Muon Spin Relaxation Studies of Cuprates in

the Normal State

by

Shayan Gheidi

M.Sc., University of Toronto, 2017

B.Sc., University of British Columbia, 2016

Thesis Submitted in Partial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophy

in the

Department of Physics

Faculty of Science

© Shayan Gheidi 2022

SIMON FRASER UNIVERSITY

Spring 2022

Copyright in this work rests with the author. Please ensure that any reproduction orre-use is done in accordance with the relevant national copyright legislation.

Declaration of Committee

Name: Shayan Gheidi

Degree: Doctor of Philosophy (Physics)

Title: Muon Spin Relaxation Studies of Cuprates in the Normal

State

Committee: Chair: Karen KavanaghProfessor, Physics

Jeff E. Sonier

SupervisorProfessor, Physics

Igor Herbut

Committee MemberProfessor, Physics

David Broun

Committee MemberAssociate Professor, Physics

Mike Hayden

ExaminerProfessor, Physics

David Hawthorn

External ExaminerProfessor, Physics and AstronomyUniversity of Waterloo

ii

Abstract

Despite immense effort dedicated to understanding the physics of high-Tc superconducting

cuprates, many questions regarding the origin of superconductivity in these materials remain

unanswered. This thesis explores the use of the muon spin relaxation/rotation (µSR) technique

as a sensitive local probe of internal magnetic fields to investigate the magnetic properties of

cuprates in their normal state.

Much of the normal state of cuprates is occupied by a “pseudogap” phase, characterized by

the depletion of the electronic density of states near the Fermi energy below a characteristic

temperature, T ∗ [1]. The origin of the pseudogap phase and its relationship with supercon-

ductivity is unclear. Polarized neutron diffraction measurements have detected intra-unit-cell

magnetic order in the pseudogap phase of several cuprates [2–11]. Investigations using local

probe techniques such as nuclear magnetic resonance (NMR), nuclear quadrupolar resonance

(NQR) and zero-field (ZF) µSR, however, have not found evidence for such magnetic order in

YBa2Cu3O6+x (Y123) and La2−xSrxCuO4 (La214) [12–18]. Here, ZF-µSR is utilized to search

for intra-unit-cell magnetic order in Bi2Sr2CaCu2O8+δ (Bi2212). Earlier studies of this material

detected weak, temperature-dependent quasistatic internal magnetic fields that appeared to be

electronic in origin in the pseudogap and superconducting phases [19, 20]. By extending that

work to include a wider range of hole-doping concentrations and using a specialized ultra-low

background µSR apparatus, the internal magnetic fields are determined to be nuclear in origin

and independent of hole-doping [21]. The weak temperature dependence of the ZF-µSR relax-

ation rate that is observed in Bi2212 is attributed to a slight modification of the nuclear field

distribution at the muon site caused by changes in the crystallographic lattice structure. These

findings reaffirm the results of previous searches for magnetic order in the pseudogap phase by

local probe techniques.

The normal state of cuprate superconductors is also believed to harbour precursor superconduct-

ing pairing correlations [22, 23]. Among mounting experimental evidence for phase fluctuating

Cooper pairing above Tc [24–29] are high transverse-field (TF) µSR measurements that reveal a

universal inhomogeneous magnetic field response above Tc in Y123, La214 and Bi2212 [30, 31].

While the data from these studies are consistent with the presence of spatially inhomogeneous

diamagnetic (i.e., superconducting) regions that proliferate with reduced temperature, it has

not been possible to determine if the source of the magnetic field broadening is diamagnetic in

iii

origin. To address this issue, muon Knight shift measurements exploiting a significant improve-

ment in high TF-µSR instrumentation were carried out on Bi2212 single crystals over a wide

hole-doping range. Aside from a hole-doping-independent temperature dependence at high tem-

perature attributed to muon diffusion, the muon Knight shift is temperature independent above

Tc and consequently insensitive to the normal state pseudogap or normal state superconduct-

ing fluctuations. Potential explanations for this, as well as limits to the experimental data are

discussed.

Keywords: high temperature superconductivity, cuprates, pseudogap, muon spin rotation/re-

laxation.

iv

Dedication

To my mother, Nahid

my father, Amir

and my brother, Shervan.

v

Acknowledgements

I am deeply grateful for support from my senior supervisor, Jeff Sonier. Jeff provided a fertile

environment for me to grow both as a researcher in physics and as a person. His encouragement

and trust in my work meant that I was always excited to share my results and projects with him

even when it was unrelated to our work. Jeff embraced me in a such a way that I always fondly

anticipated meeting up to discuss work or life. The work in this thesis would not have been

possible without the help of Sarah Dunsiger. Sarah shared a stellar amount of knowledge and

wisdom with me and I was never hesitant to approach her for questions. The ease of conversing

with Sarah made for many memorable counting room discussions. I also wish to express my

gratitude for my committee members Igor Herbut and David Broun for their feedback, support

and encouragement over the past four years. I looked forward to our spontaneous hallway chats

(whose topics ranged from politics to Bulgakov and Dostoevsky) and was always surprised at

just how much these conversations brightened my day.

The results in this thesis were also heavily dependent on help from my fellow group mem-

bers. I am indebted to Alex Fang and Kola Akintola for welcoming me to the group and teaching

me the fundamentals of µSR experiments. I thank Shyam Sundar for always having an answer

to my questions, guiding me throughout this degree and being an exceptionally reliable group

member who was always ready to cover for me. The many coffee-breaks I shared with the afore-

mentioned students are something I will miss. I am grateful for the help during beamtimes from

Andre Cote and Krishant Akella. I also thank Ali Mokhtari-Jazi and Sujit Narayanan for their

help in theory courses and for their friendship.

I thank TRIUMF’s CMMS staff Rahim Abasalti, Donald Arsenault, Bassam Hitti, Gerald Mor-

ris, and Deepak Vyas for their critical support in experimental setup and data acquisition. I am

grateful for our collaboration with Genda Gu who kindly provided our group with several Bi2212

single crystals. I also wish to thank Zaher Salman for his help during our collaboration at the

Paul Scherrer Institut.1 The many chats we had over coffee are something I cherish, and I deeply

appreciate the words of advice and guidance that were scattered throughout these discussions.

I extend my gratitude to Rob Kiefl. I was a stressed and confused undergraduate student when

I met Rob almost 10 years ago at TRIUMF and I will never forget the confidence his words of

1These results are unfortunately not included in this thesis.

vi

encouragement instilled in me. I thank Maman, Baba, and my brother Shervan for being the

pillars of support that I lean on and keeping me grounded.

vii

Table of Contents

Declaration of Committee ii

Abstract iii

Dedication v

Acknowledgements vi

Table of Contents viii

List of Tables x

List of Figures xi

1 Introduction 1

1.1 Conventional Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 High-Tc Cuprate Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Phase Diagram of Hole-doped Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The Pseudogap Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Intra-Unit-Cell Magnetic Order in the Pseudogap Phase . . . . . . . . . . . 8

1.5 Charge Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Superconducting Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6.1 Experimental Evidence for Superconducting Fluctuations . . . . . . . . . 15

1.7 Ferromagnetic Fluctuations and Ferromagnetism in the Heavily Overdoped Regime 17

2 Experimental Methods 19

2.1 Muon Spin Relaxation (µSR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Muon Production, Transport and Decay . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Total Magnetic Field Sensed by the Implanted µ+ . . . . . . . . . . . . . . . . . . . 22

2.4 Time-differential µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Zero-field µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Zero-field Relaxation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Longitudinal-field µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

viii

2.7 Transverse-field µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.1 Transverse-field Relaxation Functions . . . . . . . . . . . . . . . . . . . . . . 30

2.7.2 The Muon Knight Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7.3 High Field Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.8 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.9 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Absence of µSR Evidence for Magnetic Order in the Pseudogap Phase of Bi2212 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Ultra-low Background Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Nuclear Dipolar Field Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Normal State Inhomogeneous Magnetic Field Response 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Spatially Inhomogeneous Superconducting Fluctuations and the TF-µSR Lineshape 55

4.3 Depolarization Functions of Skewed Field Distributions . . . . . . . . . . . . . . . 56

4.4 High Transverse-field µSR Measurements in the Normal State of Bi2212 . . . . . 60

5 Conclusions 75

Bibliography 77

Appendix A Photos of Bi2212 Samples Used in Zero-field Search for Magnetic Order 86

Appendix B Calculating Depolarization Functions with Mathematica 87

ix

List of Tables

Table 2.1 Fundamental properties of muons . . . . . . . . . . . . . . . . . . . . . . . . . 19

Table 3.1 Summary of samples and the corresponding ZF-µSR measurement condi-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 4.1 Contributions to the measured muon Knight shift in Bi2212 . . . . . . . . . 66

Table 4.2 The results of fits of the Kµ vs. T data to Eq. (4.27). . . . . . . . . . . . . . 68

x

List of Figures

Figure 1.1 Superconductors exhibit zero resistivity and the Meissner effect below Tc 2

Figure 1.2 The crystal structure of Bi2Sr2CaCu2O8+δ and the CuO2 plane . . . . . . 4

Figure 1.3 Temperature dependence of the superfluid density for d-wave and s-

wave superconducting gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 1.4 Generic phase diagram of hole-doped cuprates . . . . . . . . . . . . . . . 6

Figure 1.5 Three potential scenarios for the pseudogap phase . . . . . . . . . . . . . 7

Figure 1.6 Schematic of Varma’s orbital-loop current model of the pseudogap phase 10

Figure 1.7 Temperature dependence of the ZF-µSR frequency in Y123 . . . . . . . . 12

Figure 1.8 Extent of superconducting fluctuations according to different experi-

mental probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.1 Comparison of the ranges of fluctuating internal magnetic fields de-

tectable by NMR, µSR and neutron scattering techniques . . . . . . . . . 20

Figure 2.2 The angular dependence of positron decay emission in muons . . . . . . 22

Figure 2.3 The local field sensed by a muon . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 2.4 Schematic of a zero-field µSR experiment . . . . . . . . . . . . . . . . . . 26

Figure 2.5 Simulation of a raw histogram and the calculated asymmetry in zero-field 27

Figure 2.6 Gaussian and Lorentzian Kubo-Toyabe relaxation functions . . . . . . . . 28

Figure 2.7 Schematic of a transverse-field µSR experiment . . . . . . . . . . . . . . . 31

Figure 2.8 Simulation of a raw histogram and the calculated asymmetry in transverse-

field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.9 Schematic of a muon Knight-Shift experiment . . . . . . . . . . . . . . . . 33

Figure 2.10 The normalized asymmetry as a function of applied magnetic field . . . 35

Figure 2.11 Detailed process for Fourier transforming µSR spectra and apodization 38

Figure 3.1 The pseudogap temperatures and Tcs of the Bi2212 samples . . . . . . . 40

Figure 3.2 Low field temperature dependence of bulk magnetic susceptibility for

OD70 and OD55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 3.3 Representative zero-field asymmetry spectra at 152 K and 5 K for Ag . . 41

Figure 3.4 Representative zero-field asymmetry spectra at 150 K and 4 K for OD70

and OD55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 3.5 Temperature dependence of zero-field fit results . . . . . . . . . . . . . . 44

xi

Figure 3.6 Characteristic time t1/2 required for the polarization to decay to half of

the initial amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.7 Schematic of the ultra-low background experimental setup . . . . . . . . 46

Figure 3.8 Representative zero-field spectra produced by the ULB apparatus and

corresponding fitting results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 3.9 Background contamination: the mean stretching parameter β versus the

initial asymmetry for Bi2212 samples . . . . . . . . . . . . . . . . . . . . . 48

Figure 3.10 Calculations of ∆ and δ∆ in the SrO plane . . . . . . . . . . . . . . . . . 49

Figure 3.11 Calculations of ∆ and δ∆ in the BiO and CuO2 planes . . . . . . . . . . 50

Figure 3.12 Calculations of ∆ and δ∆ in the SrO plane according to Eq. (3.6) . . . 52

Figure 3.13 Temperature dependence of the relaxation rate in Bi2212 compared to

La214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.1 Spatially inhomogeneous islands of fluctuating diamagnetism and the

modeled half-Lorentzian field distribution . . . . . . . . . . . . . . . . . . 55

Figure 4.2 Left-half Gaussian distribution and its corresponding depolarization func-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 4.3 Left-half Lorentzian distribution and its corresponding depolarization

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 4.4 An equal combination of LHL and RHL depolarization functions is equiv-

alent to an exponentially relaxed signal . . . . . . . . . . . . . . . . . . . . 60

Figure 4.5 Temperature dependence of the bulk dc magnetic susceptibility of su-

perconducting Bi2212 samples . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 4.6 Asymmetry spectra in silver at the sample and reference locations . . . 62

Figure 4.7 Temperature dependence of the exponential relaxation rate in silver at

the sample and reference positions . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 4.8 Temperature dependence of the spectrometer Knight shift and Gaussian

relaxation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 4.9 Representative asymmetry spectra for the samples with fits to the expo-

nential times Gaussian model . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 4.10 Fourier transforms of the sample asymmetry spectra at T = 160 K and

T ≃ Tc + 10 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 4.11 Temperature dependence of the exponential relaxation rate and corre-

sponding contour map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 4.12 Temperature dependence of Kµ calculated from exponential times Gaus-

sian fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 4.13 Muon diffusion may explain the temperature dependence of Kµ when

calculated from exponential times Gaussian fits . . . . . . . . . . . . . . . 69

xii

Figure 4.14 Representative asymmetry spectra for the samples fit to the left-half

Lorentzian times Gaussian model . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 4.15 Temperature dependence of the left-half Lorentzian relaxation rate and

the corresponding contour map . . . . . . . . . . . . . . . . . . . . . . . . . 71

Figure 4.16 Temperature dependence of Kµ calculated from fits to the left-half Lorentzian

times Gaussian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 4.17 Comparing the goodness of fit of the two models of interest . . . . . . . 72

Figure 4.18 An alternative magnetic field distribution associated with spatially in-

homogeneous superconducting fluctuations . . . . . . . . . . . . . . . . . 74

xiii

Chapter 1

Introduction

1.1 Conventional Superconductivity

A material is a superconductor if

(i) its electrical resistance vanishes below a critical transition temperature, Tc [see Fig. 1.1(a)],

and

(ii) below Tc it expels magnetic flux from its interior below a critical magnetic field [see

Fig. 1.1(b)]. In a type-I superconductor, superconductivity is destroyed above a critical

field Hc , whereas in a type-II superconductor an increasing amount of magnetic flux pene-

trates the material above a critical field Hc1, and superconductivity is ultimately destroyed

above an upper critical field Hc2.

The first of these physical properties was discovered by Dutch physicist Heike Kamerlingh Onnes

in 1911 by cooling mercury (Hg) below 4.2 K with liquid helium [32]. The second physical prop-

erty, known as the Meissner-Ochsenfeld effect,1 distinguishes “super” conductors from “perfect”

conductors, should the latter exist. While a perfect conductor with zero resistance prevents

changes in magnetic flux by standard electromagnetic induction, it cannot expel existing non-

zero magnetic flux like a superconductor.

Early theories formulated by the London brothers [33] and by Landau and Ginzburg [34]

provided successful phenomenological descriptions of superconductivity, but it was not until

50 years after the initial discovery of low-temperature zero resistance in Hg that a microscopic

theory of superconductivity in so-called “weakly-coupled” superconductors such as Al, Pb and

Hg was developed by Bardeen, Cooper and Shrieffer (BCS) [35]. In BCS theory, electrons near

the Fermi surface of a metal become unstable against the binding of pairs of electrons (i.e.,

Cooper pairs) caused by a weak attractive potential. In conventional superconductors, the origin

of the attractive potential is the coupling of electrons to phonons and in BCS theory this leads

1Attributed to German physicists Walther Meissner and Robert Ochsenfeld for the discovery in 1933.

1

(b) T < Tc

(a

rb.

un

its)

Temperature

0

Tc

T > Tc

(a)

H H

Figure 1.1: (a) The electrical resistivity ρ of a superconductor vanishes below Tc . (b) Illustrationof the Meissner-Ochsenfeld effect. Magnetic flux is expelled from the interior of the supercon-ductor upon cooling below Tc .

to the pairing of electrons near the Fermi surface with opposite spin (i.e., spin-singlet pairing)

and opposite momentum eigenstates. The Cooper pairs condense into a single coherent ground

state and propagate coherently in the presence of an electric field. Experiments were paramount

in providing clues as to the mechanism of superconductivity in weakly-coupled metals. One of

these was the observation of the so-called “isotope effect”, where Tc was determined to vary

with the mass M of the ions as Tc ∝ M−1/2, hinting at the role of phonons [36, 37]. A peak

near Tc in the temperature dependence of the electronic heat capacity indicated the presence

of a phase transition from a T -linear normal state to an exponentially activated temperature

dependence, suggesting the presence of an energy gap at the Fermi surface below Tc [38, 39].

This superconducting energy gap ∆ corresponds to the reduced energy of the electrons after

pairing (i.e., the energy required to break a Cooper pair is 2∆). Tunneling transport experiments

would later verify this through the direct measurement of ∆ in the electronic density of states

[40]. In BCS theory,∆ is isotropic in momentum space and corresponds to an s-wave symmetric

pairing wave function.

Despite having low values of Tc , conventional superconductors are used in a variety of tech-

nological applications. Examples include superconducting magnets that use persistent currents

to generate large magnetic fields, superconducting radio-frequency (SRF) cavities that acceler-

ate charged particles with near perfect power efficiency, and sensitive magnetometers such as

the superconducting quantum interference device (SQUID). Superconducting magnets are most

notably used in magnetic resonance imaging (MRI), one of the standard methods of imaging

the human body. However, since cooling conventional superconductors to temperatures below

Tc requires expensive helium, the more widespread use of superconductors is still limited. While

room temperature superconductivity (Tc = 287.7 K) has been recently reported in carbonaceous

sulfur hydride under a pressure of 267 GPa [41], the discovery of superconductors with more

accessible high-Tc values at ambient pressure is desired.

2

1.2 High-Tc Cuprate Superconductors

In 1986, Johannes Georg Bednorz and Karl Alexander Müller discovered superconductivity with

Tc ∼ 30 K in the perovskite-like compound La5−xBaxCu5O3−y , surpassing the previous high Tc

values of Nb3Sn (Tc = 18.3 K) and NbN (Tc ∼ 16 K) [42]. Within two years, similar copper-

oxide based compounds with Tc values above the boiling temperature of liquid nitrogen (77 K)

were discovered; most notably YBa2Cu3O6+x (Y123) and Bi2Sr2CaCu2O8+δ (Bi2212) [43, 44].

These high-Tc cuprate superconductors are layered quasi two-dimensional (2-D) compounds

consisting of planes of CuO2 separated by planes containing other atoms, commonly referred

to as “charge-reservoir” layers (see Fig. 1.2). The Cu atoms assume a +2 valence state and

are configured electronically as [Ar]3d9. This configuration corresponds to an odd number of

electrons per unit cell or a half-filled electronic band that band theory predicts to be metallic. In

spite of this prediction, strong Coulomb repulsion associated with placing two electrons on the

same lattice site results in localization of the Cu2+ 3d9 electrons and a so-called Mott insulating

state. The localized Cu spins are antiferromagnetically ordered to reduce the Coulomb energy

via virtual hopping.

Superconductivity emerges when the mobility of the localized electrons is promoted by the

introduction (i.e., doping) of charge carriers, which can be either electrons or holes. In some

cuprates, holes are introduced by cation substitution. For example, substitution of Sr2+ in place

of La3+ in La2−xSrxCuO4 (La214) introduces holes into the CuO2 plane. For other compounds,

such as Bi2Sr2CaCu2O8+δ, holes are doped into the CuO2 plane by excess oxygenation. Electron-

doped cuprates exhibit similarities to hole-doped cuprates, but also some marked differences.

For example, while an antiferromagnetic phase exists for both types of compounds, the antifer-

romagnetic order is more resilient to electron-doping than to hole-doping. The superconducting

phase spans a much wider doping range in hole-doped cuprates and charge order, which will be

discussed in a later section, is prevalent in underdoped hole-doped cuprates, while it is much

less significant in electron-doped cuprates [45]. This thesis focuses on hole-doped cuprates and

in particular, the compound Bi2Sr2CaCu2O8+δ.

