THERMODYNAMIC AND TRANSPORT PROPERTIES OFUNCONVENTIONAL SUPERCONDUCTORS AND MULTIFERROICS

By

G.R.BOYD

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

c⃝ 2010 G.R.Boyd

2

The mystery seemed to him almost crystalline now; he was mortified to have dedicateda hundred days to it. -J.L.Borges, Death and the Compass, in Ficciones

3

ACKNOWLEDGMENTS

P.J.Hirschfeld had the patience to advise me for nearly all of my time in Gainesville,

and I owe much to him. Thank you for the years of your time. I would like to thank

D.Maslov and K.Ingersent for their lucid and informative lectures. Lex Kemper and Chris

Pankow have fielded many of my questions, and I appreciate their input over the years.

The work done in conjunction with S.Graser was particularly productive, something for

which I thank him. Professors Dorsey, Stewart, Matchev, and Phillpot are also members

of my committee. P.Kumar and I have had many conversations about multiferroics and

might write a paper together one day also. I am grateful to have M.Fischer as a friend

since our time in Cargese. I would like to thank U.Paris-Sud at Orsay for their hospitality

during our visit. Je vous remercie. In the time before Florida, I feel particularly indebted

to Gerry Guralnik, Herb Fried, Brad Marston, and Mike Kosterlitz even so many years

later. B.Ovrut and V.Balasubramanian have also helped get me here in their own ways,

and this journey begun with the dedicated efforts of Art Timmons, Marla Weiss, and Ed

Berger. To all the rest of my former and current colleagues, thank you.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION TO SUPERCONDUCTIVITY . . . . . . . . . . . . . . . . . . 13

1.1 Conventional Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Experiments in Conventional Superconductors . . . . . . . . . . . . . . . 19

1.3.1 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.3 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . 231.3.4 Optical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.5 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4 Unconventional Superconductivity . . . . . . . . . . . . . . . . . . . . . . 281.4.1 Unconventional Pairing . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.2 Migdal-Eliashberg Theory . . . . . . . . . . . . . . . . . . . . . . . 32

2 SUPERCONDUCTIVITY IN CUPRATES AND PNICTIDES . . . . . . . . . . . 35

2.1 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.1 Basic Cuprate Physics . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1.4 Angle Resolved Photoemission Spectroscopy . . . . . . . . . . . . 442.1.5 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . 492.1.6 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.7 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.1.8 Electric and Thermal Conductivity . . . . . . . . . . . . . . . . . . . 532.1.9 Nernst, Kerr, µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.1.10 Remarks on High Tc Experiments . . . . . . . . . . . . . . . . . . . 59

2.2 Pnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.2.1 Optical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2.2 ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.2.3 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.2.4 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2.5 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2.6 Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5

2.3 Theoretical Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.3.1 Spin Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.3.2 Resonating Valence Bonds and Slave-Bosons . . . . . . . . . . . . 752.3.3 Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.3.4 Competing Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.3.5 BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 ANGLE-DEPENDENT THERMODYNAMICS . . . . . . . . . . . . . . . . . . . 83

3.1 Density of States and Low Temperature Specific Heat in the Vortex State 833.2 Temperature Variation of the Specific Heat in an Applied Field . . . . . . . 89

4 RAMAN SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2 Adding Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3 Model Superconducting Gaps for the Raman Response . . . . . . . . . . 1044.4 Modeling Experimental Raman Data . . . . . . . . . . . . . . . . . . . . . 108

5 MULTIFERROICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Ferroelectric Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3 Magnetic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4 Multiferroic Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.5.1 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.5.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.5.3 Adiabatic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6 Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.6.1 Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.6.2 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.6.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.7 Inhomogeneous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.8 Specfic Heat with Gaussian Fluctuations . . . . . . . . . . . . . . . . . . . 1355.9 Future Work on Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6

LIST OF TABLES

Table page

1-1 D4h Selected Singlet Representations . . . . . . . . . . . . . . . . . . . . . . . 29

1-2 D4h Selected Triplet Representations . . . . . . . . . . . . . . . . . . . . . . . . 30

7

LIST OF FIGURES

Figure page

1-1 Tin Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1-2 Lead Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1-3 Aluminium NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1-4 Conductivity of Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1-5 Thermal Conductivity in Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1-6 Strong Coupling tunnelling in Pb . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2-1 Cuprate Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2-2 Cartoon of Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2-3 BSCCO DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-4 STM BSCCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-5 Local DOS Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2-6 specific heat of YBCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2-7 ARPES FS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-8 ARPES phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-9 Fermi Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2-10 ARPES superconducting spectral peak . . . . . . . . . . . . . . . . . . . . . . 48

2-11 Pseudogap’s angular dependence . . . . . . . . . . . . . . . . . . . . . . . . . 48

2-12 NMR YBCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2-13 NMR cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2-14 Neutron Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2-15 YBCO penetration depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2-16 Doping dependence of cuprate gap . . . . . . . . . . . . . . . . . . . . . . . . 55

2-17 Linear T resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2-18 Nernst measurement in LSCO . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2-19 Kerr Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8

2-20 Polar Neutron Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2-21 122 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2-22 122 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2-23 SDW drude weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2-24 Ba-122 Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2-25 Nd-1111 Gap from ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2-26 1/T1 in 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2-27 Inelastic Neutron scattering in Pnictides . . . . . . . . . . . . . . . . . . . . . . 70

2-28 Pnictide Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2-29 c-axis Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2-30 Spin fluctuation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2-31 RVB and QCP phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3-1 Angle-Dependent DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3-2 Specific Heat vs T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3-3 Inversion of Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3-4 High field Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3-5 11-pnictide specific measurements . . . . . . . . . . . . . . . . . . . . . . . . . 93

4-1 T Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4-2 A1g Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4-3 S± with impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4-4 D wave Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4-5 Experimental Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4-6 Co-122 Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4-7 Experimental Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4-8 Intraband Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4-9 Interband Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4-10 Raman Intesity Co-122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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5-1 Cr2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5-2 Order Parameters vs T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5-3 Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5-4 Magnetoelectric susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5-5 Specific Heat for Coupled Order Parameters . . . . . . . . . . . . . . . . . . . 131

5-6 Fluctuation Contribution to Cv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5-7 One Loop Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

THERMODYNAMIC AND TRANSPORT PROPERTIES OFUNCONVENTIONAL SUPERCONDUCTORS AND MULTIFERROICS

By

G.R.Boyd

December 2010

Chair: P.J.HirschfeldMajor: Physics

Often, the phase diagram for a given material can be quite complex, presenting

evidence for multiple orders and it is the task of the condensed matter community to

describe and quantify knowledge of these properties. Significant insight can often

be gained by comparing model calculations of basic thermodynamic and transport

properties of a material with experiment. Here we consider two classes of novel

materials whose rich phase diagrams are actively under investigation: unconventional

superconductors and multiferroics. In 2006, H.Hosono discovered a new class of

iron-based superconducting materials which are not conventional superconductors.

After the initial discovery, there is a range of questions of immediate interest; foremost

among them is what is the structure and symmetry of the superconducting state, a

question which took roughly a decade to answer for the cuprate superconductors. We

present calculations that help reveal the structure of the superconducting gap using

angle dependent specific heat measurements. We then calculate the electronic Raman

scattering intensity for several polarizations of light and different models of disorder,

providing information about the anisotropy and location of nodes in the superconducting

gap. Understanding the influence of disorder is considered crucial because currently

conflicting experimental results may be due to differences in sample quality.

Recently, there has also been interest in multiferroics: materials with simultaneous

non-zero polarization and magnetic order. We present calculations of fundamental

11

thermodynamic properties, mean field behavior for the simplest ferromagnetic-ferroelectric,

characterize topological defects, and use the perturbative renormalization group to help

understand the critical point, as a beginning towards understanding the multitude

multiferroic materials with increasingly complex magnetic and polar order.

The first two chapters review conventional superconductivity and its unconventional

counterpart found in the cuprates and pnictides. The original work constituting the

body of this dissertation appears in chapters three through five. Chapter five contains a

brief introduction to the topics which are relevant for multiferroics before presenting the

original work. Portions of this thesis are based on the author’s publications and are cited

when relevant.

12

CHAPTER 1INTRODUCTION TO SUPERCONDUCTIVITY

1.1 Conventional Superconductivity

Since the primary focus of this work is on unconventional superconductivity, it

is instructive to review aspects of what we mean by conventional superconductivity

and define what is meant by a superconductor. Superconductivity refers to the

observation that in many materials, below a certain critical temperature Tc the

electrical resistance drops to zero, but this pragmatic point of view can miss subtle

physics. Perfect diamagnetism is also a signature of superconductivity. Defining the

onset of superconductivity using the resistive transition alone would not identify a

superconductor where superconducting islands form with normal non-zero resistivity

regions between islands. Diamagnetism is not sufficient either, because type-I and

type-II superconductors differ in how they respond to a magnetic field. In a type-II

material, above a minimum field Hc1, the flux can pierce the material at isolated points

although the majority of it is still superconducting.

The microscopic picture of a superconductor is that in the superconducting state the

Fermi sea becomes unstable and electrons form Cooper pairs. True identification of a

superconductor would demonstrate the existence of phase coherent paired electrons,

and would distinguish it from the case where these pairs are not phase coherent

across the sample. Cooper pairing is only one of several ways the electron sea can

become unstable and form a different ground state. Other ground states include

charge density waves or various magnetic states. In conventional superconductors,

it was discovered, notably by Frolich [1], that the effective electron-electron interaction

due to the exchange of phonons was attractive. This was an impetus for Cooper to

propose the idea that electrons form two-particle bound states [2] in this effective

potential. Below a certain energy scale (roughly the Debye frequency) the Fermi liquid is

destroyed by forming bound pairs of electrons. It is worth noting that phonon-mediated

13

superconductivity is not the only possibility, and the pairing interaction in many

unconventional superconductors is still an open question. It is even a question of

whether the formation of pairs needs to be mediated by the exchange of bosons at all.

The bound pair of electrons will have some binding energy, 2. To excite a pair into

free electrons again, breaking it apart, one needs to add more than a minimum energy

which defines the gap. The gap does not have to be a simple constant. There are

measurable consequences of this change in the spectrum. In tunneling experiments in a

pure system we should see no excitations below this scale in the density of states, and

in specific heat measurements we would no longer see the electron + phonon normal

state specific heat of the form C = γT + bT 3 but rather activated behavior of the form

C ∝ e(−/(kBT )) at low temperatures. In this thesis, we will investigate ways to measure

the quasiparticle spectrum and use it to help determine the structure of the gap.

1.2 Theoretical Description

Bardeen, Cooper, and Schrieffer provided a complete description of the superconducting

state, herein referred to as BCS theory, for the first time in 1957 [3]. We will only give an

overview of BCS theory, but fortunately there are a number of excellent sources [4]. We

use the variational wave function,

|ψ >=∏k

(uk + vkc†k↑c

†−k↓)|0 > (1–1)

to minimize the expectation value of a model Hamiltonian, with respect to the variational

parameters uk and vk

H =∑kσ

ξkσc†kσckσ +

∑kl

Vklc†k↑c

†−k↓c−l↓cl↑. (1–2)

Normalization requires

|uk |2 + |vk |2 = 1, (1–3)

14

so are free to choose u and v to be sine and cosine. Evaluating the model Hamiltonian’s

expectation value is straightforward but lengthy algebra, leading to:

< H >=∑k

ξk2 cos2 θk +

∑kl

Vkl sin θk cos θk sin θl cos θl . (1–4)

A little trigonometry later, we define the gap, k , such that

tan2θk =∑l

Vkl

sin 2θl2ξk

≡ k

ξk. (1–5)

The pair amplitude gkσ1σ2 ≡< ck,σ1c−k,σ2 > is a nonvanishing anomalous expectation

value representing the pairs of electrons. This is related to the pairing potential at

zero temperature for a spin singlet pair, by (k) = −∑

k Vkpgp. is referred to as the

gap or the order parameter. The energy gain by condensing into the superconducting

state is N(EF )2

2, where N is the density of states at the Fermi level. To break all

pairs into normal quasiparticles again, this is the energy one needs to supply. For

example, to suppress superconductivity with an applied magnetic field, Hp, one needs

µ20µ2BNFH

2p = NF

2

2. This limit is called the Clogston-Chandrasekhar limit; it is not the

only way to destroy a superconductor with a magnetic field, but it is one scenario.

The variational problem has the solution:

u2k =1

2(1 +

ξkEk

), v 2k =1

2(1− ξk

Ek). (1–6)

The single particle excitation energies are εk =√ξ2k + 2

k . The number , the gap,

represents half the binding energy of the pair. BCS theory is a mean field theory,

where the mean field is ∼< ψψ >. The fluctuations of the order parameter should

be examined to ensure they’re small enough for a mean field treatment. The small

coherence length, ξ0 = ~vFπ

, of cuprate superconductors make fluctuations relevant to

some of their observed behavior. To see how the mean field theory is constructed, we

15

take the potential to be local,

H ′ = −V0

2

∑ab

∫d3r†

a(r)†b(r)b(r)a(r). (1–7)

This Hamiltonian is treated by considering all pair-wise expectation values according to

the standard Hartree-Fock approach:∫d3r

∑a

†a(r)U(r)a(r) +

∫d3r [(r)†

↑(r)†↓(r) + ∗(r)↓(r)↑(r)]. (1–8)

So the full Hamiltonian is:

H =

∫d3r†

a(r)H0a(r)+

∫d3r

∑a

†a(r)U(r)a(r)+

∫d3r [(r)†

↑(r)†↓(r)+

∗(r)↓(r)↑(r)],

(1–9)

where H0 is given by

− ~2

2m(∇− ie

~cA)2 − µ. (1–10)

Two of the methods for performing calculations will be shown in brief but the

reader should refer to standard textbooks for more detail. The first of these is to use

standard field theory techniques such as Green’s functions, path integrals, etc., or

another technique more suited for inhomogeneous problems is the solution of the BdG

equations, named after their creators N. Bogoliubov and P. G. de Gennes.

The quadratic Hamiltonian can also be diagonalized with a Bogoliubov-Valatin

transformation,

†↑(r) =

∑n[γ

†n↑u

∗n(r)− γn↓vn(r)]

†↓(r) =

∑n[γ

†n↓u

∗n(r) + γn↑vn(r)]

(1–11)

H = Eground +∑

Enγ†naγna. (1–12)

In this picture we can see that excitations in the superconducting state are linear

combinations of quasi-particles and quasi-holes with energy En. These excitations

should be expressed as linear combinations of electron-hole pairs. To ensure that

16

γa, γ†b = δab simultaneously with †a(r),b(r

′) = δabδ(r − r ′). We require that

∑n

[u∗n(r)un(r

′) + v ∗n (r

′)vn(r)] = δ(r − r ′), (1–13)

∑n

[u∗n(r)vn(r

′)− u∗n(r

′)vn(r)] = 0. (1–14)

We calculate the equations of motion

[He ,(r ↑)] = −(H0 + U(r))(r ↑)− (r)†(r ↓), (1–15)

[He ,(r ↓)] = −(H0 + U(r))(r ↓) + (r)†(r ↑). (1–16)

After diagonalization, the Hamiltonian satisfies:

[He , γ†na] = ϵnγ

†na, (1–17)

[He , γna] = −ϵnγna. (1–18)

The result is known as the Bogoliubov-de Gennes Equations

[H0 + U(r)]u(r) + (r)v(r) = ϵu(r), (1–19)

− [H∗0 + U(r)]v(r) + ∗(r)u(r) = ϵv(r). (1–20)

These coupled differential equations should be solved self-consistently. Assuming

the quasiparticles obey Fermi-Dirac statistics, we find that the BdG solutions u and v

define our order parameter and potential self consistently by:

(r) = V0

∑n

(1− fn↑ − fn↓)un(r), v∗n (r) (1–21)

U(r) = V0

∑n

[|un(r)|2fn + |vn(r)|2(1− fn)]. (1–22)

The magnetization must also be calculated self-consistently if the magnetic response is

considered.

17

We usually express equations in reciprocal space, ψ(r) = 1

β√V

∑n

∑k e

−iωnτ+ik.rck .

Introducing Nambu notation, ψk =

ck↑

c†−k↓

, and Fourier transforming, H can be

rewritten[5]:

H0 =∑k

ψ†k

(iωn1 + ξk τ3 +k τ1

)ψk , (1–23)

from which we can read off the propagator in Nambu space:

G(k , iωn) =(iωn1 + ξk τ3 + k τ1

)−1=

1

(iωn)− (ξ2k + 2k)

iωn + ξk k

k iωn − ξk

, (1–24)

where,

τ1 =

0 1

1 0

τ2 =

0 −i

i 0

τ3 =

1 0

0 −1

. (1–25)

The density of states, DOS, is easily calculated from the Green’s function

N(ω) = −1

π

∑k

1

2Tr [ImGR(k ,ω)]. (1–26)

In an s-wave superconductor where, k = 0, the density of states is,

N(ω) = N(0)ω√

ω2 − 2θ(ω2 − 2). (1–27)

In contrast, the d-wave order parameter k = 0 cos(2ϕ) has low energy states in the

DOS

N(ω) = N(0)

∫dϕ

2π

ω√ω2 − 2

0 cos2(2ϕ)

= N(0)2ω

π0

K(ω

0

), ω < (1–28)

where K is an elliptic integral of the first kind, and for ω > , N(ω) = N(0) 2πK(0

ω).

The approach we outlined above is valid when the pairing potential N(EF )V0 << 1

is weak enough for perturbation theory to be valid. In some elemental superconductors,

this is valid, but in others like lead, it is not. There is a strong electron-phonon interaction

and the gap is energy dependent, as we will show at the end of this chapter.

18

From the gapped quasiparticle energy spectrum, we can quickly calculate

phenomenological descriptions of many experimentally accessible quantities. This is the

dictionary we want to come back to in the case of unconventional superconductors, so

that experimental work guides the theoretical description of the underlying physics.

1.3 Experiments in Conventional Superconductors

1.3.1 Specific Heat

In this section we will review experiments on ordinary superconductors. Among

these probes, we want to focus on the specific heat at constant volume more than some

of the other probes because we will revisit it later in this thesis. The specific heat at

constant volume can be derived from the entropy of the quasiparticle gas, which is that

of free fermions

S = −2kB∑k

[(fk ln(fk) + (1− fk)ln(1− fk))] (1–29)

through Cv = T ∂S∂T

. Keeping in mind that the gap is a function of temperature, (T ),

and approximating the sum (expand in ξ, low T), we can reproduce the low temperature

specific heat

Cv =2N(EF )

kB

∫dξ

eβ√

ξ2+2

(eβ√

ξ2+2

+ 1)2((ξ2 + 2(T ))

T 2− (T )

T

d(T )

dT). (1–30)

Which in the low temperature limit is:

limT→0Cv ≈ e

kBT . (1–31)

Figure 1-1 shows the activated behavior expected in the thermodynamics of a fully

gapped superconductor, a decaying exponential. Later we contrast this with the power

law temperature dependence found in unconventional superconductors, indicative of low

energy quasiparticles.

19

Figure 1-1. The low temperature limit of the specific heat in Sn compared with BCStheory. Reproduced with permission from [3]. APS c⃝1957.

1.3.2 Tunneling

A tunneling experiment puts two materials close together, with some potential

barrier in between, like an insulator or a vacuum, so that particles have to tunnel

across the barrier. One example is STM, scanning tunneling microscopy, where a

sharp conducting tip is brought within several A of a sample’s surface, and a bias

voltage is applied so that current tunnels between the material and the tip. Tunneling

is a nonequilibrium transport process, but since the current amplitude is so low, the

time between two tunneling events is much longer than typical relaxation rates, so it is

appropriate to use a quasi-equilibrium formalism. We are specifically interested in the

tunneling between a normal metal and a superconductor.

We model the tunnelling Hamiltonian[6], by H = Hn + Hs + Ht , where Hn and Hs are

the normal and superconductor Hamiltonians, and the tunneling part is given by:

Ht =∑µν

(Tµνc†1µc2ν + T ∗

µνc†2νc1µ) (1–32)

where 1 and 2 refer to the materials on either side of the junction and the Greek indices

refer to single particle states, and T is the tunnelling matrix element. The current is just

dQdt

= e < _N > where _N is the change in number with respect to time, so

20

_N = i [H,N] (1–33)

After a bit of algebra we arrive at an expression involving a convolution of the density of

states, in the sample and the tip. We have assumed a featureless density of states in the

tip; otherwise a way to deconvolve the data is essential. In this case, one arrives at the

formula:

I = e

∫dω

2π

∑µν

|Tµν |2Tr [A1(µ,ω)A2(ν,ω + eV )][f (ω + eV )− f (ω)] (1–34)

where A(ν,ω) is the spectral function and ν represents other quantum numbers (usually

momentum which is not a good quantum number in the absence of translational

symmetry, so we might have different labels for internal states, for example, in a

quantum dot). The quantity Tµν depends on the geometry of the system and the

details of the electronic states involved in tunneling. A detailed treatment and explicit

evaluation of these matrix elements shows Tµν is proportional to the derivative of

the wavefunction in the sample and the derivative depends on the orbital state of the

atoms in question[5, 7]. It should always be kept in mind that, even with atomic scale

resolution, we are looking at the tunneling voltage and not actually atoms. The patterns

one gets from experiment often have well localized peaks in the signal, and it’s tempting

to identify them with individual atoms, but the tip might be sampling a small region larger

than one atom.

Using just Fermi’s golden rule, we can get a very similar expression:

I = I (sample to tip)− I (tip to sample), (1–35)

= 2e2π

~

∫|T |2Ns(E)Nt(E + eV )[f (E)(1− f (E + eV ))− f (E + eV )(1− f (E))], (1–36)

= 2e2π

~

∫|T |2Ns(E)Nt(E + eV ), [f (E)− f (E + eV )] (1–37)

21

which we see, apart from constants, is the same as above since the imaginary part of

the Green’s function integrates to the density of states. Usually data is analyzed using

the differential conductance dIdV

.

Figure 1-2. Tunneling data from superconducting lead. Reproduced with permissionfrom [8]. APS c⃝1962. The two most salient features are thesuperconducting gap, below which there is no density of states, and thewiggles in the curve which are not captured in weak-coupling BCS theory[9].

If we assume a structureless tip, and the tunneling matrix elements do not

complicate the analysis then the voltage dependence of the tunneling conductance

essential is a thermally smeared density of states of the sample,

σ ∼ dI

dV∼∫

dω[∂f (ω − eV )

∂VN(r ,ω)] (1–38)

This is probably the most direct measure of the density of states. We reproduce

tunneling measurements from superconducting lead [8] in Figure 1-2 which shows the

density of states.

22

1.3.3 Nuclear Magnetic Resonance

In general, depending on how the experimental probe couples to the Hamiltonian

to measure the system, the occurrence of superconductivity will either cause a peak

just below the critical temperature, Tc , or a drop. This is discussed in the popular text

by Tinkham [4] and referred to as case 1 or case 2 interactions. The difference is

understood as a manifestation of quasi-particle “coherence factors.” The difference in

coherence factors is traced back to whether the interaction depends only on k − k ′ or k

and −k transitions.

The NMR spin-lattice relaxation rate can be reliably calculated using Fermi’s golden

rule, which provides a nice qualitative way to interpret data. In the original paper by

Hebel and Slichter [10], the expression is:

1

T1

∝∑k

∑k ′

|V |2f (Ei)(1− f (Ef ))N(Ef )N(Ei), (1–39)

where V is the matrix element between initial and final states. The thermal factors

f (Ei) are measures of the initial and final occupation. So one can loosely look at the

spin-lattice relaxation rate as a measure of the density of states squared, blurred by a

distribution function. By replacing the quasi-particle energy by the gapped form in an

s-wave superconductor, we would estimate that

1

T1

∝∫

dEE√

E 2 − 2

(E + ~ω)√(E + ~ω)2 − 2

eβE

(eβE + 1)2, (1–40)

which exhibits a peak in the NMR relaxation rate, called the Hebel-Slichter peak. This

peak is clearly observed in superconducting samples. Figure1-3, taken from Masuda,

Yoshika and Redfield [11], is plotted versus inverse temperature, so the peak is a dip,

but otherwise, the exponential tail and Hebel-Slichter peak are clearly in agreement

with theory. The complete expression [12] suitable for detailed calculations for the

spin-lattice relaxation rate, 1T1

, spin echo rate, 1T2

, and Knight shift are obtained from

the dynamic spin susceptibility (where the γs are gyromagnetic ratios, f is the relative

23

Figure 1-3. The spin lattice relaxation rate in superconducting aluminium as a function of1T

, plotted alongside the BCS prediction. Reproduced with permission from[11]. APS c⃝1962.

abundance of the isotope, and A are hyperfine couplings):

χab = (gµB)2

∫dte iωt < Sa(q, t)Sb(−q, 0) >, (1–41)

1

T1

= limω→0

2kBT

γ2~4∑q

|Aq|2χ′′(ω, q)

ω, (1–42)

1

T2

=2fAγ4~6

∑q

|Aq|4[χ′(0, q)]2, (1–43)

Ks =|A0|χ′(0, 0)

γeγN~2. (1–44)

The Knight shift is the shift in resonant frequency due to a local magnetic field

and is prima facie evidence for the spin state of the pair, but interpretation is not

completely transparent. At zero temperature in a pure singlet pair, the Knight shift

goes to zero. Triplet pairs can have a spin polarization, but a singlet has no net moment.

The exceptions to this interpretation need to be borne in mind. Orbital magnetism and

24

spin-orbit coupling can alter these naive conclusions by either masking the signal from

the Cooper pairs or mixing singlet and triplet states.

1.3.4 Optical Conductivity

Detailed expressions for the dynamical conductivity, σ1 − iσ2, have been derived by

Mattis and Bardeen [13] which lend themselves to simple interpretation of the physical

origins of the conductivity.

σ1σn

=1

~ω

(2

∫ ∞

dε(f (ε)− f (ε+ ~ω))(ε2 +2 + ~ω)(ε2 − 2)1/2[(ε+ ~ω)2 − 2]1/2

)+ (1–45)

1

~ω

(2

∫ −

−~ωdε

(1− 2f (ε+ ~ω))(ε2 + 2 + ~ω)(2 − ε2)1/2[(ε+ ~ω)2 − 2]1/2

), (1–46)

σ2σn

=1

~ω

(2

∫

−~ωdε

(1− 2f (ε+ ~ω))(ε2 + 2 + ~ω)(ε2 − 2)1/2[(ε+ ~ω)2 − 2]1/2

). (1–47)

Below Tc , the resistivity drops to zero. What this really means is infinite conductivity at

zero frequency, that is a delta function at ω = 0 term in the conductivity. It is worthwhile

to observe that in the absence of scattering the conductivity is infinite in the normal

state, so the presence of a finite lifetime is important to obtain sensible results. The real

part of the conductivity has two terms in the Mattis-Bardeen result, the first describing

the scattering of thermally activated quasiparticles, the second describing pairs broken

by the photons. The imaginary part of the conductivity describes the response of the

condensate itself. With this interpretation it is clear that at zero T, σ1(ω) = 0, then when

~ω ≥ 2 it rises smoothly to connect with the normal state value.

At T = 0, the integrals can be carried out by hand. For ~ω > 2, k = 2−~ω2+~ω ,

σ1σn

= (1 +2

~ω)E(k)− 2

2

~ωK(k) (1–48)

and the imaginary part of the conductivity is given by, with k ′ =√(1− k2),

σ2σn

=1

2

((2

~ω+ 1)E(k ′) + (

2

~ω− 1)K(k ′)

)(1–49)

25

Figure 1-4. The real part of the conductivity reproduced with permission from [14] as afunction of frequency. APS c⃝1959. The minima of each curve occurs at 2as a function of temperature.

K and E are elliptic integrals of the first and second kind. The temperature dependence

shows the expected peak, behaving as ln(2~ω ) for lower frequencies (if ~ω & 0.5 it’s

suppressed). At nonzero T as a function of frequency, there is a minimum in σ1 at 2(T )

as shown in Figure 1-4.

