Electronic States of Heavy Fermion Metals in High Magnetic Fields

Patrick M. C. Rourke

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

Electronic States of Heavy Fermion Metals in High Magnetic Fields

Patrick M. C. Rourke

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2009

Heavy fermion metals often exhibit novel electronic states at low temperatures, due to competing interactions and energy scales. In order to characterize these states, precise determination of material electronic properties, such as the Fermi surface topology, is necessary. Magnetic ﬁeld is a particularly powerful tool, since it can be used as both a tuning parameter and probe of the fundamental physics of heavy fermion compounds.

In CePb , I measured magnetoresistance and torque for 23 mK T 400 mK, 0 T 3 ≤ ≤ ≤ H 18 T, and magnetic field rotated between the (100), (110), and (111) directions. ≤ For H (111), my magnetoresistance results show a decreasing Fermi liquid temperature || range near H , and a T 2 coefficient that diverges as A(H) H H −α, with H 6 T c ∝ | − c| c ∼ and α 1. The torque exhibits a complicated dependence on magnetic field strength ∼ and angle. By comparison to numerical spin models, I find that the “spin-flop” scenario previously thought to describe the physics of CePb3 does not provide a good explanation of the experimental results.

Using novel data acquisition software that exceeds the capabilities of a traditional measurement set-up, I measured de Haas–van Alphen oscillations in YbRh Si for 30 mK 2 2 ≤ T 600 mK, 8 T H 16 T, and magnetic ﬁeld rotated between the (100), (110), and ≤ ≤ ≤ (001) directions. The measured frequencies smoothly increase as the ﬁeld is decreased through H 10 T. I compared my measurements to 4f-itinerant and 4f-localized elec- 0 ≈ tronic structure calculations, using a new algorithm for extracting quantum oscillation

ii information from calculated band energies, and conclude that the Yb 4f quasi-hole re-

mains itinerant over the entire measured field range, with the behaviour at H0 caused by a Fermi surface Lifshitz transition. My measurements are the first to directly track the Fermi surface of YbRh2Si2 across this field range, and rule out the 4f localization transition/crossover that was previously proposed to occur at H0.

iii Dedication

This thesis is dedicated to my grandfather, Professor Emeritus E. G. Bertram, who is a great inspiration to me as a scientist.

iv Acknowledgements

First and foremost, I would like to express my gratitude to my supervisor, Stephen

Julian, for patiently sharing with me his deep, intuitive understanding of complicated

physical phenomena, ﬁnding new research opportunities when our dilution refrigerator

was broken, and putting forth a heroic eﬀort to read this thesis and give me feedback

within my very short time constraints.

I would also like to thank our post-doc, Alix McCollam, for her tireless eﬀorts to

further the cause of our research group, and my fellow graduate students, Wenlong Wu,

Fazel Tafti, Aaron Sutton, Cyrus Turel, Patrick Morales, and Igor Fridman, for making

my time in the McLennan Physical Laboratories more enjoyable.

I have beneﬁted greatly from interesting discussions with John Wei and Hae-Young

Kee at the University of Toronto, Mike Norman at Argonne National Laboratory, Andriy

Nevidomskyy at Rutgers University, Suchitra Sebastian at the University of Cambridge,

Johnpierre Paglione at the University of Maryland, Makariy Tanatar at Ames Laboratory, and Ramzy Daou at the Universit´ede Sherbrooke, regarding wider physical concepts related to my work.

My appreciation also to Gerard Lapertot, Georg Knebel and Jacques Flouquet at CEA

Grenoble, and Suchitra Sebastian at the University of Cambridge, for the extremely pure

YbRh2Si2 and CePb3 samples, and Krystyna Biel and Marianne Khurana for cheerfully guiding me through the University of Toronto administrative jungle.

Finally, I would like to thank the Hart House Orchestra, Looks Linear, and the

Monday Night Jam Band for providing balance in my life and sharing the joy of music; and Robin Gallagher, my parents, my grandfather, and the rest of my family and friends for their unwavering support, both emotional and nutritional, without which this would not have been possible.

v Contents

1 Introduction 1

1.1 Fermiliquidtheory ...... 5

1.2 Quantumcriticality...... 7

1.3 Heavyfermions ...... 10

1.4 Densityfunctionaltheory...... 12

1.5 ThedeHaas–vanAlpheneﬀect ...... 17

1.5.1 Quantum oscillations at T =0...... 18

1.5.2 Damping factors and other complications ...... 22

1.5.3 The ﬁeld modulation measurement technique ...... 29

2 Instrumentation 35

2.1 Data Acquisition Virtual Instrument Experiment System ...... 36

2.1.1 Hardware ...... 38

2.1.2 TheDataSocketprotocol...... 41

2.1.3 Errorhandling ...... 43

2.1.4 Temperaturecontrol ...... 46

2.1.5 Magneticﬁeldcontrol...... 49

2.1.6 Samplerotationcontrol ...... 53

2.1.7 Dataacquisition...... 55

2.1.8 Automation ...... 60

vi 2.1.9 Monitoring ...... 62

2.1.10 Testresultsanddiscussion ...... 64

2.2 Graphiterotationmechanism ...... 66

2.3 Silverannealing...... 68

2.4 Glovebox ...... 71

3 Supercell K-space Extremal Area Finder 76

3.1 Basic concepts of Fermiology ...... 77

3.1.1 Comparisontobandstructure ...... 79

3.2 Algorithmdetails ...... 79

3.2.1 Overview ...... 79

3.2.2 k-spacesupercellconstruction...... 81

3.2.3 Fermisurfaceorbitdetection...... 84

3.2.4 dHvA frequency, eﬀective mass, and orbit type calculations.... 87

3.2.5 Slice-to-slice orbit matching ...... 88

3.2.6 Extremumdetermination...... 89

3.2.7 Density of states calculation ...... 89

3.3 Testresults ...... 90

3.4 CeCoIn5 results...... 92

4 UPt3 99 4.1 Materialbackground ...... 99

4.2 Theoretical models of the Fermi surface ...... 104

4.3 Discussion...... 107

5 CePb3 118 5.1 Materialbackground ...... 118

5.2 Experimentaldetails ...... 123

5.3 Experimentalresults ...... 129

vii 5.4 Discussion...... 136

6 YbRh2Si2 147 6.1 Materialbackground ...... 147

6.2 Electronicstructurecalculations ...... 152

6.3 Experimentaldetails ...... 155

6.4 Experimentalresults ...... 168

6.5 Discussion...... 177

7 Conclusions 181

7.1 Instrumentation...... 181

7.2 SupercellK-spaceExtremalAreaFinder ...... 182

7.3 UPt3 ...... 182

7.4 CePb3 ...... 183

7.5 YbRh2Si2 ...... 184 7.6 ListofPublications...... 186

Bibliography 188

viii List of Tables

3.1 Measured and calculated CeCoIn5 dHvA frequencies ...... 95 3.2 Calculated band masses, measured eﬀective masses, and resulting mass

enhancements in CeCoIn5 ...... 96

3.3 CeCoIn5 speciﬁcheatestimates ...... 97

4.1 Comparison of UPt3 band labelling schemes ...... 105

4.2 Description of the UPt3 orbits in the fully itinerant model ...... 107

6.1 Calculated and measured YbRh2Si2 eﬀective masses ...... 172

6.2 YbRh2Si2 speciﬁcheatestimates ...... 174

ix List of Figures

1.1 Generic quantum critical phase diagram ...... 8

1.2 Self-consistent density functional theory algorithm ...... 15

1.3 Landau tubes intersecting a spherical Fermi surface ...... 19

1.4 Temperature dependence of RT ...... 24

1.5 dHvA frequency back-projection illustration ...... 28

1.6 Set-up of a traditional ﬁeld modulation dHvA experiment ...... 31

1.7 Bessel function Jν(λ) for several values of ν ...... 33

2.1 Overview of the Data Acquisition Virtual Instrument Experiment System

(DAVIES)...... 37

2.2 Diagram of the DAVIES temperature control subsystem ...... 46

2.3 Diagram of the DAVIES magnetic ﬁeld control subsystem ...... 49

2.4 Diagram of the DAVIES sample rotation control subsystem ...... 54

2.5 Diagram of the DAVIES data acquisition subsystem ...... 56

2.6 Diagram of the DAVIES automation subsystem ...... 61

2.7 Diagram of the DAVIES monitoring subsystem ...... 62

2.8 DAVIES virtual lock-in vs. SR830 lock-in ampliﬁer test results...... 65

2.9 Graphiterotationmechanism ...... 67

2.10 Silver heat-sink wire annealing apparatus ...... 69

2.11 Gloveboxgashandlingsystem ...... 72

x 3.1 The band 2 Fermi surface of UPt3, tiled in several Brillouin zones . . . . 78

3.2 AnexampleSKEAFsupercell...... 82

3.3 SKEAFalgorithmﬂowchart...... 85

3.4 SKEAFexampleslice...... 86

3.5 dHvA frequency vs. magnetic ﬁeld angle for test Fermi surfaces ..... 91

3.6 CeCoIn5 crystalstructure ...... 93

3.7 CeCoIn5 Fermisurface ...... 94

4.1 UPt3 crystalstructure ...... 101

4.2 UPt3 Fermi surface sheets generated from the fully itinerant model . . . 103

4.3 Major UPt3 Fermi surface sheets generated from the partially localized model ...... 106

4.4 Predicted angle dependence of UPt3 dHvA frequencies in the fully itinerant model ...... 108

4.5 Predicted angle dependence of UPt3 dHvA frequencies in the partially localizedmodel ...... 109

4.6 Angle dependence of measured UPt3 quantum oscillation frequencies compared to the predictions of the fully itinerant model ...... 111

4.7 Angle dependence of measured UPt3 quantum oscillation frequencies compared to the predictions of the partially localized model ...... 112

4.8 Angle dependence of predicted and measured UPt3 frequencies associated withband2ofthefullyitinerantmodel...... 115

4.9 Predicted UPt3 L-2 orbits on the extended-zone band 2 Fermi surface sheet116

5.1 CePb3 crystalstructure...... 119

5.2 Ambient pressure phase diagram of CePb for H (110) ...... 121 3 || 5.3 Sketch of the spin sub-lattice and total magnetization orientations in the

spin-ﬂopscenario ...... 122

xi 5.4 CePb3 resistivityandtorquesamples ...... 125

5.5 CePb3 acsusceptibilitysample...... 127

5.6 CePb3 magnetoresistance at diﬀerent ﬁeld angles ...... 130

5.7 CePb3 resistivity vs. temperature, and quadratic fit coefficients, between 2 and 12 T, for H (111) ...... 132 || 2 5.8 Low temperature Fermi liquid fits of CePb3 resistivity vs. T , and divergence of the A coefficient near H 6 T, for H (111)...... 133 c ∼ ||

5.9 Measured CePb3 torque vs. ﬁeld, for several ﬁeld directions ...... 135

5.10 CePb ac susceptibility vs. ﬁeld, for H (110), H (111), and H (100) . . 136 3 || || || 5.11 2D spin model total energy landscape at H = 0 T and H = 4 T, for H

near(110) ...... 139

5.12 3D spin model magnetization vs. ﬁeld for various magnetic ﬁeld angles . 141

5.13 3D spin model spin angles vs. ﬁeld for various magnetic ﬁeld angles . . . 142

5.14 3D spin model torque magnitude vs. ﬁeld for H (111)...... 144 ||

6.1 YbRh2Si2 crystalstructure...... 148

6.2 “Small” and “large” Fermi surfaces of YbRh2Si2 ...... 149

6.3 Ambient pressure phase diagram of YbRh Si for H c ...... 151 2 2 ⊥

6.4 YbRh2Si2 LDA+SOC band structure and total density of states versus energy,forthelargeFScase ...... 154

6.5 The large, small and two intermediate Fermi surfaces of YbRh2Si2 . . . . 156

6.6 Predicted YbRh2Si2 dHvA frequencies vs. magnetic ﬁeld angle for the smallFScase ...... 157

6.7 Predicted YbRh2Si2 dHvA frequencies vs. magnetic ﬁeld angle for an in- termediateFScase ...... 158

6.8 Predicted YbRh2Si2 dHvA frequencies vs. magnetic ﬁeld angle for another intermediateFScase ...... 159

xii 6.9 Predicted YbRh2Si2 dHvA frequencies vs. magnetic ﬁeld angle for the large FScase ...... 160

6.10 LAP-420 batch of YbRh2Si2 single crystal samples ...... 161

6.11 YbRh2Si2 resistivity sample and normalized ρ(T ) ...... 162

6.12 Laue x-ray diﬀraction patterns for YbRh2Si2 samples A and C, with tetragonal crystal structure prediction using the LaueX program ...... 163

6.13 Pick-up coils for YbRh2Si2 samplesAandC ...... 164

6.14 Schematic diagram of YbRh2Si2 dHvA experimental apparatus ...... 166

6.15 YbRh2Si2 coil angles plotted versus CeRu2Si2 coil angle ...... 168

6.16 YbRh2Si2 quantum oscillations and FFT ...... 169

6.17 YbRh2Si2 dHvA frequency vs. magnetic ﬁeld angle ...... 171

6.18 Lifshitz-Kosevich ﬁt to the temperature dependence of the YbRh2Si2 u2 orbitdHvAamplitude ...... 173

6.19 Measured YbRh2Si2 dHvA frequencies vs. magnetic ﬁeld strength . . . . 175

6.20 Measured YbRh2Si2 eﬀective masses vs. magnetic ﬁeld strength . . . . . 176

6.21 Schematic representation of the YbRh2Si2 bands near the Fermi energy . 179

xiii Chapter 1

Introduction

Since prehistoric times, humans have obtained materials from their surroundings and

attempted to determine their properties, in order to transform them into useful tools.

While such raw materials initially consisted of stone, animal products and plant mat-

ter, the development of metallurgical techniques launched the Bronze Age (beginning

around 3200 BCE [1]) and Iron Age (beginning around 1200 BCE [1]), securing a domi-

nant role for metals in human technological and societal development ever since [2], and

underpinning the Industrial Revolution, Space Age, and current “Information Age.”

Magnetism was known to the ancient Greeks and Sumerians through the bizarre

properties of lodestones—naturally-occurring magnets composed of an iron oxide called

magnetite (Fe3O4) [1]. In fact, the early observation that a lodestone could attract small pieces of iron can be thought of as the first “experiment” to explore the effects of magnetic fields on the behaviour of metals. While lodestones were then routinely employed to magnetize iron needles for use in navigational compasses, it was not until the work of William Gilbert in 1600 CE that a comprehensive scientific study of magnetism appeared [3]. Understanding of the more fundamental nature of magnetic phenomena came in the 19th and 20th centuries with the development of theories of electromagnetism and quantum mechanics.

1 Chapter 1. Introduction 2

In modern condensed matter physics, the behaviour of conduction electrons in metals and related compounds is a major field of study. When investigating a new material, experimental condensed matter physicists attempt to determine the electronic landscape and then observe how this changes as they modify interaction energies, excite quasiparticles, explore phase transitions, and tune between different ground states in order understand the fundamental physical processes at work. This is mainly done by chemical substitution (doping), applied pressure, or applied magnetic field. Magnetic field is a particularly powerful probe: unlike doping, it is clean and homogeneous, and can be tuned in situ on a single sample, thus avoiding the complications involved in comparing different physical samples which have different dopings; unlike pressure, it can be tuned continuously at low temperatures, in an easily-controllable manner, over a wide parameter range, with very fine resolution; and it is also sensitive to material anisotropies. Converted to temperatures via Boltzmann’s constant kB, typical superconducting magnets can perturb a system in an energy range roughly equivalent to 0–15 K and larger pulsed magnets 0–

50 K or higher—much smaller than typical Fermi temperatures T = E /k 104 K F F B ∼ of simple metals, but comparable to the lower T 10–100 K Fermi temperatures often F ∼ found in heavy fermion systems. (Throughout this thesis, the applied magnetic ﬁeld H~ and magnetic induction ﬁeld B~ = µ0(H~ + M~ ) are used interchangeably, ignoring the induced magnetization M~ except where explicitly stated.)

Indeed, heavy fermions, whose internally-competing interactions tend to produce exotic states, are a particularly intriguing class of materials. All of the compounds studied during my Ph.D. are heavy fermion metals in which interesting electronic properties are induced by applied magnetic fields: in CeCoIn5, exotic superconductivity is suppressed and a quantum critical point, with associated non-Fermi liquid behaviour, is induced [4, 5]; in UPt3, a different kind of exotic superconductivity is suppressed, followed by a metamagnetic transition at much higher fields [6]; in CePb3, antiferromagnetism is transformed into field-aligned ferromagnetism, with strange behaviour at intermedi- Chapter 1. Introduction 3

ate ﬁelds [7]; and in YbRh2Si2, antiferromagnetism is suppressed to a quantum critical point [8], with associated non-Fermi liquid behaviour [9], followed by suppression of the

Kondo physics at higher ﬁelds [10]. In CeCoIn5 and UPt3, where I performed calculations in support of experiments carried out by others, the experimental focus was not on the ﬁeld-induced states themselves, but rather on generally interpreting the magnetic-

field-based de Haas–van Alphen effect. In CePb3 and YbRh2Si2, where I did both the experimental measurements and calculations, the field-induced states, transitions, and

crossovers were indeed my primary interest.

My doctoral supervisor, Prof. Stephen Julian, arrived at the University of Toronto

one month before I began my Ph.D. An enormous amount of eﬀort was therefore spent in

the early years by our small research group—Prof. Julian, our post-doc Alix McCollam,

myself, and one other Ph.D. student, Wenlong Wu—to build our lab. The delivery of

an Oxford Instruments Kelvinox 400MX 3He/4He dilution refrigerator, the centrepiece of our experimental set-up and an absolute necessity for my experiments to probe the ground states of heavy fermion metals, was delayed by 8 months due to problems at the factory. Once the dilution refrigerator arrived in Toronto, however, during the course of a few testing runs its performance degraded rapidly to an unusable state: a manufacturing defect had led to a crossover leak between the 3He condenser line and the still line. The equipment had to be sent back to the factory in England twice to ﬁx this problem (the

first “fix” was ineffective), and this delayed my experimental plans by at least two years.

In the meantime, I travelled to the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida to perform a few experiments, and computational modelling work took on a more major role in my studies. Due primarily to the dilution refrigerator defect, our lab only became fully operational within the past year, but since then the system has run smoothly and I have been able to perform a successful experiment using the instrumentation I developed.

The remainder of this chapter is devoted to brief explanations of important physical Chapter 1. Introduction 4

concepts upon which the rest of the thesis is based: Fermi liquid theory, quantum crit-

icality, heavy fermions, density functional theory, and the de Haas–van Alphen eﬀect.

This is not meant to be a comprehensive review of any of these topics, but rather should

provide theoretical grounding suﬃcient to understand the following chapters. Beyond

the introduction, the layout of this thesis follows my Ph.D. research as our laboratory

evolved from an empty room to a leading-edge low-temperature/high-magnetic-ﬁeld fa-

cility. Chapter 2 describes my new instrumentation contributions to the lab, focusing on

the hardware/software infrastructure I created to control and collect data from our dilu-

tion refrigerator and superconducting magnet set-up. Chapter 3 contains an algorithm

I developed to extract theoretical predictions for de Haas–van Alphen measurement re-

sults from electronic structure calculations [11], and an application of this algorithm to

CeCoIn5 [12]. The most significant application of this algorithm has been to UPt3, in which several new predicted orbits that agree with experiment were discovered in old band structure data, helping to resolve a controversy in the literature [13]. The results of the UPt3 work are presented in chapter 4. In chapter 5, a series of electrical resistivity, ac susceptibility and torque magnetometry measurements performed on CePb3 as a function of magnetic field strength, magnetic field angle and temperature at the

NHMFL are described. De Haas–van Alphen Fermi surface measurements of YbRh2Si2 through the Kondo suppression scale H 10 T [14, 15] are presented and discussed in 0 ≈

chapter 6. The YbRh2Si2 measurements represent the culmination of my doctoral work, since they were performed in our Toronto laboratory using the data collection system

of chapter 2 and extensively modelled with the help of the algorithm of chapter 3. The

main conclusions of chapters 2–6 are summarized in chapter 7. Chapter 1. Introduction 5

1.1 Fermi liquid theory

At the heart of modern understanding of metals lies Lev Landau’s notion of the Fermi

liquid. In the simplest view, a metal is a dense, periodic array of atoms (that is, a crystal)

in which some electrons are free to ﬂoat around—rather than being bound to a particular

atom in the lattice, they are shared throughout the entire solid. It is from these so-called

“free” or “conduction” electrons that metals derive most of their properties. However,

the task of describing the physics of such a system in detail is formidable, since there are

typically 1023 electrons and a similar number of ionic cores all aﬀecting one another ∼ via the Coulomb interaction.

An early treatment by Felix Bloch [16], now known as the independent electron approximation, ignored the interactions between electrons entirely, focusing instead on the quantum mechanical description of a non-interacting gas of fermions moving in the periodic potential generated by a set of stationary atomic cores. In this framework, the eigenstates of the system, called Bloch states, are wave functions that extend over the entire crystal and are indexed by quantum numbers ~k, the wave vector (related to the three-dimensional momentum), n, the band index, and σ, the spin. According to the

Pauli exclusion principle, a maximum of two electrons (one spin-up and one spin-down) can occupy each state. The zero-temperature ground state of the system is constructed by putting the N conduction electrons into the N/2 lowest-energy Bloch states, with the energy of the highest occupied state being called the Fermi energy, EF . For certain bands (i.e. certain values of n), all k-states may have energies below EF , and thus will all be occupied; other bands may lie above the Fermi energy and be entirely unoccupied.

In bands that cross the Fermi energy, however, there will be some occupied states and some unoccupied states, separated from one another in momentum-space (also called

~k-space or simply k-space) by a surface of constant energy En(~k) = EF , known as the Fermi surface (FS). At ﬁnite temperatures, electrons can be excited from their occupied states up to higher energy unoccupied states, but this is only possible for states within Chapter 1. Introduction 6

a relatively narrow energy range k T around E . Therefore, only a limited number ∼ B F of states which happen to lie close enough to the Fermi surface are able to contribute to the bulk electronic properties of the material, leading to the importance of Fermi surface topology in the physics of metals.

The independent electron approximation has successfully explained the results of a wide range of experiments on real metals. This is surprising because real electrons do, in fact, repel one another quite strongly, and Bloch’s treatment entirely omits this interaction. Thirty years after Bloch’s initial formulation, this mystery was explained by Landau via the concept known today as the Landau Fermi liquid [17, 18, 19], in which the success of the Bloch model is seen to be largely due to screening, which makes the Coulomb repulsion force short-ranged. Landau imagined starting from the non-interacting picture proposed by Bloch, and then slowly, adiabatically turning on the interactions between electrons. In this manner, the Bloch states of the non-interacting system continuously evolve into quasiparticle states of the interacting system, with a one-to-one correspondence between them.

While the resulting quasiparticle states are no longer eigenstates of the system, and therefore have ﬁnite lifetimes, they share many similarities with the original Bloch states.

Like the Bloch electrons, Landau quasiparticles obey Fermi statistics and are indexed by the quantum numbers ~k, n and σ. There is also a well-defined quasiparticle Fermi surface, whose volume is the same as in the non-interacting case [20]. In fact, in most respects, the complicated many-body quasiparticles states making up the Fermi liquid resemble the regular electrons treated as a Fermi gas by Bloch; much of the effect of the previously- neglected interactions is wrapped up into the quasiparticle effective mass, m∗, which can vary substantially from the bare electron mass, me (section 1.3).

One potential problem with the Landau Fermi liquid scheme is the ﬁnite quasiparticle

lifetime: if the quasiparticles decay faster than the time required to adiabatically turn

on the interactions, the smooth evolution from and one-to-one relationship with the non- Chapter 1. Introduction 7 interacting Bloch states are lost. Fortunately, for quasiparticle states close to the Fermi surface, the phase space for scattering is greatly restricted by energy conservation and the Pauli exclusion principle, leading to long quasiparticle lifetimes. Hence the Fermi liquid paradigm is valid only near the Fermi surface. This is generally not troublesome, because, as in the Bloch treatment, only the states close to the Fermi surface contribute to the electronic properties of a metal. At ﬁnite temperatures, thermal excitations broaden the phase space available for quasiparticle scattering, thus reducing the quasiparticle lifetime. This reinforces the need to work at low temperatures when experimentally probing metallic compounds, and also gives rise to one of the hallmark results of Fermi liquid theory: electrical resistivity which varies as T 2.

1.2 Quantum criticality

Fermi liquid theory (section 1.1) underpins much of modern condensed matter physics, and as such, a great deal of research continues to be devoted to studying when and how its predictions break down. The most striking example of so-called non-Fermi-liquid behaviour occurs in the vicinity of a Quantum Critical Point (QCP).

First discussed by John Hertz [21], a quantum critical point occurs when a classical phase transition is shifted to zero temperature by some non-thermal tuning parameter.

The relevant tuning parameter or parameters change from material to material, but include such quantities as hydrostatic pressure, magnetic ﬁeld, and chemical doping. A generic temperature vs. tuning parameter phase diagram showing the location of the quantum critical point is sketched in Fig. 1.1.

Starting at the left side of Fig. 1.1, at one extreme of the tuning parameter, there is a classical second-order phase transition from a high-temperature disordered state to a lower-temperature ordered state (“QM state #1” in the ﬁgure). This ordered state is often a form of magnetism, for example antiferromagnetism (AFM). Moving from low to Chapter 1. Introduction 8

QM critical (eg. T ) fluctuation regime (eg. T ) N (non-Fermi liquid) FL

QM state #1 QM state #2 (eg. AFM-ordered) (eg. non-magnetic Fermi liquid) Temperature

QM state #3 (eg. exotic SC) T = 0 K • unique ground state #1 QCP unique ground state #2 Tuning parameter

Figure 1.1: A generic temperature vs. tuning parameter phase diagram sketch illustrating quantum criticality.

high temperature across the classical phase transition, thermal fluctuations destroy the long-range order. The second moment of these fluctuations diverges at the transition temperature. As the temperature is lowered to T = 0 K, the thermal fluctuations sub- side, giving way to a unique quantum ground state (“unique ground state #1”). With increasing tuning parameter, the transition temperature of QM state #1 is suppressed to zero temperature. At T = 0 K, there are no thermal fluctuations to drive the phase transition, and it becomes truly quantum, rather than classical in nature; this point is defined as the quantum critical point.

To the right of the quantum critical point, at the opposite extreme of tuning parameter, lies a diﬀerent quantum state (“QM state #2” in Fig. 1.1), which is often a non-magnetic Fermi liquid. At T = 0 K, this state has its own unique quantum ground state (“unique ground state #2”), which diﬀers from the ground state of QM state #1.

Close to the quantum critical point, the energies of ground states #1 and #2 are sim- Chapter 1. Introduction 9 ilar enough that quantum fluctuations, governed by Heisenberg’s uncertainty principle, can flip increasingly large regions of the system back and forth between the two. These quantum fluctuations between competing ground states get stronger as the ground state energies approach one another, driving the quantum phase transition at the quantum critical point.

At elevated temperatures near the quantum critical point, the system exists in a quantum critical regime (“QM critical fluctuation regime” in Fig. 1.1) where physical properties are not derived from either QM state #1 or QM state #2, but instead are dominated by the quantum fluctuations mentioned above, and exhibit behaviour markedly different from both parent states. In particular, both QM states #1 and #2 are typically Fermi liquids (albeit with potentially different details within the Fermi liquid picture, such as Fermi surfaces, quasiparticle effective masses, etc.), whereas in the quantum critical regime, non-Fermi-liquid behaviour is manifested, for example in power-law temperature dependence of the electrical resistivity that deviates from the quadratic prediction of

Landau. As the temperature rises, thermal ﬂuctuations increasingly mix with the quantum ﬂuctuations, smearing out the energy scales involved and widening the quantum critical regime.

Another way of looking at the quantum critical picture is that at either extreme of tuning parameter there is a unique ground state, but at the quantum critical point, where both parent ground states are equally energetically favourable, the quantum uniqueness requirement is broken. Faced with the impossible choice of ground state #1 vs. ground state #2, the system often “chooses not to choose” and instead settles into an entirely new state (“QM state #3”), with its own unique ground state, built upon the quantum

fluctuations between the two competing parent ground states. This novel emergent state is usually confined to a small region of the phase diagram around the quantum critical point, where quantum fluctuations are strong and thermal fluctuations are weak, and often takes the form of exotic superconductivity, in which Cooper pairing is thought to Chapter 1. Introduction 10 be mediated by the quantum critical fluctuations themselves.

Quantum criticality has been discussed in detail elsewhere [22]. Quantum critical points have been found in heavy fermion compounds (see section 1.3) such as

CePd2Si2 [23], CeIn3 [23] and its quasi-2D analogue CeCoIn5 (chapter 3) [4, 5], and

YbRh2Si2 (chapter 6) [8]. While not conclusively conﬁrmed, quantum criticality has

been suggested as the underlying the physics of the high-Tc cuprate superconductors as well.

1.3 Heavy fermions

As discussed in section 1.1, the fermionic quasiparticles of a Landau Fermi liquid resemble

real electrons in many ways, albeit often diﬀering in mass. A “heavy fermion” material

is one in which strong interactions renormalize the quasiparticle eﬀective mass, m∗, to

be much greater than the bare electron mass, me; in some cases, more than 100 times greater. Compounds of this class tend to be intermetallics containing the f-electron

atoms cerium (Ce 4f 1), uranium (U 5f 3), or ytterbium (Yb 4f 13, the hole analogue of

Ce 4f 1). The f-electrons are the key to the heavy fermion state, because they take on

large magnetic moments and sit close to the atomic cores, but also readily hybridize

with the conduction electrons. The resulting competition between two types of magnetic

interaction, the Kondo and Ruderman-Kittel-Kasuya-Yosida (RKKY) eﬀects, leads to

the unique properties of heavy fermions compounds. Due to their rich intrinsic physics,

similarities with the dirtier high-Tc cuprates, and wide availability of high-purity single crystals, heavy fermions are the focus of intense condensed matter research, and include

all of the materials studied in this thesis: CeCoIn5 (chapter 3), UPt3 (chapter 4), CePb3

(chapter 5), and YbRh2Si2 (chapter 6).

The Kondo eﬀect was ﬁrst discovered in the context of magnetic atoms (e.g. chromium,

manganese, iron, molybdenum, rhenium, osmium) sprinkled sparsely within a non-mag- Chapter 1. Introduction 11 netic metal (e.g. copper, silver, gold, magnesium, zinc) [24]. At high temperatures, the magnetic impurity atoms act like local moments. Below the characteristic Kondo temperature, TK , the conduction electrons’ spins start to screen the local moments, by forming quasi-bound clouds of opposite spin-polarization around the magnetic impurity atoms.

At T = 0, this process is complete, and the magnetic impurities are totally screened.

In the context of heavy fermion materials, f-electrons, which are nearly localized and possess sizable magnetic moments, play the role of the magnetic impurity atoms; the f-moments are spin-screened by the conduction electrons at low temperatures.

In contrast to Kondo screening, where the conduction electrons weaken the interaction between neighbouring magnetic moments, the RKKY interaction channel uses the conduction electrons to strengthen the tendency to magnetic order. In the RKKY picture, a local moment polarizes the nearby conduction electrons, and this polarization is in turn felt by adjacent local moments, indirectly strengthening the interaction between neighbouring spins. At low temperatures, the RKKY interaction favours a ground state in which the local moments are well-deﬁned and magnetically-ordered (usually an antiferromagnetic state), and the conduction electrons behave as a Fermi liquid.

The energy scales of both the Kondo eﬀect and RKKY interaction are tied to the exchange coupling J between the mostly-localized f-moments and the conduction electrons:

T e−1/|J|N(EF ) and T J 2, where N(E ) is the density of states at the Fermi K ∼ RKKY ∼ F

energy. At low J, TRKKY > TK and the RKKY interaction dominates, giving rise to a

magnetic ground state. At high J, TK > TRKKY and the Kondo eﬀect dominates, giving rise to a non-magnetic ground state [25]. Heavy fermion materials are special in that

they tend to lie in the region where the strengths of these two channels are comparable.

This often allows the ground state to be shifted between magnetic and non-magnetic

variants via experimentally-accessible values of various tuning parameters, giving rise

to quantum criticality resembling that shown in Fig. 1.1. The strong coupling between

the conduction band and nearly-localized f-states results in heavily-renormalized quasi- Chapter 1. Introduction 12 particles, which, while still itinerant, have very large eﬀective masses. In eﬀect, these quasiparticles can be thought of roughly like a compromise between localized f-electrons and itinerant conduction electrons: they continue to move freely through the metal, but do so very slowly.

1.4 Density functional theory

As mentioned in the Fermi liquid section (1.1), a metal is a periodic array of atoms among which some electrons are “free” to move. Out of the complicated many-body interactions hosted by such a system, interesting physics such as quantum criticality (section 1.2) and heavy fermion behaviour (section 1.3) can emerge. In order to untangle and understand the novel physics arising in new metallic compounds, a wide range of theoretical and experimental techniques must be brought to bear. A powerful theoretical tool, employed extensively throughout the rest of this thesis to model intriguing metallic systems, is the so-called “density functional theory” (DFT). Through a series of clever assumptions and mathematical manipulations, DFT allows the electronic structure, including the Fermi surface, of a material to be calculated; the results can then be compared to experiment.

This section will provide a ﬂeeting glimpse of DFT, following the excellent book by

Stefaan Cottenier on the subject [26]; interested readers are encouraged to use Cottenier’s book as a starting point for further reading.

The exact many-body Hamiltonian of a metal can be written

ˆ ˆ ˆ ˆ ˆ ˆ H = Te + Tion + Ve-e + Ve-ion + Vion-ion (1.1)

where Tˆe is the kinetic energy operator of the conduction electrons (henceforth, when I talk about electrons, I am referring to the conduction electrons: all other electrons are included in the ionic cores), Tˆion is the kinetic energy operator of the ionic cores (i.e. the atoms of the crystal, which are now positively charged ions because they have Chapter 1. Introduction 13

ˆ donated some electrons to the conduction pool), Ve-e is the potential energy operator ˆ related to electron-electron interactions, Ve-ion is the potential energy operator related to ˆ electron-ion interactions, and Vion-ion is the potential energy operator related to ion-ion interactions. Since the ions are much heavier and slower-moving than the electrons, the

ﬁrst simpliﬁcation on the road to DFT is to assume that the ions are stationary. We are left with ˆ ˆ ˆ ˆ H = Te + Ve-e + Ve-ion (1.2)

ˆ ˆ in which Te and Ve-e are universal for any interacting electron system, with all of the details that vary from system to system being wrapped up in the external potential ˆ exerted on the electrons by the ions, Ve-ion.

While the assumption that the ions are fixed helps to simplify the Hamiltonian considerably, the complication of dealing with an interacting electron gas still makes it too difficult to solve in its current form. The next step, and the reason for the name “density functional theory,” comes from a theorem of Pierre Hohenberg and Walter Kohn: the ground-state expectation value of any quantum mechanical observable is a unique functional of the ground-state electron density, ρ(~r), which in turn has a one-to-one cor- ˆ ˆ respondence with Ve-ion [27]. If only we could figure out ρ(~r) from Ve-ion, we would be all set!

Unfortunately, the complicated electron-electron interactions have not yet been dealt

with. A key development, due to Walter Kohn and Lu Sham, was to temporarily sweep

these under the carpet by rearranging Eq. 1.2 to obtain the Kohn-Sham Hamiltonian [28]

ˆ ˆ ˆ ˆ ˆ HKS = Te,non-int + VH + Vxc + Ve-ion (1.3)

which describes an electron gas feeling three independent, self-consistent potentials: the

well-known Hartree potential, VˆH ; the “exchange-correlation” potential, Vˆxc; and the ˆ potential due to the ions, Ve-ion. The Kohn-Sham Hamiltonian can be used to construct Chapter 1. Introduction 14 the Schr¨odinger-like Kohn-Sham equation

HˆKSφi = Eiφi (1.4)

which can be solved for eigenvalues, Ei, and eigenvectors, φi. These, in turn, give the

ground-state electron density, ρ(~r). Since VˆH and Vˆxc depend on ρ(~r), the basic DFT computational algorithm (shown in Fig. 1.2) consists of: (1) picking a trial density, (2) calculating VˆH and Vˆxc from it, (3) solving the Kohn-Sham equation to get a new density, and then (4) repeating (2) and (3) until the density is self-consistent.

During the preceding discussion, the troublesome electron-electron interactions were conveniently hidden away in Vˆxc. However, in order to execute the self-consistency algorithm and ﬁnd the ground-state density, the exchange-correlation functional must be able to be determined from the density. This is not a simple task, and further approx- imations are required. The simplest approach, called the Local Density Approximation

(LDA), divides the material into inﬁnitesimally-small volumes, assumes that the density is constant within each local volume, and assumes that the exchange-correlation potential within each volume is that of the homogeneous electron gas (which is known), unaﬀected by the neighbouring local volumes. The exchange-correlation contributions are then inte- grated over the entire material. Despite its simplicity, LDA has been very successful, and, in fact, its name is now synonymous with DFT, even when more advanced approaches are employed. An important improvement to LDA is the Generalized Gradient Approx- imation (GGA), in which the exchange-correlation potential within each local volume is taken to depend on the density gradient with adjacent local volumes as well as the

(assumed) constant density of the given local volume. All DFT calculations that I have performed during my doctoral studies use the version of GGA by John Perdew, Kieron

Burke and Matthias Ernzerhof [29], which is standard practise for bulk intermetallic compounds.

