion of this transition in this section. Both experiments measure the voltage drop across a sample generated by an imposed transport current. In critical current

Summary of Experimental Problems and Their Possible Solutions

Problem

Manifestation

Solution

Mains interference Instrumental offset and instability Common-mode voltages

Random noise, beats, spikes Erroneous null-detection, slow drifts Linear offsets

Earth loops Electromagnetic pick-up

Linear offsets, overloads, noise Noise, spikes

Thermal gradients and instabilities

Slow drifts, Premature transitions

Mains filters, instrument repositioning Thermal stabilisation, bipolar measurements Minimising parasitic resistance, sample grounding (!), booth-strapping Redesign of experiment Wire twisting and positioning, longer signal integration Thermal anchoring, sample design

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see why interactions with the ionic lattice are absent for low enough current densities (Rose-Innes and Rhoderick, 1978). The wave function describing the superconducting ground state is a superposition of many two-electron states in which the electrons have equal but opposite momenta, say pi and –pi (Figure G2.3.9, left). A current can be described by bodily shifting this momentum distribution over a distance P/2, so that each pair has a total momentum P (Figure G2.3.9, centre). The kinetic energy of a pair can then be written as: 1 ( P / 2 + pi ) 2 + ( P / 2 − pi ) 2 2m 1 P 2 / 2 + 2 pF 2 , = 2m

Eo =

(

FIGURE G2.3.8 Kamerlingh Onnes’ historic observation of superconductivity in Hg. (From Onnes, 1912.)

measurements, the external magnetic field and temperature are kept fixed and the current is gradually stepped up, whilst in resistivity measurements, one fixes the measuring current and varies either field or temperature. It was with such a resistivity measurement that superconductivity was observed for the very first time. In 1911, Onnes and co-workers were interested in the conductivity of pure metals in the vicinity of zero temperature. Whilst measuring the resistivity of mercury, they found that the ohmic voltage suddenly disappeared at a well-defined temperature Tc (Figure G2.3.8) (Onnes, 1912). It took another 40 odd years before the development of the BCS theory enabled a full understanding of the origin of this resistance-less transport.

G2.3.3.2 Charge Transport in Superconductors The BCS theory describes superconductivity as stemming from an attractive electron–electron interaction which pairs the electrons in the vicinity of the Fermi surface into a state which has an energy 2Δ(Τ ) lower than the energy of two unpaired electrons (see Section A3.2). A simple picture describing this paired state in momentum space allows to

FIGURE G2.3.9

(G2.3.4)

)

with m the electron mass and pF the Fermi momentum. Scattering of a pair to the state with the lowest kinetic energy available will leave the two electrons unpaired, both with momentum P/2-pF (Figure G2.3.9, right). This will lower their kinetic energy but will cost a condensation energy 2Δ: E=2

( P / 2 − pF )2 + 2Δ(T )

2m PpF = Eo − + 2 Δ(T ). m

(G2.3.5)

A scattering interaction thus has to furnish a minimum energy of 2Δ(Τ ) – PpF/m in order to break up a pair. For low enough temperature and current this is unlikely, the pair retains its total momentum P and electrical transport is resistance-less. We’ll come back to this straightforward picture in Section C2.2.4, in the discussion of the so-called depairing current.

G2.3.3.3 Resistive Transitions The resistive transition thus effectively measures the establishing of the superconducting energy gap Δ(Τ ) and as such provides a straightforward probe of the thermodynamic transition. Measurements of the temperature-dependent

Schematic description of a pair-breaking scattering event in momentum space.

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FIGURE G2.3.10 Temperature-dependent resistivity of a thin amorphous Bi film. (a) shows the onset of the transition, whilst in (b) the main transition is measured in more detail. (From Glover, 1967.) The solid curve in both graphs is the same fit to the data with the Aslamazov– Larkin expression 1/R–1/Ro = A/(T–Tc).

resistance R(T) are therefore very commonly used as a convenient way to determine the critical temperature Tc. More careful measurements can reveal further details. Above the critical temperature the paired state has a higher energy than unpaired electron states, but sufficiently close to Tc, this energy difference is small enough for thermal fluctuations to cause a non-zero probability of electron pairing. In pure metallic superconductors, the temperature region where such fluctuating pairs occur is usually very narrow, so that in practice their contribution to the conductivity – the so-called fluctuation conductivity – cannot be measured. If, however, the metal is sufficiently disordered so that the coherence length decreases considerably, the fluctuation regime extends to much higher temperatures and the resistance begins to decrease well above Tc. This was first shown experimentally by Glover (1967) for an amorphous Bi film (Figure G2.3.10) and 1 year later explained theoretically by Aslamazov and

Larkin (1968), who showed that in the case of impure thin films fluctuating pairs make a contribution σfl to the total conductivity: σ fl =

1 e 2 Tc , 16 d T − Tc

(G2.3.6)

where d is the film thickness. This result was further completed by Lawrence and Doniach (1971), who showed that this two-dimensional behaviour closer to Tc crosses over to a three-dimensional dependence σfl ∝ (T − Tc)−3, and by Maki (1968) and Thompson (1970) who included interactions between pairs and normal charge carriers to explain additional fluctuation contributions. In high-Tc superconductors, with their relatively small coherence lengths, the fluctuation regime extends even further (Figure G2.3.11), and analysis of the fluctuation conductivity

FIGURE G2.3.11 (a) The temperature-dependent resistivity of an YBa 2Cu 3O7 melt-textured sample, a Bi 2Sr2CaCu 2O8 thin film and a (Bi,Pb)2Sr2Ca 2Cu 3O10/Ag tape. The dotted lines represent the extrapolated high-temperature behaviour. (b) analyses these data in terms of fluctuation conductivity. (After Cimberle et al., 1997.)

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FIGURE G2.3.12 Magnetisation loop (top panel) and fieldinduced resistive transition (bottom panel) of a Nb ribbon. The different resistance curves correspond to different measuring current amplitudes. (After DeSorbo, 1964.)

can yield a host of information regarding the pair formation (Cimberle et al., 1997; Ausloos and Varlamov, 1997). Not only the temperature dependence of the resistance, but also its field dependence can be measured to extract information regarding the thermodynamic transition. This can be seen for instance in Figure G2.3.12, which shows the resistive transition of a Nb ribbon (DeSorbo, 1964), measured by varying the external applied magnetic field μ0H whilst keeping the temperature constant. The different curves correspond to different amplitudes of the measuring current. If the current is sufficiently low, the transition occurs at the second critical field Hc2, as can be seen from comparison with the magnetisation curve measured for the same sample (top panel). Repeating the measurement at different temperatures thus allows to

FIGURE G2.3.13 et al., 1992.)

determine the temperature dependence of Hc2 in metallic Type II superconductors. Figure G2.3.12 also shows how at higher current levels the resistance develops at fields well below Hc2. This resistance does not originate in a disappearance of the superconducting gap, but rather in the interaction between magnetic flux quanta and the transport current. The current causes the flux lines to move and this motion is dissipative, a topic which is treated in subsection G2.4.5 and in Sections A4, G2.1 and G2.5. It turns out that in the oxide superconductors such flux motion related dissipation dominates the resistive transition in the presence of a magnetic field regardless of the measuring current, so that Hc2 cannot be measured resistively. Figure G2.3.13 shows the onset of resistivity in a pure and untwinned YBa 2Cu3O7 single crystal, measured either by varying the temperature T in a constant magnetic field H [Figure G2.3.13(a)] or by varying H at constant T [Figure G2.3.13(b)] (Safar et al., 1992). The hysteretic nature of the resistive transition led the authors to believe that the sudden appearance of dissipation was caused by a first-order melting transition of the so-called flux-line lattice, which occurs well below Hc2 and greatly facilitates the motion of these lines. This explanation was later confirmed by magnetisation- (Welp et al., 1996; Liang et al., 1996) and heat-capacity measurements (Schilling et al., 1996; Junod et al., 1997). In superconducting samples that contain extended inhomogeneities, another dissipation mechanism may appear well before the superconducting gap itself goes to zero. If the current is forced to pass parts of the sample where the superconducting order parameter is depressed or zero, electrical resistance might develop in these regions whilst most of the sample is still dissipation-free. Figure G2.3.14 illustrates some examples of such granular superconductivity. Figure G2.3.14(a) shows the resistive transition of Al/Ge films with different concentrations x of Al grains embedded in an amorphous Ge matrix (Deutscher, 1983). As the temperature decreases, the

(a) Temperature- or (b) field-induced resistive transition of an un-twinned YBa 2Cu 3O7 single crystal. (From Safar

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FIGURE G2.3.14 The development of a ‘foot’ in the resistive transition due to dissipation in Josephson-type weak links. (a) shows the temperature-induced transition of Al/Ge thin films (after Deutscher [1983]), (b) the temperature-induced transition of polycrystalline untextured YBa 2Cu 3O7 measured with different current amplitudes. (After Wördenweber et al., 1988.) and (c) the field-induced transition of a similar sample measured at different temperatures. (Boon et al., 1989.)

Al grains become superconducting, causing the resistivity to drop by ~80–90%. The remaining ~10–20% only disappear once the temperature is low enough to allow superconducting tunnelling to occur from one Al grain to another, through the insulating Ge. xc is the percolation threshold below which no dissipation-free path is formed regardless of temperature. This type of dissipation in superconducting junctions will also drastically influence the transport critical current and will be further treated in subsection G2.3.6. In most metallic superconductors, the coherence length ξ is sufficiently large for the superconducting wave function to extend right through ‘intrinsic’ imperfections such as metallurgical grain boundaries or small amounts of secondary phases, and this type of dissipation is absent. In oxide superconductors, on the other hand, the coherence length is much smaller and the wave function is strongly depressed at imperfections. This causes these materials to be much more susceptible to junction-related dissipation, especially at grain boundaries. Figure G2.3.14(b) shows the transition of a bulk untextured YBa 2Cu3O7 sample (Wördenweber et al., 1988). At higher current levels the sample retains ~40% of its normal-state resistance even at the lowest temperatures. Figure G2.3.14(c) shows the field-induced transition of a similar sample measured at several temperatures in a pulsed magnetic field (Boon et al., 1989). A temperature-independent resistance of ~10% of the normal-state value develops well before the main transition. In subsection G2.3.6 we will see how such residual resistance can be described with a simple model.

G2.3.4 Thermodynamic Critical Currents G2.3.4.1 Introduction Subsections G2.3.1 and G2.3.3 mentioned how dissipation in superconductors does not necessarily imply that the

superconducting state is lost. This subsection treats some situations where this is the case, i.e. where a transport current causes paired charge carriers to break up in the whole or in parts of the sample and where the dissipation has a ‘normal’ ohmic origin. Such current-induced pair-breaking is of great fundamental importance, but seldom plays a role in determining the critical current of practical superconductors. We will therefore limit ourselves to a brief discussion of the bestknown examples of thermodynamic critical currents.

G2.3.4.2 The Depairing Current The first and probably most important example is the so-called depairing current. In subsection G2.3.3 we saw how a current density in a superconductor can be described by bodily shifting the momentum distribution of the charge carriers by an amount P (Figure G2.3.9) and how a scattering event needs to furnish a minimum energy 2Δ(Τ ) – P pF/m in order to break up a pair [Equation (G2.3.5)]. Δ(Τ ) is the superconducting energy gap, pF the Fermi momentum and m the electron mass. This minimum energy becomes zero or negative when P ≥ Pc = 2Δ(Τ ) m/ pF. If the density of paired electrons is ns, we can straightforwardly write the corresponding current density Jc,depairing as: ns ePc (T ) m 2 Δ(T )ns e , = pF

J c,depairing (T ) =

(G2.3.7)

where e is the electron charge. From a practical point of view, this current density is important in the sense that it represents the absolute maximum which can be achieved in a given superconducting material. Using Ginsburg–Landau theory (see Section A3), an alternative expression can be written in

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terms of the thermodynamic critical field Hc and the penetration depth λ : J c,depairing (T ) =

4 H c (T ) . 3 6 λ(T )

subsection G2.3.6 how similar effects can occur in larger samples, especially close to Tc where the coherence length diverges as ξ(T) ∝ (Tc – T) −1/2.

(G2.3.8)

As an example, the values Hc(77 K) = 2.4 105 Am−1 and λ(77 K) = 10 −7 m for YBa 2Cu3O7 at T = 77 K, allow to calculate a depairing current density of Jc,depairing(77 K) = 2.4 1012 A/m2 (Murakami, 1992). This is two orders of magnitude larger than the best values presently obtained in thin films of this material (such films display the highest current density values due to the abundance of structural defects which, as we will see in the next subsection, obstruct the dissipative motion of magnetic flux lines). The example of YBa 2Cu3O7 is typical, in the sense that it illustrates how for most superconductors used in practical applications flux motion related dissipation kicks in well before the depairing current is reached. Nevertheless, in some cases this fundamental current-limiting mechanism does determine the critical current of superconducting specimens, notably when the size of the sample becomes comparable to the superconducting coherence length ξ(T). A clear example is shown in Figure G2.3.15, which plots the VI curves and the corresponding critical current of a microscopic Al loop measured at T = 1.3 K as a function of magnetic field (Moshchalkov et al., 1993). Since the size of the loop is comparable to the Al coherence length ξ (1.3K) ≈ 0.6 μm, the whole loop behaves coherently and its behaviour can be predicted thermodynamically by calculating the currentinduced suppression of the order parameter. We will see in

G2.3.4.3 The Silsbee Current Very soon after the discovery of superconductivity, it was noted that both a too strong magnetic field (a field in excess of the critical field Hc, see Section A2) and a too high electric current would lead to the re-appearance of resistance. In 1916, Silsbee suggested that the two might be related. He proposed that what was then known as the ‘transition’ current might be nothing else than the current that generates a self-field at the sample surface [Figure G2.3.16(a)] which is equal to the critical field Hc (Silsbee, 1916). For a cylindrical sample with radius R, it follows straightforwardly from Ampère’s law that the corresponding critical current can be written as: I c,Silsbee (T ) = 2π R H c (T ).

(G2.3.9)

This current will drive the surface of a Type I superconductor normal. As Silsbee pointed out, the situation where an inner core of the conductor would stay homogeneously superconducting under these conditions is unphysical, since such a core would ‘short-circuit’ the normal surface layer, carrying the total current in a region of even smaller radius and thus generating an even larger field at the surface of this would-be superconducting core. Instead, an intimate mixture of superconducting and normal regions (the so-called intermediate state) develops within a normal-state sheath

FIGURE G2.3.15 (a) Voltage–current characteristics of a 1-μm-sided square Al loop, measured at T = 1.30 K in various magnetic fields. (b) shows the critical current extracted from these and similar data plotted against the magnetic flux treading the loop. The solid line is calculated from Ginsburg–Landau theory without adjustable parameters. (From Moshchalkov et al., 1993.)

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FIGURE G2.3.16 (a) The self-field due to a transport current in a cylindrical wire and (b) the magnetic induction inside such a 2-mm-radius Sn wire, measured at T = 3.57 K as a function of the distance d to the wire surface. Different curves correspond to different current amplitudes. (From Makei, 1958.)

(Takács, 1998), in which the current is forced to pass both superconducting and normal regions and the latter show ohmic dissipation. A transport experiment confirming this picture is shown in Figure G2.3.16(b), which plots the magnetic induction profile measured inside a thin slot cut out in a current-carrying Sn cylinder (Makei, 1958). As soon as the transport current exceeded ~22 A, the surface field surpassed Hc(3.57 K) ≈ 2.2 mT, and the wire became resistive. From this current onwards, the induction profile developed a linear part which extended to the centre of the wire and which corresponds to the intermediate-state core region. This intermediate state should not be confused with the so-called mixed state. The former indicates the coexistence of normal and superconducting lamella in a Type I superconductor, which is partly exposed to a field exceeding Hc. The latter is used for the occurrence of quantised flux lines in a Type II superconductor between the fields Hc1 and Hc2. For Type II superconductors, a ‘modified’ Silsbee current can be defined as the current which causes the self-field at the sample surface to exceed Hc1 (Rose-Innes and Rhoderick, 1978). Currents higher than this will nucleate flux loops at the sample surface and in sufficiently pure materials will drive these loops to the sample centre. Although such flux motion will also give rise to a sample voltage, the corresponding dissipation process is not due to a thermodynamic transition from the superconducting to the normal state but will be treated separately in subsection G2.3.5. Note that it is pointless to define a Silsbee critical current density, since the Silsbee current is not a ‘pure’ material property but depends also on sample size and on sample geometry.

G2.3.4.4 Surface Currents A final example of thermodynamic critical currents is the current-induced suppression of the superconducting surface state which can exist in external magnetic fields that exceed Hc (type I) or Hc2 (type II). The existence of such a surface state was predicted in 1963 by Saint-James and de Gennes (SaintJames and de Gennes, 1963), who showed that the Ginzurg– Landau equations have a non-trivial solution at the interface between a superconductor and an insulator for fields up to √2κHc parallel to this surface, with κ = λ/ξ the Ginzburg– Landau parameter (in Type II superconductors this corresponds to a field Hc3 = 1.695 Hc2). This superconducting surface layer has a thickness of roughly the coherence length ξ and is capable of carrying a finite current (Parks, 1969). Figure G2.3.17 shows an elegant demonstration of this surface current using the Corbino disk configuration (cfr. Figure G2.3.4) (McKinnon and Rose-Innes, 1969). Jc(H) measurements were carried out both before and after cutting a radial slot from the edge to the centre of the Pb–In disk. Recall that in the intact Corbino configuration there is no surface parallel to both field and current. The slot creates such a surface, which has two effects on the critical current density. Below μoHc2(4.2K) ≈ 0.24T its presence increases Jc by a factor of nearly 2. Above the upper critical field it causes a finite Jc value to appear. The first effect is due to the so-called surface barrier, an interaction between magnetic flux lines and the sample surface which will be further elaborated in subsection G2.3.5. The second effect is due to the current-carrying superconducting surface layer along the slot, as could be confirmed

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can be made using energy arguments, such as in terms of the work done by the external sources maintaining the transport current and externally applied field (Parks, 1969) or in terms of magnetic pressure and the resulting free energy (Campbell and Evetts, 1972). Conceptually, the result of such calculations is simple. The interaction can formally be described as a Lorentz force between a vortex and the local current density J: fL = J × Φ o ,

FIGURE G2.3.17 Critical current density of a Pb/In alloy measured as a function of applied magnetic field. Two successive experiments were carried out, one using the Corbino disk configuration and a second one after cutting a radial slot from the edge to the centre of the disk. (From McKinnon and Rose-Innes, 1969.)

by the fact that this surface current disappeared again after the slot was plated with Cu. Note that the Jc value reported by the authors was calculated using the whole section of the disk. Taking into account the limited thickness of the surface layer, the local current density can be estimated to be close to the depairing current.

G2.3.5 Flux Motion G2.3.5.1 Introduction Let us first recapitulate briefly the most important features of flux lines in Type II superconductors, which are also discussed in Sections A4, G2.1 and G2.5 of this book. When a Type II superconductor is exposed to a magnetic field of amplitude larger than Hc1, the first critical field, magnetic flux will enter in the form of discreet flux quanta. The magnetic field in a flux line is generated by a cylindrical ‘vortex’ of circulating supercurrents, which extends over a radial distance of a few times λ(Τ), the magnetic penetration depth, from the centre of the vortex. The generated field is maximal at the centre of the current vortex and decays over the same typical distance λ(Τ), in such a way that the total magnetic flux per vortex equals the magnetic flux quantum ϕo = h/2e = 2.07 × 10 −15Wb. At the centre of the vortex the amplitude of the superconducting order parameter /Ψ/ (i.e. the Cooper pair density) is zero, restoring itself as one moves away from the centre over a typical distance ξ(T), the superconducting coherence length.

G2.3.5.2 Lorentz Forces and Dissipation Flux lines interact with the current vortices of neighbouring flux lines and with the imposed transport current. A rigorous treatise of how the total current pattern organises itself

(G2.3.10)

where f L is the force per axial unit length acting on the vortex, and Φo is a vector of magnitude ϕo in the direction of the applied field. The current density J is the total current density at the vortex position, originating either from neighbouring vortices, from the transport current or from a combination of both. The formula shows that the vortex will be subjected to a force which is perpendicular both to the local current density and to the applied magnetic field. If there is no other force acting on the flux line, it will move. This immediately leads us to two important conclusions. First, in equilibrium and in the absence of other forces, the (repulsive) forces between vortices will arrange them in some kind of regular pattern with a uniform density of flux lines. That way the current density due to neighbouring flux lines adds up to zero at the centre of a vortex and there is no net force. This conclusion agrees with the prediction made by Abrikosov (1957) using Ginsburg–Landau theory and has been confirmed in numerous experiments. Such regular arrangement of flux lines is known as the Abrikosov lattice. Second, when we apply a transport current to a vortex lattice, we disturb the equilibrium current pattern, and the vortices will start to move perpendicular to the transport current and to the magnetic field direction. Here, we get back to the topic of transport critical currents: a moving vortex dissipates energy. In order to understand why this is so on a microscopic level, one needs to make a careful assessment of various contributions to this dissipation process, such as the effect of the changing local magnetic field on the normal-state charge carriers inside the vortex core or the processes of pair-breaking and re-pairing at the front and back of the moving core (Bardeen and Stephen, 1965). When all is said and done, however, we once more get a conceptually simple picture. The dissipation can be modelled as stemming from a viscous drag force per unit length η which is directed opposite to the average vortex velocity vL . Once η is high enough to balance the driving Lorentz force f L , vL becomes constant. Faraday’s law tells us that this steady motion of magnetic flux leads to an electric field E: E = B × vL .

(G2.3.11)

B here is the local magnetic flux density averaged over a few flux lines, B = n Φo, with n the number of flux lines per unit

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area. The work p done by the Lorentz force per unit volume and per second is then: p = n fL ⋅ v L = (J × B) ⋅ v L = J ⋅E

(G2.3.12)

= JE so that the electric field due to the flux motion can simply be pictured as originating from an electrical resistivity, the so-called flux flow resistivity. The fact that the electric field in pure Type II superconductors submitted to a field larger than Hc1 is indeed generated by the motion of extended flux lines was clearly shown by Giaever (1965). His elegant experiment is nowadays known as the DC flux transformer. For these measurements, a Sn film of a few thousand Angstrom thickness was covered with a 100-Å-thick electrically insulating SiO2 layer. On top of this insulator a second ~1000-Å-thick Sn film was evapourated [Figure G2.3.18(a)]. The bottom and top films are referred to as the primary and secondary layer, respectively. A magnetic field was applied perpendicular to this sandwich structure, and a current was passed only through the primary layer, whilst the voltage was measured over the primary as well as the secondary layer. In the normal state above Tc, only the primary layer showed dissipation, indicating that no transport current was leaking into the secondary. Below Tc, a voltage developed in both primary and secondary layer [Figure G2.3.18(b)]. The voltage in the secondary layer could be explained by the motion of the flux lines generated by the transport current, which axially extended through the SiO2 film into the secondary layer and was pulled along by the driving Lorentz force in the primary.

Flux transformer type experiments have proven to be of particular interest when investigating the properties of vortices in oxide superconductors. Due to their anisotropic nature, single crystals of these materials behave as ‘intrinsic’ DC flux transformers. Jc along their crystallographic c-axis is generally much lower than Jc along the ab-planes. Therefore, when a current is injected by contacts on one side of a single crystal parallel to the ab-planes [say the ‘top’ face, Figure G2.3.19(a)], the current will tend to flow along this face rather than to penetrate uniformly throughout the crystal. Consequently, the ‘Lorentz’ force acting on vortices treading the crystal in the c-axis direction will be higher near the ‘top’ face than near the ‘bottom’ face. At sufficiently high current densities, this difference in driving force will ‘break’ the flux lines in segments, which will move faster at the ‘top’ than at the ‘bottom’, as can be detected by measuring the voltage on both sides of the crystal [Figure G2.3.19(b)] (Lopez et al., 1994). The magnetic fieldand temperature dependence of this effect have yielded valuable information regarding the structure of flux lines in various oxide superconductors (Brawner et al., 1996; Seow et al., 1996). We will come back to this topic at the end of this subsection.

G2.3.5.3 Vortex Pinning If the Lorentz and viscous drag forces were the only forces acting on a vortex, superconductors carrying a transport current would always exhibit dissipation when the magnetic field at some point on the surface exceeded Hc1. This is indeed what is found to be the case in very pure type II materials. As soon as the material is less pure, however, an additional force may act on the flux lines: the pinning force Fp. ‘Impure’ refers here

FIGURE G2.3.18 (a) Schematic representation of the DC flux transformer configuration and (b) V(I) curves for both primary and secondary layer measured at various temperatures. (After Giaever 1965.)

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FIGURE G2.3.19 (a) Schematic representation of the ‘intrinsic’ DC flux transformer configuration used with single crystals of oxide superconductors and (b) V(I) curve measured in La 2-xSr xCuO4 with this configuration. (From Lopez et al., 1994.)

to different types of imperfections: local deviations from the ideal chemical composition, dislocations or strain fields in the ionic lattice, inclusions of other crystallographic phases or amorphous matter, metallurgical grain boundaries, etc. The common factor to all these different types of impurities is that they may locally reduce the superconducting order parameter |Ψ |. If this happens, the region of depressed superconductivity will exert an attractive interaction on a flux line. As an illustration, let’s consider the relatively simple case of pinning by a non-superconducting spherical inclusion of size a. The increase in the superconductors’ free energy F due to the presence of this inclusion is: µ H2 4 4 ΔFinclusion = ( f n − f s ) πa3 = o c πa3 , 3 2 3

(G2.3.13)

where f n and fs are the Helmholtz free energy density of the normal and superconducting state, respectively, and Hc is the thermodynamic critical field. This energy describes the loss of negative condensation energy associated with the superconducting state throughout the volume of the normal inclusion. Also inside the normal core of the flux line the superconducting order parameter is depressed, which leads to a term: µ H2 ΔFcore = ( f n − f s )πξ 2 = o c πξ 2 , 2

(G2.3.14)

per unit length of flux line in the free energy balance. If a flux line treads the non-superconducting inclusion, the coincidence of these two non-superconducting regions leads to a lowering of the overall energy increase. In the limiting cases

of very small or very large inclusions, this energy gain can be written straightforwardly as: µ H2 4 ΔFpinning = − o c πa3 2 3

a 100 μV m−1 level as a function of magnetic field, temperature and strain. The article then concludes with a short reflection.

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G2.4.2 General Principles for Measuring I c When Testing Conductors G2.4.2.1 Four-Terminal Critical Measurements on Strand Superconductors There are many different designs for conductors, from multifilamentary NbTi, Nb3Sn and BiSCCO strands/wires and tapes to thin-film tapes of REBCO. Schematic diagrams for some of the most common are shown in Figure G2.4.2. In addition to the superconducting filaments, one can also see the other important components that are part of a technological conductor, including those parts that contribute to the fabrication of the superconducting material itself (e.g. bronze) or those parts that stabilise the conductor when carrying current (e.g. copper, silver). Some of the most important parameters for describing superconducting strands are listed in Table G2.4.1. We can usefully draw a clear distinction between the metallic/intermetallic superconductors and the oxide superconductors that helps to clarify the different approaches to fabricating them and the final forms that result (as shown in Figure G2.4.2). The resistivity of the grain boundaries in the metallic/intermetallic superconductors is typically similar to that of the grains and broadly insensitive to the angle between neighbouring grains (Byrne, 2017). However, for the oxide materials, low resistivity grain boundaries i.e. strongly coupled grains, are only achieved when the angle between the grains is very small, typically less than 5° – although low resistivity may also be possible at some higher coincidentsite-lattice magic angles (Dimos et al., 1988). Strongly coupled grains are a necessary but not sufficient requirement for high critical current densities (Wang et al., 2017). Hence, the metallic/intermetallic superconductors can be fabricated to include both high- and low-angle grain boundaries, whereas in contrast, the BiSCCO materials include a very large degree of texturing (forced grain alignment) (Larbalestier et al., 2014),

and RE-123 production uses thin-film fabrication techniques to attain almost exclusively low-angle grain boundaries to achieve high critical current densities (Iijima et al., 1992; Goyal et al., 2004; Durrell, 2009). Proper consideration of the construction of the strand is important when characterising its current-carrying capacity (c.f. Section G2.4.3). There are many excellent reviews and texts that discuss conductor design (Wilson, 1986; Uglietti, 2019). Nevertheless, the basic principles behind a critical current measurement are the same for all superconductors – those of a textbook four-terminal resistance measurement. The current is applied to the sample by means of current contacts at both ends, and the voltage is measured across a pair of taps positioned across a length of the sample between the current contacts. The current is slowly increased from zero and the voltage across the taps is monitored. Eventually a V–I (or equivalently an E–J) characteristic is measured. This process can be repeated as a function of applied magnetic field as illustrated in Figure G2.4.3, which includes a block diagram of a typical experimental arrangement and data. For the data shown, an E-field criterion of either 10 μV m−1 or 100 μV m−1 are common choices (c.f. Section G2.4.3) used to obtain the critical current density. G2.4.2.1.1 Sample Geometry and Wiring There are three sample geometries that are most commonly used for measuring critical current. Some of the different forms are shown in Figure G2.4.4. The choice of geometry mainly depends on the architecture or form of the superconductor, which includes considering the magnitude of the critical current to be measured and whether the superconductor has significant anisotropy. Here we call the geometries; ‘straight’, ‘hairpin’ (or ‘U-bend’) and ‘coil’ geometries. The straight geometry is most often used for conductors that are still in early development, where homogeneous lengths – typically sintered blocks of conductor – are the only form available. The hairpin geometry is used for samples that carry high currents, where current transfer from the current leads into the conductor is a problem; usually because it is difficult to

TABLE G2.4.1 Typical Specification Parameters and Their Values for the Nb3Sn and NbTi Strands Being Used in the ITER Toroidal and Poloidal Magnet Systems Specification Parameter

Nb3Sn Parameter Value

NbTi Parameter Value

Cryogenic measurements Critical current at 12 T (6.4 T for NbTi) and 4.22 K (10 μVm-1) > 190 A n-value at 12 T and 4.22 K (between 10 and 100 μVm-1) > 20 Hysteresis losses ±3 T (±1.5 T for NbTi) at 4.22 K < 500 mJ/cm3 Residual resistivity ratio at 273 K and 20 K (10 K for NbTi) > 100

> 306 A > 20 < 55 mJ/cm3 > 100

Room temperature measurements Strand diameter 0.820 ± 0.005 mm Strand twist pitch 15 ± 2 mm Chromium (Nb3Sn), Nickel (NbTi) plating thickness 2 (+0 or – 1) μm Cu to non-Cu ratio 1 ± 0.1

0.730 ± 0.005 mm 15 ± 2 mm 2 (+0 or – 1) μm 1.55 – 1.75

Source: Sborchia et al. (2011).

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FIGURE G2.4.2 Schematic diagrams of conductors (a) Cu-NbTi (inset [Muzzi et al., 2011]), (b) multifilamentary bronze route Nb3Sn and internal tin Nb3Sn. Adapted by Durham from SuperPower-Furukawa (2013b). (Continued)

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FIGURE G2.4.2 (Continued) (c) Ag-sheathed Bi-2212 (Larbalestier et al., 2014), (d) Bi-2223 (Sato, 2017) and (e) rare-earth-123. Adapted by Durham from SuperPower-Furukawa (2013b).

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FIGURE G2.4.3 (a) A block diagram of equipment for measuring critical current density in strand conductors (the bubbler is used to maintain the liquid helium at close to atmospheric pressure), and (b) Typical E–J characteristics generated for a Nb3Sn strand at 4.2 K and 10 T at different values of strain (Tsui and Hampshire, 2012). A 21-point smoothing process has been applied to reduce the noise from the raw data without altering the shape of the transition. Each pair of E–J characteristics at a given strain was obtained during the strain cycle from +0.4% down to −1.1% and back up to 0.4% strain. The figure shows that the E–J characteristics are a reversible function of strain over the strain range measured.

make low-resistance/high-current connections (c.f. Current transfer section in G2.4.2). The coil geometry is commonly used to test long lengths of conductor; usually for technologically mature materials that are commercially available. G2.4.2.1.2 Straight Geometry The short straight geometry is the simplest configuration. The short sample fits into the bore of a solenoid magnet, as shown in Figure G2.4.4(a) and Figure G2.4.4(b). For broadly isotropic materials, this is the standard orientation of field and current flow because the direction of current flow is perpendicular to the applied magnetic field. This orientation gives the lowest critical current, and therefore provides the most useful (limiting) case for high-field applications (Goodrich and Fickett, 1982; Grasso and Flükiger, 1997). The short sample geometry provides the least sensitive E-field criterion for the determination of J c because of the short measuring length between the voltage taps. The orientation of the sample to the current

leads also means that there is a relatively small contact area for the current to transfer into the sample compared with the other configurations. This, in turn, increases the contact resistance and heating. The reduced space around the sample limits the distance over which the current can distribute into the superconducting filaments. If the current has not transferred into the superconducting filaments, the voltage recorded across the taps is associated with the current that has remained in the normal matrix. Hence, voltage taps need to be sufficiently separated from each other to allow sensitive E-field measurements to be made, but also sufficiently separated from the current leads to avoid the problem of current transfer voltage. Current transfer into the superconducting filaments is an important problem for measurements on small samples as discussed below. No bending of the sample is required during mounting. The sample is easily supported against the Lorentz forces present when performing critical current measurements in applied magnetic fields, which is particularly

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FIGURE G2.4.4 Different geometries for performing critical current measurements. (a) Short straight sample measured in a vertical solenoidal magnet, (b) short straight sample for finding the angular dependence of critical current with magnetic field, (c) long straight sample measured in a horizontal split-pair magnet, (d) hairpin sample geometry, (e) coil geometry, (f) detail of voltage lead attachment to minimise inductive voltages for the helical geometry.

important for brittle superconductors. The width of the bore (or cryostat tail) obviously provides a limiting size. A much greater length of sample can be used in the long straight geometry, as shown in Figure G2.4.4(b). The voltage taps are in the homogenous region of the magnet and are well separated from the region where the current transfers from the current leads into the superconductor. Also, a greater length of the sample is in contact with the current leads. This larger contact area reduces contact resistance and hence heating in this area. The longer length provides a longer measuring region for the voltage taps, leading to increased E-field sensitivity. The homogeneity of the magnet determines the maximum length over which the voltage taps can be placed. For anisotropic materials, the geometry in Figure G2.4.4(c) is important because as the sample is rotated, the direction of current flow remains orthogonal to the applied field, and the anisotropy of J c results from the superconducting material’s anisotropy. Note that measurements using split-pair or Helmholtz type magnets are relatively rare (Branch et al., 2019), because such magnet systems are more expensive than vertical magnets, the forces in them are higher and hence more difficult to manage and the stored energy is generally larger, so the amount of superconducting material required for a given maximum field and bore size is larger. Figure G2.4.4(b) shows how the short sample geometry can be used with a split-pair magnet to measure the dependence of critical current on the orientation

of current flow with respect to the direction of the applied field. This geometry is used to measure the effect of the mean Lorentz force changing from a maximum value to zero. This anisotropy is usually low, typically less than a factor 2, associated with the tortuous local current density flow, the large range in the directions of flux flow and the vector nature of the local Lorentz force (Friend and Hampshire, 1993; ChislettMcDonald et al., 2019). G2.4.2.1.3 Hairpin Geometry The hairpin geometry shown in Figure G2.4.4(d) reduces the contact resistance between the leads and the conductor and hence reduces current transfer effects – similar to the long straight sample geometry but without requiring a split-pair magnet. As the current leads are not near the voltage measurement region, a longer measuring distance between the voltage taps can be used and hence a better E-field sensitivity is achieved than in the short straight geometry. The hairpin geometry is suitable either for ductile superconductors such as NbTi or for conductors that can be reacted into the required shape (e.g. Nb3Sn), but clearly remains problematic in the latter case if the reaction process is commercially sensitive (e.g. BiSCCO). This geometry is clearly not suitable for conductors that do not meet one of these two criteria because the strain produced in the sample on forming this configuration usually damages the sample. Another aspect of this geometry is the

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curved nature of the sample. If the orientation of the conductor between the voltage taps varies with respect to the direction of applied magnetic field (e.g. the conductor is of semi-circular shape), then the critical current measured will represent an angular average over the applied field. A flat-bottomed hairpin is therefore preferable to a round-bottomed hairpin. However, particular care must be taken to avoid damaging the sample when fabricating flat-bottomed samples.

this case, the thermoelectric voltages are minimised since there are no joints along the length of the voltage wires. We recommend measuring the voltage noise in measurements and comparing it with the fundamental Johnson noise associated with quantisation of charge. The root-mean-square (rms) voltage from Johnson noise is given by

G2.4.2.1.4 Coil Geometry

where kB is the Boltzmann constant, T is the temperature of the voltage leads, R is the resistance of the voltage leads and Δf is the inverse of the time constant of the voltmeter/amplifier. In high-field critical current measurements, we find the experimental noise is typically five times the fundamental noise floor.

The coil geometry [Figure G2.4.4(e)] is most commonly used for testing long lengths of conductor. It is suitable for ductile materials and conductors that can be reacted in the coil shape – most obviously, the ductile alloy NbTi, which is the workhorse material for applications below 10 T, and the brittle intermetallic Nb3Sn, which is currently the LTS material of choice for applications above 10 T. The length of conductor used is the largest of all geometries. The voltage taps can be placed the greatest distance apart (typically 0.5 m), leading to the best E-field sensitivity. The contact resistance and the current transfer effect can be largely removed for this arrangement. Although the sample is oriented at an angle to the applied magnetic field, all parts of it experience the same offset angle, which generally has little effect. For example, in Nb3Sn, a 7° pitch represents a change in I c of only 2% (Goodrich and Fickett, 1982). The coil geometry also provides the opportunity to investigate the homogeneity of the conductor by placing multiple sets of voltage taps along the sample. G2.4.2.1.5 Voltage Wiring There are many sources of voltage noise that can affect J c measurements. The largest are usually eliminated by twisting the voltage tap wires together and tying them down in order to prevent them from producing inductive voltage noise by moving in the magnetic field caused by cryogenic gas flow. The area between the voltage wires and the sample should also be minimised to reduce the inductive loop area further. This procedure minimises inductive voltages, which are produced either as the current through the conductor is increased or from the ripple in the applied magnetic field. For straight and hairpin samples, the voltage wires simply run along the surface of the conductor, as shown in Figure G2.4.4. For the coil geometry, where the distance between voltage taps can be several turns, the wire from the first voltage tap should run alongside the sample until the second voltage tap, as shown in Figure G2.4.4(f). From there, the wires should be twisted together. Other sources of voltage noise include thermoelectric, offset, ground loop, common mode and current transfer voltages (Goodrich and Bray, 1989). If these voltages stay constant during the measurement, then they can easily be subtracted from the data. This is achieved by comparing the voltage at zero current before and after the trace is measured. For the most sensitive voltage measurements, the twisted pair of voltage wires should be continuous from the sample to the voltmeter (i.e. without the use of instrumentation plugs). In

Vrms = 4kBTR ( Δf ) ,

(G2.4.1)

G2.4.2.1.6 Current Leads The design of current leads varies widely, depending on the application (Sunwong et al., 2014). The temperature gradient along sophisticated current leads, for example, can be controlled by more than one cryogenic liquid and/or the operation of a cryocooler. In general, current leads must transport sufficiently large current without thermal runaway or burnout, have low electrical resistance so they do not generate much additional heat in the system and have low thermal conductivity to minimise static boil-off. Unfortunately, a room temperature superconductor, which may have ideal properties for current leads, has not yet been discovered. Nevertheless, the high-temperature oxide superconductors can be used to reduce power consumption in current leads significantly and hence operating costs in large-scale cryogenic systems. Here, some general design principles are considered for current leads. We will provide order of magnitude values for the size of current leads to aid those new to these measurements. For some detailed analysis, the reader is referred to the excellent work in the literature (Wilson, 1986; Herrmann, 1998; Sunwong et al., 2014). In simple terms, the design of the current leads can be separated into four sections: (I) The connection from the power supply to the probe. Generally, there is very little or no gas flow, so one must ensure the room temperature wiring is rated to carry the maximum current to be used, usually the maximum current the power supply can provide. However, significant reduction in the cross-section of these leads can be achieved if one includes realistic duty cycles for the measurements. For example, we have leads rated significantly below 2000 A dc attached to our 2000 A power supplies because most of the experimental time is spent ramping the magnetic field or stabilising the temperature, which provides sufficiently long periods where current is not being carried by the leads, allowing them to cool before the next duty cycle. Typically, I c measurements only run for ~ 20% of the time and typical I c values are around 1000 A, and even when measuring the highest I c values, the current flowing is only close to 2000 A for a relatively small fraction of the measurement time. This

Characterisation of the Transport Critical Current Density for Conductor Applications

means we are able to run our current leads hot for short periods. (II) The top section of the current leads inside the probe – operating at an approximate temperature range from room temperature or higher, down to 200 K. This section is from the head of the probe (where the room temperature power supply leads are connected) down to a level close to the neck fitting at the top of the Dewar. Our experience is that this region is the most likely to burn out. Efficient use of the cryogen exhaust keeps the leads as cool as possible. The leads are usually best made of copper with a cross-sectional area similar to that of the brass leads located in the middle section of the probe. (III) In the middle section, there is a temperature difference from about 200 K down to the temperature of the cryogen. One must ensure that all the available enthalpy from the cold flowing cryogenic vapour is used to cool the leads in this region. In optimal design configurations, brass is often the preferred material. Tubes are used with sufficient bulk to provide a large surface area for efficient vapour cooling, reducing the likely risk of them burning out in comparison to, say, copper, if temporarily operated outside optimal conditions (Herrmann, 1998). An optimised constant diameter brass lead, operating from room temperature to liquid helium, can efficiently carry a current given by the condition ABrass = I Max L / 1.5 × 106 ,

(

)

(G2.4.2)

where ABrass is the cross-sectional area of the brass, I Max is the maximum current carried by the lead and L is the length of the lead (Sunwong et al., 2014). Hence, for a 1 m lead, the current density in the brass should be 150 A cm −2 when the current flowing is at I Max . Under these conditions, the leads are sufficiently large that at operating current resistive heating is optimally kept below excessive levels but are also sufficiently small to minimise static helium boil-off. The heat load (Q) into the helium cryogen is typically 1.08 W kA−1 or 1.4 l of liquid helium per hour per kA. For brass leads, vapour cooled with a nitrogen cryogen, IL/A ~ 8 × 105 A m −1 and Q = 25 W kA −1. This middle section can also have high-temperature superconducting tapes or multifilamentary conductors soldered in parallel to minimise heat generation. (IV) The leads in the bottom section of the probe, which are submerged in the cryogen, should be relatively bulky to minimise resistive losses and incorporate superconducting (usually oxide and metallic/intermetallic) wires in parallel. The leads should also be large enough to prevent film boiling of the liquid cryogen (Wilson, 1986). During film boiling, the lead is enveloped by a layer of insulating gas which results in a rapid temperature rise of the current lead, possibly leading to an undesirable increase in sample temperature (Sakurai et al., 1996). The reader should be careful with the use of flat oxide tapes (i.e. non-multifilamentary tapes) in horizontal split-pair magnets. The stray field in such magnets is large and can magnetise the tapes in the current leads. It takes a considerable torque to rotate these magnetised tapes in-field and therefore,

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probes with such tapes incorporated within their current leads can only be rotated in low fields. G2.4.2.1.7 Sample Holder – Bonding, Thermal Contraction and Resistivity Experimental testing of conductors can involve making nanovolt measurements in high magnetic fields with hundreds or thousands of amps flowing through the conductor. The brittle nature of some conductors and the high Lorentz forces present mean that the conductor must be fastened securely to prevent the sample from moving and subsequently becoming damaged. Even for ductile conductors, sample movement may also lead to additional voltage noise, variations in the measured critical current or even thermal runaway below I c. For straight and hairpin geometries, the sample is simply mounted onto the planar surface of the sample holder after reaction or fabrication. For the more complicated coil geometry, ductile samples are wound directly onto the cylindrical sample holder for testing. For brittle materials, samples are either directly reacted on the sample holder or (although less preferred) reacted on a mandrel in the furnace and then carefully transferred, after reaction, onto the sample holder. The direction of the Lorentz force is orientated so that it presses the sample against the sample holder and facilitates using less bonding agent to hold the sample in position. Bonding agents include G.E. Varnish, epoxy (such as Stycast), vacuum grease and solder. A high-strength bonding material (e.g. epoxy or solder) is required if large Lorentz forces are present; although completely covering a sample with excess bonding agent must be avoided since this will inhibit the transfer of heat from the sample to the cryogen and can reduce the maximum measurable critical current. It is also often important to try to match the coefficient of thermal contraction for the sample holder to that of the sample. This ensures that there is no additional stress applied to the sample, which may affect I c when the sample is cooled from room temperature (or the soldering temperature if solder is the bonding agent) to the cryogenic testing temperature. The coefficient of thermal contraction, for some important cryogenic materials, is shown in Figure G2.4.5. Figure G2.4.5(a) shows thermal contraction data for various superconducting compounds, composites and matrix materials (Clark et al., 1981; White, 1987; Meingast et al., 1991; Okaji et al., 1994; White, 1998; Yamada et al., 1998). It should be noted that the thermal contraction of the wire will depend ultimately on the construction of the whole composite (Ochiai et al., 1993; Osamura et al., 2014; Osamura et al., 2016). In the VAMAS work, contractions of 0.26% to 0.28% were reported from room temperature to 77 K (Goodrich and Srivastava, 1995a; Kirchmayr et al., 1995). Figure G2.4.5(b) shows the thermal contraction for various common mandrel or bonding materials (Clark, 1983; Pobell, 1996; White, 1998; Cheggour and Hampshire, 2000). This parameter for G10-CR and G11-CR is similar in the fill and warp directions. The difference between the thermal contraction of the sample holder and the sample is transmitted

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FIGURE G2.4.5 Thermal expansion as a function of temperature for (a) superconducting compounds, composites and matrix materials (Clark et al., 1981; White, 1987; Meingast et al., 1991; Okaji et al., 1994; White, 1998; Yamada et al., 1998) – the Nb3Sn wire includes a tungsten core and (b) mandrel materials and bonding agents (Clark, 1983; Pobell, 1996; White, 1998; Cheggour and Hampshire, 2000).

between the two by the bonding agent. If a large amount of bonding material is used, both the sample holder and bonding material contribute to the net stress on the sample. The stronger the bonding material, the greater the strain produced by any differential thermal contraction between the sample holder and conductor. The type of bonding agent used must therefore be considered carefully for each particular experiment (see case studies below). The most often used materials for the sample holder are the insulating composite G10, the highly resistive Ti-alloy, Ti-6Al-4V (Ti-64) and Cu-Be, which has a high elastic limit of ~1% at 4.2 K (c.f. Figures G2.4.6 and G2.4.7). Although Ti-64 alloy is widely used, at 4.2 K and below 3 T, it becomes superconducting and Ti-6Al-2Sn-4Zr-2Mo-0.2Si (Ti-6242) can be a better choice (Ridgeon et al., 2017). Cu-Be is also often used as a sample holder because one can easily solder the sample to it, minimising sample movement, and because of its high elastic limit (Cheggour and Hampshire, 2000). However, at 10 K the resistivities of the Ti-alloys are about 1.5 × 10 −6 Ω m whereas Cu-Be is much less resistive at 4.7 × 10 −8 Ω m, so one

has to be careful about current shunting through the sample holder as the superconductor becomes resistive at I c (Ridgeon et al., 2017). As discussed below, although the shunting might only produce a small error in I c, which can be corrected to first order, it can significantly affect the measured n-value. G2.4.2.1.8 Current Transfer in Composites Consider two types of current transfer: The first of these is the initial transfer of current from the current leads into the conductor. Best results are obtained if both the leads and the sample can be separately tinned (coated in solder) and then sweated (heated while applying pressure) together so that there is good electrical contact with little excess solder. This ensures that there is a low-resistance path from the copper parts of the leads to the sample/filaments of the wire, and that any heat generated can easily be conducted into the helium bath without heating the wire excessively (avoiding quenches). Measurements of J c are most reliable with the voltage taps as far away as possible from the current transfer regions near the current leads. This ensures that the properties of the conductor

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FIGURE G2.4.6 (a) The standard ITER barrel design showing the separate top and bottom copper current rings and the central titanium cylinder (Nijhuis et al., 2006). Note that only the titanium cylinder is prepared with a groove to support the sample strand. The copper rings are pinned (or supported with screws) onto the titanium cylinder. (b) The Durham barrel has copper current rings screwed into the titanium cylinder. Note that the sample strand support groove (1) from the titanium cylinder is extended into the copper rings (2). Grub screws hold the copper and titanium together (3) (c) Durham barrel with top copper ring removed showing an internal screw thread (4) for copper ring attachment. (d) Copper ring showing the screw thread (5) (for attachment to the titanium cylinder) and the outer sample support groove (2). In Durham we routinely use this barrel to measure both NbTi and Nb3Sn strands.

