Spectroscopy and Phenomenology of Unconventional Superconductors

Spectroscopy and phenomenology of unconventional superconductors

by Nicholas Robert Lee-Hone

M.Sc., McGill University, 2013 B.Sc., McGill University, 2011

Thesis Submitted in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

in the Department of Physics Faculty of Science

Copyright in this work is held by the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Declaration of Committee

Name: Nicholas Robert Lee-Hone

Degree: Doctor of Philosophy

Thesis title: Spectroscopy and phenomenology of unconventional superconductors

Committee: Chair: Igor F. Herbut Professor, Physics

David M. Broun Co-Supervisor Associate Professor, Physics

Erol Girt Co-Supervisor Professor, Physics

Malcolm P. Kennett Committee Member Associate Professor, Physics

J. Steven Dodge Examiner Associate Professor, Physics

William A. Atkinson External Examiner Professor, Physics and Astronomy Trent University

ii Abstract

The challenge of condensed matter physics is to understand how various states of matter emerge from periodic or amorphous arrangements of atoms. This is typically easier if there are no impurities in the material under investigation. However, many of the properties that make materials technologically useful stem from the very disorder and impurities that complicate their study. In this work we investigate three unconventional superconductors within the dirty BCS framework of superconductivity, with impurities treated in the self- consistent t-matrix approximation. The three materials behave quite differently and allow us to explore and reveal a rich, subtle, and poorly appreciated range of effects that must be considered when studying superconducting materials and probing the limits of applicability of current theories of superconductivity. Microwave spectroscopy measurements of FeSe, a member of the iron-based family of superconductors, were performed at frequencies below 1 GHz using a novel helical resonator. The measurements show that the material is particularly clean, with an electronic mean free path that exceeds that of all known binary compounds. They also reveal the presence of a finite gap on both superconducting bands, strongly constraining which pairing states should be considered for FeSe.

Nb-doped SrTiO3 has an extremely low carrier concentration and multiple electronic bands at the Fermi sruface. Microwave spectroscopy measurements, this time with a stripline resonator, reveal a single spectroscopic gap despite clear evidence for multiple superconducting bands. Elastic scattering introduced via Nb dopants is proposed as the mechanism by which the two bands homogenize into a single spectroscopic signature, as per Anderson’s theorem. The impurity hypothesis is tested with single-band ﬁts to the superﬂuid density and is found to be in good agreement with measured data.

Finally La2−xSrxCuO4, a member of the famous cuprate family of superconductors is explored in detail under the assumption that Born scattering with large amounts of impurities can explain a number of puzzling measurements that have caused controversy in the high-Tc community over the past few years. Many features of the experimental measurements can quantitatively be accounted for with very few free parameters and no ﬁne tuning.

iii Keywords: unconventional superconductivity, BCS, disorder, microwave spectroscopy, Mat- subara formalism, dirty superconductor, FeSe, Nb-STO, LSCO, La2−xSrxCuO4, helical resonator

iv Acknowledgements

During my time at SFU, I had the privelage of working closely on a regular basis with my supervisors Erol Girt and David Broun. Their constant guidance, encouragement, trust, and willingness to actively engage in the day-to-day lab work has been an inspiration. There are no words to describe how much I value their mentorship and friendship. From the very beginning, Colin Truncik, Natalie Murphy, Ricky Chu and Taras Chouinard made me feel at home in David’s lab. Since then many others have made my time in the research group very enjoyable; thank you in particular to Ulas Özdemir, and Steph Labbé. Tommy McKinnon, Paul Omelchenko, Eric Montoya, and Monica Arora, all helped me learn how to operate the instruments in Erol’s lab. Without their guidance I would have been lost. I have also been very fortunate to have had the opprotunity to collaborate with researchers from around the world, including but not limited to Doug Bonn, Marc Scheffler, and Dieter Suess. I have fond memories of the collaboration with Markus Thiemann, who visited us from Stuttgart for over a year. I am also grateful for the guidance and work of Peter Hirschfeld and Vivek Mishra, with whom we have fostered a very successful ongoing collaboration. Thank you to my grandparents, Roger and Margaret, and my sister Chloe for supporting me, and to my parents, Susan and Michel for allowing me to spend innumerable hours on the computer learning how to program as a teenager. It really paid off! And finally thank you to Helen, who has supported me, encouraged me, and endured me through all these years. This thesis would not have been possible without all of you.

v Table of Contents

Declaration of Committee ii

Abstract iii

Acknowledgements v

Table of Contents vi

List of Tables ix

List of Figures x

1 Introduction 1 1.1 A brief history of superconductivity ...... 1 1.2 Conventional and unconventional superconductivity ...... 4 1.3 Impurity studies of superconductivity ...... 6

