Vortices in Unconventional Superconductors and Superfluids

368 Pages·2002·16.771 MB·English

by  G. E. Volovik (auth.), Professor Dr. R. P. Huebener, Professor Dr. N. Schopohl, Professor Dr. G. E. Volovik (eds.)

Springer Series in
SOLID-STATE SCIENCES 132
Springer-Verlag Berlin Heidelberg GmbH
Springer Series in
SOLID-STATE SCIENCES
Series Editors:
M. Cardona P. Fulde K. von Klitzing R. Merlin H.-J. Queisser H. Stormer
The Springer Series in Solid-State Sciences consists of fundamental scientific
books prepared by leading researchers in the field. They strive to communica
te, in a systematic and comprehensive way, the basic principles as well as new
developments in theoretical and experimental solid-state physics.
126 Physical Properties of Quasicrystals
Editor: Z.M. Stadnik
127 Positron Annihilation in Semiconductors
Defect Studies
By R. Krause-Rehberg and H.S. Leipner
128 Magneto-Optics
Editors: S. Sugano and N. Kojima
129 Computational Materials Science
From Ab Initio to Monte Carlo Methods
By K. Ohno, K. Esfarjani, and Y. Kawazoe
130 Contact, Adhesion and Rupture of Elastic Solids
ByD. Maugis
131 Field Theories for Low-Dimensional Condensed Matter Systems
Spin Systems and Strongly Correlated Electrons
By G. Morandi, P. Sodano, A. Tagliacozzo, and V. Tognetti
132 Vortices in Unconventional Superconductors and Superfluids
Editors: R.P. Huebener, N. Schopohl, and G.E. Volovik
Series homepage - http://www.springer.de/phys/books/sss/
Volumes 1-125 are listed at the end of the book.
R.P. Huebener, N. Schopohl,
G.E. Volovik {Eds.}
Vortices
in Unconventional
Superconductors
and Superfluids
With 106 Figures
·i·
~
Springer
Professor Dr. R.P. Huebener Professor Dr. G.E. Volovik
Eberhard-Karls-Universitat Tiibingen Landau Institute for Theoretical Physics
Physikalisches Institut Kosygin Str. 2
Auf der Morgenstelle 14 117334 Moscow, Russia
72076 Tiibingen, Germany and
Low Temperature Laboratory
Professor Dr. N. Schopohl Helsinki University of Technology
Eberhard-Karls-Universitat Tiibingen P.O. Box 2200
Lehrstuhl flir Theoretische Festkorperphysik 02015, Espoo, Finland
Auf der Morgenstelle 14
72076 Tiibingen, Germany
Series Editors:
Professor Dr., Dres. h. c. Manuel Cardona
Professor Dr., Dres. h. c. Peter Fulde*
Professor Dr., Dres. h. c. Klaus von Klitzing
Professor Dr., Dres. h. c. Hans-Joachim Queisser
Max-Planck-Institut flir Festkorperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany
* Max-Planck-Institut flir Physik komplexer Systeme, Nothnitzer Strasse 38
01187 Dresden, Germany
Professor Dr. Roberto Merlin
Department of Physics, 5000 East University, University of Michigan
Ann Arbor, MI 48109-1120, USA
Professor Dr. Horst Stormer
Dept. Phys. and Dept. App!. Physics, Columbia University, New York, NY 10023 and
Bell Labs., Lucent Technologies, Murray Hill, NJ 07974, USA
Library of Congress Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Vortices in unconventional superconductors and superfluids/R.P. Huebener ... (ed.). - Berlin; Heidelberg;
New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2002 (Springer series in
solid-state sciences; 132) (Phsyics and astronomy online library)
ISSN 0171-1873
ISBN 978-3-642-07613-8 ISBN 978-3-662-04665-4 (eBook)
DOI 10.1007/978-3-662-04665-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or
parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its
current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable
for prosecution under the German Copyright Law.
Springer-Verlag Berlin Heidelberg New York
a member of BertelsmannSpringer Science+Business Media GmbH
http://www.springer.de
© Springer-Verlag Berlin Heidelberg 2002
Originally published by Springer-Verlag Berlin Heidelberg New York in 2002.