FIGURE G2.4.7 Different geometries for sample holders used to make critical current measurements under applied strain. (a) A spring barrel used to make variable-strain I c measurements in a vertical magnet (Walters et al., 1986; Taylor and Hampshire, 2005b), (b) a ‘springboard’ sample holder used to make variable-strain I c measurements in a horizontal magnet (Sunwong et al., 2013, 2014) and (c) a crossboard for simultaneously applying longitudinal and transverse strains to a sample while making Ic measurements in either a horizontal or vertical magnet (Greenwood et al., 2018, 2019b).

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alone are measured, rather than the properties of the current lead joint resistances. The voltages produced in the current transfer region, across the matrix of the superconducting wire or tape, increase linearly with current (unless, of course, there is heating in the sample). It has been shown (Ekin, 1978) that the distance required to allow for current transfer to a monocore conductor is given by L=d

0.1ρm , n ρ*

(G2.4.3)

where d is the diameter of the filament region of the wire (i.e. the area of the wire containing the superconducting filaments), n is the order of transition which describes the shape of the V–I curve for the superconductor (c.f. Section G2.4.3), ρm is the resistivity of the matrix and ρ∗ is the resistivity criterion used to define the critical current density. This equation yields typical current transfer distances of approximately 30d for Nb3Sn and 3d for NbTi (Goodrich and Fickett, 1982). The increased current transfer length for Nb3Sn compared with NbTi is due to the lower values of n (i.e. 20 for Nb3Sn compared to 40 for NbTi), and the large resistivity of the matrix (bronze for Nb3Sn compared to pure copper for NbTi). Unfortunately, as Pauli is famously quoted as saying, ‘God made the bulk; surfaces were invented by the devil’ and Equation (G2.4.3) must be considered a minimum distance. In BiSCCO 2212 and 2223, the transfer lengths are typically a millimetre at 4.2 K, and tenths of millimetres at 77 K (Polak et al., 1997). They have an unexpected temperature dependence which is attributed to a large boundary resistance at the interface between the superconductor and the silver sheath. In RE-123 tapes, detailed measurements have shown that the interfacial resistivity between the superconductor and the silver layers is 25 nΩ cm2 (Tsui et al., 2016). The current transfer lengths directly affect the choice of sample geometry. Samples are typically no longer than 35 mm in the short straight geometry. This means that measurements on conductors with high resistivity matrix (e.g. bronze route Nb3Sn) are best not made using the short sample geometry. However, because the current transfer is not an intrinsic property of the wire, the resistive voltage can be subtracted from the V–I trace (Ekin, 1989). Note that when testing conductors with twisted filaments (which is required in commercial conductors to minimise ac losses), the current contact length should be greater than the twist pitch of the sample to allow the current to enter the filaments evenly (Goodrich et al., 1982). The second current transfer process is interfilamentary and occurs along the entire length of the conductor. It is generally due to a distribution in I c. This process depends on the detailed structure and materials in the composite conductor, including the size and distribution of the filaments within the conductor matrix and is an intrinsic property of the conductor. It is particularly important for ac applications where currents can redistribute between the filaments. In dc applications, a quantitative picture of the dissipative state in inhomogeneous

high J c conductors depends on the E-field range under investigation. At low E-fields, for example, in NMR (or persistent mode) applications, the superconducting filaments may have a resistance that is still much smaller than that of the matrix. In this case, the V–I transition is largely unaffected by the matrix and the current transfer is not important. In high E-fields, sausaging of the filaments or inhomogeneities may mean the local I c in a filament is exceeded, current passes through the matrix, either back into the sausaged filament or into another filament (Polak et al., 1997). G2.4.2.1.9 Current Source and Voltmeter A current source that can smoothly ramp current up and down and a nanovoltmeter are typically used under computer control for making J c measurements. The current through the sample is usually recorded by measuring the voltage drop across a standard resistor, and the voltage across the sample by the nanovoltmeter, as shown in Figure G2.4.3. The sensitivity and response time of the nanovoltmeter is dependent on the quality of the instrument and the filters used. Voltages at about 100 nV can be measured to a few percent over an interval of around 50 ms [c.f. Equation (G2.4.1)]. Standard good practice must be observed when using filters to reduce the voltage noise on a V–I characteristic. In general, the greater the filter time used, the larger the time delay before the correct voltage reading is reached. If the response time of the filtering mode is too long, the apparent voltage will be less than the actual voltage, and the V–I transition will artificially broaden giving a false increased value for I c (and a decreased n-value – c.f. Section G2.4.4). Measurements below the 10 nV range are possible in low-noise systems with long measurement times. Noise levels can be improved if outlying points are removed and numerical smoothing implemented (Goodrich and Srivastava, 1990). The current source itself can be a source of voltage noise. For example, it has been found that a battery power supply gave a noise level during the measurement of ~2 nV, whereas a silicon controlled rectifier gave ~100 nV (Goodrich and Fickett, 1982). In the most widely used measurement technique, the current is simply ramped at a constant rate until the required voltage is generated across the sample. The measurement should be sufficiently slow that the V–I trace does not depend on the rate of increase of current. The rate of increase of current must be small enough that the generated inductive voltages vary by less than the voltage used to determine J c and/or sufficiently smooth enough that inductive voltages can be subtracted off. Other factors that limit the ramp rate or the current are possible sample movement and induced eddy currents in metallic sample holders and probe components which cause heating. In the stepped method technique, point-wise data are taken with a delay to allow the inductive voltage to decay. A third method is to ramp quickly to ~0.9I c, wait for the inductive voltage to decay and then sweep the current very slowly through the I c transition. Pulsed techniques are also used (c.f. Section G2.4.4), and at high E-fields, typically agree with dc methods

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to better than ~0.2% (Goodrich, 1991) but usually with higher noise levels. A calibrated superconducting simulator is available to assess experimental procedure (Goodrich et al., 1995). Most experimental arrangements also incorporate some form of quench protection device. This is particularly important when using unstabilised samples where there is a significant risk of burning out the sample. The protection automatically resets the current to zero when a predetermined voltage across the sample is reached. Both computer-controlled and free-standing independent analogue components have been used to provide quench protection (Goodrich and Fickett, 1982). G2.4.2.1.10 Magnetic Field The magnetic field applied to the sample during critical current measurements is often provided by using a superconducting magnet. Dedicated University and National Laboratories often include vertical solenoidal magnets producing fields of up to ~21.5 T (Jones et al., 2001) and horizontal Helmholtz split-pair magnets up to 15 T (Sunwong et al., 2014). In the VAMAS Nb3Sn project, it was recommended that the field for testing conductors should be accurate to 1% and have a precision of 0.5%; the random deviation of magnetic field should be less than 0.5% and its homogeneity should be of uniformity ±1% over the length of the sample between the voltage taps (VAMAS, 1995). Above ~1 T, high-field superconducting magnets exhibit an almost linear dependence between the field generated and the current through the magnet. At low fields, however, hysteresis can cause problems when trying to determine the lowfield properties of conductors. There are various approaches that can be taken to eliminate errors due to the remnant field. The field at the sample can be measured independently using a Hall or NMR probe. The remnant field can be reduced to typically less than ± 20 mT by degaussing the magnet. This involves sweeping the field from a high value through zero and back, reducing the amplitude and sweep rate at each reversal. An alternative is to initiate a controlled quench in the magnet. This can be done by means of a carefully designed quench heater (Clark and Jones, 1986) incorporated within the turns of the magnet, which forces the temperature of the superconducting windings above Tc. This completely destroys the remnant field. However, the dangers of damaging the magnet by quenching should be noted. At international high-field laboratories, dc fields of up to ~45 T are produced by high power resistive magnets (Miller, 2003; Pugnat et al., 2014). These conventional magnets in principle have no remnant field, although in practice the structural steel in these systems is magnetised. The water cooling and power required for this design of magnet are large, which causes mechanical vibrations in the magnet system and additional sources of voltage noise. Wholly superconducting HTS and LTS magnets have been produced. HTS magnets operating up to 26.4 T have now been produced (Yoon et al., 2016; Awaji et al., 2017) as well as many insert systems such as the 35.4 T

high-field magnet produced by a 4.4 T RE-123 insert inside the bore of a 31-T LTS magnet (Trociewitz et al., 2011) and 14.4 T RE-123 insert inside a 31.1-T resistive background magnet to obtain a dc magnetic field of 45.5 T (Hahn et al., 2019).

G2.4.3 Critical Current Density – Parameterisation and Measurement The basic process that leads to an electric field being generated along the superconductor when the critical current density is flowing, is well understood. In a superconductor, the current density is confined to the superconductor and there is a Lorentz force between the flowing current density and the fluxons. As the current density through the wire is increased, the Lorentz force increases until it is sufficiently large to unpin the fluxons. When the fluxons move, they generate an electric field in agreement with Faraday’s law (Bardeen and Stephen, 1965). Early experimental and theoretical work found universal scaling laws for the volume pinning force density that are still used to parameterise J c in technological low-temperature superconductors given by n

* Bc2 q FP = J c B = A m b p (1 − b ) , * κ1

(G2.4.4)

* Where b = B / Bc2 (T , ε ), κ1* (T , ε ) is the effective Ginzburg– * Landau parameter (Keys and Hampshire, 2003), and BC2 (T , ε ) is the effective upper critical field. The compilation of different pinning mechanisms that Dew-Hughes (1974) produced has helped shape the use of Equation (G2.4.4). He found that for a wide range of model pinning systems, consistent with general dimensionality arguments that consider the Ginzburg– Landau free-energy functional, that m = 2 or greater and n, p and q have integral or half-integral values. Experimental work provided broad confirmation of both temperature scaling and strain scaling predicted by Equation (G2.4.4), although initially with the non-physical result that n was different for data taken at fixed temperature and different strain values (n ~ 1 for strain scaling) (Ekin, 1980) compared to data taken at fixed strain for different temperatures (n ~ 2 for temperature scaling) (Hampshire, 1974). These differences in the exponent n were resolved with variable strain and variable temperature measurements on a single wire. They provided a unified scaling law consistent with Equation (G2.4.4) when the strain and temperature dependence of the Ginzburg–Landau parameter (Hampshire et al., 1985) were included (Cheggour and Hampshire, 1999; Greenwood et al., 2019a). There is now such a vast amount of J c data on low-temperature superconductors that are consistent with Equation (G2.4.4), that the language of ‘flux pinning’ has continued to be used to explain critical current density in systems using the very useful early ideas, from many different authors

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(Campbell et al., 1968; Dew-Hughes, 1974, 2001), of fluxons ripped out of free-energy wells or pinning sites. However, they do not provide a very accurate description of the dissipative state. For example, later work, notably by Kramer, included the role of fluxon–fluxon interactions that occur in the fluxline lattice, ranging from his work on flux shear models in an almost perfect flux-line lattice (Kramer, 1973, 1975) to collective pinning models with good short- and intermediaterange order in the flux-line lattice but no long-range order (Larkin and Ovchinnikov, 1984; Feigel’man et al., 1989). The important ‘pinning’ in such models then becomes associated with the strength or elastic constants of the flux-line lattice (Hampshire and Jones, 1987c), rather than a straightforward feature of the microstructure. This work highlighted what has been called the ‘grand summation problem’, which considers the difficult issue of how to sum the forces in the flux-line lattice correctly to obtain J c (Antesberger and Ullmaier, 1975). It raises questions such as whether one should consider the fluxons to be in a broadly perfect hexagonal lattice or in an amorphous structure (Kleiner et al., 1964) and what the nature of fluxon movement after depinning is. Computational solutions to the time-dependent Ginzburg–Landau (TDGL) equations for polycrystalline materials suggest that flux flow at criticality in polycrystalline materials is complex. There are broadly two distinct regions – grain boundaries which flux most easily penetrate and that provide channels along which flux traverses the superconductor at criticality and the interior of the grains where flux broadly does not move at criticality (Carty and Hampshire, 2008). Recent work has provided a useful analytic framework to describe flux movement along grain boundaries by providing high-field solutions for J c in Josephson junctions which can be considered as the basic building-block to describe grain boundaries (Blair and Hampshire, 2019). It is difficult to find a simple accurate functional form to parameterise J c in high-temperature superconductors, not least because of the anisotropy of these materials, the much larger effect of compositional variations on superconducting properties and the complexity of the microstructures that are being used to produce high J c (Carty and Hampshire, 2013; Sunwong et al., 2013). In high-field superconductors, where the coherence length can be a few nanometres, we need to understand changes at the atomic scale. These are unsolved and challenging problems even for low-temperature superconductors, where historically there has been a consensus that BCS theory describes the fundamental pairing mechanism (Bardeen et al., 1957), although recently, this has been questioned (Branch et al., 2019). For high-temperature superconductors, where there is no agreement about the origin of the superconductivity, the deficiencies in our understanding are even more severe since we do not know how the pinning sites (e.g. grain boundaries, inclusions and precipitates) affect the local superconducting properties (Uemura et al, 1989; Uemura et al., 1991). Nevertheless, despite the huge gaps in our understanding of J c , the community is firmly committed to producing, not just descriptive, but also predictive tools

Handbook of Superconductivity

that relate the structure and microstructure to the critical current density. This is not least because Ginzburg–Landau theory reminds us that J c, even in state-of-the-art high-field technological superconductors, is still typically more than two orders of magnitude below the theoretical limit at 0.5Bc2 as shown in Figure G2.4.8 (Wang et al., 2017). One may expect that this huge headroom will only close as we move away from polycrystalline superconductors towards single-crystal materials with strong pinning, as has been so effective in increasing J c in RE-123 materials (Hazelton, 2013; Wang et al., 2017). Fortunately, the technological development of superconductivity has progressed successfully even with the current (relatively low) values of J c and has not prevented the development of a multibillion dollar industry (Conectus, 2001).

G2.4.3.1 Defining the Critical Current (Ic) and the Critical Density (J c ) A generic voltage–current (V–I) characteristic is shown in Figure G2.4.9 along with the various conventions used to define the critical current. The choice of convention depends on how the data are to be used. For example, in standard highfield solenoids the engineering current density at an electric field criterion of about 10 μV m−1 will determine the performance of the magnet. For NMR applications, on the other hand, where the magnet is in persistent mode, the current density at an E-field about six orders of magnitude lower is required. The critical current is consequently dependent on the chosen criterion (Goodrich and Fickett, 1982). The most commonly used convention for defining critical current (I c) is an electric field criterion (Ec) given by Ec = V / L, where V is the voltage difference between the voltage taps, and L is the length of wire between the voltage taps. Typically, the electric field used to define I c is 10 μV m−1 or 100 μV m−1. I c never falls to zero, using such criteria. A resistive (ohmic) sample which has no curvature on the V–I trace will still cross the electric field criteria at some finite current, leading to a value of I c. For typical conductors (~1 mm diameter), this nonsuperconducting current (for example in the Cu that stabilises the conductor) ranges from 10 to 500 mA when the E-field is 10 μV m−1, which is not significant in engineering applications. It can be important for more fundamental studies where one is trying, for example, to measure the upper critical field, since low current densities obtained using this criterion can be characteristic of non-superconducting component metals. In the case of short straight geometry where the measurement distance is typically no longer than 1 cm, a voltage sensitivity of less than 100 nV is required to obtain an electric field criterion of 10 μV m−1, so often 100 μV m−1 or even a 1000 μV m−1 electric field criterion is used. The resistivity criterion (ρc ) is given by ρc = VA / IL , where V is the voltage difference between the voltage taps, L is the length of the wire between the voltage taps, A is the crosssectional area of the wire and I is the current through the wire, is also used. Values of J c are often quoted at ρc of

Characterisation of the Transport Critical Current Density for Conductor Applications

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FIGURE G2.4.8 (a) Typical critical current density values in the superconducting layer itself for the most important high-field superconductors. (b) Typical critical current density values normalised by the depairing current density at 4.2 K and zero magnetic field, for the most important high-field superconductors. Closed symbols denote parallel and open symbols perpendicular to the sample plane. The solid line without data points in the lower panel gives the field dependence of the depairing current derived from Ginzburg–Landau theory (Wang et al., 2017).

10 −14 Ω m or 10 −13 Ω m. The critical current using the resistivity criteria is eventually zero in high enough fields as long as the criterion chosen is less than the normal-state resistivity of the composite conductor. However, problems can also arise in fundamental studies if the resistivity criterion is used. For example, some superconductors have some parts of their superconducting field-temperature phase space, where flux creep dominates and the critical current is zero, showing a low but non-zero resistance region in high fields. If the (superconducting) flux flow region has a resistivity above that of the resistivity criterion, I c is zero but the conductor remains in the superconducting state (i.e. there are still paired superelectrons). If the resistivity of the sample

is below that of the criterion chosen, the analysis suggests a critical current is present although the sample is resistive. A power criterion (Pc ) defined by Pc = IV is sometimes used in large-scale applications. It can be a useful criterion in magnet design to specify the maximum allowed consumption of cryogen and thus a suitable working current. The offset method (Ekin, 1989) is a method for calculating I c that attempts to minimise the problems associated with the electric field or resistivity criterion. I c is calculated in two stages. A tangent to the V–I curve is constructed at an electric field criterion given by J c . The critical current is then defined as the current at which the tangent is extrapolated to zero voltage. This procedure provides an attempt to subtract

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in the processing of the conductor and cannot be removed. Hence, the so called ‘non-copper critical current density’, Jc, Non-Cu, can be calculated using the area of the superconductor that is not matrix material (i.e. non-Cu or non-Ag). This provides the second definition of J c and is particularly useful for wire manufacturers. In fundamental flux pinning studies, the critical current density in the superconducting layer alone is required. Measuring the cross-sectional area in NbTi or BiSCCO is relatively straightforward. However, if there are fine superconducting filaments, as in Nb3Sn or Nb3Al conductors, it can require electron microscopy or post-measurement etching techniques to determine the cross-sectional area of the superconducting layers alone (Raine et al., 2019).

G2.4.4 Voltage–Current Characteristics – Parameterisation and Measurement G2.4.4.1 Shape of the Transition – Flux Flow, Flux Creep and Damage

FIGURE G2.4.9 The upper panel shows the conventions used to define Ic – the electric field, resistivity and power criteria as well as the offset method. The lower panel shows d 2V/dI2. The dotted line is for the conductor and shunt (Willen et al., 1997). The solid line is for the superconductor alone and can be equated to the distribution in Ic.

the current flowing in the non-superconducting components from the total current which can be useful for samples where there is a large shunt in parallel with the superconductor. We do not recommend using the offset method unless it cannot be avoided. The critical current density can be calculated in three ways once the critical current has been measured. For engineering applications, the conductor is treated as a single entity. The V–I characteristic for the entire conductor, including the stabilising material and the matrix, is required. The engineering critical current density (J c,ENG ) is defined as the critical current divided by the cross-sectional area of the entire conductor (Chislett-McDonald et al., 2019). This is the important parameter when designing systems such as magnets. The second definition of current density is relevant for comparing and developing conductors of different superconducting materials. It can be seen in Figure G2.4.2 that, in contrast to NbTi and BiSCCO, Nb3Sn conductor has matrix material such as (tin depleted) bronze, which is neither superconducting nor contributes to stabilising the conductor. However, it is required

In this section, the challenges interpreting the shape of the V–I traces are discussed. Poor experimental technique can produce artefacts in the data which can be very misleading, particularly when measuring brittle superconductors. Figure G2.4.10 shows five V–I characteristics that will be discussed in this context. These data are schematic representations of actual characteristics with their important features enhanced and can be explained as follows: Figure G2.4.10(a) There is a resistive (current transfer) region at low currents due to an insufficient separation between the current leads and the voltage taps. The transition to the normal conducting

FIGURE G2.4.10 Schematic V–I characteristics illustrating (a) current transfer, (b) flux creep, (c) a thermal voltage followed by current transfer, (d) flux flow and (e) thermal runaway.

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Characterisation of the Transport Critical Current Density for Conductor Applications

state is evident at high currents [Figure G2.4.10(b)]. Flux creep is evident at low currents. At high currents, the transition rises sharply as flux flow sets in [Figure G2.4.10(c)]. At low currents, a negative thermal voltage is developed. This can occur if there is excessive heating at one end of the sample where the current is injected and can cause a significant temperature gradient between the voltage taps. At intermediate currents, the voltage can become net positive due to resistive voltage associated with current transfer in and out of a limited number of damaged filaments. At high currents, the conductor is in the flux flow state [Figure G2.4.10(d)]. There is a zero-resistance region followed by a flux flow transition [Figure G2.4.10(e)]. There is zero resistance until a quench occurs in the conductor. This is followed by heating and thermal runaway leading to an artificially high n-value (c.f. below). The explanations provided for the curves (a)–(e) are clearly not unique. If the effects of heating due to filament damage or tunnelling across cracks are introduced, for example, one can relatively easily provide a different explanation for any of the V–I traces. It is sometimes useful to reverse the direction of current flow and remeasure the sample, because one can distinguish resistive voltages that are current polarity dependent, from thermal voltages that are not. However, one has to be careful that the reversal of the Lorentz force that occurs when the current polarity is reversed does not damage the (unsupported) sample. The most important point the reader should note here from these data is that one can misinterpret curves (a), (c) and (e), which do not show the intrinsic properties of the conductor, for curves (b) and (d) which are intrinsic. From the V–I trace alone, it may not be possible to reliably distinguish between flux creep (intrinsic to the conductor) and current transfer (and heating), which can occur because of damage or poor measuring technique. It is concluded that it is essential to employ good practice in mounting samples and is preferable that more than one length of the conductor is measured to ensure the reliability of the results. The shape or functional form of the V–I (or E–J) characteristic of a conductor is a complex issue that depends on the details of how the flux is pinned in the superconductor. For technological applications, one uses an empirical parameterisation of the form E = αJ n ,

the shunt may only affect J c slightly (depending on the criterion used), the value of n can decrease significantly (Itoh et al., 1996). High n-values tend to signify more homogeneous superconductors. Typical values for n are between 10 and 100 and in engineering applications tend to be used as a figure of merit. The n-value characterises the sharpness of the E–J transition in technological superconductors (Hampshire and Jones, 1985; Hampshire and Jones, 1987a; Bruzzone, 2004); the sharper the transition, the larger the n-value (Warnes and Larbalestier, 1986b). The origin of the n-value in superconducting wires can be attributed to the distributions in the critical current and the flux flow resistivity within the filaments (Baixeras and Fournet, 1967; Hampshire and Jones, 1985; Warnes and Larbalestier, 1986b; Warnes and Larbalestier, 1986a; Hampshire and Jones, 1987a; Warnes, 1988; Edelman and Larbalestier, 1993; Wördenweber, 1998). In some simple cases, non-uniformity of the filaments can be the most important factor that determines the n-value (Taylor et al., 2002), in others intrinsic effects are important (Wördenweber, 1998). Experimentally, we have found that the n-value approaches 1 as I c tends to zero for all strands measured – and that this occurs for values of resistivity that are far below the normalstate resistivity of either the superconducting filaments or even the copper stabiliser in the strands. This result together with the similar inverted quasi-parabolic strain dependence observed for both the n-value and J c provided the motivation to describe n empirically using (Taylor and Hampshire, 2005a) (G2.4.7)

n − 1 = rJ C s ,

where r and s are constants, and s is typically 0.4 (Taylor and Hampshire, 2005a; Lu et al., 2008; Lu and Hampshire, 2009, 2010). However, we have found that strands with almost identical critical current density can have very different n-values, which suggest that J c may not always be uniquely correlated with n but may also depend on how the current leaves and re-enters a filament to by-pass a region of low J c . We conclude that detailed understanding of the connectivity between the superconducting regions and the low resistivity normal regions is still required to provide further insight into n-values.

(G2.4.5)

G2.4.4.2 The Functional Form of the E–J Characteristic

or less frequently J − J0 E = E0 exp , J E

(G2.4.6)

where α, E0, n, J 0 and J E are all experimental or materials constants. The n-value is often called the order of transition or index. It is important to note that n characterises the entire conducting path between the voltage taps, including the stabilising material. If the conductor is mounted on a metallic (low resistance) sample holder, although current sharing through

For conductors at low temperature, where thermal activation is not important and all the fluxons are moving across the sample or a fixed fraction of the flux is moving across the sample, the E–J transition of a superconductor can be described by (Baixeras and Fournet, 1967) J

E( J ) = ρ

∫ ( J − J ) f ( J )dJ , i

o

i

i

(G2.4.8)

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where J i is the local critical current, f ( J i ) is the distribution of critical currents in the sample and ρ is the resistivity describing flux flow in regions of the superconductor where the current exceeds the local critical current. In the analysis, the current density terms represent the current density flowing in the superconductor and therefore, current flowing in the shunt and the stabilising material of the conductor must be subtracted to obtain the correct f ( J i ) function (Willen et al., 1997). This has been done by measuring the V–I characteristic above Bc2(T) (Hampshire and Jones, 1985). For high-temperature superconductors (HTS), this can be achieved by etching off the matrix (Willen et al., 1997; Cai et al., 1998). By invoking the central limit theorem, it can be assumed that f ( J i ) can be described by a normal distribution of the form f ( Ji ) =

1 J − J 2 1 β exp − β i c , 2 Jc 2π J c

(G2.4.9)

where J c is the average J c of the distribution, σ ( J i ) is the standard deviation of the critical current distribution and β = J C / σ ( J i ) (Baixeras and Fournet, 1967; Hampshire and Jones, 1985). Good agreement is found between experimental data and Equations (G2.4.8) and (G2.4.9). Detailed variable temperature, variable field measurements have been completed on NbTi (Hampshire and Jones, 1985; Hampshire and Jones, 1987a), Nb3Sn (Hampshire and Jones, 1987b) and V3Ga (Hampshire et al., 1989) conductors and scaling laws found for ρ, β, n and J c . Equating the empirical equation [Equation (G2.4.5)] to the more physical one [Equation (G2.4.9)] at J = J c (Hampshire and Jones, 1987a) gives 1

2 2 β = n . π

(G2.4.10)

This equation is consistent with the empirical finding that homogeneous materials have high values of n (it is not valid for very low n values [n < 5] at low E-fields (Edelman and Larbalestier, 1993; Ryan, 1997). The distribution of critical current densities can be derived explicitly making no assumption about the form of the distribution using (Warnes and Larbalestier, 1986b; Edelman and Larbalestier, 1993) 1 d2E f ( Ji ) = 2 . ρ dJ

(G2.4.11)

A graphical representation of this analysis is shown in Figure G2.4.9. However, the distribution obtained is sensitive to noise and the algorithm used to calculate the second derivative (Goodrich et al., 1992). It is difficult to measure the distribution at high current because of heating and the local hot spots that occur at high E-field values. This is particularly problematic if the stabilising material is etched out of the conductor (Willen et al., 1997).

Generalised scaling laws have been developed (Yamafuji and Kiss, 1997) for superconductors operating at high temperatures, where thermal activation and flux pinning can both operate. In some special cases, detailed E–J characteristics have been calculated, including the formulism for the vortex glass–vortex liquid phase transition in weak-pinning highTc superconductors. In general, any realistic mathematical framework is more complex at high temperatures than at low temperatures and a distribution in Tc and Bc2 (T ) will also play a role. Scaling in the E–J characteristics has been observed in both low-temperature (Cheggour and Hampshire, 1997) and high-temperature superconductors (Yamafuji and Kiss, 1997). Hence, for all superconductors, for technological applications, we can use the empirical approach described using Equations (G2.4.5) and (G2.4.7). If one wants to understand and parameterise the underlying mechanism, at low fields and temperatures, one often finds that J c is high and thermal activation may play little role. In these cases, sausaging in the conductor filaments, variations in microstructure and composition can be important and one can expect the resultant distribution in J c to be described using Equation (G2.4.9). Alternatively, at the highest fields and temperatures, the distribution in fundamental properties, thermal activation, percolation and regions of very low J c may dominate the properties of the conductor. In this case, the phase transition formalism is probably most appropriate. These two formalisms are still both being developed to better understand and optimise the distribution and magnitude of J c in conductors.

G2.4.4.3 Magnetic Field and Temperature Corrections to I c and n-Values In addition to the applied field, there are magnetic field corrections associated with the self-field generated by the sample, by the return current leads, as well as a field correction ( ΔBoffset ) if the voltage taps are not in field centre on the axis of the magnet bore (associated with the inhomogeneity of the magnet). To first order, the self-field contribution from a straight conductor with current flowing orthogonal to the direction of the applied field is antisymmetric and hence in high fields averages to zero. The self-field correction orthogonal to the direction of the applied field, for example, from the return current in the probe, must be added in quadrature to the applied field, so is also rather small. Hence usually, the difference between the average magnetic field along the wire between the voltage taps and the nominal applied field ( ΔBaverage ) is usually the most important field correction required for technological measurements. Thereafter one can, for example, use Equation (G2.4.4) (or those given later in this article) where the values of I1c ∂∂IBc are calculated as a function of field from the scaling

( )

T

law, and the field correction to I c is then ∂I ∂ I c ( B ) = c ( ΔBaverage ). ∂B T

(G2.4.12)

Characterisation of the Transport Critical Current Density for Conductor Applications

For the field correction to the n-values [∂n ( B )], the value of ∂n ∂ Ic is taken from Equation (G2.4.5) and the correction for the n-value is given by ∂n ∂n ( B ) = ( ∂ I c ( B )) , ∂ I c T

(G2.4.13)

Temperature corrections can also be required, associated with the helium bath being at a different temperature ( ΔT ) to ‘standard boiling point of helium’. The temperature corrections to I c and n are simply obtained from Equations (G2.4.12) and (G2.4.13) after the variables B and T are cycled.

G2.4.5 Case Studies for Testing Different Materials This section has two parts. The first part surveys some of the most common/important types of measurements reported in the literature. The second part considers specific issues associated with measuring NbTi, Nb3Sn, BiSCCO and RE-123 conductors. All of these materials either have been or are currently part of international collaborative studies for the standardisation of measurement techniques. Know-how, mistakes and typical errors found when measuring each material are also discussed.

G2.4.5.1 Different Types of Measurement – I c as a Function of Field in Liquid Cryogens – J c ( B) Techniques Very accurate and stable temperature control can be achieved when making critical current measurements by direct immersion of the conductor in a liquid cryogen, which makes this a popular approach for standard fixed-temperature measurements. However, there are some general points to be aware of when making such measurements. The vapour pressure of the gas above the cryogenic liquid can be monitored to give the temperature of the bath as suggested by the VAMAS Technical Working Party in their standard method for I c determination of Nb3Sn wires (VAMAS, 1995). Bath temperatures can be determined from standard tables (Press, 2012). In Durham, we have found that UK Government barometric data for weather predictions from local weather stations provide a reliable source of pressure measurements (Varley, 2016). Temperature errors can occur when using liquid nitrogen if air (primarily oxygen) has dissolved in the nitrogen due to associated changes in the vapour pressure – temperature relation. If atmospheric pressure is used, care must be taken that there is not a build-up of pressure in the Dewar as the critical current transition is reached. Indeed, for very accurate measurements, we use bubblers rather than mechanical valves for helium exhaust as shown in Figure G2.4.3. It is prudent to monitor the temperature of the sample during the V–I measurement to ensure there is no heating. The type of thermometer used should be chosen

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based on its properties in high magnetic field (although magnetoresistance is not a concern per se because we are looking for a constant reading during the measurement rather than absolute values), its sensitivity and its response time. In the coil geometry, metal (Ti-alloy (Ridgeon et al., 2017) and Cu-Be (Cheggour and Hampshire, 2000)) and glass fibre reinforced plastics/epoxy (FRP) are commonly used for the sample holder (c.f. Figure G2.4.6). It is common practice that the wire is wound on to a spirally grooved sample holder to help prevent it from moving during the measurement; the spiral groove should be at an angle of about 7° to reduce the effects of placing the sample at an angle to the applied magnetic field whilst ensuring the self-field effects are not too large; copper ends are usually fitted to the sample holder and act as current contacts to which the ends of the sample are soldered and the direction of the Lorentz force is inwards. The groove should preferably be of V-cross-section with a depth approximately equal to the diameter of the wire although thin-walled sample holders should also be grooved to guide the wire (VAMAS, 1995). An important consideration when using metallic sample holders is the current sharing that occurs in the dissipative state. Current sharing can produce a large reduction in the n-value particularly if a solder bond is used (Itoh et al., 1996). For standard testing of commercial wires for engineering purposes, current sharing is best avoided by using either Ti-alloy or FRP. However, although the technologically important J c,ENG may not be accurately measurable on soldered metallic sample holders, in more fundamental studies that require J c for the superconducting layer alone, even the current sharing through the stabilising matrix within the conductor itself should be subtracted from the measured data. In such experiments, one can use soldered metallic sample holders and treat current sharing through the normal shunt and stabilising material as a single path in parallel with the superconductor. Indeed, in such studies it has been found useful to electroplate a very thin copper layer onto the Ti-alloy sample holder to ease the soldering process, prevent unstabilised conductors from burning out and ensure that localised damaged sections of the wire do not quench the entire conductor during testing. However, electroplating Ti-alloys usually includes using HF acid, which requires additional health and safety considerations (Muriale et al., 1996). The FRP sample holders should be machined from plate stock so the axis of the tube/cylinder is along the normal direction of the FRP plate. It is less preferred to use rolled FRP tubes with the axis of the tube in ‘fill’ direction because the thermal contraction is more anisotropic than with machined plate and depends on the dimensions on the tube (Goodrich et al., 1990). When using solder, in particular, for attaching voltage taps, it is important not to thermally shock the conductor. For brittle superconductors, a hot-air gun may be used to warm the conductor uniformly to just below the melting point of the solder before using the soldering iron to attach leads. The temperature of the gun can be set using a thermocouple-based thermometer. Low-temperature solder such as In0.52Sn0.48 (M.P. ~ 118°C)

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can be used to minimise thermal shock (Wiejaczka and Goodrich, 1997; Tsui et al., 2016). It should also be noted that standard PbSn solder (M.P. ~ 190°C) is superconducting at 4.2 K up to ~ 0.2 T (Tsui et al., 2016). In low fields, I c can be strongly dependent on whether the applied field has been increased or decreased to obtain the required value, i.e. I c is history dependent (Küpfer and Gey, 1977; Marti et al., 1997; Goodrich, 1999/2000). One way to mitigate this problem is to reproduce the magnetic behaviour of the application the sample is intended to be used in. If this is not possible, then the dependence of I c on magnetic field sweep rate and direction should be measured. In general, one tries to match the thermal contraction of the sample holder to that of the conductor. The properties of any conductor are determined by all its component parts. The sample holders suggested by the standards testing community have been chosen as appropriate for a reasonable range of the common commercial conductors. G2.4.5.1.1 J c ( B,T ) Experimental Techniques The variation of critical current as a function of both field and temperature is critical in assessing conductors for cryogen-free applications and for many large-scale systems for which forced flow helium is used. Over limited temperature ranges, vapour pressure thermometry is best used. With liquid helium, it is possible to vary the temperature between ~ 1.8 K and 5.22 K and with nitrogen from about 55 K to about 85 K. The temperature of the liquid is varied by either decreasing (pumping) or increasing (pressurising) the vapour pressure above the liquid. Techniques have also been developed to measure conductors in an isothermal environment so the temperature can be varied continuously over any temperature range above 1.8 K (Frost et al., 1992; Friend and Hampshire, 1995). Typically, the sample is in intimate thermal contact with a copper thermal block that incorporates a heater and thermometry. A temperature controller maintains the temperature constant as the V–I characteristics are measured. Purpose built and commercial variable temperature cryostats can also provide the required isothermal environment. Both dc continuous and pulsed methods have been used to obtain the V–I characteristics (Goodrich et al., 1998; Kuroda et al., 1998). G2.4.5.1.2 J c ( B , T , ε ) Experimental Techniques Although the popular press, and funding agencies, usually consider room temperature superconductivity as the holy grail in applied superconductivity, it is very probable that a new LTS superconducting material (say Tc = 10 K) with high J c and high upper critical field (say J c = 104 A mm−2 at 4.2 K and 30 T) that is ductile would have a much larger impact on the development of new markets for superconducting applications (Chislett-McDonald et al., 2019). At present, all superconductors that are used to produce fields significantly above 10 T are brittle. Hence in these materials, the effect of applied strain

Handbook of Superconductivity

on critical current is of great interest, especially for high-field applications where the wires are subject to large Lorentz forces (Branch et al., 2019). Historically the short straight geometry (Ekin, 1980; Kamata et al., 1992) was first used to measure the effect of tensile uniaxial strain at 4.2 K on LTS. In order to apply compressive uniaxial strain the sample needs to be supported. Broadly there are two different types of bending springs used for this purpose (c.f. Figure G2.4.7). A U-shaped bending spring (in a vertical magnet system) has been used (ten Haken, 1994) to which a flat-bottomed type hairpin sample can be soldered. Compression or extension is applied to the spring by means of a force applied to its legs which either separates them or brings them together. Strain is measured by a strain gauge either mounted directly to the sample, or in close proximity on the spring. More recently, this design has been adapted for measurements on long straight samples in split-pair magnets in order to overcome the problems of current transfer (Fukutsuka et al., 1984) and to facilitate more accurate measurements on anisotropic materials. Alternatively, a helical bending spring can be used which accepts coil type samples (Walters et al., 1986). One end of the spring has a torque applied to it whilst the other is held fixed. This mechanism can apply either tension or compression to the sample. In general, variable temperature measurements are made by placing the bending spring in an isothermal environment (Cheggour and Hampshire, 1999; ten Haken et al., 1999). Measurements on a range of springs (bending beams) made of different materials have demonstrated that results can be obtained that are independent of the spring material (Taylor and Hampshire, 2005b). Several detailed variable-temperature, variable-strain measurements have been completed on a wide range of conductors including Nb3Sn (Ekin, 1980; Taylor and Hampshire, 2005c), Nb3Al (Takeuchi et al., 1997; Keys, 2001), PbMo6S8 (Goldacker et al., 1989), BiSCCO (Ekin et al., 1992; Goldacker et al., 1995; Richens et al., 1997; Sunwong et al., 2011) and RE-123 (Branch et al., 2019). Typical data are shown in Figure G2.4.11 (Keys et al., 2002). The effect of transverse stress and strain on critical current has also been measured (Ekin, 1987; Kamata et al., 1992; ten Kate et al., 1992; (ten Haken, 1994). In the earliest work varying stress, samples were placed between a fixed position pressure block and a movable pressure block. The stress is applied by compressing the sample. Although it is not possible to measure the strain in this configuration, it is possible to obtain equivalent stresses for uniaxial strains to allow direct comparison between measurements. The V–I characteristics are measured at a series of different stresses. It has been demonstrated that the effect of transverse stress is critically dependent on how the stress is applied. If the stress is localised, the effect on J c is far more marked than if the stress is uniformly applied. This can be seen by comparing the degradation of J c between round and flat conductors (Ekin, 1987; Jakob et al., 1991).There are some limited transverse strain measurements of critical current using samples being developed for this

Characterisation of the Transport Critical Current Density for Conductor Applications

197

series of pulsed measurements can be used to construct the superconducting V–I characteristics (Goodrich and Srivastava, 1992). Highly sensitive instruments with rapid response times are required for these measurements. An important advantage of the pulsed method is that if the current contacts are poor (highly resistive), they have limited effect on the measurement because there is not sufficient time during the pulse for the heat to diffuse to the part of the conductor between the voltage taps. Measurements have also been performed using pulsed magnetic fields (Hole, 1995). In this case, a constant current is applied to the sample, and the voltage generated is recorded. This method has the advantage of allowing measurements to be taken up to the highest fields (> 50 T). The rapid change in magnetic field, however, causes eddy current heating that can prevent accurate measurements on some samples. G2.4.5.1.4 Persistent-Mode Experiments The V–I characteristics at very low electric fields can be found by measuring the decay of the current in a persistent-mode coil (Ryan et al., 1997). A current is induced into a superconducting coil, and the decay of the self-field measured as a function of time. The decay can be converted into an equivalent resistance for the conductor. This is currently the only method which gives measurements on long lengths of conductor down to an electric field of a few pV m−1 relevant for NMR applications. Results obtained on NbTi and Nb3Sn are in broad agreement with a normal distribution of critical currents described using Equations (G2.4.9) and (G2.4.10).

G2.4.5.2 Measurements on Specific Materials FIGURE G2.4.11 The three-dimensional critical current surface as a function of magnetic field, temperature and strain: (a) Nb3Al strand (Keys et al., 2002) (b) Rare earth-123 (Branch et al., 2019).

purpose, as shown in Figure G2.4.7 (Greenwood et al., 2018; Greenwood et al., 2019b). Critical current measurements have also been performed as a function of stress to estimate the strength of a conductor. In these cases, the sample was loaded to a certain stress level and then unloaded to zero stress at room temperature. The conductor was then cooled to 4.2 K, and the critical current was measured. The variation of critical current as a function of repeated applied stress at room temperature has been measured and the strength distribution for the composite calculated (Ochiai et al., 1993). G2.4.5.1.3 Pulsed Methods for I c Measurements The most common pulsed method involves increasing the current from zero to a predetermined level in 1–10 ms, measuring the voltage drop along the conductor and ramping back to zero again. The voltage is also measured before and after the pulse to allow for the subtraction of any offset. A

In this second part, we first consider know-how accrued in the community that is relevant for measuring different types of materials. The approach for measuring a ductile superconductor such as NbTi can be different to that for measuring a brittle RE-123 HTS tape. For each material we also consider how we parameterise J c ( B,T , ε ) data. There are a number of different approaches in the literature. We have not chosen to try to outline them all here, but rather describe the most complete approach that we favour but also point to other literature. It is understandable that there are different approaches because there are competing issues; in Durham, we tend to analyse comprehensive J c ( B,T , ε ) datasets that include the full range of operation for the material. Such data are usually very hard-earned, so we tend to parameterise the data knowing that parameterisation will be interrogated using interpolation. Some of the data in the literature can be more limited, and perhaps to be used in computation. Then, quite reasonably, the authors are more concerned about parameterisations that include extrapolation (Bordini et al., 2013; Ekin et al., 2016). This field is important, dynamic and vibrant. We point to some of the open questions and research at the end.