1.3.5 Thermal Conductivity

The electronic thermal conductivity in an s-wave superconductor can be calculated

using the Boltzmann equation. In the normal state, the expression is,

κn =N(0)vf ℓ

3

∫ ∞

0

dεε2

T

(−∂f∂ε

)(1–50)

The calculation, first performed for by Bardeen, Rickayzen, and Tewordt [16],

essentially just gaps out the low energy quasiparticles, so that

κsκn

=

∫∞dE E2

T∂f∂E∫∞

0dε ε

2

T∂f∂ε

(1–51)

The results are compared with a measurement on aluminium in Figure 1-5, and are in

excellent agreement.

26

Figure 1-5. The thermal conductivity of superconducting aluminium reproduced withpermission from [15] APS c⃝1962.

Although the statement that a superconductor is a perfect diamagnet is often

made, the electromagnetic field always enters the sample for some distance called the

penetration depth. The penetration depth can be calculated from the static limit of the

electromagnetic response. The result of that calculation is [4],

1

λ2(T )=

1

λ2(0)[1− 2

∫ ∞

dEE√

E 2 − 2

(− ∂f

∂E

)]. (1–52)

In SI units e2

mns(T ) = 1

µ0λ2(T )= K

where 1

λ2= ns is the superfluid density. For an s-wave

superconductor this expression is exponential at low T. The result for an anisotropic

superconductor is

λ2(0)

λ2(T )=

1

2

∫dω

∫dk

4πtanh(

βω

2)Re

2k

(ω2 − 2k)

3/2. (1–53)

Schawlow and Devlin [17] measured the penetration depth, and showed that

to within the experimental resolution, the temperature dependence could be fit to

λ(T ) = 1√1− T

Tc

4, but this misses the asymptotic low temperature exponential form

∼√

Te−

T . Gorter and Casimir [18] advanced a two fluid model for the superconductor

in which the two fluids are normal quasiparticles and the condensate.

27

1.4 Unconventional Superconductivity

1.4.1 Unconventional Pairing

The symmetry of the superconducting order parameter can be read off directly

from the expectation value it represents. The crystal environment requires that the

expectation value, < ck,sc−k,s ′ >, transforms as a representation of the crystal point

group G and from the spin as SU(2). In ordinary superconductivity the U(1) gauge

symmetry of the normal state is broken and in unconventional superconductors

additional point group symmetries are broken as well. If a material has inversion

symmetry both parity and time-reversal are good quantum numbers for the pairs. States

may then be classified according to even parity spin singlet or odd parity spin triplet

pairs. In the noncentrosymmetric case there is an admixture of singlet and triplet pairs.

Unconventional superconductors do not possess the full symmetry of the normal state.

It is standard practice not to use the representations of the crystal group G, but rather

to refer to the standard spherical harmonics of the rotation group SO(3). A conventional

superconductor is therefore called an s-wave superconductor. We take “unconventional

superconductor” to mean that the order parameter is not s-wave, i.e. that the order

parameter has a lower symmetry than that of the crystal. In the cuprate literature, the

d-wave order parameter is often encountered in two ways, cos(kx)− cos(ky) and cos(2θ).

The former is a basis function of a representation of the two dimensional point group

D4h, and the latter is a d-wave angular function which can be identified with a Taylor

expansion of cos(kx) − cos(ky), where θ is measured from the a-axis on a circular

Fermi surface. The representation possessing the full symmetry of the square lattice

would be s-wave, e.g. a constant . Extended s-wave cos(kx) + cos(ky) would also be

compatible with the full symmetry of the lattice, but is named extended because it differs

from 1. The s- and d-wave differ under rotations by π2. This operation will change the

sign of the order parameter for cos(kx) − cos(ky), but not for cos(kx) + cos(ky). We list

in table 1-2 some common spin singlet representations of the square lattice. What is

28

meant by conventional then is the A1g representation. The gap is given, generally, by

a linear combination of the basis functions of a representation. For a one-dimensional

representation this is trivial but, from table 1-2, the two dimensional E-representation

is more interesting, = η1kxkz + η2kykz , where η is in general complex. For a more

complete discussion of these ideas we refer the reader to Sigrist and Ueda [19].

Table 1-1. D4h Selected Singlet RepresentationsIrrep. Basis FunctionA1g 1,(k2x + k2y ) s-wave, extended sA2g kxky(k

2x − k2y ) g-wave

B1g (k2x − k2y ) dx2−y2-waveB2g kxky dxy -waveEg kxkz , kykz

The formalism for describing the d-wave, or for that matter any pairing, is a

simple extension of BCS theory that accounts for matrix elements with more possible

anomalous pairings.

< ck,sc−k,s′ >= k =

k↑↑ k↑↓

k↓↑ k↓↓

. (1–54)

Here, can be complex. Clearly for a singlet superconductor the k↑↑ and k↓↓

components have to be zero. Likewise, we need the k↑↓ and k↓↑ components to

form the three | ↑↑>, | ↓↓>, and | ↑↓ + ↓↑> states of a triplet.

Owing to the antisymmetry of Fermions, the s-wave singlet has a gap

k =

0 k↑↓

k↓↑ 0

=

0 k

−k 0

≡ d0(k)iσ2 (1–55)

In the triplet case we have a more complicated d-vector since a triplet could correspond

to any of three wave functions, in general:

(k) = d(k).σiσ2 =

−dx + idy dz

dz dx + idy

(1–56)

29

The most general form of a gap becomes

ab(k) = ((k)1 + d(k).σ)iσ2 (1–57)

Table 1-2. D4h Selected Triplet RepresentationsIrrep. Basis FunctionA1u d = xkx + y ky

A2u d = xky − y kx

B1u d = xkx − y ky

B2u d = xky + y kxEu zkx , zky

It’s a simple matter to work out which component corresponds to which spin

wavefunction by using a basis with (1,0) or (0,1) for up and down spins, so that the outer

product of two up spins in a triplet contribute the component

| ↑↑>∼

1

0

1

0

†

=

1 0

0 0

(1–58)

So

1√2| ↑↑ − ↓↓>=

1 0

0 −1

,1√2| ↑↑ + ↓↓>=

1 0

0 1

,1√2| ↑↓ − ↓↑>=

0 1

1 0

(1–59)

Using what we know about unconventional gaps, if we introduce momentum

dependence (k) = 0g(k), the density of states is given, in three dimensions, by

N(E) = NF

∫d

4π

E√E 2 − 2

0g(k)2

(1–60)

where g(k) contains the k-dependence of the superconducting gap. Generally, the clean

density of states for low E will take the form of a power law, N(E) ∝ E n. This important

result can be used to interpret the phenomenology of many experiments, see table 1-3.

For cuprates, the clean density of states is linear in E. Since the spin lattice relaxation

rate varies as the density of states squared, this means we would expect a temperature

30

dependence of 1T1

∝ T∫dEE 2

(− df

dE

)∼ T 3. For both the specific heat, C

T, and the

change in penetration depth, λ, a linear density of states, N(E) ∼ E , results in a linear

in T behavior in the region where T/ is small. To see this, we must realize that the

number of states for a given energy comes from a region√ξ2 + 2

k < ω2, meaning the

part of the Fermi surface where |k | < ω. For example, in a three dimensional material

with a p-wave gap, 0 sin(θ), the specific heat has point nodes at when θ = 0 and θ = π,

so including only a small region sized ξ/k around the nodes,

Cv ≈N0

T 2

∫dξ

ξ2

cosh2( ξ2kBT

)

∫ 2π

0

dϕ

∫ ξ/

0

dθθ

4π∝ T 3 (1–61)

Table 1-3Experiment Leading Clean Low T Power Law

if N(E) E n

then CV T n+1

λ T n

1T1

T 2n+1

It is important to keep in mind the importance of impurity effects, especially at

low temperature because the formation of an impurity band modifies the low energy

behavior of many quantities. Within the impurity band the density of states is roughly

constant. The self-consistent T-martrix approximation is sufficient for the appearance

of an impurity band. For transport questions, the lifetime of the quasiparticles will have

a particular energy dependence which can also influence what is observed. Strong,

or unitary, and weak, or Born-limit, scatterers have been shown to have the following

approximate energy dependence [20] above the impurity band:

1

τBorn(E)=

1

τN

N(E)

N(0), (1–62)

1

τunitary(E)=

1

τN

1

(∑

k Gk)2N(E)

N(0)≈ N(0)

N(E), (1–63)

31

which in d-wave materials leads to 1τ(E)

∝ Eor 1E

scaling if E << 0 respectively. Simple

Boltzmann equation estimates suggest that transport coefficients are determined largely

by expressions like [21]: ∫dω(−∂f

∂ω)N(ω)τ(ω), (1–64)

so cataloging the associated response for any particular disorder model is crucial to

accurate understanding of experiment.

1.4.2 Migdal-Eliashberg Theory

Migdal-Eliashberg theory is a natural development of BCS theory to include a more

realistic treatment of the phonons. There were already anomalies in the data measured

from Hg and Pb compared to the weak coupling BCS approach, so it was necessary

from the point of view of experiment. Mercury, we recall, was the original material

studied by Kamerlingh Onnes in 1911. We have already shown data in Figure 1-2

which has a few wiggles present in the density of states, where BCS theory predicts the

smooth form ω√ω2−2

. The result of including a more realistic electron-phonon treatment

was to predict a tunneling current of the approximate form[22],

I (V ) =

∫dωRe

(|ω|√

ω2 − (ω)2

)[f (ω)− f (ω + V )] (1–65)

The frequency dependent gap will account for the wiggles.

I follow the account in Jones and March [23]. In Nambu space the self energy

including both electron-phonon and Coulomb interactions is,

(k , iωn) = −1

β

∑n,k ′

τ3G(k ′, iωm)τ3(∑λ

|gkk ′λ|2D(k − k ′, iωn − iωm) + Vc(k − k ′)) (1–66)

A convenient parameterized form of the self energy is

= [1− Z(k ,ω)]ω1 + χ(kω)τ3 + ϕ(k ,ω)τ1 (1–67)

32

which in turn results in a Green’s function.

G(k ,ω) =ωZ(kω)1 + ε(k ,ω)τ3 − ϕ2(kω)τ1ω2Z 2(kω)− ε2(k ,ω)− ϕ2(kω)

(1–68)

Here, ε(k ,ω) = ξk + χ(kω) and (kω) = ϕ(kω)Z(kω)

.

Figure 1-6. The BCS DOS is the dashed line. The experimental data on Pb and thecalculations are the solid and long-dashed lines respectively, showing howEliashberg theory can account for the energy dependence in the tunnelingsignal. Reproduced with permission from [9]. APS c⃝1963

A number of comments need to be made. These equations can be simplified,

and solved self-consistently, resulting in excellent agreement between theory and

experiment, in particular reconciling the discrepancy between the weak coupling theory

and the data in strongly coupled superconductors like mercury and lead. Scalapino,

Schrieffer, and Wilkins [9, 24] carry this out in two papers, and compare to Pb data.

The entire method relies on an exceptional observation by Migdal, now called Migdal’s

theorem [25], that the vertex corrections for the electron-phonon coupling are always

smaller by a factor√

me

Mi, the ratio of the mass of the electron to that of the ion. This

means that the electron phonon interaction can always be taken to be the bare

interaction, = γ(1 + O(√

⌉M⟩

)) where γ is the bare electron-phonon interation[25].

In the case of particle hole symmetry and the infinite bandwidth the Green’s function

in Eq 1–66 can take the non-interacting form. One of the most important results of

33

Migdal-Eliashberg theory is an understanding of the isotope effect, which examines the

change in the critical temperature with shifting isotope mass:

β = −dlnTc

dlnM. (1–69)

One modern issue theorists must cope with is the absence of an equivalent theorem

for other interactions besides the electron-phonon interaction [26, 27], limiting known

controlled techniques essentially to weak coupling versions of proposed interactions.

Furthermore, attempts at predicting Tc for the high-Tc compounds were unsuccessful

using this framework, suggesting new physics at work. I am not aware of schemes which

use Migdal-Eliashberg theory for phonons simultaneously with other interactions beyond

what is already cited.

34

CHAPTER 2SUPERCONDUCTIVITY IN CUPRATES AND PNICTIDES

2.1 Cuprates

2.1.1 Basic Cuprate Physics

After nearly 30 years of high temperature superconductivity research already, there

are a large number of reviews, for example [28–35]. The entire family of copper oxide

superconductors consist of one or more planes of copper oxide separated by layers of

rare and alkaline earth ions. The lattices are all typically of the perovskite form, or very

similar to it. The original motivation that led to the discovery of high-Tc superconductivity

was a study of Jahn-Teller distortions, so small octahedral tilts differing from the perfect

perovskite structure are not uncommon. The Cu form a square lattice with one O

between each copper atom, and the O form the perovskite cage. Sometimes the apical

oxygen is not present, but the pattern of square CuO2 layers is ubiquitous. Examples

are shown in Figure 2-1. Among the cuprates there are details specific to certain

compounds. For example, YBCO has CuO chains alongside two CuO2-layers which

order in several ways, and for LSCO there is a dip in the superconducting dome at 18

doping. BSCCO cleaves better than other cuprates. Nonetheless there is a remarkable

similarity throughout the entire set of high Tc compounds.

Figure 2-1. A representation of several cuprates with their symmetry classifications.(reproduced with permission from www.tfm.phys.cam.ac.uk)

35

The cuprate phase diagram is somewhat controversial even today. The generally

agreed upon results are shown in Figure 2-2, but careful omissions were built in which

will be addressed along the experimental data which suggests a particular interpretation.

The basic phases are an antiferromagnet, a superconductor, a metal at high doping, and

two strange phases referred to as the strange metal and pseudogap, whose positions

are harder to demarcate. It is suggested by some experiments that the entrance into

the pseudogap phase makes a line crossing through the superconducting dome thereby

indicating a quantum critical point. On the other hand, according to a different set

of experiments the line passes over the dome and such a quantum critical point is

suggested not to exist.

Antiferromagnet

superconductorsuperconductor

bad metal

Fermi

Liquid

Hole DopingElectron Doping

T

Figure 2-2. A cartoon of the cuprate phase diagram

The undoped copper oxides are unambiguously antiferromagneic Mott insulators.

Band structure calculations [36] (LSDA=Local Spin Density Approximation) originally

predicted the parent compounds to be metals, until correlations (LSDA+U) were

included such that the correct insulating phase was obtained. Typical parameters

in the antiferromagnetic phase give an exchange J of around 1000K, and a Neel

temperature around 300K. The on-site Coulomb repulsion is of order 8-10 eV, which is

larger than the bandwidth. Cuprates are generally charge transfer insulators [37] with

the charge transfer gap significantly smaller than U, around 3-5 eV typically. Hoppings

36

are estimated to be about 0.43 eV [28] for the Hubbard model but as high as 1.3 eV for

the p-d orbitals. As holes are added to the copper oxide plane the antiferromagnetism

is rapidly destroyed. At low temperature outside the antiferromagnetic phase there

is a region sometimes referred to as the spin glass phase, where frozen short range

magnetic order is present, then as hole concentration increases we enter the superconducting

dome. The term spin glass should be used loosely. The phase lacks long range order,

and there are upturns in the resistivity at low T in this phase, but the extent to which

the name implies glass physics can be misleading. The superconductivity is now

widely accepted to have a d-wave gap, though this took about a decade to establish.

It is confirmed experimentally by a host of probes now: STM (scanning tunneling

microscopy), ARPES (angle resolved photoemission spectroscopy), tri-grain boundary

experiments [38, 39], the list goes on. The highest transition temperature as a function

of doping is referred to as optimally doped, and occurs around 16% doping. One bit

of dirty laundry in this subject is that doping is usually unknown and Tc -fits to a widely

applied function are often used: Tc(p) = Tmaxc (1 − 86(p − .16)2), where p is holes

per CuO2 unit. The superconducting dome resides between about 5-25 % doping.

Sometimes p is the same as the concentration of a dopant, x, in the chemical formulas

but not always. Underdoped and overdoped refer to doping less than or greater than

the optimal value (p=.16) respectively. Most of the anomalous properties of the cuprates

show up to a greater extent in the underdoped compounds. At high enough hole doping,

it is believed that the normal state is again a Fermi liquid, borne out by transport or NMR

data. A large region of the phase diagram outside the superconducting region is clearly

not a Fermi liquid, and is referred to as the strange or bad metal phase characterized

by the approximately linear in T resistivity. The pseudogap phase appears below a

characteristic temperature towards the underdoped side of the superconducting dome.

Whether these are crossovers or true phases is still a matter of debate, but it is clear

there are not sharp thermodynamic signatures of a phase transition as one would

37

see for magnetic second order transitions. On the other hand, some probes indicate

a sharp transition line between pseudogap and bad metal phases, so one is again

forced to reconcile experiments which will be discussed below. The electron doped side

of the phase diagram is similar in overall structure but there is a notable asymmetry

between the electron and hole doped materials. Antiferromagnetism survives to larger

doping on the electron-doped side, and the pseudogap region is not as pronounced.

Superconductivity and antiferromagnetism are much closer to one another on the

electron doped side. While resistivity measurements indicate an anomalous metal phase

on the hole doped side, they provide evidence for a Fermi liquid on the electron doped

side at all dopings.

The typical size of the gap at (0,π) where it is maximum is around 25-40 meV at

optimal doping. The materials are extreme type-II superconductors, so from the slope of

Hc2 at Tc we can estimate from BCS theory ξ0 ≈ 15 − 30A. Alternatively, one can use

Fermi velocity and gap measurements to estimate the coherence length, which comes

out to around 2 nm. Penetration depth measurements from optics and µSR are of order

1500A. These are in-plane estimates. It is also true that cuprates are very anisotropic

materials, with very high resistivity and much longer penetration depths along the c-axis

compared to the ab-plane.

A great deal of attention over the years has been paid to certain models [40] which

are claimed to capture the essential physics of the cuprates. The Hubbard model is one

such model. It is argued whether a 3-band (Cu + O) or 1-band (Cu) model is necessary

[28, 36]. The simplest form of the Hubbard Hamiltonian is:

HHubbard − t∑<ij>σ

c†iσcjσ + ciσc

†jσ + U

∑ni↑ni↓.

To date the only known solutions are in one or infinite dimensions, though there

are many approaches to the model in other dimensions. At large U, the Hubbard

Hamiltonian is mapped to the t-J Hamiltonian. A Schrieffer-Wolff transformation [41] is

38

applied to the large U limit of the Hubbard model. The t-J Hamiltonian is, with J = 4t2

Uis:

H = −t∑<ij>

c†iσcjσ + ciσc

†jσ + J

∑<ij>

Si · Sj −ninj

4.

The Hilbert spaces are different even though there is a formal relationship between

these two expressions: double occupancy has been projected out in the t-J model,

which has a drastic effect on the quasiparticles since any interaction accounting for

virtual transitions to doubly occupied sites is not included.

Starting with the spin operator,

Si =1

2

∑c†iaσabcib

and using the grand orthogonality theorem for group representations

∑R∈SU(2)

D(R)abD(R)cd = δabδcd +−→σ ab · −→σ cd =

4

2δacδbd ,

we can put the t-J model into the Hubbard Hamiltonian’s form, however one must be

careful to recognize the difference in the Hilbert spaces for the models. The t-J model

has projected out doubly occupied states.

2.1.2 Tunneling

Tunneling experiments in the cuprates were not initially easy because of issues

involving sample quality and control of the tunneling barrier [7], but once these

struggles were overcome, STM became an invaluable tool in investigating some of these

materials. Due to a weak bilayer coupling in BSCCO, it cleaves nicely, and is well suited

for these experiments. This also means that there are a few layers of atoms between

the copper oxide plane and the surface, which raises the question of how the surface

layers alter the tunneling signal. BSCCO is also unique in displaying supermodulation, a

periodic but incommensurate change in the density of the BiO layer with relatively long

wavelength. Figure 2-3 from Huang et al. [42], shows a typical cuprate density of states,

extracted from dIdV

curves. We bear in mind that interpreting the measurement as the

39

density of states could be different from understanding it as a convolution of spectral

functions from the sample and tip. In sharp contrast with s-wave superconductors, there

is a non-zero density of states observable down to zero bias. This is an indication of

nodes in the superconducting gap. Generally the density of states can be well fit from

the form expected in d-wave BCS theory, though much useful information has been

inferred from the scattering rate which would enter this expression ∼ E√(E+i(E))2−(E ,θ)2

[43].

A

-3 -2 -1 0 1 2 30.0

0.5

1.0

1.5

2.0

2.5

Ω

D

NHΩL

N0

B

Figure 2-3. Density of States extracted from tunneling data in BSCCO next to a cleand-wave DOS calculation. Scattering would broaden the peaks, create a lowenergy impurity band, and remove the sharp features. Reproduced withpermission from [42]. APS c⃝1989.

One of the most remarkable experimental results in the high Tc superconductors

is the existence of the pseudogap phase. A clear indication of what is meant by

pseudogap is shown in Figure 2-4, where we see a continuous evolution of the

differential conductance even as we cross Tc , where one would expect the superconducting

gap to close. A gap, however, remains so the phase is called the pseudogap. Extensive

temperature dependence measurements [44] suggest there is a universal low energy

shape to part of the pseudogap, for low bias, and a temperature dependent higher bias

part, which is attributed to anti-nodal excitations as those occur at higher energy. The

angular dependence of the pseudogap is reported by ARPES [45], where it is seen to

40

roughly follow a d-wave form. The observation of two energy scales in microscopy is

reflected in other experiments, like Raman scattering [46], however see also [47].

Another feature which caught the attention of the superconducting community was

the dip at high energy compared to the superconducting coherence peak. (See Figure

2-4 at 4.2K, where it is prominent) It is proposed that the peak-dip-hump structure can

be explained with band structure, modified Migdal-Eliashberg coupling to a phonon [48],

or another bosonic mode like the 41 meV mode observed by neutrons [49].

A B

Figure 2-4. Density of States for BSCCO with Tc = 83K. Reproduced with permissionfrom [50]. APS c⃝1998. A gap in the spectrum remains above Tc . The phasediagram as seen by STM. Reproduced with permission from [7]. APSc⃝2007.

In the superconducting state, tunneling into a Zn impurity, which substitutes for a

Cu atom in-plane, reveal a discrepancy between theory and experiment. It has been

addressed by a number of authors by trying to take into account the matrix elements

frequently ignored in the calculation of tunneling conductance. The center of a scattering

resonance always coincides with the site of a surface Bi atom, below which a Zn is

41

located. The extended tails from the impurity are oriented with the gap maxima instead

of the gap minima, and the local maxima in the density of states on specific atomic

sites is at odds with the simplest theory [51]. A number of explanations involving matrix

elements tunneling through the 4.5A layer have been proffered [52], which can be used

to obtain reasonable agreement. If this is correct no fundamentally new physics is

needed, but this ignores other explanations including a nonlocal Kondo-coupling instead

of point-like impurity [53] which also resolve some of the issues. Festkorperphysik ist

Schmutzphysik (attributed to Pauli).

A Experiment B Theory

Figure 2-5. (left) Logarithmic plot of the intensity around a Zn impurity in BSCCO.Reproduced with permission from Pan et al. [54]. c⃝Nature Publishing Group2000. (right) Theoretical prediction around unitary scatterer in d-wavesuperconductor. That in the experiment a maximum intensity is found at thecenter in contrast to the theoretical result. Reproduced with permission from[55]. APS c⃝1996.

From the separation of the coherence peaks and Tc , the BCS ratio 20

kBTc= 4.3 for

d-wave weak coupling theory, is far exceeded in the cuprates. Within Migdal-Eliashberg

42

theory, this value can depart from the weak coupling result. The magnitude of the

gap deduced from this scale is also found to vary strongly from position to position on

the sample surface, and the origin of the strongly inhomogeneous structure is a key

question in these compounds. McElroy et al. [43] captured data which suggest the

oxygens correlate with the local gap amplitude. The inhomogeneity is correlated with

the width of the superconducting transition. Furthermore, as a function of doping the

average spectral gap value decreases with increasing doping, ranging from about 100 to

20 meV. Typically the ratio 20

kBTcexceeds the d-wave BCS ratio 4.3.

2.1.3 Specific Heat

A Experiment B Theory

Figure 2-6. (left) T-dependence of the coefficient γ in Cv = γT .(right) Pseudogap scalefrom specific heat measurements. Reproduced with permission from [56].

In Fermi liquid theory the electronic contribution to the electronic specific heat

should be linear in temperature, C = γT . Loram et al.[56] have done extensive work

on the specific heat in the high-Tc superconductors. Figure 2-28 shows the doping and

temperature dependence of the specific heat coefficient in YBCO. We can also infer

from this data the behavior of the entropy by integrating∫dT

Cv (T )T

. As doping falls below

43

p=0.19 the specific heat jump γ(Tc) decreases rapidly, and the entropy extrapolates to

unphysical values. The general trend is reflected in many experiments. For overdoped

samples, favorable comparisons can be made with d-wave BCS theory, but as we

underdope puzzling behavior appears. There is a gap opening in the system above Tc , a

loss of entropy, and all the symptoms of pseudogap formation.

2.1.4 Angle Resolved Photoemission Spectroscopy

Angle Resolved Photoemission Spectroscopy, ARPES, is a relative newcomer

to superconductivity research, and has been mainly applied to research after the

conventional superconductors were studied. Once energy resolution became fine

enough (current claimed resolution for synchrotron based ARPES is ∼ 0.1 to 1 meV)

ARPES became a useful bridge between theory and experiment because it is commonly

thought that it directly measures the spectral function. To derive this, however, is not

straightforward[57]. Photoemission currents are directly proportional to the intensity of

incident light, for frequencies above the emission threshold. We are interested, in the

expectation value of the current operator, under the perturbation

HI = −1

c

∫dr A(r , t).j(r)

where A is the electromagnetic vector potential. The ARPES intensity is formally a

three-current correlation function

< Ja(r0, t0) >=1

~2c2

∫dtdt ′drdr ′Aµ(r , t)Aν(r

′, t ′) < Jµ(r , t)Ja(r0, t0)Jν(r′, t ′) >

Due to the complicated nature of this expectation value, there are a series of approximations

made to bring this into a simpler form. These approximations are referred to as the

independent-particle approximation, the three-step approximation, and the sudden

approximation. In this simplified model the basic measured quantity is proportional to

the spectral function convolved with the Fermi distribution within the limits of resolution

44

of the device:

I (k||,ω) ≈ |⟨nal |A.k |initial⟩|2∫

dk ′∫

dω′A(k ′,ω′)f (ω′)R(ω,ω′)

Here A is the spectral function while A refers to the vector potential, f is the Fermi-Dirac

distribution, and R is the energy resolution of the apparatus. In the presence of a finite

self energy, the spectral function generically takes the form:

A(k ,ω) =1

π

′′k(ω)

(ω − ϵk −′k(ω))

2 + (′′k(ω))

2

It is perhaps more useful to think of a Fermi liquid with finite self energy. An equivalent

representation of the spectral function is then

A(k ,ω) =Z

π

k(ω − ξk)2 + 2k

+ Ainc(k ,ω)

where Z = (1 − ∂∂ω)−1 (in a translationally invariant situation), and Ainc(k ,ω) represents

the incoherent background. The next important fact to keep in mind is that ARPES is

a surface probe. The momentum k⊥ is not conserved because translational invariance

is lost across the surface. If the surface does not cleave well or significant surface

reconstruction occurs, these effects will play a role in understanding the observation.

That said, if those effects are understood or unimportant, ARPES provides very valuable

information: electronic dispersions, lifetimes, the Fermi surface, the superconducting

gap. For a review in the cuprates see Damascelli et al. [45].

For a non-interacting Fermi gas the spectral function of the electron is a delta

function [5]. With interactions we expect a good Fermi liquid to have a sharp quasiparticle

peak corresponding to a broadened delta function with some kind of incoherent

background signal. ARPES in the cuprates does identify a quasiparticle peak [45],

although it is quite broad along with a significant incoherent part. What is also

interesting is that the weight in the quasiparticle peak strongly decreases with

decreasing doping.

