Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
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Superconductivity
K.H. Bennemann, J.B. Ketterson: Superconductivity
Volume 1: Conventional and Unconventional Superconductors
Volume 2: Novel Superconductors
K.H. Bennemann
J.B. Ketterson
Editors
Superconduc tivity
Conventional and
Unconventional Superconductors
Volume 1
123
Professor Dr. K.H. Bennemann
FU Berlin
Department of Physics
Arnimallee 14
14195 Berlin, Germany
e-mail: khb@physik.fu-berlin.de
Professor Dr. John B. Ketterson
Northwestern University
Department of Physics & Astronomy
Sheridan Road 2145
60208-3112 Evanston, USA
e-mail: j-ketterson@nwu.edu
ISBN 978-3-540-73252-5
DOI 10.1007/978-3-540-73253-2
e-ISBN 978-3-540-73253-2
Library of Congress Control Number: 2008926416
© 2008 Springer-Verlag Berlin Heidelberg
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Preface
Superconductivity is a classical problem of physics.
Since the discovery by Kamerlingh–Onnes in 1911
it has undergone tremendous advances in recent
years. Complimentary theoretical and experimen-
tal studies have advanced our understanding of sys-
tems with strong electronic correlations and the as-
sociated superconducting phase transitions. New ex-
perimental techniques and field-theoretical meth-
ods have emerged. The symmetry of the supercon-
ducting order-parameter in certain material classes
seems different from the usual singlet, s-wave form
found in conventional superconductors. There is
strong evidence of a Cooper-pairing mechanism dif-
fering from the conventional one arising from the
electron–phonon interaction. In view of the simulta-
neous occurrence of an enhanced magnetic response
in many novel superconductors (cuprates, ruthen-
ates and heavy-fermion metals), pairing due to ex-
change of spin-excitations has been proposed and
studied.Withthis mechanism,triplet Cooper-pairing
can also occur, inaddition to singlet Cooper-pairing.
This may be the case in Sr
2
RuO
4
and heavy-fermion
metals like UPt
3
.
Most notable of course was the discovery of high
temperature superconductivity in the cuprates. This
discovery resulted in an enormous activity. Combin-
ing this with the discovery of superconductivity in
other materials such as heavy-Fermion metals,MgB
2
,
various organics, fullerenes, magnetic metals under
pressure, and presumably non s-symmetry triplet
Cooper-pairing in ruthenates (Sr
2
RuO
4
)wehavean
enrichment of the phenomenon superconductivity.
Many of the new superconductorsare being called
unconventional, either with respect to the pairing
mechanism or the symmetry of the underlying order
parameter. A central problem is the definite iden-
tification of the pairing mechanism. Yet in spite of
the new superconductors continuing studies of con-
ventional electron–phonon based superconductivity
remain interesting.
With the aid of historical hindsight,we expect the
field to continueto advance in unexpected directions.
Nonetheless it is our view that a state-of-the-art trea-
tise on superconductivityis justified at this time,not
only to summarize the present understanding, but
also to introduce newcomers to the field, as was the
case with the two justly-famous 1968 “bibles of su-
perconductivity” by our former colleague R.D. Parks
for an earlier generation. It is hoped that the present
book will help to point the way for future work, re-
search and discoveries.
The treatise starts with a historical introduction
and with the basic phenomenological and micro-
scopic theory of conventional superconductors. This
should be particularly useful for students and be-
ginning researchers. This and the following chap-
ters reflect the somewhat refined present state of
the understanding of phonon-mediated supercon-
ductivity. However, many of the results, in partic-
ular the Cooper-pairing theory itself, have much
wider validity. The phenomenological models and
many-body theories are also useful for the study
of other pairing mechanisms. As an example see
the treatise on electron–phonon superconductivity,
studies using Ginzburg–Landau theory and fluc-
tuation phenomena. Note that appropriately gen-
eralized Ginzburg–Landau theories are useful for
analyzing multi-component and competing order
parameters in unconventional superconductors and
for non-equilibrium behavior in general.
VI Preface
Several chapters discuss recent developments of
superconductors containing impurities. Of particu-
lar interest are nanostructured superconductors.The
important interplayof magnetism and superconduc-
tivity is also discussed. Later chapters (see Volume
II) focus on novel superconductors, in particular on
cuprate superconductivity with its almost univer-
sally accepted d-wave symmetry order parameter. A
wide variety of important experimental results on
high superconductivity are described. In particular,
experiments determining the symmetry of the su-
perconducting order-parameter are discussed. The
theory analyses the pairing mechanism, important
properties and in particular vortices in high-T
c
su-
perconductors. Note that the study of vortex matter
is also of interest for applications. Several chapters
present a theory for cuprate superconductivity as-
suming Cooper-pairing due to the exchange of spin-
excitations. A critical comparison with experiments
is presented. Moreover, superconductivity in vari-
ous novel superconductors like ruthenates, heavy-
Fermions and organics is discussed. Many of the
properties involve an interesting interplay of mag-
netic and superconducting excitations and interac-
tions. The use of neutrons is very important for the
analysis of the elementary excitations and of mag-
netic activity, in particular regarding novel super-
conductors. Still important open problems are dis-
cussed. This should stimulate in particular current
and future research.