Unlike conventional superconductors, the superconducting gap structure in cuprates is dx2−y2

symmetric (d-wave) [46–48] corresponding to an energy gap with momentum dependence

∆(~k) =∆0(cos kx a− cos ky a). (1.1)

As shown in Fig. 1.3(a),∆(~k) undergoes a sign change from adjacent lobes and is characterized

by line nodes along kx = ±ky , where ∆(~k) = 0. The proximity of the superconducting phase to

the antiferromagnetic phase and the d-wave gap structure are consistent with Cooper pairing

mediated by antiferromagnetic fluctuations [48]. However, despite theoretical attempts to unify

antiferromagnetism and superconductivity [49], a microscopic theory of high-Tc superconduc-

tivity in cuprates is still lacking. The presence of quasiparticle excitations facilitated by the line

nodes in the energy gap leads to significant modifications of superconducting properties. For ex-

3

a

bc

Bi

Sr

Ca

Cu

O

30.7 Å

5.4 Å

5.4 Å

Figure 1.2: The crystal structure of Bi2Sr2CaCu2O8+δ with the CuO2 planes highlighted. TheCu2+ 3d9 electrons are localized and antiferromagnetically ordered.

ample in clean materials, the temperature dependence of the superfluid density ns(T ), which is

proportional to the inverse square of the London penetration depth2 1/λ2(T ), exhibits a unique

T -linear behaviour at low temperatures [see Fig. 1.3(b)]. The quasi two-dimensional nature of

cuprates also leads to large anisotropy in physical properties, since superconductivity is con-

fined within the weakly coupled CuO2 planes. The superconducting coherence length ξ0, which

is approximately the spatial size of a Cooper pair, is highly anisotropic and small compared

to conventional superconductors. For example, the in-plane coherence length ξab in optimally

doped Y123 is 18.5 Å [50], whereas ξ0 = 830 Å in Pb [51]. The small coherence length of high-

Tc cuprates allows for spatial variations of the superconducting order parameter over a shorter

length scale.

1.3 Phase Diagram of Hole-doped Cuprates

A generic temperature (T) versus doping (p) phase diagram of hole-doped cuprate supercon-

ductors is shown in Fig. 1.4. Here, p corresponds to the number of charge carries per CuO2 layer

in the unit cell. As discussed earlier, stoichiometric undoped cuprates are antiferromagnetic Mott

insulators with Néel3 temperatures TN that can exceed room temperature. When holes are in-

troduced, TN rapidly diminishes and the three-dimensional (3-D) long-range antiferromagnetic

order is destroyed by p ≈ 0.05, while short range antiferromagnetic correlations persist to higher

2The London penetration depth λ = (mc2/4πnse2)1/2 is the characteristic exponential length scale over which

external magnetic fields decay inside a superconductor.

3The Néel temperature is the characteristic onset temperature of antiferromagnetic order.

4

0.0 0.5 1.0

0.0

0.5

1.0

ns(T

)

T/Tc

dx

2-y

2

s-wave

−∆0

(b)

kx

ky

(a)

- 2 . 0 0 0

- 0 . 6 6 6 7

0 . 6 6 6 7

2 . 0 0 0

∆0

Figure 1.3: (a) A dx2−y2 symmetric superconducting gap described by ∆(~k) = ∆0(cos kx a −cos ky a). The dashed lines indicate the nodes along kx = ±ky where the gap has zero amplitude.The change in color indicates a change in the sign of the order parameter. (b) Temperaturedependence of the normalized superfluid density for d-wave and s-wave superconducting gaps.

p. At roughly the same doping that long-range antiferromagnetic order is destroyed, the super-

conducting state emerges. The superconducting transition temperature increases with p up to

an optimal doping value at which Tc is maximum (p ≈ 0.16) and then decreases with further

doping, eventually vanishing near p ≈ 0.275. The transition to the superconducting phase in

the T vs. p cuprate phase diagram is often referred to as a “dome” due to the parabolic shape

of Tc vs. p. Superconducting samples with p lower than optimal (OP) doping are referred to as

underdoped (UD), while those with greater p are considered overdoped (OD).

In the heavily-overdoped regime, hole-doped cuprates generally exhibit standard Landau-

Fermi liquid-like properties, such as a large Fermi surface [52] and an approximately T2 tem-

perature dependence of the electrical resistivity [53]. In the underdoped regime, on the other

hand, the superconducting state is preceded by a pseudogap phase. The pseudogap phase is

characterized by a depletion of the electronic density of states near the Fermi energy and will

be discussed in detail in a later section. In between the Fermi liquid-like overdoped regime and

the pseudogap dominated underdoped region of the T vs. p phase diagram lies a funnel-like

“strange metal” region. This strange metal region is characterized by an anomalous normal state,

where the temperature dependence of the in-plane resistivity can be decomposed into linear and

quadratic components. The magnitude of the T -linear component is finite and scales with Tc ,

while the T2 component is independent of hole-doping [54]. The temperature dependence of the

resistivity evolves smoothly from T -linear near optimal doping to T2 in the heavily-overdoped

regime [55]. Linear-in-temperature resistivity is observed in a number of strongly-correlated

Fermi systems and is a signature of systems exhibiting a quantum critical point (QCP). A QCP

results from the suppression of an ordered phase transition all the way down to zero temperature

by a non-thermal perturbation (e.g., hole-doping) [56].

5

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0

50

100

150

200

250

300

TN

TC

T*

Strange Metal

Pseudogap

Fermi

Liquid

Tem

pera

ture

(K

)

p

d-wave

Superconductor

An

tife

rrom

agn

eti

c

TCDW

Charge Order

SDW

Figure 1.4: Generic temperature versus doping phase diagram of hole-doped cuprates in theabsence of an applied magnetic field.

Although long-range antiferromagnetic order vanishes near p ≈ 0.05, short range spin cor-

relations survive to higher doping concentrations. Evidence for static magnetic moments at low

temperatures have been revealed by zero-field (ZF)µSR, existing up to p ∼ 0.09 in Y123 [57, 58]

and p ∼ 0.13 in La214 [59, 60]. Static magnetism has been shown to coexist with supercon-

ductivity at low temperatures above p = 0.05 by ZF and transverse-field (TF) µSR measure-

ments [58, 61, 62], while elastic neutron scattering measurements have provided evidence of

magnetic-field-driven static magnetic order beneath the superconducting dome in the vicin-

ity of p = 1/8 hole-doping in La214 [63] and Y123 (p = 0.07) [64]. Recent nuclear magnetic

resonance (NMR) and ultrasound measurements have used large magnetic fields to suppress su-

perconductivity in La214, revealing an “antiferromagnetic spin glass” phase at low temperature

extending to higher hole-doping concentrations than previously reported (i.e., p ∼ 0.19) [65].

The termination of spin glass magnetism and the pseudogap phase near the same hole-doping

concentration suggests a possible link between the antiferromagnetic Mott insulating phase, the

pseudogap phase and superconductivity, and supports the existence of a QCP beneath the su-

perconducting dome. An earlier µSR study also found evidence of field-induced spin-glass-like

magnetism in the vortex-core region of La214 and Y123 [66]. The magnetism was observed

for samples on the underdoped side of an apparent “low-temperature normal-state metal-to-

insulator crossover (MIC)”, supporting the presence of a quantum phase transition driven by

competing magnetic and superconducting orders [66]. The µSR study observed this in both

Y123 and La214 whereas the combined NMR and ultrasound study observed it only in La214

[65].

6

p*

Tc

Tc

Tc

T*T*

Tem

pera

ture

, T

Hole concentration, p

T*

pc

(a) (b)

Tem

pera

ture

, T

Hole concentration, ppc

(c)

Tem

pera

ture

, T

Hole concentration, p

Figure 1.5: Three potential scenarios for the pseudogap phase in hole-doped cuprates. (a) BothT ∗ and Tc terminate at a common hole-doping concentration, p∗. (b) T ∗ terminates at a quantumcritical point (QCP) at a critical hole-doping concentration pc in the overdoped superconductingphase. (c) T ∗ terminates at Tc in the overdoped regime.

1.4 The Pseudogap Phase

In much of the hole-doped cuprate phase diagram, superconductivity emerges below the pseu-

dogap phase. Understanding the pseudogap phase and its relationship with the superconducting

state is considered essential to explaining the mechanism of high temperature superconductiv-

ity in cuprates. The pseudogap in cuprates is a partial suppression of the electronic density of

states near the Fermi energy that appears below a characteristic temperature, T ∗ [67, 68]. A

number of experimental techniques detect some signature of the pseudogap. While it is agreed

that T ∗ decreases linearly with increasing p, the exact location of where in the phase diagram

T ∗ terminates is a topic of debate that has led to significantly different conclusions on the ori-

gin of the pseudogap and its relationship with superconductivity [69]. Figure 1.5 shows three

potential scenarios for the termination of the pseudogap phase in the cuprate phase diagram.

Different experimental techniques sensitive to manifestations of the pseudogap provide vary-

ing degrees of support for one or more of these possibilities. The third scenario was originally

compatible with experimental techniques whose signal vanishes below Tc , including transport

measurements such as resistivity, Hall resistance and thermopower. However, this scenario is no

longer seriously considered, as large pulsed magnetic fields have been used to suppress super-

conductivity allowing the relevant signals to be measured below Tc .

In Fig. 1.5(a), the pseudogap phase simultaneously terminates with the superconducting

phase at a critical doping, p∗. This scenario suggests the pseudogap phase is a precursor phase

of superconductivity in which Cooper pairs form, but lack long-range phase coherence until the

sample is cooled below Tc . While there is some experimental evidence for phase incoherent

Cooper pairing above Tc in high-Tc cuprates [24, 25, 27, 29–31], an explicit link between T ∗

and the temperature above Tc at which the Cooper pairs exist has not been made.

7

In Fig. 1.5(b), the pseudogap persists below Tc in the overdoped regime and terminates at a

quantum critical point (QCP) at a critical hole-doping concentration pc [70]. In this picture, the

superconducting and pseudogap phases compete and the pseudogap temperature T ∗ denotes a

true phase transition. Several theories suggest that the pseudogap is associated with the onset

of a competing orbital-loop-current phase [71, 72]. Experimental support includes detection

of the onset of a strange intra-unit-cell (IUC) magnetic order near T ∗ by polarized neutron

diffraction (PND) [2–11] and the detection of broken time-reversal symmetry below T ∗ via the

polar Kerr effect [73, 74]. The existence of a QCP is supported by the observation of a strong

peak in the electronic specific heat coefficient γ near pc and a log(1/T) temperature dependence

of γ for T → 0 [75]. The observation of low frequency quantum oscillations in underdoped

Y123 provide evidence for small electron pockets [76, 77], which, when contrasted with the

large Fermi surface observed in overdoped Tl2Ba2CuO6+δ (Tl-2201) [78], suggests there is a

reconstruction of the Fermi surface near a QCP within the superconducting phase [79]. On

the other hand, some have argued that the pseudogap phase exists for all temperatures below a

universal critical doping pc = 0.19, and that T ∗ is simply an associated energy scale or “crossover

scale” [80, 81].

The detection of two distinct energy gaps (i.e., superconducting gap and pseudogap) be-

low Tc in underdoped to optimally doped Bi2212 by spatially resolved scanning tunneling mi-

croscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) provides compelling

evidence for electronic phase separation in real and momentum space. The ARPES measure-

ments detect an antinodal temperature independent gap, whose magnitude decreases with dop-

ing (pseudogap), and a near-nodal temperature dependent gap with an onset at Tc that is inde-

pendent of doping (superconducting gap) [82–85]. Both gaps exist below Tc . In the underdoped

regime, the two gaps manifest as a discontinuous Fermi “surface”, often referred to as Fermi arcs

[86]. Real space “gap maps” of underdoped Bi2212 (p ≈ 0.14) generated by STM also show the

pseudogap and superconducting gap occupying segregated nanometer-size regions on the sur-

face of the sample [87]. When a magnetic field is applied, a pseudogap has been identified

within the vortex cores below Tc in under and overdoped Bi2212, providing further evidence of

competing pseudogap and superconducting phases [88]. An STM study of Bi2212 in the hole-

doping range p = 0.16− 0.22 indicates that pairing gaps nucleate in nanometer-size regions at

temperatures above Tc and these regions proliferate as the temperature is lowered [25]. How-

ever, since both ARPES and STM are surface techniques, it is unclear whether such electronic

inhomogeneity also occurs in the bulk.

1.4.1 Intra-Unit-Cell Magnetic Order in the Pseudogap Phase

Varma has theoretically proposed that the pseudogap is marked by the formation of orbital-

loop currents within the copper oxide planes, as illustrated in Fig. 1.6 [72, 89, 90]. The loop

currents cause a spontaneous magnetization that breaks time-reversal symmetry, but preserves

the translational symmetry of the lattice. The loop current model of Varma is compatible with

8

Fig. 1.5(b), whereby the pseudogap temperature T ∗ decreases below Tc with increased hole-

doping and vanishes at a QCP within the superconducting phase. The detection of a peculiar IUC

magnetic order by polarized neutron diffraction (PND) experiments in several underdoped high-

Tc cuprates is the primary experimental support for this loop current model [2–11]. The PND

measurements detect the onset of long-range magnetic order near T ∗ in Bi2212, Hg1201 and

Y123 characterized by an ordered moment of ∼0.1 µB. An orbital-like magnetic order has also

been detected in underdoped La214 by PND, but it is short range, two-dimensional, and occurs

far below T ∗. The existence of magnetic order in the pseudogap phase is also supported by the

detection of the polar Kerr effect in Y123 [73, 74]. Contrastingly, investigations using the local

probe techniques ZF-µSR [12–14, 19–21, 91], NMR and nuclear quadrupolar resonance (NQR)

[15–18] have found no evidence for magnetic order below T ∗. Furthermore, an independent

PND study of Y123 found no evidence of magnetic order in the pseudogap phase [92]. In an

attempt to reconcile the results from local probe and PND measurements, Varma’s theory was

later modified to incorporate fluctuations between different loop current configurations [90]. It

was argued that the fluctuation rate between the different loop current configurations is outside

of the dynamic range of NMR and ZF-µSR experiments but detectable by PND. The following

sections review NMR and ZF-µSR searches for pseudogap-related magnetism in different high-Tc

cuprate superconductors.

NMR Searches for Magnetic Order in the Pseudogap Phase

Lederer and Kivelson have shown that any form of magnetic order consistent with the results of

the PND experiments should result in an observable NMR signal in Y123 [93]. They calculate

the magnetic fields at most nuclear sites in Y123 to be on the order of hundreds of Gauss.

Yet, NMR studies have not found any evidence of magnetic order in the pseudogap phase of

Y123 or Hg1201. Furthermore, 89Y NMR measurements of Y2Ba4Cu7O15−δ (Y247) also found

no evidence of magnetic order down to 100 K [15]. The size of the static magnetic field at the

Y site was determined to be ≤ 0.15 mT,4 which is far smaller than predictions of theory [94].

A later study used NQR to search for evidence of orbital currents in underdoped YBa2Cu4O8

(Y124, Tc = 81 K). No evidence was found and the authors concluded that any static or dynamic

magnetic fields at the Y and Ba sites must be less than 0.07 and 0.7 mT, respectively [16].

The magnetic moments detected by PND in Y123, Hg1201, Bi2212 and La214 are tilted

away from the crystallographic c-axis by a non-negligible angle. While this is incompatible with

the original 2-D loop current patterns proposed by Varma (see Fig. 1.6), 3-D models have been

proposed in which the loop currents circulate away from the CuO2 plane towards the apical

oxygens [95, 96]. Yet, 17O NMR measurements found no evidence of static IUC magnetic order

at the apical oxygen site in underdoped Hg1201 (Tc = 74 K) [17].

4The upper limit on a fluctuating magnetic field was determined to be ≤ 0.7 mT.

9

a

b

c

o

oo

oCu Cu

Cu Cu𝐼 𝐼𝑚Figure 1.6: Schematic of the θ‖ phase of Varma’s loop current model, where orbital-loop currentsin the CuO2 plane generate intra-unit-cell magnetic order [89].

ZF-µSR Searches for Magnetic Order in the Pseudogap Phase

It was originally suggested by Varma that µSR “would be a way to look for interstitial fields”

generated by orbital-loop currents [72]. However, like NMR, the vast majority of ZF-µSR studies

have found no evidence for static or fluctuating (within the muon time window) magnetic order

in the pseudogap phase of cuprates. Early ZF-µSR measurements on optimally doped and un-

derdoped Y123 single crystals provided evidence for weak magnetism in the pseudogap phase

characterized by the onset of an enhanced spin relaxation rate near T ∗[97]. Extrapolation of

the onset temperature to zero temperature suggested the existence of a QCP hidden beneath

the superconducting dome. However, a subsequent study aimed at determining the origin of

the magnetism discovered a striking similarity in the temperature dependence of the ZF-µSR re-

laxation rate and the 63Cu linewidth obtained from NQR measurements [91]. The temperature

dependence of the 63Cu linewidth is explained by the development of charge correlations in the

CuO2 plane induced by a charge density wave (CDW) state in the CuO chain layers [98]. Muons

are spin 1/2 particles and therefore lack a quadrupolar moment that couples to electric field

gradients (EFG). However, changes in the EFG produced by charge correlations can modify the

quantization axis of I > 1/2 nuclear spins,5 which in the absence of an applied magnetic field is

along the direction of the maximum local EFG. This can lead to a modified nuclear dipolar field

distribution at the muon site. Hence, the temperature dependence of the ZF-µSR relaxation rate

in Y123 is apparently strongly influenced by the effects of charge ordering in the CuO chains.

5Both 63Cu and 65Cu nuclei have I = 3/2.

10

The temperature dependence of the ZF-µSR relaxation rate was subsequently investigated

in comprehensive measurements of six different Y123 crystals over a wide hole-doping range

(p = 0.097− 0.205) [14]. One of the samples was a large underdoped single crystal in which

weak IUC magnetic order had been detected in the pseudogap phase by PND, while the others

were mosaics of small high-quality single crystals grown at the University of British Columbia

(UBC). The temperature dependence of the ZF-µSR relaxation rate for all samples was shown

to be qualitatively similar, without any outstanding features related to T ∗ (see Fig. 1.7). Above

T ∼ 150 K, the ZF-µSR relaxation rates in all samples exhibited a significant reduction with

increasing temperature, signifying motional narrowing associated with diffusion of the muons.

The only other notable feature in the temperature dependence of the ZF-µSR relaxation rate

is an increase below T ≈ 50 K observed in all samples. A modification of the nuclear dipole

contribution to the ZF-µSR signal caused by unbuckling of the CuO2 planes is the likely source

of the enhanced relaxation rate below 50 K [99]. A small precession signal indicative of mag-

netic order corresponding to a volume fraction of just ∼ 3% was detected in the large single

crystal investigated by PND. However, while the magnitude of the average internal magnetic

field sensed by the muon agrees with that expected for the magnetic order detected by PND, the

small volume fraction and the absence of a similar signal in the other higher quality samples

suggests the signal arises from an impurity phase.

Other unique aspects of Y123 may contribute to the temperature and/or hole-doping depen-

dence of the ZF-µSR relaxation rate. For example, it has been shown that magnetic correlations

arise from oxygen vacancies in CuO chains [100]. Moreover, multiple muon sites have been

identified in Y123 whose population ratio is dependent on p [101, 102]. It is difficult to disen-

tangle these additional contributions to the behaviour of the relaxation rate of the ZF-µSR signal

from any potential magnetic order in the pseudogap phase. Consequently, Y123 is not the ideal

high-Tc cuprate superconductor to study with ZF-µSR in order to search for pseudogap phase

related magnetism. Recent µSR studies by Zhang et al. [103] and Zhu et al. [104] claim to have

detected “slow magnetic fluctuations and critical slowing down” and “fluctuating magnetism”

in the pseudogap phase of Y123. However, these studies do not address the many confound-

ing contributions to the temperature dependence of the ZF-µSR relaxation rate that have been

discussed so far [105].

Zero-field µSR studies by independent groups have found no evidence of magnetic order

in the pseudogap phase of La214. MacDougall et al. performed ZF-µSR measurements on three

La214 samples over a wide doping range (p = 0.13, 0.19, 0.30) [12]. No evidence for magnetic

order was observed in any of the samples and the ZF-µSR relaxation rate was found to be inde-

pendent of hole-doping. An independent ZF-µSR study of the pseudogap phase of several La214

single crystals in the hole-doping range p = 0.15 to 0.24 was performed by Huang et al. [13].

Despite accurately accounting for the effects of the dipolar and quadrupolar interactions of the

muon with the host nuclei by numerical calculations, no evidence of magnetic order in the pseu-

11

0 3 6 90.0

0.5

1.0

c)

b)

Time (µs)

P(t) 161 K

20 Ka)

0.00

0.01

0.02

f)

e)

d)

p = 0.11

y = 6.6

0.00

0.01

0.02

ν1 (M

Hz)

ν 1 (

MH

z)

p = 0.097

0.00

0.01

0.02

p = 0.14

0 50 100 150 200 2500.00

0.01

0.02

Temperature (K)

p = 0.11

y = 6.57

0 50 100 150 200 2500.00

0.01

0.02

g)

Temperature (K)

p = 0.205

0.00

0.01

0.02

p = 0.172

Figure 1.7: (a) Representative ZF-µSR asymmetry spectra in an underdoped Y123 single crystal(p = 0.11). (b)-(g) The temperature dependence of the ZF-µSR frequency6ν1 in Y123. Thisfigure is from Ref. [14].

dogap phase was found. Except for a reduction above T ∼ 150 K attributed to muon diffusion,

the ZF-µSR relaxation rate was observed to be independent of hole-doping and temperature.