45

Figure 2-7. The Fermi surface from ARPES in BSCCO. Reproduced with permissionfrom [58]. APS c⃝2002.

ARPES measurements unambiguously confirmed the Fermi surface[58], shown

in Figure 2-7, and the unconventional symmetry of the superconducting gap. The

importance of these measurements should not be understated but we will focus on more

current topics. ARPES has found a highly anisotropic suppression of spectral weight that

persists far above Tc .

Figure 2-8. The demarcation of the pseudogap temperature according to ARPES fromLee et al. [59] APS c⃝2006.

46

Figure 2-9. Laser-ARPES data on BSCCO, showing the Fermi arc phenomena, and acontroversial claim to see the closure of the arc. Reproduced withpermission from [60]. c⃝Nature Publishing Group 2009.

An open question here is the appearance of Fermi arcs instead of a Fermi surface

[61] which seems to violate Luttinger’s theorem [62]. Just above Tc , the Fermi surface

appears as an arc, shown in Figure 2-9, which grows linearly in temperature up to T∗

when it becomes the full Fermi surface we expect [63]. In high magnetic fields quantum

oscillation experiments suggest that these arcs are actually Fermi pockets with little to

no spectral weight in the ARPES intensity on the back surface, giving the appearance

of arcs where truly a Fermi surface exists. The suggestion is that the field may have

reconstructed the Fermi surface, either by stabilizing periodic order or some other effect.

Figure 2-9 illustrates the pitfalls well: there is clearly a nice spectral weight along the

arc, and some smaller subsidiary peak. This could be a shadow band, or a surface

effect, especially since the intensity is flat and faint, so how is one to interpret the result?

Roughly, there is an arc, and roughly we are measuring the spectral function, but in an

instance like this we become concerned with the more subtle aspects in the original idea

that the three current correlation function can be thought of as the spectral function.

At Tc , a superconducting quasiparticle peak appears at (π, 0). It displays a

peak-dip-hump structure reminiscent of the STM result. The pseudogap is observed as

a the quasiparticle peak recedes from the Fermi energy below T ∗. There is a polarized

ARPES measurement [64] to suggest that this state is a time-reversal breaking state,

47

Figure 2-10. (π, 0) quasiparticle peak for a sample with Tc=91K, showing the peak diphump structure. Reproduced with permission from [45]. APS c⃝2003.

Figure 2-11. Angular dependence of the pseudogap in LSCO and a fit to the d-waveform. Reproduced with permission from [45]. APS c⃝2003.

48

but this is still under debate. What is unusual about the pseudogap is that it is not simply

an isotropic suppression of the spectral weight, but rather it seems to follow a d-wave

form, lending credibility to the idea that d-wave superconducting fluctuations exist above

Tc : for example superconducting fluctuations are known to be sufficient to produce

fermi arcs[65]. Of course, this is also still under debate because a d-density wave or

angle-dependent quasiparticle weight Z(k ,ω) in exotic theories can also capture this

feature [66].

2.1.5 Nuclear Magnetic Resonance

Figure 2-12. The Knight shift in 89Y NMR in YBCO. At x=1 the Knight shift is nearlytemperature independent, evolving to strongly temperature dependent as xis changed, all above Tc . Reproduced with permission from [67]. APSc⃝1989.

If the cuprates were good metals, the susceptibility would be constant above

Tc . In underdoped YBCO as T is lowered, the susceptibility falls for T well above Tc .

This measure provided some of the first evidence for the pseudogap. Once in the

superconducting state, a d-wave BCS model for the spin lattice relaxation rate predicts

a rather small Hebel-Slichter peak which is not observed in experiment, but correctly

captures the low T asymptotic behavior. The absence of a Hebel-Slichter peak is

generally explained away with impurity scattering, but it is a curiously absent feature

nonetheless. The low temperature spin lattice relaxation rate scales as T 3, as one

49

expects, reflecting a linear density of states. Multiple bands, disorder, and other nodal

gaps are issues in the pnictides where similar 1T1

behavior is observed.

Figure 2-13. A schematic (in French) of how NMR spectroscopy works, courtesy ofJ.Bobroff. The arrows on the left represent magnetic moments inducedabout an impurity. This local magnetism influences the nearby nuclei,whose signal is depicted on the right. The NMR spectra is a histogram ofthe local magnetic fields throughout the sample.

In Figure 2-12, we show the Knight shift in YBCO samples. The Knight shift

unexpectedly develops a strong temperature dependence with underdoping. This

suppression occurs as if a gap were opening in the excitation spectrum of the system,

signaling the pseudogap. In a normal metal we expect the Pauli susceptibility ∝

µ2BN(EF ). This is true until about room temperature, 300K or so. As we lower the

temperature from there, the susceptibility has lost approximately 80% of its value by

the time Tc is reached, indicating a removal of density of states. The same paper

[67] demonstrates that a Korringa law, i.e. that 1T1TK2

sis a constant, holds which is an

indication of Fermi liquid behavior.

NMR widths show a distribution of magnetic hyperfine effects or equivalently a

distribution of T1 relaxation times, due to the inhomogeneity in the local magnetic

environment, often due to defects [34]. A depiction of a typical spectrum can be

explained using Figure 2-13, where satellite moments add to the distribution of line

widths. This impurity physics can be used to understand the evolution of NMR spectra in

the cuprates as a function of disorder.

50

2.1.6 Neutrons

For a recent review of neutron scattering in the cuprates see Tranquada [49].

Neutron measurements of the spin excitations in the insulating parent compounds

are well described by linear spin wave theory. With increasing hole doping the

antiferromagnetic peak at (π,π) develops incommensurability. There seems to be a

universal pattern, shown in Figure 2-14, to the acoustic spin excitations, which take an

hourglass shape in Q-energy space. At the resonance energy around 41 meV, there

are antiferromagnetic (π,π) excitations. As you raise the energy there are two branches

of excitations away from (π,π), and similarly for two branches as the energy is lowered

[68]. The incommensurability seems directly tied to doping, at least at low hole doping.

In electron doped cuprates to my knowledge the excitations remain commensurate [69].

These observations are suggestive of antiferromagnetic or nearly antiferromagnetic

excitations in the superconducting state. The resonance mode’s explanation is an open

question.

Figure 2-14. Q-Energy depiction of spin excitations in cuprates. Reproduced withpermission from [49].

At optimal doping a spin-gap, of approximately 8 meV, develops in the excitation

spectrum in contrast to lower doped samples where the spin gap is reduced [49]. The

gapped spectral weight gets pushed to higher energies.

51

One of the most interesting features revealed by neutrons is a state which shows

up at 1/8th doping in LBCO and LSCO, which has been studied intensively. There is

definite observation that the superconductivity is completely suppressed at p=.125

in those two materials, and that simultaneously a periodic modulation of spin and

charge, resembling stripes, shows up. The appearance of quasi-one-dimensional

physics and spin charge separation in these mesoscale structures suggests that this

order is somehow intertwined with the superconductivity [70]. This perspective is

supported by numerical experiments on the t-J and Hubbard models where the holes

naturally segregate and can form stripe like structures. There is a plateau in YBCO’s

superconducting dome where in LSCO it is strongly suppressed, and it is suggested that

the stripe order is somehow fluctuating or non static.

As a parting remark, we observe that the fully momentum and energy integrated

strength of magnetic scattering decreases with hole doping until its death at optimal

doping. This tantalizing bit of information suggests that there is a coexisting magnetic

glass or inhomogeneous local magnetism responsible for the host of strange underdoped

properties

2.1.7 Penetration Depth

Figure 2-15. Temperature dependent penetration depth, showing linear behavior for lowT in contrast to the exponential s-wave behavior.Reproduced withpermission from [71]. APS c⃝1993.

52

Initially the high Tc compounds were difficult to grow in large clean single crystals–

sample quality was a major issue. Once samples became adequate reliable experiments

could elucidate the underlying nature of the superconducting ground state. Hardy et al.

[71] used a cavity perturbation technique to measure the penetration depth in optimally

doped high quality YBCO crystals, clearly exhibiting linear temperature dependence,

shown in Figure2-15. This is indicative of point nodes in a two dimensional gap over

a circular Fermi surface or line nodes on a cylindrical (3D) Fermi surface. Previous

work on thin films often reported T2 behavior, which can be attributed to impurities in

a d-wave superconductor. Recall that at the time, establishing the d-wave symmetry

of the gap was a major goal. Absolute values of the penetration depth are hard to

establish because experiments are usually measuring a change in the superfluid density.

Nonetheless, a consistent subtraction must be used, which provides the best idea of

what the penetration depth should be. µSR can provide absolute values, however these

depend on details of the modeling of the vortex state. YBCO typically comes out with

1400-1600 A penetration depths, and BSCCO is about twice the size at 2100A. LSCO is

even larger with 4000A.

2.1.8 Electric and Thermal Conductivity

In the superconducting state, if we assume that a d-wave BCS superconductor is an

accurate description, something remarkable happens in transport. P.A.Lee [72] showed

that in the low temperature zero frequency limit, the intercept of the conductivities

approach a constant, universal value, because the result is very insensitive to disorder.

If we take the simple result that

σ(T ) ∝ N(T )v 2F τ(T ) (2–1)

then account for the density of states of a d-wave superconductor, N(E) ∝ E , including

that for weak scatterers τ ∝ 1E

it seems like the conductivity is independent of energy.

While this is a very naıve argument, it spurs on further investigation. In the absence

53

of vertex corrections, the same would be true for the electrical and spin conductivities,

but they are more strongly renormalized by vertex corrections than the thermal current

which remains to a good approximation, independent of the scattering rate [73]. Defining

v2 =∂k

∂k, the universal values in the absence of vertex and Fermi liquid corrections are

shown in equation 2–2. These values would need to be scaled by the number of planes

per unit cell to compare with measured values.

σ0 =e2

~π2vFv2, κ0

T=

k2B

~3vF+v2vF v2

, σs =s2

~π2vF+v2vF v2

(2–2)

Neglecting vertex corrections, we provide the recipe for obtaining the conductivities.

Generally one must calculate ”the bubble,”

Imij() =∑k

1

β

∑iωn

g2vi vjTr [G(k , iωn)τG(k , iωn + im)τ ] (2–3)

The parameter g refers to the charge or spin for those currents respectively; τ is a Pauli

matrix: τ0 for charge current, τ1 and τ3 for spin or heat currents. For thermal transport

g = ω + 2, although the precise definition of thermal current can pose subtle issues

because we are referring to a local temperature gradient, see for example Catelani and

Aleiner [74]. From Eq 2–3, σ = − Im

for the charge and spin currents while for the

thermal current, κT= Im

T 2 . At the lowest temperatures, we expect impurity scattering to

play an important role. As temperature is raised, inelastic processes will also contribute

to the transport processes, so accurate models of the conductivities will necessarily

include a treatment of both effects.

One of the useful aspects of the universal values is providing another way to

measure the Fermi velocity and superconducting gap, to compare with experiments like

ARPES and STM. vFv2

is about 14 in YCBO and more like 20 in BSCCO [75]. The doping

dependence of the gap, using a Fermi velocity from a second measurement (typical

values are 105m/s), can also be obtained as shown in Figure 2-16.

54

Figure 2-16. Doping dependence of the gap according to thermal conductivitymeasurements. Reproduced with permission from [76]. APS c⃝2003.

The normal state transport properties are much more strange. We venture no guess

at explaining them. In the phenomenology of the Fermi liquid, the resistivity scales at T2

with temperature. The resistivity is approximately linear [77] for high T in underdoped

materials with deviations at low T, linear in T near optimal doping, and then crossed over

from T to T2 in the overdoped region for all hole-doped materials [78]. Quantum critical

theories [79] can give exponents different from T 2, but cannot reproduce these results.

A convincing explanation of this result is a major open question.

The dynamic conductivity does not follow the Drude form σ01+iωτ

, but data can fit

to a Drude-like form Re(σ) = σdc1+(ωτ)y

or σ01+iωτ(ω)

[80]. Quantitative or not, the utility of

the expression is in providing reasonable measures of the scattering rate which can

help identify the microscopic interactions which control transport properties. One of

the most striking results from transport measurements is a clue about the nature of

scattering in the normal state. As soon as Tc is reached, the scattering rate drops

precipitously, indicating that gapping the electrons also removed scattering phase space.

This is strong evidence for electronic pairing in the superconducting state. The residual

55

Figure 2-17. The anomalous linear-in-T resistivity over a wide range of temperatures inthe cuprates. Reproduced with permission from [78]

resistivity per plane is a measure of the scattering rate of the conduction electrons by

the impurity potential. It can be obtained by extrapolating the high-T ρ(T ) to T=0.

The frequency integrated conductivity in underdoped cuprates is found to be

proportional to x, the doping, indicative that the holes contribute to the conductivity

directly. The same spectral weight can be compared to what is expected from single-particle

picture band calculations. Both ab-initio and experimental methods are open to some

criticisms, but as a general trend, cuprates will prove to be strongly correlated by this

measure, as well as materials like vanadium oxide, but in contrast to simple metals. A

clear signal of the formation of the pseudogap is seen in c-axis conductivity while it is

difficult to identify in the a-b plane data [30].

2.1.9 Nernst, Kerr, µSR

The Nernst-Ettinghausen effect is the connection between a transverse electric

field created by the a thermal gradient in the presence of a magnetic field or a thermal

gradient induced by an electric field.

eN(H,T ) ≡ E

|∇T |

56

Specifically, Nernst refers to the detected electric field due to an applied thermal

gradient and Ettinghausen is when a current density produces a thermal gradient, all in

a perpendicular magnetic field

QE ≡ |∇T |JH

.

The usual explanation invokes the existence of vortices above Tc . Several authors

have worked on the theory of this transport process [81, 82]. In a normal metal the

signal is supposed to vanish in the absence of electron-hole asymmetry [83, 84]. On

the other hand, the Nernst effects have also been observed in semimetals like Bi [84]

where it is large, and in graphene [85], where the signal is attributed to three physical

effects: increasing the scattering time, increasing the cyclotron frequency and reducing

the Fermi energy.

Figure 2-18. Nernst Signal in nV/(oK -T) for LSCO. Reproduced with permission from[86]. APS c⃝2006.

If we are to associate the Nernst signal with fluctuating superconducting vortices

above Tc , where the Cooper pairs are not long range phase coherent, then the data [86]

shown in Figure 2-18 clearly favors the interpretation that there is no quantum critical

57

point under the superconducting dome as the signal extends all the way around the

superconducting region.

Figure 2-19. Onset of Kerr signal in YBCO. Reproduced with permission from [87]. APSc⃝2008.

In contrast with this set of experiments, is work done by Kapitulnik [87] in YBCO.

The Kerr effect is the rotation of the polarization of light by scattering from a time-reversal

symmetry breaking perturbation. The setting for the Kerr effect is usually ferromagnets

or any uniaxial material which has a tendency to rotate the incident light, so it is

surprising in an apparently non-magnetic phase to observe any Kerr rotation at all.

Mineev [88] has shown how time reversal symmetry breaking can be calculated from the

electromagnetic kernel for unconventional superconductors. The data shown in Figure

2-19 clearly intersect the superconducting dome for YBCO, favoring the interpretation

that time reversal symmetry is broken in the pseudogap state and suggesting that the

superconducting dome obscures a quantum critical point.

This is supported by polarized neutron measurements [89] which claim to observe

a magnetic order that preserves translational symmetry, and µSR experiments [90, 91].

Briefly, µSR works the following way. The time evolution of the muon-spin polarization

is dependent on the local magnetic field distribution, and is measured by detecting

58

Figure 2-20. Onset temperature of magnetic order observed by neutrons. Reproducedwith permission from [89]. APS c⃝2006.

the muon-decay positrons because their decay is preferentially aligned with the

sample’s magnetic moment. The µSR magnetic moment is consistent with the neutron

experiments. However, the volume fraction of the sample in which these moments

exist, about 3%, in which these moments exist do not support theories that ascribe the

pseudogap to a state characterized by loop-current order [92], as claimed by the neutron

paper, rather that dilute impurities and correlations induce this effect. Furthermore,

the muon studies find that the doping dependence of the signal does not track the

onset of the pseudogap. There is a counterclaim [93] that the muon charge disturbs the

observation of a magnetic moment, but this contradicts µSR which observe the magnetic

moment in the spin-glass phase [94] consistent with NMR experiments in the spin-glass

phase.

2.1.10 Remarks on High Tc Experiments

In each of the experiments discussed there is an open question. An overview of

the theoretical proposals is given at the end of this chapter. The pseudogap’s nature

is yet to be revealed, but not for lack of effort. The anomalous resistivity is outside of

the scope of Fermi liquid phenomenology. In the tunneling microscopy the pattern

around a single impurity is not what comes out of the most straightforward theory,

59

so a critical examination of the tunneling matrix elements which supposedly account

for the differences is important if we are to have a complete understanding. Some

of the work not shown here, notably by the group of J. C. Davis, shows mesoscopic

inhomogeneity, and often tweed-like patterns in the local conductances which is

attributed to local electronic nematic order. In ARPES, the biggest open question in

my opinion is the nature of the Fermi arcs vs. Fermi pockets observed by quantum

oscillation experiments. It’s hard to see the quantum oscillation data and not believe that

there is underlying Fermi liquid physics, but the magnetic field is these experiments is

strong enough to suggest that zero field ground state is different from the state in several

Tesla, for example a field induced spin density wave. The caveats in ARPES analysis

also means that there could be a very significant loss of spectral weight on the backside

of the pockets due to surface effects or incoherence. The neutron data’s gross features

can be captured within an RPA picture, but fails to reproduce the pseudogap seen at

low doping. The normal state resistivity is a wide open question. Related to it, is the

proposal of a marginal Fermi liquid which would give a strictly linear resistivity, but this

picture lacks a microscopic foundation. Quantum critical theories give a resistivity that

varies as T 4/3, so perhaps new flavors of quantum criticality will capture this scaling.

Quantum criticality is also promising for the neutron scattering and Kerr effect, but it has

trouble with Nernst measurements and the explanation of diamagnetism above Tc .

2.2 Pnictides

In 2006 Kamihara et al.[95], in the group of Hideo Hosono, discovered superconductivity

in a new class of iron-based compounds, now referred to by any of the following

names: pnictides, iron-arsenides, iron-based superconductors. The original compound,

LaOFeP, only exhibited Tc of around 5K, so it was two years before a similar compound,

LaFeAsO1−xFx , whose Tc was 26K [96], caused excitement in the physics community.

Lower critical temperatures are assumed to be due to electron phonon coupling as in the

elemental superconductors. In the iron arsenides it has reached of order 55K [97], which

60

is an indication of unconventional superconducting mechanisms. A number of useful

reviews exist, for example Johnston [98] or Sadovskii [99].

Figure 2-21. Illustration of BaFe2As2 structure.

The iron-arsenide family of materials is subdivided into several classes referred to

as the 1111’s, 122’s, 111’s, and 11’s referring to their chemical formulae. For example

LaFeAsO is a 1111, while BaFe2As2 is a 122, LiFeAs a 111, and FeSe is a 11. It is

perhaps obvious that the beginning of understanding a new material is listing the

elements composing the compound and the crystal structure. This question is usually

resolved with X-ray or elastic neutron scattering. One of the ancillary questions that

can be addressed is whether there are any structural transitions in the material. The

two dominant structures in the pnictides are tetragonal (I4/mmm) and orthorhombic

structures. The parent compounds show a magnetic phase, and with doping, just

as in the cuprates, superconductivity appears. One exception is LiFeAs which is

61

a nonmagnetic stoichiometric superconductor with a Tc of 18K. Depending on the

specific compound there may or may not be a region of significant overlap between the

superconducting and magnetic phases. The tetragonal structure is the high temperature

phase, and at lower temperature and low dopings there is a tetragonal-orthorhombic

structural transition as in the cuprates, shown in Figure 2-26. One of the other

remarkable structural features which is similar to the cuprates is the presence of a

square lattice of Fe and high anisotropy in transport measurements, which immediately

leads to the speculation that only these FeAs plane layers contain the physics as in

the cuprates. To critique this idea, we should check first whether the band structure

has c-axis dispersion, and try to compare the phase breaking length compared to

the inter-layer spacing. Band structure calculations and experiments measuring the

dispersion can help answer the first question. To my knowledge, the second question

has not been addressed, but there is an emerging consensus that the so-called 122

compounds three dimensionality evolves as the band structure is altered by doping, and

considering these materials to be ideally two dimensional may not be justified.

Figure 2-22. Ba(Fe1−xCox )As2 phase diagram from a combination of specific heat,resistivity, and other measurements. Tα is the structural phase transition,Tβ is the magnetic phase transition, and the dome representssuperconductivity. Reproduced with permission from [100]. APS c⃝2009.

The anisotropy in the 1111 compounds is far greater than in the 122s, and the

correlations in the 11 materials seem to be more important than in the other family

62

members. I intend to demonstrate that while it may be possible to apply the same

physics to different pnictide materials, as we vary the composition throughout this

broad class of compounds, experiments report different –seemingly conflicting– results

regarding the superconducting state. The diversity of results obtained appears to be

much greater than in the cuprates, even accounting for the early stage of materials

growth development. One can speculate along the following lines: each instance of a

pnictide might live in a different region of parameter space (t,U for a Hubbard model,

in addition to impurity scattering) thereby displaying different properties under one

common Hamiltonian, or there could be significant variation in sample quality and the

effect of impurities might result in the diversity of observations.

2.2.1 Optical Conductivity

Model building is a delicate task that requires experimental input. The pnictides

have similarities with the cuprates at first glance: square planar layers, non-phonon

induced superconductivity, a magnetic phase in the parent compound. So an important

question is to address the role of correlations in pnictides. This question can be

addressed by a combination of band structure, ARPES, and optical conductivity

measurements. The general dynamic conductivity will depend on impurity scattering

and inelastic scattering, so important inelastic processes and a realistic model for

disorder are necessary to achieve an understanding of the conductivity over a broad

frequency range. The integrated Drude weight for a given material can be compared

with the same quantity as obtained from theory to provide a qualitative estimate of the

importance of correlations. Coupled with density of states information, it also helps

determine whether the materials are insulating or conducting in the magnetic state.

These measurements favor the interpretation that the pnictides are itinerant systems

with moderate to weak correlations as we will show.

By summing the area underneath the optical conductivity (f-sum rule), we can

construct the spectral weight and plasma frequency (the spectral weight of free carriers,

63

only the Drude part).

K(∞) ≡ 2

π

∫ ∞

0

dωσ1(ω) (2–4)∫ ωcuto

0

dωσ1(ω) = πϵ0ω2p

2(2–5)

In practice this is done with a cutoff for both theory and experiment, but it accurately

segregates strongly, moderately, and weakly correlated materials [101]. The extremes

of this measure are that for a strongly correlated insulator the ratio Kexp

Kband→ 0, while for

good conductors Kexp

Kband= 1. For LOFP Kexp

Kband≈ .5 while for LOFA Kexp

Kband≈ .3 − .4, implying

that pnictides are moderately but not strongly correlated[101]. This perspective has been

corroborated by x-ray absorption and resonant inelastic x-ray scattering studies[102]

which indicated weak correlations in the pnictides. Several models yield U ≈ 2eV

and J ≈ 50meV [103] for the pnictides (bandwidth around 4eV), to be compared

with U ≈ 10eV in cuprates. Other information which can be gleamed from optical

measurements are carrier concentration, which in pnictides are measured to be a few

1021/cm3, and estimated (in plane) scattering rates, at 300K 400-900 cm−1, at 10K tens

of cm−1.

Optical measurements[105] in the undoped 122s show a Drude peak, implying

the presence of itinerant carriers even in the spin density wave state, in contrast to the

insulating cuprate antiferromagnetism. It also contrasts with some measurements of the

11 compounds, with no observed Drude peak in the FeTe system, which tends to be

more correlated than the other pnictides. The Stoner factor is peaked in the vicinity of

(π,π), consistent with the picture that near nesting drives the formation of this ground

state, but it’s hard to rule out whether the carriers participating in conduction are the

same as the part of the material responsible for the spin structure. Starting at low

doping, in the SDW state, the Drude weight steadily increases with doping. [104].

64

A B

Figure 2-23. (left) Optical spectroscopy in undoped BaFe2As2, i.e. in the spin densitywave state. (right) Evolution of the Optical spectrum in Ba(Fe1−xCox )2As2with doping. Reproduced with permission from [104]. APPS c⃝2010.

2.2.2 ARPES

Angle resolved photoemission in the pnictides has observed both the Fermi

surface and the gap. It is important to understand how surface effects might manifest

themselves, however. In particular, since the perpendicular component of the photon

momentum is not usually resolved, the data will reflect one slice in the k⊥ plane. There

is a small but important change in the band structure at the surface according to

density functional calculations [106], and in the superconducting state, the surface

scattering could be playing an important role in averaging out finer gap structures. The

success of ARPES here is the overall confirmation of the correspondence between the

calculated and observed Fermi surface shapes [107]: two hole pockets around the

point k = (0, 0) and two electron and two electron sheets by the M point k = (π,π). This

two dimensional topology seems to be common in all the pnictides as shown in Figure

2-24. The third direction could have significant variation among different pnictides. The

pnictides are compensated semi-metals, meaning that the hole pocket and electron

65

pocket dip through the Fermi energy in such a way that the filling of the electron pocket

spills into the hole pocket [98]. The multiband semimetal nature of the pnictides can

be used to explain the unusual normal state susceptibility [108], which increases with

temperature (except in the 11-pnictides which increase or decrease) contrary to one

band Fermi liquid expectations.

A B

Figure 2-24. (A) Calculated (DFT) Fermi Surface of Barium-122 at 10% Co-doping.(reproduced with permission from the author) (B) ARPES NdFeAsO1−xFx .Reproduced with permission from [107]

The consensus from ARPES on the nature of the superconducting gap is that the

superconductivity is fully gapped in the ferropnictides, allowing for only a weak variation

around the Fermi surface [109]. A representative scan is shown in Figure 2-25. Small

variations in the gap are suggested from the data, but the error bars are comparable

to the variation. This claimed behavior stands in sharp contrast to the low temperature

power laws seen in bulk probes which suggest gap nodes, so the surface sensitivity of

angle resolved photo emission is possibly important in the interpretation.

66

Figure 2-25. Variation of the superconducting gap in NdFeAsO0.9F0.1 around the holepocket. Reproduced with permission from [107]

2.2.3 NMR

Cooper pairs are created by binding two spin 12

electrons. In the absence of

spin-orbit coupling, the total spin remains a good quantum number, so we can make a

clean distinction between a spin 0 and spin 1 Cooper pair. The Knight shift is the first

indication of the spin of a Cooper pair, as discussed previously. The Knight shift goes

to zero as T → 0 in a spin singlet superconductor because an applied field cannot

polarize the condensate. A nonzero Knight shift at T=0 is more difficult to interpret

because of the effects of possible spin orbit coupling and the relative orientation of the

components and the applied field. Both 31P and 75As NMR extrapolated to T=0 indicate

singlet Cooper pairs in the pnictides [110, 111].

Nuclear magnetic resonance in the superconducting state of the iron-arsenides

observed a T 3 spin lattice relaxation rate [111–113], which is consistent with nodal

lines in the gap. For a fully gapped density of states, one would expect exponential low

temperature behavior no matter how many bands exist. The next issue is how impurities

modify those notions. If disorder is pair-breaking in an s± gap, meaning the phase of

the superconducting gap on two Fermi sheets has opposite sign, then a low energy

67

impurity band could contribute to some of these probes. In nodal superconductors the

low energy impurity states will also modify power laws. The strongest conclusion we can

draw is the possibility of line nodes in the superconducting gap.

A B

Figure 2-26. Clear evidence for T3 spin-lattice relaxation rate in the pnictideLa(O1−xFx )FeAs. Reproduced with permission from [112]. (B) Knight Shift,57Fe, in the same compound providing evidence for the singlet nature ofCooper pairs. Reproduced with permission from [111]. APS c⃝2008.