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G2.4.5.2.1 NbTi NbTi is probably the most important technological superconductor. It is a ductile material with a high upper critical field and hence the material of choice for most MRI applications up to about 10 T. It is usually measured using the coil geometry. The insulation on most technological NbTi wires can be removed mechanically using emery paper or chemically using a mixture of phenol/methylene chloride to prevent mechanical damage during sample preparation. Both Ti-alloy and FRP are often used for the measurement barrel. In Durham, we routinely use Ti-alloy barrels for measuring NbTi as shown in Figure G2.4.6. The FRP used is G-11CR (US notation) or EP GC 203 (European notation) which has a thermal contraction similar to NbTi. If the sample is held in place using varnish, a strong epoxy (such as Stycast), vacuum grease or no bond at all, there is little effect on I c (Goodrich and Srivastava, 1995b). The critical current values for samples on a Ti-alloy sample holder are similar to values from samples mounted on G10 (Ogawa et al., 1996). There is now routinely excellent agreement between laboratories measuring NbTi. Critical currents were measured at 2, 4, 6 and 8 T on FRP for temperatures from 3.90 K to 4.24 K using electric field criteria from 5 to 20 μV m−1. The total uncertainty of the reported critical current values at any of the four magnetic fields was no greater than 2.6% (Goodrich et al., 1984). In subsequent series of measurements, the uncertainties for the critical currents (measured at an electric field criteria of 10 μV m−1) were found to be 1.71% and 1.97%, and the difference between the interlaboratory averages and certified critical currents were 0.2% and 0.3% for 6 and 8 T, respectively (Wada et al., 1995). In the IEC/TC90 experiments, the coefficient of variation for samples mounted on FRP (standard deviation divided by the average value) was lower than 2% for I c measurements from 1 to 7 T (Ogawa et al., 1996). It is also possible to use the NbTi strands used for ITER as an excellent cheap reference material, because there are large amounts of material available that have been extensively and accurately measured (Raine et al., 2019). There is only limited variable-strain J c data in the literature. It shows that NbTi is reasonably insensitive to strain (Goodrich et al., 1984; Chislett-McDonald et al., 2019). This insensitivity means that very accurate measurements will be required to characterise the strain dependence comprehensively that include proper account of the relatively small changes in Bc2 that occur for the different criteria used to define Bc2; geometrical changes in the sample on applying the strain; and the small anisotropy in Bc2 associated with the drawing of the wire during fabrication. The strain insensitivity also means that the Ginzburg–Landau parameter is rarely needed in the parameterisation of J c as a function of field and temperature [c.f. Equation (G2.4.4)] and is usually taken to be of the simple form n

* b p (1 − b ) , FP = A Bc2 q

(G2.4.14)

where p = 1 and q = 1 have long been used to characterise the high-field behaviour Friend and Hampshire (1993) but p = 0.52 and q = 0.63 have recently been proposed so that low-field data are also accurately included in the parameterisation (ChislettMcDonald and Hampshire, 2019), A is a constant, n ~ 2 and the effective upper critical field is taken to be of the form * Bc2 (T ) = Bc2* ( 0) 1 − t ν ,

(

)

(G2.4.15)

where t = T / Tc , T is the temperature, Tc = 9.45 K and for the current density orthogonal and parallel to the applied field, * Bc2 ( 0) and ν can be taken to be 15.8 T and 1.56 (orthogonal) and 14.9 T and 2.15 (parallel). However, for applications at * the very highest fields close to Bc2 , where the strain dependence of NbTi becomes important and for more physically meaningful values of n, one should probably use the parameterisation discussed below for intermetallic compounds (Chislett-McDonald et al., 2019). G2.4.5.2.2 Nb3Sn + Nb3Al The large sensitivity of Nb3Sn and Nb3Al to mechanical strain means that care must be taken when reacting and mounting samples to avoid degrading J c prior to measurement. In the past, Nb3Sn conductors were reacted on a stainless steel reaction mandrel (that had been oxidised), or sometimes a carbon reaction mandrel, of a geometry that closely matched the sample holder. The sample was then carefully transferred on to the sample holder taking care not to unduly strain the wire during this operation. A G10-CR (US notation) (EP GC 201: European notation) which (machined from plate) is similar to G-11CR (used for NbTi), but has a differential thermal contraction better matched to Nb3Sn was used (Kirchmayr et al., 1995). Other materials with similar thermal expansion properties to Nb3Sn conductors are non-magnetic stainless steels, copper or nonmagnetic copper alloys (see Figure G2.4.5) have also been used. As part of the VAMAS report, similar I c’s as a function of magnetic field were found using G10 and stainless steel sample holders. Vacuum grease was suggested as the bonding agent in the VAMAS test as it is easy to remove from the sample after use. Stycast (a high strength epoxy) was also tested and gave similar I c’s for stainless steel and G10-CR sample holders (Kirchmayr, 1995). The IEC/TC90 round robin test on Nb3Sn composite conductors advised explicitly against using a solder bond on the stainless steel. Alumina ceramic, like stainless steel, has also been used as a dual-purpose mandrel/sample holder, either by itself or as a coating on stainless steel (Goodrich and Srivastava, 1995b) because there was less chance of bonding between the holder and the conductor, although the thermal expansion is not very well matched (Wada et al., 1995). However, following the huge international effort directed at fabricating the toroidal field coils for ITER, a dedicated barrel based on a very resistive Ti-alloy, as shown in Figure G2.4.6, on which the sample can be both reacted and measured has eliminated the need to transfer the sample from

Characterisation of the Transport Critical Current Density for Conductor Applications

the reaction mandrel to the measurement barrel. As a result, I c measurements on Nb3Sn samples typically vary between laboratories by ~ 3%, and most of the variation is probably associated with variability in the temperature of the reaction process in the furnaces (Raine et al., 2019). In Durham, we use a version of the ITER-like barrel for almost all I c measurements at cryogenic temperatures. For both Nb3Sn and Nb3Al measurements, the surface of the reaction barrel is first heavily oxidised for 3 hours at 300°C in air, to prevent any diffusion bonding between it and the conductor. The wire is then wound on the barrel for the heat treatment during which brittle A15 compound is formed. The sample is soldered onto the copper rings and the measurements made. We have found that for both Nb3Sn (Keys and Hampshire, 2003; Taylor and Hampshire, 2005c) and Nb3Al (Keys et al., 2002), the volume pinning force density is very approximately given by 5/2

* Bc2 1/2 FP = J c B = A (1 − b )2 , 2b 1/2 ( 2πφ0 ) µ 0 κ1*

(G2.4.16)

where A is a dimensionless constant ~1/250 for Nb3Sn and ~1/100 for Nb3Al. A simple dimensionality energy argument suggests that the equation is reasonable. Accurate fits to J c ( B,T = 4,2 K ) and J c ( B,T ) data using Equation (G1.4.4) lead to values for n, m, p and q that are not always integral and half-integral. This can be attributed to the large distribution of fundamental parameters in this high J c technological material. Incorporating both microscopic (BCS) and phenomenological (GL) theory (Bardeen et al., 1957; Helfand and Werthamer, 1966; McMillan, 1968; Allen and Dynes, 1975; Orlando et al., 1979; Kresin et al., 1984; Kresin, 1987; Carbotte, 1990), considering the effects of strong coupling (Keys and Hampshire, 2003) and after some considerable care to look at the correlations been the free-parameters (Keys et al., 2002), we found that when parameterising J c data as a function of magnetic field, temperature and strain, Equation (G2.4.4) can be rewritten m

* J c ( B,T , ε I ) = A(ε I ) Tc* ( ε I ) 1 − t 2 Bc2 (T , εI )

n − m −1 p −1

b

[1 − b]q,

(G2.4.17) where the superconducting parameters are described using * Bc2 (T , εI ) = Bc2* ( 0, εI ) 1 − t ν , 1/u

A( εI ) A( 0 )

1/w

* Bc2 ( 0, εI ) = * Bc2 ( 0,0 )

(G2.4.18)

Tc* ( ε I ) = * , Tc ( 0 )

(G2.4.19)

* Bc2 ( 0, εI ) = 1 + c ε 2 + c ε3 + c ε 4 , 2 I 3 I 4 I * Bc2 ( 0,0 )

(G2.4.20)

and

199

where J c is the engineering critical current density, and ε I is the intrinsic strain, determined by the applied strain, ε A , and the applied strain at the peak in J c , ε M , according to (Rupp, 1977; Luhman et al., 1978; Markiewicz, 2004b) εI = ε A − ε M .

(G2.4.21)

By making measurements in magnetic fields up to 28 T in the Grenoble High Field Magnet Laboratory on three different Nb3Sn samples, so that the upper critical fields of the three strands were measured directly over much of the phase space, we concluded that one can fit J c ( B,T , ε ) data for many Nb3Sn strands using some universal parameters (n = 2.5, m = 2, ν = 1.5, w = 2.2, u = 0, c2 = −0.77462, c3 = −0.59345, c4 = −0.13925 ) and reduce the number of free parameters to 6, namely A ( 0 ) , * Tc* ( 0 ), Bc2 ( 0,0), p, q and ε M. More recently, Bordini has recom* mended an exponential dependence for Bc2 ( 0, εI ) of the form * Bc2 ( 0, εI ) = exp −Ca1 * Bc2 ( 0,0 )

(

J2 +3 J 2 +1

(

J 2 + exp −Ca1

)

2

I12 + 3 I12 +1

),

I12

(G2.4.22)

I1 = (1 − 2ν ) ε a + l0 + 2t0 and where Bordini gives 2 and t0 = −νl0 + 0.1, where J 2 = 13 (lo − t0 + (1 + ν ) ε a ) Poisson’s ratio, ν can be taken to be ν = 0.33, and l0 and t0 are the longitudinal and transverse residual strains in the strand (Bordini et al., 2013). The reader will find many different approaches to parameterising J c ( B,T , ε ) data in the literature (Ekin et al., 2016). In general, we use a fitting procedure that minimises the percentage error between the J c fits and the data. This has the advantage that the free parameter values obtained do not depend on whether one fits FP or J c and is broadly consistent with the errors in our experiment. Indeed, we recommend that when commercial packages are used to find the free parameters, one checks that they are not dependent on whether one fits FP or J c . Tabulations of J c ( B,T , ε ) and n-value data taken in Durham have been made available as well as the scaling-law parameterisation of J c ( B,T , ε ) and n ( B,T , ε ) for many Nb3Sn strands on the Internet (Durham-Superconductivity-Group). Alternative nine-parameter fits which, as with Bordini’s work [Equation (G2.4.22)], explicitly incorporate the three-dimensional nature of strain into the scaling law (ten Haken et al., 1995; Markiewicz, 2004a, b, 2008) have also been suggested – although the fits often include a 1/κ term rather than the 1/κ 2 found, for example, by Dew-Hughes (Dew-Hughes, 1974; Kramer, 1973, 1975). The polynomial fits using Equations (G2.4.17) to (G2.4.21) were motivated by finding the best fit to the data. They are useful for precise interpolation, but poor for extrapolation. Bordini’s Equation (G2.4.22) is better if there are only limited data available and one wishes to extrapolate the fits to beyond the strain range of the data. It also provides more physically meaningful parameters. Both approaches can be used for NbTi and A15 superconductors (ChislettMcDonald et al., 2019).

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FIGURE G2.4.12 (a) The strain dependence of the critical current density ( J c ) of a (RE)BCO tape with the field applied along the tape normal and the strain applied using a Cu–Be ‘springboard’ (Sunwong et al., 2014; Branch et al., 2019). (b) Equivalent data for a bronze route Nb3Sn wire where the strain was applied using a helical spring (Walters et al., 1986; Cheggour and Hampshire, 2000). For (1) different applied magnetic fields and (2) different temperatures are considered for REBCO (Greenwood et al., 2019a) and for Nb3Sn (Branch et al., 2019). The solid lines are parabolic fits to the data, and the arrows indicate the strain at which J c reaches its maximum value at each field and temperature. J c ( ε ) is only parabolic for the range of strains shown in the figure.

Characterisation of the Transport Critical Current Density for Conductor Applications

G2.4.5.2.3 Multifilamentary BiSCCO – 2212 and 2223 and RE-123 Because it is often not possible to wind high-temperature superconducting tapes into small coils for measurements in small-bore magnets without generating a large strain, most interlaboratory comparison measurements on high-Tc tapes use the short straight geometry (Wiejaczka and Goodrich, 1997). Some BiSCCO conductors are not fully dense. For the Ag-sheathed BiSCCO tapes and wires, it was observed that the thermal expansion was dependent on thermal cycling, which was attributed to yielding due to an internal stress between the Ag-sheath and the BiSCCO (2223) filaments (Yamada et al., 1998). Since cryogenic liquid can seep into the conductor during measurements, it is essential to warm the samples slowly after measurement so the cryogen can escape without blistering the silver matrix (Wiejaczka and Goodrich, 1997). Equally the RE-123 samples can delaminate. Hence, when measuring either BiSCCO or RE-123, one can mitigate against damage by encapsulating the sample. Brass, G10, Ti-alloy and CuBe have all been used as sample holders. However, it has been reported that BiSCCO samples are more likely to separate from the substrate when mounted on brass (due to differential thermal contraction) which can damage the samples. Hence, the samples can be bonded and encapsulated to a G10 sample holder using a glass-filled epoxy. One can also use low-temperature solder with copper-plated Ti-alloy or CuBe to encapsulate and bond the sample. For all sample holders, current contacts and voltage taps are soldered directly to the sample using a low-temperature solder. However, although some damage is likely caused by soldering, successful measurements on short samples have been made by soldering the current leads and using silver paint/epoxy for voltage taps (Sneary et al., 1999). For systems analysis, where a rough and simple approximation to the angular dependence is sufficient, we have found J c of HTS tapes can be roughly described as a function of field (B), temperature (T), angle of the field with respect to the tape (θ) and strain using (Lee et al., 2015) Bcosθ B J c ≈ α (T , ε ) 1 − exp − , Bc2 β (T )

(G2.4.23)

where the free parameters c (T ), and β (T ) can be taken as functions of temperature alone. Variable strain can also be

(

d

)

parameterised using α (T , ε ) = α (T ) 1 − c (T )( ε − ε 0 (T )) as demonstrated from fits to variable strain J c data from Sunwong et al. (2013) and Sugano et al. (2010). Recent work suggests that unlike the magnetic field and temperature dependence of J c , the strain dependence should be considered an emergent property because the overall strain dependence of a conductor can be different to its component parts. The evidence for weak emergence comes from finding that the field and temperature dependence of the strain at which J c reaches its peak value is not constant, but is a weak function of magnetic field and temperature, as shown in Figure G2.4.12 (Branch et al., 2019).

201

In round robin testing, the coefficient of variation for a BiSCCO sample premounted by a central laboratory (NIST) were 4.4% (77 K) and 3.2% (4.2 K) (Wiejaczka and Goodrich, 1997). The IEC/TC90 has developed a dc critical current test method for Ag-sheathed BiSCCO conductors IEC (2006a), and a public database of critical current data of HTS conductors is now available for magnet designers (Wimbush and Strickland, 2017).

G2.4.6 Concluding Remarks The development of reliable techniques for testing conductors is essential for underpinning a mature technology based on superconductivity and a better understanding of the underlying science. In light of the new complex composite conductors that are continuously being developed, the international community is committed to improving the testing of conductors. Domestic and international programmes for laboratory intercomparisons of mechanical, thermal and electromagnetic measurements are in progress. This review has provided an introduction to testing the current-carrying capacity of conductors. We hope that it provides a good introductory review for scientists new to such work and helps them with making reliable measurements.

Acknowledgements The authors wish to thank the U.K Engineering and Physical Sciences Research Council (EPSRC) and Siemens for their support. The authors thank Prof. K. Osamura and Dr. L. Goodrich for their help in providing literature for this article and members of the superconductivity group in Durham particularly A. Blair, P. Branch and J. Greenwood. Figure G2.4.2(a) and G2.4.2(b) are courtesy of Dr. B. Mendis and L. Bowen in the Durham University Microscopy Facility. Figure G2.4.2(e) is courtesy of SuperPower, Inc., a Furukawa Company, copyright 2015 (with adaptations by Durham). This work was funded by the RCUK Energy Programme under grant EP/I501045 and EP/L01663X/1. The data are available at: http://dx.doi. org/10.15128/r1pn89d656x

References Allen PB, Dynes RC (1975) Transition temperature of strongcoupled superconductors reanalyzed. Physical Review B 12:905−922. Antesberger G, Ullmaier H (1975) Experimental contribution to the summation problem of pinning forces in type-II superconductors. Physical Review Letters 35:59−62. Awaji S, Watanabe K, Oguro H, Miyazaki H, Hanai S, Tosaka T, Ioka S (2017) First performance test of a 25 T cryogenfree superconducting magnet. Superconductor Science and Technology 30:065001.

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Aymar R, Barabaschi P, Shimomura Y, ITER design team (2002) The ITER design. Plasma Physics and Controlled Fusion 44:519−565. Baixeras J, Fournet G (1967) Pertes par deplacement de vortex dans un supraconducteur de type II non ideal. Journal of Physics and Chemistry of Solids 28:1541−1547. Bardeen J, Cooper LN, Schrieffer JR (1957) Theory of superconductivity. Physical Review 108:1175−1204. Bardeen J, Stephen MJ (1965) Theory of the motion of vortices in superconductors. Physical Review 140:A1197−A1207. Blair AB, Hampshire DP (2019) The critical current density of superconductor-normal metal-superconductor Josephson junctions in high magnetic fields. Submitted to Archive. Bordini B, Alknes P, Bottura L, Rossi L, Valentinis D (2013) An exponential scaling law for the strain dependence of the Nb3Sn critical current density. Superconductor Science and Technology 26:075014. Branch PJ, Tsui Y, Osamura K, Hampshire DP (2019) Weaklyemergent strain-dependent properties of high field superconductors. Nature Scientific Reports 9:13998. Bruzzone P (2004) The index n of the voltage–current curve, in the characterization and specification of technical superconductors. Physica C: Superconductivity and Its Applications 401:7−14. Byrne PJP (2017) An ab-initio study of superconductors using perturbations. Thesis: Durham University. Cai XY, Polyanskii A, Li Q, Riley GN, Larbalestier DC (1998) Current-limiting mechanisms in individual filaments extracted from superconducting tapes. Nature 392:906−909. Campbell AM, Evetts JE, Dew-Hughes D (1968) Pinning of flux vortices in Type II superconductors. Philosophical Magazine 18:313−343. Carbotte JP (1990) Properties of Boson-Exchange Superconductors. Reviews of Modern Physics 62:1027−1157. Carty GJ, Hampshire DP (2008) Visualising the mechanism that determines the critical current density in polycrystalline superconductors using time-dependent Ginzburg-Landau theory. Physical Review B 77:172501. Carty GJ, Hampshire DP (2013) The critical current density of an SNS junction in high magnetic fields. Superconductor Science and Technology 26:065007. CERN (2013) The superconducting magnet of the big european bubble chamber (BEBC) http://cds.cern.ch/ record/850617/files/. Cheggour N, Hampshire DP (1997) The role of vortex melting and inhomogeneities in the transport properties of Nb3Sn superconducting wires. Institute of Physics Conference Series 158:1275−1278. Cheggour N, Hampshire DP (1999) Unifying the strain and temperature scaling laws for the pinning force density in superconducting niobium-tin multifilamentary wires. Journal of Applied Physics 86:552−555.

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Cheggour N, Hampshire DP (2000) A probe for investigating the effect of magnetic field, temperature and strain on transport critical currents in superconducting tapes and wires. Review of Scientific Instruments 71:4521. Chislett-McDonald SBL, Hampshire DP (2019) Update to the PROCESS Nb-Ti critical surface model based on the “Durham Scaling Law” CCFE Internal Report Chislett-McDonald SBL, Tsui Y, Surrey E, Kovari M, Hampshire DP (2019) The magnetic field, temperature, strain and angular dependence of the critical current density for Nb-Ti. European Conference on Applied Superconductivity. Glasgow. Clark AF (1983) Thermal expansion. In: Materials at Low Temperatures (Reed RP, Clark AF, eds), pp 75−132. Metals Park, OH: American Society for Metals. Clark AF, Fujii G, Ranney MA (1981) The thermal expansion of several materials for superconducting magnets. IEEE Transactions on Magnetics 17:2316−2319. Clark RG, Jones H (1986) The combination of dilution refrigerators and superconducting/hybrid magnets in physics research. ICEC 11:414−418. Conectus (2001) Worldwide Markets for Superconductivity & Market Shares for Low-Tc/High-Tc Superconductors. In: http://conectus.org/market/. Dew-Hughes D (1974) Flux pinning mechanisms in type II superconductors. Philosophical Magazine 30:293−305. Dew-Hughes D (2001) The critical current of superconductors: an historical review. Low Temperature Physics 27:713−722. Dimos D, Chaudhari P, Mannhart J, LeGoues FK (1988) Orientation dependence of grain-boundary critical currents in YBa 2Cu3O7-d bicrystals. Physical Review Letters 61:219−222. Durham-Superconductivity-Group http://wwwduracuk/ superconductivitydurham/. Durrell JH (2009) Importance of low-angle grain boundaries in YBCO coated conductors. Superconductor Science and Technology 22:013001. Duyn JH (2012) The future of ultra-high field MRI and fMRI for study of the human brain. Neuroimage 62:1241. Edelman HS, Larbalestier DC (1993) Resistive transitions and the origin of the n value in superconductors with a gaussian critical-current distribution. Journal of Applied Physics 74:3312−3315. Ekin JW (1978) Current transfer in multifilamentary superconductors. I Theory. Journal of Applied Physics 49:3406−3409. Ekin JW (1980) Strain scaling law for flux pinning in practical superconductors. Part I: Basic relationship and application to Nb3Sn conductors. Cryogenics 20:611−624. Ekin JW (1987) Effect of transverse compressive stress on the critical current and upper critical field of Nb3Sn. Journal of Applied Physics 62:4829−4834. Ekin JW (1989) Offset criterion for determining superconductor critical current. Applied Physics Letters 55:905−907.

Characterisation of the Transport Critical Current Density for Conductor Applications

Ekin JW, Cheggour N, Goodrich L, Splett J, Bordini B, Richter D (2016) Unified Scaling Law for flux pinning in practical superconductors: II. Parameter testing, scaling constants, and the Extrapolative Scaling Expression. Superconductor Science and Technology 29:38. Ekin JW, Finnemore DK, Li Q, Tenbrink J, Carter W (1992) Effect of axial strain on the critical current of Ag-sheathed Bi-based superconductors in magnetic fields up to 25 T. Applied Physics Letters 61:858−860. Feigel’man MV, Geshkenbein VB, Larkin AI, Vinokur VM (1989) Theory of collective flux creep. Physical Review Letters 63:2303−2306. Friend CM, Hampshire DP (1993) Transverse and longitudinal critical current densities in Nb46wt%Ti multifilamentary wire from 2 K up to Tc in magnetic fields up to 15 Tesla. In: Applied Superconductivity (Freyhardt HC, ed), pp 23−26: DGM Informationsgesellschaft mbH. Friend CM, Hampshire DP (1995) A probe for the measurement of the transport critical current density of superconductors in high magnetic fields and at temperatures between 2 and 150 K. Measurement Science and Technology 6:98−106. Frost AJ, Jones H, Belenli I (1992) Design, construction and development of an apparatus for the transport-current characterization of high-temperature superconductors at a range of temperatures and magnetic fields. Cryogenics 32:1014. Fukutsuka T, Horiuchi T, Monju Y, Tatara I, Maeda Y, Moritoki M (1984) Effects of hot isostatic pressing on the superconducting properties of Nb3Sn multifilamentary wires. Advances in Cryogenic Engineering 30:891−898. Goldacker W, Keßler J, Ullmann B, Mossang E, Rikel M (1995) Axial tensile, transverse compressive and bending strain experiments in Bi(2223)/AgMg single core tapes. IEEE Transaction on Applied Superconductivity 5:1834. Goldacker W, Specking W, Weiss F, Rimikis G, Flukiger R (1989) Influence of transverse, compressive and axial tensile stress on the superconductivity of PbMo6S8 and SnMo6S8 wires. Cryogenics 29:955−960. Goodrich LF (1991) High Tc superconductor voltage-current simulator and the pulse method of measuring critical current. Cryogenics 31:720. Goodrich LF (1999/2000) Critical-current measurements methods for oxide superconductor tapes and wires. Goodrich LF, Bray SL (1989) Integrity tests for high-Tc and conventional critical-current measurement systems. Advances in Cryogenic Engineering (Materials) 36A:43. Goodrich LF, Bray SL, Stauffer TC (1990) Thermal contraction of fibreglass-epoxy sample holders used for Nb3Sn

203

critical-current measurements. Advances in Cryogenic Engineering (Materials) 36:117. Goodrich LF, Ekin JW, Fickett FR (1982) Effect of twist pitch on short-sample V-I characteristics of multifilamentary superconductors. Advances in Cryogenic Engineering 28:571−580. Goodrich LF, Fickett FR (1982) Critical current measurements: a compendium of experimental results. Cryogenics 22:225−241. Goodrich LF, Medina LT, Stauffer TC (1998) High criticalcurrent measurements in liquid and gaseous helium. Advances in Cryogenic Engineering 44:873−880. Goodrich LF, Srivastava AN (1990) Software techniques to improve data reliability in superconductor and low-resistance measurements. Journal of Research of the National Institute of Standards and Technology 95:575−589. Goodrich LF, Srivastava AN (1992) Comparison of transport critical current measurements methods. Advances in Cryogenic Engineering (Materials) 38:559−565. Goodrich LF, Srivastava AN (1995a) Thermal contraction of materials used in Nb3Sn critical current measurements. In: Cryogenics 35:S29-S32. Goodrich LF, Srivastava AN (1995b) Thermal contraction of materials in Nb3Sn critical current measurements. Cryogenics 35:S29−S32. Goodrich LF, Srivastava AN, Yuyami M, Wada H (1992) n-value and second derivative of the superconductor voltage-current characteristic. IEEE Transactions on Applied Superconductivity 3:1265−1268. Goodrich LF, Vecchia DF, Pittman ES, Ekin JW, Clark AF (1984) Critical current measurements on a NbTi superconducting wire standard reference material. pp. 1−54. Boulder: N.B.S. special publication. Goodrich LF, Wiejaczka JA, Srivastava AN, Stauffer TC, Medina LT (1995) USA interlaboratory comparison of superconductor simulator critical current measurements. IEEE Transaction on Applied Superconductivity 5:548. Gould D, Wada H (1995) Introduction. Cryogenics 35:S7. Goyal A, Paranthaman MP, Schoop U (2004) The RABiTS approach: using rolling-assisted biaxially textured substrates for high-performance YBCO superconductors. MRS Bulletin 29:552−561. Grasso G, Flükiger R (1997) Development of rectangular Bi(2223) wires with reduced anisotropy. Superconductor Science and Technology 10:223. Greenwood JR, Surrey E, Hampshire DP (2018) Biaxial Strain Measurements of Jc on a (RE)BCO Coated Conductor. IEEE Transactions on Applied Superconductivity 28. Greenwood JR, Surrey E, Hampshire DP (2019a) The effects of percolation and distributions of critical parameters on Jc(B,T,e). Submitted to Archive. Greenwood JR, Surrey E, Hampshire DP (2019b) The biaxial strain dependence of Jc of a (RE)BCO coated conductor

204

at 77 K in low fields. IEEE Transactions on Applied Superconductivity 29. Hahn S, Kim K, Kim K, Hu X, Painter T, Dixon I, Kim S, Bhattarai KR, Noguchi S, Jaroszynski J, Larbalestier DC (2019) 45.5-tesla direct-current magnetic field generated with a high-temperature superconducting magnet. Nature 570:496. Hampshire DP, Jones H (1985) Critical current of a NbTi reference material as a function of field and temperature. Magnet Technology 9:531−535. Hampshire DP, Jones H (1987a) Analysis of the general structure of the E-I characteristic of high current superconductors with particular reference to a NbTi SRM wire. Cryogenics 27:608−616. Hampshire DP, Jones H (1987b) A detailed investigation of the E-J characteristic and the role of defect motion within the flux-line lattice for high-current-density, high field superconducting compounds with particular reference to data on Nb3Sn throughout its entire field-temperature phase space. Journal of Physics C: Solid State 20:3533−3552. Hampshire DP, Jones H (1987c) Flux flow in high-temperature type II superconductors governed by the activation of Frank-Read sources and the resultant motion of core dislocations. Journal of Physics C: Solid State 21:419−427. Hampshire DP, Jones H, Mitchell EWJ (1985) An in-depth characterisation of (NbTa)3Sn filamentary superconductor. IEEE Transactions on Magnetics 21:289−292. Hampshire DP, Clark AF, Jones H (1989) Flux pinning and scaling laws for superconducting V3Ga. Journal of Applied Physics 66:3160−3167. Hampshire RG (1974) The critical current density of Nb-60 at% Ti and Nb-25 at% Zr superconductors in small magnetic fields. Journal of Physics D: Applied Physics 7:1847−1863. Hazelton D (2013) Presentation Spring MRS 2013: 2G HTS Conductor Development at SuperPower for Magnet Applications. http://wwwsuperpower-inccom/system/ files/2013_0402+Spring+MRS+Meeting_Hazeltonpdf. Helfand E, Werthamer NR (1966) Temperature and purity dependence of the superconducting criticial field, HC2. II. Physical Review 147:288−294. Herrmann PF (1998) Current leads. In: Handbook of Applied Superconductivity (Seeber EB, ed), p 801. Bristol: Institute of Physics. Hole C (1995) Pulsed magnetic field characterisation of technological high temperature superconductors. Oxford University Thesis. IEC (2006a) Superconductivity - Part 3: Critical current measurement - DC critical current of Ag-sheathed Bi2212 and Bi2223 oxide superconductors. In: International Standard, pp IEC 61788-61783:62006: International Electrotechnical Commission.

Handbook of Superconductivity

IEC (2006b) Superconductivity - Part 2: Critical current measurement - DC critical current of Nb3Sn composite superconductors. In: International Standard, 1 Edition, pp IEC 61788-61782:62006: International Electrotechnical Commission. IEC (2006c) Superconductivity - Part 1: Critical current measurement - DC critical current of Nb-Ti composite superconductors. In: International Standard, pp IEC 61788-61781:62006: International Electrotechnical Commission. IEC (2015) Superconductivity - Part 21: Superconducting wires - Test methods for practical superconducting wires General characteristics and guidance. In: International Standard, pp IEC 61788-61721:62015. Iijima Y, Tanabe N, Kohno O, Ikeno Y (1992) In-plane aligned YBa 2Cu 3O7-x thin-films deposited on polycrystalline metallic substrates. Applied Physics Letters 60:769−771. ITER Organisation (2019) ITER, the way to new energy. (accessed 2020, 14/Nov/2020). Available: www.iter.org Itoh K, Tanaka Y, Osamura K (1996) Round robin test for the method of critical current measurement of Nb3Sn composite superconductor. ICMC:1787−1790. Jakob B, Pasztor G, Bona M, Asner A (1991) Reduced sensitivity of Nb3Sn epoxy-impregnated cable to transverse stress. Cryogenics 31:390−391. Jones H, Siertsema WJ, Richens PE, Newson M, Saleh PM, Hickman AL (2001) An update on the Oxford high magnetic field facilities in the year 2000. Physica B 294–295:598−604. Kamata K, Suzuki Y, Moriai H, Inoue K, Itoh K, Takeuchi T, Tachikawa K, Watanabe K, Muto Y, Katagiri K, Noto K, Okada T (1992) Superconducting properties and strain effects in high fields for bronze processed multifilamentary (Nb,Ti)3Sn wires and composite processed ultrafine Nb3Al wires. Series A - Physics, Chemistry and Metallurgy A 37:99. Keys SA (2001) Temperature and strain scaling laws for the critical current density in Nb3Sn and Nb3Al conductors in high magnetic fields. In: Department of Physics, p 127. Durham: University of Durham. Keys SA, Hampshire DP (2003) A scaling law for the critical current density of weakly and strongly-coupled superconductors, used to parameterise data from a technological Nb3Sn strand. Superconductor Science and Technology 16:1097−1108. Keys SA, Koizumi N, Hampshire DP (2002) The strain and temperature scaling law for the critical current density of a jelly-roll Nb3Al strand in high magnetic fields. Superconductor Science and Technology 15:991−1010. Kirchmayr H (1995) Effects of specimen bonding materials. Cryogenics 35 S95. Kirchmayr H, Siddall MB, Smathers DB (1995) Effects of mandrel materials. Cryogenics 35:S93−S94.

Characterisation of the Transport Critical Current Density for Conductor Applications

Kleiner WH, Roth LM, Autler SH (1964) Bulk solution of Ginzburg-Landau equations for Type II superconductors: upper critical field region. Physical Review 133:A1226−A1227. Kramer EJ (1973) Scaling laws for flux pinning in hard superconductors. Journal of Applied Physics 44:1360−1370. Kramer EJ (1975) Microstructure – critical current relationships in hard superconductors. Journal of Electronic Materials 4:839−879. Kresin VZ (1987) On the critical-temperature for any strength of the electron phonon coupling. Physics Letters A 122:434−438. Kresin VZ, Gutfreund H, Little WA (1984) Superconducting state in strong coupling. Solid State Communications 51:339−342. Küpfer H, Gey W (1977) Dependence of the critical current density of superconductors on the past history of the magnetic field and temperature. Philosophical Magazine 36:859−884. Kuroda T, Murakami Y, Itoh K, Yuyama M, Wada H, Mao D (1998) Temperature dependence of critical current density of Nb3Al multifilamentary wires fabricated by Nb-tube and its improved processes. Cryogenics 38:785. Larbalestier DC, Jiang J, Trociewitz UP, Kametani F, Scheuerlein C, Dalban-Canassy M, Matras M, Chen P, Craig NC, Lee PJ, Hellstrom EE (2014) Isotropic roundwire multifilament cuprate superconductor for generation of magnetic fields above 30 T. Nature Materials 13:375−381. Larkin AI, Ovchinnikov YN (1984) Collective pinning. Physica B+C 126:187−192. Lee PJ, Larbalestier DC (1987) Development of nanometre scale structures in composites of Nb-Ti and their effect on the superconducting critical current-density. Acta Metallurgica 35:2523−2536. Lee TS, Jenkins I, Surrey E, Hampshire DP (2015) Optimal design of a toroidal field magnet system and cost of electricity implications for a tokamak using high temperature superconductors. Fusion Engineering and Design 98-99:1072–1075. Lu XF, Hampshire DP (2009) The magnetic field, temperature and strain dependence of the critical current of a Nb3Sn strand using a six free-parameter scaling law. IEEE Transactions on Applied Superconductivity 19:2619−2623. Lu XF, Hampshire DP (2010) The field, temperature and strain dependence of the critical current density of a powderin-tube (PIT) Nb3Sn superconducting strand. Supercond Sci Technol 23:025002. Lu XF, Taylor DMJ, Hampshire DP (2008) Critical current scaling laws for advanced Nb3Sn superconducting strands for fusion applications with six free parameters. Superconductor Science and Technology 21:105016.

205

Luhman T, Suenaga M, Klamut CJ (1978) Influence of tensile stresses on the superconducting temperature of multifilamentary Nb3Sn composite conductors. Advances in Cryogenic Engineering 24:325−330. Lvovsky Y, Stautner EW, Zhang T (2013) Novel technologies and configurations of superconducting magnets for MRI. Superconductor Science and Technology 26:093001. Markiewicz WD (2004a) Elastic stiffness model for the critical temperature TC of Nb3Sn including strain dependence. Cryogenics 44:767−782. Markiewicz WD (2004b) Invariant formulation of the strain dependence of the critical temperature Tc of Nb3Sn in a three term approximation. Cryogenics 44:895−908. Markiewicz WD (2008) Comparison of strain scaling functions for the strain dependence of composite Nb3Sn superconductors. Superconductor Science and Technology 21:054004. Marti F, Grasso G, Huang Y, Flükiger R (1997) High critical current densities in long lengths of mono- and multifilamentary Ag-sheathed Bi(2223) tapes. IEEE Transaction on Applied Superconductivity 7:2215. McMillan WL (1968) Transition temperature of strong coupled superconductors. Physical Review 167:331−344. Meingast C, Kraut O, Wolf T, Wuhl H, Erb A, Muller-Vogt G (1991) Large a-b anisotropy of the expansivity anomaly at Tc in untwinned YBa 2Cu3O7. Physical Review Letters 67:1634−1637. Miller JR (2003) The NHMFL 45-T hybrid magnet system: past, present, and future. IEEE Transaction on Applied Superconductivity 13:1385−1390. Mitchell N, Bessette D, Gallix R, Jong C, Knaster J, Libeyre P, Sborchia C, Simon F (2008) The ITER magnet system. IEEE Transactions on Applied Superconductivity 18:435−440. Muriale L, Lee E, Genovese J, Trend S (1996) Fatality due to acute fluoride poisoning following dermal contact with hydrofluyoric acid in a Palynology laboratory. Annals of Occupational Hygiene 40:705−710. Muzzi L, De Marzi G, Zignani CF, Vetrella UB, Corato V, Rufoloni A, Della Corte A (2011) Test results of a NbTi wire for the ITER poloidal field magnets: a validation of the 2-pinning components model. IEEE Transactions on Applied Superconductivity 21:3132−3137. Nijhuis A, Wessel WAJ, Ilyin Y, Den Ouden A, Ten Kate HHJ (2006) Critical current measurement with spatial periodic bending imposed by electromagnetic force on a standard test barrel with slots. Review of Scientific Instruments 77:054701. Ochiai S, Osamura K, Watanabe K (1993) Estimation of the strength distribution of Nb3Sn in multifilamentary composite wire from change in superconducting current due to preloading. Journal of Applied Physics 74:440−445.

206

Ogawa R, Kubo Y, Tanaka Y, itoh K, Ohmatsu K, Kumano T, Sakai S, Osamura K (1996) Standardisation of the test method for critical current measurement of Cu/Cu-Ni/ Nb-Ti composite superconductors. ICMC:1799−1802. Okaji M, Nara K, Kato H, Michishita K, Kubo Y (1994) Thermal expansion of some advanced ceramics applicable as specimen holders of high Tc superconductors. Cryogenics 34:163. Orlando TP, McNiff EJ, Foner S, Beasley MR (1979) Critical fields, Pauli paramagnetic limiting, and material parameters of Nb3Sn and V3Si. Physical Review B 19:4545−4561. Osamura K (1998) Present status of international standardization. ISTEC Journal 11:26. Osamura K (2015) Standardization of Test Methods for Practical Superconducting Wires. IEEE/CSC and ESAS European Superconductivity News Forum 33:1−11. Osamura K, Sato K, Furuto Y (1997) Standardisation of the test methods for industrial superconductors by IEC/ TC90. Cryogenic Engineering 32:663. Osamura K, Machiya S, Hampshire DP (2016) Mechanism for the uniaxial strain dependence of the critical current in practical REBCO tapes. Superconductor Science and Technology 29:065019. Osamura K, Nonaka S, Matsui M, Oku T, Ochiai S, Hampshire DP (1996) Factors suppressing transport critical current in Ag/Bi2223 tapes. Journal of Applied Physics 79:7877−7883. Osamura K, Machiya S, Hampshire DP, Tsuchiya Y, Shobu T, Kajiwara K, Osabe G, Yamazaki K, Yamada Y, Fujikami J (2014) Uniaxial strain dependence of the critical current of DI-BSCCO tapes. Superconductor Science and Technology 27:085005. Pobell F (1996) Matter and Methods at Low Temperatures. Berlin: Springer. Polak M, Zhang W, Parrell J, Cai XY, Polyanskii A, Hellstrom EE, Larbalestier DC, Majoros M (1997) Current transfer lengths and the origin of linear components in the voltage-current curves of Ag-sheathed BSCCO components. Superconductor Science and Technology 10:769−777. Press CRC (2012) CRC Handbook of Chemistry and Physics, 93 Edition: CRC Press. Pugnat P, Barbier R, Berriaud C, Berthier R, Debray F, Fazilleau P, Hervieu B, Manil P, Massinger M, Pes C, Pfister R, Pissard M, Ronayette L, Trophime C (2014) Progress Report on the 43 T Hybrid Magnet of the LNCMI-Grenoble. IEEE Transaction on applied superconductivity 24. Raine MJ, Boutboul T, Readman P, Hampshire DP (2019) Large quantity measurements of Nb3Sn and Nb-Ti superconducting strands for the European contribution to ITER’s toroidal and poloidal field coils. Submitted to Superconductor Science and Technology August 2021.

Handbook of Superconductivity

Richens PE, Jones H, van Cleemput M, Hampshire DP (1997) Strain dependence of critical currents in commercial high temperature superconductors. IEEE Transactions on Applied Superconductivity 7:1315−1318. Ridgeon FJ, Raine MJ, Halliday DP, Lakrimi M, Thomas A, Hampshire DP (2017) Superconducting properties of titanium alloys (Ti-64 and Ti-6242) for critical current barrels. IEEE Transactions on Applied Superconductivity 27:1−5. Rupp G (1977) Improvement of critical current of multifilamentary Nb3Sn conductors under tensile-stress. IEEE Transactions on Applied Superconductivity 13:1565−1567. Ryan DT (1997) Critical currents of commercial superconductors in the picovolt per metre electric field regime. Thesis: Oxford University. Ryan DT, Jones H, Timms W, Kiloran N (1997) Critical current measurements at electric fields in the pV.m−1 regime. IEEE Transactions on Applied Superconductivity 7:1455−1458. Sakurai A, Shiotsu M, Hata K (1996) Boiling phenomenon due to quasi-steadily and rapidly increasing heat inputs in LN2 and LHe I. Cryogenics 36:189−196. Sato K (2017) BSCCO wire. http://global-seicom/super/hts_e/ indexhtml. Sborchia C, Fu Y, Gallix R, Jong C, Knaster J, Mitchell N (2008) Design and specifications of the ITER TF coils. IEEE Transactions on Applied Superconductivity 18:463−466. Sborchia C, Barbero Soto E, Batista R, Bellesia B, Bonito Oliva A, Boter Rebollo E, Boutboul T, Bratu E, Caballero J, Cornelis M, Fanthome J, Harrison R, Losasso M, Portone A, Rajainmaki H, Readman P, Valente P (2011) Overview of ITER Magnet System and European Contribution. IEEE/NPSS 24th Symposium on Fusion Engineering:1−8. Schlachter SI, Goldacker W, Frank A, Ringsdorf B, Orschulko H (2006) Properties of MgB2 superconductors with regard to space applications. Cryogenics – 2005 Space Cryogenics Workshop 46:201−207. Sneary AB, Friend CM, Vallier JC, Hampshire DP (1999) Critical current density of Bi-2223/Ag multifilamentary tapes from 4.2K up to 90K in magnetic fields up to 23T. IEEE Transactions on Applied Superconductivity 9:2585−2588. Sugano M, Shikimachi K, Hirano N, Nagaya S (2010) The reversible strain effect on critical current over a wide range of temperatures and magnetic fields for YBCO coated conductors. Superconductor Science and Technology 23:085013. Sunwong P, Higgins JS, Hampshire DP (2011) Angular, temperature and strain dependencies of the critical current of DI-BSCCO Tapes in High Magnetic Fields. IEEE Transactions on Applied Superconductivity 21:2840−2844.

Characterisation of the Transport Critical Current Density for Conductor Applications

Sunwong P, Higgins JS, Hampshire DP (2014) Probes for investigating the effect of magnetic field, field orientation, temperature and strain on the critical current density of anisotropic high-temperature superconducting tapes in a split-pair 15 T horizontal magnet. Review of Scientific Instruments 85:065111. Sunwong P, Higgins JS, Tsui Y, Raine MJ, Hampshire DP (2013) The critical current density of grain boundary channels in polycrystalline HTS and LTS superconductors in magnetic fields. Superconductor Science and Technology 26:095006. SuperPower-Furukawa (2013a) SuperPower 2G HTS wwwsuperpower-inccom/ Coated Conductors. content/2g-hts-wire. SuperPower-Furukawa (2013b) Soldering HTS 2G tapes. ht t p : // w w w s u p e r p ow e r-i n c c o m /s y s t e m /f i l e s / SP_Soldering+Instructions_2013FEC_v2pdf. Tachikawa K (1995) Critical current measurement method for Nb3Sn multifilamentary composite conductors. Foreward Cryogenics 35 S5. Takeuchi T, Iijima Y, Inoue K, Wada H, ten Haken B, ten Kate HHJ, Fukuda K, Iwaki G, Sakai S, Moriai H (1997) Strain effects in Nb3Al multifilamentary conductors prepared by phase transformation from bcc supersaturated solid solution. Applied Physics Letters 71:122−124. Taylor DMJ, Hampshire DP (2005a) Relationship between the n-value and critical current in Nb3Sn superconducting wires exhibiting intrinsic and extrinsic behaviour. Superconductor Science and Technology 18:S297−S302. Taylor DMJ, Hampshire DP (2005b) Properties of helical springs used to measure the axial strain dependence of the critical current density in superconducting wires. Superconductor Science and Technology 18:356−368. Taylor DMJ, Hampshire DP (2005c) The scaling law for the strain dependence of the critical current density in Nb3Sn superconducting wires. Superconductor Science and Technology 18:S241−S252. Taylor DMJ, Keys SA, Hampshire DP (2002) E-J characteristics and n-values of a niobium-tin superconducting wire as a function of magnetic field, temperature and strain. Physica C 372:1291−1294. ten Haken B (1994) Strain effects on the critical properties of high-field superconductors. In: Physics Department: University of Twente. ten Haken B, Godeke A, ten Kate HHJ (1995) The influence of compressive and tensile axial strain on the critical properties of Nb3Sn conductors. IEEE Transactions on Applied Superconductivity 5:1909−1912. ten Haken B, Godeke A, ten Kate HHJ (1999) The strain dependence of the critical properties of Nb3Sn conductors. Journal of Applied Physics 85:3247−3253. ten Kate HHJ, Weijers HW, van Oort JM (1992) Critical current degradation in Nb3Sn cables under transverse pressure. IEEE Transactions on Applied Superconductivity 3:1334−1337.

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Trociewitz UP, Dalban-Canassy M, Hannion M, Hilton DK, Jaroszynski J, Noyes P, Viouchkov Y, Weijers HW, Larbalestier DC (2011) 35.4 T field generated using a layer-wound superconducting coil made of (RE) Ba 2Cu3O7−x (RE = rare earth) coated conductor. Applied Physics Letters 99:202506. Tsui Y, Hampshire DP (2012) Critical current scaling and the pivot-point in Nb3Sn strands. Superconductor Science and Technology 25:054008. Tsui Y, Surrey E, Hampshire DP (2016) Soldered Joints - an essential component of demountable high temperature superconducting fusion magnets. Superconductor Science and Technology 290:075005. Uemura YJ, et al (1989) Universal correlations between TC and ns/m (carrier density over effective mass) in highTC cuprate superconductors. Physical Review Letters 62:2317−2320. Uemura YJ, Le LP, Luke GM, Sternlieb BJ, Wu WD, Brewer JH, Riseman TM, Seaman CL, Maple MB, Ishikawa M, Hinks DG, Jorgensen JD, Saito G, Yamochi H (1991) Basic similarities among cuprate, bismuthate, organic, chevrel-phase, and heavy-fermion superconductors shown by penetration-depth measurements. Physical Review Letters 66:2665−2668. Uglietti D (2019) A review of commercial high temperature superconducting materials for large magnets: from wires and tapes to cables and conductors. Superconductor Science and Technology 32:053001. VAMAS TWP (1995) VI. Standard method for Ic determination - VI-1: recommended standard method for determination of d.c. critical current of Nb3Sn multifilamentary composite superconductor. Cryogenics 35 S1:s105−s112. Varley R (2016) The UK Met Office. http://wwwmetofficegovuk/ public/weather. Wada H, Goodrich LF, Walters C, Tachikawa K (1995) Second intercomparison of critical current measurements. Cryogenics 35:S65−S80. Walters CR, Davidson IM, Tuck GE (1986) Long sample high sensitivity critical current measurements under strain. Cryogenics 26:406−412. Wang G, Raine MJ, Hampshire DP (2017) How resistive must grain-boundaries be to limit JC in polycrystalline superconductors? Superconductor Science and Technology 30:104001. Warnes WH (1988) A model for the resistive critical current transition in composite superconductors. Journal of Applied Physics 63:1651−1662. Warnes WH, Larbalestier DC (1986a) Analytical technique for deriving the distribution of critical currents in a superconducting wire. Applied Physics Letters 48:1403−1405. Warnes WH, Larbalestier DC (1986b) Critical current distributions in superconducting composites. Cryogenics 26:643−653. White GK (1987) Experimental techniques in low-temperature physics, 3 Edition. Oxford: Oxford University Press.