Chapter 1. Introduction 15

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Figure 1.2: Basic self-consistency algorithm of density functional theory, used to ﬁnd the ground-state electron density. Chapter 1. Introduction 16

In order to solve the Kohn-Sham equation (Eq. 1.4), one needs a set of basis functions, in which the Hamiltonian is diagonalized and from which the eigenvectors are constructed. The basis functions are deﬁned in the crystalline unit cell, with periodic boundary conditions at the cell boundaries, and are hopefully reasonably similar to the true eigenvectors (to reduce computational load) without introducing bias into the calculations. Among the most popular ways of constructing basis sets are the “Augmented

Plane Wave” (APW) method and its derivatives. The APW method divides the unit cell into two kinds of region: “muﬃn tin” spheres surrounding each atom, and interstitial space between the spheres. Outside the spheres, the basis functions are plane waves, whereas inside the spheres they are spherical harmonics, constrained so that the value of each ~k-indexed basis function is continuous across boundary between muﬃn tin and

interstitial regions.

The WIEN2k [30] code, employed in my calculations presented in this thesis, extends

the APW method by including additional “local orbital” basis functions, relativistic cor-

rections, and spin-orbit coupling, and allowing combined use of APW and an APW vari-

ant called the “Linearized Augmented Plane Wave” (LAPW) method. While WIEN2k

is an all-electron code, the electron states are divided into two types: those with energies

less than a user-controlled parameter Ecut are called “core” states, which do not partici- pate in chemical bonding, and must therefore be confined entirely within the muffin tin spheres through the choice of sufficiently large muffin tin radius, RMT , for each atom;

those with energies greater than Ecut are called “valence” states, and are allowed to leak out of the muﬃn tin spheres. Furthermore, the potentially-inﬁnite series of ~k-indexed

basis functions is truncated at a cut-oﬀ value called Kmax, which, for technical reasons,

min min is speciﬁed by the user through the parameter RMT Kmax, where RMT is the radius of the smallest muﬃn tin in the unit cell. The ground-state eigenvectors of the system are built,

and the eigenvalues (that is, the band energies) calculated, from these basis functions

on a three-dimensional grid of k-points in the Brillouin zone: accurate Fermi surface Chapter 1. Introduction 17 construction (chapter 3) requires many k-points, but this comes at a cost of increased

computational resources.

Given suﬃcient computational power, density functional theory is a powerful tool in

the study of metals, but it does have limitations. Primarily, it does not capture the

complex dynamics important to quantum criticality (section 1.2) and heavy fermions

(section 1.3). Thus, it is typical that calculated band masses are much smaller than the

heavily-renormalized quasiparticle eﬀective masses measured experimentally in heavy

fermion compounds, and calculated Fermi surfaces agree with experimentally-measured

Fermi surfaces in shape but not necessarily in size.

1.5 The de Haas–van Alphen eﬀect

While Density Functional Theory calculations (section 1.4) provide a route to modelling

the unrenormalized band structure, and therefore the Fermi surface, of a metal, such the-

oretical models must be compared to experimental measurements in order to determine

the electronic landscape of real materials. In this sense, measurements of quantum os-

cillatory phenomena such as the de Haas–van Alphen eﬀect (dHvA) are very important,

since they represent the most direct way to experimentally measure the Fermi surface

topology and quasiparticle eﬀective mass of real metals.

Measurement of the de Haas–van Alphen eﬀect constitutes the central experimen-

tal technique of this thesis, supported by specialized instrumentation (chapter 2) and

computational work (chapters 3 and 4), and as such is introduced in some detail here.

For simplicity, my description will focus on the idealized case of a metal with quadratic

dispersion and spherical Fermi surface, discussed in the independent electron picture and

neglecting electron spin, with comments about applicability to more complicated cases

inserted as appropriate. A full exposition of the physics of quantum oscillations, includ-

ing history of the technique and derivations of important equations, can be found in the Chapter 1. Introduction 18 comprehensive book by David Shoenberg [31], upon which this section is based.

1.5.1 Quantum oscillations at T = 0

As described in section 1.1, at zero temperature the Fermi surface of a metal separates occupied and unoccupied electron states in k-space. In zero magnetic ﬁeld, these states

are equally spaced throughout the Brillouin zone. However, in the presence of a magnetic

ﬁeld, the conduction electrons undergo helical cyclotron motion in real space. Such

motion is quantized, constraining the corresponding k-space states to lie on and orbit

around concentric tubes, called “Landau tubes,” which are aligned along the direction

of the magnetic ﬁeld and indexed by the quantum number n 0. As shown in Fig. 1.3, ≥ only the tube portions within the Fermi surface may host occupied states, but since

these tube sections must accommodate the same number of occupied states as originally

contained within the Fermi volume when no magnetic ﬁeld was present, each tube is

highly degenerate.

The cross-sectional area, an, of each Landau tube perpendicular to the direction of the magnetic ﬁeld is directly proportional to the magnetic ﬁeld strength H [32]:

1 2πe a = n + H (1.5) n 2 ~

Thus, if the applied field is gradually increased, the tubes’ diameters increase until, one by one, they completely exit the Fermi volume. Since the tube sections remaining within the the boundaries of the Fermi surface must still contain all of the occupied states, their degeneracy increases with the field strength as well. The result is that, as the field is increased and a Landau tube approaches the edge of the Fermi surface, the density of states at the Fermi surface increases; however, once the tube leaves the Fermi surface, it cannot host occupied states and therefore rapidly depopulates, leading to a sudden drop in the density of states at the Fermi surface. Chapter 1. Introduction 19

Figure 1.3: Landau tubes intersecting a spherical Fermi surface. Only the sections of tube inside the Fermi surface have occupied states. In this case, only the orbit around the equator is extremal, so only one dHvA frequency would be observed.

If the magnetic ﬁeld strength H is swept over a wide enough range, several Landau tubes will successively exit the Fermi volume and depopulate, creating an oscillation in the density of states at the Fermi energy. Algebraic manipulation of Eq. 1.5 shows that the oscillation is periodic in 1/H, with a frequency F that is directly proportional to the

cross-sectional area of the Fermi surface perpendicular to the ﬁeld direction, at the point

where the Landau tubes exit the Fermi volume—that is, the Fermi surface extremal area

Aext:

F =(~/2πe)Aext (1.6)

Quantum oscillations therefore constitute a direct probe of the Fermi surface. If the

Fermi surface has multiple sheets, or if a single sheet has multiple extremal areas for a given ﬁeld direction, several oscillations with diﬀerent frequencies will be superimposed.

Such multiple frequencies can be resolved by performing a Fast Fourier Transform (FFT) Chapter 1. Introduction 20 in 1/H. By rotating the ﬁeld and observing the ﬁeld-angle dependence of any detected frequencies, Eq. 1.6 allows the Fermi surface topology to be determined.

For the simple spherical Fermi surface shown in Fig. 1.3, only one frequency would be observed, corresponding to the cross-sectional area at the equator, and this frequency would stay the same no matter which direction the magnetic ﬁeld pointed. A cylindrical

Fermi surface would also only give rise to a single frequency, but in contrast to the spherical case, this frequency would increase like 1/ cos φ as the ﬁeld is rotated an angle φ

away from the cylindrical Fermi surface axis. As seen in chapters 3, 4 and 6 of this thesis,

Fermi surfaces of metallic systems of current interest to the condensed matter community

tend to be much more topologically complicated, with many frequencies appearing and

disappearing at various magnetic ﬁeld angles.

From a practical point of view, these so-called “quantum oscillations” show up exper-

imentally in any physical quantity that depends on the density of states at the Fermi sur-

face. First measured by Wander de Haas and Pieter van Alphen in bismuth in 1930 [33],

quantum oscillations manifested in the magnetization of a metal are known as the de

Haas–van Alphen eﬀect (dHvA). Similarly, the Shubnikov–de Haas eﬀect (SdH) refers to

quantum oscillations detected via magnetoresistance measurements [34, 35]. Although

less common than dHvA and SdH measurements, due to technical challenges of detecting

small signals in the presence of large backgrounds, quantum oscillatory phenomena have

also been observed in physical quantities such as magnetostriction [36], speciﬁc heat [37],

optical reﬂectivity [38], and ultrasonic velocity [39] and attenuation [40].

Because the oscillations are periodic, but not in general sinusoidal, they may be

expanded in a Fourier series of sine waves indexed by the “harmonic number” p. If oscil-

lations corresponding to several diﬀerent frequencies are superimposed, each oscillation

will have its own Fourier series, and these will be summed over the set of fundamen-

tal frequencies F . Thus, for our idealized metallic system at T = 0, the oscillatory { } Chapter 1. Introduction 21 magnetization M˜ , for example, may be written

∞ F 1 π M˜ = C √H sin 2πp (1.7) − 0 H − 2 ± 4 p=1 X{F } X where e2V F C0 = 3 (1.8) 3 2 ∗ √ ′′ 2π p m F π The positive 4 term in the Eq. 1.7 occurs when the extremal area Aext is a minimum, π V whereas the negative 4 term occurs when Aext is a maximum. In Eq. 1.8, is the volume of the sample, and m∗ is the eﬀective quasiparticle mass, which is equal to the

bare electron mass me in our simpliﬁed example. The second derivative of the oscillation frequency, F ′′, is equal to ~ 2 ′′ ∂ A F = 2 (1.9) 2πe ∂k ~ H A=Aext

where the derivative is of the Fermi surface cross-sectional area perpendicular to the ˆ ′′ field, taken along the field direction kH~ , and evaluated at the extremum Aext. Since F is therefore related to the curvature of the Fermi surface along the field direction, its

presence in Eqs. 1.8 and 1.7 can be understood: if F ′′ is very small (and therefore 1/√F ′′

is large), as in the case of a cylindrical Fermi surface with the ﬁeld aligned along the

cylinder axis, the surface area of the outermost Landau tube within the Fermi surface

remains quite large until the tube leaves the Fermi volume, resulting in a strong oscillation

eﬀect; on the other hand, if F ′′ is larger (and therefore 1/√F ′′ is smaller), as in the case of a spherical Fermi surface, the surface area of the outermost Landau tube within the

Fermi surface decreases gradually as H increases, so there is less of a jump when the tube leaves the Fermi volume, resulting in a weaker oscillation eﬀect. Chapter 1. Introduction 22

1.5.2 Damping factors and other complications

While section 1.5.1 introduced the physics of quantum oscillations primarily for the sim- pliﬁed case of a metal in the independent electron approximation, with quadratic dispersion and spherical Fermi surface, it is possible to generalize Eq. 1.7 to describe the oscillatory magnetization of real materials. Landau tube depopulation happens at the

Fermi surface, in the vicinity of which Landau’s Fermi liquid theory (section 1.1) holds, and it turns out that Eqs. 1.6–1.9 apply equally well to Landau quasiparticles as to independent electrons, so long as m∗ is taken to be the renormalized quasiparticle eﬀective mass rather than the bare electron mass. Except in extreme cases, the above theory of quantum oscillations applies even to heavy fermion materials whose strong electron- electron interactions lead to heavily renormalized eﬀective masses (section 1.3), as long as Landau’s Fermi liquid paradigm remains valid.

Relaxing the assumption of quadratic dispersion and spherical Fermi surface also does not change the theory developed above. The cross-section of each Landau tube comes from the intersection of a surface of constant energy in k-space and the plane

perpendicular to the ﬁeld direction. Thus, for a material with a complicated dispersion

E(~k) and Fermi surface, the Landau tube cross-sections may not be circles, but still

satisfy Eq. 1.5 and match the Fermi surface cross-section (which is derived from the

same dispersion relation E(~k)) as they leave the Fermi volume.

The most important modiﬁcations to Eq. 1.7 needed to describe real experiments

come in the form of multiplicative damping factors, which reduce the quantum oscilla-

tion amplitudes without aﬀecting the frequencies. Stronger damping is manifested in

smaller values of the damping factors, and therefore smaller oscillation amplitudes. Dis-

cussed individually below, the damping factors come from ﬁnite temperature eﬀects (RT ),

impurity scattering (RD), and considerations of electron spin (RS). When combined with

Eq. 1.7, RT , RD, and RS yield the famous Lifshitz-Kosevich equation for the quantum oscillatory magnetization [41], which is the primary tool used to analyze and interpret Chapter 1. Introduction 23 the results of dHvA experiments:

∞ F 1 π M˜ = R R R C √H sin 2πp (1.10) − T D S 0 H − 2 ± 4 p=1 X{F } X

Temperature dependence

At T = 0, quantum oscillations occur as successive Landau tubes exit the Fermi volume

and abruptly depopulate. As the temperature is raised, thermal excitations with energies

on the order of kBT excite some quasiparticles from occupied states just below the Fermi energy to previously-unoccupied states just above the Fermi energy, thus increasingly

“smearing out” the Fermi surface that was a sharp cut-oﬀ between occupied and unoc-

cupied states at T = 0. This results in the Landau tubes depopulating more gradually as they pass through the blurred Fermi surface, reducing the oscillation amplitude by the damping factor [41] 2 ∗ 2π pm kBT/e~H RT = 2 ∗ (1.11) sinh (2π pm kBT/e~H)

At the heart of Eq. 1.11 is a comparison between the thermal energy (kBT ), which characterizes the blurriness of the Fermi surface, and the cyclotron energy (e~H/m∗),

which characterizes the separation between adjacent Landau tubes: higher temperatures

or smaller magnetic ﬁelds increase the damping, whereas lower temperatures or larger

magnetic ﬁelds decrease it. Along these lines, heavier eﬀective masses m∗ decrease the

cyclotron energy, thereby increasing the damping and enforcing a requirement of very

low temperatures for de Haas–van Alphen measurements on heavy fermion materials.

Furthermore, since the sharpness of the oscillations due to the abrupt tube depopulations

requires strong higher-harmonic content in the Lifshitz-Kosevich equation Fourier series,

a “softening” or “smoothing out” of the oscillations has a more pronounced damping

eﬀect on the higher-p harmonics.

The temperature dependence of RT is illustrated in Fig. 1.4 for several values of the Chapter 1. Introduction 24

1.0

0.8

T 0.6

pm* = 1 0.4 pm* = 2.5 pm* = 5 NormalizedR pm* = 10 H = 16 T 0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 Temperature (K)

Figure 1.4: The temperature dependence of the damping factor RT , normalized to 1 at T = 0. Curves show increased damping as the parameter pm∗ (in units of the bare electron mass, me) is increased from 1 to 10.

combined parameter pm∗. The increased damping for heavier quasiparticles is a double-

edged sword: while it constrains measurements to low temperatures, ﬁts of the temper-

ature dependence of measured dHvA oscillation amplitudes to the “standard Lifshitz-

Kosevich form” of Eq. 1.11 allow direct extraction of the eﬀective mass, averaged around

individual extremal orbits on each Fermi surface sheet. The capability to obtain such

detailed eﬀective mass information is a major advantage of this experimental technique,

since most other types of measurement give only results that are averages over the entire

Fermi surface.

Impurity dependence

While elevated temperatures cause the Landau tubes to depopulate more gradually by

smearing out the Fermi surface, scattering of the quasiparticles into nearby states by Chapter 1. Introduction 25 impurities in the crystal causes a similar eﬀect by smearing out the Landau tubes themselves. In analogy to the preceding discussion, the damping strength is determined by a comparison between the energy kBTD, which characterizes the blurriness of the tubes, and the cyclotron energy (e~H/m∗), which characterizes the separation between adjacent tubes. The Dingle temperature TD, ﬁrst treated by Robert Dingle [42], depends on the quasiparticle scattering relaxation time τ as

~ TD = (1.12) 2πkBτ

The damping factor due to scattering, called the Dingle factor, is then

R = exp 2π2pm∗k T /e~H (1.13) D − B D in which the oscillation harmonic index p enters for the same reason it did in Eq. 1.11.

∗ Substituting Eq. 1.12 and the cyclotron frequency ωc = eH/m into Eq. 1.13 gives

R = exp ( πp/ω τ) (1.14) D − c

which can be re-written in terms of the cyclotron period tc = 2π/ωc:

R = exp ( pt /2τ) (1.15) D − c

Eq. 1.15 can then be roughly understood quasi-classically as a comparison between

the time it takes for an electron to complete a cyclotron orbit and the average time

between scattering events: In one extreme, if the electron can go around the orbit many

times before scattering, strong quantum oscillations will result; in the other extreme,

if the electron scatters so often that it cannot complete even one orbit, the quantum

oscillations will be damped out entirely. This is also apparent when RD is expressed in Chapter 1. Introduction 26

terms of the cyclotron radius, rc, and quasiparticle mean free path, l0:

R = exp ( πpr /l ) (1.16) D − c 0

Thus, in addition to the constraint of low temperatures, exceptionally pure single crystal samples are required to maximize the observable oscillations in a de Haas-van

Alphen experiment.

Spin dependence

Up to this point, I have assumed that the energies of spin-up and spin-down quasiparticles are the same, and have therefore not made any distinction between them. In reality, however, an applied magnetic ﬁeld lifts this degeneracy and Zeeman-splits the quasiparticle energy levels by

∆E = gµBH (1.17)

where g is the spin-splitting factor (approximately equal to 2 for free electrons) and µB is the Bohr magneton. Using a pseudo-spin treatment, the form of Eq. 1.17 applies even when spin-orbit coupling is strong [43], although the value of g may deviate from 2 in f-electron compounds [44, 45]. The magnetic field thus splits the Fermi surface into a larger majority branch, whose spins are aligned along the field direction, and a smaller minority branch, whose spins are anti-aligned with the field. This leads to two sets of quantum oscillation frequencies, one from each Fermi surface spin-branch, that may be out of phase with one another. In the simplest case, when g is not field-dependent and the spin-splitting is linear in H, as in Eq. 1.17, the phase difference between majority- and minority-spin oscillations may be obtained by a comparison between the Zeeman energy

∆E, which characterizes the Fermi surface splitting, and the cyclotron energy (e~H/m∗), which characterizes the separation between adjacent Landau tubes. The spin-damping Chapter 1. Introduction 27 factor is then simply

∗ RS = cos (πpm ∆E/e~H) (1.18)

By substituting in Eq. 1.17 and the formula deﬁning the Bohr magneton, µB = e~/2me,

it can be seen that RS is actually ﬁeld-independent (as long as ∆E remains linear in H):

π R = cos gpm∗/m (1.19) S 2 e

An additional complication, which is important to my study of YbRh2Si2 (chapter 6), comes from the fact that gradually changing the Fermi surface size as the ﬁeld is

swept, while simultaneously varying the size of the Landau tubes, causes a Doppler shift

eﬀect. The result is that the ﬁeld dependence of a measured dHvA frequency Fm, is

not directly equal to the “true” frequency Ft, obtained from Eq. 1.6, but rather it is the

back-projection to H = 0 T of the line tangent to Ft [46]. This is illustrated in Fig. 1.5

for a case where the Fermi surface spin-splitting is linear at low ﬁelds such as HA, but

deviates from linearity at higher ﬁelds such as HB.

Speciﬁcally, at a given ﬁeld H = Hi, the measured frequency from one spin-branch F (where σ = , indicates the majority or minority spin) is related to the true mσ,i {↑ ↓}

frequency Ftσ,i by ∂F F = F H tσ (1.20) mσ,i tσ,i − i ∂H H=Hi

For the example shown in Fig. 1.5, at fields H HA and below, where the true spin-split ∼ frequencies vary linearly with field, both spin-branches project back to the zero-field

un-split frequency F0, so only one frequency would be measured experimentally, with

spin-damping factor RS given by Eq. 1.19 (provided both spin branches have the same

∗ effective mass m ). On the other hand, at higher fields such as HB, once the true frequencies have deviated from field-linear behaviour, separate frequencies for each spin-

branch may be resolved in the experimental data. In such a case, the damping factor RS Chapter 1. Introduction 28

F t↑,B

F t↑,A F 0

F t↓,A True frequency F t↓,B H H A B F m↑,B

F =F m↑↓ ,A 0

Measured frequency F m↓,B 0 H H A B Magnetic field

Figure 1.5: An illustration of the de Haas–van Alphen frequency back-projection relation between measured and true frequencies, Fm and Ft respectively, for an example wherein the Fermi surface splitting is linear in ﬁeld for H H and below, but deviates from ∼ A linearity as H is raised from HA to HB. Frequencies deriving from the Fermi surface majority-spin branch are labelled with a , those from the minority-spin branch with a , and the frequency obtained from the un-split↑ Fermi surface at H = 0 is denoted F . ↓ 0 Chapter 1. Introduction 29 that was developed above no longer applies.

Non-linear Fermi surface spin-splitting, as described above, is fairly common in heavy

fermion metals because of their very ﬂat bands. In fact, the ﬂat dispersion and compli-

cated interactions characteristic of heavy fermions can lead to spin dependence of other

quantities as well, such as the scattering rate or quasiparticle eﬀective mass. In partic-

ular, when frequency spin-splitting cannot be resolved experimentally, spin-dependent

eﬀective masses can cause additional constructive or destructive interference between the

two spin-branches, leading to deviations from the Eq. 1.11 RT temperature dependence at extremely low temperatures. Such deviations from the standard Lifshitz-Kosevich

temperature dependence of the oscillation amplitudes were carefully searched for in the

measurements presented in this thesis, and only found in CeCoIn5 (section 3.4). Because

my doctoral involvement with CeCoIn5 has focused on band structure calculations, and constitutes a minor part of my thesis, a full theoretical account of spin-dependent eﬀective

masses falls outside of the scope of this dissertation, and may be found elsewhere [12, 47].

1.5.3 The ﬁeld modulation measurement technique

As mentioned above, quantum oscillations can show up in many diﬀerent physical quan-

tities. However, the detection method with arguably the highest signal-to-noise ratio,

due to intrinsic background rejection, is the ﬁeld modulation technique for measuring

the quantum oscillatory magnetization, originally introduced by David Shoenberg and

Phil Stiles [48]. As shown in Fig. 1.6, the ﬁeld modulation method consists of surround-

ing the sample with one of a pair of balanced, counter-wound pick-up coils and adding

a sinusoidally-varying magnetic ﬁeld (H~ AC ) to the large, monotonically increasing (or

decreasing) ﬁeld (H~ DC , the ﬁeld implicitly referred to as H in the preceding discussion).

H~ AC is in general much smaller than, and aligned parallel to, H~ DC . Voltage generated in the pick-up coils is ampliﬁed, signal-conditioned and detected by one or more lock-

in ampliﬁers, phase-locked to the second or higher harmonic of the HAC modulation Chapter 1. Introduction 30 frequency.

For simplicity, in the following derivations I will assume a single dHvA frequency F ,

treating only the ﬁrst harmonic p = 1, although the results obtained may be generalized

straightforwardly to multiple superimposed frequencies and harmonics. Deﬁning C 1 ≡

RT RDRSC0, the Lifshitz-Kosevich equation (Eq. 1.10) for our simpliﬁed case may be written 2πF M˜ = C √H sin + φ (1.21) − 1 H where 3π if Aext is a minimum φ = − 4 (1.22)  5π  if Aext is a maximum  − 4

The total sample magnetization, including the static spin-susceptibility χ0, is then

χ H 2πF M = 0 C √H sin + φ (1.23) µ − 1 H 0 where µ0 is the permeability of free space. Adding a small ﬁeld oscillating in time at a frequency ω, such that H = H + H = H + h cos ωt, with h H , gives DC AC 0 0 0 ≪ 0

χ0 (H0 + h0 cos ωt) 2πF M(t)= C1 H0 + h0 cos ωt sin + φ (1.24) µ0 − H0 + h0 cos ωt p Expanding both √H + h cos ωt and 1/ (H + h cos ωt) to ﬁrst order in h /H 1, 0 0 0 0 0 0 ≪ and using the trigonometric identity sin(u v) = sin u cos v cos u sin v, Eq. 1.24 can be − − written

χ0H0 χ0h0 cos ωt h0 cos ωt M(t) + C1 H0 + ≈ µ µ − 2√H × 0 0 0 2πF 2πFp sin + φ cos h cos ωt × H H2 0 − 0 0 2πF 2πF cos + φ sin h cos ωt (1.25) − H H2 0 0 0 Chapter 1. Introduction 31

Figure 1.6: The set-up of a traditional field modulation dHvA experiment. The metallic, single-crystal sample is wrapped in one of a pair of counter-wound pick-up coils and placed in a magnetic field composed of a large DC component H~ DC = H~ 0 and small AC component H~ AC = ~h0 cos ωt (H~ DC H~ AC ). The voltage induced across the pick-up coils is pre-amplified and then detected|| by a lock-in amplifier tuned to an even harmonic of ω. If multiple lock-in amplifiers are used in parallel, several different harmonics on one or more samples may be measured at once. A more detailed version of this diagram for the specific case of my YbRh2Si2 experiment is shown in Fig. 6.14. Chapter 1. Introduction 32

Expanding the sinusoidal terms which contain h0 in their arguments as a Fourier series with Bessel function coeﬃcients gives

χ0H0 χ0h0 cos ωt h0 cos ωt M(t) + C1 H0 + ≈ µ µ − 2√H × 0 0 0 ∞ p 2πF 2J (λ) sin + φ cos νωt (1.26) × ν H ν "ν=1 0 # X

where λ 2πFh /H2, J is the νth Bessel function of the 2nd kind, and φ = φ νπ/2 ≡ 0 0 ν ν −

(using φ deﬁned in Eq. 1.22). A plot of Jν(λ) for several values of ν is shown in Fig. 1.7. Since J (λ) < 1 for all λ and h /√H < 1, terms involving J (λ)h /√H will be 1 | ν | 0 0 ν 0 0 ≪ and can therefore be eliminated, giving

χ H χ h cos ωt ∞ 2πF M(t) 0 0 + 0 0 2C H J (λ) sin + φ cos νωt (1.27) ≈ µ µ − 1 0 ν H ν 0 0 ν=1 0 p X

The voltage V induced in the balanced pair of pick-up coils by the sample magnetization is dM V = C (1.28) coil dt

where Ccoil = µ0ηVN/ℓ, η is the ﬁlling factor of the sample in the pick-up coils, V is the sample volume, and N/ℓ is the number of turns of wire per unit length of the pick-up

coil. Combining Eqs. 1.27 and 1.28 gives

χ C ∞ 2πF V 0 coil ωh cos ωt + 2C C H ωνJ (λ) sin + φ sin νωt (1.29) ≈ − µ 0 coil 1 0 ν H ν 0 ν=1 0 p X After careful amplification and signal conditioning, using a lock-in amplifier to perform phase-sensitive detection at the ν = 2 or higher harmonic of the field modulation frequency ω gives

2πF Vlock−in, ν≥2 2ωνJν(λ)CcoilC1 H0 sin + φν (1.30) ≈ H0 p Chapter 1. Introduction 33

0.6

0.5

0.4

0.3 ) λ ( ν 0.2

0.1

0.0

-0.1 ν = 1 Besselfunction J ν -0.2 = 2 ν = 4 ν -0.3 = 6

-0.4 012345678910 λ = 2 πFh / H 2 0 0

2 Figure 1.7: Bessel function Jν(λ = 2πFh0/H0 ) for ν = 1, 2, 4, 6.

which is exactly N V 2µ ηV ωνJ (λ) M˜ (1.31) lock−in, ν≥2 ≈ − 0 ℓ ν ν where M˜ ν is the M˜ from Eq. 1.21, but with φ replaced with φν.

In principle, Eq. 1.31 is very powerful, since it allows direct measurement of the oscillatory magnetization M˜ when ν 2, independent of the static susceptibility χ ν ≥ 0 and other background eﬀects. In practise, detecting at ν 2 also excludes other major ≥ sources of noise, such as imperfectly balanced pick-up coils and pick-up coil vibrations, which tend to appear in the coil voltage at ν = 1. Furthermore, harmonic distortion introduced by pre-ampliﬁers and other electronics often occurs at odd harmonics of the modulation frequency (since these devices tend to clip the raw signal symmetrically about

0 volts, approximating a square wave), so dHvA measurements typically focus on the even harmonics ν = 2, 4, 6,... Chapter 1. Introduction 34

In addition to the experimental constraints imposed by the damping factors RT , RD,

and RS, the pre-factors in Eq. 1.31 serve as a guide to setting up good experimental conditions for a dHvA measurement: the sample should have a large volume V, be

wrapped tightly (for large ﬁlling factor η) in coils which have a large density of turns N/ℓ,

2 and λ = 2πFh0/H0 should be tuned so that the Bessel function Jν(λ) is large. It appears that using ever-higher ν’s and ω’s would also increase the signal, but there are practical limits on these due to other eﬀects. As seen in Fig. 1.7, high values of ν require high values of λ to keep J (λ) large; since the dHvA frequency spectrum F of a given material is ν { }

fixed and the highest possible DC magnetic fields H0 are used to fight damping factors,

λ must be tuned by modifying the modulation ﬁeld h0. Unfortunately, the modulation

field induces eddy currents in both the sample (heating it up and affecting RT ) and main magnet (causing field non-linearities and possible superconducting magnet quenches) proportional to ωh0, thus limiting ω and h0 to relatively small values. Overheating and increased harmonic distortion can also occur at high h0 in the power supply feeding the modulation coils.

Because λ scales with the dHvA frequency F , tuning the Bessel function so that the detected amplitudes of some oscillations are minimized, while others are maximized, can help to unravel a complicated frequency spectrum F . While λ is also proportional to { } 2 1/H0 , and therefore gradually “slips” in value during a field sweep, careful choice of h0 and the swept field range allow different parts of the dHvA frequency spectrum to be emphasized at different harmonics ν. Thus, as shown in Fig. 1.6, it is often advantageous to have a bank of lock-in amplifiers all measuring the same sample in parallel, with each focusing on a different value of ν. Furthermore, if measurements are to be performed on several samples at the same time, an expensive and bulky armada of lock-in amplifiers is required. Streamlining this process was one of my major goals in the construction of the

Data Acquisition Virtual Instrument Experiment System (DAVIES) software/hardware infrastructure, discussed in the next chapter. Chapter 2

Instrumentation

As an experimental physicist, high-quality instrumentation is very important to any study

I perform, and for my work with heavy fermion metals, precision control of low temperatures and high magnetic ﬁelds is paramount. In our lab, this instrumentation needed to be built from the ground up, since my supervisor arrived at the University of Toronto at the beginning of my Ph.D. with no preexisting measurement infrastructure in place.

During the lab-building phase of my doctoral work, I created a computer software suite to control our dilution refrigerator / superconducting magnet facility and collect data from all of our experiments (section 2.1). The most novel aspect of this software suite is the data acquisition subsystem (section 2.1.7), in which the physical lock-in ampliﬁers traditionally used for phase-sensitive detection in condensed matter laboratories (section 1.5.3) are replaced with software-based “virtual lock-ins,” signiﬁcantly expanding the measurement capabilities of our facility.

In addition, I designed a graphite rotation mechanism to allow samples to be rotated relative to the magnetic ﬁeld direction at low temperatures (section 2.2), I implemented a set-up for annealing silver heat-sink wires (section 2.3), and I built a glove box for handling air-sensitive compounds, including a gas circulation system and home-made oxygen detector (section 2.4). I have also written the software to perform the speciﬁc

35 Chapter 2. Instrumentation 36 heat measurements of my fellow graduate student Wenlong Wu [49], but that is outside of the scope of this thesis.

2.1 Data Acquisition Virtual Instrument Experiment

System

The Data Acquisition Virtual Instrument Experiment System (DAVIES) is a collection of computer programs (called Virtual Instruments or “VIs”) written in the LabVIEW programming language that, along with a variety of home-made and commercial hardware devices, is used to control our dilution refrigerator and superconducting magnet and perform all of our experiments (with a focus on those done in the dilution refrigerator, but easily adapted to 4 K dipping probe measurements, etc.). Since my supervisor opted not to purchase the Oxford Instruments IGH Intelligent Gas Handling system, modern magnet power supply, and associated software that usually accompany an Oxford

Instruments Kelvinox 400MX 3He/4He dilution refrigerator such as ours, and automatically measure, control and log the temperature (through thermometers and heaters) and magnetic ﬁeld, I was required to build these functions from scratch in DAVIES.

Rather than following a hierarchical program-subprogram structure, DAVIES is made up of 24 peer-level modular component VIs, spread across two computers, which communicate extensively with one another via the DataSocket networking protocol (section 2.1.2). A schematic overview of the DAVIES organizational infrastructure is shown in Fig. 2.1; the hardware (section 2.1.1), DataSockets (section 2.1.2), error handling (section 2.1.3), temperature control (section 2.1.4), magnetic ﬁeld control (section 2.1.5), sample rotation control (section 2.1.6), data acquisition (section 2.1.7), automation (section 2.1.8), and monitoring (section 2.1.9) subsystems are discussed and shown in more detail below.

Note that I have already written a comprehensive DAVIES user guide, which includes Chapter 2. Instrumentation 37

Figure 2.1: Overview of the Data Acquisition Virtual Instrument Experiment System (DAVIES). The subsystem colour scheme (black for hardware, magenta for DataSockets, red for error handling, blue for temperature control, purple for magnetic ﬁeld control, orange for sample rotation control, dark green for data acquisition, light green for automation, and cyan for monitoring) and connecting line colour scheme (dashed magenta lines for communications via DataSocket variables, thin dashed red lines for error handling, thick dark blue lines for direct high-speed digital connections to hardware, and black lines for cables carrying analog signals) shown here are used in the Fig. 2.2, 2.3, 2.4, 2.5, 2.6, and 2.7 detailed subsystem diagrams. Chapter 2. Instrumentation 38 important operating instructions, screen shots of each VI, and detailed descriptions of every front panel object and parameter available to the user [50]. The material presented in this thesis is not intended to duplicate this document, but rather complement it by focusing on the “under-the-hood” architecture of the system.

2.1.1 Hardware

The hardware components used to physically execute the will of the DAVIES VIs are listed below. For each device, the short-form name used throughout the rest of this chapter is italicized.

Max: Dell Optiplex GX620 desktop computer, with dual-core Pentium D 3.2 • GHz CPU and 1.5 GB of RAM, running LabVIEW 8.5, TightVNC server, and

DataSocket server. DAVIES is a multi-threaded application (with thread priorities

of each VI set by me) [51], so it can take full advantage of the dual-core CPU. Max

is located in a cabinet close to the dilution refrigerator, with most of the rest of

the instrumentation, and is typically controlled through a VNC window from Dew.

All hardware connections (except to the web cam) are made to Max, and all VIs

except DewNerveCentre.vi, Route66.vi, RealtimeAnalysis.vi, RTAConfig.vi,

and DataSocketSniffer.vi run on Max.

Dew: Dell Optiplex GX280 desktop computer, with Pentium 4 3.2 GHz CPU and • 1 GB of RAM, running LabVIEW 8.5, TightVNC server, and web cam software.

The web cam is connected to Dew, and Dew is connected to Max through the

router. DewNerveCentre.vi, Route66.vi, RealtimeAnalysis.vi, RTAConfig.vi,

and DataSocketSniffer.vi run on Dew; all other VIs run on Max.

router: D-Link DI-704P Broadband Ethernet Router that forms the local network • through which Max and Dew communicate. Chapter 2. Instrumentation 39

PXI box: National Instruments PXI-1042 PCI eXtensions for Instrumentation • (PXI) chassis and National Instruments PXI-8331 MXI-4 PXI-to-PC interface card,

connected to Max. An important limitation of this system is that each card in the

PXI box may only be accessed by one VI at any given time (i.e. diﬀerent VIs may

simultaneously access diﬀerent cards, but no two VIs can access the same card).

AIdaq: National Instruments PXI-6143 16-Bit, 250 kilosample/second/channel on • 8 analog 5 V range input channels, Simultaneous Sampling Multifunction DAQ ± card in the PXI box, with National Instruments BNC-2210 connection block. This

PXI card is associated with TControl.vi, which writes the measured values out to

DataSockets for use by other VIs.

AO13bit: National Instruments PXI-6722 13-Bit, 182 kilosample/second/channel • on 8 analog 10 V range output channels, Static and Waveform Analog Output ± card in the PXI box, with National Instruments BNC-2110 connection block. This

PXI card is associated with AOcore.vi, which reads desired output voltages from

other VIs through DataSockets and sends them to the card.

DMM : National Instruments PXI-4071 7 1 -Digit FlexDMM Digital Multi-Meter • 2 and 1.8 megasample/second Isolated Digitizer card in the PXI box. This PXI card

is associated with HControl.vi.

multiplexer: National Instruments PXI-2503 24-Channel Relay Multiplexer/Matrix • card in the PXI box, with National Instruments TB-2605 Isothermal Terminal

Block. This PXI card is associated with HControl.vi.

AIgood: National Instruments PXI-4472 24-Bit, 102.4 kilosample/second/channel • on 8 analog 10 V range input channels, Dynamic Signal Acquisition card in the ± PXI box. This PXI card is associated with LockinSub.vi.

SigGen: National Instruments PXI-5421 16-Bit, 100 megasample/second maximum • Chapter 2. Instrumentation 40

12 Vpeak−to−peak into 50 Ω load, Arbitrary Waveform Generator card in PXI box. This PXI card is associated with PFgen.vi.

AVS bridge: RV-Elektronikka Oy AVS-45 automatic resistance bridge, manufac- • tured in 1986 [52].

