SPIN FLUCTUATION THEORY FOR UNCONVENTIONAL SUPERCONDUCTIVITY IN ...

124 Pages·2014·3.39 MB·English

SPIN FLUCTUATION THEORY FOR UNCONVENTIONAL SUPERCONDUCTIVITY IN
ANTIFERROMAGNETIC METALS
By
WENYA W. ROWE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2014
⃝c 2014 Wenya W. Rowe
2
ACKNOWLEDGMENTS
I thank my advisor Professor Peter J. Hirschfeld for his continued support and
encouragement which have help me to grow. His patience, humor, enthusiasm, and
broad knowledge have been invaluable for my doctoral study.
I am also grateful for the time and support from members of my committee, Profes-
sors K. Muttalib, P. Kumar, G. Stewart, and S. Phillpot. I would like to thank Professors.
D. Maslov, K. Muttalib, and K. Ingersent for their lucid and rich lectures, their patience
and extra guidance.
My work has mostly been done in collaboration with researchers in other institutes.
I express my special appreciation to Professor Ilya Eremin at the Ruhr University in
Bochum for his guidance over the years. Thanks to Dr. J. Knolle who helped me with the
calculations of the spin susceptibility and the mean field energy. Thanks to Professor
B. M. Andersen for his help with the potential calculations. And thanks to A. Rømer for
her meticulous comparison of the results. I would like to thank the Ruhr University in
Bochum for their hospitality during my short visits and during the last year of my doctoral
study in Germany.
I thank Drs. G. Boyd, A. Kemper, V. Mishra and M. Korshunov, former members of
the Hirschfeld group. They provided encouragement and informative advices during the
beginning of my research years. I thank Dr. A. Kreisel, Y. Wang and P. Choubey for their
helpful discussions about all matters.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1 Spin fluctuation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.1 Kondo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.2 Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Spin fluctuations and superconductivity . . . . . . . . . . . . . . . . . . . 14
1.3 Unconventional superconductivity . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Iron-pnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.3 Heavy fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.4 Organic and fullerene superconductors . . . . . . . . . . . . . . . . 23
2 ANTIFERROMAGNETIC STATE . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1 Ferro- and Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Itinerant electron magnetism . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Mean field phase diagram including AF and superconductivity . . . . . . . 28
3 DYNAMIC SPIN SUSCEPTIBILITY . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Theory and calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Spin excitations in the pure antiferromagnetic state . . . . . . . . . . . . . 39
3.2.1 The dynamic spin susceptibility in the antiferromagnetic state . . . 39
3.2.2 The effect of next-nearest hopping, t′ on the spin excitations . . . . 42
3.2.3 The effect of the dopants on spin excitations . . . . . . . . . . . . . 44
3.3 Spin excitations in the coexistence state . . . . . . . . . . . . . . . . . . . 48
4 THE PAIRING INTERACTION ARISING FROM ANTIFERROMAGNETIC SPIN
FLUCTUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 The pairing interaction in the antiferromagnetic background . . . . . . . . 55
4.2 The pairing symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Angular dependence of the coherence factors . . . . . . . . . . . . 63
4
4.2.2 Angular dependence of the pairing potentials . . . . . . . . . . . . 67
4.2.2.1 Charge and longitudinal interaction . . . . . . . . . . . . 67
4.2.2.2 Transverse interaction . . . . . . . . . . . . . . . . . . . . 69
4.2.2.3 Interband interactions . . . . . . . . . . . . . . . . . . . . 70
4.2.3 LAHA expansion of gap equation . . . . . . . . . . . . . . . . . . . 71
4.2.4 Comparison with numerical evaluation . . . . . . . . . . . . . . . . 74
5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
APPENDIX
A MEAN FIELD QUANTITIES IN THE PURE ANTIFERROMAGNETIC STATE . 81
A.1 Antiferromagnetic order parameter equation: derivation . . . . . . . . . . 81
A.2 The electron filling: derivation . . . . . . . . . . . . . . . . . . . . . . . . . 82
B DERIVATIONS IN THE COEXISTENCE STATE OF ANTIFERROMAGNETISM
AND SUPERCONDUCTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.1 Antiferromagnetic order parameter equation in the coexistence state with
superconductivity: derivation . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.2 Filling level of electrons in the coexistence state: derivation . . . . . . . . 83
B.3 Mean field energy in the coexistence state: derivation . . . . . . . . . . . 85
C DERIVATTIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE PURE AN-
TIFERROMAGNETIC STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
C.1 Transverse dynamic spin susceptibility in the antiferromagnetic state:
derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
C.2 Umklapp term for the transverse dynamic spin susceptibility . . . . . . . . 92