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White GK (1998) Thermal expansion. In: Handbook of Applied Superconductivity (Seeber EB, ed), p 1107. Bristol: Institute of Physics Publishing. Wiejaczka JA, Goodrich LF (1997) Interlaboratory comparison on high temperature superconductor critical-current measurements. Journal of Research of the National Institute of Standards and Technology 102:29−52. Willen DWA, Zhu W, Cave JR (1997) Selection of the offsetcriterion voltage parameter and its relation to the second differential of a superconductor’s voltage-current curve. Institute of Physics Conference Series 158:1013−1016. Wilson MN (1986) Superconducting Magnets. Oxford: Oxford University Press. Wimbush SC, Strickland NM (2017) A public database of hightemperature superconductor critical current data. IEEE Transaction on Applied Superconductivity 27:8000105.

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Wördenweber R (1998) Mechanism of vortex motion in hightemperature superconductors. Reports on Progress in Physics 62:187−236. Yamada N, Nara K, Okaji M, Hokata T, Kaneko T, Sadakata N (1998) Effect of thermal cycles on thermal expansion of silver-sheathed Bi2223 tape at 10-310 K. Cryogenics 38:397. Yamafuji K, Kiss T (1997) Current-voltage characteristics near the glass-liquid transition in high-TC superconductors. Physica C 290:9−22. Yoon S, Kim J, Cheon K, Lee H, Hahn S, Moon S H (2016) 26 T 35 mm all-GdBa 2Cu3O7–x multi-width no-insulation superconducting magnet. Superconductor Science and Technology 29:1−6.

G2.5 Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep Michael Eisterer

G2.5.1 Introduction The most famous and important property of superconductors is dissipation-free current flow. This ability is, however, limited not only by temperature and magnetic field, but by the current density itself. The maximum critical current density a superconductor can sustain without loss is commonly referred to as the critical current density, Jc. Jc is thus a very important property of the superconductor both for theory and application and defines its practical merit. Although the absence of dissipation is a fascinating concept, it is problematic as a definition, because losses or the accompanying voltages cannot be measured with arbitrary accuracy; hence the derived value of Jc depends on the sensitivity of the experiment. Two main methods are widely used to assess Jc. Resistive measurements, which are discussed in Chapter G2.2, and magnetic techniques, which derive the current density, J , from the magnetic field, B, or moment, m, they generate: ∇ × B = µ0 J

m=

1 r × Jd 3r 2

∫

(G2.5.1) (G2.5.2)

In principle, the local current density can be calculated from the generated spatial field distribution by Ampere’s law [Equation (G2.5.1)], but this possibility is restricted in practice to thin layers where the current distribution does not change along the thickness. In contrast, the local current density distribution cannot be derived from Equation G2.5.2, since m is only one three-dimensional vector. In most cases, only the component parallel to the applied field is assessed experimentally. Additional constraints for the geometry of current flow are needed to estimate the average current density within the sample from m. They are obtained from the Bean model (Bean, 1962, 1964) which forms – together with

Equation (G2.5.2) – the basis for the determination of Jc from magnetization measurements. Both main experimental techniques (resistive and magnetic) for the assessment of Jc have their advantages and disadvantages. For practical conductors (wires, tapes, cables) resistive transport current measurements are generally favorable, since they focus directly on the quantity of interest: the critical current of the conductor, Ic. However, it is hard to derive any information on the current limiting mechanism or the current distribution, and Ic is essentially determined by the weakest section along the conductor. In this respect, much more information can be extracted from specific magnetic measurements, e.g. scanning Hall probe microscopy. Some practical arguments favor magnetic measurements, in particular, the ease of sample handling since these techniques are contact free, whereas low-resistance contacts have to be attached to the superconductor for resistive measurements and current leads, in many cases for high currents, are needed to feed the current into the superconductor. These resistive parts of the current loop dissipate energy, thus potentially compromising the temperature stability and consequently the accuracy of the measurement. Since the critical current densities of many superconductors are of the order of 1–10 MA/cm 2 at low magnetic fields and temperatures, the resulting critical currents (1–10 kA in a wire with a superconducting cross-section of 0.1 mm 2) easily exceed the capability of typical experimental set-ups. For tiny samples, such as single crystals, measurements under those conditions are practically impossible [unless they are patterned by using the focused ion beam (FIB) method, e.g. Moll et al. (2010)]. On the other hand, it is quite easy to induce huge currents inductively for magnetic measurements. This chapter will introduce standard and advanced magnetic characterization techniques, discuss their proper evaluation and point out experimental pitfalls. The focus will be 209

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on the experimental methods, whereas the underlying physics will be explained only to the extent needed to understand experimental issues. It will be shown that the available stateof-the-art methods can not only determine Jc with high accuracy, but also provide information on flux dynamics, current limiting mechanisms, homogeneity of the transport properties and current anisotropy.

G2.5.2 Theoretical Background and Definition of Jc To our present knowledge, DC currents are dissipation-free in the Meissner state, where magnetic fields do not enter the bulk of the superconductor, but are shielded in a surface layer whose thickness is characterized by the magnetic penetration depth λ. These loss-free surface currents are limited by the thermodynamics of superconductivity, and the systems become either normal conducting or enter the mixed state when the magnetic field which is generated by the currents exceeds the critical field, Hc, or the lower critical field, Hc1, respectively. The thermodynamic limit for the current density itself is called depairing current density, Jd, at which the kinetic energy of the charge carrier becomes high enough to lift them above the superconducting energy gap, thus leading to pair breaking. This limit can be experimentally attained in superconductors with very small cross-sections, where the generated field remains moderate despite a very high current density, or with short current pulses (Kunchur et al., 1994; Lang et al., 2007).

G2.5.2.1 Critical Current Density in the Mixed State This chapter will focus on the critical current density in the mixed state, which is determined by the pinning strength of the defect structure, and, in some cases, by current limitation at grain boundaries. In Type II superconductors, magnetic flux starts to enter the superconductor above Hc1 in the form of vortices. The center of the vortices consists of a normal conducting core, and its movement mediates dissipation. The movement is driven by the Lorentz force acting on the vortices when a macroscopic current density is applied, but can be impeded by vortex pinning by defects in the microstructure of the superconductor. However, since the pinning force is finite, the vortices start to move and dissipation sets in, when the Lorentz force density, FL , exceeds the maximum pinning force density, Fpmax . The limiting equilibrium, called Bean’s critical state, defines the critical current density J c0 : FL = J c0 × B = − Fpmax

and Jc0 can only reach about 20% of the depairing current density Arcos and Kunchur (2005), when all flux lines are perfectly pinned along their entire length. Although Jc0 is the current density predicted by many pinning models [e.g. collective (Larkin and Ovchinnikov, 1979) or dilute (Campbell and Evetts, 1972) pinning], it is not the critical current density, Jc , assessed experimentally. The fragile equilibrium assumed in Equation (G2.5.3) is disturbed by thermal activation processes. The first model for this mechanism was proposed by Anderson and Kim (1964). A single vortex is assumed to jump out of its pinning potenU − 0

tial U0 at a rate proportional to e kBT , with the Boltzmann constant k B, and the temperature T. When a current is applied, the pinning potential for jumps in the direction of the Lorentz force is reduced to U0(1 − J/Jc0), while it increases to U0(1 + J/Jc0) for jumps in the opposite direction. The net jump rate in the direction of the Lorentz force becomes proportional to

e

−

J U 0 1− J c0 kBT

−e

−

J U 0 1+ J c0 kBT

= 2e

U − k 0T B

sinh

JU 0 J c0 kBT

(G2.5.4)

For a constant jump distance, the electric field, E, is simply proportional to the jump rate E ( J ) = 2ρc J c0 e

U − k 0T B

sinh

JU 0 J c0 kBT

(G2.5.5)

with ρc being the resistivity at Jc0. This relation is illustrated on a double logarithmic scale in Figure G2.5.1. Two limiting cases are observed: When U0 is much larger than k BT, hopping against the Lorentz force can be neglected near Jc0, and an exponential behavior of the voltage is obtained in the flux creep regime: E ( J ) = ρc J c0 e

U − k 0T B

U0

J

e kBT Jc0 , J J c0

(G2.5.6)

(G2.5.3)

Hence, the defect structure, not the superconducting material, itself determines the critical current density. However, the pinning force depends on fundamental material properties,

FIGURE G2.5.1 Current–voltage characteristics within the Kim– Anderson model.

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

In the opposite limit, J Jc0 (or U0 ≈ k BT), the hyperbolic sine can be linearized and ohmic behavior results in the so-called TAFF (thermally assisted flux flow) regime: U0

E ( J ) = 2ρc

U 0 − kBT e J =: ρTAFF J , J J c0 kBT

(G2.5.7)

Currents are thus never dissipation-free in the mixed state of a Type II superconductor within this model, although U0/k B can become several 1000 K at low temperatures and fields rendering losses at low current academic, since they are neither measurable nor of any practical relevance. Other models use a more complex dependence of the pinning potential on the current density rather than the linear expansion of the Kim– Anderson model, e.g. a logarithmic behavior U0log(Jc0/J), which diverges for J → 0. The electric field thus converges much faster to zero than the current density in this case. Nevertheless, the current density remains a function of the electric field, J = J(E,B,T), and E disappears only in the limit J → 0. This leads to the conclusion that the popular definition of Jc by the disappearance of losses is totally inappropriate from a scientific point of view. In practice, Jc is defined by an electric field criterion Ec: J c ( B,T ) := J ( Ec , B,T )

(G2.5.8)

A big advantage of resistive transport current measurements is the knowledge of Ec, which is chosen (typically 0.1 or 1 µV/ cm) within the experimentally accessible range of the directly assessed E(J) characteristics (I–V curves). E(J) often follows a power law, which can be used for the extrapolation to an acceptable loss level of the envisaged application. Ec can be chosen to a much lesser extent in magnetic measurements, if at all, and its estimation is more complex. The electric field is either given by the induction resulting from the change of the applied magnetic field, or by the decay of the current loops (relaxation, see below). With a few exceptions, Ec is generally much lower in magnetic than in resistive measurements, in many cases extremely low, with the advantage that the loss level is lower than needed for any application. A rough estimate of typical Ec values in various experimental techniques is given in Table G2.5.1. In many cases, Ec does not only depend on the experiment, but also on magnetic field,

temperature, sample geometry and properties of the superconducting material. An estimation is easiest for vibrating sample magnetometer (VSM) measurements during field sweeping. The maximum electric field (at the surface) of a rectangular sample with base area a∙b in a magnetic field applied perpendicular to the base is given by E=

U dφ ab dB 1 =− =− (G2.5.9) 2( a + b ) 2( a + b ) dt 2( a + b ) dt

U denotes the electric potential, which is induced by the field given approximately by the field sweep rate (changes change dB dt of the self-field are neglected) of the ramping magnet. For typical values (a and b between 0.5 and 4 mm, dB = 0.01–1 T/ dt min) one finds Ec≈10 −4 −0.1 µV/cm. In the field step mode similar, but slightly higher values than for the superconducting quantum interference device (SQUID) apply. In this case, dB dt is given by the magnetic relaxation rate or the creep rate S, which is typically between 1 and 20% per decade. The time scale of SQUID measurements is minutes, thus dB is roughly dt BS divided by 100–1000 s. The average B for calculating the magnetic flux is B */3 with B *≈µ0Jcd/2, where d is the smallest sample dimension (see Section G2.5.3.3). Assuming Jc in the range 108–1011 Am−2 and d between 0.1 µm (thin film) and 1 mm, Ec becomes 10 −8–10 −3 µV/cm for realistic combinations of these parameters. In any case, the electric field decreases from the surface toward the center of the sample, however, the currents near the surface contribute the most to the magnetic moment, and these rough estimates are thus representative for the derived currents. In AC susceptibility measurements, the electric field can be tuned in principle over an extremely wide range by changing the frequency and amplitude of the applied magnetic field. However, natural restrictions occur in practice, and the vast majority of experiments correspond to the values reported in the table. The extreme differences in electric field will result in largely different Jcs obtained from different techniques. One example, a comparison of SQUID and resistive measurements on a thin film is depicted in Figure G2.5.2. While both techniques lead to similar results at low fields (large U0/k BT), the disagreement steadily increases with field, since U0 decreases.

TABLE G2.5.1 Typical Electric Field Criteria Corresponding to Various Experimental Techniques. s Denotes the Creep Rate Which Depends on the Material, Magnetic Field and Temperature (cf. Section G2.5.6) Experimental Technique Resistive transport SQUID VSM (sweep mode) VSM (step mode) AC susceptibility Scanning Hall probe

Ec (µV/cm) 0.1–10 10−8–10−3 10−4–0.1 10−7–10−2 1 Birr

FIGURE G2.5.2 Comparison of resistive and magnetic measurements. Data extracted from (Zhang et al., 2004).

An instructive example of the wide range of the E(J) characteristics that can be assessed by a combination of resistive and magnetic techniques is shown in Figure G2.5.3. A negative curvature is observed in the double logarithmic representation, indicative of a glassy state of the vortex lattice (Fisher et al., 1991), while the positive curvature predicted by the Kim– Anderson model is usually observed at higher fields.

G2.5.2.2 Irreversibility Field Birr The irreversibility field is named after the field where the magnetization loops close and the magnetization becomes reversible. It is thus obvious how to derive it from magnetization measurements. Within Bean’s model, the transition to a reversible behavior is equivalent with Jc becoming zero, which is generally used as the definition for Birr. In practice, it

FIGURE G2.5.3 Electric field as a function of the current per sample width (Thompson et al., 2008).

(G2.5.10)

Jcriterion is arbitrarily chosen in resistive measurements, typically resulting from the applied current and the sample crosssection. In principle, the definition in Equation (G2.5.10) also applies to magnetization measurements, with Jcriterion reflecting the experimental accuracy of the magnetometer. Since two criteria define Birr, one for the electric field and one for the current density, a comparison of experimental data is often problematic. If the magnetization is recorded in a static magnetic field (for instance in a SQUID), Ec becomes extremely small which results in much smaller irreversibility fields than those obtained in a VSM during field ramping or from resistive measurements, as is evident from Figure G2.5.2. Thermally assisted flux flow (TAFF) generally prevails above the irreversibility fields obtained from resistive measurements, therefore linear I–V curves (ohmic behavior) are observed above Birr. However, at the “magnetic” Birr, corresponding to much lower electric fields, the I–V curves recorded in resistive measurements are still non-linear, since they reach much higher electric fields, where the J(E)-dependence is curved (cf. Figure G2.5.1). Irreversibility and non-linear behavior depend strongly on the electric field.

G2.5.3 Magnetization Loops and Bean Model The magnetization, M, is defined as the difference between the magnetic flux density inside a material, B, and the magnetic field, H, i.e. B = µ 0 ( H + M ) = µ 0 µ r H, with the relative magnetic permeability µ r . In diamagnetic, paramagnetic and nonhysteretic ferromagnetic materials, µ r is a material-dependent constant at low magnetic fields resulting in a linear relation between B and H. H corresponds to the applied magnetic field only in infinitely long samples which are aligned with the field. In real samples, the stray field of the sample has to be considered (see Section G2.5.3.4), which renders the magnetization dependent on the sample shape. The field in the sample is no longer constant. The magnetization is a valid concept for superconductors in the mixed state only if flux pinning is negligible. In hysteretic superconductors, a local relation between B and H does not exist, and the magnetization increases with the sample size. Although the magnetization can be formally defined by dividing the magnetic moment by the sample volume, M = m / V , it is preferable to use the magnetic moment, which is assessed by magnetometers and results from the currents inside the superconductor according to Equation (G2.5.2). The magnetic properties of the material in the normal state and, in most cases, the equilibrium magnetization of the superconductor are neglected in the following. The field inside

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

the sample is discussed in terms of the local flux density B, which results from the applied field H and the currents within the samples.

G2.5.3.1 Hysteresis Loops Hysteresis loops are by far the most frequently used technique to assess a superconductor’s critical current density from its response to an external magnetic field. The evaluation is based on Equation (G2.5.2) and the Bean model, which restricts the geometry of current flow. Magnetic flux enters a superconductor only from the surface, where vortices emerge and are pushed inward by the repulsive interaction with the external field. Vortex pinning balances this magnetic pressure, leading to a flux density gradient and the corresponding shielding current [cf. Equation (G2.5.1)]. According to the Bean model, the current density can be either zero in areas where the field has not penetrated or otherwise Jc, where the force balance corresponds to the critical state expressed by Equation (G2.5.3). In simple geometries (cylinders, cuboids etc.). the currents flow parallel to the nearest surfaces with their direction depending on the magnetic history. Figure G2.5.4 shows an archetypical hysteresis loop, where the contribution of the critical currents dominates the reversible magnetization, and self-field effects are small. The insets show the ideal field profiles in an (infinitively) long cylinder aligned with the applied magnetic field. The reversible magnetization (cf. Chapter A1.2) is neglected, which corresponds to the limit Hc1 → 0 (equivalent to the Ginzburg–Landau parameter κ:=λ/ξ → ∞, with the magnetic penetration depth λ and the coherence length ξ). Under these assumptions the currents flow only

213

azimuthally, and Equation (G2.5.1) reduces to ∂∂Brz = −µ 0 J ϕ . (The axis of the cylinder and the applied magnetic field, H, are assumed to be parallel to the z-axis.) The azimuthal current density, J ϕ , is zero when B = 0, or ±Jc otherwise. The field profile hence becomes linear with the slope being proportional to Jc . The hysteresis loop in Figure G2.5.4 starts after zero-field cooling (zfc). The superconductor is initially in the Meissner state, the initial slope is therefore referred to as the Meissner slope. It only depends on the geometry of the sample, provided the smallest sample dimension is much larger than the magnetic penetration depth (see Section G2.5.3.4). If the field exceeds the lower critical field, Hc1, vortices enter the superconductor in a surface layer of thickness δ = H/Jc. (This relation holds only for negligible Hc1, as assumed in the following). In most cases, Hc1 cannot be identified easily, since the linear behavior does not change as long as the Bean penetration depth, δ, is much smaller than the respective dimension of the sample. In addition, surface barriers impede flux penetration in a certain field range above Hc1. The magnetization starts to deviate from the Meissner slope, when δ approaches the Bean penetration field H* =JcR, where the vortices reach the center of the superconductor and the so-called virgin magnetization curve, which can be obtained only upon zero-field cooling, ends. The magnetization would remain constant upon further increasing the field, if Jc was field independent, which is in general not the case. Jc does change with field and consequently the slope of the field profile and the magnetic moment also change, as sketched in the insets of Figure G2.5.4. The magnetic moment is directly proportional to Jc above H*. This is the main idea for determining Jc from hysteresis loops. The proportionality factor is a function of the sample geometry

FIGURE G2.5.4 Hysteresis loop of a highly non-reversible Type II superconductor. The insets illustrate the critical state in an infinitely long cylinder aligned with the field. The slope of the field profiles is directly proportional to Jc. The grey and striped areas symbolize magnetic moments of opposite orientation.

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and is obtained by the generic relation between the currents and the magnetic moments they generate: m=

1 1 r × Jd 3r = J c r × e J d 3r 2 2

∫

∫

(G2.5.11)

The resulting formulae for particular sample geometries are given in Table G2.5.2. After the maximum field, Hmax, is reached and the field is decreased again, currents of opposite orientation are induced, rendering the formulae in Table G2.5.2 invalid. Initially, currents are reversed in a thin surface layer δ that grows linearly with decreasing field resulting in a constant slope of the reverse leg, which is the same as for the virgin curve, namely the Meissner slope. The magnetization crosses zero despite currents flow in the entire superconducting volume when the magnetic moments of the positive and negative currents compensate each other. When the field is reduced by 2H* with respect to Hmax, the field profile in the sample is fully reversed and the formulae given in Table G2.5.2 become valid again. Reversing the profile takes twice as long as its initial establishment for the same Jc, since the field profile at Hmax–H* reaches Hmax–H* at r=0 and r=R and has its maximum of Hmax–H*/2 at r=R/2 as can be anticipated from the field profile along the reverse leg sketched in Figure G2.5.4. However, Jc is usually much smaller at Hmax than at zero field, which reduces H* accordingly. At H=0, the field in the sample is called remnant or trapped field, the corresponding magnetization is referred to as remanent magnetization. With the exception of the virgin magnetization curve and the reverse leg, the irreversible magnetic moment only changes sign between the increasing and the decreasing field branch mirr + ( H ) = −mirr − ( H )

(G2.5.12)

where mirr+ and mirr- denote the irreversible magnetic moment on the increasing and decreasing field branch, respectively. Increasing and decreasing refers to the absolute value of the magnetic field. (The sign of H is determined by the choice of the coordinate system, which of course has no influence on the superconducting properties.) In the idealized case under consideration, one branch would be sufficient for a calculation of Jc(B). In particular, mirr+ at negative fields can be evaluated in the entire field region with the formulae of Table G2.5.2. In practice, the use of at least two field branches is favorable or even mandatory for the reasons outlined in the next subsections.

G2.5.3.2 Reversible Magnetization The equilibrium magnetization (and the corresponding reversible moment, mrev) was neglected so far, which is justified if pinning is strong and κ is large. An exemplary magnetization loop (clean single crystal) with a significant contribution of mrev is shown in Figure G2.5.5. It changes the symmetry of the magnetization loop, since mrev is antisymmetric with respect to the y-axis: mrev ( − H ) = −mrev ( H )

(G2.5.14)

The different symmetries of the reversible and irreversible moment offer an opportunity to calculate them both from the measured moment, m ( H ) = mrev ( H ) + mirr ( H ):

mirr ( H ) =

m− ( H ) − m+ ( H ) m− ( H ) + m+ ( − H ) = 2 2

(G2.5.15)

and the hysteresis loop is symmetric about the y-axis: mirr + ( − H ) = mirr − ( H )

(G2.5.13)

mrev ( H ) =

m− ( H ) + m+ ( H ) 2

TABLE G2.5.2 Relations between the Critical Current Density and the Irreversible Magnetic Moment [Defined in Equation (G2.5.15)] for Idealized Sample Geometries (SI Units). Their Validity Requires a Homogeneous Material and a Fully Established (or Fully Reversed) Field Profile Geometry

Relation between Jc and the Irreversible Component of m Jc =

Cylinder of radius R and height h, axis parallel to H Cylinder of radius R and height h, axis normal to H, h>2R

Jc =

3mirr πR 3h

3mirr 4 R 3h (1 − 3πR /16h )

Cuboid with dimensions a × b × c, H parallel to c, a≥b

Jc =

4mirr ab 2c(1 − b / 3a)

Cuboid with dimensions a × b × c, H parallel to c, b≥a

Jc =

4mirr a 2bc(1 − a / 3b)

Sphere of radius R

Jc =

8mirr π2R4

(G2.5.16)

215

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

FIGURE G2.5.5 Hysteresis loop with a dominant contribution of the equilibrium magnetization.

Note that mirr is positive by definition [(Equation (G2.5.14)], while mirr+, mirr− and mrev can take positive and negative values. Equation (G2.5.14) allows for a reliable determination of mirr and consequently Jc, even if mrev is large. On the other hand, the assessment of mrev becomes difficult if mirr dominates because of the self-field effects discussed in the next subsection. In principle, mrev provides valuable information on fundamental mixed-state parameters, such as λ and κ (e.g., Brandt, 1995), which are not discussed in this chapter.

FIGURE G2.5.6 Self-field in the central plane of cylinders with various aspect ratios. The current flows azimuthally. D=2R and h denote the diameter and height of the cylinders, respectively. The field was normalized by J c D / 2 for D/h=1 and 0.1 and by J c ( h / 2 ) ln ( 20 ) for higher aspect ratios. The factor ln(20)≈3 was chosen to obtain the same value at r=0 for D/h=10 and in the limit h → ∞.

sample edges. The return field at the sample edge increases with aspect ratio, but its orientation is of course always opposite to that at the sample center. The central field, which is equivalent to H* is given by H* =

G2.5.3.3 Self-Field The critical current density in a superconductor is a function of the local induction B, not the applied field H. Since the measured magnetic moment arises from currents flowing in the sample, which in turn generates field gradients, it is in principle impossible to derive Jc(B) exactly. At best a representative B for an average Jc can be derived. While the latter can be obtained directly from mirr by the equations given in Table G2.5.2, the calculation of the representative B, which is the sum of the applied field and the self-field of the sample, has to be performed numerically for realistic but nevertheless simplified sample geometries. Figure G2.5.6 illustrates selffield profiles in the central plane of a cylindrical sample for various aspect ratios assuming a constant current density and the applied field aligned with the axis of the cylinder. If the height of the cylinder, h, is much larger than the diameter, D=2R, the self-field increases linearly from about zero at the edge (r=R) to its maximum at r=0 with a constant slope of ±µ0Jc. This is still a reasonable approximation for an aspect ratio D/h of 1, although the entire profile is shifted toward negative fields due to the return flux which generates a field of opposite orientation outside of the sample. This effect is called demagnetization, and reverses the self-field near the

Jch h2 + D 2 + D ln 2 h

(G2.5.17)

which converges to the well-known estimate H * = J c R = J c D2 for large h. In the opposite limit of a very thin sample with a fixed sheet current density J c h, the field at the center and the edges diverges logarithmically with h → 0. Apart from these divergences, the field which is typically generated in the sample is of the order of Jca/2, where a denotes the smallest sample dimension. Although the maximum self-field inside the sample sets the scale, it is just an upper limit for the needed correction, and the representative field has to be obtained by a weighted average of the field over the entire sample volume. The weight should be chosen equivalent to the weight of the currents contributing to the generated magnetic moment (Wiesinger et al., 1992), which increases with the area enclosed by the current loop. Note that the evaluation of the magnetic moment by Equation (G2.5.14) is a first-order self-field correction, if the field does not change sign within the sample, since the selffield has opposite sign on the increasing and decreasing field branch. Thus, at high fields, an elaborated self-field correction is not needed, in particular, if the field of the sample is negligible compared to the applied field. At low magnetic fields on the other hand, such a correction often becomes very important, since Jc depends on the absolute value of the local field

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Ba and the average of |Ba| is always different from zero when currents inside the sample are inducing field gradients. For a mm-sized sample having Jc of the order of MA/cm2, the selffield amounts to several tesla.

G2.5.3.4 Demagnetization The return flux also occurs in the Meissner state, where the self-field ensuring B=0 inside the sample is generated by surface currents instead of the bulk currents discussed in the previous section. Since the self-field opposes the applied field in the Meissner state, the return flux increases the applied field at the edges, which can formally be described by an effectively applied field, Heff, being higher than the applied field, Ha: H eff =

Ha 1 − Dm

the curves with different b onto one master curve (the original one for a=b) is illustrated in the right panel of Figure G2.5.7. The data approach the expected asymptotic. behavior for thin disks of diameter D, Dm=1-πh/2D (h → 0, line graph, Dh 4PD = π4 Dh ). The magnetic moment of any bulk superconductor in the Meissner state is universally m=−

1 H aV 1 − Dm

(G2.5.19)

with the sample volume V, if all sample dimensions are much larger than the magnetic penetration depth. The slope of m(H) in the Meissner regime and on the reverse leg in hysteretic superconductors (just below Hmax) becomes ± 1− 1Dm V .

(G2.5.18)

The demagnetization factor, Dm, only depends on the sample geometry. It is zero in the limit of infinitely long samples aligned with the field, and one for vanishing sample thickness. For a few geometries analytical expressions do exist for the calculation of Dm, but the demagnetization factor has to be calculated numerically for the most frequent geometries of superconducting samples. The shape of most samples can be modeled by cuboids, the demagnetization factors as a function of the aspect ratio are plotted in the left panel of Figure G2.5.7 (see also Chen et al., 2005). The aspect ratio is defined as the larger lateral dimension, a, divided by the height h of the cuboid, which is assumed to be parallel to the applied magnetic field. The demagnetization factor depends on the smaller lateral dimension b as well, but this dependence can be approximated by a redefinition of the aspect ratio ha → Dh 4PD with D being the larger lateral dimension and P the perimeter of the lateral area. This expression becomes ha a2+ab and reduces to the original definition of the aspect ratio for a=b. The collapse of

FIGURE G2.5.7 Demagnetization factors of cuboids with lateral dimensions a and b and the height h parallel to the field. 1−Dm is drawn to illustrate the convergence of Dm to the one for h→0, which leads to a diverging field enhancement according to Equation (G2.5.18).

G2.5.3.5 AC Susceptibility AC susceptometers generate small alternating magnetic fields and measure the response of the sample. In most commercial systems, the in-phase (m’) and the out-of-phase (m’’) component of the first harmonic (i.e. the frequency of the applied field) are recorded. The latter is directly proportional to the losses occurring in the sample, and is therefore nearly zero in the Meissner state at low frequencies. The temperature dependence of the AC susceptibility is frequently used to determine the superconducting transition temperature. In a homogeneous superconductor whose dimensions are much larger than the magnetic penetration depth, m’ rapidly changes from its value in the normal conducting state to a constant value below Tc, which corresponds to χ=−1 and is given by Equation (G2.5.19) (m’ replacing m and Ha being the effective value of the AC field). A broad transition is indicative of inhomogeneities or granularity. In small very thin samples or in samples consisting of decoupled grains, m’ changes down to low temperatures, which is a consequence of the temperature dependence of λ. In any case, Hc1 becomes smaller than the applied AC field close to Tc and a peak in m’’ (“loss peak”) occurs. When an additional DC field, which exceeds Hc1, is applied, the loss peak position is given by the condition 2H*≈HAC (amplitude of the AC field). This can be easily understood by considering that m’’ represents the moment measured when the applied AC field crosses zero. At the loss peak, the field profile is akin to that sketched at the decreasing field branch in Figure G2.5.4. At higher temperatures, the slope of the field profile and hence the moment becomes smaller. At lower temperatures, the profile is not fully reversed, and currents of opposite orientation reduce the net moment (cf. the profile on the reverse leg in Figure G2.5.4). In the high temperature regime, the fully established Bean profile can be exploited to calculate Jc from the relations in Table G2.5.2. However, due to the experimental limitations of HAC, this method can be applied only for fields and temperatures where Jc is small, i.e. in the vicinity of the irreversibility line. For instance, the temperature, Tirr, where the loss peak occurs, is used to

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

define Hirr(Tirr)=HDC (the applied DC field). From 2H* =HAC a criterion for Jc is obtained and the one for the electric field can be estimated easily from the geometry of the sample, e.g. µ 0 H AC ωab / 2( a + b ) [analogous to Equation (G2.5.9)] for a rectangular sample exposed to an AC field with circular frequencyω. Campbell proposed a method (1971) to derive Jc from AC susceptibility measurements in the limit of small penetration depths δ. m’ is recorded as a function of H AC , which is closely related to the reverse leg of the magnetization (cf. Figure G2.5.4). Although the reverse leg appears to be linear (Meissner slope), it is actually not and the deviation from linearity is inversely proportional to Jc, since the magnetic field and the shielding currents progress toward the center with δ = H AC / J c . This technique requires AC fields of at least a few mT to overcome the limit of elastic motion of the vortices within their pinning potentials as well as the possibility to compensate the pick-up system so that the signal in the Meissner state becomes zero, i.e. the dominant linear part of the reverse leg is suppressed. Since the electric field increases with the applied AC field, the method was extended from the static Bean model (Jc) to the real J(E) characteristic, which can be obtained from these measurements (Eisterer and Weber, 2000).

G2.5.3.6 Surface and Geometric Barriers Bean’s critical state model is based on vortex pinning within the bulk of the superconductor, which defines the bulk critical current density. However, the surface or the geometry of the sample can impede flux entry or exit and thus provide another source of irreversibility, because this barrier exhibits an asymmetry between the increasing and the decreasing field branch (Bean and Livingstone, 1964; Brandt, 1999; Zeldov, 1994). Although the corresponding surface currents contribute to a macroscopic transport current as well, their contribution to the magnetic irreversibility is usually not the quantity of interest, because they are not a bulk property of the material, but depend on the sample geometry. These effects are not helpful for a proper determination of the bulk Jc , because it is difficult to eliminate or separate them from the bulk properties. Fortunately, the surface and geometric contributions are negligibly small in most cases. They add to the overall vortex pinning significantly only in very clean materials (single crystals with very weak bulk pinning) or materials with a very large surface-to-volume ratio, such as powdered samples or thin films (Harrington et al., 2009). The effect of a large surface barrier on the magnetization loop is illustrated in the upper panel of Figure G2.5.8. The curve is highly asymmetric between the increasing and the decreasing field branch. The magnetic moment is nearly zero on the latter except in the low-field region. In the lower panel, the magnetic moment is decomposed into irreversible and reversible components according to Equations (G2.5.15) and

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FIGURE G2.5.8 Hysteresis loop with a significant contribution of the surface barrier (upper panel). Decomposition into the apparent reversible and irreversible contributions (lower panel).

(G2.5.16), respectively. At first glance the resulting mrev could result from the equilibrium magnetization Mrev of the superconductor, which is however not the case. One indication is the nearly vanishing m on the decreasing field branch, which is typical for surface effects, and is not expected in general if Mrev becomes important (cf. Figure G2.5.5). It could be however caused coincidentally by a compensation of the equilibrium magnetization and bulk pinning, and a careful analysis is necessary for drawing conclusions. If mrev stems from Mrev, the slope around H=0 has to be in agreement with Equation (G2.5.19), the peaks should be at ±Hc1(1−Dm) and Mrev(H) has to assume the value of Hc1 there and follow the expected behavior (e.g. Brandt, 2003) at high fields with deviations smaller than the self-field from Mirr. The reversible contribution shown in Figure G2.5.8 is partly caused by surface and/or geometric barriers which are different for flux entry and exit, thus Equations (G2.5.12) and (G2.5.13) are not valid, and the barrier effects contribute to the reversible magnetic moment as defined by Equation (G2.5.15). Since m irr is by definition [Equation (G2.5.15)) symmetric upon field reversal, it cannot predict asymmetric components of the overall magnetic hysteresis. The calculated m irr also does not correspond to the bulk Jc. The irreversible properties (m irr+, m irr−) can only be assessed if the equilibrium magnetization is known or can be modeled [e.g. by the interpolation formulae given by Brandt (2003)] and subtracted. However, the separation into bulk and surface pinning remains an issue. The best way to avoid problems with surface or geometric barriers is by choosing samples with a small surface-to-volume ratio and avoiding very smooth and flat surfaces, whenever possible.

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G2.5.3.7 Anisotropy and Angle-Resolved Magnetization Measurements The Bean model assumes an homogeneous, isotropic current density. Many superconductors are anisotropic, and the resulting Jc anisotropy is of interest for theory and application. Jc potentially varies with the orientation of the current within the crystal lattice and, at fixed current orientation (e.g. parallel to the ab-planes), with the crystal orientation with respect to the applied field. The anisotropic Bean model is a useful extension for anisotropic superconductors if the current flow remains parallel to the sample surfaces. Depending on the current orientation it assumes to different critical current densities, J ca and J cc, in a cuboidal sample. Here a and c refer to the edges of the cuboid which are perpendicular to the applied magnetic field. (They may not be equivalent to the crystallographic directions.) The irreversible magnetic moment is given by

mirr =

1 a 2 c J ca a J ca for J c abc 1 − ≥ 4 c J cc 3a J cc

(G2.5.20)

mirr =

1 c 2 a Ja a J cc J c a bc 1 − for ≤ cc a 4 c Jc 3c J c

(G2.5.21)

J ca and J cc cannot be simultanously derived from a single magnetization loop. In case of a very thin sample or a very high current anisotropy, the second term in the parenthesis of the equation can be neglected, and either J ca or J cc is obtained. The validity of this approach can only be guaranteed if the order of magnitude of the current anisotropy is known. Many superconductors are uniaxial systems with J ca = J cb ≡ J cab ≠ J cc (here a, b and c refer to the crystallographic axes, which are assumed to be parallel to the edges of the cuboid). In this case, both J cab and J cc can be derived for a crystal with distinctly different a and b. The crystal has to be measured twice, with a and b parallel to the field, respectively. Then the two equations for the two resulting moments can be solved for J cab and J cc. The equation for H||b is either Equation (G2.5.20) or (G2.5.21) with J ca replaced by J cab. For the other field orientation (H||a), the roles of a and b in the equations have to be exchanged. The correct choice of Equation (G2.5.20) or (G2.5.21) has to be made self-consistently. Anisotropy effects can be studied by means of angleresolved magnetization measurements in a vector magnetometer. Beside the anisotropy of the material, the form anisotropy has to be considered. Both effects potentially lead to a tilt of the intrinsic field B with respect to the applied field H. At fields below Hc1, the demagnetization effect amplifies field components which are perpendicular to large surfaces, hence the magnetization rotates toward the smallest sample dimension. This is also true in the opposite limit of a fully penetrated sample. In thin samples, the currents remain parallel to the large surface in a wide angular range, e.g. in

FIGURE G2.5.9 Geometry of current flow in thin samples induced by oblique magnetic fields (Mishev et al., 2015). The angle between the surface normal and the applied magnetic field is denoted by θ. The surface normal is parallel to the crystallographic c-axis in typical superconducting single crystals. When the crystal is rotated about an axis parallel to a (a), the Lorentz force is maximum (J c B) for J c = J ca and J cb Bcosθ for J cVLF = J ca . (b) Rotation about an axis parallel to c.

samples with an aspect ratio of 10, the currents align with the large surface if the angle between this surface and the magnetic field is above 10°. Although all currents flow parallel to the ab-planes in this case, they are not the same along a and b, because of a different Lorentz force density J × B. The situation is illustrated in Figure G2.5.9. The Lorentz force varies as cos(θ) for the currents flowing perpendicular to the rotation axis ( J cVLF), while the force on the parallel currents (Jc) remains constant. If the length of a differs significantly from the length of b, the maximum and the variable Lorentz force currents can be separated by measuring the sample twice (rotated by 90° about c), using the procedure outlined above [Equations (G2.5.20) and (G2.5.21)]. The maximum Lorentz force currents are usually the quantity of interest, flux cutting phenomena however, can be investigated by means of J cVLF ( θ ) (e.g. Clem et al., 2011). The complication of the variable Lorentz force currents for the assessment of Jc can be avoided in films and tapes by choosing the sample length appropriately for the geometry of the coil set of the VSM. The idea is to place the variable Lorentz force currents at positions where the sensitivity of the instruments becomes negligible (Hengstberger et al., 2011). The in-plane critical current of a thin sample can be assessed in a wide angular range, which is only limited by the penetration field needed to fully magnetize the sample. Since only the field component which is perpendicular to the film induces currents, the penetration field diverges as cos(θ) −1 near the parallel orientation. The penetration field of a 1-µm-thick film with a width of 4 mm and a critical current density of 10 MA/cm (which is not unusual in HTS films or coated conductors at low temperatures) is of the order of 0.5 T at θ=0°, raising to several tesla at 80°. Since

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twice the penetration field is needed to fully magnetize a superconductor, a significant angular range centered about the parallel field orientation (θ=90°) becomes inaccessible in this case.

G2.5.3.8 Granularity The evaluation based on the critical state model assumes homogeneous properties of the material so that the geometry of the current loops is given by the sample geometry. This assumption can be violated in samples consisting of grains that are not perfectly coupled. In the limit of uncoupled grains, the magnetic moment is just the sum of the moments of all the grains, and for known grain geometry, the relations given in Table G2.5.2 can be used for an evaluation of the intragranular currents. For a cubic sample consisting of cubic grains (grain size a) this leads to J cintra = N

intra 6mirr 6m = 3 irr ag4 a ag

(G2.5.22)

where N = a3 / ag3 denotes the number of grains, ag the grain size and J cintra the critical currents within the grains generatintra ing the moments mirr . For coupled grains, intergranular currents with the corresponding (average) current density J cinter lead to a moment given by the sample geometry according to the formulae in Table G2.5.2. J cintra is equivalent to J cinter in case of strong coupling, and the situation becomes equivalent to single crystals. For weaker coupling, the currents flow on the length scales of both the grains and the sample, and their intra inter contributions add up: mirr = mirr . An upper limit for + mirr the ratio between the contributions of inter- and intragrain currents is given by inter mirr a J cinter = intra mirr ag J cintra

(G2.5.23)

for a cubic sample and cubic grains. If this ratio is either small or large, J cintra or J cinter can be derived approximately. Figure G2.5.10 shows a magnetization loop (dashed line) where both contributions are similar in magnitude leading to one peak in decreasing field (B>0), which corresponds to the intergranular current, and a second peak in increasing field intra (BH*), where a first-order correction is made by Equation (G2.5.15). A discussion of the low-field behavior of Jc without self-field correction is questionable. The self-field is closely related to H*, which restricts the field range where Jc can be evaluated by the standard methods outlined in subsection G2.5.3.1. The following points are related to the properties of the magnetometer, and refer to the history of the applied field. They are relevant if the disturbance is of the order (a few percent) of the penetration field H*. For instance, in wires with very fine filaments in the µm range, H* is below 1 mT for a Jc of the order of 104 A/cm2, which results in a significant influence of field changes of the order of 100 µT on the measured magnetic moment. This field change corresponds to a relative error of the order of 10 −5, if the applied field is in the range of a few tesla. • Field overshooting Since superconducting solenoids have a large inductance, the current source may overshoot (or undershoot) the desired current before the field becomes

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stable. This overshoot can be of the order of a few mT, and may reverse the field profile at the sample edges (H overshoot < H ∗) or the entire field profile (H overshoot > 2 H ∗). In the first case, the net magnetic moment is reduced, in the latter case, the moments on the increasing and decreasing field branches are exchanged (the magnetic overshoot < H ∗ ). moment changes orientation around H H overshoot can be reduced by reducing the field sweep rate. • Field inhomogeneity The magnetic field varies spatially within a solenoid. Since the sample is moved over a distance of several centimeters in a SQUID magnetometer, the Bean profile is changed by the varying magnetic field. Although field gradients in state-of-the-art systems are minimized, some imperfections remain, in particular, if the sample is large so that the edges are not close to the axis of the solenoid. For fields in the tesla range, field inhomogeneities of the order of 10 −3 (field specifications are often much better but refer to the field along the axis of the magnet) lead to field variations in the mT range, which easily exceed H* near the irreversibility line, where H* together with Jc converge to zero. The sample normally moves symmetrically about the center of the magnet where the field is highest. Thus, the field first increases, then decreases. This leads to an asymmetry in the raw measurement signal (voltage at the sensor as a function of sample position, cf. Figure G2.5.11), which can be saved optionally in most SQUID systems and inspected if necessary. Alternatively, the scan length can be changed. A larger scan length inherently increases the field inhomogeneity, and the derived moment changes, if field inhomogeneity is an issue. • Sample vibrations – shaking field The sample rod and holder are often fragile, and they can potentially move to some extent in an unwanted direction. This movement may lead to a tilt of the sample, which generates a field component transverse to the desired field orientation. This field component can be quite significant, e.g. at 5 T a tilt of 1° leads to a transverse field of above 80 mT. In particular, if the sample holder starts to swing or oscillate, the resulting transverse AC field (“shaking field”) can destroy the Bean profile in the sample, even if it is smaller than H* Mikitik and Brandt (2003). • Sample size A SQUID magnetometer as well as a VSM is calibrated by means of a small sample, typically made of nickel or palladium. For much larger samples, the calibration becomes invalid, leading to inaccurate results. The resulting error depends on the geometry of sample and the pick-up coil arrangement and can be minimized by re-calibrating the magnetometer with a nickel specimen having the sample geometry.

Handbook of Superconductivity

G2.5.6 Flux Creep It was assumed so far that the current density Jc does not change during the measurement. As discussed in Section G2.5.2, current transport in Type II superconductors is not entirely loss-free in the mixed state, and J(Ec) is measured at a representative electric field. Implications of a changing current density caused by a decreasing electric field as well as possibilities to derive J(E) are discussed in this section.

G2.5.6.1 Relaxation Measurements The magnetic moment is measured as a function of time during relaxation measurements (see for instance, the review article by Yeshurun et al., 1996). The electric field decreases exponentially with the current density (E ∝ e αJ ∝ e βm) in the Kim–Anderson model for J J c0 , with α = Jc0Uk0BT given implicitly by a comparison with Equation (G2.5.6). J and m (hence α and β) are related via the geometry of the sample (cf. Table G2.5.2). The electric field is balanced by the inductivity of the current loops (E ∝ − ddtφ ∝ − dJdt ∝ − dm dt ): e βm ∝ −

dm dt

(G2.5.25)

The solution of this equation can be written as kT m = m0 K − B ln (t + t0 ) U0

(G2.5.26)

m0 is the magnetic moment resulting from Jc0, K and t0 are not independent from each other and defined by the electric field at t = 0. (For formal correctness all arguments occurring in logarithms are divided by their respective units.) The magnetic moment decreases with the logarithm of time for t t0 , which is widely observed in experiments (e.g. the uppermost curve in Figure G2.5.12). Moreover, since J ≈ J c0 in the flux creep regime, m0 can be approximated by the moment at the beginning of the measurement, m(t=0), and the activation barrier U0 can be obtained from the slope of m(ln(t)). The relaxation rate is usually given as percent per decade or by the normalized creep rate S := −

dlnm 1 dm kBT 1 1 dm =− ≈− = = dlnt m dlnt m (t = 0 ) dlnt U 0 n − 1

(G2.5.27)

The relaxation rate is thus Sln(10) ≈ 2.3S. The n-value denotes the exponent of the power law E ∝ J n , which is widely observed in resistive measurements. The relation between S and n [e.g. Eisterer and Weber (2000)] is useful for a comparison of these two techniques. The magnetization has a plateau for t t0 in the logarithmic representation (Figure G2.5.12), and t0 increases as the initial electric field decreases. Only the apparently linear part t > t0 of the curve should be used for determining the relaxation rate.