2 Theory 8 2.1 BCS theory ...... 8 2.2 Self-consistent t-matrix approximation ...... 12 2.2.1 Anderson’s theorem ...... 22

2.2.2 Gap equation and Tc ...... 24 2.2.3 Abrikosov-Gor’kov formula ...... 25 2.2.4 Critical scattering rate ...... 27 2.3 Experimental observables ...... 28 2.3.1 Density of states ...... 28 2.3.2 Superﬂuid density ...... 30 2.3.3 Optical conductivity ...... 34 2.3.4 Heat capacity ...... 35

3 Electrodynamics of superconductors 39 3.1 London theory ...... 39 3.2 Electrodynamics of superconductors ...... 41

vi 3.3 Surface impedance ...... 44

4 Experimental methods 48 4.1 Cavity perturbation ...... 48 4.2 Obtaining calibrated surface impedance from cavity perturbation ...... 50 4.3 Helical resonator ...... 51

5 Multiband superconductivity 56 5.1 FeSe ...... 56 5.1.1 Nodes or no nodes, that is the question ...... 59 5.1.2 Experiment details ...... 61 5.1.3 Results ...... 63 5.1.4 Anisotropic two-band model ...... 66 5.1.5 Superﬂuid density ﬁts ...... 71 5.1.6 Outlook ...... 73 5.1.7 Contributions ...... 74

5.2 Nb-SrTiO3 ...... 74 5.2.1 Experiment details ...... 77 5.2.2 Results ...... 79 5.2.3 Anderson’s theorem in two-band superconductors ...... 82 5.2.4 Is Nb-STO really a dirty superconductor? ...... 85 5.2.5 Contributions ...... 90

6 Disorder in La2−xSrxCuO4 91 6.1 Dirty d-wave BCS theory on a realistic Fermi surface ...... 97 6.1.1 Superﬂuid density ...... 99 6.1.2 Optical conductivity ...... 100 6.2 Results ...... 101 6.2.1 Finding the missing spectral weight ...... 104 6.3 Outlook ...... 111 6.4 Contributions ...... 112

7 Conclusions 113

Bibliography 115

Appendix A Evaluating elliptic integrals 130 A.1 Arithmetic geometric mean ...... 131 A.2 Reference implementation ...... 131

Appendix B Scanning tunnelling spectroscopy 134

vii B.1 Temperature dependence of tunneling ...... 135

viii List of Tables

Table 4.1 Details of the 9 attempts at creating a helical coil resonator...... 53

Table 5.1 Material properties and sample dimensions...... 77

ix List of Figures

Figure 1.1 Timeline of superconducting transition temperatures...... 5

Figure 2.1 Real space and k-space representation of BCS pairing...... 9 Figure 2.2 Schematic of the excitation spectrum of a superconductor...... 11 Figure 2.3 Temperature dependence of the gap for dirty superconductors. . . . 26 Figure 2.4 Evaluation of the Abrikosov-Gor’kov formula...... 27 Figure 2.5 Density of states for dirty s-wave and dirty d-wave superconductors. 29 Figure 2.6 Superfluid density calculation starting from the density of states. . 31 Figure 2.7 Density of states and superfluid density for clean superconductors. 32 Figure 2.8 Superfluid density suppression for an s-wave superconductor. . . . . 33 Figure 2.9 Superfluid density suppression for a d-wave superconductor. . . . . 34 Figure 2.10 Conductivity for a dirty d-wave superconductor with impurities. . . 36 Figure 2.11 Heat capacity for s-wave and d-wave superconductors with impurities. 37

Figure 3.1 Electromagnetic ﬁelds outside and inside a superconductor. . . . . 42

Figure 4.1 Cavity perturbation apparatus and plot of the S21 parameter. . . . 49 Figure 4.2 Schematic of the novel helical coil resonator inside a Cu enclosure. . 52 Figure 4.3 Picture of the ﬁrst 8 helical resonators attempts...... 53 Figure 4.4 Comsol simulations of the helical resonator...... 55

Figure 5.1 Crystal sturcture of ﬁve common types of iron-based superconductors. 57 Figure 5.2 Schematic of the FeAs lattice, Fermi surface, and band structure. . 59 Figure 5.3 Tunneling conductance data from several sources...... 60 Figure 5.4 Possible gap symmetry scenarios for multiband superconductors. . 62 Figure 5.5 Surface resistance of FeSe plotted on linear and logarithmic scales. 64