The theoretical studies illustrate our present un-
derstanding of novel superconductors and should be
helpful for further theoretical development towards
a general theory that plays the role of the BCS the-
ory in the past. The connections with superfluid
3
He,
the best understood unconventional Fermion super-
fluid, and with Bose–Einstein condensation are also
discussed.
This treatise on superconductivity is a follow up
of the previous books on the physics of supercon-
ductors published a few years ago. Several important
new chapters are presented and most of the other
chapters have been largely rewritten, add new re-
sults and are generally updated. Thus it is hoped that
this book, like Park’s famous books on conventional
superconductors, describe our present more or less
definite knowledge on superconductivity one of the
most interesting problem in physics.
It is hoped that the book will remain useful for a
long time, as was the case with the Parks treatise. A
parallel goal is that the book will be helpful for both
students and researchers.
We wishto thank the authorsfor their careful work
andgreatpatience.Special thanks go to Christof Ben-
nemann for his essential help in preparing the book.
Finally we thank Dr.Ascheron from Springer for his
patience and help during editing.
Berlin, February 2008 K. H. Bennemann and J. B. Ketterson
K.H.Bennemann is Professor of Physics at the Physics Department FUB (Berlin,Germany), Sloan Fellow,and interna-
tionally known for his research on magnetism,superconductivity, and non-equilibrium phenomena.
J.B. Ketterson is Professor of Physics at Northwestern University (Evanston, Ill. USA), internationally known for his
research in the area of Solid State Physics, in particular of Low Temperature Physics, on Helium. Both are Editors of
the well-known books The Physics of Solid and Liquid Helium.
Contents
Volume I. Superconductivity: Conventional and Unconventional Superconductors
ListofContributors .................................................................................... XV
1 History of Superconductivity:Conventional, High-Transition Temperature
and Novel Superconductors
K. H. Bennemann and J. B. Ketterson ............................................................ 3
1.1 Introduction .................................................................................... 3
1.2 NovelSuperconductors.......................................................................... 9
1.3 GranularSuperconductors,MesoscopicSystems,JosephsonJunctions ........................... 18
1.4 Outlook ......................................................................................... 23
References ............................................................................................. 25
2 Phenomenology and Microscopic Theory: Theoretical Foundations
L. Pitaevskii .................................................................................... 27
2.1 Off-Diagonal Long-Range Order and Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Off-Diagonal Long-Range Order in Superconductors . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 TheLondonEquation ........................................................................... 33
2.4 ThermodynamicsofSuperconductorsinaMagneticField....................................... 35
2.5 TheIntermediateStateofSuperconductors...................................................... 37
2.6 TheGinzburg–LandauTheory .................................................................. 38
2.7 SurfaceEnergyattheBoundaryBetweenNormalandSuperconductingPhases ................. 42
2.8 SuperconductorsoftheSecondKind ............................................................ 45
2.9 QuantizedVortexLines.......................................................................... 46
2.10 Vortex–VortexInteractions...................................................................... 49
2.11 CooperPairing.................................................................................. 52
2.12 EnergySpectrumofaSuperconductor .......................................................... 54
2.13 ThermodynamicPropertiesofSuperconductors ................................................ 57
2.14 ElementsoftheTheoryofGreen’sFunctions .................................................... 62
2.15 Green’sFunctionofaSuperconductor........................................................... 65
2.16 TemperatureGreen’sFunctions.................................................................. 68
2.17 TemperatureGreen’sFunctionsforaSuperconductor............................................ 70
References ............................................................................................. 71
VIII Contents
3 Electron–Phonon Superconductivity
F. Marsiglio and J. P. Carbotte ................................................................... 73
3.1 Introduction .................................................................................... 74
3.2 TheElectron–PhononInteraction:Overview .................................................... 74
3.3 ThePhonons .................................................................................... 92
3.4 TheCriticalTemperatureandtheEnergyGap................................................... 108
3.5 ThermodynamicsandCriticalMagneticFields.................................................. 118
3.6 ResponseFunctions ............................................................................. 122
3.7 Anisotropy and MgB
2
........................................................................... 139
3.8 Summary........................................................................................ 145
Appendix............................................................................................... 148
References ............................................................................................. 153
4 Coexistence of Singlet Superconductivity and Magnetic Order
in Bulk Magnetic Superconductorsand SF Heterostructures
M. L. Kuli´c and A. I. Buzdin ..................................................................... 163
4.1 Introduction .................................................................................... 164
4.2 FerromagneticSuperconductors................................................................. 169
4.3 AntiferromagneticSuperconductors(AFS) ...................................................... 180
4.4 MagneticSuperconductorsintheMagneticField................................................ 183
4.5 JosephsonEffectwithBulkMagneticSuperconductors .......................................... 188