The q-averaged static susceptibility,∑

q χ(q, 0), which occurs in 1T1T

, is equivalent

to the local susceptibility in real space. In Nakai et al. [112], there is no divergence

with magnetic ordering as you would expect in a second order phase transition. These

authors fit the spin dynamics to phenomenological forms deriving from self-consistent

spin fluctuation theory. The conclusions of Nakai et al. support other measurements

in saying that we have an itinerant system which has a spin density wave instability,

in contrast to the antiferromagnetic Mott insulator in the cuprates. This is one of the

68

major differences between the cuprate and pnictide families. One of the other important

contrasts is the wide variety of behavior in the pnictides. In LaFeOAs, the picture of

itineracy is supported by data, but in the 11’s, there seems to be a spin-glass region,

and a local moment description of the magnetism could be appropriate for low doping.

This is also consistent with the notion that the 11’s are more correlated than the 1111’s.

2.2.4 Neutrons

The normal state neutron data in the magnetically ordered state for low frequency

can always be fit by spin-wave analysis [98], since that is the appropriate long

wavelength description of those excitations, so there are primarily several features

to examine from neutron data. The measured ordered moments are 0.35µB in the

1111s and 0.87µB in the 122s [114]. One of the original proposals for a model of

the magnetism in these materials was a J1-J2 Heisenberg model [115], under the

assumption of a new strongly correlated material before experiment had weighed in

on the debate. The main problems with this model are that the system has itinerant

carriers and that the required size of the magnetic moment was too large compared to

experiment, so it was argued that correlation effects reduce the moment [103]. Another

issue is that spin wave damping is observed in neutron scattering, so one can either add

some inelastic process to damp the spin waves, like a spin-phonon coupling, or decay

into the Stoner continuum should be favored.

Zhao et al. [116] reported neutron data and a fit using a three-dimensional

Heisenberg model for the spin structure in CaFe2As2. The magnetic moment tends to be

right around half a Bohr magneton in the 1111s, and a little larger in the 122s. These fits

are shown in Figure 2-27. No global fit using spin wave theory can be obtained [117]. An

itinerant picture is favored by Diallo et al. [118], who use the same model to fit their very

similar neutron data, but report substantial damping due to a Stoner continuum. This

picture is also supported by non-zero Drude weight in the optical data, density functional

calculations [119], and Stoner-enhancement of the susceptibility in the region of the

69

Figure 2-27. Fits to inelastic Neutron data for two cuts through the Brillouin (a,b) zoneand the integrated intensity in part (c). [116] c⃝Nature Publishing Group2009.

70

antiferromagnetic ordering vector. A mixed interpretation, neither completely localized

nor completely itinerant, as in Ke et al. [120] for example, seems to be the way out of

these contradictory results.

On entering the superconducting state there is a spin resonance, only below

Tc , believed to exist since scattering connects two regions of the Fermi surface have

superconducting gaps of opposite sign. It is thought to be an effect from the coherence

factors [121], and is observed in d-wave superconductors as well. The energy of the

resonance is of order 10 meV, although the wavevector where it occurs seems to

depend on the material, see Johnston [98] for a chart. These observations lend support

to a sign change in the gap across the Fermi surface, which is beyond a simple s-wave

picture.

2.2.5 Penetration Depth

Penetration depth measurements have been fit both to exponential T -dependence

[122] and low-T power laws [123]. It is possible that these differences reflect genuinely

different ground states in different materials, so we should take care to keep track of the

doping and material on which the measurements were taken. Another lesson from the

cuprates in this context is to wait until relatively clean single crystal measurements can

be taken. Fortunately the field is already at this point and reliable data appears to be

available.

Again, in the simple one band picture, the linear result is the same as a d-wave

order parameter, from which we infer line nodes in the superconducting gap, but this

ignores the multiband nature of the pnictides. Systematic studies of the penetration

depth in the ferropnictides versus doping[124–126] reveals the general trend that

at optimal doping these materials seem to lack nodes (which is not to say the gaps

are isotropic). As we overdope, the power law behavior indicative of very small or

nodal superconducting gaps shows up, with Tn having n between about 1 and 3 In the

underdoped spin density wave coexistence region results are less clear. An interesting

71

A PrFeAsO B Ba(Fe1−xNix )2As2

Figure 2-28. (left) PrFeAsO exhibiting exponential fully gapped behavior. Reproducedwith permission from [122] (right) Linear temperature dependence of thec-axis penetration depth in Ba(Fe1−xNix )2As2. Reproduced with permissionfrom [123]. APS c⃝2009, 2010.

feature of some early data, was a low T upturn in the change in penetration depth. This

is ascribed to the existent of local moments in the heavy metal, for example Nd in the

1111 [127].

2.2.6 Heat Transport

Heat transport measurements in the superconducting states with nodal gaps,

should yield a linear term in the T → 0 limit of the thermal conductivity. This is generally

taken to be a strong indication for nodes. In d-wave superconductors this linear term is

universal, in the sense that its magnitude, to leading order, is independent of disorder

[72, 73]. The universal thermal conductivity won’t be present if there is competing

order, if the scattering rate is momentum dependent, or for an anisotropic s-wave gap

[128]. For a-b plane κT

there is a negligible residual linear term for all dopings [129] in

Ba(Fe1−xCox )2As2, evidence that there are no nodes in the gap.

However Figure 2-29 shows c-axis thermal conductivity in Ba(Fe1−xCox )2As2[130].

The presence of zero energy quasiparticles is indicative of nodes in the superconducting

gap. This is one of the key indicators for three dimensionality, because this effect is not

72

Figure 2-29. Heat transport along the c-axis in Ba(Fe1−xCox)2As2, which shows anonzero thermal conductivity at T=0 from the extrapolation. This indicatesthe presence of zero energy quasiparticles. Reproduced with permissionfrom [130]. APS c⃝2010.

observed by the same group in the a-b plane transport measurements. Untangling the

effects of doping on the Fermi surface and the evolving superconducting gap will be

challenging, but necessary to come to quantitative grips on this data.

We can put this in perspective with the seemingly conflicting penetration depth

studies. It is quite easy to get lost when trying to track which compounds for which

dopings do or do not indicate the presence of low energy power-laws. One reason for

the differences is that even within a single compound, the structures seem to be doping

dependent. The morphological changes in both the Fermi surface and superconducting

gap result in a variety of conclusions regarding gap structure. From that perspective,

all the data seems to make sense: some unconventional, possibly nodal sign changing

extended s-wave state which is sensitive to small changes in electronic structure

[106, 131].

73

2.3 Theoretical Suggestions

Theory has the irksome task of trying to create a framework which simultaneously

explains all of the experiments. It can be subdivided by phase and material but the

task is still daunting. We focus on the superconductivity. Early density functional

calculations found that the electron phonon coupling was too weak in the pnictides

to explain the Tc ’s with Eliashberg theory, so alternatives –especially electronic

mechanisms– were proposed. The magnetism can be understood, in particular the

neutron data, from an itinerant picture, and in cuprates, pnictides, heavy Fermions,

and organic superconductors magnetic phases abound, so spin fluctuations are a

natural candidate. Zero temperature changes in the ground state with pressure, field, or

doping also suggest looking at quantum critical scenarios. The low superfluid stiffness

and ’cheap vortices’ in cuprates, along with diamagnetism above Tc [132], suggest

a phase-disordered but paired material. There are yet more proposals. Any person

assessing the field must become acquainted with each of these approaches and their

flaws if a legitimate attempt at understanding is to be made.

2.3.1 Spin Fluctuations

The term ”spin fluctuations,” refers to an effective interaction between two electrons

mediated by fluctuating spin polarization of the electronic medium. An immediate

difficulty is that the objects which are polarizing, the electrons, are interacting with

themselves. The electron phonon interaction has two separate scales on hand because

the vibrating ions are much heavier than the electrons. No such clear separation of

scale occurs here at the level of the bare Hamiltonian, so it is difficult to understand what

to do when the objects being paired mediate their own pairing. Diagrammatically, the

approach here is to start with perturbation theory in the low U limit of the Hubbard model

[133], shown for the singlet channel in Figure 2-30.

↑↓ =U

1− (Uχ0(k − k ′,ω))2+

U2χ0(k + k ′,ω)

1− (Uχ0(k + k ′,ω))2

74

Figure 2-30. Diagrams contributing to the singlet channel of spin fluctuations effectivepairing interaction due to Uni↑ni↓

Summing all the diagrams gives rise to an interaction of the form Vkk ′ τ .τ → −32Vkk ′ ∼

32

U2χ01−(Uχ0)

in the singlet channel. For reference the omitted diagrams contributing to the

triplet channel sum to Vt =−U2χ0

1−(Uχ0)2. For a gap to be consistent with a given interaction it

must satisfy the equation:

k = −∑p

Vsf (k , p)p

2Eptanh(

Ep

2T)

The minus sign in the gap equation is crucial. If the interaction is repulsive (> 0), then

the only way to solve the gap equation is to have the gap at momentum p, opposite in

sign to the gap at momentum k. In the d-wave case, where for cuprates the nesting

vector at which the susceptibility peaks is (π,π), k = 0(cos(kx) − cos(ky)) satisfies

k = −k+(π,π).

2.3.2 Resonating Valence Bonds and Slave-Bosons

The undoped cuprates are Mott insulators. The hole doped region of the phase

diagram is a doped Mott insulator, and there is still no complete theory for this kind of

material. An idea which has been around since the discovery of the high-Tc compounds

is that there is an intermediate phase between the antiferromagnet and Fermi liquid,

where doping destroys the antiferromagnetic order and randomized singlets decorate

the lattice. Experimentally it is clear that the Neel state is destroyed with sufficient

75

doping, even if it’s more robust on the electron doped side. P.W. Anderson proposed this

resonating valence bond spin liquid to simultaneously accommodate the hole kinetic

energy and exchange energy. Triplet excitations out of this state still have a spin gap,

but in plane the holes remain gapless for charge transport. To be more precise, the

ground state is a phase coherent superposition of all singlet configurations on a given

lattice. The nomenclature comes from the Kelkule structure for benzene where the

double-bond resonates between two equivalent patterns. In the superconducting state,

it is said that these singlets condense into the superconducting pairs. The pseudogap

temperature scale is associated with the condensation of singlets. The superconducting

carriers become phase coherent at a lower temperature, forming the superconducting

dome. If we want to compare this critically to experiments, then we would take issue

with experiments which show the pseudogap scale cutting the dome. Either singlets

form before the superconducting pairs or they could form simultaneously, but it is

difficult to imagine superconducting pairs forming that are not singlets. Any region of the

superconducting dome extending beyond the pseudogap line is not consistent with this

simple idea.

There is no known Hamiltonian in dimension 2 or higher to date for which the

RVB state is a solution. The RVB proposal is a variational ansatz. As such it is often

investigated numerically [134] (variational Monte Carlo), and these calculations have not

been extended to finite temperature, according to Edegger et al. [135], representing an

important direction for future work. In lieu of referencing the communities’ work for the

more than thirty years, which seems a hopeless task, we refer the interested reader to

the reviews by Edegger et al. [135] and Lee et al. [59]. To implement the RVB idea, one

needs to propose a Hamiltonian, usually the t-J or a version of the Hubbard model, then

the variational wave function

|RVB >= PNPG |BCS >

76

is used. Where PNPG are the Gutzwiller and particle number operators. PG projects out

double occupancy and PN fixes N, since this is not defined for the BCS wavefunction,

|BCS >=∏

k(uk + vkc†k↑c

†−k↓)|0 >. The numerical details of such are procedure are in the

aforementioned citations.

Early on it was realized how to formulate a slave-boson version of the theory [136,

137], which has been reviewed in Lee et al. [59], resulting in a gauge theory of strongly

correlated electron systems. This approach usually goes under the name renormalized

mean field theory. If the slave boson method [138, 139] is used, there is a formulation

based on a fermionic representation of the spins and a bosonic representation of the

holes as well as an opposite formulation with bosonic spinons, further more is it possible

to attach flux to these operators.

c†iσ = f

†iσbi or f

†i biσ

The language associated with this transformation is that electrons fractionalized into

a holon (b) and spinon (f), like spin charge separation in one dimensional physics.

The connection between the RVB proposal and this approach is that the slave-boson

transformation enforces the no-double-occupancy constraint for the t-J model, provides

a singlet order parameter, and that the variational RVB wavefunction was found to be a

self consistent solution to the mean field equations. The constraint of single occupancy

introduces an effective gauge field by requiring a constraint via a Lagrange multiplier.

∑σ

f†iσfiσ + bibi = 1.

In one approach to the construction of a slave-boson Hamiltonian we arrive at:

H = −∑ij

(χij f†iσfjσ + c .c .)− µ+

∑i

f†iσfiσ +

∑ij

[ij(f†i↑f

†j↓ − f

†i↓f

†j↑) + c .c .]

An unresolved issue is that these theories predict the superconducting dome starting at

zero doping. An open question is how to destabilize the antiferromagnetic phase, make

77

a transition to the spin gap region, then begin the superconducting dome since at low

enough T the singlets would always form Cooper pairs.

Figure 2-31. Phase Diagrams according to the RVB, see [59](left) and QCP (right), see[79], proposals.

2.3.3 Quantum Criticality

Quantum criticality is a major theme in a variety of condensed matter physics

problems [79]. The term refers to a phase transition, at T=0 with finite T consequences,

which occurs as an external parameter other than temperature is tuned. Typical

examples of the control parameter in practice are disorder, pressure, and magnetic

field. The quantum critical point in the strictest sense refers to transitions at T=0, though

we expect effects at finite temperature from critical fluctuations due to this zero T critical

point. The discussion of quantum criticality dates back to the seminal paper by J.Hertz

[140], which led to a paradigm that goes by the name Hertz-Moriya-Millis theory [141].

This approach to magnetic quantum criticality constructs a Landau-Ginzburg-Wilson

action for the spin sector of an on site Hubbard interaction, integrates out the fermions

with the Hubbard-Stratonovich identity, and studies the critical behavior using standard

renormalization group techniques. We have learned in the intervening years, that

there are a number of subtle points in this analysis: it is not always safe to integrate

out the fermions or any other slow modes [142] because nonanalytic corrections to

the free energy occur at higher orders invalidating this approach. Only a posteriori

can the validity of this approach be examined. This leads naturally to considering the

renormalization group for both the order parameter and the fermions simultaneously

78

[143]. Another consideration is the effect of Berry’s phases in the action, which can

have an important impact [144] on the quantum to classical mapping. Yet another

concept important in classifying the existing approaches to quantum criticality is that of

local quantum criticality [145], which was studied in the Kondo-lattice model of heavy

fermions. At a local quantum critical point, we have a new local degree of freedom, i.e.

local moments, which are critically correlated in time but not space. Many experiments

observe ωT

scaling in the uniform susceptibility, whose proposed explanation is local

criticality. It is a question specific to a material, whether the local moments or the long

range order dominate the physics, but in the case that the local moments become critical

before the magnetic order parameter does. Different language to describe the same

idea is to ask when the f electrons delocalize with respect to when they could order. The

RKKY interaction can cause them to order when they are localized, but below the Kondo

temperature these moments are screened, so it becomes a question of which effect wins

out, ordering before screening or screening therefore no order.

Quantum criticality arises in the cuprates because of the nature of the pseudogap

and the non-superconducting state, as we have seen in Kerr, neutron, and transport

measurements. The controversy is that from one point of view, the pseudogap’s onset

temperature as a function of doping does not intersect the superconducting dome, and

from another point of view it does. In the later case, the funnel shaped region of this

putative quantum critical point describes the quantum critical state. In the quantum

critical scenario the Fermi liquid is the high doping low T phase, and the pseudogap

represents some kind of as of yet unidentified order, and the strange metal is a quantum

critical metal.

Characterizing something as a Fermi liquid is the statement, justified perturbatively

and recently with renormalization group approaches, that there is adiabatic continuity

between the noninteracting electron gas and one with interactions. The restriction

on the phase space of interacting electrons means that they behave as an electron

79

gas with renormalized parameters. As we approach the Fermi surface, assuming it

exists, the excitations in a Fermi liquid (the quasiparticles) are infinitely long lived.

A number of phenomenological consequences are associated with Fermi liquids.

The resistivity should scale at T2, linear electronic specific heat, the susceptibility is

temperature independent (to a good approximation), and certain famous ratios are

constant: Korringa, Wilson, Wiedemann-Franz. In heavy fermion, cuprate, pnictide, and

organic materials there are scores of observations which do not follow the Fermi liquid

paradigm.

2.3.4 Competing Order

An explanation of the pseudogap is an open question in the physics of high-temperature

superconductivity. Among many scenarios, one proposal is that the pseudogap is due

to a hidden broken symmetry. In the pseudogap phase, the density of states is depleted

at low energies, as if some of the degrees of freedom of the system were developing a

gap. To explain this, some have proposed an unconventional density wave [146] in the

particle-hole channel. To avoid a second order phase transition at the pseudogap onset

temperature T*, the state cannot break any continuous symmetries, only discrete ones,

or we would anticipate a Goldstone mode which has yet to appear in experiment. Such a

description would posit a Hamiltonian of the form

H =∑kσ

ξkσc†kσckσ +

∑k,σ

iW (k)c†kσck+Qσ +∑k

kc†k↑c

†−k↓

where Q is the nesting vector. For a conventional density wave, W ∼ constant, but we

could have unconventional states like a D-density wave with W (k) = cos(kx) − cos(ky).

No change in density occurs if < Wk >= 0, so it may be difficult to detect a transition.

The thermodynamics would resemble that of an unconventional superconductor

because of the appearance of a gap in the spectrum enters many calculations in

roughly the same way. The state is able to violate time reversal symmetry, would

possess gapless low energy quasiparticles, and has many interesting magneto-transport

80

properties [147]. The existence of spontaneously broken time reversal symmetry

has been claimed in Kerr rotation and neutron experiments, alongside the theoretical

proposal that a hidden order causes these effects. In particular, another explanation

of the Nernst data could be due to such a density wave [148, 149], but see also [82].

Coexisting charge order can reconcile the underdoped thermal current’s departure

from universality [150], although an explanation involving magnetic correlations around

impurities is also possible [151]. Firmly understanding what the exact consequences of

different varieties of unconventional density wave order are is still an open question.

Another kind of spontaneous order is a tendency for doped Mott insulators to phase

separate into mesoscale structures. Inhomogeneities in manganites are common, and

especially around 18

th doping for a few cuprates, X-ray and neutron data reveals stripe

order in cuprates. How this connects to superconductivity is a difficult open question.

It might be a competing phase coincidentally in the middle of the phase diagram and

special to a few compounds or it might reveal universal aspects of high Tc physics.

It should be noted that when stripes occur, Tc is suppressed, implying a competition

between the two phases.

2.3.5 BEC-BCS crossover

Superconductivity can loosely be thought of as two distinct phenomena. First,

we form Cooper pairs and the pairs later Bose-condense. Since pairing does not

imply a supercurrent we could form a gap without seeing a supercurrent. When long

range phase coherence occurs, there is also a supercurrent. A rule of thumb to gauge

when the Cooper pairs would condense is when the interparticle spacing becomes

is comparable to the coherence length. For ordinary s-wave superconductors, the

coherence lengths extend over hundreds of angstroms (1600A in Al), whereas in high-Tc

materials they tend to be of order tens-of angstroms (1.4 nm in YBCO). This means

that the wavefunctions in s-wave compounds tend to already overlap and there is no

difference between the BEC and pairing temperatures, whereas in high-Tc materials

81

the pairs could form then wait until low T to undergo BEC. Cuprates are also more two

dimensional, where fluctuation effects are more significant than higher dimensions.

In this model the pseudogap phase is a phase with partially paired electrons with no

long range phase coherence, thus no supercurrent. The experiment by Lu Li et al.

[152] used torque magnetometry to measure diamagnetism above Tc in LSCO, YBCO,

and BSCCO which support this picture. The onset of diamagnetism is compatible with

Nernst measurements.

A quantitative implementation of the BEC-BCS crossover [153] comes from

identifying the gap equation with the scattering length in traditional BEC problems.

Starting from a model Hamiltonian,

H = ψ(−∇2

2m− µ)ψ − g ψ↑ ψ↓ψ↓ψ↑,

a Hubbard-Stratanovich transformation can bring the action to the form

S =1

g

∫ddx

||2 − Trln

(−∂t + ∇2

2m+ µ)

(−∂t − ∇2

2m− µ)

.

Then the gap equation is related to both the coupling constant g and the scattering

rate since we understand how to form both a gap equation and a relation between the

scattering length a and the coupling g.

=1

2Tr(Gτ1),

m

4πa= −1

g+∑k

1

2εk,

results in

− m

4πa=∑k

1− 2f (Ek)

2Ek− 1

2εk.

This relates scattering and interaction effects to the pairing and is the heart of the

BEC-BCS crossover idea.

82

CHAPTER 3ANGLE-DEPENDENT THERMODYNAMICS

3.1 Density of States and Low Temperature Specific Heat in the Vortex State

The low temperature phenomenology of many probes into the nature of the

superconducting gap distinguish between fully gapped and nodal superconductors,

but do not provide phase sensitive information about the location of nodes. Furthermore,

exponential behavior in temperature can masquerade as T 4 scaling over a finite

range, and the effect of impurities can further obscure the result. It is therefore of

interest when confronted with a new superconducting material to have more specific

probes. In an ideal world, experiments sensitive to the phase of the order parameter

could be performed providing some of the sought-after information, but in practice

these experiments can be quite challenging, and precise control of the crystal growth

and interface quality is necessary before they can be undertaken. This is why it took

the better part of a decade to come to a consensus about the order parameter in

the cuprates. This chapter discusses the compromise between these two extremes:

a simple thermodynamic bulk probe which is not phase sensitive, but provides

more detailed information about the location and presence of nodes than transport

experiments. A measurement of the specific heat in the presence of a magnetic field,

H, which is rotated with respect to the crystal axes of the sample [154], maps out

oscillations in the density of states relative to the nodal positions.

Above Hc1, in an s-wave superconductor, the low-temperature entropy is dominated

by the vortex core bound states. The level spacing is of order 20

EF, which is typically

small for s-wave superconductors. In superconductors with line nodes, the vortex

core contribution to the low-energy density of states is smaller than that from the

quasiparticles outside the core region. In systems with short coherence length, such as

cuprates and heavy fermion materials, the extended quasiparticle states also dominate

83

the entropy since the level spacing in the vortex core is large, and only few such levels (if

any) are occupied at low temperature.

The quasiparticles outside the core move with the superflow around each vortex, so

the effect of an applied magnetic field can be accounted for semiclassically by moving

into this reference frame, that is, Doppler-shifting the quasiparticle spectrum according

to the local value of the superfluid velocity, vs (r). This approximation is valid when

H ≪ Hc2, as long as the vortices are spaced far enough apart to consider essentially

one vortex cell, neglecting vortex-interactions, and vs (r) varies slowly over most of the

vortex cell on the scale of the superconducting coherence length. This approximation

has been scrutinized by Dahm et al. [155], who performed a careful comparison of

several common approximations for s-wave and d-wave superconductors in the vortex

state, so we can rely on quantitative certainty that for this situation, we capture all

the qualitative physics. Quantum effects [156, 157] which intervene at extremely low

temperatures will not be relevant here.

The single-particle Green’s function in the presence of a superflow velocity

field vs (r) is obtained by Doppler shifting the quasiparticle states with energy ω and

momentum (with units such that ~ = kB = 1),

G(k , vs (r),ωn) = −(iωn − vs (r) · k)τ0 +kτ1 + ξkτ3(iωn − vs (r) · k)2 − ξ2k − 2

k

(3–1)

where ωn is the fermionic Matsubara frequency, ξk is the band energy measured with

respect to the Fermi level, τi are Pauli matrices in particle-hole space. The order

parameter k will depend on the model we consider, for example 0 cos 2ϕ for d-wave or

0 sin(ϕ) for a p-wave order parameter. The Doppler shift is given by

vs(r) · k =~kF2mr

sin(ψ) sin(θ − ϕ) (3–2)

where ψ is the winding angle of the superfluid velocity in real space, ϕ the azimuthal

angle on the Fermi surface, θ is the angle between H and the a-axis, and r is the

84

rescaled distance from the center of the vortex. Note that we explicitly exclude from

consideration quasi-1D Fermi surfaces. We will furthermore restrict our consideration to

anisotropic materials, and consider essentially two dimensional planes.

Since the BCS quasiparticles are noninteracting, the density of states can be used

to compute the entropy according to the standard recipe,

S = −2∫ ∞

−∞dωN(ω)[(1− f (ω)) log(1− f (ω)) + f (ω) log f (ω)] (3–3)

where f (ω) = (exp(ω/T ) + 1)−1 is the Fermi function. To obtain the the heat

capacity at constant volume we differentiate at constant volume with respect to T,

CV (T ,H) = T

(∂S

∂T

)V

(3–4)

At low T and H, when the gap varies weakly with temperature, the temperature

the derivative acts only on the Fermi function in Eq.(3–3), and the specific heat is

approximately given by the form

CV (T ,H) ≈ 1

2

∫ ∞

−∞dωN(ω,H)

ω2

T 2sech2

ω

2T. (3–5)

which can serve as an asymptotic low-T check to the full temperature dependent result.

The local density of states is

N(ω, r) = − 1

2πIm∑k

TrG(k ,ω − vs(r) · k)

≃ N0Re

∫ 2π

0

dϕ

2π

|ω − vs(r) · kF |√(ω − vs(r) · kF )2 − |(ϕ)|2

Here N0 is the normal state density of states. The net DOS per volume is found by

spatially averaging N(ω, r) over a unit cell of the vortex lattice, which we crudely

approximate with a circle, containing one flux quantum, 0. To account for anisotropy

and three-dimensional materials, we rescale the c-axis the unit cell area by a factor

λc/λab, and flux quantization dictates that the quasiparticles now experience the

85

effective field H⋆ = (λab/λc)H. In the new coordinates the radius and the area of such a

cell are RH =√0/πH⋆ and AH = πR2

H respectively. Introducing non-dimensionalized

polar coordinates, r = RH(ρ cosψ, ρ sinψ), we find for the average field-dependent

density of states

N(ω,H,T ) =1

π

∫ 1

0

ρdρ

∫ 2π

0

dψN(ω, r) (3–6)

There are two factors which cause an angular variation: the Fermi surface

anisotropy, and the angle between the superflow and the quasiparticle momentum,

on which the superconducting gap k depends. It is this angular dependence which

is exploited to distinguish between different superconducting gaps. In the absence of

impurity scattering, there a few important energy energy scales in the problem: the

temperature T, the gap magnitude 0, and the magnetic energy or typical Doppler shift

EH = vFRH

, which we can exploit to help interpret results.

When the pnictides first achieved prominence around 2008, there were early

indications of unconventional superconductivity, as discussed in Chapter 2 of this thesis.

Since the lowest temperature specific heat variations in low field are proportional to a

product of the density of states with a thermal factor, a rapid calculation of the angular

variation of the density of states at zero energy would serve as a useful indication that

the gap varied around the Fermi surface, and it might identify the location of nodes.

The main contribution to the density of states comes from the nodal quasiparticles, so

it is a good approximation [158] for T ,EH ≪ 0 to sum over the momenta at the nodes

alone (away from nodes quasiparticle energies are fully gapped). The results of this

calculation [160] are shown in figure 3-1. The hope was that it would be possible to

perform a simple bulk probe looking for the distribution of gap nodes.

Once experiments [159] demonstrated a variation of the specific heat there was

a strong indicator for a nodal gap, in the 11-pnictide used for the experiment. The

evidence for a nodal gap was first that there is a strong angular variation of the

specific heat in field, and second that the angular variation was observed to invert

86

A B

Figure 3-1. Predicted field dependence of the density of states[158] (left) for a selectionof unconventional gaps in the 11 iron-pnictides[159].

as the temperature and field were varied, which is a specifically attributed to nodal

superconducting behavior.

