physics and applications

Superconductivity Physics and Applications Kristian Fossheim and Asle Sudbø The Norwegian University of Science and Technology Trondheim, Norway Copyright c  2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Fossheim, K. (Kristian) Superconductivity : physics and applications / Kristian Fossheim and Asle Sudbo. p. cm. Includes bibliographical references and index. ISBN 0-470-84452-3 (alk. paper) 1. Superconductivity. I. Sudbo, Asle. II. Title. QC611.92.F67 2004 537.6  23–dc22 2004002271 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-84452-3 Typeset in 10.5/13pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Biddles Ltd, King’s Lynn This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. Contents Preface xi Acknowledgements xiii I BASIC TOPICS 1 1 What is superconductivity? A brief overview 3 1.1 Some introductory, historical remarks 3 1.2 Resistivity 6 1.3 The Meissner effect: perfect diamagnetism 10 1.4 Type I and type II superconductors 13 1.5 Vortex lines and flux lines 17 1.6 Thermodynamics of the superconducting state 18 1.7 Demagnetization factors and screening 23 2 Superconducting materials 27 2.1 Introductory remarks 27 2.2 Low-T c superconductors 27 2.2.1 Superconducting elements 27 2.2.2 Binary alloys and stoichiometric compounds 29 2.3 Organic superconductors 31 2.3.1 Polymer and stacked molecular type 31 2.3.2 Fullerene superconductors 34 2.4 Chevrel phase materials 35 2.5 Oxide superconductors before the cuprates 35 2.6 High-T c cuprate superconductors 37 2.6.1 The discovery of cuprate superconductors 37 2.6.2 Composition and structure 40 2.6.3 Making high T c materials 41 2.6.4 Phase diagrams and doping 42 2.6.5 Some remarks on the original idea which led to the discovery of cuprate superconductors 47 2.6.6 Thermal fluctuations of the superconducting conden- sate. A preliminary discussion 48 vi CONTENTS 2.7 Heavy fermion superconductors 52 2.8 MgB 2 superconductor 53 2.9 Summarizing remarks 55 3 Fermi-liquids and attractive interactions 57 3.1 Introduction 57 3.2 The non-interacting electron gas 59 3.3 Interacting electrons, quasiparticles and Fermi-liquids 61 3.4 Instability due to attractive interactions 66 3.4.1 Two electrons with attractive interaction 66 3.4.2 Phonon-mediated attractive interactions 71 3.4.3 Reduction of the effective Hamiltonian 75 4 The superconducting state – an electronic condensate 79 4.1 BCS theory: a magnetic analogue 79 4.2 Derivation of the BCS gap equation 81 4.3 Transition temperature T c and the energy gap  87 4.4 Generalized gap equation, s-wave and d-wave gaps 89 4.5 Quasi-particle tunnelling and the gap 94 4.5.1 Introductory remarks 94 4.5.2 The tunnelling principle 95 4.5.3 Single-particle NIN tunnelling 97 4.5.4 NIS quasiparticle tunnelling 99 4.5.5 SIS quasiparticle tunnelling 102 4.6 BCS coherence factors versus quasiparticle-effects: ultrasound and NMR 103 4.6.1 Introductory remarks 103 4.6.2 Transition rates in ultrasound propagation and NMR 104 4.6.3 Longitudinal ultrasonic attenuation 105 4.6.4 Transverse ultrasound 108 4.6.5 Nuclear magnetic resonance relaxation below T c 113 4.7 The Ginzburg–Landau theory 115 4.7.1 Some remarks on Landau theory 115 4.7.2 Ginzburg–Landau theory for superconductors 117 4.7.3 Flux quantization 121 5 Weak Links and Josephson Effects 123 5.1 Weak links, pair tunnelling, and Josephson effects 123 5.1.1 Introductory remarks 123 5.1.2 DC Josephson effect: the Feynman approach 125 5.2 AC Josephson effect 128 5.2.1 Alternative derivation of the AC Josephson effect 129 5.3 Josephson current in a magnetic field 131 CONTENTS vii 5.4 The SQUID principle 134 5.5 The Ferrell–Prange equation 136 5.6 The critical field H c1 of a Josephson junction 138 5.7 Josephson vortex dynamics 139 5.8 Josephson plasma in cuprate high-T c superconductors 140 6 London Approximation to Ginzburg–Landau Theory (|ψ| constant) 141 6.1 The London equation and the penetration depth λ L 141 6.1.1 Early electrodynamics and the London hypothesis 141 6.1.2 Derivation of the London equation from the free energy 146 6.2 The energy of a single flux line 148 6.2.1 Energy of a flux line: alternative derivation. An exercise 151 6.3 Interacting flux lines: the energy of an arbitrary flux line lattice 154 6.4 Self energy of a single straight flux line in the London approximation 160 6.5 