USB-to-digital converters: Measurement Computing USB-based Digital I/O Mod- • ule PMD-1208-LS (pull-up conﬁguration) and USB-1208-LS (pull-down conﬁgura-

tion) Personal Measurement Devices.

Cernox: Lake Shore Cryotronics model CX-1050-AA-1.4L Cernox resistive ther- • mometer (serial number X31110), mounted on the dilution refrigerator mixing

chamber in a brass can I designed.

M70 RuO : Oxford Instruments calibrated “M70” RuO resistive chip thermome- • 2 2 ter, mounted on the dilution refrigerator mixing chamber.

diagnostic RuO ’s: Oxford Instruments RuO resistive chip thermometers from • 2 2 the generic-calibration “M0” batch, mounted on the dilution refrigerator mixing

chamber, cold plate, still, and 1K pot.

heater current source: Three circuits that I designed and built, based on National • Semiconductor LM201AN op-amp components, to drive current through the dilu-

tion refrigerator resistive heaters. 0–5 V input signals correspond to 0–6.363 mA

(20.244 mW) through the mixing chamber heater, 0–6.377 mA (20.333 mW) through

the still heater, and 0–24.649 mA (51.643 mW) through the charcoal sorption pump

heater.

DC power supplies: Two commercial power supplies (a dual-sided Tenma model 72- • 6615 and single-sided Tenma model 72-2005), used with the motor control circuit,

and one home made power supply (giving a ﬁxed 15 V relative to the central ± ground pin), used with the heater current source. Chapter 2. Instrumentation 41

heaters: Oxford Instruments resistive heaters mounted on the mixing chamber, 1K • pot, and still of the dilution refrigerator.

resistance box: Transmille model 2090 Programmable Resistance Box, computer- • controllable via RS232 interface, from 0 Ω (short circuit) to 10 kΩ in 0.1 Ω steps. ∼

magnet power supply: Oxford Instruments Magnet Power Supply 120 MKIII, man- • ufactured between 1979 and 1984.

motor control circuit: Drive circuit for the DC motor, home made by my supervisor, • Stephen Julian.

DC motor: A DC electric motor, attached to a 10-turn, 1 kΩ potentiometer. •

level monitor: Oxford Instruments ILM 211S Intelligent Level Meter, for mea- • suring liquid helium and liquid nitrogen bath levels in the dilution refrigerator /

superconducting magnet cryostat.

lambda fridge controller: Oxford Instruments CQB0100 Lambda Controller, for • measuring the temperature of the lambda refrigerator attached to the supercon-

ducting magnet.

web cam: Philips SPC 610NC PC Camera Web Cam. •

2.1.2 The DataSocket protocol

VIs: MainFrontPanel.vi, DewNerveCentre.vi, StopVIs.vi, parts of all other VIs

Hardware: Max, Dew, router

The DataSocket protocol is an eﬃcient, high-performance communications technol- ogy, based on the TCP/IP standard, designed by National Instruments for live data transfer between applications on one or more computers connected through a local network or the Internet [53]. A DataSocket server running on one computer acts as a repository for Chapter 2. Instrumentation 42

“DataSockets,” which function as self-describing shared variables in a local or distributed computing system. Each DataSocket may be a number, character string, boolean variable, or more complicated data object (such as an array or waveform), whose value on the server may be written and/or read by any compatible application on any computer that has been granted appropriate permissions and is connected to the one running the server.

In the context of DAVIES, DataSockets serve as a “central nervous system,” thread- ing and branching through all of the diﬀerent subsystems and VIs, and allowing them to communicate with and control one another. The DataSocket server is located on

Max and is easily accessed through the local network by VIs on both Max and Dew. I have predeﬁned over 500 unique DataSockets on the server, corresponding to run-state variables, control parameters, and experimental measurement values; the most active of these are updated approximately once every 100 milliseconds, which sets the fundamental timescale of the system.

In addition to their central role of facilitating data transfer between DAVIES VIs,

DataSockets also play an important part in the command and control infrastructure coordinating the entire system. As a simple example, while it is running, TControl.vi

constantly watches the boolean DataSocket corresponding to the command to start a new

temperature sweep: if the value of this DataSocket becomes “true” (whether through

the action of a user via ManualAdjustTH.vi or an automated command originating

in FullAutoSweeps.vi), TControl.vi stops what it is doing, reads the desired sweep

parameters from other DataSockets, begins the new temperature sweep, and then changes

the value of the original DataSocket back to “false” to tell the rest of DAVIES that it

has understood and executed the command. Simultaneously, at 100 ms intervals, ∼ TControl.vi writes the currently-measured temperature of the dilution refrigerator to a

numeric DataSocket, thus making this information available to the rest of the system in

real-time. Chapter 2. Instrumentation 43

Furthermore, when a given VI starts running, it sets its boolean run-state DataSocket to “true,” and just before it stops, it sets the DataSocket to “false.” If, for example, a user wants to use ManualAdjustTH.vi, he or she presses the relevant button on

MainFrontPanel.vi (the primary DAVIES execution control access point), which then checks DataSockets to determine if ManualAdjustTH.vi is already running, and if not, writes appropriate start-up parameters to DataSockets, reaches into Max’s hard drive, and opens and runs ManualAdjustTH.vi, which in turn updates its run-state DataSocket to reﬂect its new status and reads the desired start-up parameters from DataSockets.

Similarly, if the user wants to close ManualAdjustTH.vi, he or she presses the button on

MainFrontPanel.vi, which then checks DataSockets to determine if ManualAdjustTH.vi is indeed running, and if so, tells ManualAdjustTH.vi via DataSockets that it is no longer wanted; upon receipt of this command, ManualAdjustTH.vi puts itself into a safe state, exits any running loops, sets its run-state DataSocket to “false,” and then ends execution; MainFrontPanel.vi sees this through DataSockets and closes the VI. Lab-

VIEW includes functionality to directly check the execution state of a VI, but this is very resource-intensive and in DAVIES is done sparingly, such as during the clean-up routine of StopVIs.vi, which compares the DataSocket run-state values of each VI to

their actual execution states and corrects any inconsistencies. Since the LabVIEW API

responsible for programmatically opening, closing, executing and directly checking the

run-states of VIs on the hard drive / in memory can only do so within the computer

on which it is running, an additional intermediary module called DewNerveCentre.vi

watches DataSockets and allows VIs that reside on Max to perform these functions on

VIs that reside on Dew.

2.1.3 Error handling

VIs: Route66.vi, StopVIs.vi, Emergency-VI-Aborter.vi, parts of all other VIs

Hardware: Max, Dew, router Chapter 2. Instrumentation 44

The DataSocket subsystem is integral to the core functionality of DAVIES, with a vast number of DataSocket reads and writes occurring every second. Unfortunately, due to networking, processing or other irregularities, individual DataSocket functions within isolated VIs occasionally manifest non-fatal communications errors. In the default

LabVIEW error handling scheme, the occurrence of every single error triggers a pop-up dialog box that describes the error and halts execution of all VIs currently running on that computer, inevitably leading to a cascade of software failures in the rest of the

DAVIES VIs. In order to avoid this situation for the mostly-benign DataSocket hiccups,

I have removed the DataSocket functions from under the aegis of the LabVIEW error handling subsystem, and created my own error handling subsystem in its place. All other types of errors, for example those arising from hardware conﬂicts, tend to indicate that something is seriously wrong with the system, and continue be handled by LabVIEW in the default manner. My new error handling subsystem threads and branches its way through the DataSocket functions of each VI like a “circulatory system” overlaid on top of the DataSocket central nervous system, catching errors and transporting them to central

“cleansing points” within each VI. When a triggered error arrives at a cleansing point, instead of opening up a harmful dialog box, my error handling routine:

1. turns on a red light and lists the error number on the aﬀected VI’s front panel,

until the VI is once again in an error-free state;

2. writes the error information, including date and time, to a log ﬁle labelled with the

aﬀected VI’s name; and

3. informs the rest of DAVIES about the error, through DataSockets.

Of course, it is a much more serious matter when the entire DataSocket subsys-

tem/server itself encounters a severe problem and grinds to a halt. This was exactly the

case for the infamous and unpredictable “Error 66” [50, 54], which could happen as often

as several times an hour or as rarely as once in a three week period of 24 hours a day, Chapter 2. Instrumentation 45

7 days a week continuous DAVIES operation. Years of correspondence with numerous

National Instruments engineers, via telephone, e-mail and many epic on line discussion forum threads involving the wider LabVIEW community, have failed to elucidate the underlying cause of this error. However, through two key observations, I have been able to assemble an empirical solution: I noticed that for several minutes before the occurrence of an Error 66 DataSocket meltdown, the main loop of TControl.vi slows to a crawl (50–60 seconds for a single loop iteration, rather than the usual 100 ms), and furthermore that ∼ if DewNerveCentre.vi is halted, closed, and re-opened during this time, the system never progresses to a full-blown Error 66, and TControl.vi returns to its normal behaviour.

Thus, in order to address the problem, I created Route66.vi on Dew, that watches for

the TControl.vi slow-down, and upon its detection reboots DewNerveCentre.vi and writes information about the event to an error log ﬁle. Since the DataSocket server itself is unstable during these pre-Error 66 periods, the detection must proceed independently of the DataSocket subsystem: on every loop iteration, TControl.vi writes its loop time

and the current millisecond tick count of Max’s system clock to a “disk variable” text ﬁle

on Max’s hard drive; this ﬁle is periodically read through the network by Route66.vi,

which reboots DewNerveCentre.vi if a loop time greater than 45 seconds is found in the

ﬁle or if TControl.vi has failed to update the ﬁle for at least 45 seconds.

Finally, apart from the DataSocket subsystem, it is possible for individual DAVIES

VIs to overstay their welcome. In this case, the oﬀending VI should be stopped by

StopVIs.vi. StopVIs.vi attempts a series of techniques of escalating severity to end

and close the VI, beginning with a gentle suggestion via DataSockets that the VI should

clean up its act and shut down. Should StopVIs.vi fail, Emergency-VI-Aborter.vi

may be used to forcibly terminate an unresponsive VI, but this extreme measure may

cause further DAVIES instability and should only be used as a last resort. Chapter 2. Instrumentation 46

Figure 2.2: Diagram of the DAVIES temperature control subsystem.

2.1.4 Temperature control

VIs: TControl.vi, AOcore.vi, ManualAdjustTH.vi

Hardware: AIdaq, AO13bit, AVS bridge, USB-to-digital converters, Cernox, M70 RuO2,

diagnostic RuO2’s, heater current source, DC power supply, heaters

Measurement and control of the dilution refrigerator temperature is coordinated by

TControl.vi, following the schematic diagram shown in Fig. 2.2. Although the AVS

bridge is very old, and does not include a standard computer interface [52], I was able

to achieve rudimentary computer control by using USB-to-digital converters to push

digital bits into and read digital bits out of the AVS bridge circuit board. This allows

TControl.vi to determine and change which thermometer is being measured, and which

of the six excitation voltage settings and six resistance range settings are being used. Note

that the excitation voltage and resistance range can only be changed by incrementing or

decrementing to adjacent values. Once every 100 ms, TControl.vi obtains the measured ∼ thermometer resistance by using AIdaq to read an analog voltage output from the AVS bridge (originally intended to be connected to a mechanical chart recorder). The AVS bridge has an internal 15 second ﬁltering time constant, so whenever a setting (excitation voltage, resistance range, or active thermometer) is changed, TControl.vi must wait 15 seconds for the bridge resistance reading to settle. TControl.vi does its own ﬁltering of Chapter 2. Instrumentation 47 the measured thermometer resistance on top of this, using the equation

R = R [1 exp ( ∆t/t )] + R exp ( ∆t/t ) (2.1) avg new − − c avg,last − c

where Rnew is the new resistance value measured on this iteration, Ravg is the averaged

value sent out to DataSockets on this iteration, Ravg,last is the averaged value sent out to DataSockets on the previous iteration, ∆t is the time between the previous and current iterations (typically 100 ms), and t 500 ms is the averaging time constant. ∼ c ≥ Control voltages for the three dilution refrigerator heaters are communicated once every

100 ms from TControl.vi to AOcore.vi via DataSockets, and from there are sent to ∼ my home-made heater current source using AO13bit; the current source converts the control voltages to currents that are driven through the heaters. TControl.vi itself may be controlled through DataSockets, either manually by the user with ManualAdjustTH.vi or automatically by FullAutoSweeps.vi.

The AVS bridge can only measure one thermometer at a time, but is connected to

7 thermometers (the Cernox, M70 RuO2 and diagnostic RuO2 on the mixing chamber, diagnostic RuO2’s on the cold plate, still and 1K pot, and an uncalibrated Allen Bradley resistor on the sorb) using an internal multiplexer. This is a fairly severe limitation, because it means that: (1) during an experimental data collection run it is not possible, for example, to check the temperature of the 1K pot while simultaneously recording the mixing chamber temperature; and (2) switching between thermometers is time consum- ing, due to the 15 second settle time and the fact that whenever TControl.vi switches thermometers it must start at the highest resistance range and step down from there, waiting > 15 seconds between each step, until it ﬁnds the best range for the thermometer’s current resistance. Measured resistances of each thermometer, R (in Ohms), are converted to temperatures, T (in mK, the base unit of temperature in DAVIES), using Chapter 2. Instrumentation 48 calibration curves of the form

N 2 log (R) Z Z T (R) = 1000 A cos n arccos 10 − U − L (2.2) × n Z Z n=1 U L X − for the Cernox, and

N T (R) = 1000 exp A [ln(R R )]n (2.3) × − n − T =300K ( n=1 ) X

for the RuO2 thermometers, with thermometer-specific N, An, ZU , ZL, and RT =300K parameters loaded from configuration files.

Automatic Proportional-Integral-Derivative (PID) control of the dilution refrigerator

mixing chamber temperature, T , to follow a ﬁxed or sweeping set-point, Tsetpoint, is built into TControl.vi using the LabVIEW PID Control Toolset [55]. The percent of the

maximum current sent to the mixing chamber heater, u(t), is

1 t d (T T ) u(t)= k (T T )+ (T T ) dt + t setpoint − (2.4) c setpoint − t setpoint − d dt i Z0 in which Tsetpoint and T are expressed as percents of 330 000 mK. Prior to being converted to a control voltage and sent to the heater current supply, negative values of u(t) are truncated to zero, and a software noise gate with a 1% threshold is applied to prevent heater jitter based on AVS bridge measurement noise. Through empirical testing, I determined the best values for the PID constants kc = 1000 (the “controller gain”), ti = 1 minute (the “integral time”), and td = 0 minutes (the “derivative time” or “rate time”) for our equipment over a wide temperature range below 4 K, but these may be changed ∼ by the user through ManualAdjustTH.vi. In “Computerized PID” mode, TControl.vi automatically controls the measured thermometer (switching from the Cernox to the

M70 RuO2 when the temperature drops below 4.3 K, and back to the Cernox when the temperature rises above 4.5 K), excitation voltage, resistance range, and mixing chamber Chapter 2. Instrumentation 49

Figure 2.3: Diagram of the DAVIES magnetic ﬁeld control subsystem.

heater, following Eq. 2.4; in “Suspended PID” mode, PID control is turned oﬀ, and all of these items may be manually-controlled by the user through ManualAdjustTH.vi. The still and sorb heaters are always manually controlled.

2.1.5 Magnetic ﬁeld control

VIs: HControl.vi, AOcore.vi, ManualAdjustTH.vi

Hardware: AIdaq, AO13bit, DMM, multiplexer, resistance box, magnet power supply

As with the temperature control subsystem, one of my challenges in building the

DAVIES magnetic ﬁeld control subsystem was interfacing with an instrument not originally designed for computer control, in this case the magnet power supply used for driving current through our 16/18 T superconducting main magnet. The primary reason for using this particular model of magnet power supply is that it may be run in “constant voltage mode” (described below), allowing direct control of the voltage across the magnet; newer magnet power supplies lack this option. Measurement and control of the magnetic ﬁeld is coordinated by HControl.vi, following the schematic diagram shown in

Fig. 2.3. A “Set Pot” potentiometer on the front of the magnet power supply clamps the maximum current through the magnet anywhere between 0 and 120 Amps. A setpoint control voltage between 0 and 5 V, corresponding to a desired setpoint current through − Chapter 2. Instrumentation 50 the magnet between 0 Amps and the maximum current determined by the Set Pot, is communicated once every 250 ms from HControl.vi to AOcore.vi via DataSockets, ∼ and from there sent to the magnet power supply using AO13bit. The current going

through the magnet passes through a “buzz bar” inside the magnet power supply, across

which HControl.vi measures the voltage drop once every 250 ms using the DMM, ∼ with 1 mV corresponding to 1 A. (When not in use by DAVIES, the DMM functions as

the most accurate and precise multimeter in our laboratory.) The generated magnetic

ﬁeld is

H = 0.160 I (2.5) × mag

where the magnetic field H is given in Tesla, the current through the magnet Imag is given in Amps, and the conversion factor is specific to our magnet. HControl.vi applies its own rolling averaging to the measured magnetic field values before posting them to

DataSockets, using the equation

H = H [1 exp ( ∆t/t )] + H exp ( ∆t/t ) (2.6) avg new − − c avg,last − c

where Hnew is the new ﬁeld value measured on this iteration, Havg is the averaged value sent out to DataSockets on this iteration, Havg,last is the averaged value sent out to DataSockets on the previous iteration, ∆t is the time between the previous and cur-

rent iterations (typically 250 ms), and t 500 ms is the averaging time constant. ∼ c ≥ HControl.vi also has a built-in algorithm for detecting small diﬀerences between the “Set

Pot” value typed in by the user and the actual value, and re-scaling the setpoint control voltage appropriately.

In its pre-DAVIES conﬁguration, the magnet power supply had two front panel potentiometers to control the voltage across the magnet, V , between 5 and +5 V, with mag − one potentiometer being used when Vmag was negative and the other when Vmag was positive. In order to allow HControl.vi to control the magnet voltage, I have replaced these Chapter 2. Instrumentation 51 potentiometers with the programmable resistance box; since our research group only has one resistance box, HControl.vi uses the multiplexer to switch it between negative and

positive voltage cases. As an additional safety measure, I have soldered ﬁxed resistors in

parallel across the negative and positive voltage controls, limiting the maximum magnetic

field sweep rate in any field range to dH/dt = 0.5 T/min (the maximum sweep rate is | | further limited to dH/dt = 0.35 T/min across all field ranges in the DAVIES software). | |

The actual voltage Vmag across the magnet is measured by AIdaq, with values passed from TControl.vi to HControl.vi through DataSockets. HControl.vi itself may be controlled through DataSockets, either manually by the user with ManualAdjustTH.vi

or automatically by FullAutoSweeps.vi.

Constant voltage mode allows very smooth magnetic ﬁeld sweeps (which are impor-

tant for dHvA experiments) to be performed, by holding the voltage across the magnet,

Vmag, constant, and allowing the current through the magnet, Imag, to gradually ramp under the inﬂuence of the magnet inductance, Lmag, and magnet lead resistance, Rleads (since the magnet itself is superconducting, and therefore R 0): mag ≡

dI V = R I + L mag (2.7) mag leads mag mag dt

Substituting in Eq. 2.5 to replace Imag with the magnetic ﬁeld H gives

dH 0.160 V = R H + L (2.8) × mag leads mag dt

Inspection of Eq. 2.8 reveals that for a given voltage, the ﬁeld sweep rate is itself ﬁeld- dependent, with the absolute value of the rate tending to decrease toward the end of a sweep; this can lead to a large variation in sweep rate if the magnet is swept over a large

ﬁeld range. Also, some values of H are unreachable unless Vmag is suﬃciently high. The combination of these two facts can lead to limitations when extremely slow sweeps across

large ﬁeld ranges are desired. To circumvent these drawbacks, I have created an optional Chapter 2. Instrumentation 52 quasi-constant-voltage “constant rate” mode in HControl.vi that changes the voltage across the magnet once every 5 seconds in order to maintain a roughly constant dH/dt ∼ using Eq. 2.8. Detailed testing during my YbRh2Si2 experiment (chapter 6) showed that dHvA spectra obtained during constant rate mode ﬁeld sweeps were no more noisy than those obtained during constant voltage mode sweeps.

My careful testing also revealed a puzzling phenomenon: the resistance of the magnet leads, Rleads, and inductance of the magnet, Lmag, exhibit complicated time-dependent behaviour. By taking the derivative of Eq. 2.8 and rearranging for Rleads and Lmag, I obtained the equations

2 dVmag dH d H V 2 dt dt − mag dt Rleads,meas = (2.9) dH 2 d2H 6.25 H 2 dt − dt h i and

2 dVmag dH d H V 2 1 dt dt − mag dt Lmag,meas = V + H (2.10) 6.25 dH   dH 2 d2H  dt 6.25 H 2  dt − dt   h i    2 2 By implementing Eqs. 2.9 and 2.10 in HControl.vi, with dH/dt, d H/dt and dVmag/dt obtained from polynomial ﬁts to 5 second (20 data point) blocks of measured H and

Vmag values, I was able to observe Rleads,meas and Lmag,meas in real-time. I found that the lead resistance varied between approximately 7–12 mΩ, with higher values occurring at high fields and low helium bath levels, and lower values occurring at low fields and high helium bath levels, indicating that this effect is primarily due to Joule heating of the magnet leads inside the cryostat. The magnet inductance, which should depend only on the geometry of the magnet coils and therefore be independent of all other parameters, appears to vary somewhat haphazardly between roughly 40–70 Henries. For reference, the nominal inductance of our magnet, listed by the manufacturer, Oxford Instruments, is 60.8 Henries. To compensate for the time dependence of these quantities, I introduced Chapter 2. Instrumentation 53 an optional “Adaptive R and L” algorithm, usable in constant rate mode, that allows

HControl.vi to tinker with the internally-assumed values of Rleads and Lmag so that they

better match the real values of these quantities. Since Rleads,meas and Lmag,meas are based on ﬁts to a small number of scattered data-points, they are too noisy to be used directly,

so slightly more complicated adaptive schemes were required. Within the 7–12 mΩ range,

Rleads is increased by 0.05 mΩ if the measured (signed) magnetic ﬁeld sweep rate is less than the (signed) magnetic ﬁeld sweep rate currently set by HControl.vi, and decreased

by by 0.05 mΩ if the reverse is true, as long as the measured and set sweep rates diﬀer

from one another by least 2 mT/min; this can happen at most once every 10 seconds. For

the magnet inductance, Lmag,meas is coerced into the 40–70 Henries range (if necessary)

and mixed with Lmag once every 5 seconds according to the equation

L +(n L ) L = mag,meas × mag,old (2.11) mag,new n + 1

where n 0 is the “magnet induction smoothing order” chosen by the user. At present, ≥ running in constant rate mode with “Adaptive R and L” engaged is the only way to

do very slow, reproducible ﬁeld sweeps required for precise dHvA measurements in our

system.

2.1.6 Sample rotation control

VIs: MotorControl.vi, AOcore.vi

Hardware: AIdaq, AO13bit, motor control circuit, DC power supplies, DC motor

The sample rotation control subsystem is the newest part of DAVIES, having been

added less than a year prior to this writing, during my YbRh2Si2 dHvA study. The graphite rotation mechanism that I designed (section 2.2), which is attached to the dilution refrigerator mixing chamber and sits in the homogeneous region of the magnet, is connected to a series of gears and rods that allow rotation of a room-temperature Chapter 2. Instrumentation 54

Figure 2.4: Diagram of the DAVIES sample rotation control subsystem.

rotator knob to be translated into the single-axis, low-temperature rotation of three sample-containing “bobbins” relative to the ﬁeld direction. Approximately 16 full 360◦

“clockwise turns” at room temperature correspond to the entire 100◦ rotation range ∼ of the graphite rotation mechanism. The rotator knob may either be rotated by hand or by the rotation control DAVIES subsystem, coordinated by MotorControl.vi following the schematic diagram shown in Fig. 2.4.

When hooked up to DAVIES, the rotator knob is rotated by the DC motor, which in turn is driven by the motor control circuit, designed and built by my supervisor. The rail voltage required to power the motor control circuit electronics is provided by a Tenma dual-sided commercial DC power supply. The action of this circuit depends on two analog voltage signals sent from MotorControl.vi to AOcore.vi through DataSockets, and from there out to the circuit via AO13bit. When the “relay control voltage” is 0 V, a relay inside the motor control circuit is open, clamping the DC motor at the current angle; when this signal is 5 V, the relay is closed, allowing the circuit to drive the DC motor and rotate the samples. When the relay is closed, the“motor control voltage” determines how the motor should rotate: originally intended to be 5 V for clockwise rotation, 0 V − for no rotation, and +5 V for counterclockwise rotation, the actual circuit turned out to require 15 V, 10 V, and 5 V, respectively, for these actions. Since these voltages − − − Chapter 2. Instrumentation 55 are beyond the output range of AO13bit, I connected a Tenma single-sided commercial

DC power supply set to 10 V in series with the AO13bit motor control voltage signal − in order to oﬀset it into the required input range of the circuit.

For rotation angle monitoring, a 10-turn, 1 kΩ potentiometer is coupled to the DC motor (and therefore room temperature rotator knob) with 1-to-1 gearing. MotorCon- trol.vi sends an excitation current through the potentiometer (through DataSockets,

AOcore.vi, AO13bit, and then a shunt resistor) and then measures the potentiometer voltage (through AIdaq, TControl.vi, and then DataSockets) to determine the potentiometer resistance and therefore the present rotation position of the rotator knob. Upon reaching the end of the potentiometer rotation range, the potentiometer can be decoupled from the DC motor, reset to zero, and then re-coupled, as long as the user speciﬁes the number of clockwise rotator turns corresponding to the new potentiometer zero.

The MotorControl.vi motor control and angle measurement operations happen once every 100 ms, allowing very precise manual or automatic control of the sample angles ∼ relative to the magnetic ﬁeld. Moreover, considering the mechanical movement involved,

this rotation system does not generate much heat, with one full 360◦ rotator turn raising

the dilution refrigerator mixing chamber temperature by 30 mK when the mixing ∼ chamber is in the 50–100 mK range. Like the rest of the DAVIES VIs, MotorControl.vi

reports its results to, and can receive commands from, other VIs via DataSockets.

2.1.7 Data acquisition

VIs: PFgen.vi, ManualAdjustPF.vi, LockinSub.vi, BacklogAlertMonitor.vi, Get-

Param.vi, SampleAndSave.vi, ManualSweep.vi, DynamicSignalAnalyzer.vi (stand- alone)

Hardware: SigGen, AIdaq, AIgood

In addition to controlling the experimental temperature and magnetic ﬁeld (strength and direction) environment felt by the crystalline condensed matter samples studied by Chapter 2. Instrumentation 56

Figure 2.5: Diagram of the DAVIES data acquisition subsystem.

our research group, the primary function of DAVIES is to actually perform the experi-

ment, a job which falls to the data acquisition subsystem. Moreover, data acquisition in

DAVIES represents a new paradigm for dHvA measurement, by replacing the costly and

unwieldy lock-in ampliﬁers used in a ﬁeld modulation experiment (section 1.5.3) with

software-based “virtual lock-ins,” exceeding the capabilities of a typical dHvA labora-

tory. The data acquisition subsystem can be used not just for dHvA, but for any type of

experiment that is based on either phase-sensitive or DC voltage measurements, such as

measurements of resistivity. A schematic diagram of this subsystem is shown in Fig. 2.5.

In most experiments, an excitation signal is required to perturb the sample and eluci-

date a response that can be measured by detection instrumentation. PFgen.vi ﬁlls this

role in DAVIES, and can direct SigGen to produce sine waves, square waves, triangle

waves, sawtooth ramp up and ramp down waves, and DC voltages, with frequencies in

the 1 mHz–1MHz range, amplitudes in the 17 mV–10 V range ( 3 to +3 V when used − as the DC voltage oﬀset), and a noise spectrum cleaner than any other instrument I have tested in our lab, including the signal generator built into the industry-standard

SR-830 lock-in ampliﬁer. One limitation of the SigGen hardware that must be kept in mind when setting up an experiment is that connecting or disconnecting cables from it or its downstream signal chain while it is actively generating an output waveform can Chapter 2. Instrumentation 57 cause unwanted changes to the generated signal. PFgen.vi may be controlled through

DataSockets, either manually by the user with ManualAdjustPF.vi or automatically by

FullAutoSweeps.vi.

At the core of the data acquisition subsystem, and indeed of DAVIES itself, lies

LockinSub.vi, which contains the virtual lock-in functionality. During data collection,

AIgood samples the analog voltages present at all 8 of its input channels continuously at a typical rate of 50 000 samples/second. One of these channels must contain a reference excitation signal, whether generated by PFgen.vi or an external device. Since AIgood

does not possess an on-board pre-ampliﬁer, and therefore the 24 input bits are spread

over a 10 V to +10 V range, external pre-ampliﬁcation is very important. Every 100 ms, − LockinSub.vi reads a 100-ms-long block of data on all 8 channels simultaneously from

the AIgood first-in-first-out (FIFO) hardware buffer, applies up to 24 copies of a software

lock-in algorithm (based on functions from the National Instruments Lock-In Ampliﬁer

Start-Up Kit [56]) and up to 4 DC voltage averaging tasks (useful for measuring the

analog outputs of other instruments) to any combination of channels and parameters,

then reports the results to DataSockets, before going back to AIgood to read another

block. If a given read-in block does not contain enough cycles of the locked frequency,

several contiguous blocks are stacked in a rolling FIFO manner until enough cycles are

obtained.

Since this is an extremely processor-intensive set of tasks, it is possible for Lockin-

Sub.vi to fall behind in its data processing and cause a backlog of raw data in the AIgood

hardware buﬀer. This is bad not only because it means that LockinSub.vi is working

with old data, out of sync with the measurements of TControl.vi and HControl.vi, but

because if the buﬀer ﬁlls up completely (equal to a 10 second data backlog) it causes a

serious hardware error. To keep this situation under control, BacklogAlertMonitor.vi

constantly watches the AIgood buﬀer backlog through DataSockets, and causes a “data

dropout”—briefly putting LockinSub.vi into a standby mode and then restarting data Chapter 2. Instrumentation 58 collection, to clear the hardware buffer—if the backlog exceeds a user-specified level

(typically 10% of the buffer size). If, for some reason, BacklogAlertMonitor.vi fails to protect the buffer, LockinSub.vi has the capability to catch the buffer-full hardware error at the last possible moment and reboot itself. Since the maximum “safe” number of simultaneous lock-in algorithm instances that can be applied without accumulating a data backlog (24 with the current set-up, at the time of writing) depends on the available computer processing power, I have gone to great lengths to optimize the performance of all DAVIES VIs, and indeed the entire Max computer operating environment, in order to squeeze the most resources out of the system.

Multiple copies of the lock-in algorithm are organized into four “virtual lock-ins,” each of which has an independent set of user-specified parameters and can perform simultaneous phase-sensitive detection on up to 6 user-specified harmonics ν1,2,3,4,5,6 of the reference signal, for a given measurement signal. Each of these harmonics would have traditionally required its own lock-in amplifier instrument. The control parameters for each virtual lock-in are specified by the user prior to LockinSub.vi execution using

GetParam.vi. The large available parameter space, particularly in terms of filter types, roll-offs and time constants [50, 56], and the potential to extend this parameter space by further programming, make the virtual lock-ins much more configurable than their hardware counterparts. DC voltage measurements, VDC , are also filtered before being sent out to DataSockets, using the following equation:

Nb−1 V exp [ i/ (N 1)] DC,i − b − i=0 VDC,avg = X (2.12) Nb−1 exp [ i/ (N 1)] − b − i=0 X

where Nb is the number of most recent read-in blocks to average, equal to the ﬁlter time

constant (in discrete 100 ms steps) divided by the block length (100 ms), and VDC,i is the DC voltage measurement uniformly averaged over all of the samples in block i. If Chapter 2. Instrumentation 59 a DC voltage ﬁlter time constant of 0 ms or 100 ms is speciﬁed, V V , with DC,avg ≡ DC,i

VDC,i taken from the most recent read-in block.

While TControl.vi (section 2.1.4), HControl.vi (section 2.1.5), MotorControl.vi

(section 2.1.6), and LockinSub.vi are all responsible for bringing measurements of phys-

ical quantities into Max, processing them, and dumping them into the DataSockets pool

every 100–250 ms, it is SampleAndSave.vi that periodically reads the values of these ∼ DataSockets (typically once every second for my experiments, set using GetParam.vi)

and records them to a data ﬁle on Max’s hard drive. By also watching various auxil-

iary DataSockets, SampleAndSave.vi can detect when a LockinSub.vi data dropout,

TControl.vi AVS bridge settle period, or other DAVIES trouble occurs, and temporar-

ily pause itself until the problem clears up. SampleAndSave.vi does not do any av-

eraging itself, rather picking the most recent value of each DataSocket and relying on

the VIs feeding the DataSocket pool to perform appropriate ﬁltering; to this end, when

SampleAndSave.vi is actively sampling, TControl.vi and HControl.vi set their inter-

nal time constants equal to the time between SampleAndSave.vi reads, and the time

constants of the virtual lock-ins and DC voltage measurements in LockinSub.vi should

be set similarly by the user. The data, all relevant control parameters, and descriptions

of the external signal chains are recorded to the hard drive in the Network Common Data

Form (NetCDF), a self-describing, open standard, machine-independent data format [57].

Summary text ﬁles, including details of any errors or data dropouts encountered, are also

saved.

In DAVIES nomenclature, a particular round of data-taking is called a “sweep.” When

not under the control of the automation subsystem (section 2.1.8), the data acquisition

subsystem is coordinated at a high level by ManualSweep.vi. Through DataSockets,

this VI directs the activities of GetParam.vi, LockinSub.vi, BacklogAlertMonitor.vi,

SampleAndSave.vi, and RealtimeAnalysis.vi (section 2.1.9), allowing data to be col-

lected indeﬁnitely while the user manually controls the temperature, magnetic ﬁeld, Chapter 2. Instrumentation 60 rotation angle, and signal generator through ManualAdjustTH.vi, the front panel of

MotorControl.vi, and ManualAdjustPF.vi.

DynamicSignalAnalyzer.vi is the rebellious step-child of the data acquisition subsystem. This highly-conﬁgurable, stand-alone software oscilloscope (based on functions from the National Instruments Sound and Vibration toolkit [58]) is not strictly a part of DAVIES, since it does not communicate with the rest of the system via DataSock- ets, and, in fact, can cause hardware conﬂicts with core DAVIES VIs [50]. Neverthe- less, with judicious use, DynamicSignalAnalyzer.vi is a powerful tool. As an oscilloscope, DynamicSignalAnalyzer.vi can measure and display voltage traces from any of the AIdaq or AIgood input channels in real-time. From the measured signal,

DynamicSignalAnalyzer.vi calculates the power spectrum, magnitude and phase Fast

Fourier Transforms (FFTs), power spectral density, total harmonic distortion (THD), and signal in noise and distortion (SINAD), all in real-time, with user-conﬁgurable averaging parameters. Power spectra may also be saved to the hard drive for later analysis.

2.1.8 Automation

VIs: FullAutoSweeps.vi, GetParam.vi, ExtraWaitTimer.vi

Hardware: none

Full automation of all DAVIES subsystems is possible, coordinated by FullAuto-

Sweeps.vi following the schematic diagram shown in Fig. 2.6. If pre-programmed

automatic experiment control and data collection is desired, FullAutoSweeps.vi re-

places ManualSweep.vi to control LockinSub.vi, BacklogAlertMonitor.vi, Sample-

AndSave.vi, and RealtimeAnalysis.vi, and replaces ManualAdjustTH.vi, the Motor-

Control.vi front panel objects, and ManualAdjustPF.vi to control TControl.vi, H-

Control.vi, MotorControl.vi, and PFgen.vi.

FullAutoSweeps.vi organizes a series of sequential sweeps into an automation unit

called a “job,” whose control parameters are speciﬁed by the user via GetParam.vi. Once Chapter 2. Instrumentation 61

Figure 2.6: Diagram of the DAVIES automation subsystem.

a job has begun, no further input is required from the user, but individual sweeps may be skipped or the job aborted, if desired, at any point while the job is in progress. Through the ExtraWaitTime.vi interface, the time between two sweeps in the sequence may also

be changed while the job is in progress, for example if an emergency helium transfer is

required between sweeps, or if one wishes to delay data collection until vibrations from a

lawnmower or passing truck have subsided. In a simple example job, FullAutoSweeps.vi

might automatically orchestrate the following actions:

1. Starting at 30 mK and 0 T, with the sample rotator at 4 clockwise turns, quickly

sweep the temperature to 50 mK, ﬁeld to 8 T, and rotator to 5 clockwise turns.

2. Once the temperature, ﬁeld, and rotator have suﬃciently settled at their new values,

collect data using the virtual lock-ins while the ﬁeld is slowly swept from 8 to 10 T,

with the temperature and rotator held steady at 50 mK and 5 clockwise turns,

respectively.

3. Collect data using a different virtual lock-in configuration while the field is swept

at a medium pace from 10 back to 8 T, with the temperature and rotator held

steady at 50 mK and 5 clockwise turns, respectively. Chapter 2. Instrumentation 62

Figure 2.7: Diagram of the DAVIES monitoring subsystem.

4. Quickly sweep the temperature to 100 mK, ﬁeld to 4.567 T, and rotator to 9.876

clockwise turns.