A second effect due to applied magnetic field can also be used to distinguish

between nodal and fully gapped order parameters. If we have one flux quantum per

vortex then if the vortex unit cell is taken to be a circle,

πR2 =hc2e

H

87

Since most states lie outside the vortex core, if we include the presence of a magnetic

field by Doppler-shifting ω → ω − vs(r).k , and recalculate the density of states, then at

zero frequency:

N(E → 0,H) ∝ 1

Area

∫d2r ImG =

1

Area<

|0− vs(r).k |

>

which for a circular vortex cell, from power counting, can be seen to vary as:

1

πR2

∫ R

ξ

drr~kf2mr

∝ 1

πR2R,

resulting in Volovik scaling [161]:

N(0,H)T ∼ T√H.

In a full gapped superconductor, we perform the same counting as above only we need

to remove the low energy density of states if we are below the gap-scale, i.e. there is

zero density of states below 0 so the field dependence is the contribution from the

vortex core, and is linear in H. In this way, scaling helps distinguish the essential physics.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2

Cv

/ CN

T/Tc

EH = 0EH = .4 Tc

low T asymptote

Figure 3-2. Electronic specific heat of a d-waave superconductor as a function oftemperature in zero and nonzero applied field. The Volovik effect manifestsitself in the offset at zero T.

88

The main manifestation of the Volovik effect is seen in measurement of the linear

term in the electronic specific heat, CV = γ(H)T . In a clean nodal superconductor in

zero field Cv may be shown to vary as T2. Due to the Volovik effect the zero temperature

value of Cv will have an offset due to the field, which can be seen in figure 3-2. For a

single band superconductor with a nodal gap, the linear coefficient will scale as√H. As

with the other experiments we have to ask how this clean behavior changes in the dirty

limit when disorder modifies these results. When the impurity bandwidth is comparable

to the energy associated with the Doppler shift, magnetic field effects are as important

as disorder. The influence of disorder is to change the density of states into [162]:

N(H,ω = 0)

N0

∼ 0

γimp

H

Hc

ln

(π

2

Hc

H

).

With these results in hand, we now turn to the interpretation of these experiments

with more care. The nodal approximation is relaxed, and the low temperature constraint

is relaxed. Since the experiments are performed at finite temperature, it is important to

understand the effect this has on the previous results. The predictions above for the low

temperature specific heat show in particular, that the specific heat has minima when the

magnetic field points in the direction of the nodes in the order parameter, and maxima

for the field along the antinodes, but this is not true in general.

3.2 Temperature Variation of the Specific Heat in an Applied Field

CeCoIn5 is a heavy fermion superconductor whose order parameter was only

identified recently. The initial measurements of the anisotropy of the specific heat [163]

and the thermal conductivity [164] appear to give results for the gap structure which are

rotated 45 with respect to one another, dxy and dx2−y2 respectively. In Sr2RuO4, which

is nominally a p-wave superconductor, the specific heat oscillations were observed to

invert [165] as the temperature and field was varied, that is the minima and maxima as a

function of angle changed places. The enigma is then how to understand the inversion

of oscillations and why the thermal conductivity and specific heat in CeCoIn5 gave

89

contradictory results. A resolution was provided by Vorontsov and Vekhter in [166]. To

understand this, we performed the full calculation of the temperature dependence of

the specific heat for a d-wave superconductor in an applied field in the semiclassical

Doppler-shift approximation.

To include the full temperature dependence, we need to also account for the

variation of the superconducting gap with temperature. To this end, we rely on a weak

coupling interpolation formula due to Einzel [167]:

(T ) = 0 cos(2ϕ) tanh

(πTc

0

√4

3

8

7ζ(3)(1− T

Tc

)

), (3–7)

where 0 = 2.14Tc is the gap maximum.

Π4

Π2

Θ

0.88

0.9

0.92

0.94

0.96

0.98

1

HCv HΘLLHCv H0LL

0.02Tc

0.04Tc

0.18Tc

A

Π4

Π2

Θ

0.98

0.99

1

1.01

HCv HΘLLHCv H0LL

0.02Tc

0.08Tc

B

Figure 3-3. (left) C(T ,H)/C(T , θ = 0) vs. θ for a set of equally spaced temperatures,every .02Tc , from 0 to .18Tc , Eh = .2Tc . (Right) same for a set of equallyspaced temperatures from 0 to .08 Tc every .02Tc , Eh = .05Tc .

We demonstrate that this inversion is a generic feature by calculating the angle

dependent specific heat in unconventional superconductors using both the low field

Doppler-shift [168] and the Brandt-Pesch-Tewordt approximation [169] appropriate

for fields near Hc2. There will always be a difference between nodal and antinodal

quasiparticles simply because it takes zero energy to excite a nodal quasiparticle, but

once the temperature scale is of order the largest gap, there are some quasiparticles

excited from all regions of the Fermi surface, and the difference becomes less

90

pronounced. As expected, the anisotropy in the angle-dependent density of states

is washed out at higher energies. However, the difference between N(ω,H,T ) for the

two directions of the field reappears at energies of order 0, the energy scale absent in

the versions of the calculation with linearized gap, as a result of linearization.

A

0 0.2 0.4 0.6 0.8 1 1.2T / Tc

0

0.2

0.4

0.6

0.8

1

H /

Hc2

C inversion (Doppler) C inversion (BPT)H

c2

0 0.1 0.2T / T

c

0

0.2

0.4

0.6

EH

/ ∆ 0

C n

> C an

B

Figure 3-4. (left) Results for the specific heat as a function of field, angle, andtemperature in the BPT approximation demonstrating the inversion ofoscillations. Reproduced with permission from [169]. APS c⃝2008. (right)Resulting all-T all-H phase diagram from the BPT and Doppler-shiftapproaches [168].

The Doppler shift approach is only valid in the low field regime when it is appropriate

to consider one vortex cell. As the effect of a higher applied field takes hold, it is

necessary to use a different formalism. Previous work using the Brandt, Pesch, Tewordt

[170] quasiclassical approach was completed by Vorontsov and Vekhter [171] and

complement our low H calculations. The result of their work is shown in figure 3-4. We

will not review this method in detail, only just give an overview of how it is set up.

The quasiclassical Green’s functions obey the differential equations [172],(ωn + i

vF

2· (∇

i+

2π

0

A(r))

)f (iωn, θ, r) = (θ, r)g(iωn, θ, r) (3–8)

(ωn − i

vF

2· (∇

i− 2π

0

A(r))

)f †(iωn, θ, r) = ∗(θ, r)g(iωn, θ, r) (3–9)

91

Which is solved self-consistently, together with the following conditions: the normalization,

g(iωn, θ, r) = (1− f (iωn, θ, r)f†(iωn, θ, r))

1/2 (3–10)

and the field and gap equations,

(θ, r) = N02πT∑ωn>0

∫ 2π

0

dθ′

2πV (θ′, θ)f (iωn, θ, r) (3–11)

∇×∇× a(r) = −2π2

κT∑ωn>0

∫ 2π

0

dθ′

2π

k

ig(iωn, θ, r) (3–12)

where a(r) is the internal field, A(r) = 12H × r + a(r). The Brandt, Pesch, Tewordt

approximation replaces g(iωn, θ, r) by its spatial average, and the functions , f (iωn, θ, r)

are expanded in an appropriate basis, resulting in a self-consistent set of equations.

This technique was employed to obtain the high field results, beyond the validity of the

Doppler-shift approximation.

Our main finding is that the sign of the oscillations of the specific heat as a function

of the field orientation depends on the temperature and field strength. We confirmed

that at low temperatures and fields the specific heat has a minimum when the field is

along a nodal direction. However, as H and T are increased, minima of the specific heat

begin to occur for the field along the gap maxima, i.e., an inversion of the oscillation

pattern occurs. The technique should prove useful, generally, in identifying possibly

unconventional superconductors as soon as single crystals become available.

The inversion of specific heat oscillations has been observed in [173] as well

as by H.H.Wen et al. [159]. In the paper on FeSe0.45Te0.55 [159], the 2.6-2.7K data

convincingly displays four-fold oscillations with a minimum which appears when the

field is along the M direction. In a one band model in the low T low H region, this

identifies the nodes as along the same direction. The analysis is more subtle here,

because of the four bands in the pnictides. It should also be kept in mind that Fermi

surface anisotropy can contribute to the variation. On this matter, it is the inversion

92

A B

Figure 3-5. Data demonstrating the inversion of specific heat oscillations with increasingtemperature in a 11-pnictide [159](left). The heavy fermion superconductorCeCoIn5 displays a similar inversion and oscillation pattern consistent with adx2−y2 superconducting gap. Reproduced with permission from [163]. APSc⃝2001.

of the specific heat oscillations with temperature which is a clear signature of a nodal

gap, which is not possible from simple Fermi surface anisotropy. We need to estimate

the field and temperature scales in the problem to see if these minima correspond to

antinodes. The in-plane magnetic field in these experiments is 9T. Hc2 is not given in the

paper. Measurements by Tsurkan et al. [174], find that Hc2 is 85T in the a-b plane for

the same compound at slightly different doping. This means we can estimate HHc2

to be

about a tenth, which is sufficient to be in the low field regime. Tc is 14.5K in the specific

93

heat measurement, so this is about 20% of Tc . According to our model phase diagram,

the minima here should correspond to maxima in the nodes because that is the inverted

temperature regime.

This analysis is also responsible for the resolution of the heavy fermion paradox

[166]. The interpretation is that CeCoIn5 has a dx2−y2 superconducting gap, and

that the apparent contradiction between the two experiments is simply that one of

the experiments was being performed in the inverted regime when nodal directions

correspond to maxima instead of minima. It was concluded that CeCoIn5 has a dx2−y2

superconducting gap.

94

CHAPTER 4RAMAN SCATTERING

The structure of the superconducting gap is responsible for a large part of the

observed phenomenology in transport and thermodynamic experiments, so primacy

is given to its identification. This chapter continues in the same vein as Chapter

3, in focusing on experiments which elucidate the gap structure, without being as

demanding as phase sensitive measurements so that they might be performed sooner

after the discovery of a new superconductor. As we saw in Chapter 2, nuclear magnetic

resonance showed a T 3 spin lattice relaxation rate, 1T1

, reminiscent of a gap with

nodes. Angle resolved photoemission measurements on single crystals of 122-type

materials reported nearly isotropic gaps around the Fermi surface. Penetration

depth measurements have been fit both to exponential T -dependence, indicative

of a fully gapped state, and low-T power laws, suggestive of nodes. It is possible

that these differences reflect genuinely different ground states as the specific crystal

and doping is varied. However, the complex interplay of multiband effects, disorder,

and unconventional pairing leaves open the possibility that a single mechanism for

superconductivity exists in all the Fe-pnictides, and that differences in measured

properties can be understood by accounting for disorder, band structure, and doping

changes. It is likely that a consensus will be reached only after careful measurements

are performed using various probes on the same material, for the same doping,

and same crystal quality (impurity scattering rate) –systematically mapping out the

parameter space for the pnictide family. Here we continue the theme of gap identification

and focus on the role of disorder specifically.

Raman scattering is the inelastic scattering of polarized light off of a solid. Inelastic

scattering is sensitive to the excitations in a system, but it’s the polarization dependence

of Raman scattering which offers a degree of freedom that can be exploited to the

experimentalist’s advantage. Just as we saw with the angle-dependent specific

95

heat, measurement of the electronic Raman scattering in the superconducting

state can provide important information on the structure of the order parameter

through its sensitivity to symmetry, through the polarization, and gap scales. In the

angle-dependent specific heat, it was the external magnetic field which provided an

extra degree of freedom for the investigation of the gap structure. Whether a given

Raman polarization weights a given gap strongly or weakly depends on the polarization

of the light via the Raman vertex γk. This is particularly important for superconductors

where the gap is strongly momentum dependent, and was exploited successfully in

cuprate materials to help determine the d-wave symmetry in those systems, whereas in

s-wave systems the response would look the same independent of polarization. Optical

probes, as we saw in s-wave superconductors, are sensitive to the pair breaking scale

20. It comes as no surprise, then, that the energy of the peaks in the Raman intensity

are directly related to the magnitude of the gaps on the various Fermi sheets in the

unconventional case also. In addition, the presence of nodes and the dimensionality of

the nodal manifold may be determined –though not always uniquely– by comparison

with low energy power laws in the Raman intensity in different polarization states. These

power laws can be altered by disorder, so systematic irradiation or doping studies can

compare the evolution of these power laws in comparison with theoretical predictions

to determine the overall consistency of a model. Doping is not the same as disorder.

Raman scattering with use of different polarizations may therefore be a useful method

of acquiring momentum-dependent information on the structure of the superconducting

order parameter. We discuss several cases below which should allow extraction of the

dominant gap symmetry, the gap magnitudes, and possible nodal structure of the order

parameter over the Fermi surface.

4.1 Theory

The intensity of scattered light is proportional to the imaginary part of the channel-dependent

Raman susceptibility [175], which resembles a density-density correlation function

96

weighted by a vertex,

χγ,γ(ω) =

∫ β

0

dτe−iωmτ ⟨Tτ ~ργ(τ)~ργ(0)⟩ |iωm→w+iδ . (4–1)

Off-resonance, i.e. away from collective modes or direct interband transitions, the vertex

is independent of frequency, and we can write the effective Raman charge fluctuations in

the polarization with vertex γ as

~ργ =∑k,σ

∑n,m

γn,m(k)c†n,σ(k)cm,σ(k), (4–2)

where n,m denote band indices. γn,m(k) defines the momentum- and polarization-dependent

Raman vertices. Generally, the vertex is determined by matrix elements between the

conduction band and the excited states through density and current operators which

may include both intraband and interband transitions. Even in simple metals, like Al, the

vertex is often unknown or difficult to calculate, so it is standard practice to make use of

certain approximations. The polarizations of the incoming and outgoing photons impose

an overall symmetry due to the way in which excitations are created in the directions

determined by the electric field, so it becomes possible to classify the Raman vertices

into basis functions of the irreducible point group of the crystal. We will use this basis

here. The intermediate states which contribute to the vertices are still too complicated

to be amenable to analytic work. Further progress comes from requiring that the

momentum transferred by the photon be small compared to the Fermi momentum, good

in almost all cases, and the frequency of light needs to be less than the band gap. The

intermediate states present in the general Raman vertex include both the states created

from the initial state as well as states separated from the conduction band. Thus, the

non-resonant approximation is valid if the energy of the incident and scattered photons

falls in this range. The intraband (n = m) contribution to the Raman amplitudes γn for

a clean system may be expressed in terms of the effective mass and the incident and

scattered polarization vectors esi

97

γn(k) =m

~2∑α,β

e iα∂2ϵn(k)

∂kα∂kβesβ. (4–3)

For n bands crossing the Fermi level, the intraband Raman response is given by

χ~ρ,~ρ(ω) =1

N

∑k

∑n

γn(k)2λn(k,ω), (4–4)

in the absence of charge backflow effects. Here

λn(k,ω) = tanh

(En(k)

2kBT

)4 | n(k) |2 /En(k)4E 2

n (k)− (~ω + iδ)2(4–5)

is the Tsuneto function for the nth band and the band dispersion is ϵn(k), energy gap

n(k), and quasiparticle energy En(k) =√ϵ2n(k) + 2

n(k).

Since Raman scattering probes charge fluctuations in the long-wavelength limit,

the role of the long-range Coulomb interaction is quite important. In particular it is

important to account for screening. The long-range Coulomb interaction can be taken

into account by including couplings of the Raman charge density ~ρ to the isotropic

density ρ fluctuations,

χscr~ρ,~ρ = χ~ρ,~ρ −χ~ρ,ρχρ,~ρ

χρ,ρ

, (4–6)

with

χ~ρ,ρ = χρ,~ρ =1

N

∑n

∑k

γn(k)λn(k), (4–7)

and

χρ,ρ =1

N

∑n

∑k

λn(k). (4–8)

Equations (4–4)-(4–8) constitute closed form expressions for the intraband, non-resonant

contribution to the Raman response.

It is clear from Eqs. 4–6-4–8 the Raman response is in general the sum of

each band separately, and therefore a non-linear feedback occurs. Corrections

enforce number-conservation of charge density fluctuations and gauge invariance,

in particular, the screening of the charge fluctuations have been included. Due to

98

the to the dependence on the direction of the polarization, incident photons can

create anisotropic charge fluctuations and those charge fluctuations relax by the

usual scattering mechanisms: electron-impurity, electron-phonon, electron-electron

interactions, or via breaking of Cooper pairs. These charge fluctuations transform

according to the elements of the irreducible point group of the crystal. For D4h tetragonal

symmetry, in-plane charge fluctuations transform according to the full symmetry of the

lattice A1g, meaning 90 degree rotations do not change the sign as do the B1g and B2g

representations, which change sign under 90 rotations. Raman scattering probes long

wavelength charge fluctuations, therefore the B1g and B2g charge densities that average

to zero within each unit cell are not screened via the long-range Coulomb interaction.

As a consequence, the Raman charge densities for these channels do not couple to the

isotropic charge density channel, and the terms in Eq. 4–7 vanish. Thus the B1g and B2g

Raman responses for a multi-band system consist simply of the sum of the bare-bubble

contributions from each band. In contrast, A1g fluctuations need not average to zero over

the unit cell, and therefore they can couple to isotropic charge fluctuations, giving the

corrections represented by the second term in Eq.4–6.

While the the bare bubble expression for the B1g and B2g channels are generally

accurate for the cuprates, the A1g contribution is significantly more complicated due to

the issues associated with the screening of long wavelength fluctuations. In systems

with several pairing instabilities which are nearly degenerate energetically, it is possible

to have strong excitonic peaks in the A1g polarization. These can be thought of as

transitions between two nearly degenerate superconducting gaps [176]. Spurious

exciton peaks can occur in Raman calculations using only the bare bubble because

gauge invariance is not preserved. By including the long range Coulomb effects it is this

requirement of gauge invariance which often lifts the low lying collective modes to higher

energy. A prime example is the Anderson-Bogoliubov mode, which is acoustic for charge

99

neutral superfluids like Helium, but is lifted to the plasma energy once vertex corrections

and long-range Coulomb interactions are included restoring gauge invariance.

In order to focus on general features for Raman scattering in the pnictides, we

will approximate the anisotropic Fermi surfaces from more realistic band structure

calculations and treat all Fermi surface sheets as circles. This allows for simple

symmetry-based as approximations to the Raman vertices and we can expand the

vertices as Fermi surface harmonics, and has been done in the cuprates [177]. This

has important consequences for the non-vanishing of the vertex. Expanding the

polarization-dependent vertices in Fermi surface harmonics for circular Fermi surfaces,

γn(θ)A1g = an + bn cos(4θ)

γn(θ)B1g = cn cos(2θ)

γn(θ)B2g = dn sin(2θ), (4–9)

with angle-independent band prefactors an, bn, cn, dn setting the overall strength of the

Raman amplitudes for band n.

Isotropic density fluctuation vanish for q → 0, so it can be shown via Eq.4–8 that

the A1g contribution an to the Raman vertex does not contribute to the scattering

cross-section. The lowest order non-vanishing contribution to the vertex in this

expansion is found to be cos(4θ).

Taking the imaginary part of Eq.4–6 we then obtain at T = 0

Imχ~ρ,~ρ(ω) =∑n

Imχn~ρ,~ρ(ω) =∑n

πNF ,n

ωRe

∫dθγ2n(θ)

| n(θ) |2√ω2 − 4 | n(θ) |2

. (4–10)

From the denominator in the Raman response in the clean limit we see that in the case

of an isotropic gap, k = , there should always be a peak at 2, when the energy

necessary to break pairs is supplied. In an unconventional superconductor, depending

100

on the polarization, this absorption peak is replaced by a peak or other structure at 20,

twice the extremum of the gap over the Fermi surface, nonetheless a feature remains at

this significant energy scale. The peak will be broadened by scattering, but still provides

a measure of the magnitude of the gaps in the system and may be compared to those

determined from other experiments, usually ARPES and tunneling.

The B1g and B2g vertices in Eq.4–9 have zeros in k-space and therefore weight

the part of the Brillouin zone away from these zeros. This gives rise to an asymmetry

between the two polarization channels, as observed in the d-wave cuprates, where

the sharp 2 peak occurring in the B1g channel only, and a less pronounced feature

corresponding to a change in slope occurs at the same scale in the B2g channel which

weights the nodal regions most strongly. Furthermore, the existence of nodes in a gap

creates low energy quasiparticles that cause a nonzero response for all frequencies.

This is in sharp contrast to fully gapped superconductors whose T= 0 response show

a sharp gap at ω = 20 edge with no low energy quasiparticles below that. Taking

advantage of this polarization-dependence can give information about the structure of

unconventional superconducting gaps.

4.2 Adding Impurities

The presence of impurities in an unconventional superconductor may generate a

finite density of quasiparticle excitations at zero temperature since it is not subject to

Anderson’s theorem. Since these impurity-induced quasiparticles are both generated

and scattered by impurities, it is important to include the effect of disorder in our

calculation. To include the effect of disorder we self-consistently solve for the self

energies by including all scatterings off a single impurity and performing a disorder

average. This neglects crossing diagrams, responsible for the physics associated

with weak localization and interference phenomena. The T-matrix is represented

diagrammatically in figure 4-1 . The T -matrix can be defined as

G = G0 + G0T G0 + · · · (4–11)

101

Figure 4-1. Diagrams representing the self-consistent T-matrix approximation (SCTMA).The dashed lines are scattering off an impurity, and the single line is theself-consistent Greens function.

The full matrix Greens function in the presence of scattering in the superconducting

state is

G(k ,ω) =~ωτ0 + ~ϵk τ3 + ~kτ1

~ω2 − ~ϵ2k − ~2k

,

where ~ω ≡ ω − 0, ~ϵk ≡ ϵk + 3, ~k ≡ k + 1, and the α are the components of the

disorder self-energy proportional to the Pauli matrices τα in particle-hole (Nambu) space.

Using the Nambu notation, the single-particle self energy in a superconductor can

be decomposed as:

(k,ω) =∑α

α(k,ω)τα (4–12)

where τα are the Pauli matrices and τ0 is the unit matrix. Note that the band index is

implicitly contained in the k index since we restrict pairing to individual Fermi surface

sheets. Treating impurity-scattering in T-matrix approximation gives rise to the following

self-energy

(k,ω) = niTkk(ω), (4–13)

where ni is the impurity concentration and Tkk(ω) is the diagonal element of the T-matrix

Tkk′(ω) = Vkk′ τ3 +∑k′′

Vkk′′ τ3G(k′′,ω)Tk′′k′(ω) (4–14)

We define

=nin

πN0

where ni is the density of impurities, n of electrons, and N0 the density of states at the

Fermi level. For a constant potential, the case which we will consider in this paper, this

102

expression becomes the series:

(ω) = niV0τ3∑n

(∑k′

G(k′,ω)V0τ3)n

Later we will restrict Vkk′′ to be constant for particular sets of momenta, either to allow

transitions between all the Fermi sheets or to restrict transitions to remain within Fermi

sheets.

The self-energy ~(k ,ω) has to be solved self-consistently in combination with the

single-particle Green’s function

G(k,ω)−1 = G0(k,ω)−1 − (ω). (4–15)

After solving for the self-energies, we insert them into the general expression for

the Raman response. We should note, in passing, the well-known issue that there is

no calculation of the disordered-averaged response function in the general case. Weak

localization corrections from the diffuson and cooperon have been studied by Altshuler,

Aronov, and coworkers [178], as well as these T-matrix corrections which neglect the

crossing diagrams. We are choosing to replace the single particle Green’s function by its

disorder-averaged version, instead of disorder-averaging over the full response function.

We neglect vertex corrections, which must in principle be included if such a procedure

is followed. In the simplest model of an electron gas with impurities, it is the vertex

correction to lowest order which gives the transport scattering time compared to the bare

lifetime. Vertex corrections, however, were shown to have a small effect on the Raman

response in the single band d-wave case [177]. For isotropic s-wave superconductors,

vertex corrections from s-wave scatterers vanish identically. We did not include them

here, because the effect is usually small and most noticeable near the gap edge [175].

Beginning with a spectral representation:

G(k, iωn) =

∫dx

(−1π

)ImG(k, x)

iωn − x

103

we arrive at a zero-temperature long-wavelength form,

Imχγ,γ() =∑k

∫ 0

−dx

1

πγ2kTr [ImG(k, x)τ3ImG(k, x +)τ3].

In terms of retarded and advanced Green’s functions, ImG = GR−GA

2i:

Imχγ,γ() =

1

4π⟨N(ϕ)Im

∫dξγ2ϕ

∫

0

dx [FRR(x−, x)−FRA(x−, x)−FAR(x−, x)+FAA(x−, x)]⟩ϕ

(4–16)

where

F a,b = Tr[G a(k, x −)τ3Gb(k, x)τ3] a, b = A,R

We have taken N(ϕ) = N0 in this paper. The angular brackets represent an average

over the angle ϕ around the Fermi surfaces. While for simple s-wave or even d-wave

gaps the real part of the self energy has a definite sign, this is not the case in general,

so care must be taken with the implicit branch cuts with the expression in Eq.4–16. The

difference between the zero temperature response and the finite temperature response

is to replace the hard cutoff at by one smoothed by finite T.

4.3 Model Superconducting Gaps for the Raman Response

We begin by considering model one-band clean systems with gaps inspired by

proposals for the Fe-pnictides to illustrate what intuition we can gain regarding the

Raman response for various polarizations. Figure 4-2 illustrates all the prototypical

behavior in Raman scattering spectra. Based on spin-fluctuation [179], functional

renormalization group [180], and other early calculations, an anisotropic s-wave gap

with sign changes between the electron and hole pockets became a nearly ubiquitous

possibility for the superconducting gap, despite the variety of methods used to derive

it. We chose a simple parameterization of such a state. The figure shows the Raman

spectra for a model gap of the form k = 0

1+r(−1 + r cos(2θ)). First, we observe the

positions of the peaks. These correspond to gap energy scales, i.e. twice the minimum

104

-2

-1.5

-1

-0.5

0

0.5

1

2 π0

∆ α1

A

-2

-1.5

-1

-0.5

0

0.5

1

2 π0

∆

α1

B

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Im χ

(ar

b. u

nits

)

ω / ∆

Raman Response

B1gB2g

C

0

0.5

1

1.5

2

0 1 2 3 4 5

Im χ

(ar

b. u

nits

)

ω / ∆

Raman Response

B1gB2g

D

Figure 4-2. Raman scattering intensities for:(A) A weakly nodal gap. The x-axis is thedistance around a circular Fermi surface (B) The same, but without nodes.(C) Anisotropic s-wave gap with nodes (D) Anisotropic s-wave gap withoutnodes

and maximum of k around the Fermi surface, just as in an s-wave superconductor pair

breaking does not occur until a scale 20. A key difference between these two examples

is that there is no Raman response below 20 in the fully gapped case. The nonzero

response down to ω = 0 is a clue signaling the presence of nodes. Then by examination

of the polarization dependence we see that there is a sharp peak in one polarization

which is not present in the other, signaling the anisotropy of the gaps in k-space.

The channel which shows the peak at the highest frequency gives the predominant

symmetry of the energy gap, in figure 4-2 this corresponds to the B1g channel. This

is the sensitivity of the Raman vertex to the presence of gap nodes. A useful rule of

105

thumb is to think of Raman as the vertex-weighted convolution of two densities of states,

N(ω)N(ω +). If the vertex weights the node heavily, then it’s convolving two, say, linear

density of states, so the intensity is proportional to ω and significant weight is at low

frequencies. Compare this to the vertex weighting anti-nodes, where we would convolve

two square-root divergences at the gap edge. Comparatively, two diverging density of

states would have a much larger peak, and this is reflected in figure 4-2, the larger peak

occurs for the B1g polarization. Inelastic processes which have not been included in this

approach will become important at large frequency. In actual Raman data, a description

of inelastic scattering would be necessary to model the high frequency intensity.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2

N(ω

)/N

0

ω / ∆0

Γ= 0.001∆0Γ= 0.01∆0Γ= 0.05∆0

Γ= 0.1∆0

A

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3

Im χ

ω / ∆0

Γ= 0.001∆0Γ= 0.01∆0Γ= 0.05∆0

Γ= 0.1∆0

0 0.01 0.02 0.03 0.04 0.05

0.3 0.6 0.9

B

Figure 4-3. (left) Quasiparticle density of states N(ω) for isotropic s± state, normalized tonormal state density of states N0 vs. ω/0, where ±0 is the value of thegap on hole and electron sheets. N0 is assumed constant on all Fermisheets. Shown are various interband impurity scattering rates in units of0. (Right) Raman response Im χ(ω) vs. ω/ for both B1g and B2g

polarizations for an s± state for various interband impurity scattering rates ,in units of .