Interaction between two parallel flux lines 162 6.6 Interaction between two flux lines at angle α 164 6.7 General flux-line lattice elastic matrix in the London approximation 166 7 Applications of Ginzburg–Landau Theory (|ψ| spatially varying) 171 7.1 The temperature-dependent order parameter | ψ(T) | 171 7.2 The coherence length ξ 172 7.2.1 Relations between λ, ξ and H c 174 7.3 Two types of superconductors 175 7.4 The structure of the vortex core 180 7.5 The length ξ and the upper critical field B c2 182 7.5.1 Isotropic systems 182 7.5.2 B c2 ,ξ, and λ in anisotropic superconductors 184 7.6 Ginzburg–Landau–Abrikosov (GLA) predictions for B c2 /B c1 188 7.7 Surface superconductivity and B c3 190 8 More on the Flux-line System 199 8.1 Elementary pinning forces and simple models 199 8.1.1 The concept of a pinning force 199 8.1.2 Pinning force and flux gradient 200 8.2 Critical state and the Bean model 201 viii CONTENTS 8.3 Flux-line dynamics, thermal effects, depinning, creep and flow 204 8.3.1 TAFF, flow and creep: Definitions 204 8.3.2 Flux flow 205 8.3.3 Thermally activated flux creep: Anderson model 206 8.4 Single particle TAFF 207 8.5 FLL elasticity and pinning 208 8.5.1 Collective effects 208 8.5.2 Collective creep: inverse power law U(J) 212 8.5.3 Logarithmic U(J) 213 8.5.4 The vortex solid–liquid transition 215 8.5.5 Lindemann criterion and melting of a clean flux-line system 218 8.5.6 Modelling non-linear vortex diffusion 224 8.6 Flux-line entry at B c1 : thermodynamic and geometric restrictions 228 8.6.1 The critical field B c1 228 8.6.2 The Bean–Livingston barrier 229 8.6.3 Geometric barriers 232 8.7 Critical current issues 233 8.7.1 Critical current in the Meissner state 233 8.7.2 Depairing critical current 234 8.7.3 Reduction of J c at grain boundaries 235 8.7.4 Relaxation of magnetic moment and the irreversibility line 236 8.7.5 How can J c be increased? 242 II ADVANCED TOPICS 247 9 Two-dimensional superconductivity. Vortex-pair unbinding 249 9.1 Introduction 249 9.2 Ginzburg–Landau description 250 9.3 Critical fluctuations in two-dimensional superfluids 251 9.4 Vortex–antivortex pairs 255 9.5 Mapping to the 2D Coulomb gas 257 9.6 Vortex-pair unbinding and Kosterlitz–Thouless transition 264 9.7 Jump in superfluid density 271 10 Dual description of the superconducting phase transition 279 10.1 Introduction 279 10.2 Lattice formulation of the Ginzburg–Landau theory 282 10.2.1 Lattice Ginzburg–Landau model in a frozen gauge approximation 284 CONTENTS ix 10.3 Preliminary results 286 10.4 Vortex-loops as topological defects of the order parameter 289 10.5 Superconductor–superfluid duality in d = 3 295 10.6 Zero-field vortex-loop blowout 298 10.6.1 Definitions 299 10.7 Fractal dimension of a vortex-loop tangle 306 10.8 Type I versus type II, briefly revisited 309 III SELECTED APPLICATIONS 315 11 Small scale applications 317 11.1 More JJ-junction and SQUID basics 318 11.1.1 Introductory remarks 318 11.1.2 RSJ – the resistively shunted Josephson junction 318 11.1.3 Further modelling of the Josephson junction 322 11.1.4 The autonomous DC SQUID 323 11.1.5 Simplified model of the DC SQUID 325 11.2 SQUID applications 329 11.2.1 Biomagnetism: neuromagnetic applications 329 11.3 Superconducting electrodynamics in the two-fluid model 333 11.3.1 Frequency dependent conductivity in the two fluid model 333 11.3.2 Surface impedance and AC loss 336 11.3.3 Surface resistance measurement 339 11.4 High-frequency radio technology 341 11.4.1 Microstrip filters and delay lines 341 11.4.2 Superconducting high-frequency devices 345 12 Superconducting Wire and Cable Technology 349 12.1 Low-T c wire and cable 349 12.1.1 Introductory remarks 349 12.1.2 General design considerations 350 12.1.3 Basic superconductor properties 350 12.1.4 Design of technical superconductors 353 12.1.5 Stabilization 355 12.1.6 AC losses 358 12.1.7 Mechanical characteristics 359 12.1.8 Fabrication technology 359 12.2 High-T c wire and cable 362 12.2.1 High-T c wire and tape 362 12.2.2 Full-scale high-T c cable 363 12.2.3 HTS induction heater 364 12.3 Magnet technology 366 x CONTENTS IV TOPICAL CONTRIBUTIONS 369 13 Topical Contributions 371 13.1 Spin-Triplet superconductivity, by Y. Maeno 371 13.2 π-SQUIDs – realization and properties, by J. Mannhart 374 13.3 Doppler effect and the thermal Hall conductivity of