C.3 The longitudinal dynamic spin susceptibility . . . . . . . . . . . . . . . . . 94
C.4 The longitudinal Umklapp susceptibility . . . . . . . . . . . . . . . . . . . 96
C.5 Analytic proof for the formation of the Goldstone mode . . . . . . . . . . 98
D DERIVATIONS OF DYNAMIC SPIN SUSCEPTIBILITY IN THE COEXISTENCE
STATE OF ANTIFERROMAGNETIC AND SUPERCONDUCTIVITY . . . . . . 99
D.1 Derivations of transverse dynamic spin susceptibility in the coexistence
state of antiferromagnetism and superconductivity . . . . . . . . . . . . . 99
D.2 The Umklapp term for the transverse dynamic spin susceptibility . . . . . 106
D.3 The longitudinal dynamic spin susceptibility . . . . . . . . . . . . . . . . . 110
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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LIST OF TABLES
Table page
4-1 Coherence factors, p2(k,k′) and n2(k,k′) expanded around hole pockets for
k = ((cid:25), (cid:25)) and k′ = ((cid:6)(cid:25), (cid:25)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2 2 2 2
4-2 Coherence factors, p2(k,k′) and n2(k,k′) expanded around hole pockets for
k = ((cid:0)(cid:25), (cid:25)) and k′ = ((cid:6)(cid:25), (cid:25)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2 2 2 2
4-3 Coherence factors, p2(k,k′) and n2(k,k′) expanded around electron pockets . 66
4-4 Coherence factors, p2(k,k′) and n2(k,k′) expanded around electron and hole
pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4-5 Potentials from the charge- and longitudinal spin-fluctuation contribution ex-
panded around hole pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4-6 Potentials from the transverse spin-fluctuation contribution, (cid:0)2(cid:0) expanded
s
around hole pockets in the limit of kh ! 0 . . . . . . . . . . . . . . . . . . . . . 70
F
4-7 Potentials from the charge- and longitudinal spin-fluctuation interband contri-
bution expanded between electron and hole pockets . . . . . . . . . . . . . . . 71
4-8 Angular dependence of the s-wave and d -wave symmetries on the hole
x2(cid:0)y2
pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4-9 Angular dependence of the s-wave and d -wave symmetries on the elec-
x2(cid:0)y2
tron pockets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6
LIST OF FIGURES
Figure page
1-1 The Feynman diagram in the Berk-Schrieffer approximation to the effective
electron-electron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1-2 The relative signs of superconducting gap on a cuprate-like Fermi surface . . . 16
1-3 Phase diagrams of hole-doped and electron-doped cuprates . . . . . . . . . . 18
1-4 Crystal structures of the electron-doped R Sr CuO and the hole-doped
2(cid:0)x x 4
La Sr CuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2(cid:0)x x 4
1-5 Crystal and spin structures of the electron-doped R Sr CuO and the hole-
2(cid:0)x x 4
doped La Sr CuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2(cid:0)x x 4
2-1 The Fermi surface and the band structure of electron-doped cuprates . . . . . 35
2-2 The doping-temperature phase diagram of electron-doped cuprates . . . . . . 35
3-1 Neutron scattering on Pr LaCe CuO (PLCCO) . . . . . . . . . . . . . . . . . 37
1(cid:0)x x
3-2 The band structures and imaginary part of transverse dynamic spin suscepti-
bility at half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3-3 Three possible types of Fermi surface topology in the antiferromagnetic state
in layered cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3-4 Calculated imaginary part of transverse (cid:31)+(cid:0) (q,q,Ω) . . . . . . . . . . . . . . . 46
RPA
3-5 Calculated Imaginary part of the transverse (cid:31)+(cid:0) (q,q,Ω) spin excitation spectra 50
RPA
3-6 Calculated imaginary part of the longitudinal susceptibility, (cid:31)zz (q,q,Ω) spin
RPA
excitation spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4-1 General structure of the Fermi surface of layered cuprates . . . . . . . . . . . . 64
4-2 Comparison of the analytical calculations up to (ke)2 for the longitudinal and
F
transverse pairing potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SPIN FLUCTUATION THEORY FOR UNCONVENTIONAL SUPERCONDUCTIVITY IN
ANTIFERROMAGNETIC METALS
By
Wenya W. Rowe
May 2014
Chair: Peter J. Hirschfeld
Major: Physics
The understanding of unconventional superconductivity is still a challenge for con-
densed matter physicists. To understand the interplay between antiferromagnetic order
and superconductivity is crucial for the development of unconventional superconductivity
theory, not only because the antiferromagnetic state coexists with superconductivity in
many materials such as cuprates, iron-based pnictides, heavy fermions and organic
superconductors, but also because spin fluctuations near a magnetically ordered phase
have been proposed to possibly mediate superconductivity. In Chapter 1, we introduce
the mechanism of spin fluctuations and review important categories of unconventional
superconductors. In Chapter 2 we review elements of the relevant theory of magnetism.