Now we wish to examine the fingerprints of the proposed gaps in the iron arsenides

in the Raman intensities. One of the earliest proposals was for an S±-wave gap, which

takes isotropic gaps on each Fermi surface and allows for a sign change between them.

There are two electron and two hole pockets in the real Brillouin zone, but to simplify

the discussion we consider just two Fermi sheets, one hole and one electron. Interband

106

impurity scattering is pair-breaking because of the sign difference between the gaps.

As such, we expect the creation of low energy quasiparticles when isotropic impurities

are present. The density of states and Raman intensity for this model is shown in figure

4-3. In the clean case, there are no quasiparticle excitations below 20, and a sharp

peak once pair-breaking begins at ω = 20. As isotropic disorder is added zero energy

quasiparticles are created, and we see the formation of an impurity band. If we use

the heuristic argument that the Raman intensity is somewhat similar to a convolution of

the density of states with itself at a shifted external frequency, we understand that the

consequence of the impurity band is to create excitations below the gap edge. This is

reflected in figure 4-3, where a small nonzero bump appears below the pair breaking

scale, 20. The convolution of the impurity band with itself leads to weak weight below

0 as shown in the inset of figure 4-3 and to a more significant step at 0 where the

impurity band is convoluted with the gap edge. All Raman intensities must go to zero at

zero frequency.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

N/ (

N0)

ω / ∆0

Γ=0.0∆0Γ=.01∆0Γ=.05∆0

Γ=.2∆0

A

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Im χ

ω / ∆0

B1g Γ =0.01 ∆0B2g Γ =0.01 ∆0B1g Γ =0.05 ∆0B2g Γ =0.05 ∆0

B1g Γ =0.2 ∆0B2g Γ =0.2 ∆0

B

Figure 4-4. (left) Density of states N(ω)/0 vs. energy ω/0 for a d-wavesuperconductor for various values of scattering rate /0 in unitary limit.(Right) Effect of disorder on T = 0 Raman response of d wave state vs.energy ω/0. Shown are two polarizations, B1g and B2g, for various values ofscattering rate /0 in unitarity limit.

One of the key features in the simple example extracted by calculating Raman

intensity as the impurity scattering rate rises, is that the nodes which are allowed in

107

the A1g symmetry class for the pair state are not protected. This would not be true for

representations which do not possess the full symmetry of the lattice, p and d wave

in rotation group terminology. Shown in figure 4-4 are the DOS and Raman intensity

for a single band d-wave model. The effect of scattering for a single band d-wave

superconductor, is mostly to round off sharp singularities and suppress the peaks. A

more telling sign is the lowest frequency behavior which changes from ω3 to ω [177].

The presence of symmetry protected nodes is implied when no amount of scattering

by impurities is able to average the gap into an isotropic state. This clue is important in

the pnictides where fully gapped and low temperature Tn scaling is observed in several

different probes (T3 for NMR, T for penetration depth,etc.).

4.4 Modeling Experimental Raman Data

Figure 4-5. Experimental data on Ba(Fe0.939Co0.061)2As2. Reproduced with permissionfrom [181]. APS c⃝2009.

We turn now to creating and examining models for the pnictides. These are

significantly more complicated than the previous examples just because of the fact that

they have several Fermi sheets. Another issue is how scatterers affect the compounds.

It is unclear at present whether the appropriate model for various dopant impurities is

108

closer to the Born or unitary limit, how extended the scattering is, and how inter and

intra band potentials depend on the impurity. We have considered two extreme limits to

try to extract as cleanly as possible the qualitative possibilities for impurity scattering,

understanding that any real material will be in between them in some non-idealized

blend of interband, intraband, and moderate strength scattering limits. In all cases we

have chosen an extended s-wave state whose gross features reproduce data on cobalt

doped 122 pnictides. The extended s-wave state was a natural choice given data on

penetration depth, ARPES, and NMR showing singlet exponential and T2 behavior,

implying that s-wave gaps, possibly with nodes, were present. The other choice we will

make is to put impurities in the unitary limit, simulating the large short range effective

potential found for Co dopants in Ba-122 [106] within density functional theory.

Figure 4-6. Depiction of the Brillouin zone and Raman Polarizations forBa(Fe0.939Co0.061)2As2 Reproduced with permission from [181]. APS c⃝2009.

Let’s first focus on building a clean model. The experimental result is that there

is a relatively sharp peak in the experimental B2g polarization which appears below

Tc . The B1g polarization is almost linear and does not change with T, while the A1g

A2g polarizations are nearly featureless. There is a nonzero response down to zero

frequency so we deduce that we are working with a nodal superconducting gap. To

model the signal measured in Muschler et al. [181] the gap should be large in the parts

of the Brillouin zone weighted weighted by the B2g vertex, and we need an almost

featureless B1g channel. From the position of the Fermi surfaces, we see that B2g is

109

sampling the electron pockets, while the B1g polarization samples the Brillouin Zone

diagonal where no Fermi surface sheets are present, so we should build a gap with

maxima on the β sheets. The low frequency analysis of the data shows√ behavior,

corresponding to an osculating node, a somewhat special ”accidental” case, so the node

should also live on the β-sheets. The model which captured these features was:

α1(θ) = 0

1 + r cos(4θ)

1 + rr = .75 (4–17)

α2(θ) = 0

1− r cos(4θ)

1 + rr = .75 (4–18)

β1(θ) = −0

1− r cos(2θ)

1 + rr = 1 (4–19)

β2(θ) = −0

1 + r cos(2θ)

1 + rr = 1 (4–20)

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

Im χ

ω / ∆0

Clean B1gClean B2g

Figure 4-7. Plot of the clean Raman intensity for the gap and vertices described inequations 4–20-4–21.

Note that while the gap on one of the β sheets apparently has d-wave symmetry,

it is always supplemented by an additional gap rotated by 90 on the other β sheet,

preserving the full tetragonal symmetry. It is also important to examine the normal

state response. There is little change in some of the spectra across the normal to

superconducting transition, so when the vertices were chosen, it is important to take this

110

into account. The vertices must respect the symmetry of the problem but less weight

was given to the α-Fermi surface sheets as a consequence of the observations for the

A1g polarization. The set of vertices used to model the data was:

α1 γ1g = 0. γ2g = 0. (4–21)

α2 γ1g = .25(−2) sin(θ) cos(θ) γ2g = .25 cos(2θ) (4–22)

β1 γ1g = .5(−2) sin(θ) cos(θ) γ2g = +1. (4–23)

β2 γ1g = .5(−2) sin(θ) cos(θ) γ2g = −1. (4–24)

The result of these choices in the absence of any impurities is shown in figure 4-7.

The result is normalized, so we would identify the scale 20 with the 70 cm−1 peak in

the B2g channel, and will show that the relatively weak feature at 20 in the B1g channel

is engulfed by disorder effects, or is otherwise unobservable because of scatter in the

experimental data. The next important question is how these gaps evolve as a function

of impurity scattering; this will enable us to critically examine our choice of model gap

structure.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

N(ω

)/N

0

ω/∆0

cleanΓ=0.01∆0Γ=0.05∆0

Γ=0.2∆0

0.1

0.3

0.1 0.3

A

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

Im χ

ω/∆0

B1g Γ=0.01∆0B2g Γ=0.01∆0B1g Γ=0.05∆0B2g Γ=0.05∆0

B1g Γ=0.2∆0B2g Γ=0.2∆0

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

B

Figure 4-8. (A) Density of states N(ω)/N0 for the extended s-wave state vs. ω/0 forunitary intraband scatterering rates /0. Insert shows low-energy behavior.(B) Raman intensity for the extended s-wave state vs. ω/0 for variousunitary intraband scatterering rates /0 and polarization states B1g and B2g

in the 2-Fe zone. Insert: low energy region.

111

In the first case we have restricted scattering to purely intraband scattering in the

unitary limit, shown in figure4-8. The intraband scattering’s main effect, being restricted

to each band individually, is to average the gap over each Fermi sheet, creating a more

and more isotropic state. This is seen in the Raman intensity as the low frequency edge

pulls back from zero frequency. There is no further pair breaking in this limit because

there is no scattering between pairs with amplitudes of different signs. Anderson’s

theorem should hold in this case.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

N(ω

)/N

0

ω/∆0

cleanΓ=0.01∆0Γ=0.05∆0

Γ=0.2∆0

A

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5

Im χ

ω/∆0

B1g Γ=0.01∆0B2g Γ=0.01∆0

B1g Γ=0.05∆0B2g Γ=0.05∆0B1g Γ=0.2∆0B2g Γ=0.2∆0

0 0.1 0.2 0.3

B

Figure 4-9. (A) The density of states for an extended s-wave state vs. ω/0 for unitaryisotropic scatterering rates /0. (B) Raman intensity for an extendeds-wave state for various unitary isotropic scatterering rates /0 andpolarization states B1g and B2g in the 2-Fe zone. Insert: low-energy region.

In figure 4-9, we consider unitary scatterers with equal scattering rates between

all Fermi sheets (isotropic scatterers). Here Anderson’s theorem will not hold, so

we have two competing effects: the tendency to create a low energy impurity band

alongside the averaging effect from scattering. The density of states bears a certain

similarity with the one-band d-wave model. The presence of disorder creates a low

energy impurity band which fills up as the scattering rate increases. In the unitary limit

and for this band structure this tendency always dominates the intraband averaging

effect on the superconducting gap, so although the gap becomes more isotropic as

a necessary consequence of scattering, low energy quasiparticles remain in all the

112

cases we considered. We do see a change in the low frequency slope, increasing as

the scattering rate increases. As one might hope from seeing the cusp in the B1g clean

case, that feature becomes almost completely extinguished with more scattering.

Figure 4-10. Raman intensity for the Ba(Fe0.915Co0.085)2As2 sample showing a small gap.Reproduced with permission from [181]. APS c⃝2009.

The phenomenological model cannot hope to capture the large frequency response

because additional scattering due to inelastic processes is needed. Furthermore, there

can be no way to get an exact map of the structure of the superconducting state from

this probe. Nonetheless, there are a number of firm conclusions which can be drawn

which constitute significant results. First, given the data from Muschler et al. [181] the

superconducting gap for that doping likely contains nodes. This conclusion is supported

by the presence of low energy quasiparticles, and the lack of symmetry in the responses

for the A1g and B1g polarizations. Until systematic Raman studies of the doping or

scattering rate, e.g. studies like irradiation [182], are performed for the entire pnictide

family we cannot say with certainty what the effect of disorder is, but we have a model

where increasing the scattering for intraband scatterers apparently creates gaps. This

is consistent with the data of Muschler et al. [181] on a second sample with higher Co

content, where a clear gap of order several meV was observed, shown in Fig. 4-10.

On the other hand, the early indication is that irradiation reduces the exponent T n in

the low T penetration depth. There are also indications from the doping dependence of

penetration depth [183] that optimally doped 122s are fully gapped while the overdoped

113

side is nodal which has been interpreted in terms of interband scattering. This would

point more towards a situation where it is band structure changes which cause the gap

geometry to evolve, but sharp conclusions also require understanding the exact effect

of dopants which is currently under investigation. It would be an overgeneralization to

say that these things are true for all dopings and all pnictides, but we can say make

these conclusions for this compound. The idea of an anisotropic extended s-wave state

which evolves with doping is starting to take root in the pnictide community. It explains

the discrepancy between ARPES, NMR, penetration depth, and Raman experiments.

What is left to do is understand which factors among scattering rates, band structure

changes, and the unknown pairing interaction are dominant in controlling the details of

the superconducting gap.

114

CHAPTER 5MULTIFERROICS

5.1 History

Multiferroics couple polarization, magnetization, and the elastic response in

a material. Interest in the electric control of the magnetic properties of a system

for applications requires the magnetic and polar orders be coupled as strongly as

possible. There are technological implications for a material for which polarization and

magnetization can be switched by external magnetic and electric fields, so there has

been a lot of recent interest in multiferroics. Smolenskii and Chupis [184] summarized

the subject until the early 1980s and more recently the article by Wang et al. [185]

reviews recent advances. Many physical systems can be described by the interplay of

more than one order parameter, but it is the new context of multiferroics which leads

back to this classic problem.

In this chapter, after a brief overview of ferroelectricity and magnetism, we will

consider the simplest possible description of the coupling of magnetic and polar orders,

and discuss the thermodynamic consequences independent of any specific free energy.

In the next section, we specify the simplest form for the free energy and examine

the susceptibilities at both transition temperatures. We find that in the absence of a

fluctuation-induced response, the magnetoelectric susceptibility is only nonzero when

both P and M are nonzero. Finally we examine the fixed points for the theory and the

effect of inhomogeneities.

5.2 Ferroelectric Background

The earliest known ferroelectrics are Rochelle salt, NaKC4H4O4· 4H2O, and KDP,

KH2PO4, in the earlier part of the 20th century. BaTiO3 was discovered afterwards, and

opened up a new era in ferroelectric research. A wonderful resource on ferroelectrics

until the 1977 is the text by Lines and Glass [186]. More recently, the volume by Rabe et

al. [187] is useful for developments since 1977.

115

Of the 32 crystal classes, 11 are centrosymmetric. Only noncentrosymmetric

crystals are consistent with a nonzero polarization which breaks inversion symmetry.

Of the 21 remaining classes, 20 exhibit a polar response when stress is applied [186].

A true ferroelectric has a switchable P, consequently this is differentiated from only a

spontaneous nonzero polarization with the term pyroelectric. Piezoelectricity is a polar

response from applied stress. Piezoelectricity is the basis for the digital quartz watch,

and ADP, similar to KDP, was a transducer in WWII era submarines (sound induces

a polarization). Some memory applications make use of ferroelectric random access

memory and frequently camera flashes make use of ferroelectrics as a capacitor.

KDP and BaTiO3 exhibit two different types of ferroelectric behavior which forms the

basis for understanding the formation of a macroscopic polarization. In KDP, we have

the so-called order-disorder type ferroelectric, where the placement of hydrogen within

a phosphate tetrahedra is random at high temperature and macroscopically orders at

low temperature. In BaTiO3, it is the displacement of one of the ions which gives rise to

the macroscopic polarization, so this is called a displacive ferroelectric. In this picture, a

phonon becomes unstable, the so-called soft mode, and a change in the unit cell creates

a polarization. In the context of multiferroics, we can also have improper ferroelectrics,

where polarization is not independent. It is, instead, induced by some other order,

usually magnetic.

The theoretical understanding of these materials comes from several perspectives:

Ginzburg-Landau-Devonshire theory [188], ab-initio computations [189], and microscopic

models [190]. The strength of a macroscopic approach is the simplicity and wide range

of applicability, while its drawback is that no microscopic understanding is gained. The

foundation for a microscopic picture of displacive transitions was given by Anderson

[190] and Cochran [191]. Those authors established that a transverse optical phonon,

with help of the Lyddane-Sachs-Teller relationship [192] ω2

L

ω2

T

= ϵ(0)ϵ(∞)

, has a frequency

which vanishes as (T − Tc) near the ferroelectric transition. Taking the polarization as

116

the order parameter, a Landau expansion of the free energy in powers of P is the basis

of the Landau-Devonshire theory [188]. To establish the order-disorder type ferroelectric

mechanism experimentally, a variety of structural probes are needed to study proton

motion. In KDP this was accomplished by inelastic and elastic neutron measurements

[186]. A less rigorous but useful indicator of relaxor ferroelectricity is to look at the

sharpness of the peak in a response function like permittivity, which will be broad in a

relaxor and sharp in a displacive ferroelectric. The dynamics of an order-disorder type

ferroelectric also have the effect that increase of frequency decreases the permittivity

peak, which is not true of a second order structure phase transition with one unique Tc .

The calculation of polarization from first principles only became possible in the

1993. The basic problem, that the charge distribution cannot determine P [187], is as

follows: imagine NaCl which has no macroscopic polarization. If the unit cell contains a

vertical pair of ions, the polarization in that unit cell points vertically. Redefining the unit

cell horizontally causes the polarization per cell to point horizontally. This arbitrariness is

unacceptable, and a formulation independent of the unit cell is necessary. The solution

was given by King-Smith and Vanderbilt [193] who provided a way to calculate the

polarization using the Wannier functions topologically, from the Berry phase.

Ferroelectrics are similar to superconductors in their critical behavior, in that mean

field theory usually works very well [186]. Additional evidence for this, is the width of

domain walls, usually only a few lattice spacings, 10-30 A [194], in contrast to magnetic

domain walls which are more appropriately called mesoscopic in size, typically µm.

The lowest order free energy describing a second order transition from a paraelectric

to a ferroeelctric is given by

FE =P2

2χE0+ bEP

4. (5–1)

Many ferroelectric transitions are first order, and thus must be supplemented by a term

P6. The details of this approach can be found in Lines and Glass [186].

117

5.3 Magnetic Background

For a comprehensive review of magnetism there is the series by Rado and Suhl

[195], as well as a number of useful textbooks such as White [196] or Mattis [197].

Magnetism arises from the spins of electrons. Two extreme points of view are that of

completely localized and completely itinerant electrons. In the former case, we can

take the positions to be fixed and write the hamiltonian as H = −∑

ij Jij Si .Sj . In the

latter case our notions are based on the Stoner model, see [5], where a high density of

states leads to a divergence of the static susceptibility. In the intermediate regime, are

models which combine local and itinerant electrons, such as the Anderson model where

localized electrons hybridize with a conduction band and will form a moment depending

what the specific parameters are. Also in this category is antiferromagnetism due to

superexchange, or ferromagnetism due to double-exchange [198, 199].

Multiferroics are insulators, mostly transition metal oxides. We will focus on the

macroscopic approach. There are a number of smaller magnetic interactions which

are important in an accurate description of the physics of insulating magnets. First,

is the crystal field anisotropy. The crystal environment determines a local potential

which is not isotropic and biases the alignment of spins. This results in preferred axes

or planes for the spins, and can be captured by adding terms such as az(S .z)2 to the

Hamiltonian. The sign of az would favor or disfavor alignment along the z-axis, resulting

either in Ising or XY spins. Another important interaction for multiferroic physics is the

Dzyaloshinskii-Moriya [200, 201] interaction,

∑ij

Dij .(Si × Sj) (5–2)

which is a consequence of the spin-orbit interaction. Dij is proportional to x × rij where x

is the vector perpendicular to rij and from rij to the ligand (oxygen usually). It is typically

weak, but increases in strength for increasing atomic number Z, and results in the

weak-ferromagnetism of many compounds.

118

The order parameter is taken to be the magnetization which is zero above the

critical temperature and nonzero below it. Strictly, we should use representation analysis

[202] to construct a basis which transforms as the different irreducible representations

compatible with the lattice symmetry, and from this construct the Landau expansion, but

since the interaction respects the symmetry of the lattice, a simplified approach with only

one fourier component is adequate. The general approach starts with the identification

of the propagation vector k associated with the magnetic structure, and which space

group symmetry operations leave it invariant. The propagation vector is an experimental

input along with the space group of the crystal and the positions of the magnetic ions.

The little group of k, Gk , determine the irreps (irreducible representations) of Gk . A

standard group theory technique [203] using a projection operator, projects out a basis,

which is not in general unique. The different irreps classify the magnetic structures

compatible with symmetry, and the order parameter(s) are the basis functions. We

restrict this exposition to a magnetic structure with some wavevector usually determined

from experiment, which is the minimum of the exchange in reciprocal space.

Now we want to demonstrate the connection between these approaches. We begin

with the standard expression for the free energy,

F = U − TS (5–3)

We can express the internal energy as the fourier transform of the exchange,

∑q

J(q)Sq.S−q (5–4)

where we take classical spins. The entropy is added to the energy and expanded

in powers of Sq. We want to consider a simple magnetic structure with just one

fourier component, SQ0. It is worth noting that an external field couples to the uniform

magnetization not the staggered magnetization so in addition to the fourier component

which is responsible for magnetic order in this example, SQ0, we must also include the S0

119

component.

F ≈ (J(0) + q2J ′(0))S0.S0 + (J(Q0) + (q −Q0)2J ′(Q0))SQ0

.SQ0(5–5)

−T

(−α

∑q=0,Q0...

Sq.S−q − β(Sq.S−q)2 + ...

)This is a general approach for constructing the Ginzburg Landau free energy from a

local Hamiltonian. In the case of a ferromagnet the component we keep only S(0), which

after averaging is the magnetic moment M,

FM =M2

2χM0

+ bMM4. (5–6)

5.4 Multiferroic Introduction

Figure 5-1. Temperature dependent magnetoelectric response of Cr2O3 [204].

The earliest known magnetoelectric, is Cr2O3 which exhibits a linear magnetoelectric

effect. The magnetoelectric effect is when a magnetization (polarization) is induced by

an external electric (magnetic) field. Dzyaloshinskii [201] first pointed out that, on

symmetry grounds, it was possible to induce a magnetic moment with an electric field

or vice versa, possibly in the antiferromagnet Cr2O3. The atoms in Cr2O3 belong to the

rhombohedral group [205] D63d . There are four spins in the unit cell, and the magnetic

120

structure of the ground state needs to respect the symmetry of the direct product of the

lattice and transformations acting on the spins themselves, D63d×R, where R inverts

the sign of the spins. The axial vectors si , for each of the four spins, form a reducible

representation of this group. Linear combinations of these spins can be constructed

which form the basis of an irreducible representation. Using these symmetry arguments,

he concluded that the free energy would include a term,

F = F0 − α∥EzHz − α⊥(ExHx + EyHy)

Astrov et al. [206] confirmed this prediction. Rado[207], offers an explanation for

the observed temperature dependence in Cr2O3 [204], by pointing out that the cross

susceptibility’s temperature dependence can be understood from α ∝ χL, where L is the

staggered magnetization. Due to the symmetry of the antiferromagnet, the susceptibility

couples quadratically to field. In the free energy the two invariants to lowest order must

combine in such a way that when L is parallel to the external field χ⊥ vanishes, implying

F ∝ L2B2 − (L.B)2. We observe that Curie-Weiss susceptibilities vary as 1T

whereas

the spontaneous mean field order is growing as√

TTN

− 1, and the competition between

these two trends results in the temperature dependence of α∥, while in α⊥ we just

observe the√

TTN

− 1 of L.

From a technological point of view, the mutual control of electric and magnetic

properties is a desirable property for microelectronics. During the 1960s and 1970s

several (52 are listed in [184]) multiferroic materials were discovered, such as BiFeO3,

BiMnO3, and the boracite Ni3B2O13I. Recently, the investigation of rare earth manganite

multiferroics, notably TbMnO3 [208], reignited interest in multiferroic research. A broad

class of these multiferroics follow the chemical formulas, RMnO3 and RMn2O5 [209],

where R is a rare earth ion. The appearance of spiral order in the rare-earth manganites

accompanies the presence of a polarization in these, whereas the spinel multiferroics

HgCr2S4 and CdCr2S4, are both ferromagnetic and ferroelectric. In the macroscopic

121

approach, if inversion symmetry is absent, the term M.(∇×M) [210], is known to induce

spiral order. In this more recent class of spiral-magnet multiferroics, the ferroelectricity is

referred to as improper. The primary order, the magnetism, is said to induce the electric

polarization which occurs as a consequence of the magnetism. This is to be contrasted

with proper ferroelectrics, where a spontaneous polarization exists of its own accord.

Mostovoy [211] provided a general argument based on symmetry and Ginzburg-Landau

theory for why this happens. Mostovoy studies a free energy described by

Fm + Fe + Fem = Fm +P2

2χe+ γP.(M(∇.M)− (M.∇)M) (5–7)

The magnetic free energy is not given. The author observed that in the absence of a

spontaneous polarization (akin to including P4 term), the polarization is only induced by

the magnetic order and is given by,

P = −γχe(M(∇.M)− (M.∇)M) (5–8)

This analysis clarifies the role of magnetic structure in determining a polarization, and

in particular demonstrates a generic mechanism for magnetism to allow polarization

without reference to a lattice or microscopic details. More detailed group theoretical

arguments [212] can sometimes disagree with the predicted orientations of the two

order parameters, especially for more complicated invariants, but the point is that

trilinear couplings now allowed by symmetry are generally responsible for the induced

ferroelectricity. A description for the origin of magnetic order is frequently attributed to

the presence of a Dzyaloshinskii-Moriya [200, 205] interaction [213, 214]. Those authors

have shown using a microscopic model that a polarization P ∝ rij × (Si × Sj) arises

from superexchange between the spins, where rij connects the spins. Magnetostriction

can provide another microscopic mechanism coupling the elastic, magnetic, and

polar responses. The spin-lattice interaction can also induce an interaction P2M2.

The magnetostriction’s magnitude does not have the same inherent weakness of the

122

Dzyaloshinskii-Moriya interaction because the exchange J is typically a larger energy

scale.

To quote from J.Sethna [215], ”First you must define the broken symmetry. Second

you must define an order parameter. Third you are told to examine the elementary

excitations. Fourth, you classify the topological defects.” Another novel aspect of

multiferroic physics is in the low energy excitations as a consequence of coupling the

lattice and magnetic order. A new, hybrid, low energy excitation called an electromagnon

will also be present in the excitation spectrum, resulting in the coupling of spin-waves

and polar lattice vibrations. These excitations allow for spin waves to be excited by

applied electric fields. They have been recently observed [216, 217] in RMnO3 and

RMn2O5 compounds.

The topological defects will be intertwined just as the electromagnons intertwine

the magnetic and polar excitations. GdFeO3, for example, is a weak ferromagnet and a

ferroelectric [218]. Magnetostriction through the exchange interaction between the Gd

and Fe spins couples the two orders. The domain walls for electric and magnetic orders

will respond to one another [219], so it is interesting to investigate the novel aspects of

how a magnetic domain responds to an applied electric field.

5.5 Thermodynamics

5.5.1 Maxwell Relations

To begin consider the simplest consequences of coupling two order parameters,

using the Helmholtz F (P,V ,M,T ) and the Gibbs G(E,P,B,T ) free energies which

are functions of the volume V, magnetization M, temperature T and Electric field E,

polarization P, pressure P, and magnetic field B. In those ensembles,

dF = −SdT − PdV + EdP+ BdM (5–9)

dG = −SdT + VdP − PdE−MdB (5–10)

123

The differentiability of the free energies means that cross derivatives are equal and

it follows that,

∂M

∂E|B =

∂P

∂B|E (5–11)

Which is the statement that the cross susceptibility χ12 is symmetric in Eq. 5–29. By

looking at the other cross derivatives, we find, suppressing the tensor indices:

∂2C

∂E2= T

∂2χE∂T 2

∂2C

∂B2= T

∂2χM∂T 2

(5–12)

and∂C

∂B∂E= T

∂2χ12∂T 2

, (5–13)

∂2χ12∂P2

=∂2(Vκ)

∂B∂E,∂2χE∂P2

=∂2(Vκ)

∂E2,∂2χM∂P2

=∂2(Vκ)

∂B2. (5–14)

Here χ is the differential susceptibility (the subscript E for electrical and M for

magnetic), κ is the compressibility, V is the volume and C the specific heat at constant

volume.

The convexity of the free energy, for stability, also leads to the inequality

∂2F

∂M2

∂2F

∂P2− (

∂2F

∂M∂P)2 ≤ 0 (5–15)

which in turn implies that

χ212 ≤ χeχm. (5–16)

This inequality was first derived by Brown et al [220].