quasipar- ticles in d-wave superconductors, by N.P. Ong 376 13.4 Nanometer-sized defects responsible for strong flux pinning in NEG123 superconductor at 77 K, by M. Muralidhar and M. Murakami 378 13.5 Hybrid Magnets, by H. Schneider-Muntau 380 13.6 Magneto-Optical Imaging of Vortex Matter, by T.H. Johansen 385 13.7 Vortices seen by scanning tunneling spectroscopy, by Oystein Fischer 388 13.8 Resistivity in Vortex State in High-T c Superconductors, by K. Kadowaki 392 13.9 Coated conductors: a developing application of high tempera- ture superconductivity, by James R. Thompson, and David K. Christen 395 References Chapter 13 397 V HISTORICAL NOTES 399 14 Historical notes on superconductivity: the Nobel laureates 401 Heike Kamerlingh Onnes 401 John Bardeen 402 Leon N. Cooper 403 J. Robert Schrieffer 404 Ivar Giaever 405 Brian D. Josephson 407 J. George Bednorz 408 K. Alex M ¨ uller 410 Alexei A. Abrikosov 411 Vitaly L. Ginzburg 413 Pierre-Gilles de Gennes 414 Philip W. Anderson 415 References 417 Author index 423 Subject index 425 Preface Writing this textbook was motivated by the opinion of the authors that the time has come for an updated look at the basics of superconductivity in the aftermath of progress during the last couple of decades, both through the discov- eries of new superconductors, and the ensuing theoretical development. High-T c superconductor research since 1986 represents an almost unlimited source of information about superconductivity. This is an advantage in the sense that there is ample material with which to fill new books, but a disadvantage in the sense that only a very small fraction of all the efforts that were made, and the results that came out, can be discussed here. In this sense the situation is entirely new: The older texts, like those of de Gennes and Tinkham could discuss or refer to almost all aspects of superconductivity of importance in the 1960s and 1970s. With tens of thousands of papers published after 1986, there is no possibility to take such an approach any more. We apologize to the numerous researchers in the field whose work we could not mention. This situation leaves it even more to the taste of the authors to choose. First and foremost we have wanted to review the basics of superconductivity to new students in the field. Secondly, we wanted to allow those who take a serious interest in the subject at the PhD level, to follow the ideas to old heights like in the BCS theory, or to new heights like in the theory of the vortex system in high-T c cuprates. Superconductivity is now a far richer subject thanks to the discovery of high-T c cuprates by Bed- norz and M ¨ uller. Suddenly, superconductivity became an arena for the study of critical behaviour in three-dimensional superconductors, an unthinkable sit- uation in the low-T c era. Our book seeks, among other things, to clarify this new aspect of superconductivity. In addition, we wanted to respect the wish of students to learn where physics meets the real life of applications. We have concentrated the material here to the central topics, basically how to describe and exploit the properties of Josephson junctions on the small scale, and on the large scale to give some insight into the makings of wires and cables. A special feature of this book is the inclusion of a chapter containing Topical Contribu- tions from distinguished scientists in various areas of superconductivity research and development, from the smallest to the largest scale. Each of these scientists were invited to contribute their leading edge knowledge to give a clear idea of xii PREFACE the state of the art in several important sub-fields as of September 2003 when the writing of this book came to a conclusion. Kristian Fossheim National High Magnetic Field Laboratory, Tallahassee, Florida and The Norwegian University of Science and Technology Trondheim, Norway Asle Sudbø The Norwegian University of Science and Technology Trondheim, Norway [...]... 