5. Once the temperature, ﬁeld, and rotator have suﬃciently settled to their new values,

collect data using the DC voltage channels while the temperature is slowly swept

from 100 to 567 mK, with the ﬁeld and rotator held steady at 4.567 T and 9.876

clockwise turns, respectively.

6. Sweep the temperature back to 30 mK, ﬁeld to 0 T, and rotator to 4 clockwise

turns, then end the job.

2.1.9 Monitoring

VIs: RealtimeAnalysis.vi, RTAConfig.vi, DataSocketSniffer.vi, LevelMeterRea- der.vi, TControl.vi, HControl.vi

Hardware: level monitor, lambda fridge controller, web cam

From a user’s point of view, an important part of running an experiment is being able to monitor what is going on (including remotely), while the experiment progresses. The focal point for such monitoring while a data collection sweep is underway is RealtimeAnalysis.vi. Starting from the beginning of each sweep, RealtimeAna- lysis.vi records all of the SampleAndSave.vi-sampled experimental quantities (the sample number, temperature, magnetic field, heater powers, virtual lock-in X, Y , R, θ Chapter 2. Instrumentation 63 and locked frequency parameters for all samples and harmonics, DC voltage measurements, etc.) into a large internal multi-dimensional array. Any quantity in this array can be plotted as a function of any other quantity in the array, in real-time as the data is being collected. Due to performance constraints, the size of the RealtimeAnalysis.vi internal array is limited to a maximum of 9600 samples, equivalent to 160 minutes of continuous data collection at the typical SampleAndSave.vi sampling rate of one sample per second. Once the array fills up, RealtimeAnalysis.vi automatically enters “rolling” mode, in which the oldest samples in the array are dropped as new samples are added— i.e. during long sweeps, the arrays holds the last 160 minutes of data. As the name of the VI suggests, all or parts of the plotted data can also be analyzed in real-time, using configurable, built-in routines: FFT generation, in 1/(the x-axis scale), for dHvA measurements; power law fitting, for resistivity measurements; linear fitting, in 1/(the y-axis scale), for ac-susceptibility measurements; and polynomial fitting, with selectable orders up to 5.

Compared to RealtimeAnalysis.vi, DataSocketSniffer.vi is much simpler, allow-

ing real-time read-out of current DataSockets, without any plotting, analysis, or logging

abilities. However, DataSocketSniffer.vi is also more ﬂexible than RealtimeAnalysis.vi,

and can simultaneously display the values of up to 25 numeric and 1 text-string DataSock-

ets, updated on a 100 ms timescale, at any time—not limited to the values sam- ∼ pled by SampleAndSave.vi during a sweep. Since RealtimeAnalysis.vi and DataSoc- ketSniffer.vi are fairly resource intensive, during the DAVIES development cycle I moved them from Max to Dew in order to free up more processing power on Max for

LockinSub.vi. A beneﬁcial by-product of this move is that, in the event of an emergency, such as Max crashing in the middle of a sweep and corrupting the NetCDF data

ﬁle, the contents of the RealtimeAnalysis.vi array can be dumped to a text ﬁle, thus preserving the collected data.

More generally, TControl.vi and HControl.vi show the values that they are measur- Chapter 2. Instrumentation 64 ing on their respective front panels, but unless SampleAndSave.vi is running, these are not permanently recorded. Additionally, LevelMeterReader.vi measures and displays the liquid helium and liquid nitrogen bath levels in the dilution refrigerator / superconducting magnet cryostat, and the temperature of the lambda fridge, which is used to provide extra cooling to the magnet to allow it to operate in the 16–18 T range without quenching. Max and Dew may both be accessed remotely, through several layers of security, using the Virtual Network Computing (VNC) protocol. Although, unlike the

Oxford Instruments IGH Intelligent Gas Handling system, our dilution refrigerator gas handling panels use manual valves which must be turned by hand, their pressure gauges and warning lights may be monitored remotely via a web cam connected to Dew.

2.1.10 Test results and discussion

Since our group’s dilution refrigerator was non-functional for the bulk of my Ph.D. (and therefore the magnet was warm and thus also unusable), I had to do much of the development and testing of DAVIES via room temperature mock-up. The virtual lock-ins of the data acquisition subsystem represent the most radical departure from a traditional measurement set-up, so it was important that I test and benchmark them relative to a Stan- ford Research Systems SR830 lock-in ampliﬁer, the industry-standard phase-sensitive detector used in condensed matter laboratories worldwide (especially for dHvA measurements). In order to carry out this comparison in a controlled environment, I set up a resistance measurement of the programmable resistance box, and modiﬁed HControl.vi to change this resistance as a function of time in a reproducible manner. To ensure the presence of higher-harmonic content for the lock-ins to measure, I used PFgen.vi to send a 4 Hz square-wave excitation current (whose Fourier series is a sum of odd harmonic terms) through the resistance box. The voltage across the resistance box was then measured at the ν = 5 harmonic of the excitation signal in parallel by both a DAVIES virtual lock-in and the SR830. The SR830 front panel analog outputs (which scale the displayed Chapter 2. Instrumentation 65

-0.04 DAVIES virtual lock-in -0.04 DAVIES virtual lock-in SR830 lock-in SR830 lock-in

-0.08 -0.08

-0.12 -0.12

-0.16 -0.16 (a) (b) -0.20 -0.20 = 5 = harmonic5 (arb. signalunits) = harmonic5 (arb. signalunits) ν "low filtering" ν "high filtering" -0.24 -0.24 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (seconds) Time (seconds)

Figure 2.8: DAVIES virtual lock-in vs. SR830 lock-in amplifier test results, for a sinusoidally-varying signal with a discontinuous jump just before t = 70 s. A 4 Hz square-wave excitation signal was sent through the programmable resistance box, and phase sensitive detection proceeded on the ν = 5 harmonic. In panel (a), the SR830 filter roll-off was set to 6 dB/oct and the virtual lock-in filter roll-off was set to an equivalent 20 dB/dec. In panel (b), the SR830 filter roll-off was set to 24 dB/oct and the virtual lock-in filter roll-off was set to an equivalent 80 dB/dec.

measurements up into a 10 to +10 V output range) were connected to AIgood and − measured as LockinSub.vi DC voltage measurements, with no further ﬁltering applied.

SampleAndSave.vi recorded the data once every second, and the filter time constants of both the virtual lock-in and the SR830 were correspondingly set to 1 s. Several runs with varying degrees of filter roll-off were performed, but in each case equivalent roll-off values were used in the virtual lock-in and SR830. Since, unlike the SR830, the DAVIES virtual lock-ins can use one of several types of filter, four virtual lock-ins configured with different filters were run in parallel on the same input signal. The best results, shown here, come from the default Finite Impulse Response (FIR) filter.

The results of these tests are shown in Fig. 2.8. The resistance box resistance was programmed to vary sinusoidally, with a discontinuous jump just before t = 70 s. For the data shown in Fig. 2.8(a), the SR830 ﬁlter roll-oﬀ was set to its lowest setting,

6 dB per octave, and the virtual lock-in filter roll-off was set to an equivalent 20 dB per decade. For the data shown in Fig. 2.8(b), the SR830 filter roll-off was set to its Chapter 2. Instrumentation 66 highest setting, 24 dB per octave, and the virtual lock-in filter roll-off was set to an equivalent 80 dB per decade (the highest roll-off setting currently available in the virtual lock-ins is 200 dB per decade). In the “low filtering” case, the SR830 measurement is clearly noisier than that of the virtual lock-in, while simultaneously giving a poorer, more rounded reproduction of the real data discontinuity. In the “high filtering” case, both measurements are smoother than their “low filtering” counterparts, but while the virtual lock-in still reasonably reproduces the discontinuity, the SR830 response is very rounded; the “high filtering” SR830 measurement also shows a pronounced filter phase lag. Thus, in both filtering regimes, the virtual lock-in more faithfully reproduces the actual signal.

In terms of performance in an actual measurement, during my YbRh2Si2 dHvA study

(chapter 6), the signal from YbRh2Si2 “sample C” was usually measured in parallel by the SR830, whose results were recorded by DAVIES through LockinSub.vi DC voltage

measurements. As with my resistance box test results, the virtual lock-in and SR830

datasets were consistent with one another, albeit with the SR830 continuing to exhibit

the drawbacks shown in Fig. 2.8.

2.2 Graphite rotation mechanism

When my supervisor moved from the University of Cambridge to the University of

Toronto, he brought with him a rotation mechanism that allows samples to be rotated

relative to the magnetic ﬁeld direction, in situ inside a dilution refrigerator. To avoid

excessive heating due to eddy currents induced by the modulation ﬁeld used during a

dHvA measurement, this tail was constructed out of polycarbonate plastic. Unfortu-

nately, the large nuclear moments of the hydrogen atoms in plastic have a tendency to

undergo adiabatic demagnetization during magnetic ﬁeld sweeps, which releases heat into

the samples.

Since carbon atoms have much smaller nuclear moments than hydrogen atoms, gra- Chapter 2. Instrumentation 67

Figure 2.9: Panel (a): labelled components of the graphite rotation mechanism. Panel (b): assembled rotation mechanism, with the purple arrow indicating the direction of motion of the rotation arm, and the blue arrows indicating the axes of rotation of the bobbins—rotation arm “up” motion corresponds to bobbin clockwise rotation, and “down” to counterclockwise rotation. phite is less susceptible to this phenomenon than is plastic. Therefore, in order to lower the minimum eﬀective sample temperature in our experimental set-up, I designed a new rotation mechanism made mostly of graphite, based on the layout of the old plastic rotation mechanism. (In addition to the plastic one, a graphite rotation mechanism had been previously used at the University of Cambridge, but this was not available for us to use in Toronto.) The components of the new graphite rotation mechanism were built by machinist Mark Aoshima in the University of Toronto mechanical workshop, following my designs, and are shown in Fig. 2.9(a).

The backbone of the rotation mechanism is a sturdy graphite frame. This frame contains three bays, each of which can hold one graphite “bobbin;” the bobbins serve as rigid mount points for the pick-up coils and samples (as in Fig. 6.13, for example), and rotate relative to the frame. The bobbins are held in place by“screw axles,” which are polycarbonate screws that ﬁt into threaded holes on either side of each frame bay. Chapter 2. Instrumentation 68

At the tip of each screw axle is a small, smooth axle that protrudes into a hole in the side of a bobbin, fixing the bobbin into the rotation mechanism and defining an axis around which the bobbin may freely rotate. Each screw axle also has a small hole drilled along the rotation axis, through which measurement and heat-sink wires may be routed out of the bobbin. Finally, a flexible polycarbonate “rotation arm” runs the length of a groove cut into the frame, and is coupled to all three bobbins using “rotation pins” made from unjacketed, 750 µm diameter plastic optical fibre. My testing has shown that the rotation arm is just as flexible at liquid nitrogen temperature (77 K) as at room temperature. Each rotation pin is oriented perpendicular to the rotation arm, and sits inside holes drilled in the two bobbin “shoulders” and an eyelet on the rotation arm.

The fully-assembled rotation mechanism is shown in Fig. 2.9(b). The cylindrical part of the frame (top right corner of Fig. 2.9(a)) is glued inside a hollow quartz tube whose other end is anchored to the dilution refrigerator mixing chamber. The small rounded piece at the other end of the frame (top left corner of Fig. 2.9(a)) is attached to a centring mechanism (bottom of Fig. 2.9(b)), which keeps the rotation mechanism in the middle of the radiation shield and homogeneous region of the magnet. The rotation arm is connected to a longer arm inside the quartz tube, and from there to a gearing system

(built by my supervisor) reaching up to room temperature. This gearing system translates rotations of a rotator knob located outside of the dilution refrigerator (section 2.1.6) to vertical motion of the low-temperature rotation arm (purple arrow in Fig. 2.9(b)), and therefore rotation of the sample-carrying bobbins about the axes of their screw axles

(blue arrows in Fig. 2.9(b)).

2.3 Silver annealing

Although reducing magnetocaloric heat coming from the nuclear moments in the surrounding rotation mechanism material is helpful, the samples themselves are metallic Chapter 2. Instrumentation 69

Figure 2.10: Sketch of silver heat-sink wire annealing apparatus.

and thus experience intrinsic Joule heating from eddy currents induced by the modulation ﬁeld. Therefore another important way that samples are kept cold is by good thermal linking to the dilution refrigerator mixing chamber. This is achieved by soldering one end of a pure silver heat-sink wire to the sample, and attaching the other end to the mixing chamber. Small diameters of wire are used close to the samples in the high

ﬁeld region, to reduce eddy current heating in the wires themselves, changing to larger diameters on approach to the ﬁeld-cancelled region near the mixing chamber. Annealing the silver wires to remove carbon and other impurities and encourage single crystal for- mation can result in a dramatic increase in electrical conductivity, and therefore thermal conductivity, indicated in previous work by a jump in the Residual Resistivity Ratio

(RRR ρ(T = 300 K)/ρ(T 0 K)) from 150 to 8400 [59]. ≡ → In order to anneal silver wires for use in our experiments, I set up an annealing station, sketched in Fig. 2.10, based on the design of a similar facility in my supervisor’s previous laboratory at the University of Cambridge [59]. Prior to annealing the wires, I cleaned the key parts of the system:

1. The alumina trough was boiled in a strong acid solution of 33% hydrochloric acid,

33% nitric acid and 33% distilled water, then rinsed in distilled water.

2. The annealing apparatus was set up as shown in Fig. 2.10, with valves 1 and 2 Chapter 2. Instrumentation 70

closed, valve 3 open, and no silver wires inside the quartz tube.

3. The quartz tube was pumped down to pressure of 7 10−6 millibar using the ∼ × turbo pump, and then the alumina trough was baked at 900◦C for 12 hours with

the turbo pump engaged.

4. The annealing apparatus was disassembled, the alumina trough was re-boiled in the

strong acid solution, and the quartz tube was rinsed in the strong acid solution.

5. The alumina trough and quartz tube were rinsed in distilled water.

Once the system was clean, I proceeded to the main task of annealing. 25 cm long pieces of 1 mm, 0.25 mm and 0.05 mm diameter silver wire were placed into the alumina trough, which in turn was placed into the quartz tube, and the annealing system was reassembled as shown in Fig. 2.10. I then performed the following steps to anneal the silver wires:

1. With valves 1 and 2 closed, and valve 3 open, the quartz tube was held at 200◦C

and pumped out overnight with the turbo pump.

2. Valve 3 was closed, and valves 1 and 2 opened, to ﬂush the quartz tube with 20%

nitrogen / 80% oxygen gas. During this time, the roughing pump built into the

turbo pump was left on.

3. Valves 1 and 2 were closed, and then valve 3 gradually opened, until the roughing

pump reduced the pressure inside the quartz tube to 3 10−5 millibar. ∼ ×

4. Steps 2 and 3 were repeated, to ensure high purity of the gas inside the quartz

tube.

5. Valve 3 was closed and the pump turned oﬀ, sealing the quartz tube under a rareﬁed

20% nitrogen / 80% oxygen atmosphere. Chapter 2. Instrumentation 71

6. The temperature was ramped from 200◦C to 600◦C at a rate of 3.3◦C per minute,

then held at 600◦C for 10 hours.

7. The temperature was ramped from 600◦C to 900◦C at a rate of 3.3◦C per minute,

then held at 900◦C for 12 hours.

8. The temperature was ramped from 900◦C to 25◦C (room temperature) at a rate of

1.2◦C per minute.

At the end of this annealing process, the diameter of the largest wire was still roughly

1 mm, the diameter of the medium wire had decreased from 0.25 mm to 0.20 mm, and the smallest wire had completely evaporated. Since small diameter heat-sink wires are important in the high-ﬁeld region of the magnet, I successfully repeated the steps above using three 0.05 mm diameter wires and a maximum annealing temperature of 800◦C instead of the original 900◦C.

2.4 Glove box

One of my doctoral experimental studies focused on the heavy fermion material CePb3 (chapter 5). Since pure crystals of this material react violently with air [60], all sample preparation activities (section 5.2) needed to be performed in an inert helium gas environment. To facilitate this, I constructed a glove box and associated gas handling system, shown in Fig. 2.11. The shell of the glove box is based on a large piece of plexiglass bent by our post-doc, Alix McCollam, using a home-made plexiglass-bending set-up. I cut additional plexiglass panels and fused them to the starting piece by injecting a strong solvent into the joints between the panels. I then cut holes for, and mounted, the heavy- duty, shoulder-length gloves (appearing in Fig. 2.11 as beige shapes inﬂated by glove box overpressure), installed sealed feedthroughs for electrical power and spot-welder elec- trodes, and positioned a microscope above the glove box, looking down into the interior Chapter 2. Instrumentation 72

Figure 2.11: Sketch of glove box gas handling system.

through the top panel. Throughout the construction process, I found and fixed leaks by spreading soapy water over the glove box, overpressurizing the box from the inside, and then watching to see if/where bubbles were blown. In order to help remove air and purify the helium gas inside the glove box, I added a gas handling system, sketched in Fig. 2.11, using valves, vacuum fittings, hoses, a pressure gauge, a liquid nitrogen cold trap, and an Oxford Instruments GFS/VPZ0233 oil-free gas flow diaphragm pump scavenged from around the physics department.

To ensure the safety of the air-sensitive samples I wanted to study, it was important that I be able to monitor the oxygen level inside the glove box in real-time, while the samples were being prepared and were otherwise unprotected. Unfortunately, commercial oxygen detectors cost hundreds to thousands of dollars, and were deemed too expensive. Chapter 2. Instrumentation 73

I therefore built my own “$1 oxygen detector” using a hearing aid battery, a small resistor, and an ammeter. A zinc-air hearing aid battery requires oxygen from the air in order to drive a chemical reaction and maintain a voltage across its terminals; when these terminals are shorted together, the “short-circuit” current is proportional to the diﬀusion rate of oxygen into the cell [61, 62]. Since the geometry of the holes that allow air into the cell is ﬁxed, the short-circuit current is proportional to the oxygen concentration in the surrounding atmosphere. In my case, the current was measured using a laboratory ammeter in series with a 24 Ω resistor. The type 675 zinc-air hearing aid batteries used in my oxygen detector had capacities of 500–600 mAh and a typical short-circuit current ∼ of 36 mA.

Over the course of a sample preparation run, three principal glove box gas handling system operation modes can be used:

Purge mode: Valves 1, 2, 3, and 4 are open, valves 5, 6 and 7 are closed, and the • pump is on. In this mode, pure helium gas is pumped into the glove box from

the cylinder, and the mixture of helium and air in the glove box is pumped out to

atmosphere, resulting in a decrease in air concentration over time inside the glove

box.

Circulation mode: Valve 1 is barely open, valves 2, 3 and 5 are open, valves 4, 6 • and 7 are closed, and the pump is on. In this mode, the glove box gas mixture is

circulated through the liquid nitrogen cold trap. Air and water vapour molecules

are captured in the cold trap, thus purifying the helium gas inside the glove box.

Isolation mode: Valves 2 and 3 are closed, isolating the glove box from the rest of • the gas handling system. Unless the cold trap is being cleaned, all other valves are

typically kept closed, and the pump turned oﬀ.

If the cold trap ﬁlls up and blocks (as indicated by a high and rising pressure on the pressure gauge between the pump and trap), it may be cleaned, without disruption to Chapter 2. Instrumentation 74 the activities underway in the glove box, by performing the following steps:

1. Make sure the system is in isolation mode, with all valves closed and the pump

turned oﬀ.

2. Open valves 4, 6 and 7, then turn on the pump. The contents of the cold trap are

now being pumped out to atmosphere, via the glove box bypass line.

3. Take the cold trap out of its liquid nitrogen bath, and heat it with a heat gun until

its temperature is greater than 100◦C.

4. Open valve 1 to purge the pump and glove box bypass line with helium gas.

5. Close valves 6 and 7, and turn oﬀ the pump.

6. Open valve 5 to purge the cold trap with helium gas.

7. While purging with helium gas, put the cold trap back into its liquid nitrogen bath.

8. Close valves 4, 5 and 1.

Once the samples, tools, oxygen detector, and other equipment are in place and

the glove box sealed, the system is ﬁrst operated in purge mode, without any liquid

nitrogen in the cold trap bath. Pushed to its limits, this purging process can take the

oxygen concentration in the glove box from 21% (the starting concentration of the ∼ surrounding atmosphere) to 2%. Next, the liquid nitrogen bath is ﬁlled, and the gas ∼ handling system switched to circulation mode, further reducing the glove box oxygen

concentration to 1%. The glove box is then placed into isolation mode, with a slight ∼ overpressure (so that any leaks will leak helium out, rather than air in), and sample

preparation work may commence. If, at any point, the oxygen concentration rises too

high, the system may be brieﬂy switched back into purge or circulation mode to reduce it

into an acceptable range. By heavily overpressurizing the glove box (inﬂating the gloves

and preventing any work from being done inside) and placing it in isolation mode, the Chapter 2. Instrumentation 75 oxygen concentration rise is reduced to 0.2% per day, allowing longer-term storage of ∼ sensitive materials. Chapter 3

Supercell K-space Extremal Area

Finder

The instrumentation developments presented in Chapter 2 allow for precise experimental measurement of de Haas–van Alphen (dHvA) oscillations. However, as noted in sections 1.4 and 1.5, full characterization of a given material’s Fermi surface relies on comparisons between measurements and electronic structure calculations. While user- friendly computer software that employs advanced density functional theory techniques to calculate band energies (section 1.4) is widely available [30], programs to extract de

Haas–van Alphen frequency and eﬀective mass predictions based on these calculated band energies are not. In fact, the task of determining extremal orbits on calculated

Fermi surfaces of arbitrarily-complex topology, for arbitrary magnetic ﬁeld directions, is not trivial. This chapter, following an article I have written on the subject [11], describes the Supercell K-space Extremal Area Finder (“SKEAF”) algorithm that I have developed to solve this problem in general, and thus bridge the gap between calculation and experiment.

76 Chapter 3. Supercell K-space Extremal Area Finder 77

3.1 Basic concepts of Fermiology

For readers new to Fermi surface measurements, a few deﬁnitions are required beyond those provided in Chapter 1. Since, at zero temperature, the Fermi surface separates occupied from unoccupied k-space states, an orbit refers to a path along this surface

that an electron may trace out under the inﬂuence of a magnetic ﬁeld; in my case, I am

dealing with orbits that are constrained to lie on the intersection of the constant-energy

Fermi surface and a plane perpendicular to the applied magnetic ﬁeld. A closed orbit

is one that forms a closed loop around some part of the Fermi surface. An extremal

orbit is a particular closed orbit whose cross-sectional area is either locally maximum

or locally minimum, compared to adjacent orbits on the same Fermi surface sheet at

the same magnetic ﬁeld angle. Only extremal orbits are detected as quantum oscillation

frequencies in a dHvA experiment; I, II and III in Fig. 3.1 are examples of extremal

orbits that occur at various angles in UPt3. The Brillouin zone (or equivalently, the reciprocal unit cell, depending on the particu-

lar volume of k-space enclosed) is the basic unit of momentum space, which repeats in all

directions as does the real-space unit cell of a crystal. As shown in Fig. 3.1, sometimes

a Fermi surface links up with copies of itself in neighbouring Brillouin zones, forming

one inﬁnitely large, complicated shape. On such a surface, it is possible to have open

orbits: orbits that continue forever in one direction, never coming back on themselves to

form a closed loop. IV in Fig. 3.1 is an example of an open orbit. While open orbits

cannot be detected in quantum oscillation measurements, their presence can be inferred

from other experiments, such as angle-resolved magnetoresistance. Near-open orbits are

extremal orbits that are closed, but would become open at a slightly diﬀerent magnetic

ﬁeld angle. III in Fig. 3.1 is a near-open orbit, because a small change in magnetic ﬁeld

angle will transform it into an open orbit similar to IV . Finally, when determining the

orbit type of a particular extremal orbit, an electron orbit is one that encloses occupied

states, separating them from the unoccupied states outside the orbit; conversely, a hole Chapter 3. Supercell K-space Extremal Area Finder 78

Figure 3.1: The band 2 Fermi surface of UPt3, tiled in several Brillouin zones (modiﬁed from [13]). I is a simple closed orbit, easily identiﬁed as extremal by visual inspection; II is a less-obvious extremal orbit that crosses Brillouin zone boundaries; III is a near-open extremal orbit; and IV is an open orbit. Chapter 3. Supercell K-space Extremal Area Finder 79 orbit is one that encloses unoccupied states, and is associated with the motion of a “hole” through a Fermi sea of electrons rather than motion of a single electron itself.

3.1.1 Comparison to band structure

In order to understand the physical implications of dHvA data, the measured frequencies and masses are compared to those predicted by electronic structure calculations. How- ever, since real compounds often possess complicated Fermi surfaces, including open / near-open, nested and non-central orbits, the task of extracting predicted dHvA orbits from calculated band energies is non-trivial.

One previous approach involved fitting the calculated band energies to a Fourier series of lattice-specific star functions, which was then evaluated at various points in k-space. An orbit centre and plane would be manually specified, then the algorithm would perform a series of “stepping” and “return-to-surface” operations as a function of rotation angle around the orbit centre to determine the cross-sectional area via Simpson’s- rule integration [63]. This approach was later expanded to included trapezoidal-rule integration for orbits which are multivalued at certain angles around the orbit centre [64].

While such an approach works well when one knows which orbits are extremal (and the locations of the associated orbit centres), topologically-complicated Fermi surfaces can have extremal orbits that are not obvious from visual examination of the surfaces.

3.2 Algorithm details

3.2.1 Overview

My Supercell K-space Extremal Area Finder (SKEAF) algorithm is designed to exploit the processing capabilities of current desktop computers in order to automatically extract extremal orbits, eﬀective masses and density of states contributions from calculated Fermi surfaces of arbitrary topology, without requiring manual guidance or bias from the user. Chapter 3. Supercell K-space Extremal Area Finder 80

The code [65] implementing this algorithm is written in the Fortran 90 language, and reads files defined in the Band-XCrysDen-Structure-File (BXSF) format. BXSF files specify band energies on a three-dimensional grid within a parallelepiped Reciprocal

Unit Cell (RUC) deﬁned by three reciprocal lattice vectors: ~a, ~b, and ~c [66, 67]. Typical

input ﬁles prepared for my program contain on the order of 20 000 k-points, whose band

energies have been calculated by an electronic structure program such as WIEN2k [30].

Upon reading the input ﬁle, the following steps are performed:

1. A cubic k-space Super Cell (SC), considerably larger than the original reciprocal

unit cell and aligned with the desired magnetic ﬁeld vector, is constructed (sec-

tion 3.2.2). A coordinate transformation maps the super-cell k-point grid back into

the reciprocal unit cell. Band energies at each of the super-cell grid points are de-

termined from those provided in the reciprocal unit cell via Lagrange interpolating

polynomials. The density of k-points in the super-cell grid is typically much greater

than that of the reciprocal-unit-cell grid.

2. The super-cell grid is divided into slices 1 k-point thick, perpendicular to the mag-

netic ﬁeld vector. On each slice, the program scans through the k-points, locating as

orbits the Fermi surface outlines (section 3.2.3). The cross-sectional area, eﬀective

mass, and orbit type (hole or electron) are calculated for each orbit (section 3.2.4).

3. Orbits are matched from slice to slice, so that each orbit is associated with a

particular Fermi surface sheet (section 3.2.5).

4. On each Fermi surface sheet, the orbits which are extremal are singled out (sec-

tion 3.2.6). The orbit data for similar orbits found on separate sheets are averaged,

and the results output to a ﬁle.

5. If automatic rotation is enabled, steps 1–4 are repeated for each new magnetic ﬁeld

vector. Chapter 3. Supercell K-space Extremal Area Finder 81

6. The electronic density of states contribution for the band is calculated by com-

paring the reciprocal-unit-cell occupied states k-space volume at energy iso-levels

slightly above and below the Fermi energy to those at the actual Fermi energy

(section 3.2.7).

3.2.2 k-space super cell construction

The SKEAF algorithm operates in a large cubic k-space Super Cell (SC), which is aligned

with the magnetic ﬁeld direction and contains many tiled copies of the Reciprocal Unit

Cell (RUC). The ﬁrst task is to construct this cell (Fig. 3.2) and populate it with band

energies.

The reciprocal lattice vectors are deﬁned in the BXSF ﬁle relative to orthogonal

k-space axesx ˆRUC ,y ˆRUC ,z ˆRUC , such that, for example

~a =(ax,ay,az)= axxˆRUC + ayyˆRUC + azzˆRUC (3.1)

According to convention, these axes are typically chosen so that ~a lies alongx ˆRUC (that

is, ay = az = 0); for a cubic or tetragonal reciprocal unit cell, ~b usually lies alongy ˆRUC

and ~c usually lies alongz ˆRUC . Since the super cell is aligned with the magnetic ﬁeld (ˆz H~ ), it has a diﬀerent coordinate system, and a coordinate transformation is needed SC || to map the super-cell points back to the original reference frame:

xˆ v2u + t vwu ws xˆ SC − − RUC    2    yˆSC = vwu w u + t vs yˆRUC (3.2)    − −           zˆSC   ws vs t   zˆRUC              where s = sin φ, t = cos φ, u = 1 cos φ, v = sin θ, w = cos θ, φ is the polar angle of − the magnetic ﬁeld measured fromz ˆRUC down toward thex ˆRUC –ˆyRUC plane, and θ is the azimuthal angle of the magnetic ﬁeld measured fromx ˆRUC towardy ˆRUC . Both coordinate Chapter 3. Supercell K-space Extremal Area Finder 82

Figure 3.2: An example super cell, aligned with the magnetic ﬁeld H, and drawn to-scale with the small reciprocal unit cell contained within. The super cell is cut into slices perpendicular to the magnetic ﬁeld, which are populated with grid points. A typical super cell would contain 600 slices, each holding 600 600 grid points: far more than shown here. The axes for the reciprocal-unit-cell and× super-cell coordinate systems are inset. Chapter 3. Supercell K-space Extremal Area Finder 83 systems share the same origin.

No matter what the shape of the reciprocal unit cell, the super cell is always a cube with sides deﬁned to be 4 longer than the longest reciprocal lattice vector. This way, × for any magnetic ﬁeld orientation, the super cell will contain enough tiled reciprocal unit cells to be able to track orbits that cross the zone boundaries. The super cell is situated so that 1/4 of the side length lies along xˆ while 3/4 of the side length lies along − SC

+ˆxSC , and similarly for the other dimensions. The density of k-points in the super-cell grid is typically 100–200 greater that of the reciprocal-unit-cell grid. ×

The k-points in the super cell are transformed to the reciprocal-unit-cell reference frame via Eq. 3.2, and translated into the volume of the original reciprocal unit cell.

However, since the calculated band energies are positioned uniformly on a reciprocal- unit-cell k-point grid deﬁned by the reciprocal lattice vectors, and these in turn are not necessarily at right angles to one another, a further transformation is necessary to allow the super-cell points to be compared to those read from the input ﬁle. A point

~p = pxxˆRUC + pyyˆRUC + pzzˆRUC maps to a point ~q = qaaˆ + qbˆb + qccˆ in the input array as

ax bx cx −1   ~q = M ~p, where M = ay by cy (3.3)      az bz cz     

Once the super-cell k-points have been appropriately mapped back to the space of the

input array, their band energies may be derived from those originally calculated by the

electronic structure program. Since few re-mapped super-cell grid points will coincide

exactly with reciprocal-unit-cell grid points, a series of third-order Lagrange interpolating

polynomials [68] on a 4 4 4-point grid are used to determine each super-cell k-point × × band energy. Chapter 3. Supercell K-space Extremal Area Finder 84

3.2.3 Fermi surface orbit detection

Upon population with band energies, the super cell is cut into 1-k-point-thick slices perpendicular to the magnetic ﬁeld direction. In each super-cell slice, the program steps through the two-dimensional k-point array, locating all Fermi surface orbit outlines (details of this process are shown in Fig. 3.3). Fermi surface points and associated energy slopes are determined around each orbit. Orbits that run into the super cell boundaries are ignored—this limits open orbits (the super cell is large enough that copies of non-open orbits will be found elsewhere).

Note that during the “Record Fermi surface point between (x, y) and (xg, yg)” algorithm step (Fig. 3.3), the Fermi surface point is not simply placed halfway between the “stepped on” point, (x, y), and the “glanced at” point, (xg, yg), but rather linearly interpolated between the two so that its position on the slice is given by

To further clarify the core Fermi surface detection algorithm presented in Fig. 3.3, the action of this procedure on a small portion of an example slice is shown in Fig. 3.4.

The algorithm has already been stepping around the slice by the time it arrives at the depicted portion. The operations performed, in order, are:

1. Arrive at (2,3). E(2, 3) is not <= EF .

2. Step right to the next point, (3,3). E(3, 3) < EF .

3. Glance from (3,3) to the right at (4,3). E(4, 3) >EF , so Fermi surface point a and the energy slope are recorded between (3,3) and (4,3).

4. Glance around (3,3) clockwise to (3,2). E(3, 2) >EF , so Fermi surface point b and the energy slope are recorded between (3,3) and (3,2). Chapter 3. Supercell K-space Extremal Area Finder 85

Start at bottom

left k-point (x,y)

= (1,1)

Yes

Is current

Set unchecked(x,y)

point

to FALSE

unchecked?

Glance at point

around current point,

Is E(x,y) <= clockwise from point

Step right to next

E ? (inside FS) we just glanced at

point (x+1, y)

Yes

Record FS point

Is FS point = to

between (x,y) and

1st FS point on

(x ,y ); record No

g g Are we at the current orbit? Glance to the right

energy slope

end of a row?

(x ,y ) = (x+1,y)

g g

Yes

Set

Is E(x ,y ) > E ?

g g F unchecked(x ,y ) to Go to start of next

g g

(outside FS)

row (1,y+1) FALSE

Step to this glanced

Are we at the end No

point (x,y) = (x ,y )

g g of the slice (top

right k-point)?

Yes

Is current Done

point on SC current

End border? orbit

Go back to first grid

point that was Glance at point

detected to be around current point,

inside current orbit clockwise from point

we just stepped from

Figure 3.3: Flow chart showing the Fermi surface orbit detection algorithm for ﬁnding Fermi surface outlines (orbits) within a given slice. Coordinates (x, y) refer to grid points in the two-dimensional array of k-points on the slice. Chapter 3. Supercell K-space Extremal Area Finder 86

Figure 3.4: Small portion of an example slice. “+” grid points have energies greater than the Fermi energy; “ ” grid points have energies less then the Fermi energy. Typical slices hold 600 600 grid− points, so orbits usually contain many more points than the trivial example shown× here.

5. Glance around (3,3) clockwise to (2,3). E(2, 3) >EF , so Fermi surface point c and the energy slope are recorded between (3,3) and (2,3).

6. Glance around (3,3) clockwise to (3,4). E(3, 4) is not > EF , so step from (3,3) to (3,4).

7. Glance around (3,4) clockwise from (3,3) to (2,4). E(2, 4) > EF , so Fermi surface point d and the energy slope are recorded between (3,4) and (2,4).

8. Glance around (3,4) clockwise to (3,5). E(3, 5) >EF , so Fermi surface point e and the energy slope are recorded between (3,4) and (3,5).

9. Glance around (3,4) clockwise to (4,4). E(4, 4) >EF , so Fermi surface point f and the energy slope are recorded between (3,4) and (4,4).

10. Glance around (3,4) clockwise to (3,3). E(3, 3) is not > EF , so step from (3,4) to (3,3). Chapter 3. Supercell K-space Extremal Area Finder 87

11. Glance around (3,3) clockwise from (3,4) to (4,3). E(4, 3) >EF , so a Fermi surface point and the energy slope are recorded between (3,3) and (4,3). This Fermi surface

point is the same as the ﬁrst one found on the orbit (Fermi surface point a, in step

3): this orbit is done!

3.2.4 dHvA frequency, eﬀective mass, and orbit type calcula-

tions

For each orbit (i.e. Fermi surface outline) found in a particular slice, the dHvA frequency,

eﬀective mass and orbit type are calculated. Since the frequency is proportional to the

cross-sectional Fermi surface area (Eq. 1.6), it is obtained from the area of the polygon

formed by the Fermi surface points (e.g. points a–f in Fig. 3.4):

~ 1 N−1 F = (xF S,i yF S,i+1 xF S,i+1 yF S,i) (3.5) 2πe 2 − i=1 X where the sum is over the points on the orbit, with the last point on the orbit the same as the ﬁrst: (x , y ) (x , y ). Note that although the “frequencies” are F S,N F S,N ≡ F S,1 F S,1 calculated for every orbit, cross-sectional areas only produce experimentally-measurable dHvA frequencies when they are extremal; the extremal orbits are singled out later

(section 3.2.6).

The eﬀective mass calculation draws upon the energy slopes at the Fermi surface. As one averages around a Fermi surface outline, two diﬀerent geometric cases are encountered. If the glance directions for point i and point i + 1 are parallel (e.g. points c and d in Fig. 3.4), then dE dE 2 dE 2 = + (3.6) dk dk dk i s || i ⊥ i

dE where dk is the energy slope parallel to the glance direction at point i (i.e. across the || i dE Fermi surface), and dk is the energy slope perpendicular to the glance direction at ⊥ i Chapter 3. Supercell K-space Extremal Area Finder 88 point i. If the glance directions for point i and point i + 1 are perpendicular (e.g. points

a and b in Fig. 3.4), then

dE dE 2 dE 2 = + (3.7) dk dk dk i s || i || i+1

The eﬀective mass averaged around the orbit, given in units of the electron mass me, is

N−1 2 2 ~2 dA ~2 (xF S,i+1 xF S,i) +(yF S,i+1 yF S,i) m∗ = = − − (3.8) 2πm dE 2πm q dE e e i=1 dk i X

Note that Eq. 3.8 is only correct for extremal orbits, when the energy gradient lies within

the slice.