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

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through the sample. Thus, the effect is called somewhat misleadingly dynamic relaxation, but it is a direct consequence of the dependence of the shielding currents on the electric field, which is induced in this case by the changing external field. The absolute value of the electric field at the sample edge is F given by dB dt P , where F is the cross-section perpendicular to the applied field, and P its perimeter. dB dt is the field sweep rate. The change of the sample’s self-field can usually be neglected. Plotting the logarithm of J (or m) as a function of the logarithm of E (or the field sweep rate), the local slope just becomes n−1 which can be compared with transport measurements or the creep rate obtained from relaxation measurements via Equation (G2.5.27).

G2.5.7 Measurements of the Magnetic Field Profile (Local Jc) FIGURE G2.5.12 Decrease of the magnetic moment with the logarithm of time (Gurevich and Küpfer, 1993). All curves were recorded at the same field and temperature, and differ by the initial electric field, which was adjusted by changing the field sweep rate. The logarithmic decrease with time can be best seen in the uppermost curve, whereas the initial plateau becomes more prominent at a lower initial electric field.

The Kim–Anderson model assumes a linear activation barrier U=U0(1 − J/Jc0), which is not true in the general case. (Maley et al., 1990) proposed a technique for an experimental determination of U(J). From the generalization of Kim and Anderson’s ideas, E ∝ exp( −U ( J ) / kBT ), they obtained U ( J ) = T ( A − kBTln|dJ / dt|). dJ / dt can be easily calculated from dm / dt, and the constant A is chosen in such a way that relaxation data obtained at different temperatures overlap. This procedure provides the activation barrier U(J) in a wide range of J from relaxation measurements at different temperatures. The assessment of J(E) from relaxation measurements is straightforward. J is obtained from m with the usual relations (Table G2.5.2), E follows from Faraday’s law. The change of flux through the sample caused by the decaying self-field (Section G2.5.3) has to be either calculated numerically or, if ∗ the sample is not very thin, can be approximated by µ 0 F3 dHdt , with H* given by Equation (G2.5.17). F denotes the transverse (with respect to the magnetic moment) cross-section of the sample. This has to be divided by the perimeter to get an upper limit for E, which decreases toward the center. Since the outermost current loops contribute most to the magnetic moment, this estimate is sufficient for most purposes.

G2.5.6.2 Dynamic Relaxation The magnetic moment changes with the field sweep rate when measured in field sweep mode, as usually done in a VSM. Since it becomes smaller upon reducing the sweep rate, it “relaxes” already during field ramping, when flux lines are moving

Panels A and C in Figure G2.5.13 illustrate the profile resulting from critical currents flowing homogeneously along the sample geometry. The case of totally decoupled grains is shown in panels B and D, where all current loops are restricted to grains. If the typical grain size is smaller than the distance between the sample and the plane where the profile is recorded, the contributions of currents flowing in neighboring grains cancel each other in a first-order approximation (as sketched in panel D), and only the currents at the edges of the sample (dashed arrows) add to the net signal. The situation is analogous to a homogeneously magnetized material, where atomic dipoles instead of the dipoles generated by

FIGURE G2.5.13 Theoretical field profiles above a cubic sample (upper panels) for a perfectly connected or single crystalline sample (panel A) and for a sample of totally decoupled grains (B). Sketch of the corresponding current distributions [lower panels, (Hecher et al., 2016)]. The sample extends from x=−1 to x=1.

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FIGURE G2.5.14 Field profiles recorded above a cubic sample. The field profile in (A) is a superposition of the two ideal field profiles presented in Figure G2.5.13. The field distribution in (B) results predominantly from intergranular currents.

intragranular currents sum up to a surface current. The resulting field profile has steep gradients at the edges and a smooth local minimum above the center of the sample (panel B). Figure G2.5.14 presents two field profiles recorded by scanning Hall probe microscopy. The field in the right panel is dominated by intergranular currents. An evaluation of Jc from the magnetic moment (Figure G2.5.10, decreasing field branch, B > 0) as described in Section G2.5.3 yields correct values. The field profile in the left panel of Figure G2.5.14 (recorded on the increasing field branch, B < 0, Figure G2.5.10) results from inter- and intragranular currents, as evidenced by the pronounced change in slope near the sample edges. A correct evaluation of the corresponding current densities from magnetization loops is not possible in this case. On the other hand, their average values can be obtained by fitting a superposition of the two ideal field profiles shown in the upper panels of Figure G2.5.13 to the experimental data.

G2.5.7.1 Calculation of the Local Current Density from the Field Profile (Inversion) If the sample is thin, the local current density (averaged over the thickness) can be obtained free of any assumptions by an inversion of the Biot–Savart law: B( r ) =

J × r′

∫ r − r′

3

d 3r ′

(G2.5.28)

Most algorithms use the grid of the measurement for the discretization, by assuming rectangular current loops in the sample centered on each point of the measurement grid (Figure G2.5.15). The current I j (more precisely, the sheet current density) of each current loop generates a magnetic field Bi at the position of each measured point. This leads to a system of linear equations Bi = ∑ j M ij I j, which can be solved numerically. The

FIGURE G2.5.15 algorithms.

Sketch of the current grid used in most inversion

matrix elements M ij result from the Biot–Savart law [Equation (G2.5.28)]. For rectangular current loops, they are given by: Mij =

µ0 ΔxΔy atan Δz Δx 2 + Δy 2 + Δz 2 4π

dy dx Δy = yi − y j + Δz = h 2 2 dy Δz = h + c dx Δx = x i − x j − Δy = yi − y j − 2 2 Δx = x i − x j +

(G2.5.29)

x, y denote the coordinates of the measured points, dx and dy are the grid width in x- and y-direction, h is the gap between the sample surface and the measurement plane, and c is the sample thickness. (The indefinite integral has to be evaluated at the eight corners of the rectangular current loop.) Note that the matrix M ij often becomes very large because its number of elements is equal to the square of the number of measured points. Special algorithms were developed for solving these huge systems of linear equations, for instance, the method of conjugated gradients using the fast Fourier transform (Wijngaarden et al., 1996; Jooss et al., 1998) or a method based on the particular geometry of the matrix (Toeplitz block Toeplitz matrix), which is a consequence of the translational invariance of the Biot–Savart law (Hengstberger et al., 2009). For obtaining the (average) local current density between two points k and l, the corresponding Ik and Il have to be subtracted from each other and the result divided by dx (for currents along y) or dy. An example of a field profile above a coated conductor and the calculated local current density (absolute value) is shown in Figure G2.5.16. The method is not only powerful for the observation of inhomogeneities, but also for quantifying the current anisotropy or the low-field behavior of the critical current (Lao et al., 2017). Two issues result from the discretization. First, the distance between the measurement plane and the sample surface should be similar to the grid width, i.e. h ≈ dx ≈ dy. If h is too small, the current discretization becomes too coarse for modelling a continuous current distribution. If h is too large, the condition

Magnetic Measurements of Critical Current Density, Pinning, and Flux Creep

225

FIGURE G2.5.16 Magnetic field profile above a coated conductor recorded by scanning Hall probe microscopy (upper panel). Map of the absolute values of the local current densities obtained by inversion (lower panel).

number of the problem becomes large, and small errors in the measured values lead to huge errors in the derived current densities (Eisterer, 2005). (The condition number, K ≥1, predicts the worst case error in the derived current density as its product with the experimental error: ΔJ ≤ KΔB.) In addition, h should not be much smaller than the sample thickness to maintain a reasonable condition number ( 1).

G2.6.3 Experimental Setup G2.6.3.1 AC Susceptometer AC susceptibility measurements are often performed with a coaxial mutual-inductance coil system. It typically consists of a primary excitation coil, a secondary pick-up coil, and a secondary compensation coil. The latter two are connected in series opposition. The reason for having three coils is that in the absence of a sample there is no measured signal because the signal in the secondary coils is compensated. This general arrangement is applied to different situations, the most common being that the two secondary coils are concentric with the primary coil Gömöry (1997), see Figure G2.6.1.

FIGURE G2.6.1 Simple scheme of an AC susceptometer in magnetometric measurement. The primary coil is the source of the external applied field. The sample is surrounded by a pick-up and a compensating coil, forming the secondary coils, connected in series opposition. The signals from the pick-up coil are collected in a lock-in amplifier to separate the real and the imaginary part. Commonly, the lock-in amplifier also controls the primary coil that sets the reference phase.

When a magnetized sample is present in the AC susceptometer, the induced magnetization due to the AC magnetic field created by the primary coil results in an induced electromotive force ε detected by the secondary coils system that satisfies Couach and Khoder (1991) ε=

− ∂Φ = µ 0 αVωH 0 χ m , ∂t

(G2.6.8)

where α is a constant depending on the secondary coils design, and V the sample volume. However, ε is off-balanced with respect to the primary field. Thus, both ε and the measured susceptibility χ m are complex numbers, which can be decomposed into a part in-phase with the signal and another out-of-phase. In a magnetometric measurement (Figure G2.6.1), the pickup coil measures averages of magnetization (magnetic moment per unit volume) of the sample over its volume M (t ) V , which depends on time. The general complex susceptibility is defined as this magnetization divided by H0 and the real and complex part of the AC susceptibility are the first harmonics of the Fourier transform [see Equations (G2.6.4) and (G2.6.5)] χ′ =

µ 0ω πH 0

2π

∫ M (t ) 0

V

cos(ωt )dt ,

(G2.6.9)

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χ′′ =

µ 0ω πH 0

2π

∫

M (t ) V sin(ωt )dt .

(G2.6.10)

is useful to define the magnetometric demagnetizing tensor Nm as

0

Hd,vol = − N m M vol , Both components are measured by using a lock-in amplifier. If the current fed to the primary coil is used as a reference for the lock-in amplifier, then the in-phase voltage (resistive component) is directly proportional to the imaginary component of the susceptibility, χ′′ , whereas the out-of-phase voltage (inductive component) is proportional to the real part of the susceptibility, χ′ : χ′ =

− ε ′′ , µ 0 αωH 0V

(G2.6.11)

χ′′ =

−ε′ . µ 0 αωH 0V

(G2.6.12)

In fluxmetric measurements, the pick-up coils measure averages of the magnetic flux threading a cross-sectional surfaces of the sample. Average magnetization is defined as M (t ) S = µS0 Φ (t ) S − H 0 cos(ωt ), which also depends on time. Equations (G2.6.9)–(G2.6.12) are then used with this magnetization. More details on AC susceptometer design, construction, and calibration can be found in Goldfarb, Lelental, and Thompson (1991); (Rillo et al. 1991); Couach and Khoder (1991); Gömöry (1997).

G2.6.3.2 Demagnetizing Corrections In an AC susceptibility measurement, in order to extract the intrinsic properties of the samples from the value of the measured χ it is necessary to take into account the demagnetizing effects arising from the finiteness of the measured sample. This is so because the field involved in the χ′ and χ′′ definitions [Equations (G2.6.4) and (G2.6.5)] is the applied magnetic field. In this way, Equation (G2.6.4) and (G2.6.5) actually define extrinsic susceptibilities, χEXT, since they represent the ratio of the magnetization with respect to the applied magnetic field. An often more useful intrinsic susceptibility χINT requires knowing the relation between the applied field and the actual local magnetic field at all points of the sample. Although the applied magnetic field in the sample is usually known, the actual local field is the sum of the applied plus the demagnetizing fields Hd created by magnetic poles in the magnetized sample. Demagnetizing field strongly depends on the sample geometry, which is not easy to calculate or to determine experimentally. Therefore, χEXT is what is measured in an AC susceptometer, but this value is characteristic of the sample (including its geometry), whereas χINT is characteristic of the material. In order to consider the effect of demagnetizing fields in magnetometric measurements such as AC susceptibility, it

(G2.6.13)

where the subscript ‘vol’ refers to the average over the sample volume. In general, Hd,vol and Mvol are not parallel to each other, even if the magnetic material is linear, homogeneous, and isotropic (LHI), because the sample geometry may not be symmetric. For the cases that Bvol, Md,vol and Hd,vol are parallel to the applied field Ha, one can ignore the vectorial nature of these quantities. Then, the demagnetizing tensor is reduced to a scalar, called demagnetizing factor and can be defined as Chen, Brug, and Goldfarb (1991) N m = − H d,vol / M vol .

(G2.6.14)

In this case, the external and internal susceptibilities can be related as χ EXT =

χINT . 1 + χINT N m

(G2.6.15)

If the Nm dependence on χINT for a given geometry is known, this relation allows the extraction of χINT from the χEXT obtained by AC susceptibility measurements. Demagnetizing factors have been calculated for bodies with practical geometries such as cylinders Chen, Brug, and Goldfarb (1991); Chen, Pardo, and Sanchez (2002a), ellipsoids Chen, Pardo, and Sanchez (2002b) and rectangular prisms Chen, Pardo, and Sanchez (2005).

G2.6.4 Different Models for the Superconductors G2.6.4.1 Superconductors Described with the London Equation One of the simplest models for describing the induced shielding currents in the superconductor is the London model London and London (1935). It describes phenomenologically the magnetization of Type I superconductors and that of Type II below the first critical field Bc1. It can be characterized by the so-called London equation for the magnetic flux density, B, B + λ 2 ∇ × ∇ × B = 0,

(G2.6.16)

where λ is a length scale called the London penetration depth. The solutions of this equation, together with the appropriate boundary conditions yield the B-field and current density, J = µ 0−1∇ × B, distributions inside the material. From these distributions the magnetization is obtained and the susceptibilities can be calculated. The London equation describes reversible screening currents and, thus, χ′′ = 0 and only the fundamental component χ′ can be different from zero.

231

AC Susceptibility

As an example, consider a superconducting slab that occupies the region x ∈[ −W ,W ], and is much larger (mathematically infinite) in the y and the z direction. A uniform field is externally applied in the z direction. With this symmetry, there is neither z nor y dependence of the current and field distribution inside the material. In this case, the London equation becomes Bz ( x ) − λ 2

∂2 Bz ( x ) = 0, ∂x 2

(G2.6.17)

with the boundary conditions (there are no demagnetizing fields in this case) Bz ( x = ±W ) = µ 0 H a . The solution for the field and current distribution at x < W is cosh λx , cos λx

(G2.6.18)

H a sinh λx . λ cos λx

(G2.6.19)

Bz ( x ) = µ 0 H a

J y (x ) = −

For x > W , Bz(x) = µ0Ha and there are, obviously, no currents. When W λ , one finds the classical exponential field decay, and when W λ , the sample hardly reacts to the applied field ( J y → 0 and Bz ( x ) → µ 0 H a ). The magnetization Mslb,L is given by Equation (G2.6.1) (‘slb’ and ‘L’ subscripts stand for ‘slab geometry’ and ‘London model’, respectively), M slb,L =

1 2W

W

λ

W

∫ J (x )x dx = − H 1 − W tanh λ . y

a

(G2.6.20)

−W

A factor 1/2 in the magnetization equation [Equation (G2.6.1)] has been omitted because of the contribution of the U-turn of the currents at infinite. As expected from the linear nature of the London equation, the magnetization is proportional to the applied field, so the susceptibility is χ ′slb,L =

M slb,L W λ = −1 − tanh , Ha W λ

(G2.6.21)

which tends to −1 for λ 0 . Actually, there is a maximum in the χ′′ as a function of the frequency (or δ = 2ρf / µ 0 ω ). For high frequencies, δ → 0 and we recover the ideal diamagnetic

(G2.6.29)

The boundary conditions for this equation are H(±W) = H0. The solution is H ( x ) = H 0 sech ka cosh kx ,

(G2.6.30)

where k = (1 + i ) 2ων = 1δ+i , defining δ as a length scale δ = 2ρf / µ 0 ω , comparable with the skin depth in normal metals. The currents and the magnetization can be found as J y = − H 0 Re[e iωt k sech ka sinh kx ],

Mz =

1 2W

(G2.6.31)

W

∫ xJ dx = H χ′ cos ωt + H χ′′sin ωt , y

−W

0

0

(G2.6.32)

FIGURE G2.6.3 Real and imaginary components of AC susceptibility of a slab, a long cylinder, and a sphere as a function of the skin depth normalized to the semiwidth of the slab or the radius for the long cylinder and the sphere.

AC Susceptibility

233

screening χ ′ → 1/ ( N m − 1), χ′′ → 0. For low frequencies, δ W , R and the superconductor is transparent to the magnetic field, χ′ = χ′′ → 0. Equation (G2.6.28) is an example of heat-diffusion equation. Actually, several solutions for different geometries and boundary conditions can be obtained using the known results from the conduction of heat in solids Carslaw and Jaeger (1986). Formally, the equation is also the same as that of eddy current problems, so one can also take profit of the solutions for more complex shapes as well Smythe (1989).

G2.6.4.3 Critical-State Model In hard Type II superconductors, flux lines are strongly pinned and they cannot move easily. A gradient of density of magnetic flux is established, and a (critical) current density circulates in the superconductor. The critical-state model (CSM) Bean (1962); Bean (1964) approximately describes the magnetization response of the superconductor in this strongly pinned case. According to the CSM, any variation of the local magnetic field will produce a constant current density Jc to flow in the superconductor, its sign being determined by the variation of the local field Navau, Del-Valle, and Sanchez (2013). Jc is, thus, a crucial parameter in this model. In general, Jc may depend on the internal magnetic field, although we consider here for simplicity that it is independent of it (see Section G2.6.5.2 for further comments). The model assumes zero reversible magnetization and zero lower critical field Bc1. Within this model, the current distribution (and thus the magnetization) is a hysteretic function of the applied field. It can be calculated as follows. From a given internal field distribution, a variation of the external applied field results, in general, in an exterior region with constant critical-current density Jc where the magnetic field satisfies ∇ × H = ± J c and an interior region in which the field is frozen. The critical-current region increases from the surface of the superconductor inwards as long as the sweeping sign of the external field variation is not changed. As an example, consider a slab of width 2W and an applied field Ha along the z direction. Starting from a virgin sample (no currents inside), when an external field is applied over the sample, currents with ±Jc enter the regions x 0 ≤ x ≤ W , where the field decays linearly from the surface inwards down to zero at |x| = x0. Inside the region |x| < x0, the current density is zero, as well as the magnetic field. x0 is calculated to be W − Ha /Jc. When the applied field exceeds JcW, there is no room for currents to penetrate deeper (x0 has become zero) and the sample becomes fully saturated. Further increase of the applied field does not result in further magnetization. The magnetization is easily calculated as H a2 − Ha + 2 J cW M ini = J W − c 2

If an AC external field is applied, it can be seen Brandt (1998) that when the current density is independent of the local field, the reversal magnetization Mrev (the branch of the magnetization loops that goes from the maximum applied field H0 to −H0) and the returning curve Mret (applied field from −H0 to H0) can be obtained from the virgin Mini magnetization as M rev ( H a ) = M ini ( H 0 ) − 2 M ini ([H 0 − H a ]/ 2),

(G2.6.40)

M ret ( H a ) = − M ini ( H 0 ) + 2 M ini ([H 0 + H a ]/ 2).

(G2.6.41)

This property is valid not only for slabs, but for other common geometries like long cylinders, as long as critical state with constant Jc is considered. The above equations can be used to find the different components of the AC susceptibility. In particular, the fundamental harmonics for an applied field of the form Ha = H0 cos ωt can be found as (subscript ‘CS’ stands for ‘critical state’) χ′slb,CS =

2 πH 02

− H0

∫M

H0

rev

−H (H ) dH , 2 2 H0 − H

χ′′slb,CS = −

2 πH 02

(G2.6.42)

− H0

∫M

rev

(H )dH .

(G2.6.43)

H0

In the present slab case, one can perform all the calculations and find (defining Hp,slb = JcW and h = H0/Hp,slb) χ′slb,CS = h−2 if h < 1, 2 2 −1 h− 2 3(h − 2)h cos ( h ) − 2 h − 1(h(3h − 4) + 4) if h > 1. 6πh 2

2h 3π χ′′slb,CS = 2(3h − 2) 3πh 2

(G2.6.44)

if h < 1, (G2.6.45) if h > 1.

Several other geometries have been analytically solved. For the case of a long cylinder of radius R, defining now Hp,cyl = JcR and h = H0/Hp,cyl, one has χ′cyl,CS = 5h 2 − + h −1 if h < 1, 16 −1 h − 2 2 2(h − 2) h − 1(h(15h − 8) + 8) − 3h (h(5h − 16) + 16)cos h if h > 1. 48πh 2

(G2.6.46)

if H a < J cW , (G2.6.39) if H a > J cW .

2(h − 2)h − 3π χ′′cyl,CS = 2(2 h − 1) 3πh 2

if h < 1, if h > 1.

(G2.6.47)

Handbook of Superconductivity

234

Of particular interest from the experimental point of view are the cases of a thin strip (subscript ‘st’) or thin disk (subscript ‘dsk’), with thickness d much smaller than the width of the strip 2W or the radius of the disk R. In these cases, the expressions are complicated and only for some cases fully analytical in terms of standard functions. In particular, for thin strips Brandt and Indenbom (1993) W χ′st,CS = d χ ′′st,CS =

h

∫

−

−h

2h′ tanh ( h −2h′ ) h

2

(h − h′ )(h + h′ )

dh′ , (G2.6.48)

W 4log(cosh h) − 2hW tanh h , (G2.6.49) d h2

and for thin disks Clem and Sanchez (1994) −h

−h′ 8R [S(h) − 2S(h − h′ / 2)] 2 dh′ , χdsk ′ ,CS = − 3πd h − h′ 2 h

∫

(G2.6.50)

−h

8R [S(h) − 2S(h − h′ / 2)]dh′ , (G2.6.51) χdsk ′′ ,CS = − 3πd

∫ h

where h = H0/Hp being Hp = Hp,st = Jcd/π for thin strips and Hp = Hp,dsk = Jcd/2 for thin disks, and S( x ) =

1 −1 1 sinhx . (G2.6.52) + cos cosh x cosh 2 x 2x

The results of all these expression can be seen in Figure G2.6.4. The overall behavior is similar in all cases. Note, however, the different normalizations used for the different geometries. The real part of the AC susceptibility at low fields tends to the ideal diamagnetic value. It is important to note that for the thin samples, where the demagnetizing effects are very important, the susceptibility tends to a value of the order of the ratio width/thickness (χ0,dsk = 8R/3πd for the disk, χ0,str = πW/2d for the strip), whereas when there is no demagnetizing effect, the low fields χ′ tends to −1. In the critical-state model, the low-field limit corresponds to the high critical-current density limit, since the typical fields at which the sample becomes saturated are of the order of Jc times the width (or radius) for the long samples in longitudinal applied fields and Jc times the thickness for the thin samples in perpendicular field Navau, Del-Valle, and Sanchez (2013). Another important characteristic is the presence of a maximum in the imaginary component of the susceptibility. That maximum ( χ′′m ) and its position with respect to the applied field (Hm), which depends on the geometry, are of special importance in the measurements, as we will see below. In particular, the position and value for the maximum are χ′′ = (0.238732, 0.212207, 0.371440 W/t, 0.321028 R/t) and Hm = (1.33333 JcW, JcR, 2.46421 Jct/π, 3.05194 Jct/2) for the corresponding geometries (slab, long cylinder, thin strip, thin disk).

FIGURE G2.6.4 Real and imaginary components of the AC susceptibility for a slab, a long cylinder, a thin strip and a thin disk as a function of the amplitude magnetic field H0. Susceptibilities for the thin samples are multiplied by β being β = (1, 1, d/W, d/R) for (slab, long cylinder, strip, disk) where W is the semiwidth for the slab and strip, R the radius for the long cylinder and disk, and d the thickness for thin strip and disk. The amplitude field is normalized to the typical penetration field Hp = Jc(W, R, d/π, d/2) for (slab, long cylinder, strip, disk).

There are other geometries in which one can calculate the susceptibility based on the critical-state principles. As a general behavior, the real susceptibility goes from an initial negative value at low fields (high-current densities) of the order of unity for long samples and of the order of width/thickness for thin samples. The imaginary part has a maximum of values of the order or 0.2–0.5 for long samples and 0.2–0.5 times the width/thickness for thin samples. These maxima are achieved at amplitude fields of the order the typical penetration field, which is of the order of Jc times the width for long samples and Jc times the thickness for thin films Brambilla and Grilli (2013); (Chen et al. 2010; Chen et al. 2011; Navau et al. 2008; Navau et al. 2005; Fabbricatore et al. 2000); Mawatari (1996).

G2.6.5 Practical Applications G2.6.5.1 Determination of Superconductor Parameters AC susceptibility is a very useful tool to extract relevant superconducting properties from the experimental data. In this section we will discuss some examples. G2.6.5.1.1 London Penetration Depth (λ) In the superconducting samples where the London model is valid, the determination of the London penetration depth λ (or the Pearl penetration depth Λ in the cases of samples with thickness d < λ) is an important issue. λ can be obtained from AC susceptibility measurements if the scale of the samples (or internal structure dimensions such as grains) is comparable with λ (Sargánková et al. 1996); Guerrin, Alloul, and Collin (1994). As explained in Section G2.6.4.1, χ′ depends on λ and

AC Susceptibility

this dependence is specific for each geometry of the sample. So, by fitting the experimental temperature-dependent data χ′(T ) to the theoretical expressions χ′(λ ), one can find the appropriate λ(T ) dependence. In the cases where theoretical expressions of χ′(λ ) are unknown, as for example for thin rectangles or thin strips, one can use more sophisticated methods to determine λ by AC susceptibility measurements (Chen et al. 2008; Chen et al. 2012). G2.6.5.1.2 Linear Resistivity (ρf ) and Critical Temperature (Tc) In the linear diffusion model (Section G2.6.4.2), the resistivity ρf is the only material parameter controlling the flux dynamics, so the temperature dependence of the AC response arises basically from ρf (T). Since the transition from normal to superconducting state is always accompanied by a resistivity drop, the AC measurements of χ′(T ) can be used to determine the critical temperature Tc because χ′(T ) = 0 for T ≥ Tc . By using Equations (G2.6.33), (G2.6.35), and (G2.6.37) for slab, long cylinder, and sphere, respectively, together with the experimentally measured χ′(T ), one can determine ρf(T). To ensure that the conditions for linearity are maintained, one can compare the susceptibilities measured at different applied AC field amplitudes or various frequencies. G2.6.5.1.3 Critical-Current Density (Jc) In Section G2.6.4.3 we have seen that when the flux pinning is the main mechanism controlling the flux dynamics, then the critical-state model is the relevant modelling framework. In the simplest case of this model, where the critical-current density Jc remains constant throughout the sample (Bean’s approximation), one can roughly estimate Jc with a very simple method by realizing that the maximum (peak) of χ″ is reached when the AC field amplitude just exceeds the magnitude of the full penetration field Hp. Therefore, in the geometries where the Hp is known, either by means of exact expressions or approximated ones Navau, Del-Valle, and Sanchez (2013), Jc can be obtained. Experimentally, Jc(T) can be obtained by mapping the temperature dependence of the maximum in χ′′(T ) as a function of the applied AC field amplitude (Widder et al. 1995). Campbell Campbell (1971) introduced another method to obtain the spatial dependence of Jc from AC susceptibility measurements, when a DC field is superimposed. Jc can be extracted from the slope obtained when plotting of the AC field amplitude H0 versus depth, where the latter is obtained from the derivative of the magnetization with respect to H0 Campbell (1971). G2.6.5.1.4 Critical Magnetic Fields (Bc1 and Bc2) When the superconductor completely shields the magnetic B-field (Meissner state), the critical magnetic field B c1 can be determined by recognizing that, in conjunction with χ′′ = 0, χ′ = −1 for µ0H0 < Bc1 and χ′ = − µB0c1H0 for µ 0 H 0 ≥ Bc1

235

for long samples. Otherwise, χ′ = N m1 −1 as we have seen in Section G2.6.4.1. Similarly to the use of the drop in the linear resistivity that accompanies the transition from normal to superconducting state to determine Tc (Section G2.6.4.2), AC susceptibility data can be used to find the critical magnetic field Bc2 by analysing how a superimposed DC field shifts the onset temperature in the linear regime (Küpfer et al. 1987). However, this technique to determine Bc2 has fallen into disuse in favor of other more exact methods (Vlakhov et al. 1994).

G2.6.5.2 Beyond the Models’ Assumptions There are situations in which the general principles of AC susceptibility and the simple models to interpret the measured data presented above do not have full validity. For example, in Type II superconductors the flux lines can be unpinned thermally or by other means, yielding a flux movement and a dynamics which are not accounted in the previous models. An important set of results comes from considering a currentelectric field function J ( E ) = J 0 ( E / E )( E / Ec )1/p ,

(G2.6.53)

where J0 and Ec are critical-current density and a criticalelectric field. p is an exponent that ranges from p ∈[1, ∞)1. The cases p = 1 and p → ∞ correspond to the linear diffusion and the critical-state model, respectively, and, in between, the called intermediate regime (Civale et al. 1991) is established. In this last regime we find examples like the extreme flux flow Tákacs and Gömöry (1990) and the flux creep Kim, Hempstead, and Strnad (1962); Anderson (1962). In the extreme flux-flow model the diffusion is no longer linear, and a dependence of the AC field amplitude and frequency on the penetration depth appears. In the case of the flux creep, the flux lines are thermally activated and depinned originating a drift velocity of the flux lines that obeys the Arrhenius law. A large number of calculations of susceptibility and the associated losses have been undertaken using these models (Grilli et al. 2014). Another characteristic example is the surface pinning appearing when some surface barriers prevent the entering or exiting of the flux lines, influencing both real and complex susceptibilities Bean (1964); (Cesnak et al. 1984). It is not rare to find that the response of the superconductor is due not only to one single mechanism but to several of them. Since usually they interact with each other, the study of the global solution is often not straightforward. In the linear- and the critical-state models the assumption that the different parameters (ρf and Jc) are independent of the internal magnetic field is, in general, not fulfilled since they represent averaging of microscopic magnitudes that are affected by local magnetic fields. In the AC susceptibility technique it is frequent to superimpose to the AC applied field a constant DC external field HDC of magnitude greatly larger that the amplitude of the AC field. In that regime, ρf and Jc

236

can be considered roughly constant being ρf(HDC) and Jc(HDC). Repeating the experiment at different HDC fields, one can find the dependencies of the different parameters at different fields. Any model is necessarily a simplification of the ultimate real aspects involved in the response of a material. The agreement between experimental data and the results predicted by some of the models can thus help understanding of the physical mechanism involved in the material response. Varying the externally controllable variables (amplitude of the AC field, temperature, DC extra applied field, geometry, …) one can obtain the necessary clues to understand the different physical mechanisms involved in the magnetic response of the material, showing the power and possibilities of the AC susceptibility technique.

References Anderson, P. W. 1962. “Theory of flux creep in hard superconductors.” Physical Review Letters 9: 309–311 (October). Bean, C. P. 1962. “Magnetization of hard superconductors.” Physical Review Letters 8 (6): 250–253 (March). Bean, C. P. 1964. “Magnetization of high-field superconductors.” Reviews of Modern Physics 36 (1): 31–39 (January). Brambilla, R., and F. Grilli. 2013. “Thin-film superconducting rings in the critical state: the mixed boundary value approach.” Zeitschrift für angewandte Mathematik und Physik 66 (1): 1–29 (November). Brandt, E. H. 1998. “Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves.” Physical Review B 58 (10): 6506–6522 (September). Brandt, E. H. 2000. “Ac response of thin-film superconductors at various temperatures and magnetic fields.” Philosophical Magazine B 80 (5): 835–845 (August). Brandt, E. H. 2007. “Supercurrents and vortices in SQUIDs and films of various shapes.” Physica C: Superconductivity and its Applications 460-462: 327–330 (September). Brandt, E. H., and J. R. Clem. 2004. “Superconducting thin rings with finite penetration depth.” Physical Review B 69 (18): 184509 (May). Brandt, E. H., and M. Indenbom. 1993. “Type-IIsuperconductor strip with current in a perpendicular magnetic field.” Physical Review B 48 (17): 12893–12906 (November). Campbell, A. M. 1971. “Interaction distance between flux lines and pinning centres.” Journal of Physics C: Solid State Physics 4 (18): 3186. Carslaw, H. S., and J. C. Jaeger. 1986. Conduction of Heat in Solids. 2nd. Oxford Science Publications. Cesnak, L., F. Gömöry, J. Kokavec, and S. Takàcs. 1984. “AC losses in multilayer superconducting tapes.” Cryogenics 24 (3): 119–126 (March). Chen, D. X., J.A. Brug, and R.B. Goldfarb. 1991. “Demagnetizing factors for cylinders.” IEEE Transactions on Magnetics 27 (4): 3601–3619 (July).

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Chen, D. X., C. Navau, N. Del-Valle, and A. Sanchez. 2008. “Effective penetration depths of a thin type-II superconducting strip.” Superconductor Science and Technology 21 (10): 105010 (October). Chen, D. X., C. Navau, N. Del-Valle, and A. Sanchez. 2012. “Alternating-Current Susceptibility, Critical-Current Density, London Penetration Depth, and Edge Barrier of a Type-II Superconducting Film.” In Superconductivity: Theory, Materials and Applications Books, 535–564. Nova Science Publishers. Chen, D. X., C. Navau, N. Del-Valle, and A. Sanchez. 2014. “Determination of London penetration depth from ac susceptibility measurements of a square superconducting thin film.” Physica C: Superconductivity 500: 9–13 (May). Chen, D. X., E. Pardo, and A. Sanchez. 2002a. “Demagnetizing factors of rectangular prisms and ellipsoids.” IEEE Transactions on Magnetics 38 (4): 1742–1752 (July). Chen, D. X., E. Pardo, and A. Sanchez. 2002b. “Demagnetizing factors of rectangular prisms and ellipsoids.” IEEE Transactions on Magnetics 38 (4): 1742–1752 (July). Chen, D. X., E. Pardo, and A. Sanchez. 2005. “Demagnetizing factors for rectangular prisms.” IEEE Transactions on Magnetics 41 (6): 2077–2088 (June). Chen, D. X., A. Sanchez, C. Navau, and N. Del-Valle. 2010. Analytic expressions for critical-state ac susceptibility of rectangular superconducting films in perpendicular magnetic field. Chen, D.-X., G. Via, C. Navau, N. Del-Valle, A. Sanchez, S.-S. Wang, V. Rouco, A. Palau, and T. Puig. 2011. “Perpendicular ac susceptibility and critical current density of distant superconducting twin films.” Superconductor Science and Technology 24 (7): 075004 (July). Civale, L., T. K. Worthington, L. Krusin-Elbaum, and F. Holtzberg. 1991. “Nonlinear A. C. Susceptibility Response. Near the Irreversibility Line.” In Magnetic Susceptibility of Superconductors and Other Spin Systems, edited by R. A. Hein, T. L. Francavilla, and D. H. Liebenberg, 313–332. Springer US. Clem, J. R., and E. H. Brandt. 2005. “Response of thinfilm SQUIDs to applied fields and vortex fields: Linear SQUIDs.” Physical Review B 72 (17): 174511 (November). Clem, J. R., H. R. Kerchner, and S. T. Sekula. 1976. “ac permeability of defect-free type-II superconductors.” Physical Review B 14 (5): 1893–1901 (September). Clem, J. R., and A. Sanchez. 1994. “Hysteretic ac losses and susceptibility of thin superconducting disks.” Physical Review B 50 (13): 9355–9362 (October). Couach, M., and A. F. Khoder. 1991. “Ac Susceptibility Responses of Superconductors: Cryogenic Aspects, Investigation of Inhomogeneous Systems and of the Equilibrium Mixed State.” In Magnetic Susceptibility of Superconductors and Other Spin Systems, edited by R. A. Hein, T. L. Francavilla, and D. H. Liebenberg, 25–48. Springer US.

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Fabbricatore, P., S. Farinon, F. Gömöry, and S. Innocenti. 2000. “Ac losses in multifilamentary high-Tc tapes due to a perpendicular ac magnetic field.” Superconductor Science and Technology 13 (9): 1327–1337 (September). Genenko, Y. A., S. V. Yampolskii, and A. V. Pan. 2004. “Virgin magnetization of a magnetically shielded superconductor wire: Theory and experiment.” Applied Physics Letters 84 (19): 3921 (April). Goldfarb, R. B., M. Lelental, and C. A. Thompson. 1991. “Alternating-Field Susceptometry and Magnetic Susceptibility of Superconductors.” In Magnetic Susceptibility of Superconductors and Other Spin Systems, edited by R. A. Hein, T. L. Francavilla, and D. H. Liebenberg, 49–80. Springer US. Gömöry, F. 1991. Magnetic Susceptibility of Superconductors and Other Spin Systems. Edited by Robert A. Hein, Thomas L. Francavilla, and Donald H. Liebenberg. Boston, MA: Springer US. Gömöry, F. 1997. “Characterization of high-temperature superconductors by AC susceptibility measurements.” Superconductor Science and Technology 10 (8): 523–542 (August). Grilli, F., E. Pardo, A. Stenvall, D. N. Nguyen, and F. Gömöry. 2014. “Computation of losses in HTS under the action of varying magnetic fields and currents.” IEEE Transactions on Applied Superconductivity 24 (1): 78–110 (February). Guerrin, L., H. Alloul, and G. Collin. 1994. “Susceptibility measurements of the temperature dependence of λ(T) in aligned YBa2Cu3O7 powder samples.” Physica C: Superconductivity 235-240: 1797–1798 (December). Kim, Y. B., C. F. Hempstead, and A. R. Strnad. 1962. “Critical persistent currents in hard superconductors.” Physical Review Letters 9: 306–309 (October). Küpfer, H., I. Apfelstedt, W. Schauer, R. Flükiger, R. MeierHirmer, and H. Wühl. 1987. “Critical current and upper critical field of sintered and powdered superconducting YBa2Cu3O7.” Zeitschrift für Physik B Condensed Matter 69 (2-3): 159–166 (June). London, F., and H. London. 1935. “The electromagnetic equations of the superconductor.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 149 (866): 71–88 (March). Mawatari, Y. 1996. “Critical state of periodically arranged superconducting-strip lines in perpendicular fields.” Physical Review B 54 (18): 13215–13221 (November). Navau, C., N. Del-Valle, and A. Sanchez. 2013. “Macroscopic modeling of magnetization and levitation of hard type-II superconductors: The critical-state model.” IEEE Transactions on Applied Superconductivity 23 (1): 8201023–8201023 (February).

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Navau, C., A. Sanchez, N. Del-Valle, and D.-X. Chen. 2008. “Alternating current susceptibility calculations for thinfilm superconductors with regions of different criticalcurrent densities.” Journal of Applied Physics 103 (11): 113907 (June). Navau, C., A. Sanchez, E. Pardo, D.-X. Chen, E. Bartolomé, X. Granados, T. Puig, and X. Obradors. 2005. “Critical state in finite type-II superconducting rings.” Physical Review B 71 (21): 214507 (June). Pearl, J. 1964. “Current distribution in superconducting films carrying quantized fluxoids.” Applied Physics Letters 5 (4): 65–66. Plourde, B. L. T., D. J. Van Harlingen, D. Y. Vodolazov, R. Besseling, M. B. S. Hesselberth, and P. H. Kes. 2001. “Influence of edge barriers on vortex dynamics in thin weak-pinning superconducting strips.” Physical Review B 64 (1): 014503 (June). Rillo, C., F. Lera, A. Badia, L. A. Angurel, J. Bartolome, F. Palacio, R. Navarro, and A. J. van Duyneveldt. 1991. “Multipurpose Cryostat for Low Temperature Magnetic and Electric Measurements of Solids.” In Magnetic Susceptibility of Superconductors and Other Spin Systems, edited by R. A. Hein, T. L. Francavilla, and D. H. Liebenberg, 1–24. Springer US. Sarg´ ankov´ a, I., P. Diko, J. D. Tweed, C. A. Anderson, and N. M. D. Brown. 1996. “The preparation of uniform grain size ceramics from sedimented single-crystalline powder fractions.” Superconductor Science and Technology 9 (8): 688–693 (August). Smythe, W. R. 1989. Static and Dynamic Electricity. 3rd. Taylor & Francis. Tàkacs, S., and F. Gömöry. 1990. “Influence of viscous flux flow on AC magnetisation of high-Tc superconductors.” Superconductor Science and Technology 3 (2): 94–99 (February). Vlakhov, E. S., K. A. Nenkov, M. Ciszek, A. Zaleski, and Y. B. Dimitriev. 1994. “Superconducting and magnetic properties of melt-quenched Bi-2223 superconductors doped with Pb and Te.” Physica C: Superconductivity 225 (1-2): 149–157 (May). Vodolazov, D. Y., and I. L. Maksimov. 2001. “Distribution of the magnetic field and current density in superconducting films of finite thickness.” Physica C: Superconductivity 349 (1-2): 125–138 (January). Widder, W., L. Bauernfeind, M. Stebani, and H. F. Braun. 1995. “Nonlinear AC susceptibility and critical current of (Y1xPrx)Ba 2Cu3O7δ ceramics.” Physica C: Superconductivity 249 (1-2): 78–86 (July).

G2.7 AC Losses in Superconducting Materials, Wires, and Tapes Michael D. Sumption, Milan Majoros, and Edward W. Collings

G2.7.1 Introduction The AC losses of superconducting materials, in particular, in the form of wires and tapes, as well as the losses of the cables or coils made with them, have been a topic of interest since the advent of the first technical superconductors in the early 1960s. The subject area is wide, covering bulk superconductors, wires (strands), cables, coils, and devices as well as experimental, analytical, and computational approaches to their understanding. This chapter will focus on the AC losses of superconducting wires and tapes exclusively, although the conditions imposed on these wires and tapes from the cables, coils, and devices that they are used in will be considered. Additionally, the focus of this chapter will be on simple analytical theory and straightforward treatments, without the inclusion of numerical methods (to the extent possible). This is consistent with the main theme of this article, which is to give a broad-brush understanding of the loss in superconducting wires, keeping complicating details and secondary issues to a minimum. Presently, AC losses are of great interest for YBCO coated conductors, as well as wires and tapes from Bi2212 and Bi2223, MgB2, LTS, and the pnictide materials. Such wires and tapes are of interest for use as single strands, in cables, and in coils and devices including (i) time ramping magnets, (ii) motors and generators, (iii) fault current limiters, (iv) transformers, (v) superconducting magnetic energy storage devices, and (vi) transmission cables, as well as other applications. A considerable amount of research on loss in HTS conductors has been published, in addition to the work on LTS which proceeded it. As we being our review of the basic loss equations, there is one point which can be made right up front: most of the loss properties of these conductors has little to do with specific superconducting materials properties. Mostly the loss is a function of the (i) field, temperature, and current environment that the conductor is exposed to, (ii) the geometry of the conductor

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(is it a flat or round, is it filamented?), and (iii) the properties of the normal-state material that is used to form the total wire or tape composite. In fact, we will gain a substantial benefit from realizing this, and considering the AC loss work in LTS and HTSC as two parts of a whole. Thus, below we give some background on loss studies in multifilamentary NbTi and Nb3Sn conductors before starting in the next section to consider the various loss components and loss configurations of LTS and HTS conductors with a unified approach.

G2.7.2 Background Arguably the beginning of thinking about AC loss in superconductors stems from the work of C.P. Bean, who first described the critical state in 1962 [1]. Further work put the critical state model in its modern form [2, 3]. While Bean’s critical state model assumes a field-independent critical current density, Jc, there are various extensions or modifications which allow us to include a field and temperature dependence of Jc. The best known of these is the Kim model [4], where Jc = α/(B+B*), where α and B* are parameters, and B is the applied field. Advances in the processing of multifilamentary (MF) superconducting strand, particularly during the late 1960s and early 1970s [5] were accompanied by a burgeoning interest in large-scale applications of superconductivity [6]. Both DC- (high-field magnets for particle acceleration, magnetohydrodynamic-based power, fusion, etc.) and AC applications were being engineered. Among the AC applications being considered worldwide were generators, power transmission lines, transformers, and motors. Accompanying the development of AC applications during this period, the study of AC loss in low-temperature superconductors was being actively pursued. A group at the then Westinghouse Research and Development Center was very active in superconducting AC applications. An excellent and thorough description of loss,

AC Losses in Superconducting Materials, Wires, and Tapes

developed in the context of low-Tc superconductors including multifilamentary conductors under various excitation modes, is given in a book by Carr [7]; this work was recently updated to include HTS and coated conductors [8]. Now, just as before, the emerging availability of highperformance superconducting strand is again stimulating the design and eventual construction of superconducting machinery, but this time with HTS. With HTS conductors rapidly approaching maturity, and recognizing the presence of multifilamentary MgB2 and the developing pnictide conductors, AC applications are again being developed, including power transmission lines, transformers, motors, generators fault current limiters, and other devices. Much of the AC loss treatment for LTS is translatable to HTS conductors, the main difference is due to geometric differences in the conductors and the properties of the nonsuperconducting components, the choice of which is often related to differences in the processing conditions One other difference in the AC loss treatment for HTS conductors has been the development of personal computers and advanced numerical and finite element software packages which did not exist in the time of the development of LTS loss analysis. However, our position is that numerical methods and finite element methods should be seen (1) as a correction to a basic analytic approach, or (2) an approach to be used when the case under consideration is too complicated for a simple analytical approach.

G2.7.3 Losses Due to TimeVarying External Field There are some applications of superconductors where AC loss is not critical, but for many it is an important consideration. For this latter class we can come up with a set of representative applications which, while not being exhaustive, cover most cases. These applications are (i) ramping magnet (research magnet, SMES, inductive fault current limiters), (ii) fault current limiters, resistive type, (iii) motor/generators, (iv) transformers, and (v) transmission lines. These applications illustrate the basic field and current conditions applied to superconductors, and by this their various loss components. In general terms, losses in superconductors are driven by time changing fields, time changing currents, or a combination of the two. Of course at its most basic level all of these loss types are due to an interaction of fields and currents within the sample with regard to which it is useful to distinguish these different components on the basis of whether field or current or both are being externally applied to the conductor. The most basic loss component in superconducting materials and conductors is hysteretic loss. This is the dominant loss when magnetic field, rather than current, is the excitation, and the frequency or field ramp rate is low. Typical applications where this loss type is relevant are ramping magnets, motor/generators, and transformers.