2000/2001, and upon my return to Trondheim, Norway, at the Physics Department of the Norwegian University of Science and Technology (NTNU) I greatfully acknowledge the warm hospitality of the NHMFL, and in particular Jack Crow, Director, and Hans Schneider-Muntau, Deputy Director of the NHMFL I also wish to express my thanks to my wife Elsa for her patience and her willingness to type, from day to day, my handwritten... their special thanks to scientific colleagues in the US, Europe and Japan for their topical contributions in Chapter 13: D K Christen, Ø Fischer, T.H Johansen, K Kadowaki, Y Maeno, J Mannhart, M Murakami, M Muralidhar, N P Ong, H Schneider-Muntau and J R Thompson PART I Basic Topics Superconductivity: Physics and Applications Kristian Fossheim and Asle Sudbø c 2004 John Wiley & Sons, Ltd ISBN 0-470-84452-3... for important discovery And many more were to come in other condensed matter systems below ambient temperatures Superconductivity: Physics and Applications Kristian Fossheim and Asle Sudbø c 2004 John Wiley & Sons, Ltd ISBN 0-470-84452-3 4 WHAT IS SUPERCONDUCTIVITY? Even the choice by Onnes of the metal to study, mercury, while appearing perhaps surprising today, was a judicial and natural one at that... type I for Sn and Pb and similar conductors which had been studied in Leiden Only upon discovering, understanding, and further developing type II materials could critical current density be raised to much higher, practical values From the 1960s on a gradual development of superconductor technology has taken place, until today when superconducting magnets are commonplace in laboratories and hospitals... considerable attention through the 1920s and 1930s A breakthrough came in 1933 when Meissner and Ochsenfeld [3] showed that in magnetic fields below a certain threshold value the flux inside the superconductor was expelled, and that this defined a new thermodynamic state and was not a consequence of infinite conductivity The phenomenon became known as the Meissner effect, and laid the foundation for a thermodynamic... phase diagram for type I and type II superconductors in the H, T − plane 1.5 Vortex lines and flux lines We want to stress a very important conceptual difference between what we call vortex lines and flux lines Often, these terms are used interchangeably This is permissible in most conventional type II superconductors, where the ratio between the upper and lower critical fields Hc2 and Hc1 , is not very... meet these great scientists and fine individuals in their own settings, and hear their own story told In the Historical Notes, Chapter 14 a few and brief references to these encounters are mentioned The rest of the material will have to be presented elsewhere I also want to thank Ulrik Thisted, my PhD student who undertook the task of weeding out the mistakes in my drafts, and Torbjørn Hergum for various... when superconducting magnets are commonplace in laboratories and hospitals all over the industrialized world And the remarkable SQUID technology developed for measurement of small magnetic fields, has taken on a wide range of applications, and is showing promise in ever new fields by use of both old and new superconductors 1 For an account of early work on superconductivity, see Dahl [2] SOME INTRODUCTORY,... accidental would be unjustified The necessary technical basis and opportunity for the discovery had been solidly established in the same group by the liquefaction of the inert gas helium in 1908 And research, both experimental and theoretical, on the electrical conductivity of metals at temperatures approaching the absolute zero, was ongoing and considered an important issue by leading physicists They... important magnetic properties of type II superconductors two decades earlier, and had shown promise for superconductors to carry larger current densities than materials previously studied by Onnes, and by Meissner and others 1957 was also the year when the beautiful quantum theory, known as the BCS theory, was published by Bardeen, Cooper and Schrieffer [7], finally explaining the fascinating properties of superconductors . Cataloging-in-Publication Data Fossheim, K. (Kristian) Superconductivity : physics and applications / Kristian Fossheim and Asle Sudbo. p. cm. Includes bibliographical references and index. ISBN 0-470-84452-3. Schneider-Muntau and J. R. Thompson. PART I Basic Topics Superconductivity: Physics and Applications Kristian Fossheim and Asle Sudbø c  2004 John Wiley & Sons, Ltd ISBN 0-470-84452-3 1 What is Superconductivity? A. Superconductivity Physics and Applications Kristian Fossheim and Asle Sudbø The Norwegian University of Science and Technology Trondheim, Norway Copyright c 