5.5.2 Phase Diagram

The Ehrenfest definition of the order of a phase transition needs to be considered

carefully where there is more than one mechanical field. The thermodynamic phase

boundaries for a first order transition with a latent heat are described by the Clausius-Clapeyron

equation. For a second order phase transition, the corresponding equation is the

Ehrenfest equation. The Clausius-Clapeyron equations for TM(E), which can be

124

obtained from dGi = −SidT + VdP − PdE − MdB by choosing two points on either

side of the critical line in the T-B plane,

∂TM

∂E= −P1 − P2

S1 − S2. (5–17)

For completeness, we present all the Ehrenfest equations for two mechanical fields, in

this context E and B: (∂TE

∂E

)2

=TEχEC

,

(∂TM

∂E

)2

=TMχEC

(5–18)

(∂TE

∂B

)2

=TMχMC

,

(∂TM

∂B

)2

=TMχMC

. (5–19)

In a ferromagnet (or ferroelectric) in any finite field there is no phase transition. And

yet, if the transition is second order at a finite field, as it can be for an antiferromagnet,

the evolution of Tc should be described by the Ehrenfest equations above. When the

two order parameters are coupled, as we will show below, χM is zero at TE but χE is

non-zero at TM . The ferromagnetic transition at TM(B,E) is second order in E, unlike

the uncoupled result.

5.5.3 Adiabatic Processes

Adiabatic processes are used in cooling, such as demagnetization at low temperatures.

It is a question of efficiency whether magnetic or electric cooling is useful. The cooling

arising from an adiabatic process when the entropy depends on two external fields is

described by dS=0 which upon substituting the Maxwell relations becomes:

dP

dTdE+

dM

dTdB+

dV

dTdP +

Cv

TdT = 0 (5–20)

dT

dE= − T

Cv

dP

dT(5–21)

We will defer any explicit calculation of the integral which depends on the explicit

temperature dependencies of the specific heat and also of the order parameter.

125

5.6 Free Energy Functional

A system with more than one vector order parameter has been considered many

times before, for example [221–224] and references therein. The free energy can be

written as: F = FE + FM + Fint

FE =P2

2χE0+ bEP

4 (5–22)

FM =M2

2χM0

+ bMM4 (5–23)

Fi = k(M.P)2 (5–24)

χ−1E0 = aE0

(T

TE0

− 1

)(5–25)

χ−1M0 = aM0

(T

TM0

− 1

)(5–26)

We are considering a ferromagnet described by FM and a ferroelectric represented

by FE with the interaction between them given by Fi . The coefficients of the free energy

here are all constants with the exception of the susceptibilities which are assumed

to be described by the Curie-Weiss form for local moments. For specificity we take

TE0 > TM as the transition temperatures respectively for the electric polarization and

the magnetization. In general, the interaction between the magnetization and electric

polarization must be a scalar and could be in the form k1M2P2 + k2(M.P)2 which

is required by the time reversal invariance. This term determines the relative angle

between M and P. Once the relative angle is determined, the free energy becomes of

the form in Eq.5–24 with its coefficients renormalized. The biquadratic form is also what

we could expect from the next term in the magnetostriction picture already mentioned.

The equations of state are the conventional thermodynamic equations for a

Helmholtz energy:

E =∂F

∂P(5–27)

B =∂F

∂M(5–28)

126

For a homogeneous system, we derive the thermodynamic properties using free energy

above. In a multiferroic, the linear response includes cross susceptibilities χ12, defined

by

δM = χMB+ χ12E

δP = χ12B+ χEE(5–29)

The free energy functional in the conventional form includes the external fields

−P.E and −M.B, and thus the ground state is a minimum of f. The dimensionless free

energy f = F/FE(0), is written in terms of: p = P/P0, m = M/M0, ℓ = FM(0)/FE(0),

ϵ = EFE0

P0, β.mℓ = B.M/FE0.

f =F

FE0= fe+fm+2km

2p2 = 2(T

TE0

−1)p2+p4−p.ϵ+ℓ(2( T

TM0

−1)m2+m4−m.β)+2k(m2p2)

(5–30)

The order parameters in the scaled variables are solutions of,

ϵ

4= (

T

TE0

− 1)p + p3 + km2p (5–31)

β

4= (

T

TM0

− 1)m +m3 +k

ℓmp2 (5–32)

5.6.1 Order Parameter

We consider TE0 > TM0. The solutions are:

p0 = m0 = 0 T > TE0 > TM

m20 = 0, p20(T ) = (1− T

TE0) TE0 > T > TM .

m20(T ) =

1− kℓ

1− k2

ℓ

(1− TTM

), p20(T ) = 1−k1− k2

ℓ

(1− TTE) TE0 > TM > T

TE

TE0= 1−k

1−k TE0TM0

TM

TM0

= 1−k/ℓ1− k

ℓ

TM0

TE0

(5–33)

127

As shown in Fig. 5-2, the mean field order parameter P appears at TE0 continuously.

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

TemperatureHKL

Ord

erP

aram

eter 1 - k

1 - k2

l

P0

M0

Figure 5-2. (Color Online) The polarization and magnetic moment versus temperature. Pshows a jump at TM when M condenses.

We have chosen TE0/TM0 > 1. We find TE/TE0 > 1 and TM/TM0 < 1. The

electrical transition takes place at TE0, which is unchanged by the interaction, and TM0

is re-normalized by interactions. The physical magnetic transition takes place at TM , not

TM0. As shown in Fig. 5-2, there is a kink in p0 at TM . This kink is due to the fact that the

scale temperature for p0, for T < TM is TE > TE0.

5.6.2 Susceptibility

Following the equation of state, we can derive the electric, magnetic, and the cross

susceptibilities as defined in Eq. 5–29. They both take a Curie-Weiss form, also have a

discontinuity at the other transition temperature. Figures (5-3-5-4) show these features

for a specific choice of the parameters(k= .4,kℓ= .3). The cross susceptibility diverges

at TM . As expected, it vanishes above TM . This excludes the possibility of a fluctuation

induced response, which we neglect in our study. The ferroelectric susceptibility, exhibits

a divergence at its transition temperature and an anomaly at the magnetic transition

temperature. The size of this jump depends on the energetics of the ferroelectric and

magnetic free energies, and occurs because the electric susceptibility below TM . The

magnetic susceptibility demonstrates paramagnetic behavior above it’s transition and

diverges at TM . It shows a cusp at the ferroelectric transition temperature coinciding with

128

the smooth onset of P. The inverse electric susceptibility, χ−1E , is zero at its transition

temperature and jumps at the magnetic transition temperature. The inverse magnetic

susceptibility, χ−1M , is zero at TM , and shows a change in slope at the ferroelectric

transition temperature due to the onset of P. This results in cusp-like behavior rather

than a jump.

0 10 20 30 40 50 60 700

1

2

3

4

TemperatureHKL

ΧM

,Χ

E

TMTE0

0 10 20 30 40 50 60 700

1

2

3

4

5

6

TemperatureHKL

Χ-

1 M,Χ-

1 E TM TE0

Figure 5-3. (Color Online) (Left) The ferroelectric susceptibility, the blue solid line, andthe magnetic susceptibility, the dashed red line. (Right) The inversesusceptibilities, the blue solid line, χ−1

E , and the thin dashed line representingthe inverse magnetic susceptibility, χ−1

M . An auxiliary dashed line is alsopresent to underscore the slope change in χ−1

M at TE0.

0 10 20 30 40-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

THKL

Χ12Ha

rb.u

nitsL

TM

Figure 5-4. (Color Online) The cross-susceptibility becomes nonzero only below TM andhas the opposite sign of the other susceptibilities.

129

The formal expressions are,

χE(T > TE0) =1

4( TTE0

− 1), χE(TM < T < TE0) =

1

8(1− TTE0

)(5–34)

χE(T < TM) =1

4D[(

T

TM0

− 1) + 3m20 +

k

ℓp20] (5–35)

χM(T > TM) =1

4ℓ(( TTM0

− 1) + kℓp20)

(5–36)

χM(T < TM) =1

4D[(

T

TE0

− 1) + 3p20 +k

ℓm2

0] (5–37)

χ12 =1

4D(−2k

ℓm0p0) (5–38)

with

D = [(T

TE0

− 1) + 3p20 + km20][(

T

TM0

− 1) + 3m20 +

k

ℓp20]−

(2km0p0)2

ℓ(5–39)

We briefly remark that the jump in χE is due to the fact that both D and the

numerator vanish as T approaches TM from below. The results have been plotted in

Figs. (5-3-5-4) for certain specific values of the parameters representative of the general

behavior. These parameters include the two bare transition temperatures(TE0=39K,

TE0=28K), the interaction parameter k and the free energy ratio l . In all of the results

discussed so far, k2 < l . The ground state is different in the opposite case and leads to

a correspondingly different phase diagram. The cross susceptibility χ12 is non-zero only

below TM .

5.6.3 Specific Heat

The free energy describes two second order phase transitions which are coupled.

In a mean field like analysis such as ours or in [222], that corresponds to two discontinuities

130

at TM and TE0. The algebraic results are:

CV

F0= 0 T > TE0,

2T

T 2E0

TM < T < TE0, T (2

ℓ− k2)(

1

T 2M0

+ℓ

T 2E0

− 2k

TM0TE0

) T < TM

(5–40)

0 10 20 30 40 500.00

0.01

0.02

0.03

0.04

0.05

TemperatureHKL

Cv

F0

Figure 5-5. (Color Online) The specific heat with the same parameter choice as thesusceptibility.

5.7 Inhomogeneous Effects

There are three gradient terms in the free energy, one for each of the order

parameter as well as those in the interaction term. In the following we first consider

the stability of a uniform ground state with respect to inhomogeneous perturbations.

This is followed by a discussion of the fluctuations.

A general interaction between P and M, involving space gradients (and a scalar with

respect to inversion and time reversal) can be written as a combination of the elements:

PiMjMk∂l . The gradient can operate on either M or P while they are also related by a

total derivative, a surface term. Thus all the symmetry allowed interactions are

z1M.((M.∇)P) + z2M2(∇.P) + z3(P.M)(∇.M) + z4P.[(M.∇)M] (5–41)

Since all terms are quadratic in M, there is a shift in TM proportional to the gradient of P.

The first pair of terms can be integrated by parts and turned into the second pair, up to a

total derivative that depends on the boundary. The last two terms, involving derivatives

of M, are also known as Lifshitz invariants and were introduced by Mostovoy [211].

131

These terms are linear in P and therefore spontaneously bring about a broken symmetry

value of P. In other words, as noted by Mostovoy, we have an effective local electric field

Eint here, proportional to the magnetization gradients.

Eint = z3M(∇.M) + z4[(M.∇)M] (5–42)

Which leads to a polarization P = χeEint . Replacing the polarization with its induced

response, we find a free energy for magnetization:

F =M2

2χM− 1

2χE |z3M(∇.M) + z4[(M.∇)M]|2 (5–43)

The spatial profile of the ground state is determined by the momentum dependent

χM(q), the q for which it is a maximum. The ferromagnetic ground state would be

unstable if the effect of the Lifshitz invariant were to move the ground state to finite

q. The effective free energy for M, resulting from the Lifshitz invariant is quartic in M,

therefore a ferromagnet is stable as long as bM & χE(q)z2i . Since χE is divergent at TE ,

a ferromagnetic ground state is unstable near that temperature. It may depend on the

details whether the ferromagnetic ground state can be expected to be stable anywhere.

The thermodynamic fluctuations, lead to a qualitative change in all temperature

dependent properties. There is a downward shift in the critical temperature and a

change in asymptotic mean field behavior, resulting in a different critical exponent.

Another important feature of inhomogeneous terms is the possibility of defects. A

possible way to cause improper ferroelectricity is to have defects induce the other order

parameter, so our analysis would be incomplete without considering how this might

be accomplished. In a ferromagnet there are several types of domain walls, namely

the so-called Bloch and Neel walls, which differ in how the magnetization rotates from

one ground state to the other. There is an important distinction between them in these

materials due to the Lifshitz invariant. Take, without loss of generality, the Bloch wall to

132

be described by

MB = (cos(θ(z)), sin(θ(z)), 0)

and the Neel wall to be

MN = (0, cos(θ(z)), sin(θ(z)))

The Bloch wall is divergence free, furthermore the expression M× (∇×M) vanishes for

the Bloch wall. This is not so in the Neel case. So the general expression

b1P.M(∇.M) + b2P.M× (∇×M)

which is equivalent to the Lifshitz invariant, when crystal symmetry sets b1 = b2 in for

example a cubic environment, vanishes for the Bloch wall. This means that while a Neel

wall has the possibility of inducing polarization, but it does so at a cost in energy.

It is quite natural in this light, to ask how inhomogeneous structures are influenced

by an electric field, as Dzyaloshinskii [225] did for magneto-electric materials. We

follow Dzyaloshinskii’s approach in the new context including the symmetric interaction,

cP2M2, to demonstrate that the length scale associated with a Neel wall is tunable with

electric field. We switch to a geometry where

M = (cos(ϕ(x)), sin(ϕ(x)), 0)

which is still completely general. The first choice that is made is to apply an electric field

preferentially along the y-axis, and the next is to include a crystal field anisotropy along

y, −aM2y . In this situation the free energy is

F = A(∂xϕ)2 − a sin2(ϕ)− (b1 sin

2(ϕ) + b2 cos2(ϕ))E0∂xϕ+ cE 2

0 sin2(ϕ)

We have made the assumption that we’re in the linear response regime so that P =

χeE and absorbed the susceptibility into our interaction parameters. Dzyaloshinskii

considered this action with c=0 in his earlier paper [225]. The equation of motion is a

133

form of the well-known sine-Gordon equation,

A∂2xϕ = −(a − cE 20 ) sin(ϕ) cos(ϕ)

The classical minima of the effective potential occur at ±π2. Integrating our equation

of motion with this in mind

−A(∂xϕ)2

2− 1

2(a − cE 2

0 )(sin2(ϕ)) = K

K must be −12(a− cE 2

0 ) for these boundary conditions. The equation can be integrated in

terms of elementary techniques with help of the substitution, tan(ϕ2), resulting in

ϕ(x) = 2 arctan(tanh(

√a − cE 2

0

A

x

2))

The length scale which controls the domain wall width is ξ = 2√

A

a−cE20

. This

implies there is a critical electric field, perhaps unrealistically large, where the magnetic

structure on a macroscopic scale is essentially that of the defect. We compare the scale

in Dzyaloshinskii’s paper, a threshold electric field for the proliferation of domain walls

Et = 2√aA

πb, to our scale here Ec =

√ac. The anisotropic energy scale a is common

notation to both papers, and b refers to the Lifshitz term’s coupling typically caused by

spin-orbit terms. A is of order the ferromagnetic exchange.

These defects are ad hoc in the sense that an ansatz was used to produce them.

A more fundamental investigation will not rely on this assumption. Consider a free

energy which induces P along a particular axis coupled via the Lifshitz invariant to the

magnetization.

F =

∫d3x

(∇M)2

2+ λP.(M(∇.M)− (M.∇)M)

Introduce a polarization potential, P = ∇ϕ, which is defined everywhere outside point

defects, normalize the free energy to the value of the magnetic moment,n = MM0

, and

134

integrate by parts:

F

M20

=

∫d3x

(∇n)2

2− λϕ∇.(n(∇.n)− (n.∇)n) +∇.(ϕ(n(∇.n)− (n.∇)n))

The third term is a total divergence, so it does not intervene in solving the Euler-Lagrange

equations. Usually this is thrown away without regret, but in the presence of defects it

could contain important topological information. We have not included higher powers

of P or M. As they are normalized, they would simply add constants to the free energy

which only play a role in the total energy of a state, not in the variational problem.

With this in mind we proceed by neglecting the total divergence. The expression

12∇.(n(∇.n)−(n.∇)n) is the Gaussian curvature, K, for a surface with normal, n = ∇h(x ,y)

|∇h| .

The term is familiar as the saddle-splay energy in liquid crystals where n is the director.

F

M20

=

∫d3x

(∇n)2

2− λϕ2K

This provides a geometric understanding of the Lifshitz term, as a way to balance

between the first term favoring uniform magnetization, and a curvature which depends

on the polarization’s fictitious potential ϕ. The ϕ-term induces Gaussian curvature K to

lower the free energy. The full consequences of this characterization of this point of view

are a topic for future work.

5.8 Specfic Heat with Gaussian Fluctuations

We turn back to the consequences of inhomogeneous terms. The fluctuation effects

arise from gradient terms, γe(∇P)2 and γm(∇M)2, in the free energy (Eq. 5–30). Near

TE0 the corresponding correlation length diverges. The subleading length scale derived

from γm starts to play an important role in the temperature range TM < T < TE0, but the

relative role of γm and γe changes as one moves between TE0 and TM . Near TM there

is an effective free energy similar to equation 5–43, which includes z3 and z4, however

a Gaussian calculation notes the effect of these terms as quartic in M and therefore

135

negligible. These terms are also irrelevant in the RG sense. In d-dimensions, (x 7→ x ′

b), zi

scales as b2−d .

The fluctuation contribution to the specific heat was computed in the gaussian

approximation, and shown in Fig. 5-6.

0 10 20 30 40 500.00

0.05

0.10

0.15

TemperatureHKL

Cv

F0

ýe ,ým=H140,140L

ýe ,ým= H700,1000L

Figure 5-6. The most singular contribution to the specific heat from gaussianfluctuations, for arbitrarily chosen correlation lengths

We first include gradients, γe(∇P)2 and γm(∇M)2 in the free energy. Then, we

expand about the saddle point, m0 + δm, p0 + δp, and neglect cross-terms since this is

meant to be an estimate. There are two independent contributions in that approximation,

one for the gaussian integral over the polarization and one from the magnetization. The

most singular terms in the specific heat take the approximate form

Cv uct ≈ T 2G(T )F (T )d2−2

Where the functions G and F follow a pattern. Let Oi denote the ith order parameter, M

or P. Then G(T ) = a/Tc+12O1dO1

dT+2kO2

dO2

dTand F (T ) = ( T

Tc−1)+6b1O

21(T )+kO2

2(T ).

The correlation length will take the form, ξ2i = γiF (T )

. The width of the critical region

around either Tc in three dimensions [226] is given by the expression T1/2G = kB

4Cmf ξ3

i

,

where Cmf is the size of the jump in the specific heat from mean field theory. It should

be noted that there is no exact Ginzburg criterion [226]. It is a semi-quantitative measure

of the critical region. We plot the behavior of these decoupled fluctuations in Fig. 5-6. It

136

will take a measurement to determine to what extent the system can be treated within

mean field theory. We expect the dispersion of polar phonons to be relatively weak, and

relying on experience in ferroelectrics, the polar transition could even be first order.

a)

b)

c)

Figure 5-7. The diagrams at one loop. a) two-point function b) vertex for one orderparameter c) vertex coupling two order parameters. Dotted or solid linesrefer to the propagators for each order parameter separately.

It is instructive to consider the renormalization group flow and analyze the fixed

points of our free energy. The critical behavior of two coupled order parameters each

with O(n) symmetry has already been computed in other contexts [224], and the results

can be profitably reapplied in this new context. The necessary diagrams are shown in

fig. 5-7. The five flow equations, resulting from integrating out a shell of momentum from

b

to , are two copies of:

′ri = b2[ri + 4(ni + 2)ui

∫

b

ddq

ri + q2+ 2njk

∫

b

ddq

rj + q2] (5–44)

′ui = bϵ[ui − 4(ni + 8)u2i

∫

b

ddq

(ri + q2)2− 4njk

2

∫

b

ddq

(rj + q2)2] (5–45)

and the flow for the coupling of both order parameters:

′k = bϵ[k−16k2∫

b

ddq

rm + q21

re + q2−4(nm+2)kum

∫

b

ddq

(rm + q2)2−4(ne+2)kue

∫

b

ddq

(re + q2)2]

(5–46)

137

There are six fixed points determined by the flow of the quartic terms ui and k. In

four of these, the order parameters decouple, k = 0, and are unstable except for the

possibility that the system breaks into two independent vector models. The remaining

two non-trivial fixed points, the Heisenberg-Heisenberg and so-called biconical fixed

points, interchange their stability as a function of how many components the order

parameters have. Fluctuation driven first order transitions [227] can occur if there is a

runaway flow in the RG (when there is no fixed point accessible under RG iterations),

and are absent in this model. At the double-Heisenberg fixed point, umc = uec = kc .

Linearizing, the eigenvalues are all negative for ne + nm ≤ 4 and ϵ > 0, but changes its

stability if the components of the order parameter change. For the case n=3, that is for

three dimensional vectors P and M considered here, the biconical fixed point (umc = uec )

is stable, and the double Heisenberg fixed point is unstable, meaning that the simple

coupled-free energy we began with also controls the critical behavior in the case that it

is not mean-field like. If we have two easy planes instead and n=2, or very strong crystal

field anisotropic such that n=1 effectively, then the double-Heisenberg fixed point is

again stable. The anomalous dimension as in all one-loop scalar field theories vanishes,

but we can anticipate corrections at higher order.

5.9 Future Work on Dynamics

One of the ongoing questions we wish to understand is the dispersion of the low

energy excitations. The dispersion can be sensitive to microscopic interactions. One

approach to understanding the dynamics is through an effective equation of motion. For

ferroelectric order, the equation of motion is essentially that of a phonon:

P = −f δFδP

.

138

This equation could be modified to include damping with a term proportional to _P.

Linearizing, P = P0 + δp, where p0 is the static component,

δp = −4f((T

Te

− 1) + cq2)(P0 + δp) + P30 + 3P2

0δp

)we reproduce what one would expect for a simple model of optical phonons.

ω2 = 4f [cq2 + 2(1− T

Te

)]

The interaction M2P2 will alter only the linear term, inducing a larger optical gap

proportional to M20P0. The Bloch equation for the semi-classical precession of a

magnetic moment in an effective field reads:

_M = γM × δF

δM.

After linearization, the Bloch equation reproduces the known ferromagnetic result:

−iωδm = γ(−cq2(m0 × δm)

)even in the presence of ferroelectricity. This result’s utility is that inhomogeneous

couplings are distinguished from it in that they alter the ferromagnetic spectrum. Zvezdin

and Mukhin [228] consider the dynamics of an antiferromagnet coupled via the Lifshitz

invariant to the ferroelectric, as do de Sousa and Moore [229]. The mere presence

of the inhomogeneous term alters the spectrum of the antiferromagnetic magnons.

Determining the effect of Lifshitz invariants on the spectrum as well as the spectrum for

other multiferroics is the subject of future work.

5.10 Conclusions

We have made some general considerations mostly based on thermodynamics

and symmetry about multiferroics. If the goal when analyzing a model is to understand

its ground states, excitations, and topological defects, then we have worked towards

gaining an understanding of the phase diagram and topological defects while excitations

139

can be left for future work. We showed that nonzero magnetoelectric response

only exists when both P and M are nonzero and that this effect is bounded by

the electric and magnetic susceptibilities. We determined the universality class

of ferroelectric-ferromagnets to be governed by the number of components in the

order parameter, in which case we would observe critical behavior controlled by the

double-Heisenberg fixed point or biconical fixed point. We examined the effect of

inhomogeneous terms, in particular on topological defects. The electric field was shown

to have an influence on the width of domain walls. There are a number of interesting

directions in which this work could be extended: generalizations to more exotic magnetic

or polar orders, inclusion of elastic interactions, numerical or exact solutions for new

defects, and the examination of dynamics.

140

CHAPTER 6CONCLUSION

Logic and mathematics can stand on their own, but it is comparison to the external

world which can falsify, never completely verify, our suppositions in physics. We can

extend our models insofar as they are consistent with what nature presents. In this

thesis, the observed properties of superconducting cuprates and pnictides were

presented in detail in Chapter 2. Along with this came some of the main theoretical

trends regarding a synthesis of this body of evidence. No consensus has been reached

yet, so this material serves as motivation for the active scientist. In Chapter 3, we

presented a method for determining the underlying microscopic state of unconventional

superconductors based on a bulk thermodynamic probe. While it is not sensitive

to every detail, it can discriminate between largely different superconducting gap

structures, and rule out conventional behavior when oscillations of the specific heat,

including the inversion of these oscillations, occur beyond effects from Fermi surface

anisotropy. In Chapter 4 we summarized model calculations and experimental work

on Raman scattering in the iron-arsenide superconductors, including the effect of

impurities. Those considerations led to a proposal for the superconducting gap in

Ba(Fe0.939Co0.061)2As2, which can provide a basis for further work comparing the

evolution of the Raman intensity to the model as the impurity scattering rate is changed.

In Chapter 5, we looked at an altogether different topic, multiferroics. We calculated

the thermodynamic consequences of the coupling of two order parameters coupling to

different external fields, including a fundamental bound on the magnetoelectric response

which does not depend on any particular choice of free energy. We then looked at

a simple model for coupled order parameters where the magnetic moment breaks

time-reversal invariance and the polarization breaks inversion symmetry. We calculated

the static thermodynamic responses for this model, and studied the inhomogeneous

ground states, e.g. topological defects, that can arise in that situation. Finally we used

141

the renormalization group to understand the critical behavior of the model, allowing the

universality class to be understood. The utility a macroscopic approach provides is to

confirm or deny our notions of what the essential physics is in a given system.

142

REFERENCES

[1] H. Frohlich, Phys. Rev. 79, 845 (1950).

[2] L. N. Cooper, Phys. Rev. 104, 1189 (1956).

[3] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

[4] M.Tinkham, Introduction to Superconductivity (Dover, 1996), ISBN 0-486-43503-2.

[5] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed MatterPhysics: An Introduction (Oxford Graduate Texts) (Oxford University Press, 2004),ISBN ISBN: 0198566336.

[6] J. Bardeen, Phys. Rev. Lett. 9, 147 (1962).

[7] O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod.Phys. 79, 353 (2007).

[8] I. Giaever, H. R. Hart, and K. Megerle, Phys. Rev. 126, 941 (1962).

[9] J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10, 336 (1963).

[10] L. C. Hebel and C. P. Slichter, Phys. Rev. 113, 1504 (1959).

[11] Y. Masuda and A. G. Redfield, Phys. Rev. 125, 159 (1962).

[12] E. Yusuf, B. J. Powell, and R. H. McKenzie, Phys. Rev. B 75, 214515 (2007).

[13] D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958).

[14] M. A. Biondi and M. P. Garfunkel, Phys. Rev. 116, 862 (1959).

[15] C. B. Satterthwaite, Phys. Rev. 125, 873 (1962).

[16] J. Bardeen, G. Rickayzen, and L. Tewordt, Phys. Rev. 113, 982 (1959).

[17] A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959).

[18] C. J. Garter and H. B. G. Casimir, Phys. Z. 35, 963 (1934).

[19] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).

[20] C. J. Pethick and D. Pines, Phys. Rev. Lett. 57, 118 (1986).

[21] P. J. Hirschfeld, J.Korean.Phys.Soc. 33, 485 (1998), URL http://arxiv.org/abs/

cond-mat/9809092.

[22] W. L. McMillan and J. M. Rowell, Phys. Rev. Lett. 14, 108 (1965).

[23] W.Jones and N. March, Theoretical Solid State Physics volume 2 (Dover, 1985),ISBN 0-486-65016-2.

143

[24] D. J. Scalapino, J. R. Schrieffer, and J. W. Wilkins, Phys. Rev. 148, 263 (1966).

[25] Fetter and J.D.Walecka, Quantum Theory of Many-Particle Systems (Dover,2003), ISBN 0-486-42827-3.

[26] J. Rech, C. Pepin, and A. V. Chubukov, Phys. Rev. B 74, 195126 (2006).

[27] A. Abanov, A. V. Chubukov, and J. Schmalian, Advances in Physics 52, 119(2003), URL http://www.informaworld.com/10.1080/0001873021000057123.

[28] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).

[29] D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod. Phys. 77, 579 (2005).

[30] D. N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721 (2005).

[31] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).

[32] A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006).

[33] O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod.Phys. 79, 353 (2007).

[34] H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, Rev. Mod. Phys. 81, 45(2009).

[35] N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).

[36] W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989).

[37] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (1985).

[38] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).

[39] D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995).

[40] P. Anderson, Science 235, 1196 (1987).

[41] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).