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G2.7.3.1 Hysteresis Loss G2.7.3.1.1 The Nature of Hysteresis Loss As a starting point, we will discuss the nature of the field and current profiles within the superconductor and the magnetization-versus-field (M–H) loops that result from the critical state model, Figure G2.7.1. The Bean critical state assumes that superconducting currents flowing within the superconductor shield the superconductor from additional flux as the field is increased from low values to high [the shielding branch of the M–H loop, Figure G2.7.1(a), region 3], and trap field in the conductor as field is reduced from high to low [the trapping branch, Figure G2.7.1(a) region 6]. Let us consider a superconducting slab infinite in the y and z directions, of width 2a in the x direction, and with field H0 applied along the y direction, Figure G2.7.1(b). If we consider the first, increase of the field from zero (the shielding branch) currents flow as shown in Figure G2.7.1(b) to shield the superconductor from the applied field. As noted above, the critical state model assumes that all current flows either at some positive critical value, +Jc, at –Jc, or zero. Since, from Maxwell’s equations, dH/dx = J (in 1-D), the requirement that J = ±Jc leads to a constant gradient in field within each of these regions. Before we go further, it is important to define whether the currents that flow within the superconductor are to be considered as contributing to the magnetization, M, or to a modification of H. In principle, M refers to local (atomic) moments only, and if we kept this definition, screening current in superconductors would be counted as modifying the applied H field, and not the magnetization. On the other hand, it is typical (and useful) to count the screening currents as contributing to the magnetization of the superconductor, in which case, as shown in Figure G2.7.1(c), the H field within the superconductor is constant [dotted line, Figure G2.7.1(c), in the absence of sample shape related demagnetization effects]. Thus, currents are seen as changing B (but not H), and the difference in B and H leads to a macroscopic magnetization, M. In particular, B = μ(H+M), and since for most cases μ = μ0, then M = B/μ0-H. Starting from a virgin sample which has been cooled from above Tc to some temperature below Tc under zero applied field, the applied field H is zero, as is the response M and the resultant B. If we apply a small positive H field (here we ignore the Meissner effect), bulk currents start to flow, and field gradients form as shown in Figure G2.7.1(c) (curve 1), and we are at point 1 in Figure G2.7.1(a). If we apply a slightly larger field, we will obtain the field profile shown as curve 2 in Figure G2.7.1(c), and we are at point 2 in Figure G2.7.1(a). Figure G2.7.1(d) shows the current distribution for this case. If we continue to increase the field, we will reach a point when the field penetrates to the center of the conductor, Hp = Jca, where a is the half-width of the conductor (a slab in this case). The dotted line represents μ0H within the sample, and the solid line the local B. The average difference between B/μ0 and H is Jca/2, so that M = Jca/2. If we continue to increase the field beyond this, we will increase

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FIGURE G2.7.1 Schematic of the (a) M-B loop, (b) slab, infinite in the y and z directions, B applied along y with shielding currents (along z) shown, (c) B-field profiles of Bean critical state at various points along the M-B loop, (d) current and field profiles for point “2”, (e) current and field profiles for point “5”. Note that in (b) the y-axis is into the page, while in (c) and the top two illustrations of (d) and (e) the z-axis is out of the page. The bottom two illustrations in (d) and (e) show the magnitude of the current flowing along the z-axis which generates the field profiles (along y) in the upper illustrations of (d) and (e).

both B and H at the same rate, and M will stay constant, Figure G2.7.1(c), curve 3, and point 3, Figure G2.7.1(a). When we begin to reduce the field, the bulk currents will begin to trap the field. This is shown in Figure G2.7.1(e), where a flux gradient opposite to that for increasing fields begins at the outer surface of the slab and propagates inward as a function of field reduction. The currents in these regions have opposite polarity from the inner regions of the slab. This field profile inversion continues until the applied field is lower than the maximum applied field by 2 × the penetration field, at which point we are on the trapping branch of the M–H curve, region 6, Figure G2.7.1(a), Figure G2.7.1(c), curve 6. If we take the “height” of the M–H loop to be the trapping minus the shielding magnetization, then we can define a ΔM = Jca. This will be our starting point for the calculation of losses below. This very practical approach has ignored the origin of the critical state. That is, we might ask, why does the current density flow only at ±Jc or 0? One description of this is given in Tinkham [9], a very nice and detailed one is given in (Campbell and Evetts) [10]. However, a simple explanation can be made as follows. If we consider a Type II superconductor with no defects, then the magnetic field will be present in the form of fluxons within the body of the superconductor

above the lower critical field. These fluxons will move freely when pushed by the Lorentz force, which will be present if we apply a current while a field is present, given in per unit volume terms as F = J × B. In this case, the superconductor will be not lossless, since a force through a distance is work. But, if defects are introduced, fluxons can be pinned, and lossless current can flow in the presence of a magnetic field. If we take the simplest version of a pin, which is a void or region of nonsuperconductivity, the fluxon will be attracted to the pin with an energy equal to the condensation energy of the superconductor times the volume of the pin, or the fluxon within the pin, whichever is smaller. If we assume that we have a certain density of pins and they all operate independently (direct summation) and that the fluxon array has some rigidity, we will now have a restoring force against the Lorentz force, up to a certain maximum (related to the energy of the pin and their density). For low currents, the Lorentz for will be below this critical force, and for currents above it, the fluxons will move, the critical point gives us the critical current. If we now reconsider the case of a superconductor in a zero applied field, and (ignoring the Meissner currents) we increase the applied field, the superconductor will begin to shield the applied field, as described above. Currents will

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flow (at the critical current density) in a region at the outer surface of the superconductor and a flux gradient will be formed [Figure G2.7.1(c)]. A small increase in applied field will cause a local B gradient which exceeds Jc, such that dB/ dx = μ0J > μ0Jc, driving the field inward until the fixed gradient dB/dx = μ0Jc is regained. This allows the field to penetrate deeper into the conductor with every increase in the field. As we continue to increase the field and further cycle it, we trace through the field and current distributions as given by the critical state model, Figure G2.7.1. Thus, the critical state model is seen to arise from a simple consideration of the nature of flux pinning and critical currents. We can now use this model to describe the hysteresis losses of superconductors of various geometries.

peak to peak change). Throughout this text, losses are per unit volume unless otherwise specified. When H 0 ≥ H P, the slab becomes fully penetrated and the per cycle, per volume loss is given by (Ref [11], p. 163) Hp 2 Hp 2 2 J a − Q = 2µ 0 J c aH 0 1 − c = 2µ 0 H 02 3 H0 H0 3 H0

(G2.7.2)

Thus, when the applied field is just above the penetration field and both linear and quadratic loss terms are seen. But as H0 » Hp, Q ≈ 2µ 0 H 0 H P = 2µ 0 H 0 J c a = 2 B0 J c a

(G2.7.3)

G2.7.3.1.2 Semi-Infinite Slab Consider the critical state of a slab (t w , and infinitely long ) of superconductor of width 2a = w in a field H, applied along t (the y direction) as shown in Figure G2.7.2. This is in fact the same as Figure G2.7.1(b), with our perspective rotated 90° about the x-axis, and L and t are sufficiently large that the semi-infinite description still applies. For low applied fields the sample is only partially penetrated, and the loss per cycle, per volume (Q ) is (Ref [11] p. 163) Q=

2µ 0 H 03 3H p

(G2.7.1)

where the penetration field, Hp is given by Hp = Jc w/2= Jca, and H0 is the maximum applied field (1/2 the amplitude of a

where the magnetization at full penetration is given by ΔM = Jca (where ΔM is the height of the M–H loop) such that Q ≈ 2B0ΔM. G2.7.3.1.3 Cylinder in Perpendicular Field A cylinder in a perpendicular applied field, with an applied field above full field penetration the penetration field is given by Bp = (1/π)μ0Jcd, where d is the diameter of the cylinder. The height of the M–H loop is ΔM = (4/3π)Jcd, and the loss at full penetration (Ref [8], p. 84) Q = (8/3π )B0 J c d

(G2.7.4)

G2.7.3.1.4 Cylinder in Parallel Field For a cylinder in parallel applied field the penetration field is Bp = μ0JcR0, the height of the M–H loop is ΔM = (1/3)Jcd, and the loss at full penetration by (Ref [8], p. 102) Q = (2/3)B0 J c d

(G2.7.5)

G2.7.3.1.5 Hysteresis Losses in Rotating Fields For a cylinder in a transverse applied field fixed in amplitude but rotating around the conductor z-axis, the loss, Qrot is just (Ref [8], p. 91, 105, [12]) Qrot =

FIGURE G2.7.2 A superconducting sample whose cross-section is rectangular. Although difficult to show because of the large aspect ratio, let us take the thickness, t = 2t h » w. Here we have chosen w to be the smallest dimension for ease of comparison of slab, round wire, and strip loss equations.

4 π Qtime varying = B0 J c d 2 3

(G2.7.6)

where we have set dH0/dt =2πf and I = 0 in Equation (7.62) of Ref [8]. Here Q time varying refers to the loss of Equation (G2.7.4), where the field is applied in a fixed direction, but varies with time. The factor of π/2 can be more readily appreciated by noting that for a fixed direction field varying as a triangle wave with time the dB/dt = 4H0/τ = 4H0f, while for a rotating field dB/dt = H02πf, which leads to the above ratio of π/2. Since per cycle hysteretic loss does not depend on the shape of the waveform, this ratio holds for a comparison of sinusoidally time-varying fields vs rotating fields as well.

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G2.7.3.1.6 Hysteresis Loss for an Infinitely Long, Thin Superconducting Strip The result for semi-infinite slabs which are thick along the direction of the applied field but have zero demagnetization are well described by Equations (G2.7.1)–(G2.7.3). However, if the sample thickness, t, becomes much thinner that the sample width (2a), the situation will be different. For H0 » Hp, the loss equations are identical to those of the semi-infinite slab [Equations G2.7.1)–(G2.7.3)]. However, for smaller applied fields the field lines and the flux penetration into the slab are modified, as shown in Figure G2.7.3. Thus, as H0 drops below the penetration field, which is itself modified, the loss expressions are significantly modified by a kind of demagnetization effect. Brandt [13, 14] and Müller [15] showed that Q = 2Nµ 0 H 0 J c a

(G2.7.8)

(G2.7.9)

The penetration field in this case is given by Hp =

J ct w 5 w + 1 Ln + 1 = H Ln 2π d t π t

(G2.7.10)

where Hd = 0.4 Jct is a characteristic field. We note from Ref [16], that for H0/Hd > 3 H N ≈ 1 − 2 d Ln ( 2 ) H0

For YBCO coated conductors which are twisted, the loss will clearly be lower because the amount of tape exposed to a perpendicular field is reduced. For tapes with twist pitches much longer than their width, this field average is just the average of the spatial variation of the function, in this case a trigonometric function, the average of which can be seen to be 2/π. This is shown in greater detail in (Ref [8], p. 189).

G2.7.3.2 Composite Superconductors, Eddy Current, and Coupling Current Losses

and H H H H 2H g 0 = d d ln cosh 0 − tanh 0 H d Hd H d H 0 H 0

G2.7.3.1.7 Losses for a Twisted Thin Strip

(G2.7.7)

where H H N = 0 g 0 Hd Hd

ΔM = NJca. Expressions for magnetization at various points on the M–H loop are given in [13]. All of the expressions above are per cycle, per unit volume. Note that none of these equations contain any materials parameters of the superconductors other than the Jc. What is dominant instead is the geometry of the sample.

(G2.7.11)

Therefore, when H0 » Hd, the loss of the strip is the same as that of a slab with the same width and Jc. At lower field amplitudes, the loss is modified by N such that the M–H loop height

Above we considered hysteretic losses, losses generated when the excitation is an externally applied magnetic field. The per cycle loss generated by this mechanism is frequency independent. We will now consider two additional kinds of loss generated by externally applied fields – eddy current loss and coupling current loss. In these two cases the per cycle losses grow linearly with frequency. Typical applications where these loss types are relevant include fast ramping magnets, motor/ generators, and transformers. G2.7.3.2.1 Stabilization and Eddy Currents For a number of reasons, all practical superconducting wires are composites of a superconducting component and a normal metal component, typically Cu or a Cu alloy. An outer sheath of Cu makes it easier to inject the current into the superconductor, and easier to solder connections. For LTS the primary role of the normal metal component was stabilization. That is, small disturbances could lead to energy deposition in the strand and a thermal runaway cycle involving flux motion (Ref [11], ch 7). The application of an outer stabilizing sheath allowed the currents to share into the normal metal during such an event to allow recovery to the superconducting state, and also to prevent burn-out of the superconducting material. The presence of a high conductivity outer sheath leads to normal metal eddy currents generated by time-varying applied fields according to Pe =

FIGURE G2.7.3 conductor).

Field penetration into a thin slab (coated

2 π2 ( µ 0 H 0 )wf kρn

(G2.7.12)

where k = 6 for a flat tape (e.g., the stabilizing layer on an unstriated YBCO tape) or 4 for a circular sheath (the stabilizer surrounding a round monofilament), [17]. In which these losses are power losses per unit volume, and ρn is the resistivity of the normal metal. In order to obtain per cycle losses we

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divide by f, leaving a linear frequency dependence of per cycle loss, in contrast to the lack of a frequency dependence of hysteretic loss. Thus plotting the experimentally determined per cycle loss vs f allows the hysteretic loss to be found from the intercept, and the eddy current loss from the slope. For very high frequencies, the conductor will reach a skin-effect range, where the interior of the conductor will be shielded from the applied field. The skin depth is given by (Ref [8], p. 130) δ=

2ρ ωµ 0

(G2.7.13)

where ω = 2πf, and the skin depth begins to appear for δ ≈ √2X, where X is the metallic stabilizer thickness. For δ « X, the conductor will be screened and only eddy current losses will be present, and only in a region of depth δ.

FIGURE G2.7.4 Schematic of coupling currents in a multifilamentary strand showing an (a) untwisted pair/strand, (b) twisted pair/ strand.

G2.7.3.3 Filamentation and Coupling Loss G2.7.3.3.1 Coupling Current Losses In addition to stabilizing the strand with an outer sheath of normal metal, it is common to subdivide the superconductor within the strand into filaments. The reason for this is twofold. In the first instance, the subdivision into filaments means that the strand is more stable against flux jumps (Ref [11] ch 7), since the tendency to flux jump is proportional to the stored energy of the flux profile within the strand, (or magnetization), and as we saw above, this is proportional to the width of the strand perpendicular to the field. This was the driving force to subdivide the superconductor in NbTi strands, without it no NbTi-based MRI conductor would work. In addition to this reason, and especially relevant to our present interest, we can see from Equations (G2.7.3), (G2.7.4), and (G2.7.6) that a reduction in the diameter of a round conductor or the width of a flat tape or slab will reduce hysteretic loss proportionately. Thus, filamentation is a loss reduction strategy. In round strands, the filaments are separated from one another by a normal metal matrix. Filaments in YBCO coated conductors (first demonstrated ~2002 [18–21]) are less well developed, but are typically formed by laser ablation, chemical etching, or mechanical means, in which case there may be no metal in between, or a metal overlayer applied after the fact may lead to a normal metal matrix. The expression for coupling current losses is (per unit volume) at low ramp rates and sinusoidal waveforms is Pcoup =

2 1 fLp B0 nρeff

(G2.7.14)

where n = 2 for a round MF strand (Ref [8], p. 127, and Ref [22]) and n = 4 (Ref [8], p. 188) for a striated flat tape. Equation (G2.7.14) shows the same f 2 dependence of power loss, or f dependence of per cycle loss as do normal metal eddy currents. However, the conductor width (or diameter) is replaced

by the strand twist pitch, Lp. This is because the normal metal eddy currents which would form are in effect amplified by the ease of transport through the strands. Thus the eddy currents flow along the strands over the length of a twist pitch, crossing over through the normal metal matrix at both ends of the twist region, as shown in Figure G2.7.4. Figure G2.7.4(a) shows two filaments of an untwisted strand of length L subjected to an applied dB/dt. Supercurrents flow along the filaments and across the matrix in order to fulfil Kirchhoff’s law. For an untwisted sample of length L, the loss is approximately described by Equation (G2.7.14) with Lp replaced by 2L. Figure G2.7.4(b) shows a schematic of a twisted strand and the associated current paths; again, the current flowing down the superconducting filaments cross over via the normal matrix in order to obey Kirchhoff’s law. The length the current path down the filaments in one direction is set to be the Lp/2, since the EMF developed in each loop is driven by the (dB/dt)A, and from the perspective of the twisted pair, dB/dt changes sign with period Lp/2 along the length. This result indicates that filamentizing round or tape conductors, or striating coated conductors, is not sufficient to reduce losses, strand twisting is also required. As stated above, the per cycle experimental losses plotted vs f allow the hysteretic losses to be determined by the intercept and the coupling currents by the slope. But in order to distinguish the coupling currents from the normal metal eddy currents, the twist pitch must be varied. We note here that the coupling losses predicted by Equation (G2.7.14) saturates to the hysteretic loss of the monolithic conductor, this corresponds to a complete recoupling of the filaments with respect to external field. The exact nature of this saturation depends on the shape of the waveform, but for a sinusoidal waveform (Ref [11], p. 179) Qcoup =

B02 fL2p 1 2ρ 1 + ω 2 τ 2

(

)

(G2.7.15)

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244

which agrees with Equation (G2.7.14) for low frequencies and where B0 is again ½ the peak to peak field swing, ω = 2πf, and τ = (μ0/2ρ)(Lp/2π)2. While the detailed shape of the applied waveform (assuming field sweeps well above penetration) does not affect the hysteretic loss component, waveform shape does affect the coupling current loss. As described in (Ref [11], p. 180) a triangular waveform where the period is large compared to τ, the losses are modified by a prefactor of 8/π2 [compare Equations (8.53) and (8.58), Ref [11]) which should be applied to both Equations (G2.7.12) and (G2.7.14). Other waveforms have different prefactors. As noted above, at sufficiently high frequencies the conductor will enter the skin-effect regime. For skin depth, δ, much less than the thickness of the outer stabilizer, the whole of the filamentary region will be shielded from the applied field, and only eddy current loss will be present, and only in the region of thickness δ at the outer boundary of the conductor. If δ is not smaller than the outer sheath thickness, but is smaller than the whole conductor radius, a different kind of skin effect will be present; in this case, the loss per unit volume will be given by (Ref [8], p. 132) Lp δ P = 2π µ 0 H 02 f 1 + R0 2πR0

2 −1

(G2.7.16)

The critical frequency for skin depth is given by f c2 =

1 ρ 2π µ 0 R02

(G2.7.17)

G2.7.3.4 Total Loss in External Applied Fields, Effective Resistivity In order to find the total loss for a composite conductor in an applied field, assuming we are well below normal metal screening and filamentary recoupling frequencies, we merely sum the hysteretic, normal metal eddy current, and coupling current components. This linear addition fails at higher frequencies where saturation effects occur. The eddy current [Equation (G2.7.12)] and coupling current [Equation (G2.7.14)] loss is inversely proportional to ρn. For Cu, this value at 4.2 K can easily be 100 lower than its value at room temperature (1.7 μΩcm), and 7 times lower at LN2 temperatures. Metallic alloys can reach resistivities as high as 100–200 μΩcm, leading in principle to a reduction of the loss of these terms by about four orders of magnitude. Most superconductors (except YBCO) are fabricated by wire drawing, this limits us to drawable alloys. Typical alloys include Cu–Ni (up to 30 at %Ni, used in some specialty low loss NbTi strands), Monel (used in some MgB2 strands as an outer shell), as well as pure Ni and Fe and Fe alloys (used for some MgB2 strands). For Bi2212 and Bi2223 strands Ag is the interfilamentary matrix, with Ni or Ag sometimes used as an alloying element. From a practical point of view, matrix resistivity values of about 30–50 μΩcm have been the limit based on drawability

FIGURE G2.7.5 Current flow paths for (a) high filament/matrix interface resistance and (b) low filament/matrix interface resistance in superconducting composites.

requirements. A second approach to loss mitigation is the use of resistive barriers. These were used for NbTi strands [23], and then again for Bi2223 strands [24]. Most YBCO strands to date use pure Cu stabilizing layers since the external field loss is dominated by the hysteretic loss component. It must be noted that the important quantity which controls the loss is a modification of the bulk normal metal resistivity, the effective resistivity, ρeff. This effective resistivity is ρn as modified by the addition of size effects (important for NbTi fine filament conductors), and more importantly a path effect related to the quality of the superconducting filament/ normal metal interface. If the filament matrix resistivity is low, the superconducting filaments will act like a short circuit as the currents flow across the composite, reducing the resistance by a factor (1−λ)/(1+λ) times the bulk resistivity [8], see Figure G2.7.5. This factor may approach 5 or so (for λ = 0.7). On the other hand, a high interface resistance will increase the effective resistance by the factor (1+λ)/(1−λ). This factor can lead to significant uncertainties in the effective resistivity and the associated loss. ρeff = ρn 11−λ +λ for low filament/matrix interface resistance Equation (G2.7.18a) ρeff = ρn 11+λ for high filament/matrix interface resistance −λ Equation (G2.7.18b) Similar equations which take into account a filament with a significant diffusion barrier are given in [25]. G2.7.3.4.1 Coupling and Eddy Currents for a Wire with Various Matrix Conductivity Zones In a series of articles Turk et al [26, 27] developed the equations to describe coupling loss in round strands with various regions and various conductivities with those regions. A nice description and compilation of some of these expressions is given again in the thesis of C Zhou [25], from which we take the following representative case for a round strand with an outer

AC Losses in Superconducting Materials, Wires, and Tapes

245

magnetic field is due to the transport current flowing in the conductor itself. Transport current loss is important for resistive fault current limiters, and as a loss component in motor/ generators and transformers. In the self-field of an AC transport current of amplitude I0, a round or elliptical strand experiences a power loss per unit length given, according Norris [28] (also see Ref [8], p. 102), by 2 P µ 0 f 2 I0 I0 I0 1 I0 I c 1 − ln 1 − + − = L π I c I c I c 2 I c

(G2.7.20)

where i = I0/Ic and there is no applied field. For I0/Ic « 1, P/L ∝ Ic2(I/Ic)3. For a strip, this becomes ([27], Ref [8], p. 192) 2 P µ 0 f 2 I0 I0 I0 I0 I0 I c 1 − ln 1 − + 1 + ln 1 + − = L π I c I c I c I c I c (G2.7.21)

FIGURE G2.7.6 Generic round conductor defining the various zones and properties of Equation (G2.7.14), after Zhou [25].

stabilizer, filamentary zone, and inner core, Figure G2.7.6. The power loss given as 2

2 2 r dB Lp 1 r02 − rf2 1 rf2 − rc2 1 rc2 Pcoup = f + + 2 2 2 ro dt 2π ρms r0 + rf ρtf rf ρmc rf2

+

2 1 dB r04 − rf4 4ρms dt r02

(G2.7.19)

Here r0 = R0 is the strand radius, rf is the outer radius of the filamentary region, rc is the outer readius of a filament free core, ρmc is the resistivty of the core, ρtf is the resistivity in the filamentary zone, and ρms is the resistivity of the outer sheath, see Figure G2.7.6. This agrees with Equation (G2.7.14) in case rf = r0, rc = 0, and we use dB/dt = 4B0f for a triangular waveform, and include a prefactor of π2/8 to convert to loss for a sinusoidal waveform. This particular expression is one of a set, for other variants, look to [25, 26] and [27].

G2.7.4 Transport Loss and Combined Transport and Field Loss G2.7.4.1 Transport Current Loss The next loss component to consider is the transport loss. This loss is dominant when no external field is applied, and the

For I0/Ic ≪ 1, P/L ∝ Ic2(I0/Ic)4. Note that these last two equations are given in terms of power per unit length (W/m), as opposed to those of hysteretic loss above, which were given in terms of per cycle loss per unit volume. These could be converted to the latter by removing f from the RHS and normalizing by the area of the conductor.

G2.7.4.2 Combined Loss from Current and Field Excitation In the above we have considered separately the losses due to externally applied magnetic fields (hysteretic, eddy current, and coupling losses) as well as losses due to transport current only. For a number of applications, both transport current and magnetic field are applied to the conductor, and some or all of these components are present. Such applications include motors/generators and transformers as particularly relevant examples, but in fact many devices will have some level of mixed excitation. As a first approximation, the total loss can be considered as the sum of the individual components. However, when both field and current excitations are present, each of these loss components is modified. In general, we might consider cases involving (i) DC current and AC field, (ii) AC current and DC field, and (iii) AC current and AC field. We will consider cases (i) and (iii) below. G2.7.4.2.1 DC Current and AC Field: Dynamic Resistance and Loss Modification When DC current is present in a conductor which is exposed to AC field: (i) the AC loss from the applied field is modified, and (ii) a new loss term associated with a dynamic resistance loss appears. To see why this happens consider applying a DC current to a conductor where an applied AC field is well above the penetration field. Since the conductor is already carrying its critical current throughout its cross-section, a non-zero voltage must be applied to drive an additional DC

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246

current, which leads to a loss component called the dynamic resistance. A modification of the external field loss can also be expected as well since the flux profiles will be modified by the applied current. As shown in (Ref [8], p. 93) for applied fields well above the penetration field and with applied DC currents, the power loss per unit length of conductor is given by

cross-sectional area of the conductor than is afforded by the unpenetrated region. It should be noted that the penetration field is modified from the no current case, the value for a round wire with current applied being approximated by (Ref [8], p. 95) I H p ( I ) = H p ( 0) 1 − Ic

2 3/2

Rdyn 4 dH 0 y1 P = I2 + µ 0 I c R0 1 − 3π dt R0 L L

(G2.7.22) G2.7.4.2.2 AC Transport Current in AC Applied Field

Where Rdyn is the dynamic resistance, given by Ly dH 0 Rdyn = µ 0 1 I dt

(G2.7.23)

The first term is the loss due to the dynamic resistance (the power input coming from the current supply), and the second term is the now modified term which describes the loss due to externally applied field. Here y1 /R0 is the fraction of the radius which is used to carry transport current, Figure G2.7.7. It can be seen that for small applied currents, as y1 goes to zero, the loss becomes just that of an applied external field. It can also be shown that as y1 approaches R0, the external field loss component vanishes, and the loss is just that of DC flux flow resistance. In the above equations y1 is defined implicitly by y y I 2 y1 = 1 − 1 + sin −1 1 R0 R0 I c π R0

Here again conductor losses are modified significantly from either a pure external field configuration or a transport only configuration when a conductor is exposed to an AC field while carrying an AC transport current. This condition has been considered by Carr for a 1−D slab with field applied perpendicular to the width of 2a and a current flowing down the length of the slab. In this case (Ref [8], p. 75), H(t) = H0sin(ωt), and J = J0 sin ( ωt ), where J is the transport current per unit width of sheet (such that Jc = 2aJ c ), and the loss is given by Q =

Q =

2

(G2.7.24)

However, Equation (G2.7.24) can be made explicit for y1 /R0 when approximated as 2

(G2.7.26)

I y1 I I = −0.0113 + 0.9711 − 0.6063 + 0.6097 Ic Ic Ic R0

3

(G2.7.25) The above expressions were for applied fields well above the penetration field. Under partial penetration the DC signal appears only when the transport current demands a larger

2 2H 0 µ 0 J 03 J0 + 1 3 if H 0 < J 0 6 J c 2

2 J J 4µ 0 H 03 J 1 + 3 0 if 0 < H 0 < c 3 Jc 2 2 H 0 2

(G2.7.28)

J 3 1 J 2 µ 0 2 0 Q = µ 0 H 0 J c 1 + − J c 1 − 0 3 3 J c J c µ + 0 2

2 µ J 02 Jc − J0 − 0 1 6 Jc H 0 − J 0 2

(

)

3 Jc − J0 J 02 Jc 2 if H 0 > J c 2 1 H 0 − J0 2 (G2.7.29)

(

)

Carr has also considered this case for a monofilamentary round wire [29], and the results were seen to be quite similar. In the regime of large applied fields, we can see that the leading term will dominate (Ref [7], p. 98), and we find the following simple approximation for combined in phase field loss and current loss for a round wire 1 I 2 Qh + i ≈ Qh 1 + 0 3 I c

FIGURE G2.7.7 Current distribution for a superconductor in an externally applied field carrying a DC current, after [8].

(G2.7.27)

(G2.7.30)

This condition has also been considered for a multifilamentary wire, in which case the transport currents tend to couple the filaments together in the absence of an applied time-varying field. However, in a time changing applied field, the transport losses again decouple, and Equation (G2.7.30) can be used, taking the hysteretic loss to be that of the individual filaments. For thin strips, the loss expressions are different. A paper by Schönborg [30] develops the losses for a thin strip for under

AC Losses in Superconducting Materials, Wires, and Tapes

in-phase AC field and AC current conditions. For a thin strip in the low-current, high-field regime, the losses are given by P=

2 fµ 0 I c2 1 −1 q0q0′ + a0 − ( q0C + q0′ D ) * 2coth π CD 4

−1 q0 −1 q0′ 1 cosh + cosh + (C − D ) * a0 a0 2 −1 q0 −1 q0′ q0cosh + q0′ cosh a0 a0 +

1 1 (C − D )2 − {(C − D )(C + D )} 4 2

(G2.7.31)

D = (1 − po )2 − ao2 (G2.7.32)

I0 2 Ic

( ) ( ) Bo Bf

(G2.7.33)

and where Bf = μ0Jc/π. For a thin strip in the high-current and low-field regime, P=

fµ 0 I c2 π

Q µ 0 I c2 = L π 1 − β2

)

I 0 1 − β2 2 I c

(

)

I0 2 Ic 1 − β 1 − 2

(

I I + 1 − 0 1 − β 2 ln 1 − 0 1 − β 2 I c Ic

(

)

(

)

)

(G2.7.35)

where β = R i /R0, the ratio of the inner and outer radii of the hollow tube. If h = R0-R i « R0, then this can be approximated by

and w I0 w 1− B tanh 0 a′ = p= Bf 2 Ic 2 cosh

pitches, current directions, and other factors. Nevertheless, here we offer some very basic results to give some flavor and serve as a comparison to the single conductor losses which are the focus of this chapter. A transmission line can be idealized as a gapless thin wall hollow tube of radius R0 and superconductor wall thickness h, Figure G2.7.8(a). While such a gapless tube is unrealistic in practice, the losses can be simply calculated, the analysis instructive, and the per cycle per unit length loss can be simply expressed in the form [38–41]

(

where q0 =1 + p0, q0’= 1-p0, p0 = 2p/w, a0 = 2a’/w, C = (1 + po )2 − a02

247

2 1 −1 q0 q0′ + a0 −2coth − 4 ( q0C − q0′ D ) * CD

−1 q0 −1 q0′ 1 cosh − cosh + (C + D ) * a0 a0 2 1 2 −1 q0 −1 q0′ q0 cosh + q0′ cosh − (C + D ) (G2.7.34) a0 a0 4 Schönborg’s theoretical results [Equations (G2.7.31)–(G2.7.34)] have the same general character as the experimental results [31]. However, while the in-field loss agrees reasonably well with Brandt’s theory for a strip, the transport AC loss does not agree with the Norris strip results. In both the round wire and thin strip expressions above, we are describing current and field in-phase. However, these are the most relevant conditions of interest. While no analytic expressions exist for round wires or thin strips (coated conductor geometry) for out of phase AC current and AC field conditions, such expressions can be found for 1-D slabs. These rather unwieldy expressions can be found in [32, 33], but here we may be entering the province where numerical methods are the best approach. A few examples of experimental and numerical work in this area are given by [34–37]. G2.7.4.2.3 Transmission Configuration Loss Losses in transmission cables is a topic unto itself that includes details about tape widths, tape spacings, layer numbers, layer

Q µ 0 hI c2 I 0 = L 3πR0 I c

3

(G2.7.36)

Realistic transmission lines are wound with tapes separated by gaps. The introduction of these gaps [Figure G2.7.8(b)] gives a per gap Q/L of [38] Qg µ 0 J 2 w g2 π I0 π I0 π I 0 + tan = (G2.7.37) 2ln cos 2 I c 2 I c 2 I c L π where J is a kind of sheet current defined such that Ic = 2J a [28] (note that this differs from the definition of Jc above). If I0 « Ic, then Equation (G2.7.37) can have the approximate from Qg µ 0 I c2 w g 2 π 3 I 0 = L 96w 2 I c

4

(G2.7.38)

FIGURE G2.7.8 Field distribution for a thin wall transmission configuration showing (a) main field component and (b) the field disturbance near the gaps, after [38].

248

Although the total loss is the sum of Equations (G2.7.36) and (G2.7.38) or Equations (G2.7.35) and (G2.7.37), the latter term is typically dominant and can be usefully compared to the Norris loss, Equations (G2.7.20) and (G2.7.21). After doing so, we find that the ratio of transmission line loss to Norris loss is of order (π/96)(wg /w)2~ 1/3000 per tape. This result shows why the transmission configuration is favored. Essentially, the transmission configuration restricts fields to be perpendicular to the thin direction of a tape, while in a simple Norris configuration, self-field also hits the wide side of the tape, increasing the loss. As mentioned above, these simple equations are for a single layer of conductor, not a true transmission cable, and are provided for comparison to the other common single tape geometries.

G2.7.5 Summary and Further References In the above work, we focused on general analytic expressions for loss for round, tape, and strip conductors of HTS and LTS. Excellent reviews from an analytical approach with greater detail can be found in Refs [22, 33, 42–44]. We have not delved into measurement techniques, but see, for example, Ref [45]. Work from a numerical point of view can be found in [46–51]; Ref [46] is a particular good overview for those interested in a numerical approach and has an extensive reference list. The focus of this article has been practical, as befits a handbook. The purpose has been to give a background and a straightforward understanding of the basic principles underlying AC loss in superconductors, and the analytic expressions which can be used to quantify them under a wide set of circumstances. We have restricted ourselves to conductors (wires and tapes) and did not include the cables wound from them. We reviewed the basic loss equations for the field and current excitation modes associated with the most common applications of superconductors. The loss expressions for hysteretic, eddy current, coupling current, and transport loss were given for slabs, round wires, and thin strips. The influence of waveform shape, rotation, and skin depth were considered. Finally, we gave expressions for the case of mixed excitation, including DC current and AC field, and AC current and AC field. The treatment was not and could not be exhaustive, but it may serve as a useful guide for AC loss calculations.

List of Symbols Roman a = Half width of slab or strip A = Area of loop a’ = Parameter in equation for loss in cases of combined AC field and AC current, Equations (G2.7.31)–(G2.7.34) a0 = Parameter in equation for loss in cases of combined AC field and AC current, Equations (G2.7.31)–(G2.7.34)

Handbook of Superconductivity

B = Magnetic flux density, T Bf = μ0Jc/π Bp = Penetration field B0 = For a time-varying field (sinusoidal or triangle wave), the peak applied magnetic field (1/2 the peak to peak value) = μ0H B* = Parameter in Kim model C = Parameter in equation for loss in cases of combined AC field and AC current, Equations (G2.7.31)–(G2.7.34) d = Wire or filament diameter D = Parameter in equation for loss in cases of combined AC field and AC current, Equations (G2.7.31)–(G2.7.34) f = Frequency F = Per unit volume Lorentz force fc2 = Critical frequency of the skin depth g = A function defined at an intermediate step in calculating the hysteresis loss for coated conductor, defined in Equation (G2.7.9) and used in Equation (G2.7.8). h = thickness of the tube for an un-gapped superconductor in simple transmission configuration [Equation (G2.7.35)] H = Field strength, A/m Ha = Same as H0 Hd = Important field for the Brandt model of hysteretic loss of thin strips similar to a penetration field for round or slab conductors Hp = Penetration field, A/m H0 = Maximum of externally applied field. For a sine wave, 1/2 the peak to peak height Ic = The critical current of a superconductor I0 = For a sinusoidal applied current, the peak current (1/2 the peak to peak value) J = Current density Jc = Critical current density Jc = A sheet Jc where Jc = 2aJ c (units A/m) J0 = For an applied AC current, the maximum applied value (1/2 the peak to peak value), defined as a sheet parameter (units A/m) J = A sheet current density where I = 2a J (units A/m) k = Prefactor in Equation (G2.7.12) L = Length of the conductor Lp = Twist pitch of the conductor M = Magnetization ΔM = M–H loop height n = Geometry prefactor for coupling losses in an applied for round (n = 2) or strip (n=4) conductors N = Loss prefactor which accounts for the influence of very thin coated conductor geometry (0 < R < 1) Pcoup = Power loss due to coupling currents Pe = Power loss due to normal metal eddy current p0 = Parameter in equation for loss in cases of combined AC field and AC current, Equations (G2.7.31)–(G2.7.34) Q = Per cycle loss, per unit volume unless otherwise specified Qcoup = Coupling current loss per cycle due to coupling currents Qd = Loss contribution per gap to the loss in a transmission loss configuration

AC Losses in Superconducting Materials, Wires, and Tapes

Qh = Hysteretic loss per unit cycle Qh+I = Loss per cycle due to a combination of hysteretic and coupling currents Qrot = Q for a field where H0 is fixed in time, but the direction of application rotates around the z-axis of the conductor Qtime varying = Q for a field applied in a fixed direction, but with an amplitude that varies with time (sinusoidal unless otherwise specified). q0 = Parameter in Equations (G2.7.31)–(G2.7.34) for loss in cases of combined AC field and AC current q0′ = Parameter in Equations (G2.7.31)–(G2.7.34) for loss in cases of combined AC field and AC current rc = Radius of core region (region with no filaments) Rdyn = Dynamic resistance of a superconductor with an externally applied AC field and an impressed DC current (units Ω) rf = Radius of filamentary region r0, R0 = Wire radius t = Thickness of the tape Tc = Superconducting transition temperature t h = Half thickness of the tape w = Width of slab or strip = 2a wg = Width of gap between tapes [Equation (G2.7.36)] X = Metallic stabilizer thickness [see skin depth section and Equation (G2.7.13)]. y1 = Width of the strand that is carrying current in an applied AC field with a DC applied current [in a state of dynamic resistance), Equations (G2.7.23) and (G2.7.24)].

Greek β = R i /R0 δ = Normal metal skin depth λ = Local SC fill factor μ = Magnetic permeability μ0 = Permeability of free space = 4π × 10 −7 H/m ρ = Resistivity ρn = Normal-state resistivity ρeff = Effective resistivity, modified by current path effects from ρn ρmc = Resistivity of metal in the core (in an inner region with no filaments) ρms = Resistivity of metal in outer sheath (outside the filamentary region) ρn = Normal-state resistivity of the matrix ρtf = Resistivity of metal surrounding filaments (inside the filamentary region) τ = Time constant for coupling loss ω = Angular frequency of temporal or spatial oscillation = 2πf

References 1. Bean C P 1962 Phys. Rev Lett 8 250 2. London H 1963 Phys. Lett. 6 162 3. Bean C P 1964 Magnetization of high-field superconductors Rev. Mod. Phys. 36 31–39

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4. Kim Y B, Hempstead C F, and Strnad A R 1962 Phys. Rev. Lett. 9 306 5. Foner S and Schwartz B B (ed) 1981 Superconductor Materials Science, Metallurgy, Fabrication, and Applications (New York: Plenum) 6. Foner S and Schwartz B B (ed) 1974 Superconducting Machines and Devices, Large Systems Applications (New York: Plenum) 7. Carr W J Jr 1983 AC Loss and Macroscopic Theory of Superconductors (London: Gordon and Breach) 8. Carr W J Jr 2001 AC Loss and Macroscopic Theory of Superconductors, 2nd ed. (London: Taylor and Francis) 9. Tinkham M. 1996 Introduction to Superconductivity, 2nd ed. (Mcgraw Hill) 176–178 10. Campbell A M, Evetts J E 1972 Critical Currents in Superconductors - Monographs on Physics (London: Taylor & Francis Ltd.) 11. Wilson M N 1983 Superconducting Magnets, 1st ed. (New York: Oxford University Press) 12. Hlásnik I 1981 Review on AC losses in superconductors IEEE Trans. Appl. Magn. 17 2261 13. Brandt E H and Indenbom M 1993 Type-II superconductor strip with current in a perpendicular magnetic field Phys. Rev. B 48 12893–12906 14. Brandt E H 1996 Superconductors of finite thickness in a perpendicular magnetic field: strips and slabs Phys. Rev. B 54 4246–4264 15. Müller K H 1997 Physica C 281 1–10 16. Sumption M D, Lee E, Cobb C B, Barnes P N, Haugan T J, Tolliver J, Oberly C E, and Collings E W 2003 Hysteretic loss vs. filament width in thin YBCO films near the penetration field IEEE Trans. Appl. Supercond. 13 3553 17. Carr W J Jr 1998 Adv. Cryog. Eng. (Mater.) 44 593–600 18. Carr W J and Oberly C E 1999 Filamentary YBCO conductors for AC applications IEEE Trans. Appl. Supercond. 9 1475–1478 19. Cobb C B, Barnes P N, Haugan T J et al 2002 Physica C 382 52–56 20. Polák M, Kempaský L, Chromik S et al. 2002 Physica C 372-376 1830–1834 21. Tsukamoto O, Sekine N, Ciszek M, and Ogawa J 2005 A method to reduce magnetization losses in assembled conductors made of YBCO coated conductors IEEE Trans. Appl. Supercond. 15 2823–2826 22. Campbell A M 1982 A general treatment of AC losses in multifilamentary superconductors Cryogenics 3–16. 23. Collings E W 1986 Applied Superconductivity, Metallurgy, and Physics of Titanium alloys, Vol 2 (New York: Plenum Press) 434 24. Krelaus J, Nast R, Eckelmann H, and Goldacker W 2000 Novel, internally stranded Bi cuprate conductor concept for ac applications: ring-bundled barrier (RBB) tapes produced by the powder-in-tube assemble and react (PITAR) method Supercond. Sci. Technol. 13 567–575

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25. Zhou C 2014 Intra Wire Resistance and Strain affecting the Transport Properties of Nb3Sn Strands in CableIn-Conduit Conductors Thesis, University of Twente 26. Turck B 1979 J. Appl. Phys. 50 5397 27. Turck B 1882 Cryogenics 22 466 28. Norris W 1970 Calculation of hysteresis losses in hard superconductors carrying ac: isolated conductors and edges of thin sheets J. Phys. D: Appl. Phys. 3 489–507 29. Carr W J 1979 AC Loss from the combined action of transport current and applied field IEEE Trans. Appl. Magn 15 240. 30. Schönborg N 2001 Hysteresis losses in a thin hightemperature superconductor strip exposed to ac transport currents and magnetic fields J. Appl. Phys. 90 2930–2933 31. Ashworth A P and Suenaga M 1999 Measurement of ac losses in superconductors due to ac transport currents in applied ac magnetic fields Physica C 313 175–187 32. Takács S 2007 Hysteresis losses in superconductors with an out-of-phase applied magnetic field and current: slab geometry Supercond. Sci. Technol. 20 1093–1096 33. Wang Y 2013 AC loss, in Fundamental Elements of Applied Superconductivity in Electrical Engineering (Singapore: John Wiley and Sons) 141–208 34. Zannella S, Montelatici L, Grenci G, Pojer M, Janšák L, Majoroš M, Coletta G, Mele R, Tebano R, and Zanovello F 2001 AC losses in transport current regime in applied AC magnetic field: experimental analysis and modeling IEEE Trans. Appl. Supercond. 11 2441–2444 35. Mawatari Y and Kajikawa K 2007 Hysteretic ac loss of superconducting strips simultaneously exposed to ac transport current and phase different ac magnetic field Appl. Phys. Lett. 90 022506 36. Nguyen D N, Sastry P V, Knoll D C, Zhang G, and Schwartz J 2005 Experimental and numerical studies of the effect of phase difference between transport current and perpendicular applied magnetic field on total ac loss in Ag-sheathed (Bi, Pb) SrCaCuO tape J. Appl. Phys. 98 073902. 37. Amemiya N, Miyamoto K, Murasawa S, Mukai H, and Ohmatsu K 1998 Finite element analysis of AC loss in non-twisted Bi-2223 tape carrying AC transport current and/or exposed to DC or AC external magnetic field Physica C 310 30–35. 38. Majoros M 1996 Physica C 272 62

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39. Vellego G and Metra A 1995 An analysis of the transport losses measured on HTSC single-phase conductor prototype Supercond. Sci. Technol. 8 476–483 40. Salmon D R and Catterall J A 1970 J. Phys. D: Appl. Phys. 3 1023 41. Gömöry F, Gherardi L 1997 Transport AC losses in a round superconducting wire consisting of two concentric shells with different critical current density Physica C 280 151–157 42. Campbell A 1998 Introduction to AC losses, in Handbook of Applied Superconductivity (London: IOP Publishing) Ed: Seeber 173–185 43. Campbell A 1998 Hysteresis loss in superconductors, in Handbook of Applied Superconductivity (London: IOP Publishing) Ed: Seeber 186–204 44. Duchateau J L, Turk B, and Ciazynski D 1998 Coupling current losses in composites and cables: analytic calculations, in Handbook of Applied Superconductivity (London: IOP Publishing) Ed: Seeber 205–231 45. Hlásnik I, Majoroš M, and Janšák L 1998 AC losses in superconducting wires and cables, in Handbook of Applied Superconductivity (London: IOP Publishing) Ed: Seeber 46. F Grilli, E Pardo, A Stenvall, DN. Nguyen, W Yuan, and F Gömöry 2014 Computation of losses in HTS under the action of varying magnetic fields and currents Appl. Supercond. 24 8200433 47. EMJ Niessen and AJM Roovers 1998 Numerical calculations of AC losses, in Handbook of Applied Superconductivity (London: IOP Publishing) Ed: Seeber 232–248 48. Gömöry F, Vojenčiak M, Pardo E, Solovyov M, and Šouc J 2010 AC losses in coated conductors Supercond. Sci. Technol. 23 034012 49. Amemiya N, Sato S, and Ito T 2006 Magnetic flux penetration into twisted multifilamentary coated superconductors subjected to ac transverse magnetic fields J. Appl. Phys. 100 123907 50. Grilli F, Brambilla R, Sirois F, Stenvall A, and Memiaghe S 2013 Development of a three-dimensional finiteelement model for high-temperature superconductors based on the H-formulation Cryogenics 53 142–147 51. Grilli F, Brambilla R, and Martini L 2007 Modeling high-temperature superconducting tapes by means of edge finite elements IEEE Trans. Appl. Supercond. 17 3155–3158