There are a number of reasons why an additional ZF-µSR study of Bi2212 could yield differ-

ent results. First, Bi2212 lacks CuO chains and can be reliably overdoped by excess oxygenation,

unlike Y123 [106, 107] or La214 which require cation substitution that may generate cation-

induced disorder. Unlike La214, the intra-unit-cell magnetic order detected by PND is believed to

be long-range and three-dimensional [9]. Overdoped Bi2212 samples (up to p ≈ 0.22) also have

T ∗ temperatures that are below the onset temperature of muon diffusion (T ∼ 150 K) [83, 108]

and are also free of static, short-range spin correlations that are present in underdoped samples.

6The depolarization of the ZF-µSR signal in [14] was modeled by a cosine function, the frequency of which tracksthe signal’s weak relaxation rate.

12

A previous low-temperature ZF-µSR study of underdoped and optimally doped Bi2212 single

crystals (p = 0.094 and 0.16) found no evidence of magnetic order down to 25 mK and placed

a narrow limit of 109 to 1011 Hz on the fluctuation rate of any potential IUC magnetic order

[19].

A subsequent study by Pal et al. investigated the temperature dependence of the ZF-µSR

relaxation rate in optimally and overdoped Bi2212 single crystals (p = 0.16, 0.198) up to

T ∼ 160 K [20]. No signature of magnetic order was observed but a weak, temperature de-

pendent relaxation rate was observed, apparently electronic in origin. Longitudinal-field (LF)

µSR measurements determined that the local internal fields responsible for the temperature

dependent relaxation are quasistatic on the muon time scale and on the order of 1 G. However,

since the samples were optimally and slightly overdoped, the pseudogap temperature is above

the onset temperature of muon diffusion and the study could not determine if the onset tem-

perature of the temperature dependent spin relaxation signal coincides with T ∗. This is the first

open question addressed by this thesis: Is the weak quasistatic magnetism detected in by Pal et

al. [20] related to the pseudogap phase?

1.5 Charge Order

In cuprates, there is charge ordering associated with an incommensurate self-arrangement of

valence electrons in the CuO2 planes [109]. Neutron scattering measurements were the first

to report evidence for alternating static charge and spin modulations in what was dubbed a

“stripe order” phase in La1.6−xNd0.4SrxCuO4 (x = 0.12) [110]. Thus far, static stripe order

has only been observed in the doped La214 family. Later, it was revealed that a ubiquitous

fluctuating short-range CDW order also exists in hole-doped cuprates in zero magnetic field,

including Y123 [111–114] and Bi2212 [115, 116]. Note, in Y123 this is distinct from the CDW

order that occurs in the CuO chains. The CDW order is primarily present in the underdoped

regime (0.08 < p < 0.17) and is detected at a characteristic temperature TCDW, which is above

Tc but below T ∗ (see Fig. 1.4). Similar to the superconductivity in cuprates, the fluctuating CDW

order is quasi two-dimensional with the coherence length in the CuO2 planes greatly exceeding

that in the perpendicular direction.

The short-range CDW order seems to compete with superconductivity. Below Tc , the coher-

ence length and the intensity of the CDW signal observed by resonant soft X-ray scattering are

reduced [113, 117, 118]. Also, Tc is suppressed near 1/8 hole-doping concentration where the

charge ordering is strongest [111, 113, 115, 117, 119]. When superconductivity is suppressed

via large magnetic fields, both the X-ray scattering intensity and correlation length of the charge

order are enhanced and 3-D CDW order appears, characterized by a divergent correlation length

as T → 0 [111, 120, 121]. The presence of 3-D CDW order hints at the existence of a charge-

order driven QCP near optimal doping, and some authors have argued for the existence of

charge density fluctuations over a more extensive region of the phase diagram [122, 123]. Re-

13

cently, short-range charge density fluctuations have been observed by resonant X-ray scattering

persisting well above the pseudogap temperature T ∗ over an extended region of the cuprate

phase diagram [109, 124].

1.6 Superconducting Fluctuations

The superconducting order parameter ψ is a complex wavefunction possessing an amplitude

and a phase

ψ =p

nse−iθ , (1.2)

where ns is the superfluid density and θ denotes the phase of the order parameter. The suscep-

tibility of the superconducting phase to fluctuations is characterised by an energy scale known

as the “phase stiffness” [23]

V0 =~

2ns(0)a4m∗

, (1.3)

where ~ is the reduced Planck constant, ns(0) is the superfluid density at zero temperature, a

is a length scale related to the coherence length ξ0 and m∗ is the effective mass of an electron.

Equation (1.3) can be expressed in terms of the zero-temperature magnetic penetration depth

λ(0) as

V0 =(~c)2a

16πe2λ2(0). (1.4)

If the effects of phase fluctuations are neglected, the upper bound on the superconducting tran-

sition temperature Tc is given by

Tmaxθ = AV0, (1.5)

where A is a dimensionless parameter on the order of 1 describing short-range physics. For

a simple cubic system A = 2.2 [125], while for near-zero coupling between two-dimensional

planes A = 0.9 [23, 126]. For Bi2212 and Y123, the zero-temperature in-plane magnetic pen-

etration depth is approximately λab(0) = 1850 Å and 1600 Å [127, 128], respectively, which

corresponds to Tmaxθ/Tc ∼ 1.5. In contrast, Tmax

θ/Tc ranges anywhere from 300 in the heavy

fermion compound UBe13 [129] to 2 × 105 in the conventional superconductor Pb [51]. Due

to the lower superfluid density in cuprates, phase fluctuations play a critical role in determin-

ing Tc . For these materials, Cooper pairing apparently occurs at temperatures above Tc , but the

material only becomes superconducting when long-range phase coherence is achieved at Tc . In

conventional superconductors, on the other hand, Tc is controlled by the breaking of Cooper

pairs and phase fluctuations are unimportant [23].

Early experimental evidence that Tc marks the onset of phase coherence and not Cooper

pairing in the underdoped cuprates came from µSR experiments. Uemura and coworkers used

TF-µSR to demonstrate that Tc is linearly proportional to the superfluid density ns(T → 0) in

underdoped cuprates [22, 130, 131]. The TF-µSR relaxation rate is a measure of the width of

the internal magnetic field distribution, which in the vortex state of a superconductor is pro-

14

portional to the inverse squared London penetration depth 1/λ2, which in turn is proportional

to the superfluid density ns. The linear relationship between Tc and ns(T → 0) provided early

experimental evidence that Tc is dependent on the superfluid stiffness in underdoped cuprates.

1.6.1 Experimental Evidence for Superconducting Fluctuations

A vortex-motion contribution to the Nernst signal7 of La214 (p = 0.05 − 0.17) at tempera-

tures up to several times Tc provided initial evidence for phase-fluctuating Cooper pair corre-

lations in high-Tc cuprates [28]. A field-enhanced diamagnetic signal detected above Tc ob-

served by torque magnetometry (TM) was found to track the Nernst signal closely providing

corroborating evidence of superconducting fluctuations (SCFs) in underdoped to overdoped

Bi2212 (p ≈ 0.086− 0.214) [24]. It was later argued that there are multiple contributions to

the Nernst signal and that the onset temperatures of SCFs are significantly lower than previously

believed [27]. A study of the penetration depth in YBa2Ca3O6.95 discovered a temperature de-

pendence of the form λ(T )∝ (1 − T/Tc)−0.33 over the temperature interval 0.001 < (Tc −

T )/Tc < 0.1, consistent with 3-D X Y critical fluctuations [132]. Microwave absorption mea-

surements for magnetic fields up to 16 T show evidence of SCFs in a narrow range of temperature

above Tc in underdoped Y123 (Tc = 57 K) [26]. Yet, scanning tunneling microscopy measure-

ments have provided direct visual evidence of pairing gaps in nanoscale-sized regions on the

surface of Bi2212 (p ≈ 0.14 − 0.23) at temperatures up to 70 K above Tc at optimal doping

[25]. These regions grow rapidly with decreasing temperature until macroscopic superconduc-

tivity is achieved at Tc . The onset temperature at which pairing gaps are detected in more than

10% of the surface also roughly scales with Tc in the optimally to overdoped regime, indicating

the presence of SCFs in the overdoped regime as well. The detection of SCFs in the overdoped

regime, where Tc is reduced despite the availability of additional charge carriers hints at the

potential competition between superconductivity and another phase. The lack of consensus as

to the extent above Tc at which Cooper pairing occurs is likely due to the sensitivity of the

various experimental techniques and the potential inhomogeneous nature of these correlations.

Figure 1.8 displays the onset temperatures for SCFs inferred by different experimental methods.

In the absence of a magnetic field, the ZF-µSR signal for well underdoped high-Tc cuprates

is depolarized by the internal magnetic fields associated with remnant Cu moments [57–60].

When a large magnetic field is applied parallel to the crystallographic c axis, strong field-induced

quasistatic magnetism is observed by TF-µSR in underdoped and Eu-doped (La, Sr)2CuO4 and

La1.875Ba0.125CuO4 persisting well above Tc but absent in optimally and overdoped samples

[133]. There is also a contribution to the TF-µSR signal from paramagnetic moments in highly

overdoped cuprates. In an applied magnetic field, the paramagnetic moments contribute a Curie-

like temperature dependence to the TF-µSR relaxation rate that is enhanced with increasing p

7The Nernst coefficient is defined by ν= E/H∇T , where E is the transverse electric field generated from a samplesubject to an applied field H and thermal gradient ∇T .

15

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0

50

100

150

200

250TM, Bi2212

Wang et al. (2005)

Nernst, Bi2212

Wang et al. (2006)

Nernst, Y123

Cyr-Choinière et al. (2018)

STM, Bi2212

Gomes et al. (2007)

TF-µSR, Bi2212

Lotfi-Mahyari et al. (2013)

T (

K)

Hole-doping, p

Tc

d-wave

Superconductor

Figure 1.8: Onset temperatures of signatures of superconducting fluctuations detected in Y123and Bi2212 by different experimental techniques.

[134]. Hence, the TF-µSR contributions from Cu magnetic moments at low doping and param-

agnetic moments at high doping are recognizable. A TF-µSR study of samples at intermediate

doping found an unexpected temperature-dependent inhomogeneous magnetic field response

above Tc in La214 and Y123, which appeared to be distinct from the effects of Cu moments in the

underdoped regime and paramagnetic moments in the overdoped regime [31]. The hole-doping

dependence of the TF-µSR relaxation rate was found to be proportional to Tc even far above Tc

and to increase with the applied magnetic field. In Y123, the TF relaxation rate above Tc follows

the characteristic dip in Tc near p = 1/8 doping, suggesting the source of the relaxation above

Tc is related to superconductivity — rather than field-induced spin-density-wave (SDW) or CDW

order,8 which would enhance the relaxation rate. A subsequent extended study showed that the

TF-µSR relaxation rate in Bi2212 exhibits a similar temperature dependence above Tc and that

it also tracks Tc as a function of hole-doping far above Tc (see Fig. 1.8) [30]. Given that the

degree of chemical disorder varies substantially between the Y123, La214 and Bi2212 families,

the “universal” broadening of the internal magnetic field distribution above Tc suggests an in-

trinsic propensity toward electronic inhomogeneity in the normal state. While spatially varying

changes in the Pauli susceptibility χ0 could in principle cause the observed broadening of the

TF-µSR lineshape, this would be a small effect. Furthermore, χ0 would be dominated by the

reduction in N(EF ) due to the pseudogap phase in the underdoped samples. The field-induced

8In the absence of a magnetic field, the quantization axis of nuclear moments is defined by the maximum EFGso a CDW can modify the nuclear dipolar field distribution at the muon site. In large magnetic fields, however, themuon is effectively “blind” to EFG effects as the quantization axis of the nuclear spins is defined by the magneticfield.

16

broadening of the TF-µSR lineshape also cannot be due to the vortex liquid inferred from the

Nernst signal above Tc , since the vortices fluctuate too quickly. However, the line broadening

could originate from spatially inhomogeneous SCFs characterized by nanometer-size regions

exhibiting varying degrees of fluctuating diamagnetism. Muons stopping in different regions

of fluctuating diamagnetism experience distinct time-averaged magnetic fields, resulting in a

distribution of time-averaged fields sensed by the muon ensemble. As the islands of fluctuating

diamagnetism grow with decreasing temperature, the TF-µSR field distribution broadens [135].

While previous TF-µSR studies of hole-doped cuprates seemed compatible with this scenario,

the experiments were unable to explicitly show that the source of the inhomogeneous magnetic

field response is diamagnetic in origin. This is the second open question addressed by this thesis:

Is the universal inhomogeneous magnetic field response observed in hole-doped cuprates above

Tc [30] diamagnetic in nature?

1.7 Ferromagnetic Fluctuations and Ferromagnetism in the Heavily

Overdoped Regime

As mentioned earlier, the heavily overdoped side of the phase diagram displays features charac-

teristic of a Fermi liquid. These include an approximate T2 dependence of the electrical resistiv-

ity and an adherence to the Wiedemenn-Franz law [53, 136]. However, it is not understood why

both the superconducting transition temperature and the superfluid density are reduced with

increased hole-doping concentration above optimal doping and eventually vanish. While some

have argued that the simultaneous reduction of both Tc and ns observed in highly overdoped

La214 [137] can be explained within the framework of dirty d-wave BCS theory [138], an alter-

native proposal is that there is a competition between superconductivity and a low-temperature

ferromagnetic (FM) phase residing immediately beyond the superconducting state [139]. In

this scenario, a QCP is located where the superconducting dome ends and a FM phase emerges.

The proposed FM phase is facilitated by rather weak interlayer coupling of the CuO2 planes

and consequently only exists at very low temperatures. This initial theoretical proposal was pri-

marily motivated by the large upturn in the uniform magnetic susceptibility of Tl2Ba2CuO6±δ(Tl2201) at low temperatures, which cannot be explained by a Curie paramagnetic impurity

phase [140]. Subsequent electronic band theory calculations performed for La2−xBaxCuO4 su-

percells also show a tendency toward FM order at high doping near clusters of high Ba con-

centration [141]. While a ZF-µSR study on highly-overdoped superconducting (p = 0.24) and

non-superconducting (p = 0.33)9 La214 single crystals [143] found no evidence for magnetic

order, an enhancement and eventual saturation of the ZF-µSR relaxation rate below T ∼ 0.9 K

9Achieving hole-doping homogeneity in such heavily-overdoped samples is difficult. TF-µSR measurements haverevealed a stark difference between “as-grown” and post annealed under high oxygen partial pressure single crystalsfor p = 0.33 [142].

17

was observed in the p = 0.33 sample, indicating the development of weak static magnetism.

In-plane electrical transport measurements on the p = 0.33 single crystal also revealed a tem-

perature dependence of the resistivity of the form ρab(T ) = ρab(0) + AT5/3 above 60 K. The

T5/3 power dependence is consistent with self-consistent renormalization (SCR) theory of spin

fluctuations for 3-D weak itinerant-electron ferromagnetism [144].

Kurashima et al. have found evidence for the development of FM fluctuations in heavily

overdoped (Bi,Pb)2Sr2CuO6+δ (Bi2201) single crystals up to p = 0.282, using a combination

of thermodynamic techniques and ZF-µSR [145]. Unlike La214, the Bi2201 system is doped

with excess oxygen, ruling out effects from cation substitution-induced disorder and oxygen

deficiency in the CuO2 planes. While no hysteresis in the magnetization was observed down to

T = 0.5 K, the saturation of the magnetization below 20 K in high magnetic fields was found

to be incompatible with the paramagnetic Brillouin function, suggesting that the behaviour is a

precursor to a FM transition at lower temperatures. ZF-µSR was used to gain insight into the spin

dynamics of the system. The ZF-µSR relaxation rate was found to increase with decreasing tem-

perature below T = 200 K, with the increase becoming larger with hole-doping. This suggests

that FM spin fluctuations develop above 200 K and are enhanced with hole-doping. In-plane re-

sistivity measurements revealed a temperature dependence described by ρab(T ) = ρab(0)+AT n

above Tc . The exponent n was found to be close to 1 near optimal doping, but increases and

eventually saturates near a value of 4/3 at higher doping. The T4/3 power dependence is in

agreement with the SCR theory for 2-D itinerant FM spin fluctuations and incompatible with

the conventional T2 behaviour expected for a non-magnetic Fermi liquid [146]. The authors at-

tribute the discrepancy between the dimensionality of the itinerant FM fluctuations (as deduced

by fits to the predicted behaviour from SCR theory) in Bi2201 and La214 to the greater distance

between the CuO2 planes in the former compound.

In addition to experimental evidence for competing FM order in the heavily-overdoped

regime of hole-doped cuprates, very recently evidence for FM order below T = 4 K has been

reported for electron-doped thin film La2−xCexCuO4 (x = 0.19) [147]. The evidence for a low-

temperature FM phase comes from magnetoresistance, magnetization measurements, and the

detection of the polar Kerr effect below T = 4 K.

18

Chapter 2

Experimental Methods

2.1 Muon Spin Relaxation (µSR)

The primary experimental technique applied for this thesis project is µSR. In general, µSR stands

for muon spin relaxation, rotation or resonance, depending on the experimental configuration.

The µSR technique utilizes a nearly 100% spin-polarized muon beam and the anisotropic emis-

sion probability of the positron from the muon decay to serve as a sensitive local probe of mag-

netism in materials; capable of measuring magnetic fields as small as 0.1 G. In condensed matter,

the positive muon (µ+) behaves like a light short-lived proton whereas the negative muon (µ−)

behaves like a heavy unstable electron. The positive muon is more widely used for µSR due

to the different behaviours of the µ+ and µ− in matter. Table 2.1 lists some of the fundamen-

tal properties of the positively charged muon, the electron and the proton. The combination of

the muon lifetime and muon gyromagnetic ratio enables detection of internal magnetic fields

fluctuating at rates between 104 Hz and 1012 Hz, so that µSR bridges the ranges of sensitivity

to spin dynamics accessible by nuclear magnetic resonance (NMR) and neutron scattering (see

Fig. 2.1).

Comparison with NMR

The three letter acronym “µSR” was chosen to emphasize the close analogy with similar spin

resonance/relaxation techniques, and in particular, NMR. Both µSR and NMR are local probe

techniques that monitor the time evolution of a spin polarized probe (i.e., the host nucleus or

Table 2.1: Fundamental properties of the electron,proton and positve muon.

Mass Charge Spin Lifetime γ/2π(me) (e) (~) (µs) (MHz/T)

e− 1 -1 1/2 ∞ 28025p+ 1836 +1 1/2 ∞ 42.58µ+ 207 +1 1/2 2.197 135.5

19

10-4

10-2

100

102

104

106

108

1010

1012

1014

Neutron Scattering

µSR

Fluctuation Rate (Hz)

NMR

Figure 2.1: Comparison of the dynamic range for fluctuating internal magnetic fields detectableby NMR, µSR and neutron scattering. This figure is adapted from Ref. [148].

the µ+) to access information regarding the local magnetic and electronic properties of mate-

rials. The nearly complete spin polarization of the muon beams used for µSR is a considerable

advantage over conventional NMR, in which large magnetic fields on the order of several tesla

are required just to achieve a very small thermal equilibrium spin polarization (∼ 10−5) of the

host nuclei. The µSR signal is generated by the detection of the muon decay positrons, whereas

an NMR signal is generated by using an inductive pickup coil to detect the precession of the nu-

clear magnetization. Typically, 106 muon decays and 1019 nuclear spins are required to produce

quality µSR and NMR signals, respectively. Unlike most nuclei, the muon is a spin 1/2 parti-

cle that does not possess an electric quadrupole moment. Consequently, the muon is a “pure”

magnetic probe and is not directly sensitive to electric field gradients in materials. This can be

particularly advantageous when the interest is in probing weak internal magnetic fields. Finally,

an important distinction between NMR and µSR is that while the location of the nuclei in a solid

are known, the muon stopping sites within a crystal lattice are unknown a priori. Although the

latter can sometimes be determined experimentally or inferred using density functional theory,

in general, determining the muon sites in a crystal lattice can be a challenging task.

2.2 Muon Production, Transport and Decay

The µSR measurements reported in this thesis were carried out using the M20 and M15 surface

muon beamlines at TRIUMF. The TRIUMF cyclotron delivers 500 MeV protons that collide with

a cooled graphite or Be metal target. The collisions with the target nuclei generate positive pions

(π+) according to

p+ p→ p+ n+π+, (2.1)

p+ n→ n+ n+π+. (2.2)

20

Pions at rest decay via the weak interaction with a mean lifetime of τπ+ = 26 ns, emitting a

positive muon and a muon neutrino

π+→ µ+ + νµ. (2.3)

Due to parity violation in the weak interaction [149, 150], the spin of the emitted neutrino

is antiparallel to its linear momentum. Since the pion is spinless, conservation of linear and

angular momentum require the spin of the muon to also be oriented antiparallel to its linear

momentum, in the pion rest frame. This enables production of a spin-polarized beam of muons.