[42] Q. Huang, J. F. Zasadzinski, K. E. Gray, J. Z. Liu, and H. Claus, Phys. Rev. B 40,9366 (1989).

[43] K. McElroy, J. Lee, J. A. Slezak, D.-H. Lee, H. Eisaki, S. Uchida, and J. C. Davis,Science (New York, N.Y.) 309, 1048 (2005), URL http://www.sciencemag.org/

cgi/content/abstract/309/5737/1048.

[44] A. N. Pasupathy, A. Pushp, K. K. Gomes, C. V. Parker, J. Wen, Z. Xu, G. Gu,S. Ono, Y. Ando, A. Yazdani, et al., Science (New York, N.Y.) 320, 196 (2008),URL http://www.sciencemag.org/cgi/content/abstract/320/5873/196.

[45] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).

144

[46] LeTacon, Sacuto, Georges, Kotliar, Gallais, Colson, and Forget, Nature Physics 2,537 (2006), URL http://dx.doi.org/10.1038/nphys362.

[47] S. Blanc, Y. Gallais, M. Cazayous, M. A. Measson, A. Sacuto, A. Georges, J. S.Wen, Z. J. Xu, G. D. Gu, and D. Colson, Phys. Rev. B 82, 144516 (2010).

[48] J. F. Zasadzinski, L. Coffey, P. Romano, and Z. Yusof, Phys. Rev. B 68, 180504(2003).

[49] J. Tranquada, chapter in ”Handbook of High Temperature Superconductivity” Ed.J.R.Schrieffer (2005), URL http://arxiv.org/abs/cond-mat/0512115.

[50] C. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki, and O. Fischer, Phys. Rev. Lett.80, 149 (1998).

[51] A. V. Balatsky and M. I. Salkola, Phys. Rev. Lett. 76, 2386 (1996).

[52] J.-X. Zhu, C. S. Ting, and C.-R. Hu, Phys. Rev. B 62, 6027 (2000).

[53] J.-X. Zhu and C. S. Ting, Phys. Rev. B 63, 020506 (2000).

[54] S. Pan, E. Hudson, K. Lang, H. Eisaki, S. Uchida, and J. Davis, Nature 403, 746(2000), URL http://www.nature.com.lp.hscl.ufl.edu/nature/journal/v403/

n6771/full/403746a0.html.

[55] M. I. Salkola, A. V. Balatsky, and D. J. Scalapino, Phys. Rev. Lett. 77, 1841 (1996).

[56] Loram, Luo, Cooper, W. Liang, and J. Tallon, Journal of Physics and Chemistry ofSolids 62, 59 (2001).

[57] W. E. M. Cardona and L. Ley, Photoemission in Solids I General Principles(Springer, 1978).

[58] A. A. Kordyuk, S. V. Borisenko, M. S. Golden, S. Legner, K. A. Nenkov,M. Knupfer, J. Fink, H. Berger, L. Forro, and R. Follath, Phys. Rev. B 66, 014502(2002).

[59] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).

[60] J. Meng, G. Liu, W. Zhang, L. Zhao, H. Liu, X. Jia, D. Mu, S. Liu, X. Dong,J. Zhang, et al., Nature 462, 335 (2009), URL http://www.nature.com.lp.

hscl.ufl.edu/nature/journal/v462/n7271/full/nature08521.html.

[61] M. Norman, Physics 3 (2010), URL http://physics.aps.org/articles/v3/86.

[62] J. M. Luttinger, Phys. Rev. 119, 1153 (1960).

[63] D. S. Marshall, D. S. Dessau, A. G. Loeser, C.-H. Park, A. Y. Matsuura, J. N.Eckstein, I. Bozovic, P. Fournier, A. Kapitulnik, W. E. Spicer, et al., Phys. Rev. Lett.76, 4841 (1996).

145

[64] V. Arpiainen, A. Bansil, and M. Lindroos, Phys. Rev. Lett. 103, 067005 (2009).

[65] Q. Han, T. Li, and Z. D. Wang, Phys. Rev. B 82, 052503 (2010).

[66] P. Ghaemi, T. Senthil, and P. Coleman, Phys. Rev. B 77, 245108 (2008).

[67] H. Alloul, T. Ohno, and P. Mendels, Phys. Rev. Lett. 63, 1700 (1989).

[68] C. Stock, W. J. L. Buyers, R. A. Cowley, P. S. Clegg, R. Coldea, C. D. Frost,R. Liang, D. Peets, D. Bonn, W. N. Hardy, et al., Phys. Rev. B 71, 024522 (2005).

[69] N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).

[70] S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada,A. Kapitulnik, and C. Howald, Rev. Mod. Phys. 75, 1201 (2003).

[71] W. N. Hardy, D. A. Bonn, D. C. Morgan, R. Liang, and K. Zhang, Phys. Rev. Lett.70, 3999 (1993).

[72] P. A. Lee, Phys. Rev. Lett. 71, 1887 (1993).

[73] A. C. Durst and P. A. Lee, Phys. Rev. B 62, 1270 (2000).

[74] I. A. G. Catelani, JETP 100, 331 (2005).

[75] M. Chiao, R. W. Hill, C. Lupien, L. Taillefer, P. Lambert, R. Gagnon, andP. Fournier, Phys. Rev. B 62, 3554 (2000).

[76] M. Sutherland, D. G. Hawthorn, R. W. Hill, F. Ronning, S. Wakimoto, H. Zhang,C. Proust, E. Boaknin, C. Lupien, L. Taillefer, et al., Phys. Rev. B 67, 174520(2003).

[77] N. Hussey, Journal of Physics: Condensed Matter 20, 123201 (2008), URLhttp://stacks.iop.org/0953-8984/20/i=12/a=123201?key=crossref.

6224fd642e70d1dd46cb99b0615f2ee0.

[78] Naqib, Cooper, Tallon, and Panagopoulos, Physica C 387, 365 (2003).

[79] H. v. Lohneysen, A. Rosch, M. Vojta, and P. Wolfle, Rev. Mod. Phys. 79, 1015(2007).

[80] P. J. Turner, R. Harris, S. Kamal, M. E. Hayden, D. M. Broun, D. C. Morgan,A. Hosseini, P. Dosanjh, G. K. Mullins, J. S. Preston, et al., Phys. Rev. Lett. 90,237005 (2003).

[81] I. Ussishkin, S. L. Sondhi, and D. A. Huse, Phys. Rev. Lett. 89, 287001 (2002).

[82] V. Oganesyan and I. Ussishkin, Phys. Rev. B 70, 054503 (2004).

146

[83] E. H. Sondheimer, Proceedings of the Royal Society A: Mathematical,Physical and Engineering Sciences 193, 484 (1948), URL http://rspa.

royalsocietypublishing.org/cgi/content/abstract/193/1035/484.

[84] K. Behnia, M.-A. Measson, and Y. Kopelevich, Phys. Rev. Lett. 98, 076603 (2007).

[85] J. G. Checkelsky and N. P. Ong, Phys. Rev. B 80, 081413 (2009).

[86] Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510 (2006).

[87] J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. Hardy,R. Liang, W. Siemons, G. Koster, M. M. Fejer, et al., Phys. Rev. Lett. 100, 127002(2008).

[88] V. P. Mineev, Phys. Rev. B 77, 180512 (2008).

[89] B. Fauque, Y. Sidis, V. Hinkov, S. Pailhes, C. T. Lin, X. Chaud, and P. Bourges,Phys. Rev. Lett. 96, 197001 (2006).

[90] J. E. Sonier, V. Pacradouni, S. A. Sabok-Sayr, W. N. Hardy, D. A. Bonn, R. Liang,and H. A. Mook, Phys. Rev. Lett. 103, 167002 (2009).

[91] J. E. Sonier, J. H. Brewer, R. F. Kiefl, R. H. Heffner, K. F. Poon, S. L. Stubbs, G. D.Morris, R. I. Miller, W. N. Hardy, R. Liang, et al., Phys. Rev. B 66, 134501 (2002).

[92] C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).

[93] A. Shekhter, L. Shu, V. Aji, D. E. MacLaughlin, and C. M. Varma, Phys. Rev. Lett.101, 227004 (2008).

[94] J. Sonier, private communication.

[95] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, andH. Hosono, 128, 10012 (2006).

[96] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, 130, 3296 (2008).

[97] Zhi-An Ren, Wei Lu, Jie Yang, Wei Yi, Xiao-Li Shen, Zheng-Cai Li, Guang-CanChe, Xiao-Li Dong, Li-Ling Sun, Fang Zhou, Zhong-Xian Zhao, preprint (2008),arXiv:0804.2053.

[98] D. Johnston, Advances in Physics 59, 803 (2010), URL arxiv.org/abs/0808.

2647.

[99] M. Sadovskii, Physics Uspekhi 51, 1201 (2008).

[100] J.-H. Chu, J. G. Analytis, C. Kucharczyk, and I. R. Fisher, Phys. Rev. B 79, 014506(2009).

[101] M. M. Qazilbash, J. J. Hamlin, R. E. Baumbach, L. Zhang, D. J. Singh, M. B.Maple, and D. N. Basov, Nature Physics 5, 647 (2009).

147

[102] W. L. Yang, A. P. Sorini, C.-C. Chen, B. Moritz, W.-S. Lee, F. Vernay,P. Olalde-Velasco, J. D. Denlinger, B. Delley, J.-H. Chu, et al., Phys. Rev. B 80,014508 (2009).

[103] Yildrim, Physica C 469, 425 (2009).

[104] M. Nakajima, S. Ishida, K. Kihou, Y. Tomioka, T. Ito, Y. Yoshida, C. H. Lee, H. Kito,A. Iyo, H. Eisaki, et al., Phys. Rev. B 81, 104528 (2010).

[105] W. Z. Hu, J. Dong, G. Li, Z. Li, P. Zheng, G. F. Chen, J. L. Luo, and N. L. Wang,Phys. Rev. Lett. 101, 257005 (2008).

[106] A. F. Kemper, C. Cao, P. J. Hirschfeld, and H.-P. Cheng, Phys. Rev. B 80, 104511(2009).

[107] Liu, Kondo, Tillman, Gordon, Samolyuk, Lee, Martin, McChesney, Budko, Tanatar,et al., Physica C 469, 491 (2009), URL http://arxiv.org/abs/0806.2147.

[108] B. C. Sales, M. A. McGuire, A. S. Sefat, and D. Mandrus, Physica C 470, 304(2010), URL http://arxiv.org/abs/0909.3511.

[109] Kondo, Takeshi, Santander-Syro, A. F. Copie, Liu, Tillman, Mun, Schmalian,Budko, Tanatar, et al., Phys. Rev. Lett. 101, 147003 (2008).

[110] Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, S. Kasahara, T. Shibauchi, Y. Matsuda,and T. Terashima, Phys. Rev. B 81, 020503 (2010).

[111] H.-J. Grafe, D. Paar, G. Lang, N. J. Curro, G. Behr, J. Werner, J. Hamann-Borrero,C. Hess, N. Leps, R. Klingeler, et al., Phys. Rev. Lett. 101, 047003 (2008).

[112] Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono, J.Phys.Soc.Jpn 77,073701 (2008).

[113] N. Terasaki, H. Mukuda, M. Yashima, Y. Kitaoka, K. Miyazawa, P. Shirage, H. Kito,H. Eisaki, and A. Iyo, J. Phys. Soc. Jpn. 78 (2009).

[114] Y. Su, P. Link, A. Schneidewind, T. Wolf, P. Adelmann, Y. Xiao, M. Meven, R. Mittal,M. Rotter, D. Johrendt, et al., Phys. Rev. B 79, 064504 (2009).

[115] Dai, J. H., Q. Si, J. S. Zhu, and E. Abrahams, Proc. Natl Acad. Sci. 106, 4118(2008).

[116] J. Zhao, D. T. Adroja, D.-X. Yao, R. Bewley, S. Li, X. F. Wang, G. Wu, X. H. Chen,J. Hu, P. Dai, et al., Nature Physics 5, 555 (2009), URL http://www.nature.com.

lp.hscl.ufl.edu/nphys/journal/v5/n8/full/nphys1336.html.

[117] M. Holt, O. P. Sushkov, D. Stanek, and G. S. Uhrig, Iron pnictide parent com-pounds: Three dimensional generalization of the j1-j2 heisenberg model on asquare lattice and role of the interlayer coupling jc , arxiv.org/abs/1010.5551.

148

[118] S. O. Diallo, V. P. Antropov, T. G. Perring, C. Broholm, J. J. Pulikkotil, N. Ni, S. L.Bud’ko, P. C. Canfield, A. Kreyssig, A. I. Goldman, et al., Phys. Rev. Lett. 102,187206 (2009).

[119] D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 (2008).

[120] L. Ke, M. van Schilfgaarde, J.J.Pulikkotil, T. Kotani, and V.P.Antropov, Low energy,coherent, stoner-like excitations in cafe2as2, http://arxiv.org/abs/1004.2934.

[121] T. A. Maier and D. J. Scalapino, Phys. Rev. B 78, 020514 (2008).

[122] K. Hashimoto, T. Shibauchi, T. Kato, K. Ikada, R. Okazaki, H. Shishido,M. Ishikado, H. Kito, A. Iyo, H. Eisaki, et al., Phys. Rev. Lett. 102, 017002 (2009).

[123] C. Martin, H. Kim, R. T. Gordon, N. Ni, V. G. Kogan, S. L. Bud’ko, P. C. Canfield,M. A. Tanatar, and R. Prozorov, Phys. Rev. B 81, 060505 (2010).

[124] R. T. Gordon, H. Kim, N. Salovich, R. W. Giannetta, R. M. Fernandes, V. G.Kogan, T. Prozorov, S. L. Bud’ko, P. C. Canfield, M. A. Tanatar, et al., Phys. Rev. B82, 054507 (2010).

[125] C. Martin, R. T. Gordon, M. A. Tanatar, H. Kim, N. Ni, S. L. Bud’ko, P. C. Canfield,H. Luo, H. H. Wen, Z. Wang, et al., Phys. Rev. B 80, 020501 (2009).

[126] H. Kim, R. T. Gordon, M. A. Tanatar, J. Hua, U. Welp, W. K. Kwok, N. Ni, S. L.Bud’ko, P. C. Canfield, A. B. Vorontsov, et al., Phys. Rev. B 82, 060518 (2010).

[127] L. Pourovskii, V. Vildosola, S. Biermann, and A.Georges, EPL (EurophysicsLetters) 84, 37006 (2008), URL http://stacks.iop.org/0295-5075/84/i=3/a=

37006.

[128] V. Mishra, G. Boyd, S. Graser, T. Maier, P. J. Hirschfeld, and D. J. Scalapino, Phys.Rev. B 79, 094512 (2009).

[129] M. A. Tanatar, J.-P. Reid, H. Shakeripour, X. G. Luo, N. Doiron-Leyraud, N. Ni,S. L. Bud’ko, P. C. Canfield, R. Prozorov, and L. Taillefer, Phys. Rev. Lett. 104,067002 (2010).

[130] J.-P. Reid, M. A. Tanatar, X. G. Luo, H. Shakeripour, N. Doiron-Leyraud, N. Ni,S. L. Bud’ko, P. C. Canfield, R. Prozorov, and L. Taillefer, Phys. Rev. B 82, 064501(2010).

[131] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki, Phys.Rev. Lett. 101, 087004 (2008).

[132] Y. Wang, L. Li, M. J. Naughton, G. D. Gu, S. Uchida, and N. P. Ong, Phys. Rev.Lett. 95, 247002 (2005).

[133] N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. 17, 433 (1966).

149

[134] C. Gros, Phys. Rev. B 38, 931 (1988).

[135] Edegger, Mutuhkumar, and Gros, Advances in Physics 56, 927 (2007).

[136] A. E. Ruckenstein, P. J. Hirschfeld, and J. Appel, Phys. Rev. B 36, 857 (1987).

[137] G. Baskaran, Z. Zou, and P. W. Anderson, Solid State Communications 63,973 (1987), ISSN 0038-1098, URL http://www.sciencedirect.com/science/

article/B6TVW-46P7R26-47B/2/8480cccd755aae88e9b5025ba2a0716d.

[138] S. E. Barnes, J. Phys. F 6, 1375 (1976).

[139] P. Coleman, Phys. Rev. B 29, 3035 (1984).

[140] J. A. Hertz, Phys. Rev. B 14, 1165 (1976).

[141] A. J. Millis, Phys. Rev. B 48, 7183 (1993).

[142] D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod. Phys. 77, 579 (2005).

[143] S. J. Yamamoto and Q. Si, Phys. Rev. B 81, 205106 (2010).

[144] S. Kirchner and Q. Si, Berry phase and the breakdown of the quantum to clas-sical mapping for the quantum critical point of the bose-fermi kondo model,http://arxiv.org/abs/0808.2647.

[145] Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature 413, 804 (2001).

[146] A. V. Balazs Dora, Kazumi Maki, Mod. Phys. Lett. B 18, 119 (2004), URL arXiv:

cond-mat/0407666.

[147] A. A. Nersesyan, G. I. Japaridze, and I. G. Kimeridze, Journal of Physics:Condensed Matter 3, 3353 (1991), URL http://iopscience.iop.org.lp.

hscl.ufl.edu/0953-8984/3/19/014.

[148] C. Zhang, S. Tewari, and S. Chakravarty, Phys. Rev. B 81, 104517 (2010).

[149] B. Dora, K. Maki, A. Vanyolos, and A. Virosztek, Phys. Rev. B 68, 241102 (2003).

[150] G. Miransky, The European Physical Journal B 37, 363 (1991).

[151] B. M. Andersen and P. J. Hirschfeld, Phys. Rev. Lett. 100, 257003 (2008).

[152] L. Li, Y. Wang, S. Komiya, S. Ono, Y. Ando, G. D. Gu, and N. P. Ong, Phys. Rev. B81, 054510 (2010).

[153] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

[154] I. Vekhter, P. J. Hirschfeld, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B 59, R9023(1999).

150

[155] T. Dahm, S. Graser, C. Iniotakis, and N. Schopohl, Phys. Rev. B 66, 144515(2002).

[156] M. Franz and Z. Tesanovic, Phys. Rev. B 60, 3581 (1999).

[157] L. Marinelli, B. I. Halperin, and S. H. Simon, Phys. Rev. B 62, 3488 (2000).

[158] I. Vekhter, P. J. Hirschfeld, and E. J. Nicol, Phys. Rev. B 64, 064513 (2001).

[159] B. Zeng, a. H. Mu G., T. Xiang, H. Yang, L. Shan, C. Ren, I. I. Mazin, P. C. Dai,and H.-H. Wen, Anisotropic structure of the order parameter in fese0.45te0.55

revealed by angle resolved specific heat, http://arxiv.org/abs/1007.3597.

[160] S. Graser, G. R. Boyd, C. Cao, H.-P. Cheng, P. J. Hirschfeld, and D. J. Scalapino,Phys. Rev. B 77, 180514 (2008).

[161] G. Volovik, JEPT lett. 58, 469 (1993).

[162] C. Kubert and P. J. Hirschfeld, Phys. Rev. Lett. 80, 4963 (1998).

[163] K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys.Rev. Lett. 87, 057002 (2001).

[164] Y. Matsuda, K. Izawa, and I. Vekhter, Journal of Physics: Condensed Matter 18,R705 (2006), URL http://iopscience.iop.org.lp.hscl.ufl.edu/0953-8984/

18/44/R01.

[165] K. Deguchi, Z. Q. Mao, and Y. Maeno, Journal of the Physical Society of Japan 73,1313 (2004), URL http://jpsj.ipap.jp/link?JPSJ/73/1313/.

[166] A. Vorontsov and I. Vekhter, Phys. Rev. Lett. 96, 237001 (2006).

[167] D. Einzel, Journal of the Physical Society of Japan 131, 1 (2003).

[168] G. R. Boyd, P. J. Hirschfeld, I. Vekhter, and A. B. Vorontsov, Phys. Rev. B 79,064525 (2009).

[169] A. B. Vorontsov and I. Vekhter, Phys. Rev. B 75, 224501 (2007).

[170] U. Brandt, W. Pesch, and L. Tewordt, Zeitschrift fur Physik 201, 209 (1967),ISSN 0939-7922, 10.1007/BF01326812, URL http://dx.doi.org/10.1007/

BF01326812.

[171] A. B. Vorontsov and I. Vekhter, Phys. Rev. B 75, 224501 (2007).

[172] M. Ichioka, A. Hasegawa, and K. Machida, Phys. Rev. B 59, 184 (1999).

[173] K. An, T. Sakakibara, R. Settai, Y. Onuki, M. Hiragi, M. Ichioka, and K. Machida,Phys. Rev. Lett. 104, 037002 (2010).

151

[174] V. Tsurkan, J. Deisenhofer, A. Gnther, C. Kant, H.-A. K. von Nidda, F. Schrettle,and A. Loidl, Physical properties of fese0.5te0.5 single crystals grown underdifferent conditions, http://arxiv.org/abs/1006.4453.

[175] T. P. Devereaux and R. Hackl, Rev. Mod. Phys. 79, 175 (2007).

[176] D. J. Scalapino and T. P. Devereaux, Phys. Rev. B 80, 140512 (2009).

[177] T. P. Devereaux and A. P. Kampf, Int. Journ. Mod. Phys. B 11, 2093 (1997), URLhttp://arxiv.org/abs/cond-mat/9702186.

[178] B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. Lett. 44, 1288 (1980).

[179] S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, New J. Phys. 11,025016 (2009), URL http://arxiv.org/abs/0812.0343.

[180] F. Wang, H. Zhai, and D.-H. Lee, Phys. Rev. B 81, 184512 (2010).

[181] B. Muschler, W. Prestel, R. Hackl, T. P. Devereaux, J. G. Analytis, J.-H. Chu, andI. R. Fisher, Phys. Rev. B 80, 180510 (2009).

[182] H. Kim, R. T. Gordon, M. A. Tanatar, J. Hua, U. Welp, W. K. Kwok, N. Ni, S. L.Bud’ko, P. C. Canfield, A. B. Vorontsov, et al., Phys. Rev. B 82, 060518 (2010).

[183] R. T. Gordon, H. Kim, N. Salovich, R. W. Giannetta, R. M. Fernandes, V. G.Kogan, T. Prozorov, S. L. Bud’ko, P. C. Canfield, M. A. Tanatar, et al., Phys. Rev. B82, 054507 (2010).

[184] G. A. Smolenskii and I. E. Chupis, Sov. Phys. Usp. 25 (1982).

[185] J. M. L. Wang and Z. F. Ren, Adv.Phys. 58, 321 (2009).

[186] M. Lines and A. Glass, Principles and Applications of Ferroelectrics and RelatedMaterials (Oxford University Press, 2001), ISBN ISBN-10: 019850778X.

[187] T. J. Rabe K, Ahn C., Physics of ferroelectrics: a modern perspective (Springer,2007), ISBN ISBN-10: 3540345906.

[188] A. F. Devonshire, Philos. Mag. 40, 1040 (1949).

[189] W. Zhong, D. Vanderbilt, and K. M. Rabe, Phys. Rev. Lett. 73, 1861 (1994).

[190] P. Anderson, in Proceedings of the All-Union Conference on the Physics ofDielectrics (Academy of Sciences of the U.S.S.R., 1958), p. 290.

[191] W. Cochran, Phys. Rev. Lett. 3, 412 (1959).

[192] R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 59, 673 (1941).

[193] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).

152

[194] W. Y. Shih, W.-H. Shih, and I. A. Aksay, Phys. Rev. B 50, 15575 (1994).

[195] R. S. Eds., Magnetism, A Treatise on Modern Theory and Materials. Volumes I-(Academic Press, 1963-).

[196] R. White, Quantum Theory of Magnetism: Magnetic Properties of Materials 3rdEd. (Springer Series in Solid-State Sciences) (Springer, 2010), ISBN ISBN-10:3642084524.

[197] D. Mattis, The Theory of Magnetism I: Statics and Dynamics (Springer Series inSolid-State Sciences) (Springer, 1981), ISBN ISBN-10: 3540106111.

[198] E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance: ThePhysics of Manganites and Related Compounds (Springer, 2010), ISBN ISBN-10:3642077536.

[199] P. Fazekas, Lecture Notes on Electron Correlation and Magnetism (Series inModern Condensed Matter Physics) (World Scientific Pub Co Inc, 1999), ISBNISBN-10: 9810224745.

[200] T. Moriya, Phys. Rev. 120, 91 (1960).

[201] I. Dzyaloshinskii, Sov. Phys. JETP 10 (1960).

[202] E. Bertaut, J.Phys.Chem.Solids 21, 256 (1961).

[203] D. J. Dresselhaus, M., Group Theory: Application to the Physics of CondensedMatter (Springer, 2010), ISBN ISBN-10: 3642069452.

[204] V. J. Folen, G. T. Rado, and E. W. Stalder, Phys. Rev. Lett. 6, 607 (1961).

[205] I. Dzyaloshinsky, Journal of Physics and Chemistry of Solids 4, 241 (1958),ISSN 0022-3697, URL http://www.sciencedirect.com/science/article/

B6TXR-46MF5FH-BG/2/e9da76b63bd5b09338881726c973788e.

[206] D. Astrov, Sov. Phys. JETP 11 (1960).

[207] G. T. Rado, Phys. Rev. Lett. 6, 609 (1961).

[208] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426,55 (2003), URL http://www.nature.com.lp.hscl.ufl.edu/nature/journal/

v426/n6962/full/nature02018.html.

[209] N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S.-W. Cheong, Nature 429,392 (2004), URL http://www.nature.com.lp.hscl.ufl.edu/nature/journal/

v429/n6990/full/nature02572.html.

[210] L. Landau and Pitaevskii, Course of Theoretical Physics, volume 8: Electrodynam-ics of Continuous Media (Butterworth-Heinemann, 1984).

153

[211] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).

[212] M. Kenzelmann and A. B. Harris, Phys. Rev. Lett. 100, 089701 (2008).

[213] I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).

[214] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).

[215] J. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity(Clarendon Press Oxford, 2010).

[216] A. B. Sushkov, R. V. Aguilar, S. Park, S.-W. Cheong, and H. D. Drew, Phys. Rev.Lett. 98, 027202 (2007).

[217] R. Valdes Aguilar, M. Mostovoy, A. B. Sushkov, C. L. Zhang, Y. J. Choi, S.-W.Cheong, and H. D. Drew, Phys. Rev. Lett. 102, 047203 (2009).

[218] Y. Tokunaga, N. Furukawa, H. Sakai, Y. Taguchi, T.-h. Arima, and Y. Tokura,Nature materials 8, 558 (2009), URL http://www.nature.com.lp.hscl.ufl.edu/

nmat/journal/v8/n7/full/nmat2469.html.

[219] F. Kagawa, M. Mochizuki, Y. Onose, H. Murakawa, Y. Kaneko, N. Furukawa, andY. Tokura, Phys. Rev. Lett. 102, 057604 (2009).

[220] R. M. H. W. F. Brown and S. Shtrikman, Phys. Rev. 168, 574 (1968).

[221] Y. Imry, J Phys C 8, 567 (1975).

[222] Watanabe and Usui, Prog. Theo. Phys. 73, 1305 (1985).

[223] A. D. Bruce and A. Aharony, Phys. Rev. B 11, 478 (1975).

[224] J. M. Kosterlitz, D. R. Nelson, and M. E. Fisher, Phys. Rev. B 13, 412 (1976).

[225] I. Dzyaloshinskii, Euro.Phys.Lett. 83, 67001 (2008).

[226] N. Goldenfeld, Lectures on Phase transitions and critical phenomena (Perseus,1992), ISBN ISBN-10: 0201554097.

[227] B. I. Halperin, T. C. Lubensky, and S.-k. Ma, Phys. Rev. Lett. 32, 292 (1974).

[228] Zvezdin and Mukhin, JETP 89, 328 (2009).

[229] R. de Sousa and J. E. Moore, Phys. Rev. B 77, 012406 (2008).

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BIOGRAPHICAL SKETCH

G.R.Boyd defended his Ph.D. dissertation on the 23rd of November 2010.

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