Figure 5.6 Real part of the quasiparticle conductivity, σ1(T )...... 65 Figure 5.7 Quasiparticle relaxation rate from fits to a two-fluid model. . . . . 66 Figure 5.8 Superfluid density obtained from fits to a two-fluid model...... 67 Figure 5.9 Gap function for various pairing symmetries...... 68 Figure 5.10 Fit to the superfluid density using a two-band extended s-wave model. 72 Figure 5.11 Bogoliubov quasiparticle interference measurements of the FeSe gaps. 73

Figure 5.12 Crystal structure, Fermi surface, and band structure of Nb-SrTiO3. 75

x Figure 5.13 Tunneling spectroscopy data for Nb-doped SrTiO3...... 76

Figure 5.14 Doping dependence of the electron mobility in Nb-doped SrTiO3. . 78 Figure 5.15 Microwave stripline resonator and sample...... 79

Figure 5.16 Temperature dependence of fB at f0 ≈ 2 GHz...... 80 Figure 5.17 Fits to the complex conductivity of Nb-STO...... 81

Figure 5.18 Temperature dependence of σ1/σN measured at several frequencies. 82 Figure 5.19 Measured and ﬁtted temperature dependence of the spectroscopic gap. 83 Figure 5.20 Homogenization of the spectroscopic and Matsubara gaps...... 85 Figure 5.21 Suppression of multigap signature with increasing disorder...... 86 Figure 5.22 Merging of spectroscopic features with increasing scattering rate. . 86 Figure 5.23 Results of ﬁts to Nb-STO with a dirty single-gap BCS model. . . . 88

Figure 5.24 Magnetic ﬁeld dependence of Rs and Xs at 8.02 GHz...... 89

Figure 6.1 Tetracrystal experiment with Tl2201...... 92 Figure 6.2 Crystal structures of several cuprate superconductors...... 93

Figure 6.3 Phase diagram of the high-Tc cuprate superconductors...... 94 Figure 6.4 Representative data for the mutual inductance experiment of Ref. [1]. 95

Figure 6.5 Temperature and doping dependence of ρs from Ref. [1]...... 96

Figure 6.6 Energy oﬀset, 0, and next-nearest neighbour hopping integral. . . . 98 Figure 6.7 Tight-binding energy contours from the ARPES spectra in Ref. [2]. 98 Figure 6.8 Weighting factor used in the calculation of the electronic properties. 99 Figure 6.9 Comparison of the measured and calculated superﬂuid density. . . . 102

Figure 6.10 Transition temperatures of La2−xSrxCuO4...... 103 Figure 6.11 Temperature and doping dependence of the LSCO gap maximum. . 104 Figure 6.12 Schematic of the conductivity spectrum of a d-wave superconductor. 105

Figure 6.13 The plasma frequency of La2−xSrxCuO4...... 106 Figure 6.14 Measured and calculated optical conductivity of overdoped LSCO. . 107 Figure 6.15 Measured and calculated dimensionless optical spectral weights. . . 108

Figure 6.16 Scattering rate, Tc, and Tc0 for three disorder scenarios...... 109 Figure 6.17 Calculated superﬂuid density of LSCO...... 110

Figure 6.18 Absolute superﬂuid density of LSCO calculated for constant ΓN . . 111

xi Chapter 1

Introduction

Festkörperphysik ist eine Schmutzphysik Wolfgang Pauli

This famous quote, attributed to Wolfgang Pauli, translates as: solid state physics is the physics of dirt. Although certainly said in disdain of the — at the time — relatively new ﬁeld of solid state physics, he could not have been more correct! Many of the materials used in our everyday lives depend crucially on the careful control of disorder and impurities. From the semiconductor devices that enable veritable supercomputers to be carried in our pockets, to the high-ﬁeld superconducting magnets that power the magnetic resonance imaging instruments in hospitals, and to the atomic defects in materials that are poised to unleash the revolution of quantum computing, all these depend crucially on understanding how impurities and defects interact with, and modify, the materials within which they reside.