Since the orbit-ﬁnding algorithm (section 3.2.3) always steps around the inside of

electron orbits (e.g. Fig. 3.4) and the outside of hole orbits, the orbit type is determined by

comparing the orbit area to the area bounded by the polygon formed by the “stepped-on”

grid points. If the orbit area is larger than the stepped-on area, it is ﬂagged as an electron

orbit; if the reverse is true, it is ﬂagged as a hole orbit. At this point, average coordinates,

coordinate standard deviations, maximum coordinates, and minimum coordinates for the

orbit outline are also calculated.

3.2.5 Slice-to-slice orbit matching

Once the orbit outlines have been located on all slices in the super cell, Fermi surface

sheets must be rebuilt by matching orbits on adjacent slices. In order for an orbit i on one slice to be matched with an orbit j on the preceding slice, and therefore added to orbit j’s sheet, all of the following conditions must be met:

The average x and y coordinates of orbit i are both within one standard • F S F S

deviation of the average xF S and yF S coordinates of orbit j. Chapter 3. Supercell K-space Extremal Area Finder 89

The maximum x and y coordinates of orbit i are both within two standard • F S F S

deviations of the maximum xF S and yF S coordinates of orbit j.

The minimum x and y coordinates of orbit i are both within two standard • F S F S

deviations of the minimum xF S and yF S coordinates of orbit j.

If multiple orbits on one slice satisfy the conditions for matching with an orbit on the

preceding slice, the parameter Bi is calculated for each candidate orbit, and the orbit with the lowest Bi value is chosen for the match:

2 2 B = avg (x ) avg (x ) + avg (y ) avg (y ) i F S j − F S i F S j − F S i 2 2 h+ max(x ) max(xi ) h + max(y ) max(iy ) (3.9) F S j − F S i F S j − F S i 2 2 + hmin(x ) min(x ) i + hmin(y ) min(y ) i F S j − F S i F S j − F S i h i h i 3.2.6 Extremum determination

If an orbit has a frequency (cross-sectional area) which is greater than both of the adjacent

orbits on the same sheet, or less than both of the adjacent orbits, it is taken to be

extremal. Once all extremal orbits in the super cell have been selected, they are sorted

by frequency, and those which have frequencies within 1% of the next smallest orbit are

taken to be multiple copies of the same orbit. Since the super cell contains more than

one reciprocal unit cell, it is expected that multiple copies of a given extremal orbit will

be found. The copies have their frequencies, eﬀective masses and orbit types (electron =

1, hole = -1; useful for conﬁrming that orbits of diﬀerent type have not been put on the

same sheet) averaged, with standard deviations used to provide an indication of error

bounds on these quantities.

3.2.7 Density of states calculation

Calculation of the band contribution to the electronic density of states proceeds sepa-

rately from the super cell calculations, and is only performed once for a given input ﬁle Chapter 3. Supercell K-space Extremal Area Finder 90

(compared to the many repeated super-cell iterations for diﬀerent magnetic ﬁeld directions). To determine the density of states contribution, the following steps are executed:

1. The band energies within the reciprocal unit cell are interpolated very ﬁnely. No

super cell is used.

2. The k-space volume of the reciprocal unit cell and volume per interpolated k-point

are calculated.

3. The number of occupied points (those with E~k < EF ) is counted, and the total

occupied volume, VF , calculated.

4. The number of points within an iso-surface at E+ = EF + dE, with typical dE =

0.00001 EF , is counted, and the total volume of such points, V+, calculated.

5. The number of points within an iso-surface at E = E dE is counted, and the − F −

total volume of such points, V−, calculated.

The electronic density of states (DOS) contributions are then

V V DOS = | F − ±| (3.10) ± dE

with an indication of error bounds given by the diﬀerence between + and values. −

3.3 Test results

In order to validate my algorithm, two artificial cases with exactly known results were constructed and used as program inputs. First, a spherical Fermi surface was generated, with band energies that depend on the square of the k-space distance from the Fermi surface centre. The input file contained band energies specified on a 28 28 28 grid in a × × cubic reciprocal unit cell, picked such that the effective mass m∗ 10.543 m and dHvA ≡ e Chapter 3. Supercell K-space Extremal Area Finder 91

18 2.4

16 cylinder m* = 3.457 m 2.2 input e cylinder m* = 3.460 m 14 extracted e

2.0

dHvAfrequency (kT) 1.8 10 sphere m* = 10.543 m input e sphere m* = 10.539 m 8 1.6 extracted e

6 0 1 2 3 4 5 6 7 8 910 φ dHvA frequency(kT) Magnetic field angle (degrees) 4

0 0 10 20 30 40 50 60 70 80 90 Magnetic field angle φ (degrees)

Figure 3.5: de Haas–van Alphen frequency plotted as a function of magnetic ﬁeld angle φ for spherical (bottom) and cylindrical (top) test Fermi surfaces. The cylinder axis lies along the φ = 0 direction. Data extracted by the SKEAF algorithm appear as points; exact results are shown as solid lines. The inset shows a close-up of the low-angle region, as well as the exact input eﬀective masses and corresponding band masses extracted by my algorithm for both cases.

frequency F 1.568 kT, constant for all magnetic field angles. Second, a cylindrical ≡ Fermi surface was generated, with band energies that depend on the square of the k- space distance from thez ˆRUC axis. The input file energies were specified on the same grid and reciprocal unit cell as the spherical case, and picked such that the effective mass

m∗ 3.457 m and dHvA frequency F 2.346 kT when the magnetic ﬁeld is aligned ≡ e ≡

along thez ˆRUC axis. As the ﬁeld is rotated away from thez ˆRUC axis by an angle φ, the frequency should increase as 1/ cos φ, asymptoting to an open orbit as φ 90 degrees. → Fig. 3.5 shows the results my program obtained from these two test cases. In both

cases, agreement with the input values is excellent, particularly in light of the coarseness

of the starting band energy grid. For the spherical case, SKEAF extracted a frequency Chapter 3. Supercell K-space Extremal Area Finder 92

∗ of F = 1.566 kT and mass of m = 10.539 me, varying by less than 0.002 kT and

0.005 me respectively over the angular range tested. For the cylindrical case, with ﬁeld directed along thez ˆRUC axis, SKEAF extracted a frequency of F = 2.345 kT and mass of

∗ m = 3.460 me. As the ﬁeld is rotated away from thez ˆRUC axis, the extracted frequency follows the predicted 1/ cos φ curve very closely.

3.4 CeCoIn5 results

The ﬁrst application of the SKEAF algorithm to a real material is CeCoIn5. CeCoIn5 is the heavy-fermion superconductor with the highest transition temperature, Tc = 2.3 K [69], and can be tuned to a ﬁeld-induced quantum critical point [4, 5]. It crys-

tallizes in the tetragonal HoCoGa5 crystal structure (space group # 123: P 4/mmm), shown in Fig. 3.6, with lattice constants a = b = 4.612 A˚ and c = 7.549 A,˚ and frac-

tional atomic coordinates Ce (0, 0, 0), Co (0, 0, 0.5), In1c (0.5, 0.5, 0), and In4i (0, 0.5,

0.305) [70]. In some respects, CeCoIn5 can be thought of as a quasi-2D version of CeIn3,

since it is composed of alternating 2D layers of CeIn3 and CoIn2, stacked along the c- axis. This compound hosts a wide variety of interesting physics, reviews of which can

be found elsewhere [47, 71]. I have a long personal history with CeCoIn5, going back to my undergraduate thesis project with Professor Louis Taillefer developing a technique to

enable ultrasound attenuation measurements on it [72]. During my M.Sc., under the su-

pervision of Professor John Wei, I performed dilution refrigerator point contact Andreev

reﬂection spectroscopy measurements and discovered two co-existing order parameters in

the CeCoIn5 superconducting state [71, 73, 74]. Multi-band superconductivity has since been conﬁrmed in this material by other measurements [75, 76].

CeCoIn5 electronic structure calculations predict 3 bands crossing the Fermi energy, α, β, and γ; the Fermi surface sheets corresponding to these bands are shown in Fig. 3.7.

This calculated Fermi surface topology was conﬁrmed experimentally by Settai et al. Chapter 3. Supercell K-space Extremal Area Finder 93

Figure 3.6: CeCoIn5 crystal structure.

via measurements of the de Haas–van Alphen eﬀect as a function of magnetic ﬁeld an-

gle [70]. The dHvA frequencies they measured for extremal orbits on the α and β sheets,

Fexpt,Settai, and their corresponding theoretical predictions, Fcalc,Settai, are listed in Ta- ble 3.1 for ﬁelds aligned along the crystallographic c-axis (no γ-sheet orbits were detected at this ﬁeld angle). The temperature dependence of the oscillation amplitudes was measured down to 30 mK, found to be well-described by the standard Lifshitz-Kosevich form

∗ (Eq. 1.11), and used to obtain the eﬀective masses, mSettai, listed in Table 3.2. The enhancement factor of these experimentally-determined eﬀective masses relative to the

∗ calculated band masses, mSettai/mb,Settai, is also listed in Table 3.2.

During her doctoral studies at the University of Cambridge under Prof. Julian, prior to the move to the University of Toronto, Dr. McCollam (our current post-doc) performed de Haas–van Alphen measurements of CeCoIn5 down to 6 mK [12, 47]—much lower in temperature than Settai et al. While her measured dHvA frequencies in a 13–15 T Chapter 3. Supercell K-space Extremal Area Finder 94

Figure 3.7: The Fermi surface of CeCoIn5, calculated using WIEN2k [30] and visualized via XCrysDen [66]. Three bands cross the Fermi energy: α, β, and γ. The dark blue side of each sheet is the electron-occupied (i.e. hole-unoccupied) side, whereas the light yellow side is the electron-unoccupied (i.e. hole-occupied) side.

field range (Fexpt,McCollam in Table 3.1) [77] are consistent with those from the previous study, she found striking deviations from the standard Lifshitz-Kosevich form in the temperature dependence of the oscillation amplitudes below 30 mK. The full oscillation temperature dependence, including the low-temperature deviations, can be reproduced by fitting to a modified Lifshitz-Kosevich equation, in which the effective mass for a given orbit differs on the majority-spin and minority-spin Zeeman-split branches of the Fermi

∗ surface [12, 47]. The resulting spin-dependent eﬀective masses, mMcCollam, are listed in Table 3.2 for both spin-directions ( and , although these are arbitrary labels). ↑ ↓

This discovery of spin-dependent eﬀective masses in CeCoIn5 was published in Phys- ical Review Letters [12]; my contribution was to calculate the electronic structure of

CeCoIn5, extract dHvA frequencies, band masses and density of states contributions using SKEAF, and then estimate the enhanced electronic speciﬁc heat for comparison to

CeCoIn5 experimental speciﬁc heat measurements. Using the WIEN2k software pack- age [30] and the CeCoIn5 crystal structure shown in Fig. 3.6, I obtained band energies on a 20 000 k-point Reciprocal Unit Cell (RUC) mesh (section 3.2.1). The calculations included spin-orbit coupling, the Perdew-Burke-Ernzerhof generalized gradient approx-

ˆ min imation to the exchange-correlation potential Vxc [29], RMT Kmax = 9, and a “valence Chapter 3. Supercell K-space Extremal Area Finder 95

Table 3.1: Measured and calculated CeCoIn5 dHvA frequencies. Fcalc,Settai and Fexpt,Settai are frequencies predicted and detected, respectively, by Settai et al.[70]; Fexpt,McCollam come from the doctoral measurements of Dr. McCollam [77]; and Fcalc,SKEAF are frequencies extracted by the SKEAF algorithm from my WIEN2k band structure calculations. Error bounds in Fcalc,SKEAF are determined using the process described in section 3.2.6, with an error of “0” listed when no multiple copies of a given orbit with slightly diﬀerent frequencies were found in the supercell.

Orbit Fcalc,Settai Fcalc,SKEAF Fexpt,Settai (15–17 T) Fexpt,McCollam (13–15 T) (kT) [70] (kT) (kT) [70] (kT) [77] α1 5.43 5.542(4) 5.56 5.56(2) α2 4.53 4.6064(0) 4.53 4.93(2) α3 3.90 4.0037(0) 4.24 4.47(1) βc 13.3 13.5075(3) — — β1 13.0 13.021(2) 12.0 12.30(1) β2 6.45 6.546(3) 7.5 7.53(2)

state” energy range of 7 to 1.5 Ry. The Fermi surface sheets derived from these calcu- − lated band energies are shown in Fig. 3.7, and resemble those previously published [70].

During the application of the SKEAF algorithm to the calculated band energies, as in Dr. McCollam’s experiment, only magnetic ﬁelds aligned along the crystalline c-axis were considered. Extracted frequencies are listed as Fcalc,SKEAF in Table 3.1 and band

masses as mb,SKEAF in Table 3.2; there is reasonable agreement between my values and those calculated by Settai et al., considering diﬀerences in the band structure calculation

method. Electronic density of states contributions, DOSavg = (DOS+ + DOS−)/2, from each of my calculated Fermi-energy-crossing bands were determined by the method described in Section 3.2.7, and are listed in Table 3.3. Table 3.3 also shows the band

−3 −1 speciﬁc heat contributions, γb, obtained by converting DOSavg from units of A˚ Ry to mJ K−2 per mole of cerium atoms.

Following standard practise in the ﬁeld of heavy fermion dHvA measurements [78],

∗ the “mass enhancement” (m /mb in Table 3.2) of each measured orbit relative to its calculated counterpart was determined, and then averaged over each band (αavg and β in Table 3.2). The average enhancements of the McCollam/SKEAF -spin α and β avg ↑ Chapter 3. Supercell K-space Extremal Area Finder 96

∗ Table 3.2: Calculated band masses mb, measured effective masses m , and resulting ∗ ∗ mass enhancements m /mb in CeCoIn5. mb,Settai and mSettai are masses calculated and ∗ measured, respectively, by Settai et al. [70]; mMcCollam are the spin-dependent effective masses measured by Dr. McCollam [47]; and mb,SKEAF are the band masses extracted by the SKEAF algorithm from my WIEN2k band structure calculations. Error bounds in mb,SKEAF are determined using the process described in section 3.2.6, with an error of “0” listed when no multiple copies of a given orbit with slightly different frequencies were found in the supercell.

∗ ∗ ∗ ∗ Orbit mb,Settai mb,SKEAF mSettai mMcCollam mSettai mMcCollam m m (me) [70] (me) (me) [70] (me) [47] b,Settai b,SKEAF α1 1.78 1.64(1) 15 : 13(1) 8.4 : 7.9(6) ↑: 18.4(8) ↑: 11.2(5) ↓ ↓ α2 1.09 1.0231(0) 18 : 12.5(4) 17 : 12.2(4) ↑ : 28(2) ↑ : 27(2) ↓ ↓ α3 1.48 1.3247(0) 8.4 : 8.1(1) 5.7 : 6.11(8) ↑ : 21(1) ↑: 15.9(8) ↓ ↓ βc 4.37 3.9908(0) — — — — β1 3.48 3.200(1) 48 : 36.9(5) 14 : 11.5(2) ↑: 180(20) ↑ : 56(6) β 1.73 1.39(2) 49 ↓ : 47(1) 28 ↓: 33.8(9) 2 ↑ ↑ : 160(30) : 120(20) ↓ ↓ αavg 10(3) : 9(2) ↑: 18(5) ↓ βavg 21(7) : 20(10) ↑: 90(30) ↓ Chapter 3. Supercell K-space Extremal Area Finder 97

Table 3.3: CeCoIn5 density of states contributions (DOS+, DOS−, and DOSavg) and associated band specific heats (γb) extracted by SKEAF; enhanced specific heat for each band (γ∗ and γ∗) using the band-averaged - and -spin McCollam/SKEAF mass en- ↑ ↓ ↑ ↓ hancements from Table 3.2; and combined renormalized specific heat estimates without ∗ ∗ † (γ↑+↑) and with (γ↑+↓) spin-dependent effective masses. Since there are two spin directions (even when they both have the same mass), each band contributes twice to the total specific heat. ‡ / mass enhancements for the undetected γ-surface are assumed to be averages of / α↑- and↓ β-surface enhancements. ↑ ↓

∗ ∗ ∗ ∗ Band DOS+ DOS− DOSavg γb γ↑ γ↓ γ↑+↑ γ↑+↓ (A˚−3 Ry−1) (mJmol−1 K−2) α 37.7 38.3 38.0(3) 4.26(3) 38(9) 80(20) 80(20) 120(20) β 62.1 61.1 61.6(5) 6.91(6) 140(70) 600(200) 300(100) 700(200) γ 4.7 5.2 5.0(3) 0.56(3) 8(3)‡ 30(20)† 16(6) 40(20) Total 23.5(1)† 400(100) 900(200)

surfaces are equal to the non-spin-split average mass enhancements of Settai et al. within

error bounds, but the -spin enhancements are much larger, particularly on the β-sheet. ↓

By multiplying the calculated band specific heat, γb, by the average mass enhancement for each band, and then adding up the contributions from all bands, it is possible to estimate the renormalized specific heat of a material. Such estimates can be compared to experimental specific heat measurements, and can serve as a guide to whether there are any Fermi surface pockets that contribute to the total electronic specific heat but remain undetected in the dHvA experiment. Renormalized specific heat estimates for CeCoIn5 (γ∗ in Table 3.3) were calculated using the - and -spin average mass enhancements ↑ ↓ ∗ (mMcCollam/mb,SKEAF for αavg and βavg in Table 3.2) and band specific heat contributions extracted by SKEAF (γb in Table 3.3). The total specific heat for the case of Settai et al., in which spin-dependent effective masses were not found, can be approximated by assuming that both Fermi surface spin-branches of each band have the same mass enhancement, taken to be the -spin ↑ McCollam/SKEAF enhancements (which are equal within error bounds to those of Set- tai et al., but are derived from my band structure calculations, which is important for Chapter 3. Supercell K-space Extremal Area Finder 98

internal self-consistency with γb). The resulting renormalized speciﬁc heat estimate is

∗ −1 −2 γ↑+↑ = 400(100) mJ mol K , which agrees with the experimentally measured value of γ 430 mJ mol−1 K−2 at H = 13 T [79]. On the other hand, the renormalized speciﬁc expt ∼ heat estimate for the case where the eﬀective masses (and therefore mass enhancements)

∗ −1 −2 are taken to be spin-dependent, γ↑+↓ = 900(200) mJ mol K , is much larger than γ . An overwhelming contribution to γ∗ is made by the huge -spin β-sheet effec- expt ↑+↓ ↓ tive masses, but it is possible that the deviations from Lifshitz-Kosevich temperature dependence in the β orbits are not due to spin-dependent effective masses, but rather non-Fermi-liquid effects that affect all orbits at lower fields [12, 47]. In this light, a

“hybrid” speciﬁc heat estimate that includes spin-dependence of only the α masses (for

∗ −1 −2 which the evidence for this phenomenon is strongest), γhybrid = 400(100) mJ mol K ,

∗ is more appropriate and in is agreement with both γ↑+↑ and γexpt. Chapter 4

UPt3

One of the most signiﬁcant results of my SKEAF work has been the application of the algorithm to UPt3 band structure calculations by Mike Norman, and comparison to quantum oscillation measurements on this material performed by my supervisor, Stephen

Julian, when he was at the University of Cambridge. My algorithm was able to ﬁnd new extremal orbits within the old band structure calculation, that had not been previously noticed. These new orbits agree with measured quantum oscillation frequencies, bringing the theoretical results into closer agreement with experiment, and helping to resolve a recent controversy over the fundamental itinerant nature of the UPt3 5f conduction electrons. This work has been published in the New Journal of Physics (copyright 2008 by the Institute of Physics and Deutsche Physikalische Gesellschaft e.V.) [13], and forms the basis of this chapter.

4.1 Material background

UPt3 is a heavy fermion superconductor with a nearly hexagonal crystal structure. For the electronic structure calculations discussed in this chapter, a hexagonal close packed

SnNi3-type structure (space group #194: P 63/mmc), shown in Fig. 4.1, was used. Lattice constants are a = b = 5.76390 A˚ and c = 4.90270 A,˚ and fractional atomic coordinates

99 Chapter 4. UPt3 100

are U (1/3, 2/3, zU ) and Pt (xPt, 2xPt, zPt), with xPt, zPt, and zU set to their ideal values of 5/6, 1/4, and 1/4, respectively [80, 81]. Recent high-energy x-ray diﬀraction

and transmission electron microscopy measurements on this compound have revealed

a slight trigonal distortion in the crystal structure, leading to a modiﬁed space group

(#164: P 3¯m1) and atomic coordinates xPt = 0.83692(2), zPt = 0.25210(6), and zU = 0.2507(1) [82], but this distortion should not have an appreciable eﬀect on the band structure calculations and was not included.

Historically, UPt3 is a very important compound in the study of strongly-correlated electron systems, as it is the archetype of both a multi-component superconductor and

heavy fermion material (section 1.3). Three diﬀerent superconducting phases exist be-

low T 0.5 K, with unconventional order parameters including nodes in the super- c ≃ conducting gap [83, 84]. The normal state behaves as a Landau Fermi liquid (section 1.1) with very heavy quasiparticles, as indicated by the large linear coefficient of specific heat, γ 420 mJ mol−1 K−2 [6, 85], large T 2 coefficient of the resistivity, ∼ A 0.49 µΩcm K−2 [86], enhanced magnetic susceptibility, χ 50 10−9 m3 mol−1 ∼ ∼ × (H c) and χ 100 10−9 m3 mol−1 (H c) [6], and signatures in optical reflectivity [87] || ∼ × ⊥ and inelastic neutron scattering [88].

The most direct conﬁrmation of a heavy Fermi liquid ground state in UPt3 comes from the de Haas–van Alphen (dHvA) eﬀect measurements of Louis Taillefer and Gil

Lonzarich, in which ﬁve separate extremal Fermi surface orbits were observed, with

effective masses as high as 100 me [78, 89]. All of the experimentally observed orbits match predictions made using the relativistic band structure calculations of Mike Norman and coworkers [81] for the calculated Fermi surface sheets shown in Fig. 4.2. Pioneering the technique described in section 3.4 for estimating the renormalized electronic specific heat coefficient using dHvA effective masses, Taillefer and Lonzarich found agreement with low temperature specific heat measurements [78]. Although Taillefer and Lonzarich observed dHvA frequencies predominantly when the field was near the a-axis, and did Chapter 4. UPt3 101

Figure 4.1: UPt3 crystal structure: (a) two formula units shown within the primitive unit cell, viewed perpendicular to the c-axis; (b) six formula units shown within the extended hexagonal unit cell, viewed along the c-axis. Chapter 4. UPt3 102

not detect some of the predicted Fermi surface sheets [78], Noriaki Kimura and coworkers

were later able to measure quantum oscillations across all three major rotation directions,

including along the c-axis [90, 91, 92]. These early measurements and calculations of the

UPt3 electronic structure underpin much of the modern framework for understanding heavy fermion systems.

A new theoretical development by Gertrud Zwicknagl, Alexander Yaresko and Peter

Fulde, however, has created a controversy over the correct way to interpret the experi-

mental dHvA results. In contrast to the original “fully itinerant” band structure view of

UPt3, in which all of the U 5f electrons are considered to be itinerant, Zwicknagl et al. propose a “partially localized” model in which two of the three U 5f electrons are local-

ized and therefore do not contribute to the Fermi volume [93]. A driving force behind

the adoption of this alternate proposal is that it has a built-in mechanism to explain the

observed heavy eﬀective masses via the exchange coupling between the two populations

of itinerant and localized U 5f electrons, although there are other possible mass enhance-

ment mechanisms, such as a Kondo lattice [94, 95, 96] or conduction electron coupling to

collective electronic spin-ﬂuctuations [97, 98, 99, 100], that are known to apply to other

materials and do not require partial f localization. With this in mind, the best way

to evaluate the partially localized and fully itinerant models is to rigorously determine

which one most accurately describes the actual Fermi surface of UPt3, as experimentally measured via quantum oscillation experiments. This is what my collaborators and I have

done, by re-examining the predictions of the fully itinerant model using the SKEAF al-

gorithm (chapter 3) and comparing the predictions of both models to a comprehensive

experimental data set for this material.

It is important that we conclusively resolve the controversy regarding the Fermi sur-

face of UPt3 for several reasons. First, since the exotic, multi-component superconductivity of this material arises against the backdrop of the normal state electronic structure,

its properties must be modelled using the correct Fermi surface [101]. Second, the lo- Chapter 4. UPt3 103

Figure 4.2: Fermi surface sheets of UPt3 generated from the fully itinerant model, which assumes that all three of the U 5f electrons are included in the Fermi volume. Bands 1 and 2 give rise to hole-like surfaces, whereas bands 3, 4 and 5 give rise to electron-like surfaces. Coloured lines denote some of the extremal orbits on each sheet, with the two green lines on the band 2 surface representing open orbits. The colour of each surface (cyan for band 1, brown and orange for band 2, magenta and pink for band 3, green for band 4, and red for band 5) is maintained throughout the other ﬁgures in this chapter, such that dHvA frequencies are given the colour of their associated Fermi surface sheet. The a-axis corresponds to the Γ–K direction, the b-axis to the Γ–M direction, and the c-axis to the Γ–A direction. Chapter 4. UPt3 104

calized vs. itinerant nature of f-states is a subject of heated debate in regard to many strongly correlated electron systems of interest to the condensed matter community, such as YbRh2Si2 (chapter 6). Third, from an electronic structure calculation point of view,

UPt3 pushes the limits of current techniques and can be seen as a stepping stone from magnetic d-electron metals, for which density functional theory (section 1.4) works well, to cases such as NaxCoO2, where it seems to breaks down [102].

4.2 Theoretical models of the Fermi surface

In the fully itinerant model, ﬁve bands cross the Fermi energy, giving rise to the ﬁve- sheeted Fermi surface shown in Fig. 4.2. This is a so-called “large” Fermi surface, because all valence f-states are itinerant and therefore contribute to the total Fermi volume.

Fig. 4.2(a) shows the ﬂattened, star-shaped hole-like surface of band 1, centred on the A

Brillouin zone symmetry point; Fig. 4.2(b) shows the multiply-connected hole-like surface of band 2, centred on A; Fig. 4.2(c) shows the large electron-like surface, centred on Γ, and smaller electron-like pockets, centred on K, of band 3; Fig. 4.2(d) shows the small electron-like surface of band 4, centred on Γ; and Fig. 4.2(e) shows the small electron-like surface of band 5, which is also centred on Γ. The general features of these Fermi surface sheets agree with previous fully itinerant calculations [103, 104, 105].

Since Fermi surface studies of UPt3 span two decades and multiple groups of authors, several diﬀerent band labelling schemes (Table 4.1) have arisen. For clarity, when discussing the theoretical models in this chapter I will stick to the notation of Norman et al. [81], in which the ﬁve Fermi surface sheets shown in Fig. 4.2(a)–(e) are respectively labelled as bands 1–5. The predicted extremal orbits on each sheet, which I extracted using the SKEAF algorithm, are denoted by combining the Brillouin zone symmetry point around which the orbit is centred with the given sheet’s band number. For example, in this scheme the orbit shown as a red line in Fig. 4.2(a) is named A-1. For a “non-central” Chapter 4. UPt3 105

Table 4.1: Comparison of UPt3 band labelling schemes. The “band number” notation of Norman et al. [81] is used in this chapter. The last column indicates the colour used in Figs. 4.4, 4.5, 4.6, 4.7, and 4.8 for dHvA frequencies associated with each Fermi surface sheet.

Band number Joynt & Taillefer & Kimura Zwicknagl Colour [81] Taillefer [83] Lonzarich [78] et al. [92] et al. [93] Band 1 ‘Starﬁsh’ Band 5 Band 35 Band 1 (Z1) Cyan Band 2 ‘Octopus’ Band 4 Band 36 — Brown & Orange Band 3 ‘Oyster & Band 3 Band 37 Band 2 (Z3) Magenta urchins’ & Pink Band 4 ‘Mussel’ Band 2 Band 38 — Green Band 5 ‘Pearl’ Band 1 Band 39 Band 2 (Z5) Red

orbit, whose centre does not lie on a major symmetry point (such as KH-2, the white

line in Fig. 4.2(b)), two or three symmetry points are listed, deﬁning the line or plane

on which the orbit centre lies. When discussing the experimental measurements, dHvA frequencies are labelled with lowercase Greek letters, following the general consensus in the literature [78, 90, 91, 92].

In the partially localized model, three bands cross the Fermi energy, giving rise to the Fermi surface shown in Fig. 4.3. In the context of the wider debate between so- called “large” and “small” Fermi surfaces in f-electron/hole materials (chapter 6), this radical proposal would be called a “medium” Fermi surface case, because some of the valence f-states are localized and therefore do not contribute to the total Fermi volume, while others are itinerant and do contribute to the Fermi volume. Fig. 4.3(a) shows the partially localized Z1 surface, which resembles the fully itinerant band 1 surface of

Fig. 4.2(a), except now the arms of the star reach the zone boundaries, allowing open orbits for H a in a manner similar to the fully itinerant band 2 surface of Fig. 4.2(b). || Fig. 4.3(b) shows the partially localized Z3 surface, which somewhat resembles the fully itinerant band 3 surface of Fig. 4.2(c). The partially localized Z5 surface, which resembles the fully itinerant band 5 surface of Fig. 4.2(e), is not shown in Fig. 4.3. Chapter 4. UPt3 106

Figure 4.3: An artist’s impression of the major Fermi surface sheets in the partially localized model [93]. The Z1 surface resembles the fully itinerant band 1 surface shown in Fig. 4.2(a), but has open orbits similar to those of the fully itinerant band 2 surface shown in Fig. 4.2(b). The Z3 surface is somewhat similar to the fully itinerant band 3 surface shown in Fig. 4.2(c). The Z5 surface, not shown, resembles the fully itinerant band 5 surface of Fig. 4.2(e).

Despite some vague resemblances, the full Fermi surface predicted by the partially lo-

calized model (Fig. 4.3) is quite diﬀerent from that of the fully itinerant model (Fig. 4.2).

The main goal of our study was to determine which of these two models best describes

the actual, experimentally-measured Fermi surface of UPt3. To facilitate comparison with experimental quantum oscillation data, the ﬁeld-angle dependences of the dHvA frequencies predicted by the fully itinerant and partially localized models are shown in

Figs. 4.4 and 4.5, respectively. Fig. 4.4 was generated by applying the SKEAF algorithm

(chapter 3) to the band energies of Norman et al. (i.e. the Fermi surface of Fig. 4.2) [81]; the individual extremal orbits labelled in Fig. 4.4 are described in Table 4.2. Crucially, in addition to conﬁrming the previously-known predicted dHvA frequencies of the fully itinerant model, my algorithm found new extremal orbits in the old fully itinerant electronic

structure calculation (solid lines in Fig. 4.4). These new orbits, discussed in section 4.3,

had not been previously noticed, because they are topologically complicated and often Chapter 4. UPt3 107

Table 4.2: Description of the predicted UPt3 orbits, “Calc. orbit,” extracted by the SKEAF algorithm from the fully itinerant model (Fig. 4.4), ordered roughly from lowest to highest frequency. Experimentally-detected orbits that we believe correspond to these predictions are listed in the “Expt. orbit” column, and are labelled in Figs. 4.6, 4.7, and 4.8.

Calc. Description Expt. orbit orbit L-2 Lowest frequency electron orbit (magenta line in Fig. 4.2(b)) Possibly γ ML-2 Circles one arm on band 2 (cyan line in Fig. 4.2(b)) α Γ-5 Central orbit around band 5 sphere (yellow line in Fig. 4.2(e)) γ′ AHK-2 Circles 2 arms (yellow line in Fig. 4.2(b)) η′ ′ AH-2 Circles 4 arms (purple line in Fig. 4.2(b)) α4 and α4 Γ-4 Electron orbit (red line in Fig. 4.2(d)) ǫ KH-2 Electron orbit at c-axis (white line in Fig. 4.2(b)) ζ K-3 Electron orbit (yellow line Fig. 4.2(c)) κ ◦ KH-2 Near 70 in c–a plane (not shown) α3 A-1 Hole orbit (red line in Fig. 4.2(a)) δ ΓA-3 Non-central electron orbit (cyan line in Fig. 4.2(c)) σ ALM-2 Circles 3 upper arms and one lower (not shown) α3 A-2 At b-axis (red line in Fig. 4.2(b)) λ A-2 At c-axis (blue line in Fig. 4.2(b)) λ′ Γ-3 Electron orbit (blue line in Fig. 4.2(c)) ω L-2 a–b-plane high frequency (red line in Fig. 4.9) η? L-2 a–b-plane higher frequency (blue line in Fig. 4.9) η?

non-central, making them diﬃcult to ﬁnd by visual inspection of the Fermi surface.

4.3 Discussion

In order to determine which of the two theoretical models best captures the physics of

UPt3, their predictions are compared in detail to a comprehensive experimental dataset that includes the dHvA results of Taillefer and Lonzarich [78], non-oscillatory magnetoresistance data of Louis Taillefer, Jacques Flouquet and Walter Joss [106], and new non-oscillatory magnetoresistance, Shubnikov–de Haas (SdH) and dHvA measurements performed by my supervisor, Stephen Julian, at the University of Cambridge prior to his move to the University of Toronto. Chapter 4. UPt3 108

Figure 4.4: Predicted angle dependence of UPt3 dHvA frequencies in the fully itinerant model, extracted by the SKEAF algorithm from the band structure calculations of Norman et al. [81]. Frequencies noticed for the ﬁrst time by SKEAF are shown as solid (orange for band 2, pink for band 3) lines. The labelled extremal orbits are described in Table 4.2. Chapter 4. UPt3 109

Figure 4.5: Predicted angle dependence of UPt3 dHvA frequencies in the partially localized model. Chapter 4. UPt3 110

Taillefer, Flouquet and Joss observed a sharp dip in the non-oscillatory magnetoresis-

tance for H a, and very weak ﬁeld dependence of the magnetoresistance at this angle— || strong evidence for an open orbit running perpendicular to the a-axis [106]. In addition to the dip for H a, the new magnetoresistance data contains a dip at H b, suggesting a || || second open orbit, running perpendicular to the b-axis. These two open orbits are nat-

urally found on the band 2 surface of the fully itinerant model, and are shown as green

lines in Fig. 4.2(b): the line that spans opposite arms, going through the zone centre,

is perpendicular to the a-direction, whereas the line that runs non-centrally across two

arms that are separated by only one interjacent arm is perpendicular to the b-direction.

However, since the arms of the Z1 sheet of the partially localized model (Fig. 4.3(a))

resemble those of the fully itinerant band 2 sheet, the partially localized model will also

have open orbits. On its own, the non-oscillatory magnetoresistance data therefore does

not favour one model over the other, but does provide an important constraint that any

Fermi surface modiﬁcations made to increase agreement with the quantum oscillation

data must still preserve the presence of the open orbits.

The new experimental quantum oscillation frequencies measured by my supervisor,

along with those previously published by Taillefer and Lonzarich [78], plotted as a func-

tion of magnetic ﬁeld angle, are shown as data points in Figs. 4.6 and 4.7, with lines

representing the theoretical predictions of the fully itinerant model (from Fig. 4.4) in the

former, and partially localized model (from Fig. 4.5) in the latter. The new quantum os-

cillation measurements performed by my supervisor agree with the previously-published

′ ′ ′ ′ results for UPt3 [78, 92], but also contain ten new orbits: λ , α3, α3, α4, α4, κ, γ , ζ, η, and η′.

Indeed, it is in the predicted angle dependence of quantum oscillation frequencies

that the striking diﬀerences between the fully itinerant and partially localized models are

most apparent: the partially localized model (Figs. 4.5 and 4.7) simply predicts far fewer

extremal orbits than the fully itinerant model (Figs. 4.4 and 4.6), predominantly due to Chapter 4. UPt3 111

Figure 4.6: Angle dependence of measured UPt3 quantum oscillation frequencies (data points) compared to the predictions of the fully itinerant model (lines), with band asso- ciations indicated by the colour scheme of Fig. 4.2 / Table 4.1. Frequencies noticed for the first time by SKEAF are shown as solid (orange for band 2, pink for band 3) lines. Black symbols are believed to be “breakdown” orbits in which the quasiparticles tunnel between bands 1 and 2 as they undergo cyclotron motion [78]. Triangles are the data of Taillefer and Lonzarich [78], circles are from the new SdH measurements, filled diamonds are from the new dHvA data in a 8–12 T field range, and pentagons are from the new dHvA data in a 14–18 T field range. Chapter 4. UPt3 112

Figure 4.7: Comparison between measured quantum oscillation frequencies (data points; same data as Fig. 4.6) and the predictions of the partially localized model (lines). Chapter 4. UPt3 113 the absence of the complicated band 2 sheet in the partially localized model. Speciﬁcally,

′ ′ ′ predicted orbits corresponding to the experimentally-measured α, α3, α3, α4, η, η , ζ, λ, λ′, and ǫ frequencies are missing from the partially localized model. Also, since the three frequencies in the 1.5–2.5 kT range near the b-axis (black symbols in Figs. 4.6 and 4.7) are thought to arise from magnetic breakdown orbits between bands 1 and 2 of the fully itinerant model [78], they remain unexplained in the partially localized model, which is missing band 2. Finally, the experimentally-observed δ orbit, ascribed to band

1 of the fully itinerant model or Z1 of the partially localized model, extends across all three rotation planes, indicating a closed surface; in contrast, according to the partially localized picture, this frequency should not exist for ﬁelds along the c-axis or anywhere in the a–c rotation plane, due to the open orbits on the Z1 sheet.