Pions that decay at rest near the downstream surface of the production target isotropically

emit “surface muons” that possess an average kinetic energy of 4.12 MeV and momentum of

29.79 MeV/c. These surface muons prove to be the most suitable for µSR experiments because of

their high stopping density (∼ 140 mg/cm2) in solids [151]. By collecting the decay muons over

a small solid angle using quadrupole magnets, a nearly 100% spin-polarized beam is achieved.

The polarized muon beam is guided along the beamline from the production target toward the

experimental area via dipole and quadrupole magnets that steer and focus the beam. Prior to

reaching any µSR spectrometer, the muon beam passes through a pair of so-called “separa-

tors”, which are devices consisting of crossed electric and magnetic fields. The separators act

as a velocity filter that remove unwanted positrons produced from muons decaying in flight in

the beam channel. The separators also have the capability of rotating the direction of the ini-

tial muon spin polarization by up to 90. This is essential in a transverse-field configuration,

where a large magnetic field is applied perpendicular to the muon spin polarization direction.

In this case the field must be applied parallel to the muon linear momentum to avoid creating

a Lorentz force that deflects the muon beam. After passing through the separators, the muons

enter the experimental area and stop in the sample. At the end of their lifetime (mean lifetime

τµ = 2.196 µs), the muons decay via the weak interaction

µ+→ e+ + νe + νµ, (2.4)

where e+ is a positron, νe is an electron neutrino and νµ is a muon anti-neutrino. Parity vio-

lation in the weak interaction leads to an asymmetric angular probability distribution of decay

positrons that depends on the positron energy

W (E,θ ) = 1+ a0(E) cosθ , (2.5)

where θ is the angle between the emitted positron and muon spin direction upon decay. The

function a0(E) describes the asymmetry of the decay, and is given by

a0(ε) =2ε− 13− 2ε

, (2.6)

21

where ε = E/Emax and Emax = 52.83 MeV is the maximum energy of the emitted positron.

The value of a0(ε) ranges between -1/3 for the minimum positron energy (Emin = 0 MeV) to

unity for the maximum positron energy. Sampling the distribution of positron energies [152]

equally leads to the maximum theoretically achievable asymmetry a0 = 1/3. The angular decay

positron probability distribution is illustrated in Fig. 2.2 for several positron energies. In practice,

the experimental value of a0 is somewhat lower than the theoretical value due to a variety of

factors that include imperfect alignment of the positron detectors and differences in positron

detector efficiencies.

52.8

MeV

26.4

MeV

µ+

e+

θ

0 M

eV

Figure 2.2: The angular probability distribution of decay positrons for different positron ener-gies given by Eq. (2.5). With increasing energy, the angular distribution of the decay positronbecomes more anisotropic and cardioid-like, with the positron more likely to be emitted alongthe direction of the muon spin.

2.3 Total Magnetic Field Sensed by the Implanted µ+

At the µ+ site, the presence of a magnetic field ~Bµ with a component perpendicular to the muon

spin results in Larmor precession of the muon magnetic moment with an angular frequency

ωµ = γµBµ, (2.7)

where γµ/2π= −0.0135538817(3) MHz/G is the muon gyromagnetic ratio [153].

22

The magnetic field at the µ+ stopping site is the vector sum of internal and applied magnetic

fields. The sources of the internal magnetic field include dipolar fields arising from from localized

nuclear and electronic moments (~Bdip) and contact hyperfine fields (~Bc) due to the overlap of

the muon wavefunction with the net spin density of polarized conduction electrons. In order

to account for the total dipolar field sensed by the muon, it is useful to introduce a “Lorentz

sphere” approximation,1 which basically allows the sum associated with dipolar magnetic fields

from localized magnetic moments in the entire sample to converge.2 In this approximation, the

muon is surrounded by an imaginary sphere. Within the sphere the magnetic field is treated

microscopically and the dipolar field is calculated with a restricted lattice sum. Outside of the

sphere, the dipolar magnetic field due to the localized moments is treated in the continuum

limit. The total magnetic field at the muon site is

~Bµ = ~Bext − ~Bdemag + ~BLor + ~Bc + ~Bdip, (2.8)

where ~Bext is the external applied magnetic field, ~Bdip accounts for dipolar fields from the local

moments inside the Lorentz sphere, ~Bdemag is the demagnetization or stray field produced by the

finite magnetized sample and ~BLor is the “Lorentz” field generated by the empty Lorentz sphere

in the magnetized sample (which aligns with the sample magnetization). The demagnetization

and Lorentz fields are proportional to the magnetization ~M of the sample

~Bdemag = −N ~M , (2.9)

~BLor =4π3~M , (2.10)

where N is a demagnetization factor dependent on the geometry of the sample and the factor

1/3 in the Lorentz field arises from the demagnetization factor of the Lorentz sphere. When the

magnetization is induced by an external magnetic field, then

~M = χtot~Bext, (2.11)

where χtot is the sample’s bulk magnetic susceptibility. These contributions to the total magnetic

field at the µ+ stopping site are illustrated schematically in Fig. 2.3.

2.4 Time-differential µSR

In a time-differential µSR experiment, a muon first passes through a thin plastic muon detector

that triggers the start of an electronic clock. The muon subsequently stops in the bulk of the

1CGS units are used for these calculations throughout this thesis.

2The dipole field of a localized moment falls with distance r from the muon as 1/r3, while the number of momentsgrows with increasing volume as r2. Consequently, the lattice sum is proportional to 1/r and diverges logarithmically.

23

𝜇+𝑩ext 𝑩Lor𝑴/𝑽𝑩demag

Lorentz sphere

N

S

N

S

Figure 2.3: Illustration of the Lorentz sphere approximation for a uniformly magnetized, singledomain sample with an external magnetic field applied along its long axis. Inside the Lorentzsphere the magnetic field is calculated microscopically by summing over the dipolar fields ofindividual magnetic moments. Outside the sphere, the sample magnetization generates a de-magnetization field ~Bdemag that depends on the sample geometry. The Lorentz field ~BLor arisesfrom placing the Lorentz sphere in the magnetized environment.

sample (considered to be the same time that the muon detector was triggered) and its spin

precesses according to the local magnetic field it senses. After some time t, the muon decays

and emits a positron preferentially along the direction of its spin. This positron is then detected

by one of the positron counters positioned about the sample. The positron detectors are typically

arranged in pairs, with counters on opposite sides of the sample. For example, a forward (F) and

backward (B) detector placed behind and in front of the sample, respectively. Once the positron

is detected, the electronic clock stops and the elapsed time (defined as the “event”) is recorded

in a time histogram for the detector [see Fig. 2.5(a)]. The maximum time between detection of

the incoming muon and the decay positron is typically restricted to a few muon lifetimes, (i.e.,

∆t ≈ 10 µs). During this time window, other muons triggering the muon detector are rejected

by the data acquisition electronics. A schematic diagram illustrating this sequence of events is

depicted in Fig. 2.4.

The number of counts in the time histograms of opposing positron detectors i and j are

described by

Ni(t) = Ni(0)e−t/τµ

1+ a0,i P(t))

+ Bi , (2.12)

N j(t) = N j(0)e−t/τµ

1− a0, j P(t))

+ B j , (2.13)

where a0,i is the decay positron asymmetry factor for detector i, P(t) is a function describing

the time evolution of the muon spin polarization and Bi characterizes the random background

for detector i. The pair of time histograms [Eq. (2.12) and Eq. (2.13)] can be combined to form

24

the “raw” asymmetry signal

Araw(t) =Ni(t)− N j(t)

Ni(t) + N j(t), (2.14)

where Bi = B j is assumed. In reality, Bi 6= B j , but both quantities are experimentally deter-

minable. The raw asymmetry can be corrected to account for unequal detector efficiency and

initial amplitudes by introducing the parameters α = N j(0)/Ni(0) and β = a0, j/a0,i . Inserting

Eq. (2.12) and Eq. (2.13) into Eq. (2.14) we have

A(t) = a0Pz(t) =Araw(t)(α+ 1)− (α− 1)(αβ + 1) + Araw(t)(αβ − 1)

. (2.15)

Formation of the asymmetry spectrum removes the exponential decay associated with the muon

lifetime and, provided enough events are recorded, it yields an accurate measurement of the

time evolution of the spin polarization of the muon ensemble P(t), as illustrated in Fig. 2.5.

2.5 Zero-field µSR

As the name suggests, zero-field (ZF) µSR is carried out in the absence of an applied magnetic

field. As mentioned earlier, the direction of the default (unrotated) initial muon spin polarization

is antiparallel to its linear momentum, which is defined to be along the z-axis. The ZF condi-

tion is typically achieved by removing stray magnetic fields detected with a Hall probe at the

sample position using three mutually perpendicular Helmholtz coils. A schematic of a ZF-µSR

experimental configuration and representative ZF asymmetry spectra are shown in Fig. 2.4 and

Fig. 2.5, respectively.

2.5.1 Zero-field Relaxation Functions

The general expression for the time evolution of the muon spin polarization subject to an internal

magnetic field, Bµ is given by

Pz(t) = cos2 θ + sin2 θ cos (γµBµ t), (2.16)

where θ is the angle between the spin polarization and the direction of the magnetic field.

When the muon ensemble experiences a non-uniform internal magnetic field, not all of the

muons sense the same value of the field. Consequently, they Larmor precess at different fre-

quencies resulting in a depolarization of the µSR signal. The depolarization is described by a

relaxation function Gz(t) calculated by averaging Eq. (2.16) over the appropriate field distribu-

tion, n(~Bµ) = n(Bµ,x)n(Bµ,y)n(Bµ,z) such that

Gz(t) =

∫ ∫ ∫

dBµ,x dBµ,y dBµ,z

Pz(t)n(~Bµ)

. (2.17)

25

𝜇+

𝑒+

𝑷(𝑡 = 0)

x

y

z

𝒑

1 𝜇s

0 𝜇s

Figure 2.4: Schematic diagram for a standard zero-field (ZF) µSR experiment. Muons arriveone-by-one and pass through a thin plastic scintillator muon detector, triggering the start of anelectronic clock. The muon then stops in the sample. Upon decay, a positron is emitted prefer-entially along the muon spin direction and subsequently detected by either the forward (F) orbackward (B) detector, stopping the clock. The event, which is defined by the time differencebetween detection of the incoming muon and the detection of the decay positron, is recordedin a “time-differential” histogram for the positron detector. The asymmetry spectrum is thenconstructed according to Eq. (2.14) [see Fig. 2.5].

In a system of dense, static (relative to the muon lifetime) and randomly oriented magnetic

moments (e.g., nuclear dipole moments), n(~Bµ) is approximated by a Gaussian distribution

n(Bµ,i) =γµq

2π∆2G

exp

−γ2µB2µ,i

2∆2G

(i = x , y, z), (2.18)

where ∆G/γµ is the root mean square (RMS) width of the field distribution (in magnetic field

units). Inserting Eq. (2.18) into Eq. (2.17) and carrying out the integration leads to the so-called

Gaussian Kubo-Toyabe relaxation function [154]

Gz(t) =13+

23(1−∆2

G t2)exp

−∆

2G t2

2

. (2.19)

This ZF relaxation function assumes the internal field distribution is isotropic and is usually

a good approximation of the muon spin relaxation due to the dipolar magnetic fields of the

26

0 2 4 6 8 10 0 2 4 6 8 100.0

0.1

0.2

0.3

Posi

tron

cou

nts

(arb

. u

nit

s)

t (µs)

NB(t)

NF(t)

(a)

(b)

A(t

)

t (µs)

Figure 2.5: (a) Simulation of the number of counts as a function of time for opposing positrondetectors generated using Eqs. (2.12) and (2.13) assuming a0 = 0.25. The depolarization func-tion is assumed to be the product of an exponential relaxation function and the Gaussian Kubo-Toyabe function [Eq. (2.19)] with ∆G = 0.3 µs−1 and λ = 0.1 µs−1. (b) The raw asymmetrycalculated with Eq. (2.14), which removes the exponential decay associated with the lifetime ofthe muons to reveal the time evolution of the muon ensemble’s spin polarization.

host nuclei. As shown in Fig. 2.6(a), a feature of this relaxation function is the recovery of the

asymmetry to 1/3 at late times, which for ∆G > 0.15 µs−1 is observable in a typical 10 µs time

window. This recovery to 1/3 can be understood intuitively. On average 1/3 of the randomly

distributed fields point along the direction of the initial muon spin polarization, which does not

induce spin precession for those muons.

In a system consisting of dilute, randomly oriented static moments, the internal magnetic

field distribution sensed by the muon is approximated by a Lorentzian distribution

n(Bµ,i) =γµ

π

∆L

∆2L + γ

2µB2µ,i

(i = x , y, z), (2.20)

where ∆L/γµ is the half-width at half-maximum (HWHM) of the field distribution. Integrating

Pz(t) over this field distribution leads to the static Lorentzian Kubo-Toyabe relaxation function

[see Fig. 2.6(b)]

Gz(t) =13+

23(1−∆L t)exp (−∆L t). (2.21)

Dynamic Fields and Muon Diffusion

Fluctuating magnetic fields lead to modifications of the relaxation functions described above.

For sufficiently fast fluctuations of a Gaussian distribution, (i.e., ν >∆G, where ν is the charac-

teristic fluctuation rate and∆G is the width of the Gaussian field distribution), the corresponding

27

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Pz(t

)

t (µs)

∆G = 0.15 µs

-1

∆G = 0.30 µs

-1

∆G = 0.70 µs

-1

(a) (b)

Pz(t

)

t (µs)

∆L = 0.15 µs

-1

∆L = 0.30 µs

-1

∆L = 0.70 µs

-1

Figure 2.6: (a) Gaussian and (b) Lorentzian static Kubo-Toyabe relaxation functions shown fordifferent values of ∆G or ∆L.

relaxation function is described by the so-called “Abragram” formula

Gz(t) = exp

− 2

∆2G

ν2

(e−νt − 1+ νt)

. (2.22)

In the fast fluctuation limit ν/∆G≫ 1, Eq. (2.22) is reduced to a single exponential relaxation

function

Gz(t) = e−λt , (2.23)

where λ = 2∆2G/ν.

In some materials, a positive muon may have enough thermal energy to break free from

its stopping site and diffuse to higher energy stopping sites. When this occurs in an inhomo-

geneous medium, each muon experiences multiple magnetic environments during its lifetime

and is therefore exposed to a “time-averaged” magnetic field. The distribution of time-averaged

magnetic fields is more uniform than the magnetic field distribution sensed by a static muon

ensemble, a phenomenon that is known as “motional narrowing” — a reference to the narrow-

ing of the NMR linewidth by diffusion of the host atoms (nuclei). This results in a diminished

relaxation rate that is proportional to the muon diffusion rate.

Stretched Exponential

The following phenomenological stretched exponential function may be employed when the

aforementioned relaxation functions do not adequately describe the muon spin relaxation spec-

trum:

Gz(t) = exp

−(λt)β

. (2.24)

This relaxation function makes no assumptions about the internal magnetic field distribution

sensed by the muon ensemble, but can be more effective in capturing the functional form of

the depolarization via a stretch parameter, β . Stretched exponential relaxation with non-trivial

28

stretch values (i.e., β 6= 1 or 2) are sometimes an indication of inhomogeneous spin freezing,

a distribution of local magnetic field fluctuation rates or multiple magnetically inequivalent

muon sites in the crystal lattice. Sample inhomogeneity and phase separation can also cause a

single muon stopping site in one location of the sample to be magnetically inequivalent from

another, leading to stretched exponential relaxation. While in such scenarios, the magnetic vol-

ume fractions associated with the phase separation or population ratios of the multiple muon

sites can generally be modelled accurately by multi-component fitting functions, the stretched

exponential serves as a convenient approach for phenomenologically approximating non-trivial

relaxation behaviour. As will be shown later in Chapter 3, a stretched exponential relaxation

function was used to effectively model the ZF muon spin relaxation in Bi2212, accounting for a

very small temperature dependent background signal contribution to the asymmetry spectra.

2.6 Longitudinal-field µSR

The experimental configuration for a longitudinal-field (LF) µSR experiment is identical to that

used for ZF-µSR except that an external magnetic field is applied parallel to the direction of the

initial muon spin polarization. The purpose of LF-µSR measurements is to distinguish between

static and dynamic internal magnetic fields. If the internal magnetic fields are dynamic, LF-

µSR may be used to determine the characteristic fluctuation rate and the RMS amplitude of

transverse fluctuations of the local field.

The application of a longitudinal magnetic field ~BLF leads to two muon Zeeman levels —

a higher energy eigenstate |+⟩ corresponding to the muon magnetic moment ~mµ oriented an-

tiparallel to the magnetic field, and a lower energy eigenstate |−⟩ associated with the muon

magnetic moment oriented parallel to the magnetic field. The energy difference ∆E between

the upper and lower energy levels is

∆E = −⟨ ~mµ · ~BLF⟩= −γµ⟨~Iµ · ~BLF⟩= −12γµ~BLF, (2.25)

where both ~Iµ and ~BLF are along z and Iµ,z = ~/2 is the spin of the muon such that

|∆E|= 12~ωµ. (2.26)

Transitions between the |−⟩ and |+⟩ eigenstates are induced by fluctuating magnetic fields with a

component transverse to the applied field and fluctuating at a frequency that corresponds to the

energy difference between the Zeeman energy levels. This causes “flipping” between the muon

spin states and a depolarization of the muon ensemble. Therefore, by changing the longitudinal

magnetic field and tuning the Zeeman energy levels, the spin system’s spectrum of fluctuations

can be directly probed by monitoring the LF relaxation rate.

29

In the fast fluctuation limit ν≫∆G, the fluctuation rate ν and RMS value of the fluctuating

transverse component of the local magnetic field can be determined by fitting the LF dependence

of the exponential relaxation rate λ to the following Redfield equation

λ(BLF) =2∆2

Gν

(γµBLF)2 + ν2

, (2.27)

where λZF = 2∆2G/ν is the value of λ for BLF = 0 and λ→ 0 as BLF→∞.

If the internal magnetic fields are static, then the application of a LF in the z-axis direction

modifies the internal field distribution such that n(Bz) → n(Bz − BLF). For an applied LF ∼10

times the value of the local internal fields [155], the vector sum of the applied LF and local field

at the muon site is essentially directed along z. In this case, the precessing component of the

muon spin polarization [see Eq. (2.16)] vanishes and since static magnetic fields cannot induce

transitions between the muon Zeeman levels, the muon ensemble preserves its spin polarization

over the entire time window and the muon spin is “decoupled” from the local internal fields.

2.7 Transverse-field µSR

To study the magnetic response of materials, a magnetic field can be applied transverse to the

initial muon spin polarization. At a continuous-wave muon facility (TRIUMF and PSI) dc mag-

netic fields on the order of several tesla can be applied to the sample in a transverse-field (TF)

µSR experiment. However, since the default spin polarization of the muons is anti-parallel to

their linear momentum, application of a large transverse field exerts a significant Lorentz force

on the muon beam, deflecting it from the straight line path to the sample. While this problem

can be solved by pre-steering the beam with a secondary magnet polarized opposite to that of

the primary magnet [156], a more convenient solution is to apply the magnetic field parallel

or anti-parallel to the muon linear momentum and then rotate the spin of the muons by up to

90 before implantation into the sample using the separator beamline elements described in

Section 2.2. A generic TF-µSR experimental configuration is illustrated in Fig. 2.7, and repre-

sentative TF asymmetry spectra are shown in Fig. 2.8.

2.7.1 Transverse-field Relaxation Functions

Muon spin depolarization functions in TF can be determined by the same method used to de-

termine ZF relaxation functions. The presence of a large external field results in some slightly

modified but familiar relaxation expressions. A dense system of static randomly oriented mo-

ments, such as those of nuclear moments, results in a Gaussian relaxation function

Gx(t) = e−σ2 t2

, (2.28)

30

𝜇+

x

y

z

𝒑𝑷(𝑡 = 0)

0 𝜇s

1 𝜇s

𝑩𝐞𝐱𝐭Figure 2.7: Schematic diagram for a standard transverse-field (TF) µSR experiment. An externalmagnetic field ~Bext is applied along the z-direction perpendicular to the initial muon spin polar-ization ~P(t = 0), causing the spin to precess in the x y plane. The decay positron is detected bythe left (L) or right (R) detector. Up and down positron counters above and below the sampleare also generally utilized to maximize the positron count rate.

where σ2 = γ2µ⟨B2

µ⟩ and ⟨B2µ⟩ is the width of the field distribution parallel to the external field.

A dilute system of static, randomly oriented spins leads to spin relaxation of the form

Gx(t) = e−λt . (2.29)

Since the muon spin polarization is fully perpendicular to the applied field, θ = π/2 in Eq. (2.16)

and in both of these cases the characteristic ZF feature of a 1/3 tail disappears.