Softcover reprint of the hardcover 1s t edition 2002
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
Typesetting: camera-ready copies by the authors
Cover concept: eStudio Calamar Steinen
Cover production: design & production GmbH, Heidelberg
Printed on acid-free paper SPIN: 10838764 57/3141tr -5 43 2 1 0
Preface
The physics of vortices in classical fluids has been a highly important subject
for many years, both in fundamental science and in engineering applications.
About 50 years ago, vortices started to become prominent as quantum me
chanical objects constructed from a macroscopic wavefunction. Here the key
developments are associated with the names R. Feynman, L. Onsager, L. D.
Landau, F. London, V.L. Ginzburg and A.A. Abrikosov. Recently, the physics
of vortices has undergone a further important step of diversification, namely
in unconventional superconductors and superfluids, which are characterized
by an anisotropic and/or spatially complex order parameter. It is this latest
evolutionary step of vortex physics that is addressed in this book. The indi
vidual chapters are concerned with the microscopic structure and dynamics
of vortices in diverse systems ranging from superfluids and superconductors
to neutron stars.
Each of the 20 chapters is written by one or more experts on the parti
cular subject. Each chapter provides an introduction and overview, empha
sizing theoretical as well as experimental work, and includes references to
both recent and pioneering earlier developments. In this way non-expert rea
ders will also benefit from these lecture notes. Hence, the book will be useful
for all researchers and graduate students interested in the physics of vorti
ces in unconventional superconductors and superfluids. It may also serve as
supplementary material for a graduate course on low-temperature solid-state
physics.
The idea for this book originated from a workshop held in Dresden, Ger
many from February 28 to March 3, 2000, at the Max-Planck-Institut Fur
Physik Komplexer Systeme. The editors express their special thanks to Prof.
Dr. P. Fulde and his staff from this Institute for their support and hospitality
during the workshop.
It is our privilege to thank the participants of the workshop for their
contributions during the discussions. The quality of all the lectures and the
enthusiasm shown by all the participants made the workshop a great success.
We trust that this book will be similarly well received.
Tiibingen, N. Schopohl, R. Huebener
Moscow C.E. Volovik
October 2001
Contents
1 The Beautiful World of the Vortex
G.E. Volovik ................................................... 1
Part I Vortices in Superconductors, Superfiuids,
Neutron Stars, and QFT
2 Type II Superconductors and Vortices
from the 1950s to the 1990s
A.A. Abrikosov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Preamble............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 History............................ . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Imaging........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
2.4 Pinning and Melting of the Vortex Lattice .................... 15
2.5 Other Kinds of Vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
References ..................................................... 19
3 What Can Superconductivity Learn
from Quantized Vorticity in 3He Superfluids?
G.E. Volovik, V.B. Eltsov, and M. Krusius . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.1 Unconventional Quantized Vorticity. . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.2 Special Features of 3He Superfluids . . . . . . . . . . . . . . . . . . . . . . . . . .. 24
3.3 Continuous Vortices, Skyrmions and Merons. . . . . . . . . . . . . . . . . .. 25
3.4 Transformation from Singular
to Continuous Vortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
3.5 Vortex with Composite Core ................................ 28
3.6 Vortex Sheet .............................................. 28
3.6.1 Vortex-Sheet Structure in 3He-A . . . . . . . . . . . . . . . . . . . . . .. 28
3.6.2 Vortex Sheet in Rotating Superfluid . . . . . . . . . . . . . . . . . . .. 28
3.6.3 Vortex Sheet in Superconductor. . . . . . . . . . . . . . . . . . . . . . .. 30
3.7 Fractional Vorticity and Fractional Flux ...................... 30
3.8 Broken Symmetry in the Vortex Core. . . . . . . . . . . . . . . . . . . . . . . .. 32
3.8.1 Vortex Core Transition ............................... 32
3.8.2 Ferromagnetic Core .................................. 34
VIII Contents
3.8.3 Asymmetric Double Core ............................. 34
3.9 Vortex Formation by Intrinsic Mechanisms. . . . . . . . . . . . . . . . . . .. 34
3.9.1 Nucleation Barrier ................................... 35
3.9.2 Vortex Formation in a Hydrodynamic Instability. . . . . . . .. 36
3.9.3 Formation of Continuous Vortex Lines:
Dependence of Critical Velocity on Core Size ............ 38
3.9.4 Formation of Vortex Sheet ............................ 41