Looking at the δ orbit in more detail, it can be seen that although the prediction of the partially localized model for the quantitative value of this frequency at the b-axis is closer to that measured experimentally than is the prediction of the fully itinerant model, only the fully itinerant model correctly predicts that this orbit spans all three a–c, a–b, and b–c rotation planes, including H c. Thus, the topology of the partially localized Z1 || surface is fundamentally wrong, precisely because of the open orbits required for agreement with the non-oscillatory magnetoresistance data discussed above. This appears to be a fatal ﬂaw in the partially localized model, since no amount of tinkering can make the

Z1 surface topologically agree with both the quantum oscillation and magnetoresistance data, and even shifting the open orbits to the main Z3 sheet (discussed below) would ruin the agreement between that surface and the quantum oscillation data—clearly the partially localized model is missing a major Fermi surface sheet.

In the fully itinerant model, the band 1 Fermi surface sheet is closed, and the open orbits lie on the band 2 sheet, which is a topologically-complicated surface (Fig. 4.2(b)) not found in the partially localized model. The complex shape of the band 2 sheet gives rise to a multitude of extremal orbits, shown in Fig. 4.8 along with the experimentally Chapter 4. UPt3 114 detected dHvA frequencies which we believe are associated with them. In fact, most of the newly-predicted orbits located by the SKEAF algorithm in the old fully itinerant band structure calculation of Norman et al. come from this band (solid lines in Fig. 4.8), and were not previously noticed because they follow complicated paths that are not centred on the Brillouin zone symmetry points and are not obvious by visual inspection of the surface. Moreover, from an experimental side, almost all of the newly-detected frequencies from Prof. Julian’s quantum oscillation measurements are believed to correspond to orbits on band 2, especially the complicated orbits ﬁrst noticed by SKEAF.

Thus, without the contributions made by the SKEAF algorithm, the experimental data would have been thought to be in much poorer agreement with the fully itinerant band

2 surface than is actually the case.

In terms of speciﬁc band 2 frequencies, Taillefer and Lonzarich had previously associated the experimentally-detected λ frequency with the predicted A-2 orbit that encircles the main body of the surface (the red line in Fig. 4.2(b)), and the α frequency with the

ML-2 orbit that encircles one arm (the cyan line in Fig. 4.2(b)) [78]. For the newly- detected experimental frequencies, we believe that:

λ′ is the A-2 orbit that encircles the waist of the surface (the blue line in Fig. 4.2(b)) •

α corresponds to a KH-2 orbit and α′ corresponds to an ALM-2 orbit, both of • 3 3 which encircle three upper arms and one lower arm

α and α′ correspond to the two AH-2 orbits (sketched, for example, as the purple • 4 4 line in Fig. 4.2(b)), both of which encircle four arms but with orbit centres located

at diﬀerent points along the A–H line

η would seem to correspond best with the red L-2 orbit spanning two Brillouin • zones in Fig. 4.9, but this orbit is extremal for a smaller range of angles than the

similar blue L-2 orbit in Fig. 4.9, so it may be better to associate the experimental

frequencies with the latter, rather than the former, predicted orbit Chapter 4. UPt3 115

Figure 4.8: Angle dependence of predicted and measured UPt3 frequencies associated with band 2 of the fully itinerant model. Frequencies noticed for the ﬁrst time by SKEAF are shown as solid orange lines. Chapter 4. UPt3 116

Figure 4.9: Predicted UPt3 L-2 orbits on the extended-zone band 2 Fermi surface sheet, with the cross-sectional shape of each orbit shown at right. These orbits span two Bril- louin zones, and it is believed that the blue orbit, which remains extremal over a wider angular range than the red orbit, corresponds to the experimentally-detected η frequency.

η′ corresponds to the AKH-2 orbit that encircles two arms (the yellow line on • Fig. 4.2(b))

ζ corresponds to the KH-2 electron orbit that runs around the interior of the arms • of three adjacent Brillouin zones (the white line in Fig. 4.2(b))

Turning to the remaining three bands (Figs. 4.6 and 4.7), both the fully itinerant band 3 surface and partially localized Z3 surface agree well with the angle dependence of the experimentally-observed ω orbit. In fact, the predicted partially localized Z3 frequencies are quantitatively closer to the magnitudes of the experimental values than are those predicted by the fully itinerant model, which is the primary argument used by

Zwicknagl et al. to claim that their model is better than the fully itinerant one [93], but Chapter 4. UPt3 117

this small improvement is outweighed by the serious deﬁciencies of the partially localized

model regarding the incorrect shape of the Z1 sheet and missing predictions for the

experimental frequencies explained by the fully itinerant band 2 sheet. The experimental

κ frequency is thought to correspond to the K-3 orbit encircling the small fully itinerant band 3 pockets, but it is possible that this is actually the continuation of the ǫ frequency, which had been ascribed by Taillefer and Lonzarich to the Γ-4 orbit on the fully itinerant band 4 surface [78]. Finally, Taillefer and Lonzarich ascribed the measured γ frequency,

seen mostly in the a–b rotation plane, to the Γ-5 orbit on the fully itinerant band 5

surface [78], but Γ-5 may actually correspond to the new γ′ frequency, which is strongly manifested in the SdH data and could be tracked across both the a–b and b–c rotation planes.

Taking all of the experimental evidence together, it can be seen that the partially localized model, in which two of the three U 5f electrons are assumed to be localized and

therefore do not contribute to the Fermi volume, provides a poor description of the actual

UPt3 Fermi surface. Quantum oscillation measurements show that the Z1 Fermi surface sheet is topologically incorrect, but since the partially localized model lacks a sheet like

the band 2 surface of the fully itinerant model, it is not possible to tweak this sheet

so that it simultaneously agrees with both the quantum oscillation measurements and

open orbits found by the non-oscillatory magnetoresistance measurements. Furthermore,

many new experimentally-observed quantum oscillations are not explained at all by the

partially localized model, but correspond well to complicated orbits that were noticed

on the fully itinerant model band 2 sheet for the ﬁrst time by the SKEAF algorithm.

Thus, it is clear that the fully itinerant model provides a much better description of

the real, experimentally-measured UPt3 Fermi surface than does the partially localized model, and that all three U 5f electrons contribute to the Fermi volume. Chapter 5

CePb3

This chapter details my resistivity, torque, and ac susceptibility studies of CePb3, measured as functions of magnetic ﬁeld strength, magnetic ﬁeld angle, and temperature, as well as the results of my classical spin model calculations for this material. Due to the air-sensitive nature of CePb3, I prepared the samples in the glove box that I built (section 2.4). Since our research group’s dilution refrigerator was non-functional at the time of this study, I performed these measurements at the National High Magnetic Field

Laboratory (NHMFL) in Tallahassee, Florida, with the help of Tim Murphy, Suchitra

Sebastian, and my supervisor, Stephen Julian. This work is currently unpublished.

5.1 Material background

CePb3 is a heavy fermion Kondo lattice system that has an AuCu3-type cubic crystal structure (space group # 221: Pm3m), shown in Fig. 5.1, with lattice constant a = b = c = 4.876 A˚ and fractional atomic coordinates Ce (0, 0, 0) and Pb (0.5, 0.5, 0) [107].

Since CePb3 is a cubic material, the directions ~a = (100), ~b = (010), and ~c = (001) are equivalent, and are used interchangeably throughout this chapter.

In zero magnetic ﬁeld, the system behaves as a paramagnet at high temperatures, with the free moment, µ = 2.32 µ , close to the value expected for the 4f local moments of | | B

118 Chapter 5. CePb3 119

Figure 5.1: CePb3 crystal structure.

3+ Ce ions [107]. Below TK = 3.3 K [108], these local moments are increasingly screened due to the Kondo eﬀect, although this screening remains incomplete in the T 0 K →

limit [109]. Below TN = 1.16 K, CePb3 exhibits a second order phase transition into an antiferromagnetic state [109, 110]. Neutron diﬀraction measurements have found

that the magnetic structure in this state is not simple antiferromagnetism, but rather is

characterized by an incommensurate ordering wave vector ~q = (0.135, 0.058, 0.5) [109].

With spins aligned along the (001) direction, the magnetic moments on a given plane

perpendicular to (001) are aligned opposite to those on adjacent planes stacked along

(001). Within each plane, the moment amplitudes are sinusoidally modulated along

the ( 0.135, 0.058, 0) and ( 0.058, 0.135, 0) directions, with a maximum moment ± ± ± ± amplitude of µ = 0.55 µ at T = 30 mK [109]. The linear coeﬃcient of speciﬁc heat in | | B the antiferromagnetic state is approximately 1–1.5 J mol−1 K−2 [111, 112], which is large

even by heavy fermion standards.

Early magnetoresistance measurements on polycrystalline samples revealed a resis- Chapter 5. CePb3 120

tivity that rises with ﬁeld up to H 6 T, then gradually decreases to a small value ∼ at H 11–12 T [110]. This behaviour was originally interpreted as evidence for ∼

ﬁeld-induced superconductivity in CePb3 [110], but later measurements by the same research group on polycrystalline samples [113], and by other groups on single crystal

samples [60, 114], showed that no superconductivity is present.

Prior to my work, the most detailed study of CePb3 magnetoresistance, measured as a function of temperature, magnetic ﬁeld strength and magnetic ﬁeld angle in the (110)–

(100) plane, was performed by John McDonough and my supervisor, Stephen Julian [60].

For H (100), they measured a magnetoresistance that increases with increasing field up || to a broad peak near H 6 T, then gradually decreases to a kink at H 10 T, above ∼ ∼ which the resistivity is field independent, consistent with the previous polycrystalline measurements [110, 113]. For H (110), McDonough and Julian observed more com- || plicated behaviour: with increasing field, the resistivity rises linearly to H 5 T, then ∼ declines sharply, and finally decreases more gradually to a kink at H 10 T, above which ∼ it is field independent. They interpreted the H (110) 5 T anomaly as a transition || ∼ from the low-field, low-temperature antiferromagnetic phase to a mid-field “spin-flop” phase, and interpreted the H (110) 10 T kink as a change from the spin-flop phase to || ∼ a field-induced ferromagnetic phase. These anomalies, which are also detected in ultrasound velocity [115] and differential susceptibility [116] measurements, can be tracked as functions of temperature to produce the phase diagram shown in Fig. 5.2.

Ignoring the incommensurate nature of the CePb3 antiferromagnetism, the spin-ﬂop transition proposed by McDonough and Julian consists of a discontinuous jump of the

magnetization vector M~ from alignment along (100) (the magnetic “easy” direction), to alignment parallel to the field, along (110) (the magnetic “hard” direction). This scenario is sketched in Fig. 5.3, which shows the behaviour of the two spin sub-lattices and total magnetization vector as the applied magnetic field is increased. At low fields

(Figs. 5.3(a) and (b)), the spins remain near the (010) and (010) easy axes, symmetric − Chapter 5. CePb3 121

1.4

1.2 Paramagnetic / field-induced ferromagnetic phase 1.0

0.8

0.6

Temperature (K) 0.4

0.2 Antiferromagnetic Spin-flop phase phase

0.0 0 2 4 6 8 10 12 Magnetic field (T), for H||(110)

Figure 5.2: Ambient pressure phase diagram of CePb3 for H (110), prior to my study, based on measurements of magnetoresistance [60], ultrasound|| velocity [115], and diﬀer- ential susceptibility ∂χ/∂H [116].

around (100), and gradually cant toward the magnetic ﬁeld direction, with the total

magnetization vector pinned along the (100) easy axis. At the spin-ﬂop transition, the

spins jump to be symmetric around (110), the magnetic ﬁeld direction, with the total

magnetization vector correspondingly jumping to (110) (Fig. 5.3(c)). As the magnetic

ﬁeld is further increased in the spin-ﬂop phase, the spins gradually rotate toward (110)

(Figs. 5.3(d) and (e)); the completion of this process represents the entry into the ﬁeld-

induced ferromagnetic phase (Fig. 5.3(f)), above which no further change occurs. For

H (100), the spins would simply rotate, continuously, from (010) to (1 1 0), and then || ± ± jump to (100), explaining the single anomaly seen for this ﬁeld direction as the point

where the spins are ﬁnally aligned with the ﬁeld. Although these spin re-orientations

form the basis of the spin-ﬂop interpretation, the H 5 T and H 10 T anomalies ∼ ∼ are manifested only very weakly in the experimentally-measured CePb3 magnetization

and ac susceptibility [114, 116]. De Haas–van Alphen (dHvA) measurements of CePb3 Chapter 5. CePb3 122

Figure 5.3: Sketch of the spin sub-lattice and magnetization orientations in the spin- flop scenario. The two blue arrows (“spin 1” and “spin 2”) represent the magnetization directions of the two spin sub-lattices, the green arrow (“M”) represents the total magnetization, equal to the vector addition of the sub-lattice magnetizations, and the purple arrow (“H”) represents the applied magnetic field, aligned along the (110) direction. Starting at 0 T in panel (a), the field increases from left to right, reaching 5 T between panels (b) and (c), and 10 T between panels (e) and (f). ∼ ∼

revealed a Fermi surface with at least two large, spherical sheets; the dHvA frequency

corresponding to an orbit on one of these sheets was tracked as lowa6Tfor H (100) || and found to be ﬁeld independent, unaﬀected by the H 10 T anomaly. ∼

Quantum critical points typically occur when a second order phase transition is tuned

to T = 0 K (section 1.2). However, if a line of ﬁrst order phase transitions ends in a

ﬁnite-temperature critical endpoint, it is sometimes possible to tune this critical end-

point to T = 0 K and thereby create a different kind of quantum critical point called a “quantum critical endpoint.” Such a quantum critical endpoint, which differs from a traditional second order quantum critical point in that no symmetry is broken across the transition, has been induced, for example, in the itinerant metamagnet Sr3Ru2O7 by the application of a magnetic field [117]. In the context of CePb3, if the sharp decline of magnetoresistance at H (110) = 5 T indicates a first order transition, it may be possible || to tune an associated finite-temperature critical point to T = 0 K, and therefore create a quantum critical endpoint, by changing the magnetic field angle. Chapter 5. CePb3 123

5.2 Experimental details

In order to probe the spin-ﬂop physics of CePb3 and look for signs of quantum criticality, I have conducted ambient-pressure resistivity, ac susceptibility, and cantilever torque measurements from 0–18 T, and 23–400 mK, with the ﬁeld rotated in the (110)–(100),

(110)–(111), and (111)–(100) planes. The single-crystal samples studied were grown at

Stanford University from Pb ﬂux by Suchitra Sebastian. These samples (as-grown), from the “S1071” batch, are cuboids with faces aligned along the (100) = (010) = (001) directions.

Since high-purity CePb3 crystals react violently with air [60], I performed most of the sample preparation activities in a home-made glove box that I constructed speciﬁcally for this study (section 2.4). Due to reduced dexterity in the bulky gloves, awkward microscope/sample positioning caused by focal length considerations, and the overpressure required to ensure that any glove box leaks were outward (helium) rather than inward

(air), this work was very diﬃcult. In order to allow the samples, once prepared, to be safely removed from the glove box, transported to the NHMFL, and mounted in a dilution refrigerator, I designed a set of tiny, seal-able polycarbonate boxes (shown in Figs. 5.4(b) and 5.5(b)), which were built by machinist Mark Aoshima in the University of Toronto mechanical workshop. Since my CePb3 measurements were originally intended to take place in Toronto, these polycarbonate boxes were designed to ﬁt inside the bobbins of the graphite rotation mechanism (section 2.2), and have external dimensions of 4 4 5 mm, × × internal sample space dimensions of 3 3 3.5 mm, and recessed lids containing two ∼ × × holes through which wires may be routed.

The samples arrived sealed inside a quartz tube, under an inert helium or argon atmosphere. Once the glove box was closed and the air inside it replaced with helium, I broke the quartz tube to extract the samples, and selected the 8 most platelet-like CePb3 cuboids. The thicknesses of these 8 samples were then reduced by dry-polishing, to make them more appropriate for resistivity and cantilever torque experiments. Note that the Chapter 5. CePb3 124

samples did not cleave well, and due to limitations of working in the glove box it was

only possible to polish the largest face of each cuboid. The best of these samples, shown

in Fig. 5.4(a), were chosen for resistivity (samples 1 and 2, with respective dimensions

1.78(8) 0.76(4) 0.39(6) mm and 1.83(6) 1.44(4) 0.33(5) mm) and torque (sample × × × × 3, with dimensions 1.47(3) 1.34(7) 0.36(7) mm) measurements. Attaching four leads × × (two for excitation current and two for voltage measurement) to each of the resistivity

samples was non-trivial, and many techniques were attempted, including spot-welding

25 µm diameter gold wires to the sample and soldering these to twisted pairs of 100 µm

diameter copper wire. However, I found that the only reliable method to simultaneously

and robustly secure all four leads to the sample, while minimizing sample damage, was

to solder each of the copper voltage wires to a current wire and then solder these to

opposite ends of the sample, using Indalloy ﬂux #3 [118] and 90% Sn / 10% Ag solder.

Since the current and voltage leads at each end of the sample are shorted together, this

wiring configuration may be susceptible to contact resistance effects, but such effects

are expected to be field-independent, while my study focused specifically on the field-

dependence of the resistivity. Once leads were attached to both resistivity samples, I

cleaned them with ethanol and placed them in individual polycarbonate boxes (with

copper wire twisted pairs exiting through the holes in the box lids), which were sealed

shut with Stycast 1266 epoxy.

The largest cube-shaped sample was chosen for the ac susceptibility measurement.

This sample, shown in Fig. 5.5(a), has dimensions 2.8(1) 2.2(1) 2.2(1) mm. I soldered × × a 50 µm diameter silver heat-sink wire to the sample using Indalloy ﬂux #3 and 90% Sn /

10% Ag solder; in case this delicate wire were to break during subsequent transportation

or mounting, a back-up 100 µm diameter copper heat-sink wire was also soldered to the sample. I then cleaned the sample with ethanol and placed it in a polycarbonate box, aligning the sample faces with the sides of the box. The heat-sink wires were routed out of the box through the holes in the lid, and the box was sealed shut with Stycast 1266 Chapter 5. CePb3 125

Figure 5.4: Panel (a): Polished samples 1 (for resistivity), 2 (for resistivity), and 3 (for torque), photographed inside the glove box (section 2.4) through the microscope. 400 µm and 320 µm diameter wires are laid beside the samples for a sense of scale. Panel (b): Resistivity samples 1 and 2 sealed inside polycarbonate boxes and attached to a 16-pin DIP connector with GE varnish. Panel (c): torque sample (sample 3 from panel (a)) attached to a cantilever and sealed from contact with air using GE varnish. Chapter 5. CePb3 126

epoxy (Fig. 5.5(b)).

All three sample-containing polycarbonate boxes were placed into individual glass

jars, whose threads were greased to help seal the lids. Once the lids were in place,

the jars were further sealed with silicone sealant. Thus, for transportation, both the

polycarbonate boxes and the glass jars were ﬁlled with inert helium gas. I loaded the

remaining samples, including polished sample #3 from Fig. 5.4(a), into a pyrex glass

pipette tube, which I had previously melted closed at the small end. The large end of

the pipette was attached to a section of rubber hose, which was pinched shut with a

hose clamp. Upon removal from the glove box, I attached the rubber hose to our group’s

glass-blowing pump station and melted the large end of the pipette closed while pumping

on the inside of the pipette.

For the ac susceptibility experiment, I machined a polycarbonate former into which

the susceptibility-sample polycarbonate box ﬁts, and around which two counter-wound

pick-up coils (one around the polycarbonate box “sample space,” and one around an

empty portion of the former) and one drive coil (covering both pick-up coils) were wound

(Fig. 5.5(c)). The pick-up coils each contained 400 turns of 40 µm diameter self-bonding copper wire; the drive coil contained 214 turns of 130 µm diameter superconducting

NbTi-in-Cu-matrix wire, bonded with GE varnish.

At the NHMFL in Tallahassee, I mounted the polycarbonate boxes containing the two resistivity samples on a 16-pin DIP connector, using GE varnish (Fig. 5.4(b)). The samples were oriented by eye (by looking at the sample shapes through the translucent polycarbonate box sides), such that sample 1 rotated relative to the ﬁeld in the (100)–

(110)–(010)=(100) plane and sample 2 rotated in the (110)–(111)–(001)=(100) plane.

The DIP connector was then mated with a NHMFL rotation mechanism. Inside a commercial NHMFL glove box, Dr. Sebastian removed the polished CePb3 sample 3 from the sealed pyrex pipette and mounted it on a cantilever with a drop of GE varnish, such that the (110) axis is parallel to the cantilever “tongue” (Fig. 5.4(c)); the GE varnish also Chapter 5. CePb3 127

Figure 5.5: CePb3 ac susceptibility sample. Panel (a): sample photographed inside the glove box (section 2.4) through the microscope, with copper and silver heat-sink wires visible. Panel (b): sample sealed inside a polycarbonate box. Panel (c): polycarbonate former and coils—the Cu pick-up coils are covered by the NbTi/Cu drive coil. Panel (d): entire sample-box-former-coils assembly mounted inside the dilution refrigerator rotation mechanism at the NHMFL in Tallahassee. Chapter 5. CePb3 128

coated the sample, protecting it from contact with air after removal from the glove box.

The cantilever was then inserted into a set-up that allowed the torque to be measured, via

cantilever capacitance, for rotations relative to the ﬁeld in the (110)–(111)–(001)=(100)

plane. Finally, I secured the polycarbonate box containing the ac susceptibility sample

into place in the coil-wrapped polycarbonate former using 5 minute epoxy, and mounted

the entire assembly inside a NHMFL rotation mechanism, for rotations relative to the

ﬁeld in the (110)–(111)–(001)=(100) plane.

All measurements were performed using the NHMFL SCM1 18/20 T superconduct-

ing magnet and top-loading dilution refrigerator facility, with the samples/polycarbonate

boxes immersed in the “dilute phase” liquid 3He/4He mixture inside the dilution refrigerator mixing chamber. For the resistivity experiment, 0.5 mA sinusoidal excitation currents were generated at ω1 = 76.5 Hz and ω2 = 39.99 Hz, for samples 1 and 2 respectively, using Keithley model 6221 current sources. The voltage across each of the two samples

was amplified 500 using a Stanford Research Systems model SR554 transformer pream- × plifier, and measured using a Stanford Research Systems SR830 lock-in amplifier, with a 1 second filter time constant, 18 dB/octave filter roll-off, and 500 µV scale. For the torque experiment, the cantilever capacitance was measured using an Andeen-Hagerling model AH 2700A ultra-precision capacitance bridge, measuring at 5000 Hz with a 300 ms

filter time constant. For the ac susceptibility experiment, a Keithley model 6221 current source was used to induce a sinusoidally varying h0 = 1 G modulation field in the drive coil, at a frequency of ω = 76.5 Hz. The voltage across the combined counter-wound pick-up coil pair was amplified 500 using a Stanford Research Systems model SR554 × transformer preamplifier, and measured using a Stanford Research Systems SR830 lock- in amplifier, with a 1 second filter time constant, 18 dB/octave filter roll-off, and 500 µV scale. Magnetic fields were swept between 0 T and 18 T at rates ranging from 0.05 T/min to 0.5 T/min. Chapter 5. CePb3 129

5.3 Experimental results

The experimental results shown in this section are plotted over a magnetic ﬁeld range

from 2 T, deep inside the antiferromagnetic phase, to 12 T, deep inside the ﬁeld-induced

ferromagnetic phase (see Fig. 5.2). Measured behaviours below 2 T and above 12 T simply

continue the trends shown just above 2 T and just below 12 T, respectively, except that

a superconducting transition in residual Pb from the ﬂux growth process is seen with all

three experimental techniques at low ﬁelds, and de Haas–van Alphen oscillations from

these same Pb inclusions become prominent in the torque data at high ﬁelds.

The measured magnetoresistance of CePb3 samples 1 and 2 at base temperature (T = 23 mK) is shown in Fig. 5.6, for magnetic ﬁelds along (110) and (100) (both

samples), and (111) (sample 2). Each curve is normalized to the resistivity value at

H (110) = 2 T for its sample. Absolute resistivity values were calculated from the ||

voltage measured by the lock-in ampliﬁer, Vlock−in, using the equation

Aperp WD ρ = R = Vlock−in (5.1) L IexcL

where R is the measured sample resistance, Aperp = WD is the cross-sectional area of the sample perpendicular to the direction of current flow, L is the distance between voltage contacts along the direction of current flow, W and D are the width and thickness of the sample perpendicular to the direction of current flow, respectively, and Iexc = 0.5 mA is the excitation current through the sample. For sample 1, L = 1.45(15) mm, W =

0.76(4) mm and D = 0.39(6) mm, yielding ρ(T = 23 mK, H = 2 T) = 6(1) µΩcm when

H (110). For sample 2, L = 0.7(1) mm, W = 0.6(1) mm and D = 0.33(5) mm, yielding || ρ(T = 23 mK, H = 2 T) = 8(2) µΩcm when H (110). (Note that L and W for sample || 2 are somewhat smaller than the dimensions listed in section 5.2, due to the particular geometry of the contacts and a crack in the sample running parallel to the current direction, beside the contact solder points.) These two low-temperature, H (110) = 2 T || Chapter 5. CePb3 130

1.3

Sample 1 H||(110) 1.2 H||(100) ρ(H||(110) = 2T) = 6(1) µΩcm

Sample 2 1.1 H||(110) H||(111) H||(100) ρ(H||(110) = 2T) = 8(2) µΩcm 1.0

(H||(110) = 2 = T) 2 (H||(110) ρ 0.9 (H) / (H) ρ 0.8

0.7 2 3 4 5 6 7 8 9 101112 Magnetic field (T)

Figure 5.6: Magnetoresistance of CePb3 resistivity samples 1 and 2 for H (110), (111), and (100), normalized to the value at H (110) = 2 T for each sample (6(1)|| µΩcm for || sample 1 and 8(2) µΩcm for sample 2), at T = 23 mK. The H (110) and H (100) curves agree with previous measurements [60]. || ||

resistivity values are comparable to those published previously for CePb3 [60]. The magnetoresistance for H (110) and H (100) agrees with the previous measure- || || ments of McDonough and Julian [60]: for H (110), the resistivity increases linearly at || low fields, undergoes a sharp decline near 5–6 T, and then declines more gradually to ∼ a kink at 10–11 T, above which it is field independent; for H (100), the resistivity has ∼ || a broad hump near 6 T, and is field independent above 10 T. My measurement of ∼ ∼ the H (111) magnetoresistance is the first of its kind in this material, and shows that, || for this field angle, the two features seen in the H (110) data have merged into a large, || single sharp decline near H 6 T. Although the declines near 5–6 T are very steep, c ∼ ∼ they were not found to be first order jumps at any angle, down to the lowest temperature

measured (T = 23 mK). As the magnetic ﬁeld is rotated through intermediate angles Chapter 5. CePb3 131

not shown in Fig. 5.6, the curves simply morph into one another. For H < 2 T and

H > 12 T, the magnetoresistance is nearly independent of magnetic ﬁeld angle.

Since the H (111) magnetoresistance is a new measurement, and exhibits qualita- || tively different behaviour from that found at other magnetic field angles, I have focused on this angle for measurements of resistivity temperature dependence. One major draw- back of the NHMFL SCM1 facility is the inability to do temperature sweeps at finite magnetic fields: there is no field-cancelled region, so the mixing chamber thermometer temperatures are only valid at H = 0 T, and therefore cannot be used for temperature control or read-out at elevated fields. Thus, in order to obtain temperature-dependent resistivity data I performed the following steps:

1. Sweep the ﬁeld to H = 0 T.

2. Drive a constant current through the mixing chamber heater, and wait for the

temperature to settle to a constant value.

3. Record the H = 0 T temperature, then do a data collection run while the ﬁeld is

swept from 0 T to 12 T.

4. Repeat steps 1–3 for diﬀerent heater drive currents. Field sweeps were done at 7

temperatures, ranging from 23 mK to 400 mK.

5. Extract data at fixed field values from each field sweep.

Resistivity versus temperature curves obtained in this manner are shown in Fig. 5.7(a).

The temperature dependence appears to be quadratic at low and high fields, as predicted by Fermi liquid theory (section 1.1), but looks linear in T near the H 6 T magne- c ∼ toresistance decline. As shown in Fig. 5.7(b), this behaviour can be quantified by fitting

2 the resistivity data to an equation of the form ρ(T )= AT + BT + C: far from Hc, the linear term (characterized by the B coeﬃcient) is much smaller than the quadratic term Chapter 5. CePb3 132

22 H = 2 T (a) (b) 14 H = 4 T 20 ρ(T) = AT 2 + BT + C fits at constant H H = 6 T Fit range: 23 mK to 400 mK H = 8 T 18 2 H = 10 T A [ µΩ cm / K ] µΩ H = 12 T 16 B [ cm / K] 12 H||(111) C [ µΩ cm] 14 cm)

µΩ 12 (

10 10 offset ρ 8 + BT + C BT fitcoefficient+ C + values 2 (T) +

ρ 6 8 4

(T) =AT (T) 2 ρ

6 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 3 4 5 6 7 8 9 101112 Temperature (K) Magnetic field (T), for H||(111)

Figure 5.7: Panel (a): resistivity vs. temperature curves (offset for clarity) of CePb3 sample 2, for 2 T H (111) 12 T, showing T -linear behaviour near Hc 6 T and T 2 behaviour elsewhere.≤ || Panel≤ (b): field dependence of A, B and C coefficients∼ taken from quadratic ρ(T )= AT 2 + BT + C fits of the data shown in panel (a), at fixed fields 2 T H (111) 12 T. The dominance of the linear coefficient (B) over the quadratic ≤ || ≤ coefficient (A) near Hc 6 T can be seen. The field dependence of the C coefficient matches the H (111) magnetoresistance∼ shown in Fig. 5.6. Dashed lines are guides to the eye. ||

(characterized by the A coefficient), but it becomes dominant as the field approaches Hc; the C coefficient tracks the low temperature magnetoresistance shown in Fig. 5.6.

Deviations from T 2 dependence of the resistivity can indicate a breakdown of Fermi liquid theory, often occurring in the quantum critical fluctuation regime near a quantum critical point (section 1.2). In such cases, as shown in Fig. 1.1, Fermi liquid theory applies only in a finite low-temperature range that decreases in size on approach to the quantum critical point. Following the reasoning that this situation may apply to CePb3, in Fig. 5.8(a) I have plotted the resistivity curves of Fig. 5.7(a) as functions of T 2, and fitted them, over varying low temperature ranges, to lines of the Fermi liquid form

ρ(T 2) = AT 2 + C. At ﬁelds far from H 6 T, the data is well-described by these ﬁts c ∼ over the entire measured temperature range, but as H approaches Hc, the upper limit of

fit applicability, TFL, monotonically decreases. For the H = 6 T curve, the “fit” applies only to the lowest two temperature points measured. This field dependence of TFL is Chapter 5. CePb3 133

1.0 18 100 (b) 2 2 H = 4 T (a) A(H) from ρ(T ) = AT + C fits

H = 5 T ) at constant H FL

2 90 α 16 H = 6 T Best A(H) ∝ |H - H |- fit c 0.8 H = 7 T 80 H = 8 T Maximum temperature

H||(111) cm T / fitted ~ T 14 70 FL µΩ 0.6 cm) 60 ← µΩ

( 12

50 offset ρ 40 0.4

) + + ) 10 2 30 (T ρ 20 0.2 8 Fit range: 7 to 12 T Fit range: 2 to 5.5 T H = 6.12(7) T coefficient ofresistivity( H = 6.6(6) T c

2 10 → c α = 1.18(6) MaximumtemperatureT fitted ~ (K) T α = 0.9(3) 6 0 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 101112 T2 (K 2) Magnetic field (T), for H||(111)

2 2 Figure 5.8: Panel (a): Fermi liquid ρ(T ) = AT + C fits to CePb3 sample 2 resistivity vs. T 2 curves (offset for clarity) for H (111), over shrinking low temperature ranges near || Hc 6 T. Panel (b): field dependence of the A coefficient from the fits shown in panel (a). Best∼ fits to the low-field (2 T H (111) 5.5 T) and high-field (7 T H (111) 12 T) ≤ || ≤ −α ≤ || ≤ regions show the A coefficient diverging as A(H) H Hc , with Hc 6 T and α 1. Maximum temperatures of the panel (a) fitting∝ regions | − (up| to the highest∼ temperature∼ measured, T = 400 mK), giving a rough indication of the range of Fermi liquid theory applicability, TFL, are also indicated in panel (b).

plotted in Fig. 5.8(b), up to T = 400 mK, the highest temperature measured.

Fig. 5.8(b) also shows the ﬁeld dependence of the A coeﬃcient from the ρ(T 2) =

AT 2 + C fits of Fig. 5.8(a). This coefficient diverges near H 6 T: the best fit to the A c ∼ coefficient high field (7 T H (111) 12 T) data reveals that A(H) H H −α, with ≤ || ≤ ∝ | − c| H = 6.12(7) T and α = 1.18(6); a similar fit to the low field (2 T H (111) 5.5 T) c ≤ || ≤

data yields Hc = 6.6(6) T and α = 0.9(3). Although it is diﬃcult to push this analysis much further, due to the small number of temperature points available for the initial

ρ(T 2) = AT 2 + C ﬁts, I conclude that A(H) is indeed diverging as H H −α, with | − c| H 6 T and α 1. This result is similar to the divergence observed near ﬁeld- c ∼ ∼

induced quantum critical points in other heavy fermion materials, such as CeCoIn5 [4] and YbRh Si [8], both of which were found to have a critical exponent α 1. Divergence 2 2 ∼ of the A coeﬃcient also occurs near the ﬁeld-induced quantum critical endpoint in the

non-heavy-fermion ruthenate Sr3Ru2O7 [117]. Chapter 5. CePb3 134

A divergence of the resistivity T 2 coeﬃcient suggests that a divergence of the elec-

tronic speciﬁc heat should also be observed. Although, to my knowledge, no speciﬁc

heat measurements have been performed on oriented CePb3 single crystals in magnetic ﬁelds, measurements on a polycrystalline sample (conceptually similar to single crystals

measurements averaged over all possible ﬁeld orientations) show a broad peak centred

on H 6 T [112]. Comparing my A coefficient values to the measured specific heat c ∼ 2 2 2 −2 values away from Hc gives Kadowaki-Woods ratios of A/γ = 16(1) µΩcm mol K J for the low field regime (2 H 4 T) and A/γ2 = 5(1) µΩcm mol2 K2 J−2 for ≤ ≤ the high field regime (9 H 12 T). These ratios are comparable to the “univer- ≤ ≤ sal” value, A/γ2 10 µΩcm mol2 K2 J−2, observed for many heavy fermion com- ∼

pounds [119], as well as experimentally-determined values for H >Hc in CeCoIn5, A/γ2 3 µΩcm mol2 K2 J−2 [120], and YbRh Si , A/γ2 6 µΩcm mol2 K2 J−2 [8]. ≈ 2 2 ≈

The CePb3 torque measurements, the ﬁrst of their kind for this material, are shown in Fig. 5.9. Each torque curve is plotted as a percent change from its value at H = 2 T.

Pre-normalization H = 2 T capacitance values at all measured angles are within 0.5% of 1.200 pF, and are nearly independent of ﬁeld strength and angle below 2 T. For

H > 2 T, the torque depends strongly on both field angle and field strength, in a complicated manner. This dependence is particularly striking when H (111), exhibiting || several changes of sign over the measured field range: just above 2 T, the torque gradually decreases with increasing field, then increases to a peak near H 6 T, then decreases as c ∼ the field surpasses Hc, then gradually increases to 11 T, where it has a kink and begins to rise faster. These sign changes are very sensitive to small rotations away from (111)

(Fig. 5.9(b)). As the ﬁeld is rotated further from (111), the torque curves stretch away from ∆τ = 0, before relaxing back toward ∆τ = 0 as the ﬁeld approaches (110) or (100).