Similar to the ZF situation, fast fluctuations of a Gaussian distributed internal magnetic field

leads to an exponential depolarization rate

Gx(t) = e−λt . (2.30)

In this case λ = σ2/ν, where σ2/γ2µ is the second moment of the Gaussian field distribution

and ν is the characteristic fluctuation rate.

31

0 2 4 6 8 10 0 2 4 6 8 10-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Posi

tron

cou

nts

(arb

. u

nit

s)

t (µs)

NB(t)

NF(t)

(a)

(b)

A(t

)

t (µs)

Figure 2.8: (a) Simulation of the number of counts as a function of time for opposing positron de-tectors generated using Eqs. (2.12) and (2.13) assuming a transverse applied field of B = 50 G,a0 = 0.25 and a depolarization function given by Gx(t) = e−σ

2 t2e−λt with λ = 0.1 µs−1 and

σ = 0.3 µs−1. (b) The corresponding raw asymmetry calculated with Eq. (2.14).

2.7.2 The Muon Knight Shift

As discussed in Section 2.3, contributions from dipolar, demagnetization, Lorentz, and contact

hyperfine magnetic fields lead to a local field at the muon site that can be different from the

externally applied magnetic field. The fractional difference between the local field at the muon

site and the externally applied field is known as the muon Knight shift.

In a muon Knight shift experiment, a silver (Ag) annulus is sometimes placed immediately in

front of the sample, such that a certain fraction of the incoming muons stop in the annulus, while

others pass through the annulus and stop in the sample. By sandwiching the Ag annulus between

two thin plastic scintillator detectors (an “inner” and “outer” muon detector) it is possible to

distinguish muons that stop in the annulus (only the outer detector is triggered) from those

stopping in the sample (both the inner and outer muon detectors are triggered). A schematic

diagram of such an arrangement is shown in Fig. 2.9. Consequently, muons stopping in the Ag

annulus can be used as a “reference” to accurately track the magnetic field applied to the sample.

The fractional difference between the sample and reference TF-µSR precession frequencies

Kexpt ≡νsample − νreference

νreference, (2.31)

has several contributions:

Kexpt = Kµ + KLor + Kdemag + Kspect + KAg. (2.32)

32

𝜇+𝑩𝐞𝐱𝐭

x

y

z

𝒑 𝜇+𝑷(𝑡 = 0)

Figure 2.9: A blow-up schematic (not to scale) outlining the basic configuration of the TF-µSRKnight shift experiments done for this thesis. An external field ~Bext is applied along the z-axisparallel to the µ+ momentum and transverse to the initial muon spin polarization ~P(t = 0).Incoming muons that trigger the outer muon detector and stop in the Ag annulus contribute toa reference signal, whereas muons that trigger the outer muon detector but pass through theAg annulus and trigger the inner muon counter placed in front of the sample, are assumed tostop in the sample.

The terms KLor and Kdemag are due to Lorentz and demagnetization fields

KLor + Kdemag = 4π(1/3− N)χρ

m, (2.33)

where N is the geometrical demagnetization factor of the sample, χ is the bulk dc magnetic

susceptibility, and ρ and m are the sample density and mass, respectively. The term KAg is the

muon Knight shift in Ag, which is small (94 ppm) and temperature independent [157]. Spatial

inhomogeneity of the externally applied magnetic field (i.e., a slightly different magnetic field

at the sample and reference positions) is accounted for by the spectrometer offset term Kspect.

The value and temperature dependence of Kspect can be measured by placing Ag samples at both

the sample and reference positions. The first term in Eq. (2.32) is the muon Knight shift in the

sample, given by

Kµ = Kdip + K0 + Kdia. (2.34)

In systems with localized electronic moments, Kdip is due to the associated dipolar fields at the

muon site(s) (within the Lorentz sphere). The contact hyperfine field contribution to the Knight

shift K0 arises from the Pauli paramagnetism of the conduction electrons and their Fermi contact

33

interaction with the muon. In the presence of non-zero spin density at the muon site, K0 is given

by

K0 =8π3⟨|ψ(0)|2⟩χs/n, (2.35)

where ⟨|ψ(0)|2⟩ is the density of conduction electrons at the muon site, n is the electron density

and χs is the Pauli paramagnetic spin susceptibility

χs = µ2BN(EF ), (2.36)

where µB is the Bohr magneton and N(EF ) is the local density of states at the Fermi energy

[155, 158]. Since the implanted muon is screened by the conduction electrons, the density of

conduction electrons at the muon site ⟨|ψ(0)|2⟩ in Eq. (2.35) can be significantly larger than the

value of n and so the ratio ⟨|ψ(0)|2⟩/n is referred to as the “spin density enhancement factor”.

The last term Kdia in Eq. (2.34) is due to diamagnetic screening of the muon by the orbital

component of the electron spin polarization at the muon site [159].

2.7.3 High Field Considerations

Aside from the Lorentz force on the incoming muons mentioned in Section 2.7, other compli-

cations must be considered in TF-µSR experiments involving large magnetic fields. High timing

resolution is required to resolve the high muon spin precession frequencies. This requires fast

electronics that enable small time bin widths. With increasing magnetic field, the maximum

amplitude of the muon spin precession signal is reduced [160], as follows

a(Bext)

a0(0)= exp−(2.355πγµBext∆t)2

4 ln 2

, (2.37)

where ∆t is the timing resolution of the electronics. The “NuTime” spectrometer used to carry

out the TF-µSR measurements in this thesis, allows for a timing resolution of ∆(t) = 48.8 ps.

Figure 2.10 shows the field dependence of the signal amplitude for different timing resolutions.

In a real experiment, the spatial homogeneity of the applied field is crucial. A non-uniform

applied magnetic field results in a distribution of muon Knight shifts that contributes to the

depolarization of the TF-µSR spectrum. Magnetic fields transverse to the muon spin polarization

exert a Lorentz force on the decay positrons, causing them to undergo cyclotron motion. The

cyclotron orbit radius is inversely proportional to the applied field. At 7 T, the orbit radius for

positrons emitted from a muon at rest is just 1.3 cm [161]. Consequently, the positron detectors

must be positioned as close as possible to the sample, often within a cryogenic environment.

The Complex Asymmetry Signal

In a TF-µSR experiment, more than one pair of opposing detectors can be used to construct a

more efficient TF asymmetry spectrum. Not shown in Fig. 2.7 are an additional pair of detectors

34

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

∆t = 48 ps

∆t = 390 ps

a(B

ext)

/a

0(0

)

Bext

(T)

∆t = 180 ps

Figure 2.10: The normalized amplitude of the detected TF-µSR signal as a function of externalmagnetic field for different timing resolutions.

placed immediately above and below the sample (along the x-axis). These additional up (U) and

down (D) counters increase the data acquisition rate by increasing the solid angle for detection

of the decay positrons. The asymmetry spectrum for the two pairs of opposing detectors are

Ay(t) ≡ ALR(t) and Ax(t) ≡ AU D(t). The former is out of phase with the latter by π/2 and can

be treated as the imaginary part of the complex polarization function [162]

Py(t) =

∫ ∞

0

n(B) cos(γµBt +φ −π/2)

=

∫ ∞

0

n(B) sin(γµBt +φ). (2.38)

Since the asymmetry is proportional to the time evolution of the muon spin polarization, the

complex asymmetry is

Acomplex(t) = Ax(t) + iAy(t). (2.39)

The number of counts in the individual positron detector time histograms is described by

Ni(t) = Ni(0)e−t/τµ[1+ a0,i Pi(t)] + Bi , (2.40)

where i = L, R, U , D. Rearranging Eq. (2.40) and using the relation Ai(t) = a0,i Pi(t) yields the

asymmetry function for single positron counters

Ai(t) = et/τµ

Ni(t)− Bi

Ni(0)

. (2.41)

35

The real and imaginary components of the complex asymmetry may be described in terms of

the individual histogram asymmetries, as follows

Ax(t) =12[AR(t)− AL(t)], (2.42)

Ay(t) =12[AU(t)− AD(t)]. (2.43)

The analysis of the high TF-µSR measurements in this thesis was carried out employing complex

asymmetry histograms.

The Rotating Reference Frame

A 6 T magnetic field corresponds to a muon spin precession frequency of 813 MHz. According

to Nyquist’s theorem, a discrete signal must be sampled at at least twice the signal’s highest

frequency to avoid aliasing. A sampling frequency of 1,624 MHz translates to a sampling period

of 615 ps, such that at least ∼ 13,008 bins are required to resolve the signal over an 8 µs time

period. The smallest timing resolution of the electronics employed at TRIUMF is 48 ps which,

for the same 8 µs time range, corresponds to a total of 166,666 bins. Operating at very high

frequencies means that the data acquired at such high fields does not have much room to be

packed or binned further. This leads to fitting involving an extremely large number of data points

which can, for some sophisticated functions, be time-consuming to fit.

To solve this problem, a discrete rotating reference frame (RRF) transformation can be ap-

plied to the data to shift the data to a lower frequency

ARRF(t) = A(t) cos (ωRRF t +φRRF). (2.44)

This results in the production of two frequency shifts, ω−ωRRF and ω+ωRRF. By introducing

binning at an appropriate scale — so-called “RRF-binning” — it is possible to remove (i.e., bin

out) the unwanted component oscillating at ω +ωRRF. The exact and optimal details for the

choice of RRF frequency and RRF bin size are important and can be found in Ref. [163]. It turns

out that the conditions for choosing the RRF frequency and RRF binning size are more forgiving

at very high magnetic fields since theω+ωRRF component becomes exceptionally large and can

be ignored [163]. An added benefit of fitting in a RRF is that both the data and the corresponding

fit can be visually inspected.

2.8 Fourier Transforms

Since µSR data is recorded in the time domain, most quantitative analysis also occurs in the

time domain. Unsurprisingly, Fourier transforms to the frequency domain (or equivalently the

magnetic field domain via the muon gyromagnetic ratio) still provide a valuable complementary

source of information.

36

The Fourier transform of a discrete quantity xn is defined by

Xk =

N−1∑

n=0

xne−i2πkn/N , (2.45)

where k = 0, ..., N−1. In practice, Eq. (2.45) is computed using a “Fast Fourier Transform” (FFT)

algorithm. The benefits of FFTs to visualize the frequency domain spectral shapes come at a cost.

Due to the∼ 2.2 µs muon lifetime, µSR spectra are typically restricted to a time domain of< 10

µs. As shown in Fig. 2.11(a) and (b), despite a healthy sampling rate in the time domain (which,

by the Nyquist theorem, defines the frequency response of the system), the frequency of the peak

is ill defined in the frequency domain and contains non-zero values for fields other than B0. One

can extend time domain spectra by “zero-padding” the end of the time data series and then

carrying out the FFT, as shown in Fig. 2.11(c). This leads to an increase in the number of FFT

bins or a pseudo-boost to the resolution of the raw FFT where “spectral leakage” or “ringing”

becomes apparent. This phenomenon occurs due to so-called “windowing” (or a lack of it). The

finite domain of an undamped signal can be thought of as being “windowed” or truncated by a

rectangular, or Heaviside step function that defines the domain of the signal (i.e., 0 to 10 µs)

and which has a sinc function (also referred to as the kernel) as an associated Fourier transform.

The resulting Fourier transform is thus a convolution of the coherent peak with the sinc function

leading to the sinc-like characteristic of “ringing” or ripples around the central peak.3

This issue can be remedied by windowing or “apodizing” the signal with a different, non-

rectangular function. Figure 2.11(e) shows the original time domain signal multiplied by a Gaus-

sian window function [see Eq. (2.28)] centered at 5 µs with σ = 0.3 µs−1. The resulting FFT

has significantly reduced spectral leakage, at the cost of a broadened peak. The software used

in the analysis of data in this thesis contains apodization options for the visualization of FFT

spectra.

2.9 Data Analysis

All of the µSR data analysis presented in this thesis was carried out using the software programs

musrfit and msrfit, both of which are based on the CERN ROOT framework using the Minuit2

χ2 minimization routines [164]. Some custom fitting functions were implemented with home-

made scripts written in python using the iminuit package [165].

3This effect is sometimes referred to as the Gibbs phenomenon.

37

0 5 10

-1

0

1

0 25 50 75 100

0

50

100

150

200

0 5 10 15 20

-1

0

1

0 25 50 75 100

0

50

100

150

200

0 25 50 75 100

0

50

100

150

200

0 5 10 15 20

-1

0

1

(c)

(e) (f)

(d)

(b)A

sym

metr

y

t (µs)

(a)

Fou

rier

Pow

er

B (Oe)

Asy

mm

etr

y

t (µs)

zero-padding

apodization

spectral

leakageFou

rier

Pow

er

B (Oe)

Fou

rier

Pow

er

B (Oe)

Asy

mm

etr

y

t (µs)

Figure 2.11: (a) Simulation of an undamped muon precession signal assuming an applied mag-netic field of B0 = 50 G and 300 data points. (b) Fourier transform of (a) where fields other thanB0 = 50 G possess non-zero values. (c) Padding the end of the asymmetry spectra with 5000zeros leads to (d) more FFT bins and a pseudo-interpolation in the resolution of the Fouriertransform revealing the ringing associated with a rectangular window function that defines thefinite time domain of the signal. (e) Multiplying the spectra by a Gaussian centered at 5 µs withσ = 0.3 µs−1 (i.e., apodizing the signal) leads to (f) suppressed leakage in the vicinity of thepeak in the field domain, at the cost of a broadened peak.

38

Chapter 3

Absence of µSR Evidence for Magnetic

Order in the Pseudogap Phase of

Bi2212

3.1 Introduction

Previously, ZF-µSR has been used to search for magnetic order in the pseudogap phase of opti-

mally (Tc = 91 K, p = 0.16, “OP91”) and overdoped (Tc = 80 K, p = 0.198, “OD80”) Bi2212

single crystals [20]. While no evidence of magnetic order was detected, a temperature depen-

dent relaxation rate was observed that is apparently electronic in origin. The prior measure-

ments were restricted to temperatures below the temperature above which the muon diffuses in

cuprates, T ∼ 150 K. The magnetic field distribution sensed by diffusing muons is motionally

narrowed, resulting in a decrease of the ZF relaxation rate with increasing temperature that

cannot be disentangled from potential onset temperatures. Thus, it was not obvious whether

the observed temperature dependence was associated with the pseudogap phase since the up-

per bound estimates of the pseudogap temperature T ∗ of the samples extended above 150 K

[83]. Furthermore, if the temperature dependent relaxation signal was indeed associated with

the pseudogap phase, it would be expected to evolve with the hole-doping concentration of

the sample. However, the limited hole-doping range of the study also meant that any potential

hole-doping dependence could not be measured definitively.

To circumvent the ambiguity imposed by muon diffusion and to determine if the observed

temperature dependent relaxation rate below 150 K is associated with the pseudogap phase,

ZF-µSR experiments were performed on two additional overdoped single crystals (Tc = 70 K,

p = 0.213, “OD70” and Tc = 55 K, p = 0.23, “OD55”) with T ∗ values [83] well below 150 K

(see Fig. 3.1).

39

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0

50

100

150

200

250

300

T (

K)

p

muon diffusion

Tc

T*

Figure 3.1: Hole-doping concentration (p), Tc (red circles) and T ∗ (black bars) for the samplesused in this study. The ranges of T ∗ are adapted from Ref. [83] while p is determined via theempirical relationship Tc = Tc,max[1− 82.6(p− 0.16)2] with Tc,max = 91 K [166].

3.2 Experimental Details

The Bi2212 single crystals studied in this work were grown by the traveling-solvent-floating-

zone method [167] at Brookhaven National Laboratory. The hole-doping concentration was

achieved by annealing the single crystals in an oxygen partial pressure of 2.3 atm for 72-250

hours at 400 C. Photographs and surface dimensions of the samples are shown in Fig. A.1, in

Appendix A. The superconducting transition temperatures of the samples were determined by

low-field bulk magnetization measurements (see Fig. 3.2). Sample OD55 shows a slight decrease

of the bulk magnetic susceptibility with decreasing temperature prior to the sharp drop indica-

tive of bulk superconductivity. This suggests the presence of phase-separated superconducting

regions with slightly higher transition temperatures.

The ZF-µSR measurements were carried out at TRIUMF’s M20D surface muon beamline

using the LAMPF spectrometer. Stray magnetic fields at the sample position were measured

using a Hall probe placed at the sample position and compensated for using three orthogonal

pairs of Helmholtz coils. The direction of the initial muon spin polarization was oriented parallel

to the crystallographic c-axis of the Bi2212 single crystals. The single crystals were attached to

silver (Ag) tape and suspended by attaching the tape to a ring type sample holder. The Ag tape

is non-magnetic and thin enough that incoming muons pass through it. A veto detector was

placed behind the sample to reject muons that pass by or through the sample. The OD70 sample

thickness was below the minimum thickness (∼ 180 mg/cm2) required to fully stop all muons

incident on the sample. Therefore, to minimize the number of muons that passed through the

sample, the muon beam momentum was reduced by 2.5% from its nominal value of 29.3 MeV/c

and a 0.025 mm thick film of Ag was placed in front of the sample to further degrade the muon

40

0 50 100 150-25

-20

-15

-10

-5

0

0 50 100 150-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

OD70

OD55

OD70

T (K)

χ (

10

-2 e

mu

⋅g-1

⋅Oe

-1)

T (K)

HZFC

= 5 Oe || c

ZFC

OD55

(a)

FC

(b)

χ (

10

-2 e

mu

⋅g-1

⋅Oe

-1)

HFC

= 5 Oe || c

Figure 3.2: The temperature dependence of the bulk dc magnetic susceptibility measured un-der (a) zero-field cooling (ZFC) and (b) field-cooling (FC) conditions for the OD55 (blue) andOD70 (red) samples. The two arrows in (b) indicate two distinct or a range of superconductingtransitions in the OD55 sample indicative of phase separation.

momentum. Similar experimental modifications were employed in Ref. [20], where the muon

beam momentum was reduced by 4.0% for both the OP91 and OD80 samples, but for which a

0.025 mm thick Ag momentum degrader was used only for the former. Table 3.1 summarizes

the characteristics of the samples and the experimental conditions for each measurement.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

A(t

)/a

0

t (µs)

T = 152 K

(a)

Ag Ag

(b)

A(t

)/a

0

t (µs)

T = 5 K

Figure 3.3: Normalized ZF asymmetry spectra for pure Ag at (a) T = 152 K and (b) T = 5 K.The solid curves indicate fits to Eq. (3.2) with β fixed to 1.

Normalized zero-field asymmetry spectra for a 99.998% pure Ag sample at T = 152 K and T

= 5 K are shown in Fig. 3.3. Silver serves as a good reference material since it does not possess

electronic moments and possesses very weak nuclear moments. Consequently, the ZF-µSR spec-

tra exhibit a negligible relaxation rate. Representative normalized ZF asymmetry spectra for the

41

Table 3.1: Summary of samples and the corresponding ZF-µSR mea-surement conditions.

Sample Area Areal density1∆pµ/pµ

2 Degrader thickness3

(cm2) (mg/cm2) (mm)OP914 0.25 140 -4.0 % 0.025OD804 0.39 185 -4.0 % -OD70 0.25 104 -2.5 % 0.025OD55 0.68 172 0 -OD805 0.18 216 0 -

Ag 0.56 262 -2.5 % -Ag6 0.36 262 0 -Ag5 0.18 262 0 -

1 Thickness of Ag needed to stop muons with momentumpµ≃29.3 MeV/c is ∼0.19 mm, corresponding to an areal densityof 199 mg/cm2.

2 Nominal muon beam momentum is pµ≃29.3 MeV/c.3 99.998 % pure Ag foil.4 Stacked and tiled mosaic of single crystals studied in Ref. [20].5 Measurements carried out using an ultra-low background cryostat

insert.6 Studied in Ref. [20].