G2.8 Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields Philippe Vanderbemden

G2.8.1 Introduction This chapter deals with the characterization of the magnetic properties of superconductors which are subjected to magnetic fields that have been applied along two orthogonal directions, which is commonly referred to as a “crossed” magnetic field configuration. In a classical electromagnetism context, the term “crossed” field may sound puzzling since it is learned from textbooks that field lines are locally parallel to the field vectors, hence they cannot cross. The term “crossed” refers rather to the sets of fields that have been applied to the sample, either sequentially, i.e., the superconductor is magnetized by applying and removing a field along one direction and then the material is subsequently subjected to a field along another direction, or simultaneously, i.e., two orthogonal external fields are applied, often with a time delay between them or with different characteristics, e.g., one is a DC field and the other is an AC field. Depending on the field characteristics, these configurations correspond in reality to a kind of rotation or periodic tilting of the applied field, but the somewhat artificial separation between what happens along two orthogonal directions proves often extremely helpful in understanding the observed phenomena. Hereafter, we will therefore omit quotation marks when using the terms crossed field, crossed flow or crossed flux. One can equivalently talk of mutually perpendicular fields or orthogonal fields. The notion of a crossed field is of interest when studying Type II superconductors in the mixed state, and in this chapter we will only deal with this situation. In a normal material, the concept of magnetic flux lines is rather abstract since the physical quantities, i.e., the B-field (flux density) vectors, are related to the direction and the density of flux lines, but it is not possible to define where a particular flux line is (Campbell, 2011). In Type II superconductors, however, magnetic flux lines are quantized as individual vortices and can be observed experimentally. The vortices can move either reversibly or in

an irreversible manner if they are pinned by defects. In the case of strong flux pinning, the motion properties of flux lines in several simple configurations can be described using the Bean model (Bean, 1964) or its extensions for samples of finite thickness [e.g., Brandt (1996)]. In these situations, the direction of the external applied field is fixed and the shape and orientation of the superconductor are such that induced currents flow perpendicular to the field. In a crossed field configuration, on the other hand, individual vortices may appear to cross each other or to allow other vortices to pass through them, which has given rise to the concept of flux cutting and associated models (Clem, 1982; Romero-Salazar and Pérez-Rodrı´guez, 2004). In spite of more than three decades of investigations, the properties of the flux-line lattice in such situations remains ill-understood, and several experimental observations cannot be predicted satisfactorily by the existing theories (Campbell, 2011; Clem et al., 2011). The study of superconductors in multicomponent situations (including the crossed field configuration), therefore, is of fundamental interest. In order to illustrate the relevance of the crossed field configuration to engineering applications, we first describe one of the early experimental results obtained in low-Tc Nb–Zr–Ti superconducting wires (Funaki and Yamafuji, 1982). Consider a linear array of wire segments aligned with each other such that the overall array can be considered as a squared slab, as illustrated schematically in Figure G2.8.1(a). This ensures that the aspect ratio of the sample is the same along the directions of each of the two orthogonal fields to be applied, H and h0., where the second field (h0) is usually smaller than the first (H). We first consider the superconductor subjected to the field H only. The simplest situation is when the sample, initially cooled below its critical temperature Tc in zero-field [zero-field cooled procedure (ZFC)], is subjected to an external field H of amplitude H0. Due to flux pinning, the distribution of vortices is inhomogeneous, and the overall sample 251

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FIGURE G2.8.1 (a) Crossed field configuration of the sample studied by Funaki and Yamafuji (1982). (b) Schematic illustration of the magnetization loop along the direction of the magnetizing field. (c) (Courtesy T. Matsushita) Magnetization as a function of the transverse field in the process of (top) increasing field (diamagnetic state), (middle) field cooling and (bottom) decreasing field (paramagnetic state). The right panels show the schematic distributions of the initial DC magnetic flux density inside the sample for each process (Matsushita, 2007).

magnetization M is negative, as predicted by the Bean model and illustrated schematically by the M–H loop shown in Figure G2.8.1(b). This corresponds to an initial diamagnetic state. If the field is increased further up to some given value H = Hmax and then reversed down to H0, the sample magnetization becomes positive (trapped flux due to the strong pinning) and the initial state is paramagnetic. The third possible state is when the sample is cooled down to its operating temperature under an applied field H0 [field-cooled procedure (FC)]. Then vortices are “frozen” in the superconductor and their distribution is uniform; the associated magnetization is zero. Figure G2.8.1(c) (Matsushita, 2007) shows the experimental behavior of the magnetization in the direction of H when the superconductor is subjected to oscillations of magnetic field h0 perpendicular to H, for each of the three initial conditions described above: diamagnetic (top), field-cooled (middle) and paramagnetic (bottom). As shown, the successive cycles of magnetic field h0 induce a strong irreversible decrease (in absolute value) of the sample magnetization in the magnetic and paramagnetic states, while the magnetization remains zero in the field-cooled state. This modification of the magnetization in a crossed field configuration was first called the “abnormal transverse magnetic field effect” (Funaki and Yamafuji, 1982). It was then studied extensively in several

high-temperature superconductors. These include YBa 2Cu3O7 (YBCO) ranging from single crystals to large single domains (Park and Kouvel, 1993; Fisher et al., 1996; Vanderbemden et al., 2007a), Bi- and Tl-based single crystals (Kim et al., 2004), (Uspenskaya et al., 2004), MgB2 (Luzuriaga et al., 2009), firstgeneration Bi-tapes (LeBlanc et al., 2001) and, more recently, on stacks of second-generation coated conductor tapes (Baghdadi et al., 2014; Campbell et al., 2017; Baghdadi et al., 2018; Baskys et al., 2018). The observed effects were described using various names, e.g., “collapse of the magnetic moment” (Fisher et al., 1997), “crossed field demagnetization” (Swartz et al., 2003; Baskys et al., 2018) or “vortex shaking” (Mikitik and Brandt, 2004; Liang et al., 2017). A typical example of a practical situation in which the crossed field effects play a significant role is when superconductors are used as quasi-permanent magnets or so-called “trapped field magnets”. This corresponds to the paramagnetic configuration mentioned above in the particular case H = 0 (when finite-size effects can be neglected) or H < 0 (due to the return path of magnetic flux lines). In several engineering applications (including magnetic bearings, levitation systems or rotating machines), the permanently magnetized superconductor is subjected to a magnetic field produced by another part of the device. This can be the rotating magnetic

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FIGURE G2.8.2 Schematic illustration of a so-called “trapped-field” synchronous motor. The superconductor is placed in the machine rotor and permanently magnetized before the machine is started. Under normal operation (a), the superconductor follows the rotating magnetic field generated by the stator with a constant angle (torque angle) between the rotating field and the superconductor magnetization. During transients or faults (b), the superconductor is misaligned for a short period of time with respect to this position.

field produced by the three-phase stator in a synchronous motor, as illustrated schematically in Figure G2.8.2. In (hypothetical) ideal operating conditions, the superconductor follows continuously the rotating magnetic field with a constant angle; hence the external field experienced by the superconductor is constant [Figure G2.8.2(a)]. In reality, however, the superconductor is likely to be subjected to a variable field that arises, e.g., from sudden variation of the applied torque on the shaft [Figure G2.8.2(b)]. This causes the magnetization of the sample to become misaligned with its equilibrium position for a short period of time (Qiu et al., 2005). These field variations are almost always accompanied by an additional component of the field orthogonal to the initial magnetization. The irreversible demagnetization of the superconductor that results and, ultimately, the possible failure of the device underline the relevance of crossed field effects in applications. Demagnetization phenomena in crossed magnetic fields are not restricted to superconductors and may also happen for conventional (ferromagnetic) permanent magnets. Although the physical origin is different, an irreversible decrease of the remanent magnetization can be observed for sintered Nd– Fe–B magnets subjected to an external field orthogonal to the direction of their magnetization. The effect is dependent on the shape of the magnet as well as the degree of orientation of adjacent grains within the magnet (Katter, 2005). Practical situations in which such demagnetization may arise are similar to those involving superconductors, e.g., fault conditions, change of loads or armature reaction (Ruoho and Arkkio, 2008). An efficient characterization of superconductors in the crossed field configuration often requires designing a bespoke experimental apparatus, since there is no “off-the shelf” device available to carry out such measurements on large samples. The purpose of this chapter is to describe the techniques that are useful to perform crossed field experiments, with an emphasis placed on practical aspects that are useful for designing the system and for understanding the measured

data. This chapter is organized as follows. In Section G2.8.2, the key terms involved in the literature dealing with crossed field effects are defined. Section G2.8.3 deals with experimental methods, and some key parameters will be outlined. In Section G2.8.4, next challenges in this area will be discussed.

G2.8.2 Definitions There are a number of experimental situations resulting in orthogonal flux lines. These can be divided in two categories: • the magnetic fields in both directions are each provided by an external magnet, or • one field is generated by an external magnet and the second field results from a transport current injected in the superconductor. In the framework of this chapter we will only deal with the first situation, i.e., transport currents will not be considered. The concept of longitudinal currents, however, will be mentioned shortly in order to avoid confusion between usual expressions. We first focus on the field configurations when only one field is applied. These are shown in Figure G2.8.3.

G2.8.2.1 Parallel Field The magnetic field is applied along the longest dimension of a superconducting sample, which can be assumed of infinite length. Typical examples include a field parallel to a thin slab or to a long cylinder, as illustrated in Figure G2.8.3(a).

G2.8.2.2 Transverse Field or Perpendicular Field This term is usually used when the field is perpendicular to the longest dimension of a long sample, e.g., perpendicular to the axis of a wire or perpendicular to the largest face of a superconducting strip. It is also used when the field is perpendicular

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FIGURE G2.8.3 Various magnetic field configurations. (a) Parallel field, (b) transverse field, (c) oblique field. Numbers ① ② ③ refer to the sequence of fields. In “scenario 1”, the field is increased continuously at constant angle with respect to the reference direction. In “scenario 2”, the transverse component is applied first, followed by the parallel component. In “scenario 3”, the parallel component is applied first, followed by the transverse component.

to the axis of a short cylinder. These configurations are illustrated in Figure G2.8.3(b).

G2.8.2.3 Oblique Field or Inclined Field The magnetic field is applied at some angle (≠90°) with respect to one of the symmetry axes or symmetry planes of the sample (Figure G2.8.3(c)). An oblique field can be decomposed into a parallel and a transverse component. If the two components are provided by two external magnets, it is assumed that the two components are applied simultaneously and proportional to each other so that, on increasing the field, the angle between the field and one of the reference directions (attached to the sample) is constant [scenario 1 in Figure G2.8.3(c)]. Note that

applying the field components sequentially [scenarios 2 and 3 in Figure G2.8.3(c)] leads, in general, to results that differ from each other and from an oblique field, as studied in detail by (Gheorghe et al., 2006).

G2.8.2.4 Crossed Fields The configurations where two orthogonal fields are applied are shown in Figure G2.8.4 for several superconductor shapes. These include a slab, a long cylinder (wire), a short cylinder (puck), a prism with rectangular cross-section or a long strip (and, by extension, a film or a tape). An initial magnetic field (magnetizing field) H is applied, possibly cycled, which is followed by the application of another field h (crossed field).

FIGURE G2.8.4 (a)–(f) Examples of crossed field configurations for various superconductor geometries. (a) thin slab, (b) long cylinder, (c) short cylinder, (d) prism with rectangular cross-section, (e), (f) thin strip or thin film. (g), (h) Two examples of rotating field configurations. (i) “Longitudinal field” configuration.

Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields

Naturally, the sequential application of parallel and transverse fields shown in Figure G2.8.3(c) is also a crossed field configuration. In irreversible Type II superconductors, it should be recalled that vortices may be pinned in the sample even when the magnetizing field is removed. The different experimental conditions shown in Figure G2.8.4 have a direct impact on the hypotheses that can be made in the theories used to explain them. In Figure G2.8.4(a), for instance, both fields are applied to the longest dimension of the sample, so that finite-size effects can be omitted. In Figures G2.8.4(d)–(f), the amplitude of the magnetizing field is usually chosen much larger than the full-penetration field, so bending of flux lines at low fields can be neglected, and vortices are assumed to be all perpendicular to the largest face of the sample. Although they look similar, the two configurations shown in Figure G2.8.4(e) (“longitudinal shaking”) and Figure G2.8.4(f) (“transverse shaking”) involve very different physical phenomena, the latter leading to an amazing “walking” behavior of vortices out of the sample (Brandt and Mikitik, 2002).

G2.8.2.5 Force-Free Effects Another way of looking at Figure G2.8.4 is to consider the macroscopic persistent current loops in the superconductor. These are shown by dashed blue lines in Figure G2.8.4. The different configurations can be distinguished by whether or not the crossed field is applied along the direction of the current induced by the magnetizing field. If this is the case, then the sample experiences a force-free configuration, i.e., a configuration with J parallel to B and the Lorentz force J × B is zero. In this case, the traditional Bean model picture does not apply. This arises, for example, in Figures G2.8.4(a) and G2.8.4(e) and partially in Figures G2.8.4(b) and G2.8.4(c). In contrast, the situations shown in Figures G2.8.4(d) and G2.8.4(f) do not involve force-free effects (except, strictly speaking, at the extremities of the sample), so they can be treated by classical approaches. Interestingly, those can sometimes be used for modelling more sophisticated geometries. As an example, the experimental results for a short disk subjected to crossed fields [Figure G2.8.4(c)] can be understood qualitatively by using 2D modelling of a long prism [Figure G2.8.4(d)], without taking flux cutting into account (Vanderbemden et al., 2007a and 2007b; Badía and López, 2007; Hong et al., 2008).

G2.8.2.6 Rotating Fields Related to the crossed field configurations mentioned above are the geometries in which the field is rotated [Figures G2.8.4(g) and G2.8.4(h)] either continuously or between two extreme directions. The physical phenomena involved are closely related to those in the crossed field configurations, and some theories were developed in this context (Carballo-Sanchez et al., 2001), in particular to explain anharmonic effects (Badía and López, 2002b).

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G2.8.2.7 Longitudinal Field Effects Injecting a transport current in the superconductor in the presence of an applied field is another way of obtaining orthogonal flux lines. The simplest (at first glance) configuration is a wire subjected to an external magnetic field parallel to the current, as sketched in Figure G2.8.4(i): the concentric flux rings are orthogonal to the axial flux lines parallel to the current. This geometry is often referred to as a longitudinal field configuration, or longitudinal currents, and should not be mistaken with the parallel field geometry mentioned in Figure G2.8.3(a). The longitudinal configuration and its numerous variants have been largely studied for several decades. They are closely related to flux cutting theories (Campbell, 2011) and lead to a number of physical phenomena (Matsushita, 2007), which will not be discussed here. One of the main differences between crossed “applied” fields experiments and those involving transport currents is that the associated electric fields as a function of the transport current density can be measured directly in the latter. This can be used to test theories describing these phenomena at a macroscopic level (Clem et al., 2011).

G2.8.2.8 “AC” Fields vs. “DC” Fields In addition to the geometric configuration of the crossed fields discussed in Figure G2.8.4, an important point relates to their time-dependence. In several experiments, the magnetizing field is kept constant while the crossed field is swept between two extreme values (as done in Figure G2.8.1), so one talks logically about perpendicular DC and AC fields. Strictly speaking, however, the magnetic field applied to a superconductor from the virgin state changes from zero to the final value at a finite rate. The expression “DC”, therefore, is somewhat equivocal. Note that the same ambiguity applies to the DC characterization of any magnetic material (Fiorillo, 2010). On the other hand, the term “AC” is often used to describe a periodic variation for which the associated frequency may range from a few kHz down to a few mHz. At such low frequencies, the distinction between “AC” and “DC” blurs. It is therefore recommended to describe the time dependences used in the experimental procedure as precisely as possible in order to avoid confusion. The main reason why results are affected by the sweep rate of the applied field(s) is related to the electric field arising from the time-dependent flux density: higher sweep rates lead to higher electric fields and, in turn, to higher current densities. This effect is strongly dependent on the roundness of the E–J curve, i.e., it is hardly perceptible for low-Tc superconductors but may play a non-negligible role in high-Tc superconductors.

G2.8.3 Experimental Methods This section addresses how the superconductor properties in crossed magnetic fields can be characterized. We start by describing various magnet arrangements that can be used

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to produce the crossed field configuration. Then the useful magnetic quantities that can be measured will be outlined, and the corresponding experimental methods will be described.

G2.8.3.1 Methods for Generating Crossed Fields The generation of a crossed field configuration requires either two sets of magnets with orthogonal axes (the fields can be applied simultaneously), or a relative rotation between the sample and the magnet (the fields can be applied sequentially): a stationary sample in a rotating magnet or a rotating sample in a magnetic field of constant direction. Magnets with orthogonal axes may consist, for example, of a long (solenoid) coil and one split pair [Figure G2.8.5(a)], two split pairs [Figure G2.8.5(b)] or a solenoid inside a racetrack coil [Figure G2.8.5(c)], as described in (Funaki and Yamafuji, 1982). The configurations displayed in Figures G2.8.5(a) and G2.8.5(b), as well as numerous variants, can be made using copper as well as superconducting wire. They are commercially available as “vector magnets” or “vector rotate superconducting magnets”, i.e., magnets in which the magnetic field can be rotated or tilted around two or three axes. Figure G2.8.5(e) shows an atypical configuration involving a toroidal magnet coil wound around a superconducting tube, so as to generate an azimuthal field superimposed to the axial field provided, for example, by a solenoid magnet (LeBlanc et al., 1993).

Handbook of Superconductivity

If large superconducting samples are studied, the uniformity of the applied fields is worth considering. For solenoid coils, the magnetic field strength along the axis can be calculated easily and can be found in various references, e.g., the very comprehensive book (Tumanski, 2011). In a split pair, the best uniformity is achieved when the distance between the two coils is equal to their radius, so as to obtain a pair of Helmholtz coils. Other arrangements can be considered, for example, adding a second split pair to improve the field uniformity or using square coils to improve the accessible volume. For long samples (e.g., superconducting wires or tapes), it has been shown that a pair of carefully designed racetrack coils [Figure G2.8.5(d)] enables magnetic field uniformity of 0.2% over 8 cm to be achieved (Trojanowski et al., 2014). In order to boost the field produced by a copper coil, a ferromagnetic circuit can be used, as shown in Figure G2.8.5(f). The configuration involves a yoke made of high-permeability soft ferromagnetic material. In the “medium” gap range (1–10 cm), the applied magnetic field strength is, as a first approximation, inversely proportional to the size of the gap. Higher fields, therefore, can be applied at the expense of available sample space. The above configurations can be used for DC or slowly ramped (quasi-static) magnetic fields, usually cycled at frequencies well below 1 Hz. If AC cycles of higher frequencies are used, the major concern is to avoid eddy currents. The magnetic yoke, if any, should be either made of ferrite or

FIGURE G2.8.5 Examples of magnet configurations for applying crossed fields. (a) Solenoid and split pair, (b) two orthogonal split pairs, (c) solenoid inside a racetrack coil. (d) Photograph of a pair of racetrack coils ensuring a good field uniformity over a long length (courtesy M. Ciszek). (e) Toroidal coil generating an azimuthal field. (f) Example of configuration where the coil is wound on a ferromagnetic yoke.

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FIGURE G2.8.6 (a) Schematic illustration of a DC coil picking up the AC magnetic flux generated by the AC coil, resulting in an AC current in the DC circuit. (b) Insertion of a resonant LC filter in the DC circuit to reduce the parasitic AC current.

laminated materials, e.g., 0.4-mm-thick insulated Si-doped iron sheets can be used up to a few kHz. The magnet configurations above can still be used provided there are no plain metallic piece in the coil frame or flanges, the cryostat and the sample holder. When a good thermal conductivity sample holder is required (e.g., to ensure thermal conduction between the cold head of a cryocooler and the sample space), a good compromise is to use a polycrystalline silicon rod (Vanderbemden, 1998); silicon has a medium electrical resistivity (101–103 Ωm), and its thermal conductivity at 77 K is higher than that of copper. When several magnet coils are powered simultaneously to generate a crossed field configuration in AC regime, there are additional issues related to inductive pick-up. This situation is illustrated in Figure G2.8.6(a), where one coil is powered with AC current and the other is supposed to be the DC coil. Because of the (possibly small, but finite) mutual inductance between the two coils, the DC coil picks up a part of the AC magnetic flux. The result is an induced AC voltage across the DC coil, and an AC current in the DC circuitry. This current affects the characteristics of the AC magnetic field (amplitude and phase) at the sample location. An easy way to reduce this phenomenon is to insert a LC resonant filter

placed in series with the DC coil, as schematically illustrated in Figure G2.8.6(b).

G2.8.3.2 Magnetic Quantities to Be Measured Having established the crossed field configuration, we now turn to the magnetic quantities to be measured (Figure G2.8.7). The most common techniques are: • using one or several coils placed around the sample and measuring the flux variation induced by a change of the applied field or a relative displacement between the sample and the coil; • using Hall probes (or Hall probe arrays), either stationary or moving across the sample surface to record the flux density against the surface; • using a magneto-optical indicator (MOI) based on the Faraday effect to record the flux density distribution with high resolution; • measuring the magnetic torque produced by the nonparallel applied field and magnetic moment. Below these techniques are described, then experimental issues common to all of them are discussed.

FIGURE G2.8.7 Various means of probing magnetic quantities in crossed field experiments. (a) Search coil wound closely around a sample to probe the average flux density. (b) Two search coils larger than the sample to probe the magnetic moment. The relative speed between the coil set and the sample is denoted by v. (c) Hall probe moved across the surface of the sample. (d) Magneto-optical indicator. (e) Torque magnetometer.

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G2.8.3.2.1 Methods Based on Sensing Coils Using a set of sensing coils (interchangeably called search coils or pick-up coils) to characterize the properties of superconductors in the crossed field configuration is a very convenient technique. These measurements divide into two classes: those in which the sample is moved or vibrated through the coils (as done in conventional flux extraction magnetometers or vibrating sample magnetometers), or those in which the flux variation is caused by changes of the external applied field. Depending on the radius of the search coil (R) compared to that of the sample (a), there are basically two extreme cases, as illustrated in Figure G2.8.7. In Figure G2.8.7(a), the search coil is wound around the sample, as close as possible to the sample surface (R ~ a). In this case, the measured quantity is the average axial flux Φ threading the sample surface S, from which the axial average flux density < Bz > = Φ/S can be calculated. In order to probe the axial dipolar magnetic moment mz (and hence the axial magnetization Mz = mz/V, where V is the sample volume), the search coil should be intentionally chosen much longer and/or larger than the sample (R ≫ a), as shown in Figure G2.8.7(b). This can be understood as follows. We can consider a hypothetical experiment where the search coils are fed by a constant current and check whether the magnetic field due to this current is uniform at the sample location. If this is the case, the voltage across the search coils when used for probing a magnetic parameter [as done in Figure G2.8.7(b)] is proportional to the magnetic moment (Campbell, 1991). In a crossed field experiment, either two sets of pick-up coils should be used or the sample should be rotated. The first experiments in this configuration required to add selfwound coils to the vibrating sample magnetometer’s (VSM) original design (Park and Kouvel, 1993; Hasanain et al. 1995). Today commercial VSM have options where the sample can be rotated and two pick-up coil sets are available: one measures the magnetic moment parallel to the applied field and the other measures the magnetic moment perpendicular to the applied field. These are known under the names “vector option” or “biaxial coil systems”. These systems are well suited to single crystals or films, but generally accommodate samples of small size, i.e., well below 1 cm³. If large samples are to be studied, magnetometers for large samples should be developed, as done by (Egan et al., 2015). Note that custom-built probes based on a commercial SQUID magnetometer have also been used successfully to investigate crossed field effects (Luzuriaga et al., 2009). When one wants to determine the average axial flux density, the most straightforward way is to measure the voltage across the pick-up coils wound around the sample when the external applied field Happ is changed, i.e., Happ(t). The measured voltage is proportional to N dΦ/dt, where N is the number of turns of the search coil (typically N < 50). In quasi-static conditions, i.e., where the external field is ramped slowly, the induced voltages are typically a few µV, which require great care (Philippe et al., 2014). Since the signal feeds an analog

Handbook of Superconductivity

or digital integrator to obtain the flux Φ(t), perhaps the most significant sources of error are the tiny (~ a few nV) DC offsets (e.g., thermoelectric) superimposed on the useful voltage. The integration of these offsets yields a spurious time-dependent output signal. If the external field is cycled between two extreme values, the practical consequence is that the B-Happ hysteresis loop reconstructed from Happ(t) and Φ(t) does not close. Such unusual behavior can be noticed immediately and corrective actions can be taken, e.g., thermal anchoring of solders to avoid thermoelectric effects, or subtracting the offset voltage carefully measured before and after the experiment. In a crossed field configuration, however, the important point is that the hysteresis loops do not necessarily close, as evidenced by Park and Kouvel (1993). A non-closed loop (see Section G2.8.3.3), therefore, does not provide any evidence of an experimental artefact, but might result from offset problems. The common point of methods based on sensing coils is that the measured quantity (either flux density or magnetization) is related to the volume properties of the sample. G2.8.3.2.2 Methods Based on Hall Probes Hall sensors provide an easy and straightforward way to record how the magnetic flux density against the sample is modified in the presence of crossed magnetic fields. Unlike the coil measurements described above, the flux density is not probed within the sample but next to its surface. If a miniature Hall probe is moved across the surface [Figure G2.8.7(c)], the spatial distribution of the “local” magnetic flux density is recorded. It should always be remembered that the probes have a small but finite active area and that this is located at some small but finite distance from the sample surface. A number of Hall sensors with various characteristics and sensitivities are available. The Hall voltage is directly proportional to the control current, but in some cases the dissipated power and the self-heating that results may be unacceptable. Therefore, the smallest possible drive current compatible with the measurement sensitivity should be used. In order to remove the influence of DC offsets, a sinewave control current can be used and the corresponding Hall voltage can be measured with a lock-in amplifier (Durrell et al., 2014). Hall probes are also sensitive to spurious voltage overshoots that result, e.g., from switching the polarity of magnet coil current. If this happens, appropriate protection circuits should be placed across the voltage terminals. In crossed field experiments, Hall sensors can be placed as two orthogonal arrays and allow the flux density distribution to be recorded before and after the transverse cycles are applied (Baghdadi et al., 2014). G2.8.3.2.3 Magneto-Optical Measurements The measurement of flux density distribution with excellent spatial resolution can be done with a magneto-optical imaging (MOI) system (Koblischka et al., 1996). Since this characterization technique is covered in another chapter of the handbook, only the main point relevant to the crossed field

Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields

configuration will be emphasized here. In general, the MOI system is sensitive to the component of magnetic flux density normal to the indicator film placed against the superconductor surface (Bz). In practice, however, the magneto-optical film is also weakly sensitive to the in-plane component of the flux density (By). As studied in details by (Johansen et al, 1996), the grey level values G of an MO image depend on By and Bz as follows G ∝

Bz2

(B

A

2

+ By ) + Bz2

,

(G2.8.1)

where BA is a characteristic parameter of the MO film. It is therefore recommended to perform a careful calibration of the MO film with superimposed fields of known amplitude By and Bz to determine BA and its possible influence on the crossed field measurements. The crossed field configuration on Type II high-temperature superconductors was used either to study the penetration path of a H||c field (perpendicular to the MO film) in the presence of a pre-existing longitudinal field H||ab (Indenbom et al., 1994; Uspenskaya et al., 2004), or to investigate the decrease of a M||c field as a function of cycles of a H||ab transverse field (Vanderbemden et al., 2007a). G2.8.3.2.4 Magnetic Torque Measurements When a uniform applied field H is applied at some angle with respect to the magnetic moment m of a magnetized material, the sample experiences a torque τ given by τ = m × µ0H, as illustrated schematically in Figure G2.8.7(e). Torque measurements, therefore, allow information on magnetization to be determined. One of the main advantages of this technique is its high sensitivity (noise levels of the order of 10 −9 N.m are accessible with state-of-the-art torque magnetometers). The main constraint is that measurements should always be carried out in the presence of a small but finite transverse field. Torque magnetometry can be used to determine the properties of the penetrated state in anisotropic superconductors subjected to oblique fields (Koblischka et al., 1996). In the context of crossed field experiments, magnetic torque measurements can be used to determine the magnetic moment of permanently magnetized superconductors in the presence of applied fields which are not exactly parallel to the magnetization, either to study the decay due to crossed fields or the possible remagnetization when the field is rotated back toward the magnetization axis (Vanderbemden et al., 2007b). In order to measure large, bulk materials or stacked tapes using this technique, it was demonstrated recently that a torque magnetometer able to accommodate sizable magnetic samples (>1 cm³) both at room and cryogenic temperatures, with a magnetic moment sensitivity ranging from 5 × 10 −3 Am2 up to 1.5 Am2, i.e., two orders of magnitude above the maximum magnetic moment of commercial magnetometers (Brialmont et al., 2019).

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G2.8.3.2.5 Experimental Issues Common to All Techniques There are basically two kinds of experimental issues related to measurements in the crossed field configuration. The first is related to the correct alignment of the sample surface with respect to the applied fields. The angular position of the superconductor should be carefully adjusted through carrying out measurements above Tc. The most sensitive method is to choose a calibration position where the signal should theoretically be zero and to adjust the alignment so that the measured signal is minimum. Any misalignment with the transverse field may result in an unphysical “tilt” of the Mz vs. Hy curve. The second problem relates to the magnetic torque experienced by the sample, as discussed in Section G2.8.3.2.4. If we consider a cubic superconductor (edge length a = 1 mm) having a critical current density Jc = 105 A/cm2, the magnetic moment of the fully magnetized sample, given by (Jc a4)/6, is equal to m ≈ 1.6 × 10 −4 Am2. Under a transverse crossed field µ0H = 1 T, the magnetic torque is 1.6 × 10 −4 N.m, and is equivalent to 0.32 N force pulling on the sample side. If we consider a sample edge length a = 1 cm with the same properties, the torque under the same transverse field is now 1.6 N.m and is equivalent to a 320 N force applied on the side! A strong clamp, therefore, is required to prevent the sample from being rotated by the field. Note that similar problems apply to the study of conventional permanent magnets under inclined applied fields. Careful experiments require to set the sample in a special jig placed in the magnetizing fixture (Ruoho and Arkkio, 2008).

G2.8.3.3 Some Experimental Phenomena In this section, we describe some interesting phenomena that have been observed in irreversible Type II superconductors subjected to crossed magnetic fields and we point out the experimental conditions required to put them into evidence. Such experimental data can be used to test various theories of the critical state with nonparallel arrays of vortices. The first important thing is the magnetic state of the superconductor before applying the crossed field. As pointed out from the data shown in Figure G2.8.1, the superconductor initially cooled in zero field can be driven either to a diamagnetic or paramagnetic initial state. The question that arises is whether, for a given magnetizing field, the crossed field cycles lead to a symmetric or asymmetric decrease (in absolute value) of the magnetization. Once the magnetizing field is applied and magnetization is established in the sample, depending on the field distribution and the temperature, relaxation of the magnetization (flux creep) may occur. This applies mainly to high-temperature superconducting (HTS) materials and the corresponding decay is logarithmic with time. Such a decay due to “natural” relaxation of vortices has to be distinguished from the (possible) decay caused by the crossed field, although in reality

260

Handbook of Superconductivity

FIGURE G2.8.8 (a) Schematic illustration of the decay of the magnetization due to flux creep. At t = t1, the crossed field is applied, resulting in a further decay. (b) Schematic illustration of the changes of magnetization Mz caused by a crossed field hy, showing either a monotonous decay (top) or so-called “butterfly” loops (bottom). (c) Schematic illustration of simultaneous recording of the orthogonal components of the magnetization, either perpendicular (top) or parallel (bottom) to the direction of the crossed field.

both phenomena are linked. If one wants to probe the effects due to the crossed field, a constant time interval (typically 15 minutes) should be allowed for magnetic relaxation of vortices before applying the crossed field, as shown schematically in Figure G2.8.8(a). In any case, it is recommended to make sure that the time interval between the application of the magnetizing and the crossed field is known in order to get repeatable results. Similarly, the sweep rate at which the crossed field is applied may have a crucial importance, especially in HTS materials at the liquid nitrogen temperature (77 K). The amplitude of the crossed field is also of importance, and the best way is to compare it to the full-penetration field Hp along the same direction as the crossed field. It should be emphasized here that Hp is defined in the absence of crossed field effects. A transverse field of amplitude much greater than Hp and applied in the presence of an existing magnetization – i.e., a crossed field configuration – leads to the following counter-intuitive situation: the transverse field does not penetrate fully and does not destroy the magnetization (Vanderbemden, 2007a). In fact, the crossed field does penetrate in the sample, but because the cross-section is already saturated with current, the penetration depth of the crossed field in a premagnetized sample is smaller than for a field applied from the virgin state. In many crossed field experiments, the crossed field is cycled. If relaxation effects play a role, the waveform of the crossed field is likely to have an influence on the results. This effect was investigated by (Fagnard et al., 2015) comparing

triangular and sinusoidal waves having the same period. It should be noted that the usual experimental conditions used with superconductors may differ substantially from those used in order to study the demagnetization of permanent magnets, such as short pulses of increasing amplitudes (Katter, 2005). When the field is cycled between two symmetric values, unusual phenomena can arise [Figure G2.8.8(b)]. The decrease of the magnetization Mz vs. hy can be either monotonous or exhibit so-called “butterfly” loops, as predicted by (Badía and López, 2002a) and observed experimentally in some conditions (Perez-Rodriguez et al., 2001). When the two components of the magnetization Mz and My are recorded, there are particular situations where the hysteresis loop in the direction of the transverse field (i.e., My vs. hy) fails to close after a complete cycle of magnetic field (Park and Kouvel, 1993). If a large number N of crossed field sweeps is applied, the natural question to be addressed relates to (i) the type of law relating the decaying magnetization M (or flux density B) to N and whether the decay is due to the number of sweeps or the time during which the sweeps are applied, and (ii) the steady state value of the magnetization after many sweeps. Due to the small changes to be investigated, this is not an easy problem, both from an experimental and modelling point of view. In the case of thin strips and platelets, semi-analytical modelling predicts the existence of several regimes, with an exponentiallike decrease of the magnetization vs. time at high amplitudes of the transverse field (Brandt and Mikitik, 2002; Mikitik and Brandt, 2004). For bulk samples or crystals, there are several

Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields

reports showing that the magnetization eventually stabilizes after ~ 10 cycles. When looking carefully at more of such cycles, however, a power law decrease is found, i.e., B ~ N-α, where α is related to the crossed field amplitude. Investigation of a large number (N > 100,000) cycles confirms the power law behavior but gives evidence of a slowdown of further magnetization decay (Fagnard et al., 2015). The conclusion to be drawn from these results is that the transient regime before a stationary magnetization is achieved may involve a considerable number of sweeps of crossed fields. Finally, we briefly discuss possible phenomena arising when increasing the frequency of the crossed field or applying short crossed field pulses. In such a case, the rapid movement of vortices generates losses. These may yield selfheating of the sample, a decrease of its critical current density Jc and, in turn, a decrease of the magnetization. This phenomenon is observed in the parallel field configuration (i.e., when the AC field is parallel to the magnetization) and has been well documented, for example (Tsukamoto et al., 2005). Self-heating can be expected in the crossed field configuration as well, and this can be checked by placing thermocouples against the sample surface or running magneto-thermal modelling (Fagnard et al., 2015). If the sample is cooled by a cryocooler or placed in a gaseous atmosphere, an often neglected but critically important parameter to estimate the possible impact of self-heating effects is the heat transfer coefficient (U) between the superconductor and its environment. Different thermal behavior regimes are predicted depending on the value of the dimensionless Biot number, defined as Bi = (U L/k), where L is a characteristic length of the superconductor and k its thermal conductivity. When Bi ≪ 1, the equilibrium temperature Teq reached by a cylindrical sample (volume V, outer surface A) subjected to an axial field (amplitude H, frequency f ) can be estimated analytically (Vanderbemden et al., 2010) fV 8µ H 3 1 Teq = T0 + 1 − 1 − 0 (Tc − T0 ), 3 H p AU (Tc − T0 ) 2

(G2.8.2)

where T0 and Tc are, respectively, the initial temperature of the superconductor and its critical temperature; Hp denotes the full-penetration field.

G2.8.4 Future Directions After several years of experimental and theoretical investigations, the properties of superconductors in the crossed field configuration remains a fascinating topic. Although some theories or numerical modelling can give a clearer idea of the physical phenomena involved, a unified view has not yet emerged. On the theoretical side, new theories applicable to dense vortex lattices (Campbell, 2011) and/ or including the intrinsic anisotropy of the superconductor are required. Very recently, an analytical model taking

261

into account the two distinct mechanisms of decay, current redistribution and flux creep, could be used to reproduce the experimental data (Srpčič et al., 2019). On the numerical side, finite element modelling (FEM) applicable in thin superconductors, as done, for example, by Celebi et al. (2015) should help understand the experimental data observed in films or coated conductors subjected to various crossed field configurations. It should be also of interest to investigate whether FEM results are consistent with the theoretical predictions Mikitik and Brandt, 2004). Recent computer simulations based on the critical state show good agreement between the Brandt and Mikitik theory and the decay of the magnetization in long thin superconductors (Campbell et al., 2017). As pointed out by the authors, however, the movement of vortices in such configurations is very small, hence the effects of reversible fluxoid motion need to be investigated. Additionally, three-dimensional (3D) modelling (Kapolka et al., 2018) with a very fine mesh to investigate the details of the current distribution is a considerable task. On the experiment side, the investigation of an extremely large number of cycles yields issues related to ultra-small amplitude signals to be measured accurately. It is also required to design experiments that are sensitive to the results predicted by the different theories, in particular when the anisotropy of the critical current is taken into account (Romero-Salazar, 2016). In this respect, the constant improvement of flux distribution measurement techniques with enhanced spatial resolution, possibly at the vortex scale, is of great help. It is also required to keep designing experiments that are representative of the conditions encountered in increasingly demanding practical applications. As an example, the combination of bulk superconductors and soft ferromagnetic materials was recently shown to have a beneficial of the decay of trapped magnetization (Fagnard et al., 2016; Baghdadi et al., 2018). Applications of superconductors in various engineering disciplines involve numerous experimental configurations that are worth of being investigated, e.g., when superconductors are magnetized under pulsed fields (Srpčič et al., 2018). The task is both challenging and stimulating.

References Badía A and López C (2002a) Vector magnetic hysteresis of hard superconductors Phys. Rev. B 65:104514 Badía A and López C (2002b) Magnetic flux bifurcation and frequency doubling in rotated superconductors. J. Appl. Phys. 92:6110−6118 Badía A and López C (2007) Critical-state analysis of orthogonal flux interactions in pinned superconductors. Phys. Rev. B 76:054504 Baghdadi M, Ruiz HS, and Coombs TA (2014) Crossedmagnetic-field experiments on stacked second generation superconducting tapes: reduction of the demagnetization effects. Appl. Phys. Lett. 104:232602

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Baghdadi M, Ruiz HS, and Coombs TA (2018) Nature of the low magnetization decay on stacks of second generation superconducting tapes under crossed and rotating magnetic field experiments Sci. Rep. 8:1342 Baskys A, Patel A, and Glowacki BA (2018) Measurements of crossed field demagnetisation rate of trapped field magnets at high frequencies and below 77 K. Supercond. Sci. Technol. 31:065011 Bean CP (1964) Magnetization of high-field superconductors. Rev. Mod. Phys. 1:31−39 Brandt EH (1996) Superconductors of finite thickness in a perpendicular magnetic field: strips and slabs. Phys. Rev. B 54:4246−4264 Brandt EH and Mikitik GP (2002). Why an ac magnetic field shifts the irreversibility line in type-II superconductors. Phys. Rev. Lett. 89:027002 Brialmont S, Fagnard JF, and Vanderbemden P (2019) A simple torque magnetometer for magnetic moment measurement of large samples: application to permanent magnets and bulk superconductors. Rev. Sci. Instrum. 90: 085101 Campbell AM (1991) DC magnetisation and flux profile techniques. In: Magnetic Susceptibility of Superconductors and Other Spin Systems (Hein RA, Francavilla TL, Liebenberg TL, eds), pp 129−155. Plenum Press, New York. ISBN: 978-1-4899-2381-3 (Print) 978-1-48992379-0 (Online) Campbell AM (2011) Flux cutting in superconductors. Supercond. Sci. Technol. 24:091001 Campbell AM, Baghdadi M, Patel A, Zhou D, Huang KY, Shi Y, and Coombs T (2017) Demagnetisation by crossed fields in superconductors. Supercond. Sci. Technol. 30:034005 Carballo-Sanchez AF, Perez-Rodriguez F, and Perez-Gonzalez A (2001) Magnetic response of hard superconductors subjected to parallel rotating magnetic fields. J. Appl. Phys. 90:3455−3461 Celebi S, Sirois F, and Lacroix C (2015) Collapse of the magnetization by the application of crossed magnetic fields: observations in a commercial Bi:2223/Ag tape and comparison with numerical computations. Supercond. Sci. Technol. 28:025012 Clem JR (1982) Flux-line-cutting losses in type-II superconductors. Phys. Rev. B 26, 2463–2473 Clem JR, Weigand M, Durrell JH, and Campbell AM (2011) Theory and experiment testing flux-line cutting physics. Supercond. Sci. Technol. 24:062002 Durrell JH et al. (2014) A trapped field of 17.6 T in meltprocessed, bulk Gd-Ba-Cu-O reinforced with shrink-fit steel. Supercond. Sci. Technol. 27:082001 Egan R et al. (2015) A flux extraction device to measure the magnetic moment of large samples; application to bulk superconductors. Rev. Sci. Instrum. 86:025107 Fagnard JF, Kirsch S, Morita M, Teshima H, Vanderheyden B, and Vanderbemden P (2015) Measurements on

Handbook of Superconductivity

magnetized GdBCO pellets subjected to small transverse ac magnetic fields at very low frequency: evidence for a slowdown of the magnetization decay. Physica C 512:42−53 Fagnard JF, Morita M, Nariki S, Teshima H, Caps H, Vanderheyden B, and Vanderbemden P (2016) Magnetic moment and local magnetic induction of superconducting/ferromagnetic structures subjected to crossed fields: experiments on GdBCO and modelling. Supercond. Sci. Technol. 29:125004 Fiorillo F (2010) Measurements of magnetic materials. Metrologia 47:S114−S142 Fisher LM, Kalinov AV, Voloshin IF, Baltaga IV, Il’enko KV, and Yampol’skii VA (1996) Superposition of currents in hard superconductors placed into crossed ac and dc magnetic fields. Solid State Commun. 97:833−836 Fisher LM et al. (1997) Collapse of the magnetic moment in a hard superconductor under the action of a transverse ac magnetic field. Physica C 278:169−179 Funaki K and Yamafuji K (1982) Abnormal transverse-field effects in nonideal Type II superconductors I. A linear array of monofilamentary wires. Jpn. J. Appl. Phys. 21:299−304 Gheorghe DG, Menghini L, Wijngaarden RJ, Brandt EH, Mikitik GP, and Goldacker W (2006) Flux penetration into superconducting Nb3Sn in oblique magnetic fields. Phys. Rev. B 73:224512 Hasanain SK, Manzoor S, and A Amirabadizadeh (1995) Magnetization and hysteresis of melt-textured YBa 2Cu3O7-x in a crossed flux configuration Supercond. Sci. Technol. 8:519−524 Hong Z, Vanderbemden P, Pei R, Jiang Y, Campbell AM, and Coombs TA (2008) IEEE Trans. Appl. Supercond. 18:1561−1564 Indenbom MV et al. (1994) Anisotropy of perpendicular field penetration into high-Tc superconductors induced by strong longitudinal field. Physica C 226:325−332 Johansen T H, Baziljevich M, Bratsberg H, Galperin Y, Lindelof P E, Shen Y, and Vase P (1996) Direct observation of the current distribution in thin superconducting strips using magneto-optic imaging. Phys. Rev. B 54:16264−16269 Kapolka M, Srpcic J, Zhou D, Ainslie MD, Pardo E, and Dennis AR (2018) Demagnetization of cubic Gd-BaCu-O bulk superconductor by crossed fields: measurements and three-dimensional modeling. IEEE Trans. Appl. Supercond. 28:6801405 Katter M (2005) Angular dependence of the demagnetization stability of sintered Nd–Fe–B magnets. IEEE Trans. Magn. 41:3853−3855 Kim DH, Kim HJ, Dan NH, and Lee SI (2004) Systematic reduction of magnetization by an ac transverse field and of the anomalous magnetization peak in a N2-annealed Tl2Ba 2CuO6 single crystal. Phys. Rev. B 70:214527

Characterization of Superconductor Magnetic Properties in Crossed Magnetic Fields

Koblischka MR, van Dalen AJJ, and Ravi Kumar G (1996) How is a fully penetrated state formed in an anisotropic superconductor? J. Supercond. 9:143−150 LeBlanc MAR, Celebi S, and Rezeq M (2001) Generation of quasi-reversibility in a commercial Bi:2223/Ag tape by vortex shaking with varying orthogonal magnetic fields. Physica C 361, 251–259 LeBlanc MAR, Celebi S, Wang SX, and Plechácêk V (1993) Cross-flow of flux lines in the weak link regime of highTc superconductors. Phys. Rev. Lett. 71:3367−3370 Liang F, Qu T, Zhang Z, Sheng J, Yuan W, Iwasa Y, and Zhang M (2017) Vortex shaking study of REBCO tape with consideration of anisotropic characteristics. Supercond. Sci. Technol. 30:094006 Luzuriaga J et al. (2009) Magnetic relaxation induced by transverse flux shaking in MgB2 superconductors. Supercond. Sci. Technol. 22:015021 Matsushita T (2007) Flux Pinning in Superconductors. Springer-Verlag Berlin Heidelberg Mikitik GP and Brandt EH (2004) Vortex shaking in rectangular superconducting platelets Phys. Rev. B. 69: 134521 Park SJ and Kouvel JS (1993) Cross-flux effect as a vortex pinning process in grain-oriented YBa 2Cu3O7. Phys. Rev. B 48:13995−13997 Perez-Rodriguez F, LeBlanc MAR, and Gandolfini G (2001) Flux-line cutting in granular high-Tc and semi-reversible classical type-II superconductors. Supercond. Sci. Technol. 14: 386−397 Philippe M P et al. (2014) Magnetic characterisation of large grain, bulk Y-Ba-Cu-O superconductor-soft ferromagnetic alloy hybrid structures. Physica C 502:20−30 Qiu M, Huo HK, Xu Z, Xia D, Lin LZ, and Zhang GM (2005) Technical analysis on the application of HTS bulk in “permanent magnet” motor. IEEE Trans. Magn. 15:3172−3175 Romero-Salazar C (2016) Bianisotropic-critical-state model to study flux cutting in type-II superconductors at parallel geometry. Supercond. Sci. Technol. 29:045004 Romero-Salazar C and Pérez-Rodrı ́guez F (2004) Response of hard superconductors to crossed magnetic fields: elliptic critical-state model. Physica C 404:317–321

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Ruoho S and Arkkio A (2008) Partial demagnetization of permanent magnets in electrical machines caused by an inclined field. IEEE Trans. Magn. 44:1773−1778 Srpčič J et al. (2018) Demagnetization study of pulse-field magnetized bulk superconductors. IEEE Trans. Appl. Supercond. 28:6801305 Srpčič J et al. (2019) Penetration depth of shielding currents due to crossed magnetic fields in bulk (RE)-Ba-Cu-O superconductors. Supercond. Sci. Technol. 32:035010 Swartz JP, McCulloch MD, Pecher R, Prigozhin L, Vanderbemden P, and Chapman SJ (2003). Critical state modeling of crossed field demagnetization in HTS materials. In: Applied Superconductivity (Proceedings of 6th EUCAS, Adreone A. et al., eds), pp 859−866 Trojanowski S, Ciszek M, and Maievskyi E (2014) Design of a race-track coil for measurements of AC power losses in high-temperature superconducting tapes. Metrol. Meas. Syst. 21:293–304 Tsukamoto O, Yamagishi K, Ogawa J, Murakami M, and Tomita M (2005) Mechanism of decay of trapped magnetic field in HTS bulk caused by application of AC magnetic field. J. Mater. Process. Technol. 161:52–57 Tumanski S (2011) Handbook of Magnetic Measurements, CRC Press, Taylor & Francis Uspenskaya LS, Kulakov AB, and Rakhmanov AL (2004) Anisotropic flux creep in Bi2212:Pb single crystal in crossed magnetic fields. Physica C 402:136–142 Vanderbemden P (1998) Design of an A.C. susceptometer based on a cryocooler. Cryogenics 38:839−842 Vanderbemden P et al. (2007a) Behavior of bulk high-temperature superconductors of finite thickness subjected to crossed magnetic fields: Experiment and model. Phys. Rev. B. 75:174515 Vanderbemden P et al. (2007b) Remagnetization of bulk high-temperature superconductors subjected to crossed and rotating magnetic fields. Supercond. Sci. Technol. 20:S174−S183 Vanderbemden P, Laurent P, Fagnard JF, Ausloos M, Hari Babu N, and Cardwell DA (2010) Magneto-thermal phenomena in bulk high temperature superconductors subjected to applied AC magnetic fields. Supercond. Sci. Technol. 23:075006

G2.9 Microwave Impedance Adrian Porch

G2.9.1 Introduction The response of a superconducting sample at high frequencies is characterized by its surface impedance Zs. This section deals with the measurement and interpretation of the microwave surface impedance of superconductors, the concept of which was introduced in Chapter A2.6 High-Frequency Electromagnetic Properties. To summarize briefly here, high-frequency electromagnetic fields are confined to a conducting material’s surface; they penetrate by an amount called the skin depth, which for a superconductor is the magnetic penetration depth λ. If the conductor is flat on the scale of the skin depth, the surface impedance is defined as the quantity Zs = E/H = Rs + iXs, where E and H are the magnitudes of the tangential electric and magnetic fields, respectively, at the surface. Physically, the surface resistance Rs quantifies the rate of energy dissipation within the skin depth, whilst the surface reactance Xs quantifies the peak electromagnetic energy stored within the skin depth. Consequently, the measurement and interpretation of these parameters are highly relevant for high-frequency superconductor applications (see Section H2). Most experimental techniques for evaluating surface impedance Zs involve measurement of the quality factor (Sucher and Fox, 1964) (or, equivalently, the resonant bandwidth) and resonant frequency of a microwave resonator constructed either wholly or partially from the superconductor under test; these are related to the surface resistance Rs and reactance Xs, respectively, of the sample. Examples of host resonators suitable for studying all forms of sample, i.e. bulk, single crystal, unpatterned thin films and patterned thin films, in the frequency range 1–100 GHz are discussed in this section. The nonlinear surface impedance of HTS materials is a topical area of high relevance to microwave device applications, and a separate subsection is devoted to this issue.