The ac susceptibility measurements, for H (110), H (111) and H (100), are shown || || || in Fig. 5.10(a). For all ﬁeld directions, the measured susceptibility is dominated by a strong, smoothly-varying signal that exhibits a χ(H) H2 ﬁeld dependence. This is ∝ Chapter 5. CePb3 135

4 1.5 (a) H||(110) (b) 3 H 10° from (111) to (110) 1.0 H||(111) H 20° from (111) to (100) H||(100) 0.5 2 0.0 1 -0.5 0

-1.0 (%), relative (%), value to = T 2 H at -1 relative (%), value to = H2 T at -1.5 ∆τ ∆τ H 2° from (111) to (100) -2 -2.0 H 1° from (111) to (100) H||(111) H 1° from (111) to (110) -2.5 -3 H 2° from (111) to (110) H 6° from (111) to (110) -3.0 -4 Torquechange 2 3 4 5 6 7 8 9 101112 Torquechange 2 3 4 5 6 7 8 9 101112 Magnetic field (T) Magnetic field (T)

Figure 5.9: Measured change in torque, ∆τ, normalized to each curve’s value at H = 2 T. Panel (a) shows data for H (110), H (111), H (100), and two intermediate field angles. Panel (b) shows several field|| angles|| very close|| to H (111). Torques for all measured angles collapse onto the same, nearly field independent,|| curve below 2 T, and all pre- normalized raw capacitance values at 2 T are within 0.5% of 1.200 pF. The measured torque exhibits strong and complicated dependence on both field angle and field strength, particularly near H (111). ||

likely an experimental artifact caused by vibration of the rotation mechanism: the drive coil was fitted to the sample / pick-up coil assembly rather than being fixed to and co- linear with the main magnet, thus experiencing a torque proportional to H, oscillating at the drive frequency; since the voltage induced in the pick-up coils is also proportional to H, an overall H2 dependence may result. The only distinctive feature present in the ac susceptibility above this background is a quench of the superconducting NbTi drive coil at H 14–15 T (not shown). When the H2 background is fitted and subtracted ∼ (Fig. 5.10(b)), the residuals are 100–550 smaller than the original signals, putting ∼ × an upper bound on the size of any other ac susceptibility signals that may be present in the data. (The discontinuity seen in the Fig. 5.10(b) H (111) data comes from a || measurement irregularity in the NHMFL instrumentation, and not intrinsically from the pick-up coils.) A previously-published CePb3 ac susceptibility study [116] also contained broad background signals, but additionally observed weak kinks corresponding to the Chapter 5. CePb3 136

3.0x10 -5 (a) 2.5x10 -5 (b) 2.0x10 -5

-5 10 -2 1.5x10 1.0x10 -5

-6 (arb. units) 5.0x10 χ

(arb. units) 0.0

-5.0x10 -6 background χ -1.0x10 -5 -

-3 H||(110) χ 10 -5 H||(110) ac susceptibilityac H||(111) -1.5x10 H||(111) H||(100) -2.0x10 -5 H||(100)

-2.5x10 -5 2 3 4 5 6 7 8 9 101112 2 3 4 5 6 7 8 9 101112 Magnetic field (T) Magnetic field (T)

Figure 5.10: CePb3 ac susceptibility vs. ﬁeld, for H (110), H (111), and H (100). Panel (a) shows smoothly-varying H2 behaviour of the measured|| susceptibility.|| Panel|| (b) shows the data from panel (a), with the H2 component ﬁtted and removed. The ranges of the residual H (110), H (111), and H (100) curves in panel (b) are 100 , 550 , and 200|| smaller|| than the ranges|| of their respective counterparts∼ in pa×nel∼ (a). No× strong∼ features× corresponding to those seen in the magnetoresistance (Fig. 5.6) and torque (Fig. 5.9) are detected (the discontinuity near H (111) = 7 T is an instrumentation artifact). ||

features seen in magnetoresistance (Fig. 5.6) and torque (Fig. 5.9). For a purported spin-ﬂop transition, in which the spins “ﬂop” around as illustrated in Fig. 5.3, it is surprising that the signals observed in the spin susceptibility of the system are so weak.

5.4 Discussion

In order to help interpret my results in the context of the spin-flop phenomenology, first proposed for this material by McDonough and Julian [60], I created two simplified models

(“2D” and “3D”) of possible spin behaviour in a magnetic field. In both models, collective behaviour of the CePb3 local moments is abstracted to the case of two classical spins, and the spin configuration giving lowest total energy of the system for a given magnetic field magnitude and direction is found. Similar classical spin model approaches [121, 122] have been shown to accurately describe the physics of conventional antiferromagnetic spin-flop compounds, such as MnF2 [123, 124]. For the 2D spin model, in which the Chapter 5. CePb3 137

spins and applied magnetic ﬁeld are constrained to lie in a two-dimensional (100)–(110)–

(010) plane, the total energy of the system is

E (θ , θ ) = µ H cos (θ θ ) µ H cos (θ θ )+ E cos (θ θ )+ tot 1 2 − | | 1 − H − | | 2 − H afm 1 − 2 2 2 2 + Eanis sin (2θ1)+ Eanis sin (2θ2)+ Edeg sin (θ1/2) (5.2)

where µ = 0.55µ is the maximum ordered moment seen experimentally by neutron | | B

scattering in the antiferromagnetic state [109], H is the magnetic ﬁeld strength, and θ1,

θ2 and θH are the azimuthal angles of spin 1, spin 2 and the magnetic field, respectively, measured from (100) toward (010). The terms involving µ H tend to favour alignment | | of each spin along the direction of the magnetic field, the term involving the antiferromagnetic interaction energy E J µ 2 tends to favour anti-alignment of the spins afm ∝ | | relative to one another, the cubic crystalline anisotropy terms involving Eanis tend to favour alignment of each spin along (100), (010) or (001), and the term involving ± ± ± E E ,E was introduced to slightly lift the degeneracy between the (100), deg ≪ afm anis ± (010) and (001) directions for the purposes of numerical calculation. Similarly, gen- ± ± eralized to the 3D spin model, in which the spins and applied magnetic field may point in any direction, the total energy of the system is

E (θ , θ ,φ ,φ ) = µ H [cos (θ θ ) sin φ sin φ + cos φ cos φ ] tot 1 2 1 2 − | | 1 − H 1 H 1 H − µ H [cos (θ θ ) sin φ sin φ + cos φ cos φ ]+ − | | 2 − H 2 H 2 H + E [cos (θ θ ) sin φ sin φ + cos φ cos φ ]+ afm 1 − 2 1 2 1 2 1 + 4E sin2 φ sin2 (2θ ) sin2 φ + cos2 φ + anis 1 4 1 1 1 1 + 4E sin2 φ sin2 (2θ ) sin2 φ + cos2 φ + anis 2 4 2 2 2 + E sin2 (θ /2) + sin2 [(φ π/2) /2] (5.3) deg 1 1 − Chapter 5. CePb3 138

where φ1, φ2 and φH are the polar angles of spin 1, spin 2 and the magnetic ﬁeld, respectively, measured from (001) down toward the (100)–(110)–(010) plane, and all other parameters are deﬁned as in Eq. 5.2.

I have written programs in Fortran 90, for both the 2D and 3D spin models, that determine the spin configuration corresponding to the lowest Etot value for a given input magnetic field vector (H~ ), and output the individual spin directions (~µ1 and ~µ2) of that configuration, the total magnetization (M~ = ~µ1 + ~µ2), and the torque felt by the sample (~τ = M~ H~ ). In all calculations presented here, parameter values of E = 1.8 10−23 J × afm × (similar to k T 1.5 10−23 J for CePb ), E = 1.5 10−24 J, and E = 1 10−29 J B N ≈ × 3 anis × deg × were used. Three separate implementations for finding the minimum Etot value were tried:

A brute force algorithm, in which all possible spin angles are sampled, with a • 0.1◦ resolution for the 2D spin model and 1◦ resolution for the 3D spin model,

within the allowed rotation range for each angle ( 180◦ θ , θ +180◦ and − ≤ 1 2 ≤ 0◦ φ ,φ +180◦). ≤ 1 2 ≤

A gradient-guided algorithm, in which a starting conﬁguration is assumed and the • gradient is followed from there down into a (local) minimum.

A many-dimensional downhill simplex algorithm, in which a starting conﬁguration • is assumed and a deformable, tetrahedral “simplex” structure oozes its way from

there down into a (local) minimum [68].

While the gradient-guided and simplex methods are more eﬃcient than the brute force method, they have two main drawbacks: (1) they often get stuck in local minima; and

(2) even if they find the global minimum, they do not necessarily find the bottom of the minimum, within a reasonable number of iterations, as accurately as does the brute force technique. Problem (1) is particularly bad for the parameter range of interest in my spin models, since spin flops and other spin transitions take place when the system jumps between two widely-spaced, but nearly-degenerate configurations; these algorithms Chapter 5. CePb3 139

Figure 5.11: 2D spin model total energy landscape as a function of spin 1 (θ1) and spin 2 (θ2) angles relative to the (100) direction, at (a) H = 0 T and (b) H = 4 T, ◦ −23 for θH = 44.9 (i.e. H near (110)). Parameter values used were Eafm = 1.8 10 J, E = 1.5 10−24J, and E = 1 10−29 J. × anis × deg ×

therefore have the poorest performance in the most important calculation regimes. Of course, this problem may be circumvented by picking several starting conﬁgurations and comparing the Etot values of the local minima found by each, but in order to do this without bias, for an a priori unknown energy landscape, one ends up back with a brute force method. Fig. 5.11 shows two example energy landscapes from the 2D spin model: in both cases there are several nearly-degenerate local minima (leading to problem (1)), which themselves are very shallow (exacerbating problem (2)). Because of these problems, the brute force method was deemed the most suitable for the spin model calculations, and was used to generate all of the results shown in this section.

With the field aligned along (110) in the 2D spin model, the spin-flop behaviour illustrated in Fig. 5.3 is realized: the magnitude of the total magnetization vector, M , | | increases from zero with increasing magnetic field, has a first order jump at the spin-flop phase transition, and then continues increasing until both spins are fully aligned with the Chapter 5. CePb3 140

field, at which point the magnetization saturates at a 100% value of 2 µ . With the × | | field aligned along (100), the spins start at (010) and (010), continuously rotate from − there to (110) and (1 1 0), and then jump to their final positions along (100), giving − rise to a magnetization that gradually increases with increasing magnetic field, and then jumps to its saturation value.

Of course, CePb3 is a cubic material, and the spins need not be constrained to a single plane. The total magnetization magnitude calculated in the 3D spin model for

fields along (110) and (100), shown in Fig. 5.12, is qualitatively the same as that calculated in the 2D model. However, when the field is along (110), the behaviour of the spins themselves differs markedly between the two models: for the 3D model, shown in

Fig. 5.13(b), the spins start along (001) and (001), gradually rotate from there toward − (110), and then jump to positions in the (100)–(110)–(010) plane like those shown in

Fig. 5.3(c); from there, the spins continuously rotate to (110), as shown in Figs. 5.3(d)–

(f). This allows M~ to be aligned along the field direction at all fields, unlike in the 2D model calculation illustrated in Figs. 5.3(a) and (b). For intermediate magnetic field angles, the 3D spin model predicts two magnetization jumps (Fig. 5.12), and spin-angle behaviour (Fig. 5.13(c)) combining attributes of both the H (110) and H (100) calcu- || || lations. For fields along (111), the spins trace out complicated paths in three dimensions

(Fig. 5.13(a)), but the field dependence of the total magnetization (Fig. 5.12) is nearly identical to the H (110) case. || Assuming, as do McDonough and Julian [60], that the H (110) jump in 3D spin || model magnetization (Fig. 5.12) is associated with the rapid decrease in experimentally- measured magnetoresistance near 5–6 T (Fig. 5.6), and the saturation of the magne- ∼ tization is associated with the magnetoresistance kink near 10–11 T, I explored the ∼ spin model parameter space in an attempt to get quantitative agreement between experimental and modelled feature field values. As Eanis is increased at constant Eafm, the magnetization jump gets larger, but stays at the same field, while the saturation Chapter 5. CePb3 141

100

40 θ φ ( H, H) = (45°, 55°) 5 (111) Magnetization (%) Magnetization 30 θ φ ( , ) = (45°, 90°) 5 (110) H H ( θ , φ ) = (22.5°, 90°) 20 H H ( θ , φ ) = (11.25°, 90°) H H θ φ 10 ( H, H) = (0°, 90°) 5 (100)

0 012345678910 Magnetic field (T)

Figure 5.12: 3D spin model magnetization vs. ﬁeld for H (111), H (110), H 22.5◦ from ◦ || || −23 (110) to (100), H 11.25 from (110) to (100), and H (100), using Eafm = 1.8 10 J, E = 1.5 10−24 J, and E = 1 10−29 J. Dotted|| lines are guides to the eye.× anis × deg × Chapter 5. CePb3 142

180 180 165 (a) 165 (b) θ φ 150 θ φ 150 1 1 1 1 θ φ 135 θ φ 135 2 2 2 2 θ φ θ φ M M 120 M M 120 θ φ θ φ H H 105 H H 105 90 90 75 75 60 60 45 45

Spinangles(°) 30 Spinangles(°) 30 15 15 0 0 -15 -15 H||(111) H||(110) -30 -30

012345678910 012345678910 Magnetic field (T) Magnetic field (T) 180 180 165 (c) 165 (d) θ φ 150 θ φ 150 1 1 1 1 θ φ 135 θ φ 135 2 2 2 2 θ φ θ φ M M 120 M M 120 θ φ θ φ H H 105 H H 105 90 90 75 75 60 60 45 45

Spinangles(°) 30 Spinangles(°) 30 15 15 0 H 22.5° from 0 -15 -15 (100) to (110) H||(100) -30 -30

012345678910 012345678910 Magnetic field (T) Magnetic field (T)

Figure 5.13: 3D spin model spin 1 (θ1,φ1), spin 2 (θ2,φ2), magnetization (θM ,φM ), and applied magnetic field (θH ,φH ) angles plotted as functions of the magnetic field strength, for (a) H (111), (b) H (110), (c) H 22.5◦ from (110) to (100), and (d) H (100), using E = 1||.8 10−23 J, ||E = 1.5 10−24 J, and E = 1 10−29 J. Isolated|| points afm × anis × deg × corresponding to nearly-degenerate configurations have been removed. Chapter 5. CePb3 143

field increases. As Eafm is increased at constant Eanis, the magnetization jump gets smaller, and both the jump and the saturation point move to higher fields. In fact, when either increasing E or decreasing E , both the H (111) and H (100) jumps are afm anis || || smoothed out and disappear before the H (110) jump, indicating in the model that (110) || is a magnetically “harder” direction than (111). The parameters E = 1.8 10−23J afm × and E = 1.5 10−24J are tuned to push the magnetization jump to the highest field anis × possible, while keeping the saturation point near the experimentally-measured value—in

this “best case,” the jump field, H 2.5 T, is still less than half that seen experimentally. ∼ The modelled, low-field, first-order transition decreases in size as the field is rotated from (110) toward (100), finally disappearing into a possible quantum critical endpoint at H (100). Unfortunately, this does not appear to be reflected in the experimental mag- || netoresistance data, which is not first order at any field angle or temperature. Moreover, as the field is experimentally rotated from (110) to (100), the measured low-field feature continuously broadens, behaving even less like a first order transition. Of course, the spin model includes neither dynamical classical effects or any quantum effects at all, so it is possible that fluctuations smooth out the experimental transition.

Beyond this, the measured H (111) magnetoresistance does not look like the mea- || sured H (110) magnetoresistance, in contrast to the magnetization predictions of the || model, and in fact would be better described by a magnetization curve similar to that

modelled for H (100), since it coincides with the H (110) curve at low ﬁelds, but has || || a single jump directly to saturation. The measured magnetoresistance for ﬁelds along

(100) shows no jump at all.

Looking at the torque, since in the 3D model M~ lies along H~ for all field magnitudes directed along (110) or (100) (Figs. 5.13(b) and (d)), the predicted torque is therefore zero for these field directions; this may explain the weak field dependence of the torque measured when H (110) or H (100) (Fig. 5.9(a)). For H (111), if the magnitude of || || || the spin model torque vector, τ , is reflected about the τ = 0 axis (Fig. 5.14), its field | | | | Chapter 5. CePb3 144

0.0

-0.1

-0.2

-0.3

-0.4 Negative |torque| (arb. units) (arb. Negative|torque| H||(111)

-0.5 012345678910 Magnetic field (T)

Figure 5.14: 3D spin model torque magnitude vs. ﬁeld curve for H (111), reﬂected about τ = 0, that roughly resembles the measured H (111) torque shown|| in Fig. 5.9. The |dotted| line is a guide to the eye, and the non-zero torque|| for H > 8.5 T is a computational artifact of the brute force algorithm 1◦ angle resolution.

dependence roughly resembles that of the experimental measurement: at low fields, it decreases with increasing field, then rises to a peak at “Hc,” then decreases, and finally increases again. However, as with the magnetization prediction / magnetoresistance measurements, the spin model torque peak comes from a first order jump, whereas that seen experimentally is not first order.

Finally, if the physics of CePb3 is described by the 2D spin-ﬂop model shown in Fig. 5.3, or the more generalized 3D spin model discussed above, magnetization curves such as those shown in Fig. 5.12 should give rise to very strong features in the susceptibility at the magnetization jumps (even if these are somewhat smoothed out by

ﬂuctuations). This is at odds with the weak bumps seen in previous ac susceptibility measurements [116], and lack of any features distinguishable above the (large) background in Chapter 5. CePb3 145

my own ac susceptibility results (Fig. 5.10). Indeed, the actual measured CePb3 magnetization shows no jumps, and continues monotonically increasing well above the H = 10 T

ﬁeld at which the predicted magnetization saturates [114].

While the 3D spin model captures some of the experimentally-observed behaviour, it

disagrees strongly with the measurements on numerous points. Moreover, the diﬀerence

in detailed spin angle behaviour between the 2D and 3D models shows that the origi-

nal formulation of the spin-ﬂop scenario, in which the sub-lattice magnetization vectors

for H (110) are conﬁned to the (110)–(100) plane (Fig. 5.3), is incorrect. Fundamen- || tally, neither spin model includes: (1) dynamical processes, since the individual terms

contributing to Eqs. 5.2 and 5.3 are simpliﬁed mean-ﬁeld interaction energies; (2) quan-

tum eﬀects, including the Kondo and RKKY interactions at the heart of heavy fermion

physics (section 1.3); (3) quadrupole-quadrupole interactions, which may be more impor-

tant in this material than spin-spin interactions [125]; and (4) the fact that at H = 0 T

CePb3 is not a simple antiferromagnet, but rather has a magnetic structure exhibiting incommensurate modulation of the moment amplitudes [109].

The moment amplitude modulation might, in fact, be caused by a “Kondo stripe”

phenomenon, arising from the interplay between spin-liquid and Kondo physics, and

resulting in an inhomogeneous Kondo screening modulation in real space [126, 127]. Thus

far, this concept has been formulated in detail only for the case of non-magnetic heavy

fermion systems, and it remains to be seen how it would be aﬀected by the commensurate

part of the CePb3 antiferromagnetic ordering vector, as well as applied magnetic ﬁelds. If the Kondo stripe properties are ﬁeld-dependent, a possible scenario in which the moments

dissolve into the screening cloud and then re-emerge pointing in a diﬀerent direction might

be realized, in contrast to the spin-ﬂop assumption of freely rotating moments with ﬁxed

amplitudes.

Although CePb3 has long been assumed to be the archetypal heavy fermion spin-ﬂop system, a comparison between my spin models and experimental data shows that the Chapter 5. CePb3 146

spin-ﬂop phenomenology may not be the correct basis for explaining the ﬁeld-dependent

behaviour of this material. The true microscopic mechanism underlying the physics of

CePb3 is not yet clear, and should be the subject of further experimental and theoretical investigations.

In particular, it would be interesting to extend the measurements presented in this chapter using our group’s dilution refrigerator / superconducting magnet facility, in order to: (1) carry out true temperature sweep measurements of the resistivity at ﬁxed magnetic

fields, allowing more precise investigation of the maximum Fermi liquid applicability temperature, T , and divergence of the T 2 coefficient, A, for H (111); and (2) repeat FL || the ac susceptibility measurement using the modulation coils that are fixed in place in

the cryostat (built into our main magnet assembly), in an eﬀort to decrease the H2

background and increase the sensitivity to weak features corresponding to the anomalies

seen in other properties. I hope to collaborate with Prof. Julian on these experiments in

the future.

More generally, since the indirectly-inferred magnetic structure of CePb3 at elevated magnetic ﬁelds is central to the interpretation of the ﬁeld-induced anomalies seen in the

magnetoresistance, ultrasound velocity, and diﬀerential susceptibility, a crucial experi-

ment that remains to be performed is the direct measurement of this magnetic structure

as a function of magnetic ﬁeld strength and angle by elastic neutron scattering. Other

spin-sensitive probes, such as muon spin relaxation or nuclear magnetic resonance, may

also be enlightening. Chapter 6

YbRh2Si2

My de Haas–van Alphen measurements of YbRh2Si2, described below, represent the culmination of my doctoral studies, since they were performed in our laboratory at the

University of Toronto using the DAVIES experimental control and data acquisition system, graphite rotation mechanism, and annealed silver wire of chapter 2, and analyzed using the SKEAF algorithm of chapter 3. I presented this work as a poster at the LT25 conference in Amsterdam and several talks across Europe in August 2008. The bulk of this chapter is drawn from two manuscripts I have written which summarize my experiment: an article focusing on the band structure calculations and high-ﬁeld angle dependence of the dHvA frequencies, which has been accepted for publication in the

6.1 Material background

YbRh2Si2 is a heavy fermion compound that has a ThCr2Si2-type tetragonal crystal structure (space group # 139: I4/mmm), shown in Fig. 6.1, with lattice constants

147 Chapter 6. YbRh2Si2 148

Figure 6.1: YbRh2Si2 crystal structure.

a = b = 4.010 A˚ and c = 9.841 A,˚ and fractional atomic coordinates Yb2a (0, 0, 0),

Rh4d (0, 0.5, 0.25), and Si4e (0, 0, 0.375) [128]. It has been the focus of much recent interest because it can be tuned through a ﬁeld-induced quantum critical point (QCP) which is both easily accessible to experiment and appears to exhibit a new class of “local” quantum criticality [129]. It has been argued that this scenario involves dramatic Fermi surface (FS) changes as a function of temperature and magnetic ﬁeld [10, 129, 130].

The central issue with regard to the proposed Fermi surface changes is whether or not the Yb 4f quasi-hole is included in the Fermi volume. In the “small” Fermi surface case Chapter 6. YbRh2Si2 149

Figure 6.2: “Small” and “large” Fermi surfaces calculated with the LDA + spin-orbit coupling method. Following convention from Ref. [131], the sheets are labelled donut “D,” jungle-gym “J,” and pillbox “P.” From an electron point of view, the dark blue side of each sheet is the occupied side and the light yellow side is the unoccupied side, such that D is a hole surface and P is an electron surface. Note that D is a torus in the small Fermi surface case only.

(also known as “4f localized” or “Yb3+”), it does not contribute, whereas in the “large”

Fermi surface case (also known as “4f itinerant”), it does. The main Fermi surface sheets corresponding to these two cases are shown in Fig. 6.2.

An ambient-pressure phase diagram of YbRh2Si2 summarizing the state of knowledge regarding this material in the condensed matter community prior to my study is shown in

Fig. 6.3. At zero field, T is associated with the Kondo scale, T 25 K [129]. If the Fermi 0 K ≈ energy is defined as the point where the Fermi-Dirac distribution is at half-occupancy, the Fermi surface can be discussed for T 0. At 14 K, the band structure corresponding ≥ to the small FS case is observed in angle-resolved photoemission spectroscopy (ARPES) experiments [131]. As the temperature is decreased, non-Fermi-liquid (nFL) behaviour Chapter 6. YbRh2Si2 150 is observed, for example in the linear temperature dependence of electrical resistivity and logarithmic temperature dependence of the specific-heat coefficient, down to the phase transition into a weak antiferromagnetic (AF) state at T 70 mK [9, 132]. As N ≈ the magnetic field is increased at low temperature, the weak magnetism is suppressed, leading to a quantum critical point at H 0.06 T when the field is applied in the c ≈ easy plane (H c) and H 0.66 T when the field is applied along the magnetic hard ⊥ c ≈ direction (H c) [8]. A region of Landau Fermi liquid (FL) behaviour is recovered when || 2 H >Hc, with field-dependent linear coefficient of specific heat and T coefficient of resistivity [8, 10, 130].

∗ The main focus of attention in YbRh2Si2 has been at low magnetic fields, where the T line shown in the Fig. 6.3 phase diagram corresponds to series of crossovers in the magnitude of the Hall coefficient, extrapolating to a discontinuous jump at the quantum critical point. This has been interpreted as signalling a sudden Fermi surface reconstruction from a small to a large FS [133]; such an interpretation is controversial, however, due to the sensitivity of the YbRh2Si2 Hall effect to small changes in f-occupancy [134, 135, 136] and/or changes in quasiparticle scattering as antiferromagnetic fluctuations give way to ferromagnetic fluctuations for H>Hc [137].

The additional association of crossovers in magnetization and ac susceptibility [10],

29 speciﬁc heat [9], Si-NMR Knight shift and nuclear spin-lattice relaxation rate 1/T1T [138], and ESR g-factor and line width [45] with the T ∗ line, has been interpreted as evidence for a second energy scale at the QCP [139]. The coincidence of the T ∗ 0 endpoint → with the quantum critical point, as well as the lack of FS reconstruction in a conventional spin-density wave (SDW) AF QCP scenario, has led to the proposal of new type of “local” quantum criticality involving the breakdown of Kondo screening at the QCP [129].

However, another mechanism for quantum critical Fermi surface reconstruction without

Kondo breakdown has also been proposed [140], and chemical substitution of cobalt atoms on some of the rhodium sites has shown that the T ∗ crossover line and quantum critical Chapter 6. YbRh2Si2 151

T small FS 0 10

small FS T*

1 nFL large FS

Temperature (K) 0.1 FL H AF 0

small FS large FS large FS ? 0.01 • 0.01QCP 0.1 1 10 In-plane magnetic field (T)

Figure 6.3: Ambient pressure phase diagram of YbRh2Si2 for H c. The quantum critical point (QCP) lies between the antiferromagnetic (AF) and Land⊥ au Fermi liquid (FL) regimes. Regions of “small” and “large” Fermi surface (FS) are labelled, with dashed arrows indicating supposed continuous Fermi surface crossovers. T0 is identified with the Kondo temperature, TK 25 K, and H0 10 T is the field at which Kondo fluctuations supposedly are suppressed.≈ The T ∗ line indicates≈ a crossover in the Hall effect and other quantities. Chapter 6. YbRh2Si2 152 point can be separated [141], raising doubts about the local criticality interpretation.

At higher fields, specifically at H c 10 T, crossovers of dc-magnetization, T 2 0⊥ ≈ coefficient of resistivity, specific heat, and linear magnetostriction coefficient [10] have

been interpreted as evidence for a second Fermi surface reconstruction, from the large FS

back to the small FS, as the Zeeman energy becomes comparable to the Kondo energy

scale kBTK . This has been suggested to be a continuous change [10] or a transition [130] of the total Fermi volume. An alternate interpretation of the experimental crossovers

at H0, discussed in more detail in section 6.5, involves a Lifshitz transition where one spin-split branch of the Fermi surface grows to encompass the entire Brillouin zone and

disappears, while the total Fermi volume is conserved by compensating changes in the

other Fermi surface sheets [142].

The primary goal of my study was to look for these proposed Fermi surface changes

near H0. Because dHvA is limited to high ﬁelds, I could not directly access the low-

ﬁeld QCP, but the Fermi surface behaviour near H0 sheds light on the electronic struc-

ture between Hc and H0, and therefore provides experimental constraints for theories of

YbRh2Si2 quantum criticality.

6.2 Electronic structure calculations

Since Lu and Yb diﬀer by exactly one 4f hole, an all-itinerant calculation of isostructural

LuRh2Si2 yields a Fermi surface nearly identical to the YbRh2Si2 small FS (that is, without the 4f quasi-hole contributing to the Fermi volume) [131], whereas an all-itinerant

calculation of YbRh2Si2 results in the large FS. I have performed these calculations using the WIEN2k density functional theory code, which is based on the all-electron full-

potential augmented plane-wave + local orbitals method [30]. Beginning with the crystal

structure shown in Fig. 6.1 (with Lu in the Yb position for the small FS case), I obtained

band energies on a 20 000 k-point Reciprocal Unit Cell (RUC) mesh (section 3.2.1), Chapter 6. YbRh2Si2 153 using the Perdew-Burke-Ernzerhof generalized gradient approximation to the exchange- correlation potential Vˆ [29], RminK = 8, and a “valence state” energy range of 7.1 xc MT max − to 5 Ry. The Fermi surface sheets derived from these calculated band energies are shown in Fig. 6.2.

Following the previous density functional theory work of Tae Jeong on YbRh2Si2 [143], for the large FS case I performed three separate calculations: a basic self-consistent band structure calculation (“LDA”); a calculation with fully-relativistic spin-orbit coupling included at the one-electron level (“LDA+SOC”); and a calculation including both spin-orbit coupling and additional correlation eﬀects (“LDA+U”). The resulting band

structure and total density of states, plotted as functions of energy for the LDA+SOC

(Fig. 6.4) and LDA calculations, are identical to those obtained by Jeong. My LDA+U calculation disagrees with those of Jeong, but even Jeong’s YbRh2Si2 LDA+U calculations disagree with one another [143, 144]. Since the LDA+SOC calculation best captures the experimentally measured spin-orbit splitting and the positions of the 4f derived single-particle-like excitations [143], this was the band structure that I used for the large

FS case in all further work on YbRh2Si2.

For the small FS case, Gerald Wigger and coworkers used the WIEN2k code [30] to perform a calculation including both spin-orbit coupling and additional correlation effects of YbRh2Si2, in which the Yb 4f hole was somehow forcibly localized [131]. Although such selective localization is beyond my capabilities, my calculated LDA+SOC LuRh2Si2 Fermi surfaces (Fig. 6.2) appear to be identical to those of Wigger et al. (my final calculations did not include an intra-atomic repulsion U, since this had a negligible effect on the

Fermi surfaces). In general, the large-scale features of both my small and large Fermi surfaces are in good agreement with those published previously [131, 134, 143, 145], although there are diﬀerences of detail between my calculations and the literature, and between the various calculations within the literature, which do not aﬀect the data analysis of this chapter. Chapter 6. YbRh2Si2 154

Figure 6.4: YbRh2Si2 LDA+SOC band structure and total density of states versus energy, for the large Fermi surface case. The major Brillouin zone symmetry points Γ, X, P, U, and Z are labelled in Fig. 6.2 Chapter 6. YbRh2Si2 155

In my calculations, under a rigid Fermi energy shift of approximately 30 mRy, the − small FS looks very similar to the large FS. Fermi surfaces for large, small, and two intermediate cases (the small FS with E shifts of 5 mRy and 20 mRy) illustrating F − − this transformation are shown in Fig. 6.5, while predicted dHvA frequencies extracted by the SKEAF algorithm (chapter 3) and plotted versus magnetic field angle are shown in Figs. 6.6–6.9. Similarly, under a shift of approximately +20 mRy (not shown), the calculated large FS looks like the small FS, raising the possibility that under sufficiently strong spin-splitting of the bands at high magnetic fields one of the spin-branches of the large FS could look like the small FS without requiring a localization transition.

Moreover, a Fermi energy shift of approximately 50 mRy (not shown) transforms the − donut “D” sheet of the large FS case into a sheet topologically similar to the large FS jungle-gym “J” sheet, revealing a band shape resemblance between the D and J bands.

Note that the fact that the energy of these shifts is very large is not a concern: these are the shifts given by LDA, which greatly overestimates the dispersion of the bands, typically by factors of several hundred in heavy fermion materials. A more sophisticated discussion of the possible eﬀects of spin-splitting on the actual renormalized quasiparticle bands of YbRh2Si2 will be discussed below, in section 6.5.

6.3 Experimental details

In order to probe the Fermi surface of YbRh2Si2 as the ﬁeld is increased through H0, I have conducted ambient-pressure de Haas–van Alphen measurements from 8–16 T and

30–600 mK, using the standard ﬁeld modulation technique described in section 1.5.3, extended through the use of the DAVIES virtual lock-ins described in section 2.1.7. The single-crystal samples studied were grown from indium ﬂux by Gerard Lapertot, Georg

Knebel and Jacques Flouquet in Grenoble [145]. These samples, from the “LAP-420” batch shown in Fig. 6.10, are thin platelets that are very brittle and therefore must be Chapter 6. YbRh2Si2 156

Figure 6.5: The large, small and two intermediate Fermi surfaces of YbRh2Si2 from the LDA+SOC calculation. Chapter 6. YbRh2Si2 157

Lu ∆E =0mRy F 4 PredicteddHvA frequency(kT) 2

0 90 75 60 45 30 15 0 30 150 15 30 45 60 75 90 (001) (100) (110) (001) Magnetic field angle (degrees)

Figure 6.6: Predicted YbRh2Si2 dHvA frequencies extracted from the small Fermi surface sheets (Fig. 6.5) by the SKEAF algorithm and plotted versus magnetic ﬁeld angle. The frequencies were obtained from the small FS calculation using the as-calculated Fermi energy. Blue squares come from the D band, magenta diamonds come from the J band, and cyan crosses come from the P band. Chapter 6. YbRh2Si2 158

Lu ∆E =-5mRy F 4 PredicteddHvA frequency(kT) 2

0 90 75 60 45 30 15 0 30 150 15 30 45 60 75 90 (001) (100) (110) (001) Magnetic field angle (degrees)

Figure 6.7: Predicted YbRh2Si2 dHvA frequencies extracted from a set of intermediate Fermi surface sheets (Fig. 6.5) by the SKEAF algorithm and plotted versus magnetic ﬁeld angle. The frequencies were obtained from the small FS calculation with a 5 mRy Fermi energy shift. Blue squares come from the D band, magenta diamonds come− from the J band, and cyan crosses come from the P band. Chapter 6. YbRh2Si2 159

Lu ∆E =-20mRy F 4 PredicteddHvA frequency(kT) 2

0 90 75 60 45 30 15 0 30 150 15 30 45 60 75 90 (001) (100) (110) (001) Magnetic field angle (degrees)

Figure 6.8: Predicted YbRh2Si2 dHvA frequencies extracted from another set of intermediate Fermi surface sheets (Fig. 6.5) by the SKEAF algorithm and plotted versus magnetic ﬁeld angle. The frequencies were obtained from the small FS calculation with a 20 mRy Fermi energy shift. Blue squares come from the D band and magenta diamonds come− from the J band. Chapter 6. YbRh2Si2 160

Yb ∆E =0mRy F 4 PredicteddHvA frequency(kT) 2

0 90 75 60 45 30 15 0 30 150 15 30 45 60 75 90 (001) (100) (110) (001) Magnetic field angle (degrees)

Figure 6.9: Predicted YbRh2Si2 dHvA frequencies extracted from the large Fermi surface sheets (Fig. 6.5) by the SKEAF algorithm and plotted versus magnetic ﬁeld angle. The frequencies were obtained from the large FS calculation using the as-calculated Fermi energy. Blue squares come from the D band and magenta diamonds come from the J band. Chapter 6. YbRh2Si2 161

Figure 6.10: The LAP-420 batch of YbRh2Si2 single crystal samples.

handled with care.

Because quantum oscillations are strongly damped by the presence of impurities

(section 1.5.2), it is important to conﬁrm high sample quality prior to attempting a

dHvA experiment. To this end, I performed a standard 4-wire resistance measurement

on sample #12 from Fig. 6.10 in order to determine the Residual Resistivity Ratio

(RRR ρ(T = 300 K)/ρ(T 0 K)) for my batch of samples. A large RRR value ≡ → indicates that scattering due to impurities and other crystalline defects (dominant at

low temperatures) is much smaller than the scattering due to thermal ﬂuctuations (dom-

inant at high temperatures; it is conventional to pick the value at room temperature), and

therefore that the samples are very pure single crystals. As shown in Fig. 6.11(a), 25µm diameter gold current (outer) and voltage (inner) leads were attached to the sample using quick-drying room temperature conducting epoxy; the current direction is perpendicular to the c-axis. Since my research group’s dilution refrigerator was out of commission at the time, these resistance measurements were performed in the dilution refrigerator of

Professor John Wei, at room temperature (300 K) and from 6 K down to 30 mK, in zero magnetic ﬁeld. The low-temperature resistivity vs. temperature data, normalized to the value at T = 300 K and shown in Fig. 6.11, is best ﬁt by an equation of the form

ρ(T )/ρ(T = 300 K) = AT 2 + BT + C, where A = 4.5(3) 10−4, B = 1.27(2) 10−2, × × and C = 1.00(1) 10−2. The strong T -linear term (B is two orders of magnitude larger ×

than A) conﬁrms the non-Fermi liquid character of YbRh2Si2, a result ﬁrst obtained by Chapter 6. YbRh2Si2 162

0.12 (b) 0.10

0.08

0.06 = 300= K) T ( ρ

) / 0.04 T ( ρ

0.02 Experimental data A T2 + B T + C fit

0.00 0 1 2 3 4 5 6 Temperature (K)

Figure 6.11: (a) YbRh2Si2 resistivity sample (#12 from Fig. 6.10) with leads attached using conducting epoxy. (b) Resistivity versus temperature, normalized to the value at T = 300 K, measured experimentally for the sample shown in panel (a) (circles) and ﬁtted to the equation ρ(T )/ρ(T = 300 K) = AT 2 + BT + C (solid line) over the temperature range 30 mK–6 K.

Trovarelli et al. [9]. The RRR = 1/C = 100(1), with a residual resistivity ρ0 of the order of 1 µΩcm, indicates very high sample quality, suitable for a dHvA experiment.