Bi2212 samples at T = 150 K and T ≤ 10 K are shown in Fig. 3.4. While there is some tempera-

ture dependence for all samples [cf. Figs. 3.4(a)-(d)], the spectra do not show any appreciable

dependence on hole-doping [cf. Fig. 3.4(e) and Fig. 3.4(f)]. Zero-field asymmetry spectra for

high-Tc cuprate superconductors are typically fit to a function of the form

A(t) = a0GKT(∆, t)exp (−λt), (3.1)

where GKT(∆, t) is the static Gaussian Kubo-Toyabe (GKT) function approximating spin relax-

ation due to randomly oriented nuclear dipoles that are static on the muon time scale [i.e.,

Eq. (2.19)], while the exponential relaxation function accounts for potential fast fluctuating or

dilute static electronic moments. It has previously been shown that the nuclear dipolar field

contribution to the relaxation of the ZF-µSR spectrum in La214 deviates strongly from the GKT

approximation, which is based on the assumption that the nuclear dipole field distribution is

isotropic at the muon site. Hence, the ZF-µSR asymmetry spectra for the Bi2212 samples were

fit to a phenomenological stretched exponential relaxation function

A(t) = a0 exp [−(λt)β]. (3.2)

As shown in Fig. 3.5(a), the temperature dependence of λ is similar for all of the Bi2212 samples

and there is no systematic dependence of λ on hole-doping. By comparison, the temperature

42

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

(c)

OD70

A(t

)/a

0

4 K

150 K

OP91

OD80b

OD70

OD55

A(t

)/a

0

t (µs)

T = 150 K

(e) (f)

t (µs)

T ≤ 10 K

(b)

(d)

OD55

4 K

150 K

OP91

A(t

)/a

0

10 K

150 K

(a)

OD80b

4 K

150 K

Figure 3.4: Normalized ZF asymmetry spectra at T = 150 K and T ≤ 10 K for the (a) OP91,(b) OD80b, (c) OD70 and (d) OD55 samples. The hole-doping dependence of the asymmetryspectra is shown for (e) T = 150 K and (f) T ≤ 10 K. The solid curves through the data pointsare global fits to Eq. (3.2) assuming the same initial asymmetry.

43

0.09

0.10

0.11

0.12

0 50 100 150 2001.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

0 50 100 150 2000.0

0.5

1.0

1.5

OP91

OD80a

OD80b

OD70

OD55

λ (

µs-1

)

T (K)

(a)

T*

(c)

T (K)

(b)

λ (

10

−2 µ

s-1)

Ag

Figure 3.5: The temperature dependence of (a) the ZF relaxation rate for the Bi2212 and (b) highpurity Ag samples, and (c) the stretch parameter for the Bi2212 samples from global fits toEq. (3.2). The value β = 1 was assumed for Ag. Note that the open and filled orange circlesfor OD80 in (a) and (c) correspond to data collected at different times during two separateexperiments, demonstrating the reproducibility of the results.

dependence of λ for the 99.998% pure Ag sample shown in Fig. 3.5(b) is negligible. The tem-

perature dependence of the stretch parameter β for the Bi2212 samples shown in Fig. 3.5(c)

exhibits some variation with hole-doping. Raman spectroscopy measurements indicate that the

pseudogap phase should vanish at p ∼ 0.22 [108]. Hence, the OD55 sample is not expected to

possess a pseudogap phase, which suggests the weak temperature dependence observed in the

ZF relaxation rate has another origin.

To simultaneously track the behaviour of λ and β with a single parameter, the characteristic

time for the muon spin polarization to decay to half of its initial amplitude

t1/2 =[− ln(0.5)]1/β

λ, (3.3)

is shown in Fig. 3.6(a). This parameter is plotted as a function of hole-doping for a few tem-

peratures in Fig. 3.6(b). The behaviour of t1/2 exhibits a clear dependence on temperature, but

not on hole-doping.

3.3 Ultra-low Background Measurements

The measurements discussed so far, including those presented in Refs. [20] and [103], were per-

formed using a standard “low-background” helium gas-flow cryostat insert designed for samples

44

0.15 0.18 0.21 0.24

7.0

7.5

8.0

8.5

9.0

(b)

OD55OD70

OD80

150 K

100 K

50 K

10 K

4 K

t 1/2

(µ

s)

Hole-doping, p

OP91

(a)

0 2 4 6 8 100.00

0.25

0.50

0.75

1.00

1.25 OD80a

OD80b

T = 50 K

A(t

)

t (µs)

Figure 3.6: (a) ZF-µSR asymmetry spectra for the OD80 samples at 50 K showing the character-istic time t1/2 required for the spectra to decay to half of the initial amplitude. (b) Hole-dopingdependence of t1/2 for different temperatures. The multiple symbols in the OD80 column cor-respond to independent measurements.

with an area of ∼ 0.25 cm2 to 9.6 cm2. The surface area of most of the Bi2212 single crystals

employed in this study was near the minimum required for experiments using this apparatus,

meaning that some of the incoming muons may have stopped outside of the sample and thus

were not rejected by the veto detector, and generated a background contribution to the recorded

spectra. For this reason, we repeated measurements on the OD80 single crystal using an “ultra-

low background” (ULB) cryostat insert specifically designed for smaller samples. The ULB ap-

paratus, shown schematically in Fig. 3.7, consists of an internal µ+ detector placed between the

sample and an Ag annulus. Incoming muons that trigger both the external and internal muon

detectors are assumed to stop in the sample. A scintillating plastic annulus placed immediately

after the Ag annulus is used as an additional veto to reject positron events associated with

muons that stop in the Ag annulus. The trade-off in using the ULB insert to significantly reduce

the background contribution is a much lower data acquisition rate. As indicated in Table 3.1,

the OD80 single crystal studied using the ULB has a surface area of 0.18 cm2 and a mass of 39

mg so reducing the nominal muon beam momentum or placing an Ag degrader in front of the

sample was not necessary.

Representative ZF-µSR spectra and fits to Eq. (3.2) for the ULB measurements are shown in

Fig. 3.8(a). The results of global fits to Eq. (3.2) assuming a common value of a0 are shown in

Figs. 3.8(b) and (c). Apart from some “playing off” of the fitting parameters β and λ at 75 and

25 K presumably due to covariance, the value of β is pushed up close to 1.9. This value of β is

similar to that for the thick OD55 single crystal (which did not require reduction of the muon

beam momentum or the use of an Ag degrader), indicating the small hole-doping dependence

of β visible in Fig. 3.5 is associated with a background contribution to the ZF-µSR signal. This

presumably is caused by the reduced muon beam momentum, resulting in a larger fraction of the

incoming muons being scattered at wide angles upstream of the sample and a reduced triggering

45

Backward e+ detector

Veto detector

Forward e+ detector

External

μ+ detector

Sample

ULB

Ag + Scintillating annuli

(Active collimator)

μ+

Internal

μ+ detector

Figure 3.7: Schematic diagram showing the counter arrangement used for the ZF-µSR mea-surements on Bi2212, including the relative positions of the external muon, external positron,internal muon, and internal veto counters. Here external and internal refer to outside and insidethe helium-gas flow cryostat, respectively. Muons passing through the internal µ+ counter thattrigger the veto counter are rejected. Also shown are internal Ag and plastic scintillator counterannuli with equal 3-mm-diameter holes. These serve as a second type of veto (or “active colli-mator”) in the “ultra-low background” (ULB) setup. Muons stopping in the Ag annulus do nottrigger the internal muon counter. Consequently, they do not contribute to the sample signal.The Ag annulus protects the scintillator counter annulus from muons, so that it can be used todetect and reject positrons from decay muons in the Ag annulus.

46

0 50 100 150 2000.09

0.10

0.11

0.12

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 2000.0

0.5

1.0

1.5

0 50 100 150 2001.4

1.5

1.6

1.7

1.8

1.9

2.0

(b) OD80a

OD80b

OD80-ULB

λ (

µs-1

)

T (K)

OD80-ULB

t (µs)

T = 150 K

T = 5 K

A(t

)/a

0

(a)

Ag-ULB

Ag

Ag, Pal et al. (2018)

T (K)

λ (

10

−2 µ

s-1)

(d)

T (K)

(c)

Figure 3.8: (a) Normalized ZF asymmetry spectra for the OD80 sample at T = 150 K (bluecircles) and T = 5 K (red squares) recorded using the ULB cryostat insert. Solid curves are theresults of global fits to Eq. (3.2). Temperature dependence of (b) λ and (c) β for measurementsperformed with the ULB apparatus (grey triangles) and the standard low-background inserts(orange circles). (d) Temperature dependence of the relaxation rate λ for 99.998% Ag frommeasurements performed using the ULB insert (blue circles), and the standard low-backgroundinsert from this experiment (black squares) and from Ref. [20] (pink triangles).

of the veto detector. The ULB measurements also demonstrate that the temperature dependence

of the relaxation rate is reproducible, establishing it as an intrinsic property of Bi2212.

Figure 3.9 shows the correlation of the temperature-averaged value of the stretch parameter

β with the initial amplitude of the asymmetry signal recorded for each sample. As annotated

in the figure, the amplitudes are correlated with the area and thickness of the samples. Despite

adjusting beamline elements to select lower momentum muons and using an Ag degrader to

further reduce the incoming muon momentum, the initial amplitude is smaller in the thinner

and smaller samples. The largest amplitudes correspond to the largest and thickest sample,

OD55, and the measurements of the OD80 sample using the ULB configuration where muons

not stopping in the sample are vetoed with exceptionally high efficiency. This implies that the

hole-doping dependence observed in β is due to muons that do not stop in the sample — ei-

ther through missing the sample or passing through it and failing to trigger the veto detector

47

behind the sample. Muons stopping in the plastic scintillator rapidly depolarize leading to a loss

of asymmetry, although the exact amount depends on the specific material [168, 169]. Further-

more, a large fraction of muons stopping in the plastic scintillator form muonium (a µ+ bound

to an electron) and rapidly depolarize. The samples with the largest amplitude ZF-µSR signal

have values of β closest to 2, as expected for internal magnetic fields arising from randomly

oriented, static nuclear dipole moments.

0.18 0.20 0.22 0.24 0.261.4

1.5

1.6

1.7

1.8

1.9

2.0

OD70

OD80a

172 mg/cm2

0.68 cm2

OD80 ULB

OD55

OD80b

216 mg/cm2

0.18 cm2

-4.0% pm

185 mg/cm2

0.39 cm2

-2.5% pm

0.025 mm Ag

104 mg/cm2

0.25 cm2

β

a0

-4.0% pm

0.025 mm Ag

140 mg/cm2

0.25 cm2

OP91

Figure 3.9: The mean temperature-averaged stretch parameter β versus the initial amplitude a0of the ZF-µSR asymmetry spectrum for each Bi2212 sample. The surface area and thickness ofeach sample are indicated.

3.4 Nuclear Dipolar Field Distributions

Longitudinal-field µSR measurements presented in Ref. [20] show that a magnetic field of just

∼ 10 G was enough to recover a non-relaxing signal in both OP91 and OD80. This indicates the

presence of weak static internal magnetic fields (much smaller than 10 G) which are character-

istic of nuclear dipole fields. Thus, the temperature dependence of the ZF relaxation rate could

be due to subtle changes in the nuclear dipole field distribution at the muon site. Measurements

from thermal expansion, internal friction, and powder neutron diffraction have shown that the

lattice parameters of the orthorhombic crystal structure decrease with decreasing temperature

[170–173]. While the a-axis and c-axis lattice parameters cease shrinking near ∼ 50 K and

∼ 90 K, respectively, the b-axis lattice parameter continues to shrink down to lower temper-

atures. Such changes in the lattice parameters can cause subtle changes in the nuclear dipole

contribution to the muon depolarization rate. For example, a structural change near 60 K in

Y123 is believed to be responsible for a sudden jump in the ZF-µSR relaxation rate at that tem-

perature [91, 143].

48

b (Å)

(b)

O

b (Å)

O

a (

Å)

b (Å)O

a (

Å)

b (Å)

(e)

O

a (

Å)

a (

Å)

24

24

20

42

04

24

20

4

24

20

4

0.1

8

0.1

6

0.1

5

0.1

4

0.1

2

0.1

0

0.0

9

0.0

7

0.0

6

Δ(μ

s-1)

0.1

8

0.1

6

0.1

5

0.1

4

0.1

2

0.1

0

0.0

9

0.0

7

0.0

6

Δ(μ

s-1)

0.0

72

0.0

64

0.0

56

0.0

48

0.0

40

0.0

32

0.0

24

0.0

16

0.0

48

0.0

42

0.0

36

0.0

30

0.0

24

0.0

18

0.0

12

0.0

06

0.0

00

δΔ(1

0-2

μs-1

)δΔ

(μs-1

)

(a)

a

bc

Bi

Sr

Ca

Cu

O

(c)

(d)

Figu

re3.

10:(

a)T

heor

thor

hom

bic

crys

tals

truc

ture

ofB

i221

2.T

hesh

aded

regi

onin

dica

tes

the

SrO

plan

ew

here

the

apic

alox

ygen

resi

des

and

calc

ulat

ions

of∆

for

diff

eren

tpo

tent

ialm

uon

stop

ping

site

sar

eca

rrie

dou

tfo

r(b

)T=

150

Kan

d(c

)T=

5K

.(d)

The

diff

eren

cebe

twee

n∆

in(b

)an

d(c

)at

100×

mag

nific

atio

n.(e

)T

hedi

ffer

ence

betw

een∆

atT=

150

Kan

dT=

5K

wit

hth

eca

lcul

atio

nsat

T=

5K

carr

ied

out

ata

perp

endi

cula

rdi

stan

ceof

0.15

Åou

tof

the

SrO

plan

eto

war

dth

elo

wer

BiO

plan

e.

49

b (Å)

(b)

a (

Å)

24

20

4

0.5

0

0.4

5

0.4

0

0.3

5

0.3

0

0.2

5

0.2

0

0.1

5

0.1

0

Δ(μ

s-1)

Bi

O

O O

O

b (Å)C

u a (

Å)

24

20

4

0.5

4

0.4

8

0.4

2

0.3

6

0.3

0

0.2

4

0.1

8

0.1

2

0.0

6

Δ(μ

s-1)

OO

OO

(d)

b (Å)

a (

Å)

24

20

4

0.1

05

0.0

90

0.0

75

0.0

60

0.0

45

0.0

30

0.0

15

0.0

00

-0.0

15

δΔ(μs-1

)

(c)

Bi

O

O O

O

b (Å)

Cu a (

Å)

24

20

4

δΔ(μ

s-1)

OO

OO

(e)

0.1

05

0.0

90

0.0

75

0.0

60

0.0

45

0.0

30

0.0

15

0.0

00

(a)

a

bc

Bi

Sr

Ca

Cu

O

Figu

re3.

11:

(a)

The

crys

tals

truc

ture

ofB

i221

2hi

ghlig

htin

gth

eB

iOan

dC

uO2

laye

rs(s

hade

dpl

anes

).(b

)C

alcu

late

dva

lues

of∆

inth

eB

iOpl

ane

usin

gex

peri

men

tal

valu

esof

the

latt

ice

para

met

ers

atT=

150

K,

and

(c)

the

diff

eren

cebe

twee

nth

eva

lue

of∆

at15

0K

and

5K

inth

eB

iOpl

ane

(inc

ludi

nga

sim

ilar

0.15

Åof

fset

asbe

fore

)(d

)C

alcu

late

dva

lues

of∆

for

T=

150

Kin

the

CuO

2pl

ane,

and

(e)

the

diff

eren

cebe

twee

nth

eva

lue

of∆

at15

0K

and

5K

inth

eC

uO2

plan

e(i

nclu

ding

asi

mila

r0.

15Å

offs

etas

befo

re).

50

To determine if changes in the crystallographic lattice parameters with temperature are re-

sponsible for the temperature dependence of the ZF relaxation rate, the second moment of the

distribution of nuclear dipolar fields at potential muon stopping sites is calculated. In Bi2212,

the only nuclei with non-zero nuclear moments are 209Bi with spin I = 9/2 and magnetic mo-

ment µBi = 4.1103 µN and the two isotopes 63Cu and 65Cu, both of which have nuclear spin

I = 3/2 and a weighted-average (by natural abundance) magnetic moment of µCu = 2.275 µN.

The only isotope of strontium with a non-zero nuclear moment is 87Sr with 7% natural abun-

dance and hence is ignored in the calculations. The second moment of the Gaussian distribution

of nuclear dipole fields is determined as follows

⟨(BdipN − ⟨BdipN⟩)2⟩=∆

2

γ2µ

=23

∑

i, j

µi

1

r6i, j

, (3.4)

where i indexes the nuclear species and j indexes the location of the jth nucleus. The factor

of 2/3 accounts for the modified electric field gradient (EFG) at the sites of the nuclei with

I > 1/2 due to the unscreened charge of the muon [174, 175]. A crystal lattice consisting of

3 × 3 × 3 unit cells is sufficient for the calculation to converge since the sum is proportional to

r−4 (∆∝ r−6, while the number of moments grows as r2 with increasing volume).

While the muon site in Bi2212 has not previously been determined, the muon is believed

to form a hydrogen-like bond with an oxygen atom. In La214, the muon site is believed to be

situated near the apical oxygen [13], while there are apparently two muon sites in Y123 — one

near the apical oxygen and the other near an oxygen atom in the CuO chain layer [101]. As

illustrated in Fig. 3.10(a), the oxygen atoms in Bi2212 are situated in the BiO, SrO and CuO2

planes. Calculations of∆ for potential muon stopping sites in the SrO plane for the experimental

lattice parameter values at T = 150 K and T = 5 K are shown in Figs. 3.10(b) and 3.10(c),

respectively. To facilitate a direct comparison to experimental values, the asymmetry spectra for

sample OD80 measured with the ULB apparatus were fit to

A(t) = a0GKT(∆, t), (3.5)

yielding ∆ = 0.1036(8) µs−1 and 0.1136(8) µs−1 at T = 150 K and T = 5 K, respectively.

The difference between the calculated values of ∆ at the two temperatures, denoted δ∆, is

shown in Fig. 3.10(d). The change in the lattice parameters alone is not sufficient to explain the

observed experimental change in the relaxation rate of δ∆ = 0.010(1) µs−1. However, a shift

of the muon site just ∼ 0.15 Å out of the SrO plane toward the neighbouring BiO plane does

result in a value of δ∆ matching the experimental change in relaxation rate [cf. Fig. 3.10(e)].

Figure 3.11 shows the same calculation carried out for the muon sites in the BiO and CuO2

planes. No site in a region within a ∼ 1.5 Å radius around an oxygen atom in the BiO plane

accounts for the measured relaxation rate of ∆ ∼ 0.1 µs−1 [Fig. 3.11(b)]. In the same vein,

51

O

a (Å)

2 4

b (

Å)

2

4

0

δΔ (μs-1)Δ (μs-1)

2 4

b (

Å)

2

4

0

O

0.085

0.088

0.090

0.092

0.095

0.098

0.100

0.103

0.105

0.108

0.005

0.006

0.007

0.008

0.009

0.010

0.010

0.011

0.006

(b)(a)

a (Å)

Figure 3.12: (a) Calculations of ∆ carried out according to Eq. (3.6) near the SrO layer atT = 150 K. (b) The difference between∆ at T = 150 K and T = 5 K incorporating a∼ 0.3 Å off-set toward the BiO layer at 5 K. Evidently, we find that ∆ and δ∆ are consistent with exper-imental values in the ∼ 1.5 Å radius region around the apical oxygen atom. Note that thesecalculations assume a radially directed EFG caused by the muon and that the muon spin isinitially oriented along the z-axis (crystallographic c-axis).

the sites in the CuO2 plane with ∆ ∼ 0.1 µs−1 [Fig. 3.11(d)] do not possess an accompanying

change in relaxation rate of δ∆ ∼ 0.01 µs−1 [cf. Fig. 3.11(e)].

Equation (3.4) corresponds to the second moment of the nuclear dipolar field width assum-

ing an isotropic average internal field, which would be the case for a polycrystalline sample. The

expression for the second moment of the nuclear dipolar field width perpendicular to the initial

muon spin polarization in a single crystal system containing half integer nuclear spins situated

in the radially directed EFG of the muon is given by [158, 175–177]

⟨(BdipN − ⟨BdipN⟩)2⟩=13

∑

i, j

µi

1

r6i, j

4sin2Ωi, j +

34

Ii +12

Ii(Ii + 1)(2− sin2

Ωi, j)

. (3.6)

Here Ii is the nuclear moment of the ith nucleus and Ωi, j is the angle between the initial muon

spin polarization and ~ri j . As shown in Fig. 3.12(a), calculations of∆ based on Eq. (3.6) are con-

sistent with the calculated value of∆ from Eq. (3.4) for muon sites located within a ∼ 1.5 Å ra-

dius region around an oxygen atom in the SrO plane. Similarly, an ∼ 0.3 Å shift of the muon

site out of the SrO plane toward the BiO plane also leads to a change in the width of the nuclear

field distribution δ∆ ∼ 0.01 µs−1 within the same region [cf. Fig. 3.12(b)]. I do not claim to

have identified the muon site in Bi2212 based on these calculations, but rather argue they add

additional support to the overwhelming evidence that the temperature dependence detected in

the relaxation rates is most likely nuclear in origin and not caused by localized moments that

may arise from sources such as loop currents.

52

0 50 100 150 200 250 300

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

La214

x=0.16

x=0.198

x=0.198

x=0.213

x=0.23

λ (

µs

-1)

Temperature (K)

|| c

Bi2212

Figure 3.13: The temperature dependence of relaxation rates extracted from this study onBi2212 (left) compared with corresponding results for La214 (right) from Ref. [12], shownon identical scales.