3.9.5 Vortex Formation in Ionizing Radiation. . . . . . . . . . . . . . . .. 42
3.10 Vortex Dynamics Without Pinning. . . . . . . . . . . . . . . . . . . . . . . . . .. 43
3.11 Conclusion................................................ 44
References ..................................................... 45
4 Nucleation of Vortices in Superfluid 3He-B
by Rapid Thermal Quench
Igor S. Aranson, Nikolai B. Kopnin, and Valerii M. Vinokur . . . . . . . . .. 49
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
4.2 Model.................................................... 50
4.3 Results of Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
4.4 Dynamics of Vortex/ Antivortex Annihilation. . . . . . . . . . . . . . . . .. 54
4.5 Instability of Normal-Superfluid Interface. . . . . . . . . . . . . . . . . . . .. 55
4.5.1 Stationary Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
4.5.2 Linearized Equations .... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
4.5.3 Long-Wavelength Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57
4.5.4 Large u Limit ....................................... 58
4.5.5 Estimate for the Number of Vortices. . . . . . . . . . . . . . . . . . .. 60
4.6 Generalization............................................. 61
4.7 Conclusion................................................ 62
References ..................................................... 63
5 Superfluidity in Relativistic Neutron Stars
David Langlois ................................................. 65
5.1 Introduction............................................... 65
5.2 Superfluidity and Superconductivity. . . . . . . . . . . . . . . . . . . . . . . . .. 66
5.2.1 Composition of the Interior of a Neutron Star. . . . . . . . . . .. 66
5.2.2 Energy Gaps and Critical Temperature. . . . . . . . . . . . . . . .. 67
5.2.3 Various Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . .. 68
5.3 Cooling Processes in Neutron Stars. . . . . . . . . . . . . . . . . . . . . . . . . .. 68
5.4 Rotational Dynamics of Neutron Stars: Glitches ............... 69
5.4.1 The Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
5.4.2 Role of the Vortices .................................. 70
5.4.3 Origin of the Glitches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
5.5 Relativistic Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
5.5.1 Perfect Fluid in General Relativity. . . . . . . . . . . . . . . . . . . .. 74
5.5.2 Relativistic Superfluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
Contents IX
5.5.3 Superfluid-Superconducting l\'1ixtures . . . . . . . . . . . . . . . . . .. 76
5.6 Relativistic Neutron Stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78
5.6.1 Static Neutron Star .................................. 78
5.6.2 Oscillations of Superflllid Neutron Stars. . . . . . . . . . . . . . . .. 79
References ..................................................... 80
6 Superconducting Superfluids in Neutron Stars
Brandon Carter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
6 .1 Introduction............................................... 83
6.2 Generic Category of 3-Constituent Superconducting Superfluid
lVIodels ................................................... 84
6.3 The Semi-Macroscopic Application. . . . . . . . . . . . . . . . . . . . . . . . . .. 88
6.4 Phenomenological Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
References ..................................................... 96
Part II Vortex Dynamics, Spectral Flow
and Aharonov-Bohm Effect
7 Vortex Dynamics and the Problem of the Transverse Force
N.B. Kopnin ................................................... 99
7.1 Introduction............................................... 99
7.2 Boltzmann Kinetic Equation Approach ....................... 101
7.2.1 Localized Excitations ................................. 101
7.2.2 Delocalized Excitations ............................... 104
7.3 Forces .................................................... 105
7.3.1 Flux-Flow Conductivity ............................... 107
7.4 Transverse Force ........................................... 108
7.4.1 Low-Field Limit and Superfluid 3He .................... 111
7.5 Vortex Momentum ......................................... 113
7.5.1 Equation of Vortex Dynamics .......................... 114
7.5.2 Vortex Mass ......................................... 115
7.6 Conclusions ............................................... 117
References ..................................................... 117
8 Magnus Force and Aharonov-Bohm Effect
in Superfluids
Edouard Sonin ................................................. 119
8.1 Introduction ............................................... 119
8.2 The Magnus Force in Classical Hydrodynamics ................ 121
8.3 The Magnus Force in a Superfluid ............................ 124
8.4 Nonlinear Schrodinger Equation
and Two-Fluid Hydrodynamics .............................. 125