We discuss the differences between localized and itinerant approaches to study mag-
netism. We focus on cuprates which have a simple one-band Fermi surface. We use the
Hubbard model to describe the band structure of the cuprates, and introduce the mean
field phase diagram of electron-doped cuprates. In Chapter 3, we study the dynamic
susceptibility of cuprates in the pure antiferromagnetic state and in the coexistence state
of antiferromagnetism and superconductivity. We identify the key features of particle-
hole spin excitations which are affected by the next-nearest neighbor hopping, t′. We
compare the different spin wave features between electron- and hole-doped cuprates.
We conclude that the long range commensurate antiferromagnetic state is unstable
on the hole doped side within the self-consistent-mean field theory due to the negative
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spin-stiffness. In the coexistence state, we see the spin resonance peak caused by
superconductivity as well as the Goldstone mode in the spin excitation spectrum. Last
we present the instability analysis of superconductivity from spin fluctuations in the
antiferromagnetic state. We derive the superconducting pairing potentials and the gap
equations for the spin singlet and triplet pairings. We separate the singlet potentials
into longitudinal and transverse channels and expand the them around the pocket’s
center in the small pocket limit. Our result shows on the electron-doped side the leading
symmetry is d (cid:0)wave and on the hole-doped side it is p(cid:0)wave. This implies that
x2(cid:0)y2
from the paramagnetic state to the antiferromagnetic state, the superconducting gap
has a smooth transition on the electron-doped side whereas the gap has to change
symmetry on the hole-doped side. We conjecture that this may account for the lack of
bulk coexistence of antiferromagnetic and superconducting order on the hole-doped side
of the cuprates phase diagram.
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CHAPTER 1
INTRODUCTION
1.1 Spin fluctuation models
For localized spin systems, magnetic properties can be fairly well described by
a variety of theoretical approaches[1]. For weakly ferromagnetic systems, which are
itinerant, predictions are not accurate, since statistical fluctuations of the charge and
screening of the spin moment have to be included for the itinerant electron systems.
Both thermal and quantum fluctuations can be important. Thermal fluctuations vanish
when temperature is zero, and then increase with temperature, but quantum fluctuations
are present even when the temperature is zero. Here we consider quantum spin
fluctuation theory which has been developed with Green’s functions. Spin fluctuation
theory is really a collection of methods based on a small set of models[2], the main
ones being the Kondo, Anderson and Hubbard models, each of them having several
variations.
The theoretical treatment of magnetism in metals began with Stoner theory in the
1930s. At that time, people had doubts about the possibility of describing spin waves
in itinerant systems. The RPA was later developed by Doniach and Engelsberg[3],
and applied it to Pd metal which is nearly ferromagnetic. Anderson and Brinkman [4]
used the same theory to understand the stability of the 3He A-phase liquid. Berk and
Schrieffer [5] made the important observation that spin fluctuations would suppress
s-wave superconductivity.
I now briefly review the models which have been discussed in the context of spin
fluctuations.
1.1.1 Kondo Model
The Kondo model has been used to describe isolated impurities in metals and
quantum dot systems.The magnetism comes from the partially filled d(cid:0) or f(cid:0) shell,
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