Theory of Unconventional Superconductors : Cooper-Pairing Mediated by Spin Excitations

233 Pages·2004·4.45 MB·English

by  Dirk Manske (auth.), Dirk Manske (eds.)

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Dirk Manske
Theory
of Unconventional
Superconductors
Cooper-Pairing Mediated by Spin Excitations
With84Figures
1 3
DirkManske
Max-Planck-InstitutfürFestko¨rperforschung
Heisenbergstr.1
70569Stuttgart,Germany
E-mail:[email protected]
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PhysicsandAstronomyClassificationScheme(PACS):
74.20.Mn,74.25.-q,74.70.Pq
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To Claudia, Philipp, and Isabell
Preface
Superconductivity remains one of the most interesting research areas in
physics and complementary theoretical and experimental studies have ad-
vancedourunderstandingofit.Inunconventionalsuperconductors,thesym-
metry of the superconducting order parameter is different from the usual s-
wave form found in BCS-like superconductors. For the investigationof these
new material systems, well-known experimental tools have been improved
and new experimental techniques have been developed.
This book is written for advanced students and researchers in the field
of unconventional superconductivity. It contains results I obtained over the
last years with various coworkers. The state of the art of research on high-
Tc cuprates and on Sr2RuO4 obtained from a generalized Eliashberg theory
is presented. Using the Hubbard Hamiltonian and a self-consistent treat-
ment of spin excitations and quasiparticles, we study the interplay between
magnetism and superconductivity in various unconventional superconduc-
tors. The obtained results are then contrasted to those of other approaches.
In particular, a theory of Cooper pairing due to exchange of spin fluctua-
tions is formulatedfor the caseof singlet pairingin hole- and electron-doped
cuprate superconductors, and for the case of triplet pairing in Sr RuO . We
2 4
calculatebothmanynormalandsuperconductingpropertiesofthese materi-
als,their elementaryexcitations,andtheir phase diagrams,whichreflect the
interplay between magnetism and superconductivity.
In the case of high-Tc superconductors, we emphasize the similarities of
the phase diagrams of hole- and electron-doped cuprates and give general
arguments for a dx2−y2-wave superconducting order parameter. A compar-
ison with the results of angle-resolved photoemission and inelastic neutron
scattering experiments, and also Raman scattering data, is given. We find
that key experimental results can be explained.
For triplet Cooper pairingin Sr RuO , we focus on the importantrole of
2 4
spin–orbit coupling in the normal state and compare the theoretical results
withnuclearmagneticresonancedata.Forthesuperconductingstate,results
and general arguments related to the symmetry of the order parameter are
provided.Itturnsoutthatthemagneticanisotropyofthenormalstateplays
an important role in superconductivity.
Stuttgart, May 2004 Dirk Manske
D.Manske:TheoryofUnconventionalSuperconductors,STMP202,VII–XI(2004)
(cid:1)c Springer-VerlagBerlinHeidelberg2004
Contents
1 Introduction.............................................. 1
1.1 Layered Materials and Their Electronic Structure........... 3
1.1.1 La2−xSrxCuO4................................... 4
1.1.2 YBa2Cu3O6+x ................................... 5
1.1.3 Nd2−xCexCuO4 .................................. 6
1.2 General Phase Diagram of Cuprates and Main Questions .... 7
1.2.1 Normal–State Properties .......................... 8
1.2.2 Superconducting State: Symmetry
of the Order Parameter ........................... 12
1.3 Triplet Pairing in Strontium Ruthenate (Sr RuO ):
2 4
Main Facts and Main Questions .......................... 15
1.4 From the Crystal Structure to Electronic Properties ........ 19
1.4.1 Comparison of Cuprates and Sr RuO : Three–Band
2 4
Approach ....................................... 19
1.4.2 Effective Theory for Cuprates: One–Band Approach .. 22
1.4.3 Spin Fluctuation Mechanism for Superconductivity ... 23
References ................................................. 28
2 Theory of Cooper Pairing Due to Exchange
of Spin Fluctuations ...................................... 33
2.1 Generalized Eliashberg Equations for Cuprates
and Strontium Ruthenate ............................... 33
2.2 Theory for Underdoped Cuprates......................... 46
2.2.1 Extensions for the Inclusion of a d-Wave Pseudogap .. 48
2.2.2 Fluctuation Effects ............................... 52
2.3 Derivation of Important Formulae and Quantities........... 60
2.3.1 Elementary Excitations ........................... 60
2.3.2 Superfluid Density and Transition Temperature
for Underdoped Cuprates.......................... 62
2.3.3 Raman Scattering Intensity
Including Vertex Corrections....................... 65
2.3.4 Optical Conductivity ............................. 71
2.4 Comparison with Similar Approaches for Cuprates.......... 73
2.4.1 The Spin Bag Mechanism ......................... 74
X Contents
2.4.2 The Theory of a Nearly Antiferromagnetic Fermi
Liquid (NAFL) .................................. 76
2.4.3 The Spin–Fermion Model.......................... 77
2.4.4 BCS–Like Model Calculations...................... 80
2.5 Other Scenarios for Cuprates: Doping a Mott Insulator...... 84
2.5.1 Local vs. Nonlocal Correlations .................... 84