3.5 Remarks

Calculations of the dipolar field generated from the θII orbital-loop current pattern proposed

by Varma (see Fig. 1.6) assuming an ordered magnetic moment of 0.1 µB (consistent with PND

measurements [9]) and a muon site near the apical oxygen (in the SrO plane) yields a local

magnetic field in the range of 20-70 G. This value is similar to the net dipolar field calculated at

the local probe site in other high-Tc cuprate superconductors — e.g., 41 G in La214 [12] and 70-

290 G in YBa2Cu4O8 [16]. If the orbital-loop current order is long range, this would be expected

to produce an observable oscillation in the ZF-µSR asymmetry spectra. If it is short range, it is

expected to produce a significantly larger relaxation rate. Figure 3.13 shows the temperature and

hole-doping dependence of the relevant ZF relaxation rate from this study of Bi2212 compared

to results from a ZF-µSR study of La214 [12]. Evidently, there is an overwhelming consistency

between the two results — despite the different families of cuprates involved and different

approaches to data analysis.

53

Chapter 4

Normal State Inhomogeneous

Magnetic Field Response

4.1 Introduction

As discussed in Chapter 1, it is widely believed that the normal state of cuprate materials hosts

superconducting fluctuations. While investigations employing several experimental techniques

have uncovered signatures of superconducting fluctuations above Tc , there remains a lack of

consensus as to the extent above Tc to which they survive — an issue that is likely related to

the sensitivity of the different experiments. Early high transverse-field (TF) µSR measurements

discovered an inhomogeneous magnetic field response in the normal state of several hole-doped

high-Tc cuprate compounds [30, 31, 133, 134]. The field-enhanced inhomogeneous magnetic

field response was found to track Tc and to scale with the maximum value of Tc for each cuprate

family. Given that muons are pure local magnetic field probes and that in a large magnetic field

the quantization axis of nuclear dipolar moments in the host material is essentially aligned with

the applied magnetic field, charge ordering is not the source of the observed doping-dependence

in the broadening of the internal magnetic field distribution above Tc . Thus, it was argued that

a plausible explanation is the occurrence of spatially varying superconducting fluctuations that

generate a set of distinct, time-averaged local magnetic fields that broaden the internal mag-

netic field distribution sensed by TF-µSR [30, 31]. However, experiments that were performed

were unable to determine whether the source of the broadening was diamagnetic in origin,

as expected for superconducting fluctuations. A diamagnetic signal may be detected by muon

Knight shift measurements, but can only be resolved using an extremely homogeneous external

magnetic field. The recent commissioning of an ultra-homogeneous superconducting magnet,

as part of the the so-called “NuTime” spectrometer at TRIUMF, made it possible to perform a

precision muon Knight shift study of the normal state of cuprate materials to address this issue.

54

B0

...<B(t)>

n

<B(t)>3

<B(t)>2

<B(t)>1

n(B

)

B

Figure 4.1: Muons stopping in spatially inhomogeneous islands of fluctuating superconductivityexperience varying degrees of time-averaged diamagnetic fields ⟨B(t)⟩n. Muons stopping out-side of these islands (grey regions) experience the applied magnetic field denoted by B0. Thus,the magnetic field distribution associated with spatially inhomogeneous superconducting fluc-tuations may resemble a left-half Lorentzian field distribution. As the islands proliferate withdecreasing temperature, the width of the corresponding field distribution n(B) increases. Notethat this figure does not include the field broadening effects of nuclear and electronic moments.

4.2 Spatially Inhomogeneous Superconducting Fluctuations and the

TF-µSR Lineshape

Figure 4.1 displays a model illustrating how superconducting fluctuations in distinct spatially-

separated regions can be detected by µSR. If the superconducting fluctuations are spatially in-

homogeneous, the muon ensemble senses a distribution of time-averaged local magnetic fields

⟨B(t)⟩i (i = 1,2, ...) from fluctuating diamagnetic regions, with varying degrees of supercur-

rent screening [135]. Spatially homogeneous superconducting fluctuations, on the other hand,

would produce a single time-averaged field, which does not broaden the internal field distribu-

tion detected by TF-µSR (i.e., the TF-µSR “lineshape”). Thus, it has been argued that the field

distribution associated with spatially inhomogeneous superconducting fluctuations can be rep-

resented by a left-half Lorentzian field distribution and therefore account for the exponential

relaxation of the TF-µSR spectra reported in Refs. [30, 31, 134]. In the analysis of the normal

state TF-µSR signal reported in Ref. [135], the time evolution of the muon spin polarization is

approximated by the following depolarization function

PTF(t)≈ Gnuc(t)Gel(t)

f exp (−at) cos (γµ⟨B(t)⟩t) + (1− f ) cos (γµB0 t)

. (4.1)

Here, Gnuc(t) and Gel(t) account for spin relaxation due to nuclear and electronic dipolar mag-

netic fields, respectively. The volume fraction of muons that experience the mean value of the

diamagnetic time-averaged fields ⟨B(t)⟩ is parameterized by f , while the factor (1− f ) accounts

for muons that sense the applied field B0. Given that the width of the field distribution and the

associated relaxation rate are small, exp (−at)≈ 1− at. Assuming the time-averaged fields are

55

also small, then the mean value of the time-averaged magnetic fields associated with diamag-

netic regions may be approximated as ⟨B(t)⟩ ≈ B0, in which case Eq. (4.1) can be written

PTF(t)≈ Gnuc(t)Gel(t)

f (1− at) cos (γµB0 t) + (1− f ) cos (γµB0 t)

= Gnuc(t)Gel(t)(1− f at) cos (γµB0 t)

≈ Gnuc(t)Gel(t)exp (−Λt) cos (γµB0 t). (4.2)

Thus, changes in the sample volume fraction of the fluctuating time-averaged magnetic fields

can be approximately accounted for via the exponential relaxation rateΛ = f a. While relaxation

due to fast fluctuating electronic moments is described by an exponential decay Gel(t) = exp (−st),

for the temperature and hole-doping concentration of interest, these fluctuations are too fast to

cause a detectable relaxation in the muon time window and hence Gel(t)≈ 1 [57–60].

As mentioned earlier, this question of whether the magnetic field response observed in the

normal state of cuprates by TF-µSR is diamagnetic in origin could not be addressed due to

instrumental detection limits. If the source of the line broadening is diamagnetic in origin, a

left-skewed internal magnetic field distribution will result and the average internal magnetic

field sensed by the ensemble of muons will be reduced. The previous iteration of the high mag-

netic field spectrometer at TRIUMF, known as “HiTime”, lacked the resolution to resolve a small

diamagnetic shift in the average internal magnetic field. In 2017, HiTime was outfitted with

a new homogeneous large-bore magnet (± 0.001% homogeneity over a 1.0 cm diameter by

0.6 cm length cylinder) and the spectrometer was renamed “NuTime”.

4.3 Depolarization Functions of Skewed Field Distributions

As described above, an exponential relaxation function can serve as an adequate approximation

of the depolarization of the TF-µSR asymmetry spectrum due to a left-skewed field distribution.

However, to measure a very small diamagnetic shift of the average internal magnetic field in a

muon Knight shift experiment, an accurate fitting function is essential. It is therefore necessary to

calculate the exact analytical depolarization function corresponding to a left-skewed Lorentzian

distribution of diamagnetic, time-averaged fields.

In a TF-µSR experiment, the direction of the initial muon spin polarization is perpendicular

to the applied field ~B0 and the time evolution of the muon spin polarization associated with a

distribution of magnetic fields n(B) sensed by the ensemble of muons implanted in the sample

is given by

P(t) =

∫ ∞

0

n(B) cos (Bγµ t)dB. (4.3)

56

0 2 4 6 8 10

-1

0

1

40 50 600.00

0.05

0.10

0.15

0.20

0.25

0.30

LH-Gaussian

Gaussian

n(B

)

B (G)

(b)

P(t

)

t (µs)

LH-Gaussian

Gaussian

(a)

Figure 4.2: (a) Comparison of left-half and symmetric Gaussian magnetic field distributions.(b) The corresponding TF depolarization functions calculated using Eq. (4.9) and Eq. (2.28).The calculations assume B0 = 50 G and σ = σ− = 0.3 µs−1.

Equation (4.3) is equivalent to a cosine Fourier transform, which can be written in terms of a

complete Fourier transform as follows

P(t) =ℜ∫ ∞

−∞n(B)e−iBγµ t dB

. (4.4)

The upper or lower bound of the integral is changed to B0 for a left-half or right-half magnetic

field distribution as follows

PL(t) =ℜ

2

∫ B0

−∞n(B)e−iBγµ t dB

, (4.5)

PR(t) =ℜ

2

∫ ∞

B0

n(B)e−iBγµ t dB

, (4.6)

where the factor of 2 is inserted to normalize the skewed field distribution. Consider the left-

half and right-half Gaussian magnetic field distributions, denoted by nLHG(B) and nRHG(B) and

characterized by standard deviations σ− and σ+, respectively

nLHG(B) =

2γµσ−p

2πexp

−12(γµ(B−B0)

σ−)2

for B ≤ B0

0 for B > B0,(4.7)

nRHG(B) =

0 for B < B02γµ

σ+p

2πexp

−12(γµ(B−B0)

σ+)2

for B ≥ B0.(4.8)

Computing the integrals indicated by Eq. (4.5) and Eq. (4.6) using Mathematica (see Ap-

pendix B for the detailed list of commands) results in the following normalized depolarization

57

0 2 4 6 8 10

-1

0

1

40 50 600.00

0.05

0.10

0.15

0.20

0.25

n(B

)

B (G)

Lorentzian

LH-Lorentzian

P(t

)

t (µs)

LH-Lorentzian

Lorentzian

(a) (b)

Figure 4.3: (a) Comparison of left-half and symmetric Lorentzian magnetic field distributions.(b) The corresponding TF depolarization functions calculated using Eq. (4.13) and Eq. (2.30).The calculations assume B0 = 50 G and λ = λ− = 0.3 µs−1.

functions in the time domain

PLHG(t) = e−12σ

2− t2

cos (γµB0 t) + Erfi(σ− tp

2) sin (γµB0 t)

, (4.9)

PRHG(t) = e−12σ

2+ t2

cos (γµB0 t)− Erfi(σ+ tp

2) sin (γµB0 t)

, (4.10)

where Erfi(x) is the imaginary error function. The left-half depolarization function and left-

half magnetic field distribution are illustrated and compared to their symmetric counterparts

in Fig. 4.2. This skewed Gaussian depolarization function is implemented in the software tool

musrfit for analyzing µSR data [164].

The corresponding TF depolarization functions for left-half and right-half Lorentzian (LHL

and RHL, respectively) field distributions nLHL(B) and nRHL(B) are calculated in a similar man-

ner. The left-half and right-half Lorentzian field distributions, characterized by half-width at

half-maximum (HWHM) values λ− and λ+, respectively, are defined as follows

nLHL(B) =

2λ−πγµ

(B − B0)2 + (

λ−γµ)2−1

for B ≤ B0

0 for B > B0,(4.11)

nRHL(B) =

0 for B < B0

2λ+πγµ

(B − B0)2 + (

λ+γµ)2−1

for B ≥ B0,(4.12)

58

where a factor of 2 is inserted for normalization. The TF depolarization function for the left-half

Lorentzian field distribution is given by1

PLHL(t) = e−λ− t cos (γµB0 t +φ) +1π

e−λ− tEi(λ− t)− eλ− tEi(−λ− t)

sin (γµB0 t +φ), (4.13)

where Ei(z) is the exponential integral

Ei(z) =

∫ z

−∞

et

td t. (4.14)

The depolarization function for the right-half Lorentzian field distribution is

PRHL(t) = e−λ+ t cos (γµB0 t +φ) +1π

eλ+ tEi(−λ+ t)− e−λ+ tEi(λ+ t)

sin (γµB0 t +φ). (4.15)

A two-sided skewed Lorentzian distribution with variable width on either side can be defined

using a combination of Eq. (4.13) and Eq. (4.15) by

PSkewLor(t) =λ−

λ− +λ+PLHL(t) +

λ+

λ+ +λ−PRHL(t). (4.16)

As illustrated in Fig. 4.4, a symmetric Lorentzian depolarization function as defined by Eq. (4.16)

is equivalent to an exponentially damped precession signal.

Unlike symmetric field distributions, the average field of a skewed field distribution is not

equal to the applied magnetic field B0. The average and higher moments of the skewed Gaussian

field distribution are calculated in Ref. [178]. The average field of a left-half Lorentzian field

distribution ⟨B⟩LHL, however, is undefined as the integral

⟨B⟩LHL =

∫ B0

−∞BnLHL(B)dB, (4.17)

is divergent. Thus, we use the median of the LHL field distribution BLHL as an alternative. The

median of a LHL field distribution is calculated as follows

∫ B0

BLHL

nLHL(B)dB = 1/2, (4.18)

BLHL = B0 −λ−/γµ, (4.19)

1As pointed out by Mike Hayden, the output of this transformation by Mathematica contains many redundantterms that cancel each other out or are zero. Furthermore, the depolarization function for the RHL distributionreturned by Mathematica consists of generalized Meijer G-functions which are unnecessary. While I have verifiedthat the outputs of Mathematica are correct, the expressions for PLHL(t) and PRHL(t) written here are the conciseand elegant versions provided by Mike Hayden.

59

0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

1.0

P(t

)

t (µs)

Exponential depolarization

0.5 [PLHL

(t) + PRHL

(t)]

Figure 4.4: An exponential depolarization function with λ = 0.2 µs−1 (solid black curve) com-pared to Eq. (4.16) with λ+ = λ− = 0.2 µs−1 (dashed magenta curve).

where B0 is the maximum magnetic field of the LHL field distribution. Similarly, the median

field of a RHL field distribution BRHL is given by

BRHL = B0 +λ+/γµ, (4.20)

where B0 in this case is the lowest magnetic field of the magnetic field distribution. The median

field of a skewed Lorentzian field distribution with variable width on either side is given by

BSkewLor =λ−

λ− +λ+BLHL +

λ+

λ+ +λ−BRHL. (4.21)

4.4 High Transverse-field µSR Measurements in the Normal State

of Bi2212

To study the normal state magnetic field response of cuprate materials, high TF-µSR measure-

ments were performed using TRIUMF’s M15 surface muon beamline and the NuTime spec-

trometer. The NuTime superconducting magnet was used to produce an ultra-homogeneous 6 T

magnetic field along the muon beam (z-axis) while the muon spin polarization was rotated so as

to be perpendicular to the beam (x-axis). In this configuration the muon momentum is parallel

to the magnetic field and no deflecting Lorentz force is incurred.

The Bi2212 single crystals used in this investigation were the same as those used in the

zero-field study described in the previous chapter. They were mounted with the c-axis parallel

to the applied magnetic field. The temperature dependence of the dc magnetic susceptibility

for these samples is shown in Fig. 4.5 for a field of 6 T applied parallel to the c-axis. The Pauli

paramagnetic contribution χ0 to the susceptibility is expected to increase with doping because

60

0 50 100 150 200 250 300 350

0

1

2

3

4

5

OP91

OD80

OD70

OD55

χ (

10

−7 e

mu

/g)

T (K)

Figure 4.5: Temperature dependence of the bulk dc magnetic susceptibility of the superconduct-ing Bi2212 samples carried out in a magnetic field B= 6 T applied parallel to the crystallographicc-axis. Data were collected while warming after field-cooling to a base temperature of about 4 K.

of the growing carrier concentration. While the change in χ0 is obvious when the magnetic sus-

ceptibility of OP91 is compared with that of the overdoped samples, the overdoped samples do

not show the expected trend. This is likely due to a small temperature independent diamagnetic

contribution from material used to hold the overdoped samples in place in the straw sample

holder used in the SQUID (i.e., Kimwipe™ napkins, gel capsules and other plastics). Such ad-

ditional material was not needed for the OP91 sample, since it fit snugly in the straw sample

holder. A gentle, hole-doping enhanced Curie-like upturn of the normal-state susceptibility — a

common feature of overdoped cuprates — is evident.

Calibration Measurements using Ag

Prior to each TF-µSR experiment, the difference between the muon spin precession frequency

in the sample and reference locations in the NuTime spectrometer were determined using a

piece of silver (Ag) for both the sample and the reference. The difference is caused by the field

inhomogeneity of the magnet, and contributes an offset to the inferred muon Knight shift. The

TF asymmetry spectra of Ag are well described by

A(t) = a0G(t) cos (γµBt +φ), (4.22)

where a0 is the amplitude of the signal, γµ/2π= 135.5 MHz/T is the muon gyromagnetic ratio,

B is the total average magnetic field at the muon site, φ is the initial phase angle between the

61

0 2 4 6 8-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 2 4 6 8-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Asy

mm

etr

y

t (µs)

ReferenceA

sym

metr

y

t (µs)

Sample

Figure 4.6: Representative TF-µSR asymmetry spectra for Ag at the sample (left) and reference(right) locations from decay events in up and down positron detectors recorded during the2020 beam period at TRIUMF. Solid curves indicate fits to Eq. (4.22) and Eq. (4.23). The datais displayed and were fit in a RRF of frequency 812 MHz. The smaller error bars in Ag at thereference position are due to a larger number of muons stopping in the Ag annulus (see Fig. 2.9),which improves the statistics.

initial muon spin polarization and the x-axis, and G(t) is the relaxation function

G(t) = e−λt e−(σt)2 , (4.23)

where σ accounts for relaxation due to very small, randomly oriented static nuclear moments,

and λ accounts for additional sources of spin relaxation, such as the inhomogeneity of the ap-

plied magnetic field across the sample. Figure 4.6 shows representative TF spectra for Ag both

at the reference and sample positions with fits to Eq. (4.22). The resulting temperature depen-

dence of λ for both the sample Ag and reference Ag is shown in Fig. 4.7. The relaxation in the

Ag in the sample position is consistently faster than in the reference Ag. In addition to a larger

value of λ for almost every beam period, σ is also larger, indicating the fitted value is not strictly

caused by nuclear dipole fields [see Fig. 4.8(b)].

The muon Knight shift is determined from the measured relative frequency shift

Kexpt =νsample − νreference

νreference, (4.24)

where νsample and νreference are the average muon spin precession frequencies in the sample and

reference, respectively, which are determined from fits of Eq. (4.22) to the asymmetry data.

The full details of the Knight shift experiment can be found in Chapter 2. The temperature

dependence of the relative frequency shift Kspect arising from the inhomogeneity of the external

magnetic field is shown in Fig. 4.8(a). The absolute value of Kspect is consistently < 5 ppm

above 50 K, where our measurements on Bi2212 are focused, but not necessarily independent

62

0 100 200 3000.00

0.02

0.04

0.06

0 100 200 300

0.00

0.02

0.04

0.06

0 100 200 300

0.00

0.02

0.04

0.06

0 100 200 300

0.00

0.02

0.04

0.06

(d)

(b)

(c)

2018

20202019

λ (

µs-1

)

T (K)

Ag:Ag 2017(a) Sample

Reference

λ (

µs-1

)

T (K)

λ (

µs-1

)

T (K)

λ (

µs-1

)

T (K)

Figure 4.7: Temperature dependence of the exponential relaxation rate for Ag at the sample(black diamonds) and reference (colored circles) positions for an applied field of B = 6 T andfor (a) 2017, (b) 2018, (c), 2019, and (d) 2020 muon beam periods at TRIUMF.

of temperature. Thus, our measurements of the muon Knight shift in Bi2212 are accurate to

within 5 ppm ceteris paribus.

Measurements on Bi2212

Representative TF-µSR asymmetry spectra for the Bi2212 single crystals are shown in Fig. 4.9.

The complex asymmetry for both the sample and reference were fit to Eq. (4.22) in a rotat-

ing reference frame (RRF) at a frequency of 812 MHz with G(t) assumed to be described by

Eq. (4.23). Here the Gaussian relaxation rate σ describes spin relaxation due to randomly ori-

ented Cu and Bi nuclear moments, while λ arises from “other" sources of relaxation, including

potential inhomogeneous superconducting fluctuations. The value of σ, which is considered to

be independent of hole-doping, is determined to be σ = 0.130(1) µs−1 from global fits of the

OP91 sample data over the temperature range Tc < T < 160 K. Above 160 K the muon spin

relaxation rate is diminished due to motional narrowing associated with muon diffusion.

The amplitude of the real component of Fourier transforms of the TF asymmetry spectra,

which provide visual representations of the internal field distribution detected by the implanted

muons, are shown in Fig. 4.10. The temperature dependence of λ from the fits of Eq. (4.22) to

63

0 50 100 150 200 250 300-15

-10

-5

0

5

10

2017 2018 2019 2020

-0.02

0.00

0.02

0.04

0.06

0.08

2020

2019

2018

2017

Ksp

ect (

pp

m)

T (K)

(a) Ag:Ag Sample

Reference

σ (

µs-1

)

Year

(b)

Figure 4.8: (a) Temperature dependence of the experimental relative frequency shift Kspect mea-sured with NuTime for an Ag sample and Ag reference. (b) Comparison of the fitted value of σfor the sample and reference signals recorded for different beam periods.

the data is shown in Fig. 4.11(a). The exponential relaxation rate is much larger below Tc due

to the broad field distribution of the vortex lattice, which has a linewidth that is proportional to

the superfluid density ns(T ). The interpolation of λ above Tc on the temperature hole-doping

phase diagram is illustrated in Fig. 4.11(b), where it is immediately obvious that even in the

normal state far above Tc the relaxation rates follow the same hole-doping dependence as the

superconducting dome, but deviate from this behaviour at lower temperature.