4.6 Superconductor/FerromagnetHeterostructures ................................................. 189
4.7 Conclusion ...................................................................................... 196
Appendix............................................................................................... 197
References ............................................................................................. 199
5 Theory of SuperconductingAlloys
L. P. Gor’ko v ..................................................................................... 201
5.1 Introduction .................................................................................... 201
5.2 AveragesofGreenFunctionsOverImpurities ................................................... 202
5.3 SuperconductingAlloyswithaSmallGap ....................................................... 208
5.4 ParamagneticAlloysandGaplessSuperconductivity ............................................ 216
5.5 EilenbergerEquations........................................................................... 218
5.6 FinalRemark.................................................................................... 223
References ............................................................................................. 224
6 Impurity Nanostructures and Quantum Interference in Superconductors
D. K. Morr ...................................................................................... 225
6.1 Introduction .................................................................................... 225
6.2 Review: Single Impurity Effects in Conventional and Unconventional Superconductors . . . . . . . . . 228
6.3 Formalism ...................................................................................... 231
6.4 Quantum Interference and Quantum Imaging in s-waveSuperconductors....................... 234
6.5 Quantum Interference Phenomena in d
x
2
−y
2
-waveSuperconductors ............................. 244
6.6 Molecules on the Surface of d
x
2
−y
2
-waveSuperconductors ....................................... 250
6.7 Conclusions ..................................................................................... 253
References ............................................................................................. 256
Contents IX
7 Nanostructured Superconductors
G. Deutscher .................................................................................... 259
7.1 Introduction .................................................................................... 259
7.2 Nano-structuredCompounds ................................................................... 260
7.3 TheGranularStructure.......................................................................... 260
7.4 NormalStateTransport ......................................................................... 262
7.5 SingleGrainBehavior ........................................................................... 265
7.6 WeaklyCoupledGrains:GranularInsulatorsandSuper-Insulators .............................. 269
7.7 WellCoupledGrains............................................................................. 272
7.8 CriticalTemperatureofGranularSuperconductors.............................................. 275
References ............................................................................................. 277
8 Proximity–Coupled Systems: Quasiclassical Theory of Superconductivity
V. Chandrasekhar ............................................................................... 279
8.1 Introduction .................................................................................... 279
8.2 TransportEquationsintheDiffusiveApproximation ............................................ 280
8.3 TheKeldyshGreen’sFunctions.................................................................. 283
8.4 TheQuasiclassicalApproximation............................................................... 288
8.5 Non-equilibriumGreen’sFunctionsforSuperconductingSystems ............................... 291
8.6 QuasiclassicalSuperconductingGreen’sFunctions .............................................. 294
8.7 TheDirtyLimit:TheUsadelEquation ........................................................... 297
8.8 ParametrizationoftheQuasiclassicalGreen’sFunction.......................................... 302
8.9 ApplicationsoftheQuasiclassicalEquationstoProximity-CoupledSystems..................... 304
8.10 Summary........................................................................................ 311
References ............................................................................................. 312
9 Principles of Josephson-Junction-Based Quantum Computation
S. E. Shafranjuk and J. B. Ketterson .............................................................. 315
9.1 Introduction .................................................................................... 316
9.2 Josephson-Junction-BasedQubitDevices........................................................ 319
9.3 QubitDynamics................................................................................. 329
9.4 QuantumOscillationsinTwoCoupledChargeQubits ........................................... 345
9.5 SISISTwo-QubitGatewithIntrinsicCoupling................................................... 350
9.6 Conclusions ..................................................................................... 361
Appendix............................................................................................... 362
References ............................................................................................. 365
10 Fluctuation Phenomena in Superconductors
A. I. Larkin and A. A. Varlamov .................................................................. 369
10.1 Introduction .................................................................................... 370
10.2 Ginzburg–LandauFormalism:Thermodynamics................................................ 373
10.3 FluctuationsBelowtheCriticalTemperature .................................................... 390
10.4 Ginzburg–LandauTheoryofFluctuationsinTransportPhenomena............................. 396
10.5 FluctuationsNeartheS–ITransition ............................................................ 405
10.6 MicroscopicDerivationoftheTime-DependentGinzburg–LandauEquation.................... 410
10.7 MicroscopicTheoryofFluctuationConductivityofLayeredSuperconductors................... 416
10.8 ManifestationofFluctuationsinVariousProperties ............................................. 429