X Contents
8.5 Scattering of Phonons by the Vortex
in Hydrodynamics ......................................... 129
8.6 The Iordanskii Force
and the Aharonov~Bohm Effect .............................. 133
8.7 Partial-Wave Analysis
and the Aharonov~Bohm Effect .............................. 136
8.8 Momentum Balance in Two-Fluid Hydrodynamics ............. 139
8.9 Magnus Force and the Berry Phase ........................... 142
8.10 Discussion and Conclusions ................................. 143
References ..................................................... 144
9 Lorentz Force Exerted by the Aharonov-Bohm Flux Line
Andrei Shelankov and A.F. Ioffe .................................. 147
9.1 The Magnetic Scattering .................................... 150
9.1.1 Paraxial Solution ..................................... 150
9.1.2 Deflection of the Beam ................................ 152
9.1.3 Exact Solution ....................................... 154
9.1.4 Scattering Amplitude ................................. 156
9.2 The Momentum Balance .................................... 157
9.2.1 The Force ........................................... 159
9.2.2 The AB~Line ........................................ 160
9.3 Conclusions ............................................... 161
9.4 Appendix: The Momentum~Flow Tensor
for the Schrodinger Equation ............................... 163
9.5 Appendix: The Force: Arbitrary Wave ........................ 164
References ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10 Relativistic Solution of Lordanskii Problem
in Multi-Constituent Superfluid Mechanics
B. Carter, D. Langlois, and R. Prix ............................... 167
10.1 Introduction ............................................... 167
10.2 Perfect Multiconstituent Fluid Dynamics ...................... 168
10.3 Specification of Lift Force on Vortex .......................... 169
10.4 Generalised Joukowski Theorem ............................. 170
10.5 Application to the Landau Model ............................ 171
References ..................................................... 173
11 Vortex Core Structure and Dynamics
in Layered Superconductors
M. Eschrig, D. Rainer, and J. A. Sauls ............................. 175
11.1 Introduction ............................................... 175
11.2 Nonequilibrium Transport Equations ......................... 177
11.2.1 Constitutive Equations ............................... 179
11.2.2 Linear Response ..................................... 181
Contents xr
11.3 Electronic Structure of Vortices .............................. 184
11.3.1 Singly Quantized Vortices for S-Wave Pairing ............ 185
11.3.2 Singly Quantized Vortices for D-Wave Pairing ........... 188
11.3.3 Vortices Pinned to Mesoscopic Metallic Inclusions ........ 191
11.3.4 Doubly Quantilled Vortices ............................ 193
11.4 Nonequilibrium Response ................................... 195
11.4.1 Dynamical Charge Response ........................... 197
11.4.2 Local Dynamical Conductivity ......................... 199
11.4.3 Induced Current Density .............................. 201
11.4.4 Summary ........................................... 202
References ..................................................... 202
Part III Fermion Zero Modes on Vortices
12 Band Theory of Quasiparticle Excitations
in the Mixed State of d-Wave Superconductors
Alexander S. Mel'nikov .......................................... 207
12.1 Introduction ............................................... 207
12.2 Basic Equations ........................................... 210
12.2.1 BdG Equations for a Spin Singlet Unconventional Super-
conductor in a Magnetic Field ......................... 210
12.2.2 Quasiparticles Confined Near Gap Nodes ................ 212
12.3 A Single Isolated Vortex Line:
Aharonov-Bohm Effect for Quasiparticles ..................... 213
12.4 Quasiparticle States in Vortex Lattices ....................... 214
12.4.1 Cyclotron Orbits in the Mixed State .................... 214
12.4.2 Quasiparticle Band Spectrum in Vortex Lattices ......... 215
12.4.3 Modified Semiclassical Approach for QP States
in a Vortex Lattice ................................... 219
12.5 Conclusions ............................................... 222
References ..................................................... 222
13 Magnetic Field Dependence
of the Vortex Structure Based on the Microscopic Theory
Masanori Ichioka, Mitsuaki Takigawa and Kazushige Machida ........ 225
13.1 Introduction ............................................... 225
13.2 Magnetic Field Dependence
of the Vortex Structure ..................................... 227
13.2.1 Quasiclassical Eilenberger Theory ...................... 227
13.2.2 Local Density of States ............................... 228
13.2.3 Pair Potential and Internal Field Distribution ............ 232
13.3 Site-Selective Nuclear Spin Relaxation Time ................... 233