2.5.2 The Large-U Limit ............................... 86
2.5.3 Projected Trial Wave Functions and the RVB Picture . 88
2.5.4 Current Research and Discussion ................... 90
References ................................................. 92
3 Results for High–Tc Cuprates Obtained from a
Generalized Eliashberg Theory: Doping Dependence...... 99
3.1 The Phase Diagram for High–Tc Superconductors .......... 99
3.1.1 Hole–Doped Cuprates............................. 99
3.1.2 Electron–Doped Cuprates ......................... 109
3.2 Elementary Excitations in the Normal
and Superconducting State: Magnetic Coherence,
Resonance Peak, and the Kink Feature.................... 115
3.2.1 Interplay Between Spins and Charges:
a Consistent Picture of Inelastic Neutron Scattering
Together with Tunneling and Optical–Conductivity
Data............................................ 115
3.2.2 The Spectral Density Observed by ARPES:
Explanation of the Kink Feature ................... 125
3.3 Electronic Raman Scattering in Hole–Doped Cuprates ...... 137
3.3.1 Raman Response and its Relation to the Anisotropy
and Temperature Dependence of the Scattering Rate.. 138
3.4 Collective Modes in Hole–Doped Cuprates ................. 144
3.4.1 A Reinvestigation of Inelastic Neutron Scattering..... 145
3.4.2 Explanation of the “Dip–Hump” Feature in ARPES .. 148
3.4.3 Collective Modes in Electronic Raman Scattering?.... 149
3.5 Consequences of a dx2−y2–Wave Pseudogap
in Hole–Doped Cuprates ................................ 151
3.5.1 Elementary Excitations and the Phase Diagram ...... 152
3.5.2 Optical Conductivity and Electronic Raman Response 158
3.5.3 Brief Summary of the Consequences of the Pseudogap 167
References ................................................. 169
4 Results for Sr2RuO4 ...................................... 177
4.1 Elementary Spin Excitations in the Normal State of Sr RuO 179
2 4
4.1.1 Importance of Spin–Orbit Coupling................. 179
4.1.2 The Role of Hybridization......................... 182
4.1.3 Comparison with Experiment ...................... 185
4.2 Symmetry Analysis of the Superconducting Order Parameter 187
Contents XI
4.2.1 Triplet Pairing Arising from Spin Excitations ........ 188
4.3 Summary, Comparison with Cuprates, and Outlook......... 192
References ................................................. 197
5 Summary, Conclusions, and Critical remarks ............. 201
References ................................................. 208
A Solution Method for the Generalized Eliashberg
Equations for Cuprates ................................... 211
References ................................................. 214
B Derivation of the Self-Energy (Weak-Coupling Case) ..... 215
C dx2−y2-Wave Superconductivity Due to Phonons?......... 225
Index......................................................... 227
1 Introduction
One of the most exciting and fascinating fields in condensed matter physics
is high-temperature and unconventional superconductivity, for example in
hole- and electron-doped cuprates, in Sr RuO , in organic superconductors,
2 4
in MgB , andin C compounds.In cuprates,the highesttransitiontemper-
2 60
ature (without application of pressure) T (cid:1) 134 K has been measured in
c
HgBa Ca Cu O , followed by – to name just a few – Bi Sr CaCu O
2 2 3 8+δ 2 2 2 8+δ
(δ = 0.15 ↔ T (cid:1) 95 K), YBa Cu O (x = 0.93 ↔ T (cid:1) 93 K),
c 2 3 6+x c
Nd2−xCexCuO4 (x = 0.15 ↔ Tc (cid:1) 24 K), and La2−xSrxCuO4, where, for
an optimum doping concentration x = 0.15, a maximum value of T (cid:1) 39
c
K occurs. Since 77 K is the boiling temperature of nitrogen, it is now pos-
sible that new technologies,basedfor example on SQUIDs (superconducting
quantum interference devices) or Josephson integrated circuits [1], might be
developed.However,atpresent,thecriticalcurrentdensitiesarestillnothigh
enoughformosttechnologyapplications.Arecentoverviewanaccountofthe
possible prospects can be found in [2] and references therein.
Throughout this book, we shall focus mainly on Cooper pairing in
cupratesandinSr RuO .Allmembersofthecupratefamilydiscoveredsofar
2 4
contain one or more CuO planes and various metallic elements. As we shall
2
discuss in the next section, their structure resembles that of the perovskites
[3]. It is now fairly well established that the important physics related to
superconductivity occurs in the CuO planes and that the other layers sim-
2
ply act as charge reservoirs. Thus, the coupling in the c direction provides
a three–dimensionalsuperconducting state, but the main pairing interaction
acts between carrierswithin a CuO plane. The undoped parent compounds
2
are antiferromagnetic insulators, but if one dopes the copper–oxygen plane
withcarriers(electronsorholes),thelong-rangeorderisdestroyed.Notethat
evenwithoutstrictlong-rangeorder,the spin correlationlengthcanbe large
enoughtoproducealocalarrangementofmagneticmomentsthatdiffersonly
little from that observedbelow the N´eeltemperature in the insulating state.
In the dopedstate the cuprates become metallic or,below T ,superconduct-
c
ing.
As mentioned above, in hole–doped cuprates T is of the order of 100 K
c
and in electron–doped cuprates one finds T (cid:1) 25K (as will be explained
c
later), and thus much larger values of T are obtained than in conventional
c
D.Manske:TheoryofUnconventionalSuperconductors,STMP202,1–32(2004)
(cid:1)c Springer-VerlagBerlinHeidelberg2004