G2.9.2 General Resonator Methods Modern microwave measurements up to about 110 GHz are performed using vector network analysers which, although expensive, employ synchronous detection and internal error 264

correction (Pozar, 2012) to allow highly accurate measurements of the amplitude and phase of a microwave signal. Alternative systems for resonator measurements use synthesized microwave oscillators, which are stabilized by including the resonator in a feedback loop (Luiten et al., 1996). Most resonator measurements are performed in transmission mode. Coupling to the resonator is achieved using either loop or probe terminated coaxial lines, which couple to the magnetic or electric fields of the resonator. Provided that the coupling is not too strong, the magnitude of the amplitude transmission coefficient |S21| has a Lorentzian frequency response (Petersan and Anlage, 1998). S21 ( ω ) =

S21 ( 0 ) 1 + 4Q 2

(

ω ω0

−1

)

2

(G2.9.1)

where Q is the quality factor and ω 0 is the resonant frequency. If the microwave losses in the resonator are small, then Q >> 1 and, in this limit, Q = ω 0 / ω B, where ω B is the full bandwidth at half power. Therefore, ω B can be used to quantify the losses in the resonator, part of which is due to the finite surface resistance of the resonator material. The Q factor of a high Q resonator is determined either from a single measurement of ω B and ω 0 or, more properly, by fitting the measured frequency response to Equation (G2.9.1) (Petersan and Anlage, 1998). Alternatively, very large Q factors can be measured using the relaxation method (Sridhar and Kennedy, 1988). An input signal at ω 0 is pulsed at a low frequency (typically a few kHz) and the transmitted power is measured using a high-speed oscilloscope. When the input signal switches off, the output power decays exponentially with a characteristic decay time τ = Q / ω 0, which, at 10 GHz, is greater than 10 μs when Q > 106. The measured Q factor is called the loaded quality factor QL , which decreases with increasing input and output coupling strengths Lancaster (1997). The Q factor in the limit of weak coupling is called the unloaded quality factor Q0 (> QL) and is independent of the coupling strength. It is straightforward

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FIGURE G2.9.1 The effects of changing the temperature of a resonator constructed wholly or partially from a superconductor. On cooling through Tc there is a decrease in bandwidth and an increase in resonant frequency, both as a consequence of the reduction of the surface impedance of the superconductor.

to determine Q0 from the measured QL Lancaster (1997), but such corrections are unnecessary in the limit of weak coupling since Q0 ≈ QL . All Q factors referred to in this section are unloaded values. The effect of cooling a resonator constructed wholly or partially from a superconductor is shown in Figure G2.9.1. On cooling through Tc there are two main consequences: an increase in Q (i.e. a decrease in bandwidth ω B) due to the rapid decrease in the surface resistance Rs of the superconductor, and an increase in frequency associated with the electromagnetic fields being confined to within the penetration depth λ of the surface of the superconductor (since λ is much smaller than the normal state skin depth at microwave frequencies). If the temperature of the superconductor is now changed, the corresponding changes in resonant bandwidth Δω B = ω 0 Δ(1/ Q ) and resonant frequency Δω are related to the changes in the surface resistance and surface reactance, respectively, by ΔRs = ΓΔω B , ΔX s = −2ΓΔω .

(G2.9.2)

Γ is called the resonator constant, and is a function of resonator geometry which can be calculated either experimentally or from the electromagnetic field distribution in the resonator Lancaster (1997). Although Equation (G2.9.2) only relates to changes in the surface impedance, absolute values can be calculated for specific types of resonator, as illustrated in the examples below.

G2.9.3 Measurements of Small Single Crystals A cylindrical resonator (Sridhar and Kennedy, 1988; Mao et al., 1995) is commonly used as the host resonator for small single crystal measurements, as shown schematically in Figure G2.9.2(a); other possible host resonators are split ring

FIGURE G2.9.2 (a) A schematic diagram of the TE011 cylindrical resonator method for measuring small single crystals. (b) The crystal orientation that results in the lowest perturbation of the microwave field and the induced current flow within it, taking into account anisotropy; the right-hand side diagram illustrates the cleaved crystal method for deducing all diagonal tensor components of the surface impedance.

(Bonn et al., 1991) and dielectric resonators (Wingfield et al., 1997), which are based on the same measurement and analysis principles. The method described in this section is most appropriate for HTS crystals, which typically are platelets of area 1 mm2 and thickness a few tens of μm. The host resonator for crystal measurements should preferably have a very high Q factor so that the sample contributes a large proportion of the microwave losses. For measurements in zero applied fields, the conducting parts of the resonator are fabricated from Nb or Pb plated copper; the Nb should be vacuum annealed to reduce its surface resistance, after which Q > 107 at 4.2 K is possible above 10 GHz. For measurements in large applied magnetic fields, OFHC copper resonators

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must be used, though in this instance it is better to use a selfresonant dielectric resonator to reduce the microwave field magnitudes around the lossy, normal conducting walls (Wingfield et al., 1997). The cylindrical resonator is usually operated in the TE011 mode, the electromagnetic field distribution, frequency and Q factor of which can be easily found from standard electromagnetic theory (Collin, 1992). The currents in the resonator walls in this mode are azimuthal and are therefore relatively unaffected by the joints between the cylinder end plates and curved perimeter wall. One problem is that the TE011 mode always has the same resonant frequency as the TM111 mode, but the latter can be shifted to lower frequencies using a mode trap (usually a circular groove cut near the perimeter of each end plate). Coupling to the TE011 microwave magnetic field is accomplished using loop-terminated coaxial lines, oriented in such a way that their planes are perpendicular to the field. The TE011 microwave magnetic field distribution is also shown in Figure G2.9.2(a), which has a maximum value in the centre of the resonator, where the field is approximately uniform and parallel to the axis of the cylinder. A thin crystal platelet is best oriented so that its plane is parallel with the microwave field, as shown in Figure G2.9.2(b), in which case it will not significantly perturb the field distribution, and the demagnetization factors are almost zero. Microwave currents are induced within the crystal, which form closed loops on the cross-section perpendicular to the field. The crystal is fixed to a sapphire rod using a thin layer of vacuum grease and is inserted into the position of largest microwave magnetic field for maximum sensitivity. The rod and crystal are heated using a resistance coil whilst maintaining the resonator at a constant low temperature (preferably 4.2 K or below). The excellent thermal conductivity of sapphire below 100 K ensures that there is a negligible temperature gradient across the ends of the rod, and also that the thermal response time is very short. The dielectric loss of sapphire is extremely low, so it has no measurable contribution to the resonator Q factor. It is highly desirable to be able to remove the crystal from the resonator in situ, for reasons which will become apparent below. The surface impedance of the crystal is obtained using cavity perturbation theory (Altshuler, 1963), which assumes that its presence has little effect on the electromagnetic field distribution within the resonator. Measurements of the bandwidth and resonant frequency are performed as a function of temperature with and without the crystal present. Defining Δω B to be the bandwidth change at each temperature between these two configurations, the surface resistance of the crystal is then approximately Rs ≈ ΓΔω B .

(G2.9.3)

The resonator constant Γ is best determined experimentally by measuring the bandwidth change of the crystal in its normal state Δω B,n , having previously measured its dc normal

state resistivity ρdc. The surface resistance in the normal state can then be calculated using the classical skin effect formula Rs,n = µ 0 ω 0 ρdc / 2 assuming that the crystal is thicker than 1/2 the normal state skin depth δ = (2ρdc / µ 0 ω ) , in which case Γ ≈ Rs,n / Δω B,n and the surface resistance of the crystal is approximately Rs ≈ Rs,n

Δω B . Δω B,n

(G2.9.4)

Alternatively, Γ can be found by performing measurements on different crystals of a material of known surface resistance, or can be calculated from knowledge of the electromagnetic field distribution in the resonator. The absolute value of the surface reactance of the crystal can be determined by measuring the resonant frequency difference ω 0 − ω 0,n between superconducting and normal states, in which case X s − X s,n = −2Γ ( ω 0 − ω 0,n ).

(G2.9.5)

Equation (G2.9.5) can be used to find surface reactance Xs at all temperatures since Γ ≈ Rs,n / Δω B,n and in the normal state X s,n = Rs,n = µ 0 ω 0 ρdc / 2 , both of which have been previously determined. It is important to consider the resolution in the surface impedance offered by resonator methods when measuring small single crystals, which means estimating the resonator constant Γ from first principles. If a crystal is placed in a region of maximum microwave field magnitude H0, its contribution to the resonant bandwidth is Δω B =

1 H 2 2A Rs Rs ≈ 02 Γ H µ 0Vres

(G2.9.6)

where A is the area of the crystal face parallel to the applied field, Vres is the volume of the host resonator and H 2 is the volume average of H2 throughout the resonator (less than H 02). Typical measurement limits using stable microwave sources of the resonant bandwidth and frequency are Δω B,min / ω ≈ 10−9 and Δω min / ω ≈ 10−9, respectively. The minimum measurable surface resistance is therefore Rs,min ≈

H 2 µ 0 ωVres Δω B . 2A H 02 ω

(G2.9.7)

Remembering that ΔX s = −2Γ Δω = µ 0 ω Δλ, the minimum measurable change in penetration depth is Δλ min ≈

H 2 Vres Δω . H 02 A ω

(G2.9.8)

These minimum values are reduced by measuring the crystal in the region of maximum magnetic field, and also by reducing the resonator volume. For example, reducing the dimensions of a TE011 cylindrical resonator by a factor of 2 reduces

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the volume by a factor of 8 and increases the resonant frequency by a a factor of 2. These effects combine to give improvements by factors of 4 and 8 in in Rs,min and Δλ min, respectively, assuming that Δω B,min /ω and Δω min /ω are independent of frequency. Additionally, by doubling the frequency, the surface resistance of the crystal is larger by a factor of 4 (since Rs ∝ ω 2 ), so that the fraction Rs,min /Rs is actually reduced by a factor of 16. As a numerical example, consider a TE011 cylindrical resonator of radius 1 cm and length 2 cm. This aspect ratio produces the highest Q factor of the host resonator, and also good frequency separation from other resonant modes. The resonant frequency of the TE011 mode is f = ω / 2π = 23.7GHz, whilst H 2 /H 02 = 0.136 and Vres = 6.28cm3 . For a typical HTS crystal of area 1 mm2, Rs,min ≈ 80 μΩ and Δλ ≈ 1 nm, suitable for measurements of the highest-quality crystals down to 4.2 K. HTS materials are in the London limit (λ L >> >> ξ) so, from the experimental determination of the surface impedance Zs = Rs + iXs, it is possible to calculate the microwave conductivity in the superconducting state σ = σ1 − iσ 2 = σ1 − i / µ 0 ωλ L2 , since from Equation (A.2.6.6) Rs ≈ σ1µ 02 ω 02 λ 3L / 2 and Xs ≈ μ0ω0λL . This analysis is complicated by the high anisotropy of HTS single crystals, but this can be dealt with quite easily in the parallel field geometry of Figure G2.9.2(b). For the moment, it is assumed that the crystal is untwinned (i.e. it has unique aand b-axes within the plane), with the microwave field applied parallel to the b-axis. From Figure G2.9.2(b) it is seen that the screening current flows continuously for a distance a along the a-axis parallel to the crystal face, then for a short distance c along the c-axis, then back along the a-axis on the opposite face, before completing the loop along the c-axis. Defining Rs,a and Rs,c to be the surface resistances for screening currents flowing parallel to the a- and c-axes, respectively, the effective surface resistance in this geometry for a thin crystal (with c 1), again Zs,eff ≈ Zs =Rs + iXs; however, for samples of arbitrary thickness the measured surface impedance exceeds Zs and, in the thin sample limit (i.e. x 3 the effective and intrinsic surface impedances differ little from each other.

G2.9.5 Measurements of Unpatterned Thin Films and Bulk Samples Dielectric resonator (DR) methods have been developed since the early 1990s, specifically to evaluate the surface impedance of HTS thin or thick films (Shen et al., 1992), although these methods also work well for thin film or bulk samples

of conventional materials. DRs offer improved sensitivity over cylindrical resonator end-wall techniques, where the sample replaces the end wall of a cylindrical resonator. DRs are self-resonant structures, where the electromagnetic fields are confined to within and just outside the dielectric, which is usually a single crystal cylinder of low loss material. Microwave screening currents are induced within the sample in the vicinity of the dielectric, and the surface impedance can again be determined from measurements of the Q factor (or bandwidth) and resonant frequency. For c-axis–oriented HTS thin films, the screening currents flow entirely within the abplanes and, consequently, the ab-plane surface impedance is measured. Analytical solutions for the electromagnetic fields within a DR exist only for the Hakki–Coleman configuration Hakki and Coleman (1960), shown in Figure G2.9.6, where a dielectric cylinder is sandwiched between two films of infinite extent. Sometimes it is necessary to investigate a single film, in which case the upper film is moved away from the dielectric (Ormeno et al., 1997); the full analysis of this latter structure is complicated and not considered here. The microwave field distribution in the TE011 mode in the Hakki– Coleman geometry is similar to that of the TE011 cylindrical resonator mode. In this mode, the currents in the two end plates are again azimuthal, so that the mode is unaffected by joints to the outer radiation shield. Coupling to the microwave magnetic field is achieved in a similar manner using loop-terminated coaxial lines. A thin quartz plate prevents sideway movement of the dielectric. The field magnitudes outside the dielectric are very small when its relative permittivity εr is high, so that one can ignore the microwave losses in the outer perimeter of the structure (usually OFHC copper) and assume that the films are infinite in extent, making the analysis easier. Good mode separation of the TE011 mode with other resonant modes is achieved for a dielectric whose radius equals its length, in which case the resonant frequency is approximately f (GHz) ≈

20.9 a(cm) ε r

(G2.9.13)

Handbook of Superconductivity

270 TABLE G2.9.1 The Dielectric Properties at 80 K of Some Common Materials Used as Dielectric Resonators Material

εr

tanδ

Sapphire, Al2O3 LaAlO3 Rutile, TiO2

9.4 23.5 105 (110)

10−7 ( < 10−8) 10−5 ( < 10−6) 5×10−4 ( < 10−6)

Note: The numbers in parentheses are the corresponding values at 10 K. All of the dielectrics listed are anisotropic and the values quoted are for microwave electric fields perpendicular to the c-axis for single crystal specimens. The loss tangents are typical values at 10 GHz for high-quality samples that, in practice, vary significantly with sample purity and microstructure. Generally, tanδ is approximately proportional to frequency.

where a is the radius of dielectric (assumed to be equal to its length). Equation (G2.9.13) is valid within about 1% of the exact solution for the resonant frequency when εr > 9. Some data for the permittivity of sapphire (Al2O3), lanthanum aluminate (LaAlO3) and rutile (TiO2) are listed in Table G2.9.1. These data can be used in conjunction with Equation (G2.9.13) to design a DR appropriate for the size of the films that require evaluation. Sapphire has an extremely low dielectric loss (quantified by its low loss tangent tanδ) and is commonly used to characterize large area films (e.g. 5 cm × 5 cm). Although it has anisotropic dielectric properties, if the sapphire is oriented with its c-axis parallel to the cylinder axis of the DR, then the electric field within the TE011 mode is always parallel to its hexagonal (0001) plane, which has an isotropic in-plane relative permittivity of about 9.4. Since this is fairly small, to prevent excessive losses in the radiation shield for measurements of small area ( 9; Rs,avg (measured in Ω) is the average surface resistance of the two films used in the experiment. To characterize individual films, the surface resistance of one of them should already be known; otherwise a batch of three films is required,

with measurements performed using the three possible pairs that can be selected from this batch. Equation (G2.9.14) can now be used to extract surface resistance from the measured Q factor or resonant bandwidth ω B = 2πf B. As can be seen from Table G2.9.1, the loss tangent of sapphire is typically less than 10 −7 at 77 K for frequencies below 30 GHz. The best quality HTS films at 10 GHz have surface resistance values around 100 μΩ at 77 K, so that ignoring the effects of tanδ for a sapphire DR thus leads to an overestimation of surface resistance by about 20%; this overestimation is worse when using other dielectric materials with higher values of tanδ. To account properly for the effects of dielectric loss, it is necessary to first measure tanδ at the same operating frequency and temperature, which can be accomplished using standard techniques Kobayashi and Katoh (1985). From Equation (G2.9.14), the resolution in measuring Rs at a given frequency is improved by using a high permittivity dielectric, with a radius a chosen to obtain the appropriate frequency. However, as an example, a 10 GHz sapphire DR with Δω B,min / ω ≈ 10−9 yields Rs,min < 1μΩ. This hardly needs improvement by using a higher permittivity dielectric unless unstable microwave sources are used. In practice, the experimental values of Rs are much more likely to be affected by the systematic error associated with uncertainties in tanδ, or due to uncertainties in the losses in the outer perimeter walls, both of which lead to an overestimation of Rs. Surface resistances obtained using DR techniques are effective values which are uncorrected for the finite film thickness. Unfortunately, it is very difficult to perform measurements of the penetration depth λL in any DR configuration, since most of the frequency shift observed on changing the temperature is due to thermal expansion and the temperature dependence of εr (particularly when using rutile; see Table G2.9.1). However, some modified methods exist for determining both Rs and λL [20]. Generally, for HTS thin films, λL at any temperature can be estimated from the empirical relation λ L (T ) = λ L (0)(1 − (T / Tc )2 )−1/2, where λ L ( 0 ) ≈ 140 – 160 nm in the highest quality YBa2Cu3O7 films. The correction factor of Equation (G2.9.12) can then be used to find the intrinsic surface impedance. Figure G2.9.7 shows the measured Q and surface resistance as a function of temperature of a high-quality thin film of YBa2Cu3O7 measured using a LaAlO3 dielectric resonator at 15 GHz. Further thin film data are discussed in Section G2.9.6.

G2.9.6 Measurements of Patterned Thin Films Thin films can be patterned lithographically into a number of planar resonator structures. For c-axis–oriented films the microwave currents flow along the ab-planes, so that the ab-plane surface impedance is measured using planar structures (Porch et al., 1995). The easiest planar structure to fabricate is the coplanar resonator (shown schematically in

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FIGURE G2.9.7 The temperature dependence of the Q factor and computed surface resistance of a high-quality YBa 2Cu 3O7 thin film at 15 GHz measured using a LaAlO3 dielectric resonator. The surface resistance has been corrected for the finite film thickness.

Figure G2.9.8) since it requires only a single film surface. The package used to measure the coplanar resonator is also shown in Figure G2.9.8. If the ends of the resonator are in open circuit (i.e. the microwave electric field magnitude is highest at these ends), microwave coupling is best achieved using probeterminated coaxial connectors. Coplanar resonators offer a very high-resolution measurement of the surface impedance, and are also excellent structures for studying nonlinear effects owing to the large microwave current densities at the patterned edges. Typically, for a film of thickness t and penetration depth λL , the microwave current density is very large within a distance λ L2 / t from the edge. These edge regions therefore carry a significant proportion of the microwave current, so that the surface impedance of the film material very close to the edges is evaluated, of crucial importance for planar microwave applications. Since the coplanar resonator is a quasi-TEM structure, resonances occur when the length of the line equals an integer number n of half wavelengths. The resonant frequencies are therefore fn =

nc 2 ε r ,eff

(G2.9.15)

where c is the speed of light in free space, and εr,eff is the average relative permittivity of the dielectric above and below the coplanar structure; since the former is usually a vacuum or a gas, ε r ,eff ≈ (1 + ε r ) / 2 , where εr is the relative permittivity of the substrate. As an example, an 8-mm-long coplanar resonator on MgO (εr ≈10) has a fundamental resonant frequency of 8 GHz and subsequent resonances at 16, 24, 32 GHz, etc; importantly, this allows measurements at more than one frequency using the same resonant structure. Computing the surface impedance from coplanar resonator measurements is a complicated procedure. Quite generally,

FIGURE G2.9.8 A schematic diagram of the microwave package used for measuring coplanar resonators, together with a typical resonator design which yields a fundamental frequency of 8 GHz when using an MgO substrate. The silver pads aid ground plane contact with the walls of the package.

one may write for the effective resistance R and inductance L of the line per unit length R ≈ Rs g1 ( λ L ) , L ≈ L0 + µ 0 λ L g 2 ( λ L )

(G2.9.16)

where the factors g 1 and g 2 (of units m −1) are functions of the cross-sectional geometry (in addition to λL), and account for the nonuniform current distribution on the resonator cross-section. The term L 0 is the external inductance per unit length, associated with the magnetic field outside the conductors. For thin films, the term µ 0 λ L g 2 ( λ L ) is made up mainly by the kinetic inductance of the supercurrent (associated with its nondissipative kinetic energy) and is usually much smaller than L 0 for temperatures not too close to Tc . In terms of Equation (G2.9.16), the resonant bandwidth is ωB =

R Rs ≈ g 1 (λ L ). L L0

(G2.9.17)

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Since ω ∝ L−1/2 , any change in penetration depth ΔλL will cause a fractional change in resonant frequency Δω 1 ΔL 1 µ0 d (λ L g 2 )Δλ L . (G2.9.18) ≈− ≈− ω 2 L0 2 L0 dλ L The frequency decreases for an increase in λL, since all of the terms in Equation (G2.9.18) other than the prefactor are positive. It is possible to calculate both g1(λL) and g2(λL) numerically, either using full-wave analysis (Kessler et al., 1991) or the coupled transmission line method (Sheen et al., 1991), and some approximate analytical formulae can be deduced from the numerical calculations (Rauch et al., 1992). Consequently, measurements of the resonant bandwidth and resonant frequency can now be used to deduce the surface impedance of the film edge regions. The critical parameter in the analysis is λL , which must be deduced first from measurements of the resonant frequency as a function of temperature. To ensure that the frequency shifts, owing to changes in λL with temperature, are larger than the shifts owing to the temperature dependence of εr or due to the thermal expansion of the substrate, it is necessary to choose a sufficiently narrow gap between the centre line and each ground plane. The temperature dependence of the frequency shift is shown in Figure G2.9.9(a) for two resonators of central line width 200 μm, with a gap to each ground plane of 10 and 73 μm. The fractional frequency shift of the narrow gap resonator at each temperature is the larger of the two, since its kinetic inductance makes up a larger proportion of the total inductance. Importantly, measurements of the frequency shift of two resonators of different cross-sections patterned onto the same film can be used to deduce an absolute value of λL (Porch et al., 1995), assuming that the film properties (e.g. film thickness) do not vary over its surface. The temperature dependence of λL obtained from the data of Figure G2.9.9(a) is shown in Figure G2.9.9(b), from which a value of λL(0) ≈ 160 nm has been deduced. The coplanar resonator technique offers high resolution since a small change in surface impedance gives rise to large changes in the resonator response. For example, a resonator of width 200 μm, with a gap either side of 10 μm, deposited on a 0.35-μm-thick film has a Q factor of only 20,000 at 8 GHz for a surface resistance of 25 μΩ. Importantly, almost all of the resonant bandwidth is due the surface resistance of the film for narrow gap resonators, since the loss tangent of single crystal MgO is less than 2 × 10 −6 below 50 K, so that substrate losses account for less than 5% of the bandwidth; losses in the conducting housing are negligible since the fields are highly confined to the central strip. The main measurement uncertainty when using a stable microwave source is caused by a systematic error in λL of about ±10% obtained by the technique of varying the resonator cross-section, which then results in an error of about ±25% in surface resistance. These errors refer to absolute values, since changes in each can be measured to better than 1% accuracy.

FIGURE G2.9.9 (a) The fractional frequency shift Δf(T)/f(T) = (f(12K – f(T))/f(T) for the 8 GHz mode for two YBa 2Cu 3O7 resonators of thickness 0.35 μm of different cross-sectional geometries. (b) The computed supercurrent conductivity σ 2 (T )/ σ 2 (0) = λ 2L (0)/ λ 2L (T ), with λL(0) = 160 nm.

Some data for the Q factor and computed surface resistance as a function of temperature at 8 and 16 GHz are shown in Figure G2.9.10 for YBa 2Cu3O7 thin films. The surface resistance values for the highest quality thin films (whether patterned or unpatterned) deposited by a range of techniques are similar, implying that the behaviour of these films is reflecting intrinsic properties of the superconductor. For example, the presence of a rather broad plateau in Rs between 40 and 60 K, associated with the similar peak in Rs observed in single crystals at lower temperatures, but this time in the presence of strong quasi-particle scattering, and also the observation that ΔλL(T) = λL(T) – λL(0) ∝ T2 at low temperatures, consistent with that expected for a d-wave superconductor with strong scattering.

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FIGURE G2.9.11 A collective plot of worldwide data for the surface resistance at 77 K of high-quality YBa 2Cu 3O7 (closed circles) and Tl 2Ba 2CaCu 2O8 thin films (open circles) as a function of frequency. The surface resistance of Tl2Ba 2CaCu 2O8 is slightly lower than that of YBa 2Cu 3O7 owing to its higher Tc. Both materials exhibit R s ∝ f2 up to about 100 GHz. Also shown is the typical surface resistance for thick film YBa 2Cu 3O7, which exhibits R s ∝ f 1:2–1:5 and has a much lower cross-over frequency with copper (about 10 GHz) compared to thin film YBa 2Cu 3O7 (about 100 GHz).

when deciding the choice of material for a particular microwave application.

FIGURE G2.9.10 (a) The unloaded Q factor as a function of temperature for the 8 and 16 GHz modes of the 10 μm gap YBa 2Cu 3O7 resonator whose frequency shift data are shown in Figure G2.9.9(a); since R s ∝ f2, this means that Q ∝ f/R s ∝ 1/f over the whole temperature range. (b) The corresponding surface resistance at 8 and 16 GHz, calculated using the penetration depth data of Figure G2.9.9(b).

The frequency dependence of the surface resistance is plotted in Figure G2.9.11 for both patterned and unpatterned HTS films at 77 K. As a benchmark figure, a high-quality HTS thin film has a surface resistance of less than 150 μΩ at 77 K and 10 GHz. All surface resistance values scale as ω 2 below 100 GHz, with a cross-over frequency compared to OFHC copper at 77 K of around 100 GHz; above this frequency the superconductor is more dissipative than copper. The surface resistance of high-quality YBa 2Cu3O7 and Tl2Ba 2CaCu2O8 thin films are similar below about 70 K at low microwave field levels; above 70 K, Tl2Ba 2CaCu2O8 films have lower surface resistance owing to their higher Tc. However, the nonlinear properties of the two materials at high-field levels differ significantly, and this is an important consideration

G2.9.7 Measurements of the Nonlinear Properties at High Microwave Power Levels At low frequencies, magnetic flux lines enter a Type II superconductor when the magnetic field magnitude exceeds Bc1. However, matters are complicated for a high-frequency magnetic field owing to the finite nucleation time of the flux lines Samoilova (1995). For conventional materials like Nb with a long coherence length ξ, the nucleation time is long, about 10 −7 s, which means that, at microwave frequencies above 1 GHz, it is possible for ac magnetic field levels to exceed Bc1 without the nucleation of flux lines, a result of importance when using Nb microwave cavities for high-field accelerator applications. For HTS materials ξ is very small and, whilst it has not yet been measured directly, one would expect the nucleation time to be about 10 −12 s. This means that, below about 100 GHz, flux enters a HTS sample at a microwave field of Bc1 or, equivalently, when the microwave current density exceeds the dc critical current Jc. This is the most basic form of magnetic field-induced nonlinearity in HTS materials at microwave frequencies. It presents a fundamental limit to the high power operation of any microwave HTS device (see Section H2, High-Frequency Devices). It is not

274

clear whether the critical state is then established at frequencies above 1 GHz, since the large values of flux line viscosities present in HTS means that it is attained over a much longer timescale than for nucleation, typically about 10 −9 s. Both the dielectric and coplanar resonator methods can be used to study nonlinear effects at high power levels in unpatterned and patterned films, respectively. In the former case, this is possibly due to the very high resonator Q factor (typically > 106) yielding very high stored energies and, consequently, high surface field magnitudes; in the latter case, this is possibly due to very high microwave currents induced at the patterned edges owing to the effects of current crowding. When the surface impedance of a sample becomes nonlinear, the resonator Q factor becomes field (or, equivalently, microwave power) dependent; there is also harmonic generation at odd harmonics of the excitation frequency, and frequency intermodulation. All of these effects are undesirable for microwave applications. Some results for the nonlinear surface resistance as a function of microwave current density (proportional to the microwave magnetic field) at 15 K of two YBa 2Cu 3O7 thin films are shown in Figure G2.9.12(a). These were obtained for a patterned film using the coplanar resonator technique. Although the two films have similar low power surface resistance values, the high power behaviour is dramatically different, implying that a low surface resistance value is not necessarily a good indicator of a film’s power handling capabilities. Generally, high-quality films exhibit little field dependence below Bc1; lower-quality films typically exhibit a quadratic dependence of the surface resistance on the microwave field due to the effects of granularity (i.e. weak links) at low fields, with the possibility of pair-breaking at intermediate fields. Figure G2.9.12(b) shows the skewed Lorentzian frequency responses obtained at high power levels, which have to be modelled appropriately to extract the resonant bandwidth and frequency (Jacobs et al., 1996). Above Bc1 there is a rapid increase in surface impedance associated mainly with the combined effects of flux line nucleation (and subsequent motion) and heating. Since Bc1 ∝ 1/λ2, one expects approximately that Bc1(T)∝1−(T/Tc)2 for HTS thin films, so that the microwave critical field can become small at temperatures close to Tc. Typically, for YBa 2Cu3O7 thin films at microwave frequencies Bc1(0) ≈ 20 – 30 mT, yielding a critical microwave current density Jc(0) of 107 Acm−2. When using a continuous microwave input signal, there will certainly be some sample heating when Bc1 is exceeded, owing to the large thermal boundary resistance between the film and substrate. In very granular films, local heating may be observed even for fields well below Bc1 due to the presence of localized defects with low critical current densities, e.g. a weak link due to a single, large-angle grain boundary. The effects of heating can be investigated using pulsed microwave input signals (Wosik et al., 1997), and the extent of heating can be minimized by reducing the duty cycle of the input signal.

Handbook of Superconductivity

FIGURE G2.9.12 (a) The nonlinear surface resistance at 15 K of two YBa 2Cu 3O7 thin films obtained using the coplanar resonator method at 8 GHz. Although the surface resistance values at low microwave currents are similar, the behaviour at high currents is dramatically different. (b) The skewed resonator transmission responses at high microwave current levels (i.e. high microwave input powers, measured in units of dBm, where 0 dBm = 1 mW) as a result of the nonlinear surface impedance of the thin film.

Since degraded areas and defects significantly affect the microwave power handling capability of a thin film, microwave probes based on an open-ended coaxial cable with an inner conductor of extremely small diameter have been developed recently, which allow local surface impedance measurements on the scale of 1 μm (Dutta et al., 1999). The film can be scanned with the probe so that a surface impedance map is produced, indicating the offending areas. Such tools will prove to be useful aids when fabricating planar microwave devices since it can be ensured that all patterned edges are

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Figure G2.9.13(b) shows results for third harmonic generation for HTS coplanar lines (Shen et al., 1997). Unexpectedly, the gradient changes from 3 to 2 and back again over a relatively wide range of input powers for Tl2Ba2CaCu2O8 thin films. The origin of this ‘S-shaped’ behaviour is unclear, but results in a lower TOI for Tl2Ba2CaCu2O8 films compared to YBa2Cu3O7 films, whose slope is close to 3 over a similar power range. Interestingly, the roles are reversed while measuring the surface impedance using resonator techniques at high power levels. For both patterned and unpatterned films at 70 K, the Q factor degradation occurs at lower power levels for Tl2Ba2CaCu2O8 (Shen et al., 1997), implying a lower value of Bc1 for microwave fields than for YBa2Cu3O7. The choice of material for use in high power applications is, therefore, not straightforward, but if harmonic generation and intermodulation at low power levels can be tolerated, then YBa2Cu3O7 is probably a more reliable choice for temperatures below about 70 K.

G2.9.8 Summary

FIGURE G2.9.13 (a) The experimental setup required for third harmonic generation measurements. (b) Results for the third harmonic generation for coplanar lines with input frequency of 1.3 GHz for a YBa 2Cu 3O7 line at 70 K and a Tl2Ba 2CaCu 2O8 line at 80 K. The unusual S-shaped feature in the data of the Tl 2Ba 2CaCu 2O8 line is not understood. The third-order intercept of the Tl 2Ba 2CaCu 2O8 line was +72 dBm, compared with +62 dBm for the YBa 2Cu 3O7 line. (Permission from Shen et al., 1997 © 1997 IEEE.)

well away from defective regions, which are not always apparent while using optical techniques. Measurement of the harmonic generation and intermodulation provides a far more sensitive assessment of nonlinear effects than Q degradation, and are present even at very low power levels, where the surface impedance often appears linear while using resonator techniques. Such effects are best studied using a length of coplanar transmission line (as opposed to a coplanar resonator) incorporated within the test setup shown in Figure G2.9.13(a). To observe harmonic generation, a microwave signal at frequency f is input to the transmission line and the power output at frequency 3f is measured using a spectrum analyser. To observe intermodulation, two signals of slightly different frequencies f 1 and f2 (< f 1) are input and the power output at frequency f 12 = 2f 1 − f2 is measured; in this section, only harmonic generation is discussed. The gradient of the log–log plot of the output power against the input power at the fundamental frequency f is 1. A similar plot of the output power of the third harmonic at frequency 3f against the input power at f has a gradient of 3. These two plots cross at a point called the third-order intercept (TOI), in units of dBm and which quantifies the extent of the third harmonic generation; the higher the TOI, the less harmonic generation.

This section described in detail the various methods of the evaluation of the microwave surface impedance of the common forms of superconducting sample that are currently available. Powders and single crystals are best measured using cavity perturbation techniques using high Q host resonators for the highest sensitivity. Unpatterned thick and thin films use dielectric resonators where the film(s) make up one (or both) of the end plates. Thin film measurements have to be corrected for the effects of sample thickness if this is comparable to or less than the magnetic penetration depth. In the case of patterned thin film resonators, this is coupled with a lateral current distribution whose details are sensitive to the ratio of the film thickness to penetration depth; any characterization technique based on patterned films has to account for this, and the example of the coplanar resonator (which has the advantage of requiring a single film but the disadvantage of having a highly nonuniform current distribution) has been described in detail. The assessment of nonlinear effects with increasing microwave power is very important for microwave device applications. Nonlinear measurements by two methods—surface impedance, and harmonic generation and intermodulation products—as a function of microwave input power have been discussed. The origin of nonlinearities in HTS films is still open to debate, but is closely linked to the microstructural quality of the films, with the best epitaxial thin films of Tl2Ba 2CaCu2O8 and YBa 2Cu3O7 offering similar performance in terms of their microwave power handling capabilities.

References Altshuler HM (1963) Handbook of Microwave Measurements II. New York: Polytechnic Institute of Brooklyn. Bonn DA, Morgan DC and Hardy WN (1991) Split-ring resonators for measuring microwave surface resistance of oxide superconductors. Rev. Sci. Instr. 62:1819−1823

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Collin RE (1992) Foundations For Microwave Engineering. Singapore: McGraw-Hill. Dutta SK et al. (1999) Imaging microwave electric fields using a rear-field scanning microwave microscope Appl. Phys. Lett. 74:156−158 Hakki BW and Coleman PD (1960) A dielectric resonator method for measuring inductive capacities in the millimeter range. IRE Trans. Microwave Theor. Tech. 8:402−410 Hosseini A, Kamal S, Bonn DA, Liang R and Hardy WN (1998) c-Axis electrodynamics of YBa 2Cu3O7−d. Phys Rev. Lett. 81:1298−1301 Jacobs T, Willemsen BA and Sridhar S (1996) Quantitative analysis of nonlinear microwave surface impedance from non-Lorentzian resonances of high Q resonators. Rev. Sci. Instrum. 67:3757−3758 Kamal S, Liang R, Hosseini A, Bonn D A and Hardy W N (1998) Magnetic penetration depth and surface resistance in ultrahigh-purity crystals. Phys. Rev. 58: R8933−R8936 Kessler J, Dill R and Russer P (1991) Field-theory investigation of high-Tc superconducting coplanar wave-guide transmission-lines and resonators. IEEE Trans. Microwave Theor. Tech. 39:1566−1574 Kitano H, Hanaguri T and Maeda A (1998) c-Axis conductivity of Bi2Sr2CaCu2Oy in the superconducting state. Phys. Rev. B 57:10946−10950 Klein N et al. (1990) The effective microwave surface impedance of high-Tc thin films J. Appl. Phys. 67:6940−6945 Kobayashi Y and Katoh M (1985) Microwave measurement of dielectric properties of low-loss materials by the dielectric rod resonator method. IEEE Trans. Microwave Theor. Tech. 33:586−592 Lancaster MJ (1997) Passive Microwave Device Applications of High-Temperature Superconductors. Cambridge: Cambridge University Press. Luiten AN, Mann AG and Blair DG (1996) High-resolution measurement of the temperature-dependence of the Q, coupling and resonant frequency of a microwave resonator. Meas. Sci. Technol. 7:949−953 Mao J, Wu DH, Peng JL, Greene RL and Anlage SM (1995) Anisotropic surface impedance of YBa 2Cu3O7−d single crystals. Phys. Rev. B. 51:3316−3319 Ormeno RJ, Morgan DC, Broun DM, Lee SF and Waldram JR (1997) Sapphire resonator for the measurement of surface impedance of high-temperature superconducting thin films. Rev. Sci. Instrum. 68:2121−212 Panagopoulos C, Cooper JR, Xiang T, Peacock GB, Gameson I and Edwards PP (1997) Probing the order parameter and the c-axis coupling of high-Tc cuprates by penetration depth measurements. Phys. Rev. Lett. 79:2320−2323 Petersan PJ and Anlage SM (1998) Measurement of resonant frequency and quality factor of microwave resonators: comparison of methods. J. Appl. Phys. 84:3392−3402

Handbook of Superconductivity

Porch A, Lancaster MJ and Humphreys RG (1995) The coplanar resonator technique for determining the surface impedance of YBa 2Cu3O7−δ thin films. IEEE Trans. Microwave Theor. Tech. 43:306−314 Pozar DM (2012) Microwave Engineering. Hoboken, NJ: Wiley. Rauch W, Gornik E, Sölkner G, Valenzuela A A, Fox F and Behner H (1992) Microwave properties of YBa 2Cu3O7−x thin films studied with coplanar transmission line resonators. J. Appl. Phys. 73:1866−1872 Samoilova TB (1995) Nonlinear microwave effects in thin superconducting films. Supercond. Sci. Technol. 8:259−278 Sheen DM, Ali SM, Oates DE, Withers RS and Kong JA (1991) Current distribution, resistance, and inductance for superconducting strip transmission lines. IEEE Trans. Appl. Supercond. 1:108−115 Shen YZ, Wilker C, Pang P, Face DW, Carter CF and Harrington CM (1997) Power handling capability improvement of high-temperature superconducting microwave circuits. IEEE Trans. Appl. Supercond. 7:2446−2453 Shen ZY, Wilker C, Pang P, Holstein WL, Face DW and Kountz DJ (1992) High Tc superconductor-sapphire resonator with extremely high Q-values up to 90 K. IEEE Trans. Microwave Theor. Tech. 40:2424−2432 Sridhar S and Kennedy WL (1988) Novel technique to measure the microwave response of high-Tc superconductors between 4.2 K and 200 K. Rev. Sci. Instr. 59:531−536 Sucher M and Fox J (1964) Handbook of Microwave Measurements. New York: Wiley. Wingfield JJ, Powell JR, Gough CE and Porch A (1997) Sensitive measurement of the surface impedance of superconducting single crystals using a sapphire dielectric resonator. IEEE Trans. Appl. Supercond. 7:2009−2012 Wosik J, Xie LM, Miller JH, Long SA and Nesteruk K (1997) Thermally-induced nonlinearities in the surface impedance of superconducting YBCO think films. IEEE Trans. Appl. Supercond. 7:1470−1473 Zhang K et al. (1994) Measurement of the ab plane anisotropy of microwave surface impedance of untwinned YBa 2Cu3O6.95 single crystals. Phys. Rev. Lett. 73:2484−2487

Further Reading Gallop J (1997) Microwave applications of high-temperature superconductors. Supercond. Sci. Technol. 10 A120. (A thorough introductory review, also covering the basic science at microwave frequencies.) Hein MA (1996) Microwave properties of high-temperature superconductors: surface impedance,

Microwave Impedance

circuits and systems in Studies of High Temperature Superconductors Vol. 18 Ed. A Narlikar. New York: Nova Science. (A masterly review covering all aspects of microwave superconductivity.) Lancaster MJ (1997) Passive Microwave Device Applications of High-Temperature Superconductors. Cambridge: Cambridge University Press. (A good overview of the basic science, measurements and applications of high temperature superconductors at microwave frequencies, with emphasis on the work performed at the University of Birmingham, U.K.)

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Portis AM (1993) Lecture Notes in Physics - Vol. 48: Electrodynamics of High-Temperature Superconductors. Singapore: World Scientific. (Describes in detail the various measurement techniques and interpretation of the data. Comprehensive discussions of nonlinear effects at high field amplitudes.) Shen ZY (1994) High-Temperature Superconducting Microwave Circuits