The measured relative frequency shift Kexpt is composed of the following contributions

Kexpt = Kµ + KLor + Kdemag + KAg + Kspect. (4.25)

The last four terms have the same meaning as described below Eq. (2.32). In particular, KLor and

Kdemag are the relative frequency shifts produced by the Lorentz and demagnetization fields, re-

spectively, KAg is the muon Knight shift of Ag [158], and Kspect is the relative frequency shift due

to the inhomogeneity of the magnet. Since the samples are all plate-like, the demagnetization

factor was assumed to be N = 1. Ignoring any contribution from Cu spins in the higher doping

range of the current study, Kdip = 0 and the muon Knight shift in the sample Kµ is given by

Kµ = K0 + Kdia + KSCF, (4.26)

where K0 is due to Pauli paramagnetism and is defined in Eq. (2.35) and Kdia is due to diamag-

netic screening of the muon by the electron spin polarization at the muon site [159]. The third

term KSCF in Eq. (4.26) is the effect of a potential diamagnetic response associated with super-

conducting fluctuations. Table 4.1 summarizes the factors that contribute to Kexpt. The results

of the temperature dependence of Kµ are displayed in Fig. 4.12.

64

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

OD55, T = 65 KOD55, T = 162 K

OD70, T = 160 K OD70, T = 80 K

OD80, T = 160 K OD80, T = 90 K

OP91, T = 160 K OP91, T = 100 K

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8

-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8

-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Figure 4.9: Representative TF-µSR asymmetry spectra from decay events recorded in up anddown positron detectors for the OP91 (green data), OD80 (blue data), OD70 (red data) andOD55 (cyan data) samples at T ∼ 160 K (left column) and T ≃ Tc + 10K (right column). Solidcurves are fits to Eq. (4.23). The asymmetry spectra were fit and are plotted here in a RRF offrequency 812 MHz.

65

812 813 814

0.0

0.2

0.4

812 813 814

0.0

0.2

0.4

812 813 814

0.0

0.2

0.4

0.6

812 813 814

0.0

0.2

0.4

Real

Am

pli

tud

e

f (MHz)

65 K

162 K

Real

Am

pli

tud

e

f (MHz)

80 K

160 K

Real

Am

pli

tud

e

f (MHz)

90 K

160 K

OD70 OD55

OD80R

eal

Am

pli

tud

e

f (MHz)

100 K

160 K

OP91

Figure 4.10: The amplitude of the real component of the Fourier transforms of the TF-µSRsignals for the Bi2212 samples at T ∼ 160 K and T ≃ Tc + 10 K. The Fourier transforms areapodized with the Gaussian function exp (−0.5(t/C)2), where C was chosen to be 10 µs toreduce ringing. The small shoulder that appears at 65 K on the low-frequency side of the mainpeak in the Fourier transform for OD55 is presumably caused by the minority Tc = 65 K phasedetected by the low-field SQUID measurements shown in Fig. 3.2.

Table 4.1: Contributions to the measured muon Knight shift in Bi2212 and their respectivevalues. The dc magnetic susceptibility divided by the mass of the sample is denoted by χm.

Term Description Value Ref.

KLor Lorentz field 4π3 χmρ -

Kdemag Demagnetization field 4πNχmρ -KAg Knight shift of Ag ∼ 95 ppm [158]Kspect Spectrometer shift < 5 ppm Fig. 4.8Kdia Diamagnetic shift ∼ -20 ppm [159]

66

50 100 150 200 250 300 350

0.00

0.05

0.10

0.15

(b) OP91

OD80

OD70

OD55

λ (

µs-1

)

T (K)

(a)

0.0 0.1 0.2 0.3

0

100

200

300

p

T (

K)

0.00

0.02

0.04

0.07

0.09

0.11

λ (µs-1)

Superconducting

T*

Figure 4.11: (a) The temperature dependence of the exponential relaxation rate λ for the Bi2212samples from fits of TF-µSR signals for B = 6 T to Eq. (4.23). (b) Interpolated behaviour of λas a function of temperature and hole-doping concentration. The T ∗ line indicates the upperbound estimates of T ∗ in Bi2212 reported in Ref. [83].

50 100 150 200 250 300 350 400

-20

-10

0

10

20

30

40

OP91

OD80

OD70

OD55

Kµ (

pp

m)

T (K)

Figure 4.12: The temperature dependence of the muon Knight shift Kµ for the Bi2212 samples.The open circles for OD80 and OD70 indicate data recorded during different beam periods.

67

Table 4.2: The results of fits of the Kµ vs. T data to Eq. (4.27).

E0 (meV) χ2/NDFGlobal fit 81(7) 1.78Single fit (OP91) 120(13) 1.18Ref.[20] (OP91) 127(5) -

Figure 4.13(a) shows results of global fits of the temperature dependence of the muon Knight

shift above Tc for each sample to the Arrhenius equation

Kµ(T ) = Aexp (−E0/kBT ) + B, (4.27)

where the activation energy E0 is a common fitting parameter and the temperature-independent

constant B is sample dependent. Figure 4.13(b) shows a comparison of the global fit with an

individual fit of the muon Knight shift data for the OP91 sample to Eq. (4.27). The activation en-

ergy E0 from the latter fit of the OP91 data is almost identical to the activation energy associated

with muon diffusion [see Fig. 4.13(c)] [20]. In Ref. [20], ZF-µSR spectra for the OP91 sample

were modelled by a non-analytical strong-collision dynamic Gaussian Kubo-Toyabe relaxation

function, which incorporates a hop rate that can parameterize the muon’s diffusion rate or the

average rate at which the magnetic field sensed by the muon changes. The agreement between

the activation energies determined by the ZF and TF-µSR measurements strongly suggests that

the temperature dependence of the hop rate ν and the muon Knight shift of Bi2212 OP91 are

governed by the same physics, namely muon diffusion. Hence, above∼ 160 K the muon becomes

mobile in the sample and experiences an internal magnetic field averaged over its lifetime. The

time-averaged magnetic field sensed by the muon results in changes in Kµ. Results of fits to

Eq. (4.27) are presented in Table. 4.2. The hole-doping dependence of the offset parameter B

in Eq. (4.27) is illustrated in Fig. 4.13(d). The increase of B with increasing p is as expected

for the contribution of the hyperfine field produced by the polarization of conduction electrons,

which increases with p due to the increased carrier concentration.

The TF-µSR asymmetry spectra were also fit to a depolarization function associated with a

left-half Lorentzian distribution of magnetic field

A(t) = a0PLHL(λ, B,φ, t)e−(σt)2 , (4.28)

where PLHL(λ, B,φ, t) is given by Eq. (4.13) and σ is assumed to be independent of tempera-

ture with a value of σ = 0.130(1) µs−1 determined by fits described earlier. Representative TF

asymmetry spectra with fits to Eq. (4.28) are shown in Fig. 4.14. The corresponding tempera-

ture dependence of λ and the muon Knight shift Kµ are illustrated in Fig. 4.15 and Fig. 4.16,

respectively. While the temperature dependence of λ is similar to that obtained from the fits

of the TF-µSR signals to Eq. (4.23), the behaviour of Kµ is different and correlated with the

temperature dependence of λ. Evidently, Kµ decreases with increasing λ. This is expected as

68

75 150 225 300

-10

0

10

20

30

40

50

100 200 300

-10

0

10

20

100 200 300

0.0

0.5

1.0

1.5

0.15 0.20 0.25

-10

-5

0

5

10

15

20 (d)(c)

(b)

OP91 OD80

OD70 OD55

Kµ (

pp

m)

T (K)

(a)

Global fit

Single fit

Kµ (

pp

m)

T (K)

OP91

OP91

ν (

µs-1

)

T (K)

(

pp

m)

p

Figure 4.13: (a) The temperature dependence of Kµ above Tc . The solid curves through the datapoints are the result of a global fit to Eq. (4.27) assuming a common value of E0. (b) Temperaturedependence of Kµ for the OP91 sample, showing the result of the global fit compared to a fitof the OP91 data alone to Eq. (4.27). (c) The temperature dependence of the muon hop ratedetermined by ZF-µSR [20]. (d) Hole-doping dependence of B determined from the global fitsto Eq. (4.27).

69

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

OD55, T = 65 KOD55, T = 162 K

OD70, T = 160 K OD70, T = 80 K

OD80, T = 160 K OD80, T = 90 K

OP91, T = 160 K OP91, T = 100 K

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8

-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8

-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8-0.10

-0.05

0.00

0.05

0.10

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Asy

mm

etr

y

t (µs)

Figure 4.14: Representative TF-µSR asymmetry spectra from decay events recorded in up anddown positron detectors for the OP91 (green data), OD80 (blue data), OD70 (red data) andOD55 (cyan data) samples at T ∼ 160 K (left column) and T ∼ Tc + 10K (right column). Solidcurves are fits to Eq. (4.28). The asymmetry spectra were fit and are plotted here in a RRF offrequency 812 MHz.

70

50 100 150 200 250 300 350

0.00

0.05

0.10

0.15

0.20

0.25

(b) OP91

OD80

OD70

OD55

λ (

µs-1

)

T (K)

(a)

0.0 0.1 0.2 0.3

0

100

200

300

Superconducting

p

T (

K)

0.00

0.02

0.05

0.09

0.13

0.16

λ (µs-1)

T*

Figure 4.15: (a) Temperature dependence of the LHL relaxation rate λ obtained from fits toEq. (4.28). (b) Interpolated behaviour ofλ as a function of temperature and hole-doping concen-tration. The T ∗ line indicates the upper bound estimates of T ∗ in Bi2212 reported by Ref. [83].

the median magnetic field of the left-hand Lorentzian field distribution is proportional to λ, as

shown earlier.

Discussion

Figure 4.17 shows a comparison of the reduced chi-squared2 of the TF-µSR spectra for Bi2212

to the two different depolarization functions. Neither is decisively better at describing the TF

signals than the other, which suggests within the resolution of the experiment, both models

provide enough degrees of freedom to adequately describe the data. This is not surprising and

consistent with the findings of Ref. [178] which suggest skewed functions only converge well

(for least-squares fits performed by minuit [165]) for relaxation rates that are larger than those

involved in the normal state of cuprates (e.g., σ > 0.15 µs−1 for skewed Gaussian relaxation

functions). While the temperature dependence of the relaxation rate λ is similar for the two

models, albeit with slightly different absolute values, it is interesting that the temperature de-

pendence of Kµ is different. The temperature dependence of Kµ for all samples obtained from

fits involving Eq. (4.22) and Eq. (4.23) are well described by an Arrhenius behaviour with an

activation energy consistent with muon diffusion. On the other hand, fits assuming a left-half

Lorentzian internal magnetic field distribution superimposed on a Gaussian field distribution

due to nuclear dipolar fields indicate that Kµ decreases as the temperature is reduced with little

variation among the samples. It is interesting to note that Kµ does not appear to be sensitive to

2The reduced chi-squared is defined as the weighted sum of squared deviations χ2 divided by the number ofdegrees of freedom (NDF). A χ2/NDF of 1 indicates an ideal match between the fitting model and the data.

71

50 100 150 200 250 300 350

-250

-200

-150

-100

-50

0

50

100 OP91

OD80

OD70

OD55

Kµ (

pp

m)

T (K)

Figure 4.16: Temperature dependence of the muon Knight shift Kµ obtained from fits toEq. (4.28). The open circles for OD80 and OD70 indicate data recorded during different beamperiods.

50 100 150 200 250 300

0.8

1.0

1.2

50 100 150 200 250 300 350

0.8

1.0

1.2

1.4

1.6

1.8

2.0

50 100 150 200 250 300 3500.6

0.8

1.0

1.2

1.4

50 100 150 200 250 300 3500.6

0.8

1.0

1.2

1.4

χ2/N

DF

T (K)

Exp

LHL

χ2/N

DF

T (K)

χ2/N

DF

T (K)

OD55OD70

OD80

χ2/N

DF

T (K)

OP91

Figure 4.17: Comparison of χ2/NDF for fits of the TF-µSR signals of the Bi2212 samples to theLHL (colored circles) and exponential (open circles) depolarization functions given by Eq. (4.28)and Eq. (4.23), respectively.

72

the reduction in N(EF ) due to the pseudogap phase, which should be present in Bi2212 samples

with p < 0.22 [108]. The OP91 sample definitely exhibits a pseudogap phase with a pseudogap

temperature T ∗ between 140 K and 200 K [83]. The opening of the pseudogap should reduce

Kµ. Yet, Kµ in OP91 is temperature independent between Tc and∼ 225 K (cf. Fig. 4.12). A muon

Knight shift study of the single layer cuprate Bi1.76Pb0.35Sr1.89CuO6+δ [(Bi,Pb)2201] detected

changes in Kµ due to the pseudogap [179]. The lack of sensitivity to the pseudogap in the cur-

rent study may be due to a weaker Fermi contact interaction of the conduction electrons with

the muon.

An alternative model of the magnetic field distribution sensed by muons stopping in spatially

inhomogeneous regions of fluctuating diamagnetism is shown in Fig. 4.18. In this scenario, the

field distribution associated with the spatially inhomogeneous SCFs is described by a right-

half Gaussian, which assumes the regions with the largest diamagnetic response to the applied

magnetic field are most prevalent. The corresponding depolarization function is

P(t) = f PRHG(σ+, BSCF,φ, t) + (1− f ) cos (γµB0 t +φ)e−σ2 t2

, (4.29)

where f is the volume fraction of the muons stopping in regions of the sample exhibiting dia-

magnetism, PRHG is the right-half Gaussian depolarization function [see Eq. (4.10)] with width

σ+, minimum magnetic field BSCF and phase φ. The second component of the depolarization

function accounts for muons that sense the applied magnetic field B0 with σ accounting for spin

relaxation due to nuclear dipole moments. Unfortunately, the accuracy of this model cannot be

compared to the LHL and exponential depolarization functions, as fits of the TF-µSR spectra to

Eq. (4.29) lead to parameters “playing off” one another. This is likely due to the two additional

fitting parameters ( f and BSCF) required for this model.

73

BSCF

n(B

)

B

B0

Figure 4.18: An alternative model of the magnetic field distribution sensed by muons stoppingin an environment of spatially segregated regions exhibiting varying degrees of superconduct-ing fluctuations. Muons stopping outside of the regions exhibiting superconducting fluctuationssense a Gaussian distribution of magnetic fields due to the static randomly oriented nuclearmoments centered about the applied magnetic field B0 (blue region). Muons stopping in therandom patches of superconducting fluctuations experience varying degrees of diamagnetismproportional (red region). This model assumes the most diamagnetic regions are the most preva-lent.

74

Chapter 5

Conclusions

Consistent with the results of earlier NMR [15–18] and ZF-µSR [12–14, 91] searches for mag-

netic order in the pseudogap phases of Y123 and La214, no evidence of magnetic order or

fluctuations between different orbital-loop-current configurations (within the time window of

ZF-µSR) is observed in the pseudogap phase of Bi2212 over a wide hole-doping range [21]. The

new results firmly establish that the temperature dependent ZF-µSR relaxation rate observed in

optimally doped and slightly overdoped Bi2212 in an earlier ZF-µSR study [20] is not associ-

ated with the pseudogap phase. Calculations of the width of the nuclear dipolar field distribution

for potential muon sites show that the observed temperature dependence of the ZF relaxation

rate likely arises from very small changes in the lattice parameters of the orthorhombic crys-

tal structure measured by other techniques. The results presented here stress the importance

of not always assuming the nuclear dipolar field contribution to the ZF-µSR relaxation rate is

temperature independent.

The current study does not explain why experiments using local probe techniques (µSR,

NMR and NQR) do not detect the IUC magnetic order sensed by PND in the pseudogap phase of

cuprates. It has been proposed that PND experiments actually detect ordered magneto-electric

quadrupoles at Cu sites [180]. However, the internal magnetic fields generated by magneto-

electric quadrupolar ordering should still be detectable by ZF-µSR [181]. As shown in Fig. 2.1,

the upper limit of the dynamic range for detecting fluctuating magnetic fields using ZF-µSR

and NMR is below that of neutron scattering. It is therefore possible that the IUC magnetic

order detected by PND fluctuates between different loop-current configurations at a frequency

outside of the NMR and ZF-µSR fluctuation rate windows and that any potential magnetic field

distribution associated with orbital-loop currents is motionally narrowed to the point that it

is not detected by muons. The source of this discrepancy may also originate from differences

in sample quality. As discussed in Chapter 1, ZF-µSR measurements performed on Y123 found

evidence of magnetic order with a magnetic volume fraction of just 3% in a large single crystal

in which a PND study had found evidence of magnetic order [14]. While the contribution to the

ZF-µSR signal is compatible with the expected size of internal magnetic fields at the muon sites

generated by the ordered moments observed by PND, the absence of a similar signal in the other

75

smaller, high-quality single crystals of Y123 strongly suggests that the magnetic order detected

in the large single crystal is associated with an impurity phase. On the other hand, IUC magnetic

order in the pseudogap phase has been detected by PND in several high-Tc cuprate families.

The work described in this thesis also aimed to shed further light on the origin of the univer-

sal inhomogeneous magnetic field broadening in the normal state of cuprates [30]. To search

for a possible diamagnetic response above Tc , muon Knight shift experiments were performed

on Bi2212 single crystals over a wide hole-doping range using an ultra homogeneous magnetic

field TF-µSR spectrometer, “NuTime”. Between Tc and approximately 200 K, the muon Knight

shift in Bi2212 is temperature independent, but increases with hole-doping. While the increas-

ing muon Knight shift with hole-doping indicates an increasing hyperfine contact field at the

muon site presumably due to the growing carrier concentration with hole-doping, there is no

observable reduction of the muon Knight shift due to the pseudogap phase. This suggests that

the reduction of K0 caused by a reduction of N(EF ) in the pseudogap phase is too small to be

resolved by the NuTime apparatus. Above 200 K, the muon Knight shift in all samples increases

with temperature and is well described by an Arrhenius equation with an activation energy con-

sistent with muon diffusion. There is no evidence of any temperature-dependent diamagnetic

contribution to the muon Knight shift due to superconducting fluctuations. While it is difficult

to theoretically predict the magnitude of the average diamagnetic shift expected from spatially

inhomogeneous superconducting fluctuations due to the many unknown variables, such as the

variation of λ and Tc for the various regions, the results here place an upper bound on any such

diamagnetic frequency shift in Bi2212 at ∼ 5 ppm. We find no evidence for ferromagnetic fluc-

tuations in the samples used in this study. Preliminary muon Knight shift measurements on the

non-superconducting La214 (x = 0.33) single crystal from Ref. [143] have shown a markedly

different temperature dependence that may be attributable to ferromagnetic fluctuations.

76

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Appendix A

Photos of Bi2212 Samples Used in

Zero-field Search for Magnetic Order

Figure A.1: Photos of the OD70, OD55 and OD80 (used in the ultra-low background cryostatinsert) single crystals. The samples are outlined and traced for clarity. The overlayed ruler mea-sures centimeters.

86

Appendix B

Calculating Depolarization Functions

with Mathematica

To calculate the Fourier transform (i.e., the depolarization function) associated with a field dis-tribution n(B− B0) with width s the following commands are used in Wolfram Mathematica1.Symmetric distribution:

ComplexExpand[Re[Integrate[(gmu/(Sqrt[2*Pi]*s)) E^(-(1/2)*gmu^2*(B - b0

)^2/s^2 )

Exp[-I*B*gmu*t], B, -Infinity, Infinity,

Assumptions -> t > 0 && B > 0 && b0 > 0 && s > 0 && gmu > 0]]]

which should yield

E^(-(1/2) s^2 t^2) Cos[b0 gmu t]

The skewed case is similar but the bounds of integration are just changed to be −∞ to B0 forleft-sided or B0 to∞ for right-sided field distribution:

Simplify[ComplexExpand[

Re[2*Integrate[(gmu/(Sqrt[2*Pi]*s)) E^(-(1/2)*gmu^2*(B - b0)^2/s^2 )

Exp[-I*gmu*B*t], B, -Infinity, b0,

Assumptions -> t > 0 && B > 0 && b0 > 0 && s > 0 && gmu > 0]]]]

This yields

E^(-(1/2) s^2 t^2) (Cos[b0 gmu t] -

Cos[b0 gmu t] Im[Erfi[(s t)/Sqrt[2]]] +

Re[Erfi[(s t)/Sqrt[2]]] Sin[b0 gmu t])

Note that the center term in the above result is zero and not simplified by Mathematica.

1Version 12.1

87