Thin samples, such as those shown in Fig. 6.10, are not ideal for dHvA measurements because they have a small volume and additionally are difficult to wrap tightly in pick-up coils, leading to small V and filling factor η, and therefore poor signal strength above the noise floor via Eq. 1.31. Thus, for my dHvA experiment, I picked the two thickest samples from the LAP-420 YbRh2Si2 batch: “sample A” is sample #11 in Fig. 6.10, measures 4.5(1) 2.2(1) 0.32(3) mm, and was used for field rotations in the (100)– × × (110) plane; “sample C” is a piece cleaved from sample #24 in Fig. 6.10, measures approximately 2.5(1) 1.7(1) 0.20(3) mm, and was used for field rotations in the × × (110)–(001) plane. Comparison of experimental Laue backscattering x-ray diffraction patterns (Fig. 6.12(a) and (c)) with the predicted pattern for a generic tetragonal crystal

(Fig. 6.12(b)), generated with the LaueX computer program [146], conﬁrmed that the

YbRh2Si2 platelet planes grow perpendicular to the (001) direction, and that the long Chapter 6. YbRh2Si2 163

Figure 6.12: Laue x-ray diffraction patterns for YbRh2Si2 dHvA sample A (panel (a)) and sample C (panel (c)), shown alongside the predicted diffraction pattern for a generic tetragonal crystal with the x-ray beam along the crystalline (001) direction and oriented so that the (110) direction is pointing up, generated using the LaueX computer program (panel (b)) [146]. Experimental diffraction spots in panels (a) and (c) are marked with red dots as guides to the eye.

axes of both samples are aligned along the (110) direction.

Counter-wound pairs of pick-up coils were custom wrapped to the dimensions of each sample, using 12.5 µm diameter copper wire, with the coil axes aligned along the samples’

(110) directions. The coils for sample A are each composed of 1001 turns of wire, while

the coils for sample C are each composed of 1000 turns of wire. The samples were placed

in the top coil of each pair, then the pairs were mounted in graphite rotation bobbins

using 5 minute epoxy, ﬁshing line, and pieces cut from computer punch cards (Fig. 6.13).

Due to the small size of my samples, optimizing parameters to maximize measurement

sensitivity was very important in the dHvA experiment. These measurements were per-

formed in our group’s Oxford Instruments Kelvinox 400MX dilution refrigerator (after

the second repair of the manufacturing defect), using a vibration-isolated cryostat and

16/18 T superconducting magnet with split modulation coils. All experimental control

and data collection activities were coordinated through the DAVIES computer system

(section 2.1). My study was conducted in magnetic ﬁelds from 16 T to 8 T (where the Chapter 6. YbRh2Si2 164

Figure 6.13: Counter-wound pick-up coil pairs for YbRh2Si2 dHvA sample A (panel (a)) and sample C (panel (b)), mounted in their respective graphite bobbins.

oscillations were lost into the noise ﬂoor), and temperatures from 30 mK (the lowest

“trustworthy” thermometer calibration temperature) to 600 mK (where the oscillations were lost into the noise floor). The field was swept in constant rate mode with “adaptive magnet R and L” engaged (section 2.1.5), at a typical rate of 35 mT/min for the high-field fine rotation measurements and 15 mT/min for the effective mass and lower-

ﬁeld measurements. The modulation ﬁeld frequency was ω = 4 Hz, and the modulation

ﬁeld amplitude h0 was chosen for each 2 T ﬁeld-sweep range such that the parameter λ 2πFh /H2 3–4 for F = 6 kT and 1.5–2 for F = 3 kT, ensuring a strong J (λ) ≡ 0 0 ∼ ∼ ν=2 response (Fig. 1.7) in the dHvA frequency range covering most observed oscillations.

A schematic diagram of my experimental apparatus is shown in Fig. 6.14. The bobbins containing my YbRh2Si2 samples A and C, as well as a dHvA sample of the heavy fermion compound CeRu2Si2 previously measured by Ramzy Daou [147] (for diagnostic purposes), were mounted in the graphite rotation mechanism I designed (section 2.2), which was centred in the homogeneous field region of the superconducting magnet. Sil- ver heat-sink wires, annealed using the set-up I developed (section 2.3), were soldered to each sample and attached to the dilution refrigerator mixing chamber, thus provid- Chapter 6. YbRh2Si2 165 ing a thermal short that ensured good thermalization between the samples and mixing chamber (along with its associated thermometry). Although both pick-up coils in each counter-wound pair had exactly the same number of turns, factors such as the presence of a sample in one coil of the pair but not the other, the rotation angle relative to the magnetic field, and the strength of the magnetic field changed the balance between the coils; this balance was restored for a given field range and angle by tuning room temperature compensating potentiometers connected across one coil of each pair. For the main oscillatory voltage measurement, leads connected across each combined counter- wound pick-up coil pair were attached to a low-temperature transformer cold-anchored to the dilution refrigerator 1K pot: transformer “LT1” (turns ratio 384) for YbRh2Si2 sample A, “LT2” (turns ratio 169) for YbRh2Si2 sample C, and “LT3” (turns ratio 280) for the CeRu2Si2 sample. The signals were transferred via shielded cables from the low temperature transformers into a room-temperature shielded box, where 0.1 µF tuning ca- pacitors removed unwanted high-frequency noise. The YbRh2Si2 signals then underwent additional amplification using Signal Recovery model #5184 low-noise fixed 1000 gain × pre-amplifiers, followed by ν = 1 fundamental modulation frequency elimination using

Twin-T notch filters tuned to 4 Hz, to give the best dynamic range for virtual lock-in detection on harmonics ν 2. The last of the three available Twin-T filters was used to ≥ reduce 60 Hz electrical mains noise in the YbRh2Si2 C signal. Finally, the signals from all three samples were amplified by EG&G PARC model #113 pre-amplifiers, with low- and high-frequency roll-offs respectively set to 3 and 10 Hz, and sent to the DAVIES virtual lock-ins (section 2.1.7). The #113 pre-amp gains were 500 for the YbRh Si A × 2 2 signal, 1000 for the YbRh Si C signal, and 50 for the CeRu Si signal, giving rise to × 2 2 × 2 2 total amplification factors of roughly 2 108, 2 108, and 1 104, respectively for each × × × pick-up-coil–DAVIES-input signal-chain.

One data point was recorded each second, matched by a one second virtual lock-in

(section 2.1.7) Finite Impulse Response (FIR) ﬁlter time constant, with 200 decibels Chapter 6. YbRh2Si2 166

Figure 6.14: Schematic diagram of the YbRh2Si2 de Haas–van Alphen experimental apparatus. Alternate signal paths (for measuring direct coil voltages) are shown in light orange, magenta and green. Chapter 6. YbRh2Si2 167 per decade ﬁlter roll-oﬀ. Detection proceeded simultaneously on typical modulation frequency harmonics ν = 1, 2, 3, 4, and 6 for all three samples, but the data and analysis presented in the remainder of this chapter focus exclusively on ν = 2 for the dHvA measurements and ν = 1 for the direct coil voltage measurements.

Since the gearing connecting the room temperature rotation drive to the low temperature graphite rotation mechanism suﬀered from slippage and backlash, it was important to independently determine the angles of the pick-up coils relative to the magnetic ﬁeld.

To this end, I measured the direct ν = 1 pick-up coil voltage across a single, uncompen- sated coil from each bobbin (using the alternate light orange, magenta and green signal paths shown in Fig. 6.14) as all three bobbins were rotated through the entire angular range allowed by the rotation mechanism, and extracted the real coil angles relative to the magnetic ﬁeld direction by ﬁtting the measured voltages to cos θ curves. Measured coil angles for YbRh2Si2 samples A (θY bA,coil)andC(θY bC,coil) are plotted in Fig. 6.15 as functions of the measured coil angle for the CeRu2Si2 sample (θCe,coil). For all samples,

θcoil = 0 is the angle where each coil is aligned along the field direction, corresponding to H (110) for both YbRh Si samples and near H (001) for the CeRu Si sample. || 2 2 || 2 2 ◦ YbRh2Si2 sample A covers a field rotation plane 45 from (110) to (100), and sample C covers a plane 60◦ from (110) toward (001), with portions of both planes covered more ∼ than once over the entire rotation mechanism range. Best linear fits to the data shown

in Fig. 6.15, with angles expressed in units of degrees, are

θ = 1.23(4) θ 25(1) (6.1) Y bA,coil × Ce,coil −

θ = 1.27(3) θ 48(1) (6.2) Y bC,coil × Ce,coil −

During the high-ﬁeld dHvA rotation study, the CeRu2Si2 direct voltage signal path was

left in place and used to determine the real angles of the YbRh2Si2 A and C samples via Eqs. 6.1 and 6.2. By careful comparison of the CeRu2Si2 direct coil voltages and Chapter 6. YbRh2Si2 168

60 dHvA direct coil pick-up

) coil rotation alignment ° 40 ) and ) YbA,coil

θ 20 "A" ( "A" 2 0 Si 2

-20

) relative to magnetic field ) ( -40

YbRh 2Si 2 sample "A"

YbC,coil YbRh Si sample "C"

θ 2 2 -60 "C" ( "C" Coilforangles YbRh -10 0 10 20 30 40 50 60 70 Coil angle for CeRu Si ( θ ) relative to magnetic field (°) 2 2 Ce,coil

Figure 6.15: YbRh2Si2 coil angles plotted versus CeRu2Si2 coil angle, measured relative to the magnetic field direction. Data points are derived from direct coil voltage measurements of all three samples, while solid lines are the best fits to this data (Eqs. 6.1 and 6.2). Zero degrees represents the angle where each coil is aligned along the direction of the magnetic field, corresponding to H (110) for both YbRh2Si2 samples and near H (001) for the CeRu Si sample. || || 2 2

metamagnetic transition ﬁeld (HMMT = (7.7/ cos θ) T [147]) across the rotation range, I

was able to determine that the CeRu2Si2 sample is misaligned inside its pick-up coils by 6.1◦. ∼

6.4 Experimental results

Quantum oscillations with characteristic 1/H periodicity were observed in both the

(100)–(110) and (110)–(001) rotation planes. Measured oscillatory magnetization and

the corresponding Fast Fourier Transform (FFT), with the ﬁeld aligned 5◦ from (110) ∼ in the (110)–(001) plane, is shown in Fig. 6.16. At this particular angle, two frequencies

are superimposed: a very strong one at 6.1 kT, labelled d11, and a weaker one at ∼ 3.3 kT, labelled d12. ∼ Chapter 6. YbRh2Si2 169

7 1.8 1.6 d11 YbRh Si 6 1.4 2 2 1.2 quantum 5 1.0 oscillations 0.8 4 0.6 0.4

Amplitude(arb.units) 0.2 d12 3 0.0 0 1 2 3 4 5 6 7 8 910 dHvA frequency (kT) 2

Magnetization(arb. units) 0

-1

14.0 14.5 15.0 15.5 16.0 Magnetic field (T)

Figure 6.16: YbRh2Si2 quantum oscillations measured via the de Haas–van Alphen effect in a 14–16 T magnetic field range, at base temperature, with the field rotated 5◦ from (110) toward (001). A Fast Fourier Transform (FFT) of this data in 1/H is inset,∼ with the frequencies corresponding to orbits d11 and d12 labelled. Chapter 6. YbRh2Si2 170

Fig. 6.17 shows the full ﬁeld-angle dependence of my measured quantum oscillation frequencies in the 14–16 T ﬁeld range, below 50 mK. For comparison, previously- published data from torque measurements on samples grown in the same facility [145], as well as predicted dHvA frequencies from the calculated small (Fig. 6.17(a)) and large

(Fig. 6.17(b)) Fermi surfaces of Fig. 6.2, are included. Predicted dHvA frequencies, band masses and band speciﬁc heat contributions were extracted from each calculated Fermi surface sheet using the SKEAF algorithm (chapter 3). With the exception of the extra frequency near 6 kT beyond 15◦ in the (110)–(001) plane, and persistence of observed ∼ frequencies over a larger angular range in this plane, there is reasonable qualitative agreement between the small Fermi surface calculation and experiment. Quantitative agreement is generally not obtained for heavy fermion compounds since the ﬂatness of the bands makes the FS size depend sensitively on small shifts of the Fermi energy.

The qualitative agreement between the large Fermi surface calculation and experiment is signiﬁcantly poorer, since the calculation misses numerous branches in the (100)–(110) plane.

To highlight the correspondence between experiment and theory, matching pairs of orbits have been labelled in Fig. 6.17(a) with the same letter/number combination— uppercase letters for calculated orbits and lowercase letters for experimental data. Pre- dicted major orbits on the small J surface are at higher or lower frequencies and are not shown in Fig. 6.17(a), as I was unable to observe them: all of my experimentally- detected frequencies correspond to hole orbits on the D sheet. Calculated band masses, measured cyclotron eﬀective masses and corresponding mass enhancements for measured orbits and their theoretical counterparts are listed in Table 6.1; the average mass enhancement is 10(1). Within the resolution of my experiment, no spin-splitting of the dHvA frequencies is observed, and the temperature dependence of all measured oscillations is well-described by the standard Lifshitz-Kosevich relation (Eqs. 1.10 and 1.11).

An example Lifshitz-Kosevich ﬁt to the temperature dependence of the dHvA amplitude Chapter 6. YbRh2Si2 171

8 (a) e h d3 d2 7 e h d1 d11 u1 6 d4 d5 u2 D2 h h 5 h D11 D3 h D1

D4 LDA+SOC: small FS 4 h D5 h d10 d6 h d12 u3 3 dHvA frequency (kT) d9 d7 h D12 h d8 D6 D10 h h h 2 hD9 This study (14-16 T) h D7 D8 Knebel et al. (12-28 T) 1 40 30 20 100 10 20 30 40 50 60 (100) (110) → (001)

Magnetic field angle (degrees)

8 h h (b) h

7 e

h h 6

LDA+SOC: large FS 4

3 dHvA frequency (kT) e e 2 This study (14-16 T) Knebel et al. (12-28 T) 1 40 30 20 100 10 20 30 40 50 60 (100) (110) → (001)

Magnetic field angle (degrees)

Figure 6.17: dHvA frequency vs. magnetic field angle. Closed blue circles depict the current experimental data (14–16 T field range), open squares show previously-published data (12–28 T field range) [145], and red lines are from the calculated small (a) and large (b) Fermi surfaces shown in Fig. 6.2. In panel (a), calculated orbits are labelled with upper-case red letters and numbers, whereas corresponding experimental orbits are labelled with lower-case blue letters and numbers. In both panels, hole-like calculated orbits are indicated with an h, whereas electron-like calculated orbits are indicated with an e. Chapter 6. YbRh2Si2 172

Table 6.1: Calculated band masses (mb) from the small Fermi surface of Fig. 6.2, measured effective masses (m∗) from the dHvA experiment (14–16 T field range), and re- ∗ sulting mass enhancements (m /mb), for calculated and measured orbits labelled in Fig. 6.17(a). Note that each mass was determined at the magnetic field angle where its oscillation was strongest, and therefore not every mass was measured at the same angle.

∗ ∗ LDA+SOC Expt. mb (me) m (me) m /mb orbit orbit D1 d1 1.00(6) 10.1(5) 10.1(8) D2 d2 0.82(2) 8.6(5) 10.5(7) D5 d5 0.71(3) 8.6(5) 12.1(9) D6 d6 0.62(5) 6.8(3) 11(1) D7 d7 0.72(1) 10(1) 14(1) D8 d8 0.554(4) 5.00(5) 9.0(1) D11 d11 1.69(9) 9.02(9) 5.3(3) — u1 — 9.2(1) — — u2 — 9.33(7) — — u3 — 12.7(5) — average mass enhancement: 10(1)

of the u2 orbit labelled in Fig. 6.17(a) is shown in Fig. 6.18.

As I did with CeCoIn5 (section 3.4), I have estimated the high-field YbRh2Si2 renormalized electronic specific heat (γ∗ in Table 6.2) by multiplying the band specific heat

contributions (γb in Table 6.2) by the average mass enhancement factor (10(1), from

Table 6.1). Electronic density of states contributions, DOSavg = (DOS+ + DOS−)/2, from each of my calculated Fermi-energy-crossing bands in the large and small FS cases

were determined by the method described in Section 3.2.7, and are listed in Table 6.2.

The band speciﬁc heat contributions were obtained by converting DOSavg from units of A˚−3 Ry−1 to mJ K−2 per mole of ytterbium atoms. Adding together the enhanced

speciﬁc heats for each band (twice: once for each spin direction) gives the total renor-

∗ malized electronic speciﬁc heat estimate for the small FS case, γsmall = 130(10) mJ

−1 −2 ∗ −1 −2 mol K , and large FS case, γlarge = 86(8) mJ mol K . The total renormalized electronic speciﬁc heat estimate for the quasiparticle bands of the “modiﬁed Kusminskiy Chapter 6. YbRh2Si2 173

1.2

1.1 u2 orbit, 14-16 T field range 1.0 LK fit, with m* = 9.33(7)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

dHvA amplitudeunits) (arb. 0.2

0.1

0.0 0 100 200 300 400 500 600 Temperature (mK)

Figure 6.18: Lifshitz-Kosevich fit (line) to the temperature dependence of the YbRh2Si2 u2 orbit dHvA amplitude (points). Two 14–16 T field sweeps were performed at each temperature, with the spread between isothermal amplitude points giving an estimate of the measurement error. The u2 orbit effective mass was determined in the best fit to be ∗ m = 9.33(7)me. Chapter 6. YbRh2Si2 174

Table 6.2: YbRh2Si2 density of states contributions (DOS+, DOS−, and DOSavg) and associated band speciﬁc heats (γb) extracted by SKEAF, and enhanced speciﬁc heat contributions for each band (γ∗), assuming the average 10(1) small-D mass enhancement from Table 3.2 applies to all bands.

∗ Band DOS+ DOS− DOSavg γb γ (A˚−3 Ry−1) (mJmol−1 K−2) Small D 11.9 12.0 11.95(5) 1.321(6) 13(1) Small J 45.2 45.6 45.4(2) 5.02(2) 50(5) Small P 1.6 1.7 1.65(5) 0.182(6) 1.8(2) Large D 14.2 14.1 14.15(5) 1.564(6) 16(2) Large J 24.5 23.7 24.1(4) 2.66(4) 27(3)

model” (section 6.5), composed of one spin-surface resembling the small D sheet and two

∗ −1 −2 spin-surfaces resembling the small J sheet, is γmodK = 110(10) mJ mol K . These

∗ estimates, especially γmodK , are very close to the experimental speciﬁc heat measure-

−1 ment in this material at 11.5 T (the highest ﬁeld measured), γexpt = 110(10) mJ mol K−2 [10], although strong conclusions should not be drawn from this comparison since

the unobserved J surface contributes most strongly to the estimated totals.

Since my aim was to investigate possible Fermi surface changes across H0, it is important that I was able to follow the strongest oscillations (d5, d6, d8 and d11 from

Fig. 6.17(a)) from 16 T (well above H0) to 8 T (well below H0). Fig. 6.19 shows the measured frequency of these oscillations as a function of ﬁeld strength, normalized to the

value at 15.5 T. Remarkably, all of the four D-surface orbits that I observe exhibit the

same behaviour: roughly constant above about 11 T, rising sharply as the ﬁeld is reduced

through H , and perhaps tending to saturate below 9 T. Fig. 6.20 shows the measured 0 ∼ eﬀective mass of these orbits as a function of ﬁeld strength, normalized to the value at

15.5 T. Due to low signal-to-noise ratios in the low-ﬁeld, high-temperature regime, the

eﬀective masses suﬀer from larger error bars and scatter than the frequencies (which come

from only the lowest-temperature data), but nevertheless appear to be roughly constant

above about 11 T, and rising as the ﬁeld is reduced through H0. Chapter 6. YbRh2Si2 175

1.16 (i)

1.14 f t,A 1.12 True frequency True ft,B H H H 1.10 A 0 B f (ii) 1.08 m,A

1.06 f m,B

1.04 frequency Measured 0 H H H d5 A 0 B 1.02 d6 Magnetic field d8

Normalized dHvA frequency d11 1.00

8 9 10 11 12 13 14 15 16 Magnetic field (T)

Figure 6.19: Measured dHvA frequencies as a function of the magnetic field strength, normalized to each orbit’s frequency at 15.5 T. Orbit labels correspond to those in Fig. 6.17(a). The high-field frequencies for each orbit are Fd5 = 5.35 kT, Fd6 = 3.45 kT, Fd8 = 2.56 kT, and Fd11 = 6.06 kT; error bars are smaller than the size of each data point. The inset schematically shows the correspondence between a true Fermi volume change (i) and that observed via back-projection in a dHvA experiment (ii). The frequencies I follow lowest in field show the S-shape of inset (ii). Chapter 6. YbRh2Si2 176

1.4

1.3

1.2

1.1

1.0

0.9 d5 d6 0.8 d8 Normalized effective mass d11 0.7 8 9 10 11 12 13 14 15 16 Magnetic field (T)

Figure 6.20: Measured effective masses as a function of the magnetic field strength, normalized to each orbit’s effective mass at 15.5 T. Orbit labels correspond to those in ∗ ∗ Fig. 6.17(a). The high-field effective masses for each orbit are md5 = 9.2(3)me, md6 = ∗ ∗ 6.8(2)me, md8 = 4.98(8)me, and md11 = 8.82(9)me; these differ from the effective masses listed in Table 6.1, because these masses come from Fourier transforms over a 15–16 T field range, whereas those in the table use a 14–16 T range. Chapter 6. YbRh2Si2 177

As mentioned in section 1.5.2, interpretation of the ﬁeld dependence of a measured dHvA frequency, fm, requires some care, because fm is not directly related to the ‘true’ frequency, ft, but rather it is the back-projection to H = 0 of the line tangent to ft [46]. This is illustrated in Fig. 1.5 and in the inset of Fig. 6.19. For example, at a particular

field H , f = f H (∂f /∂H) . Observed field dependence of f usually reflects A m,A t,A − A t A m changes of ∂ft/∂H rather than of ft itself. In YbRh2Si2, the magnetization roughly saturates above H [130], so the slope ∂f /∂H should be lower for H>H . Thus the 0 | t | 0 behaviour must be roughly as shown in the inset of Fig. 6.19, with a gradual change in fm corresponding to a gradual change in ft and therefore the Fermi surface. I conclude that I am observing a continuous shrinking of the measured D-surface at a rate that slows

above H0.

6.5 Discussion

It is tempting to infer that the shrinking D surface means that the Yb 4f quasi-hole is

gradually disappearing from the Fermi volume, and that H0 represents the completion of this process [10]. However, the traditional Luttinger’s theorem-based view is that at

T = 0 K continuous changes in total Fermi volume should not occur. If the Yb 4f quasi-hole were to localize across H0, a discontinuous jump in dHvA frequencies would be observed, as in CeRhIn5 under pressure [148]. Since the observed frequencies change continuously across H0, 4f localization must not occur at this ﬁeld.

An alternative scenario for the behaviour at H0 has been developed by Silvia Viola Kusminskiy and coworkers [142]. They consider the spin-splitting of the Fermi surface at high fields, and propose that H0 represents a Lifshitz transition, where a heavy, majority- spin branch of the large Fermi surface disappears, leaving only a moderate-effective- mass minority-spin branch of the large Fermi surface. The 4f-localization transition is predicted to occur at much higher fields. This idea was initially explored through static Chapter 6. YbRh2Si2 178 mean field calculations [142], and has since been theoretically confirmed using dynamical mean field theory [149].

The Kusminskiy scenario was developed for a toy, one-band model, but Fig. 6.21 shows how it might work for the actual band structure of YbRh2Si2, using a simpliﬁed sketch of the D and J energy bands along the Z–U direction in the Brillouin zone. The bands giving rise to the small Fermi surface (solid grey lines in Fig. 6.21(a)) hybridize with a virtual f-level near the Fermi energy, EF . The resulting many-body quasiparticle bands (dashed lines in Fig. 6.21(a)) have a large Fermi surface (i.e. the f-hole is now included in the Fermi volume), normally assumed to be the same as the large LDA Fermi surface

(dotted grey lines in Fig. 6.21(a)). When the magnetic ﬁeld polarizes the quasiparticle bands (dashed lines in Fig. 6.21(b)), the majority- and minority-spin branches split: the

(red) minority hole quasiparticle D-band sinks, such that its Fermi surface resembles the small D FS sheet; the (red) majority hole quasiparticle D-band rises to resemble the small J FS sheet, where it becomes the opposite-spin counterpart of the (blue) minority hole quasiparticle J-band; finally, the (blue) majority hole quasiparticle J-band rises, and H0 is the field at which the quasiparticle Fermi surface corresponding to this band has grown to encompass the entire Brillouin zone and disappears. Thus, through this process it is possible for the minority branch of the large D Fermi surface to resemble the small D Fermi surface without requiring 4f localization at H0. In fact, localization will occur at fields much higher than H0, when the upper spin-split branch of the lowest (green) quasiparticle band shown in Fig. 6.21 crosses the Fermi energy and becomes the

opposite-spin counterpart of the (red) minority hole quasiparticle D-band.

The situation is reminiscent of the case of CeRu2Si2, whose metamagnetic transition

(MMT) occurs at applied ﬁelds of HMMT = (7.7/ cos θ) T, where θ is the angle between the ﬁeld direction and the crystallographic c-axis. dHvA measurements below and above the MMT were matched to band structure predictions for large and small Fermi surface cases respectively, implying a sudden localization of the Ce 4f state at the MMT [150]. Chapter 6. YbRh2Si2 179

H Ho H>Ho ≪

EF (c) (d)

E(k) (a) (b)

ZUk ZUk

Figure 6.21: A schematic representation of the YbRh2Si2 bands near the Fermi energy, EF , following Kusminskiy et al. [142] Bands associated with the “D” and “J” Fermi surface sheets of Fig. 6.2 are labelled. Panel (a) shows the band structure without spin- splitting: solid grey lines for the small Fermi surface case, and dashed blue, red and green lines for the many-body quasiparticle bands whose Fermi surface coincides with the “large” LDA FS (dotted lines). The virtual f-hole is shown by a grey horizontal line in panel (c). Panel (b) adds spin-splitting to the quasiparticle bands, causing one minority-spin surface to resemble the un-split small D surface, and the largest majority- spin band to no longer cross EF . Panels (c) and (d) show magniﬁed views of panels (a) and (b), respectively, near EF . Chapter 6. YbRh2Si2 180

However, as in YbRh2Si2, measurements of other CeRu2Si2 bulk properties rule out the possibility of a ﬁrst order transition at the MMT, leading to the proposal of a continuous

Fermi surface Lifshitz transition [142, 151, 152].

The modiﬁed Kusminskiy model discussed above is in accord with both Luttinger’s theorem and my experimental results. A key point is that I do not resolve spin-splitting of the orbits in any of my measurements, suggesting that what I observe are indeed small-

D-like oscillations that arise from one spin direction only, as in Fig. 6.21(b). If the 4f quasi-hole had localized at H0, all of the observed orbits would be spin split. In contrast, in the Kusminskiy model, the majority-spin quasiparticle branch has a completely diﬀer-

ent topology from the minority-spin branch. As the minority-spin quasiparticle branch

shrinks to resemble the small D LDA Fermi surface, it becomes a topological torus, al-

lowing the appearance of orbits D6, 7, 8, 9, 10 and 12, which thread through the centre

of the torus. The corresponding majority-spin branch is never toroidal, and thus would

not have these orbits. Moreover, as this majority surface grows to resemble the small J

LDA Fermi surface, its topology will radically change, such that orbits D1, 2, 3, 4 and

11 also vanish.

I therefore believe that my results, combined with Luttinger’s theorem, rule out the

4f localization scenario and support the approach of Kusminskiy et al. [142]: the high-

ﬁeld Fermi surface is a spin-split version of the large YbRh2Si2 Fermi surface, rather than a direct realization of the small Fermi surface. This, in turn, implies that the Fermi

surface includes the 4f degrees of freedom for Hc

Conclusions

7.1 Instrumentation

During the construction of our laboratory, I created a comprehensive software and hardware suite, called the Data Acquisition Virtual Instrument Experiment System (DAV-

IES), that controls my research group’s dilution refrigerator and superconducting magnet facility, and collects the data for all of our experimental measurements. At the core of

DAVIES is a novel data acquisition subsystem that replaces the bank of physical lock- in ampliﬁers typically used for de Haas–van Alphen eﬀect measurements with up to 24 software-based “virtual lock-ins,” running in parallel. This system streamlines the data collection process, and exceeds the capabilities of a traditional set-up.

I have also designed a graphite rotation mechanism, which allows samples mounted on our dilution refrigerator to be rotated in situ relative to the magnetic ﬁeld direction; set up an annealing station, which allows the thermal conductivity of silver heat-sink wires to be greatly increased; and built a glove box, with a liquid nitrogen cold trap gas circulation system, which allows air-sensitive samples to be handled under an inert helium atmosphere.

181 Chapter 7. Conclusions 182

7.2 Supercell K-space Extremal Area Finder

In order to facilitate comparison between electronic structure calculations and de Haas– van Alphen measurements, I have developed a new approach for extracting quantum oscillation frequencies, effective masses, orbit types and electronic density of states contributions from calculated band energies. By employing a large, heavily-interpolated k-space super cell and exploiting recent advances in desktop computing power, my algorithm can robustly characterize complicated Fermi surfaces to locate all extremal orbits, including those that are non-central, cross Brillouin zone boundaries, or are otherwise difficult to find by visual inspection. This Supercell K-space Extremal Area Finder

(SKEAF) algorithm was initially tested on idealized spherical and cylindrical Fermi surfaces, before being applied to more complex cases corresponding to real materials.

Based on my own band structure calculations and experimental de Haas–van Alphen effect data measured by Alix McCollam, I have used this algorithm to estimate the enhanced electronic specific heat coefficient of CeCoIn5. The predicted de Haas–van Alphen frequencies and effective masses extracted by SKEAF agree with those from

∗ previously-published calculations. I find that a “hybrid” specific heat estimate, γhybrid = 400(100) mJ mol−1 K−2, in which only the effective masses corresponding to orbits on the Fermi surface α sheet are taken to be spin-dependent, is in best agreement with the measured specific heat of this material.

7.3 UPt3

I have also applied the SKEAF algorithm to a “fully itinerant” UPt3 electronic structure calculation done by Mike Norman, in which all three U 5f electrons are assumed to contribute to the Fermi volume, and compared the results to non-oscillatory magnetoresistance, Shubnikov–de Haas and de Haas–van Alphen measurements performed by my supervisor, Stephen Julian. My algorithm found new orbits within the old band struc- Chapter 7. Conclusions 183

ture calculation, that had not been previously noticed, particularly on the topologically-

complicated, multiply-connected band 2 sheet of the Fermi surface. These new predicted

orbits match frequencies observed in the quantum oscillation measurements, bringing

the theoretical predictions of the fully itinerant model into much closer agreement with

the experimental data than was previously the case. In contrast, “partially localized”

UPt3 electronic structure calculations, in which only one of the three U 5f electrons are assumed to contribute to the Fermi volume, disagrees with the experimental data on several key points. Thus, it is the fully itinerant model, rather than the partially localized model, that provides the best description of the UPt3 5f electrons and characterization of the UPt3 Fermi surface.

7.4 CePb3

In CePb3, previous measurements had been interpreted as evidence for a “spin-ﬂop” phase between 5–10 T, for magnetic ﬁelds aligned along the crystalline (110) direction. ∼ At the National High Magnetic Field Laboratory in Tallahassee, Florida, I performed

magnetoresistance, cantilever torque, and ac susceptibility measurements of CePb3 at a temperature of 23 mK, with magnetic ﬁeld swept between 0 and 18 T and rotated in the

(110)–(111)–(001)=(100) plane. I additionally measured the magnetoresistance for ﬁeld

rotations in the (100)–(110) plane, and at temperatures up to 400 mK when the ﬁeld

was along the (111) direction.

For ﬁelds in the (100)–(110) plane, and along (110) and (001)=(100) in the (110)–

(111)–(001)=(100) plane, my magnetoresistance results agree with those in the literature.

The magnetoresistance had not been previously measured for ﬁelds along (111), and my

low-temperature data for this ﬁeld direction exhibits fundamentally new behaviour: at

low ﬁelds, the resistivity increases roughly linearly with ﬁeld, declines sharply at H c ∼

6 T, then is ﬁeld independent for H>Hc. Analysis of the temperature dependence of Chapter 7. Conclusions 184

this data reveals a decreasing low-temperature range of Fermi liquid theory applicability

on approach to H , a T 2 coeﬃcient that diverges as A(H) H H −α, with H 6 T c ∝ | − c| c ∼ and α 1, and an estimate of the Kadowaki-Woods ratio consistent with the “universal” ∼ heavy fermion value.

The torque measurement is the first of its kind on CePb3, and shows a complicated dependence on magnetic field strength and angle. This field-dependence is particularly striking for field angles near (111), where numerous sign changes are observed between

2 and 12 T, including the presence of a peak at Hc. The ac susceptibility measurements show a broad background proportional to H2 at all ﬁeld angles (likely an experimental artifact), and lack any features corresponding to those seen strongly in the magnetoresistance and torque.

By comparing the results of my measurements to theoretical spin-flop predictions, as treated numerically in 2D and 3D classical spin models, I find that the spin-flop phenomenology may not be the correct basis for explaining the field-dependent behaviour of CePb3.

7.5 YbRh2Si2

Based on indirect evidence, YbRh2Si2 has been claimed to exhibit an itinerant-to-localized change of the 4f quasi-hole behaviour at the H 10 T Kondo suppression field. In 0 ≈ order to directly investigate this claim, I measured the Fermi surface of this material via the de Haas–van Alphen effect, as the magnetic field was swept between 8 to 16 T, through

H0. These measurements were performed in my group’s laboratory at the University of Toronto, using the DAVIES experimental control suite, graphite rotation mechanism, and annealed silver heat-sink wires that I created earlier in my doctoral studies. I also calculated the electronic structure of YbRh2Si2 for both the “large” (4f-itinerant) and “small” (4f-localized) Fermi surface cases, and extracted quantum oscillation predictions Chapter 7. Conclusions 185 using the SKEAF algorithm.

At high ﬁelds, I measured the quantum oscillations as the ﬁeld was rotated 45◦ from (100) to (110) and 60◦ from (110) toward (001), in a temperature range from

30 to 600 mK. The ﬁeld-angle dependence of the measured oscillation frequencies is best matched by the predictions of the small Fermi surface case, and no frequency spin- splitting was observed. The temperature dependence of the oscillations is well described by the standard Lifshitz-Kosevich equation, giving eﬀective masses roughly 10(1) larger × than the unenhanced band masses obtained by SKEAF from my small Fermi surface electronic structure calculation.

The strongest oscillations were tracked at the lowest temperatures while the field was swept from 16 T to 8 T. The measured frequencies show a smooth, continuous increase as the field is decreased through H 10 T, corresponding to a gradual increase in 0 ≈ the size of the measured Fermi surface sheet. However, since Luttinger’s theorem only allows discontinuous total Fermi volume changes at low temperatures, the measured field dependence of my de Haas–van Alphen frequencies must not arise from a localized-to- itinerant transition of the Yb 4f quasi-hole. Rather, the data is explained by a Lifshitz transition scenario, in which the 4f quasi-hole remains itinerant over the entire measured

ﬁeld range. The total Fermi volume is preserved through Fermi surface spin-splitting, and the experimental behaviour near H0 arises when the majority-spin branch of one of the bands crossing the Fermi energy at low ﬁelds rises entirely above the Fermi energy, while the Fermi surface topology of the minority-spin branch of the band we observe in our experiment gradually changes to resemble the calculated small Fermi surface for

H>H0. Chapter 7. Conclusions 186

7.6 List of Publications

† P. M. C. Rourke, A. McCollam, G. Lapertot, G. Knebel, J. Flouquet, and S. R. Ju- lian. Magnetic-ﬁeld-dependence of the YbRh2Si2 Fermi surface. Physical Review Letters, 101:237205, 2008 [15].

† P. M. C. Rourke, A. McCollam, G. Lapertot, G. Knebel, J. Flouquet, and S. R. Ju- lian. The high-ﬁeld Fermi surface of YbRh2Si2. To appear in the Journal of Physics: Conference Series [14].

† G. J. McMullan, P. M. C. Rourke, M. R. Norman, A. D. Huxley, N. Doiron-Leyraud,

J. Flouquet, G. G. Lonzarich, A. McCollam, and S. R. Julian. The Fermi Surface and f-valence Electron Count of UPt3. New Journal of Physics, 10:053029, 2008 [13].

† W. Wu, A. McCollam, I. Swainson, P. M. C. Rourke, D. G. Rancourt, and S. R. Julian.

A novel non-Fermi-liquid state in the iron-pnictide FeCrAs. To appear in Europhysics

Letters [49].

† P. M. C. Rourke and S. R. Julian. Numerical Extraction of de Haas–van Alphen Fre- quencies from Calculated Band Energies. Under peer review for publication in Computer

Physics Communications [11].

† A. McCollam, S. R. Julian, P. M. C. Rourke, D. Aoki, and J. Flouquet. Anomalous de

Haas–van Alphen Oscillations in CeCoIn5. Physical Review Letters, 94:186401, 2005 [12].

P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T.

Wei. Spectroscopic Evidence for Multiple Order Parameter Components in the Heavy

Fermion Superconductor CeCoIn5. Physical Review Letters, 94:107005, 2005 [73].

†Work performed during my doctoral studies. Chapter 7. Conclusions 187

P. M. C. Rourke, M. A. Tanatar, C. S. Turel, C. Petrovic, and J. Y. T. Wei. Rourke et al. reply. Physical Review Letters, 96:259703, 2006 [74].

P. M. C. Rourke, J. Paglione, F. Ronning, L. Taillefer, and K. Kadowaki. Elastic Tensor of YNi2B2C. Physica C, 397:1, 2003. Bibliography

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