1.1 A brief history of superconductivity

There remain many unsolved mysteries in physics, but few have stood the test of time and rigorous investigation in the way superconductivity has. Over a century after the 1911 discovery of zero resistivity below a critical temperature, Tc = 4.2 K, in mercury by H. K. Onnes [3], a full theory of superconductivity has yet to be formulated. At the time of discovery, the prevailing belief was that superconductivity was caused by the disappearance of scattering. In 1914, Onnes further discovered that superconductivity is destroyed above a critical magnetic field. Then, in 1927, Meissner noted the absence of a thermoelectric effect in superconductors [4], showing that there is a type of charge carrier that has zero entropy in the superconducting state. A jump in the heat capacity of tin was observed in 1932, strongly suggesting that superconductivity is a distinct thermodynamic state [5]. This was followed by the discovery of the Meissner (Ochensfeld) effect in 1933 [6] whereby superconductors expel magnetic flux from their bulk, confirming that the superconducting state is in fact a distinct phase of matter, and not just a metal in which scattering suddenly disappears. Gorter and Casimir hypothesized in 1934 that a superconductor is made of a mixture of

1 normal electrons and super-electrons conducting in parallel [7]. The super-electrons were assumed to flow without resistance, carry no entropy, and as temperature was lowered the fraction of super-electrons increased. Although incorrect in many respects, the two-fluid model accounts for both the thermal and electrical properties of a superconductor, and is still a useful way to think about the parallel contributions to the electrical conductivity from the electrons that have condensed to form the superconducting condensate, and from the quasiparticle excitations in the material. In 1935, the London brothers derived the electrodynamic equations of a superconductor and showed that a magnetic field impinging on a superconductor is exponentially screened from the bulk over a length scale λL, the London penetration depth. Ginzburg and Landau then explored in 1950 the consequences of allowing a spatially non-uniform wavefunction [8]. This led to the concept of a coherence length, ξ, which sets the length scale over which the amplitude of the superconducting wavefunction can vary. With this, superconductors can then be classified into type I and type II depending on the sign of the interface energy between superconducting and normal regions. In type I superconductors there can be no flux entry into the bulk, whereas in type II superconductors, where the normal–superconducting interface energy is negative, it becomes energetically beneficial to allow some flux into the bulk in the form of flux-line vortices. The theories mentioned above have all been macroscopic theories of superconductivity, having no direct connection to the microscopic world that must undoubtedly govern the superconducting state. The 1950 discovery of a relation between transition temperature √ and isotopic mass in mercury [9], with Tc ∝ 1/ M, pointed towards the electron–phonon interaction being the key to superconductivity. Around the same time, the electron–phonon interaction was shown to give rise to a low-energy attractive interaction [10]. This turned out to be a prerequisite to understanding the microscopic origins of superconductivity. In 1956, Cooper found that an arbitrarily small attractive interaction — such as the electron– phonon interaction — induces a two-particle bound state [11], now known as a Cooper pair. He remarked that this state acts like a boson, despite being formed of fermionic particles. The Fermi surface is unstable to the formation of Cooper pairs, and, being bosons, they will continue to condense into a macroscopic ground state until some equilibrium is reached. The electron–phonon interaction relies on the interchange of a virtual phonon, allowing the strong Coulomb repulsion to be overcome by having the interaction be spread out in time, i.e., a retarded interaction. This formed the basis for the microscopic theory of superconductivity. The full Bardeen–Cooper–Schrieffer (BCS) theory followed in 1957 with an inspired guess at the ground state of the BCS Hamiltonian [12, 13]. The superconducting state described by BCS theory, far from being a quiescent state, is characterized by the continuous creation and destruction of Cooper pairs as the electrons propagate through the material. BCS theory was very successful at describing the properties of most superconducting ma-

2 terials that had been measured up to this point. It also incorporated the isotope eﬀect, solidifying the case for the validity of BCS theory. Based on this success, the problem of superconductivity seemed resolved. In 1968, using the the strong-coupling theory of superconductivity, McMillan [14] was able to predict the maximum Tc that might be obtained if, for a given material, one could modify the phonon spectrum or the electron-phonon coupling constant. The calculations were applied to several superconducting compounds that were known at the time, and it was predicted that for Nb2Sn, the maximum transition temperature would be 28 K, and for

V3Si, it would be 40 K. These predictions are sometimes incorrectly interpreted as placing a hard limit on the transition temperatures of phonon-mediated superconductivity. Instead they placed a limit on the transition temperature of a specific material or class of materials.1 The work of McMillan provided a blueprint for raising the transition temperatures of a particular class of superconductor; high-Tc materials would have a high Debye temperature, and large coupling constant. In the years that followed, many attempts were made to raise the transition temperature, but it stubbornly refused to increase much beyond 20 K, lim- iting the technological applicability of superconductors. This all changed suddenly in 1986 when Bednorz and Müller synthesized a new superconductor, La5−xBaxCu5O5(3−y), with a transition temperature of 30 K [15], much higher than the best materials so far. Within a year the transition temperature had been raised to 93 K [16] — well above the boiling point of nitrogen at 77 K — in a quaternary compound containing yttrium, barium, copper and oxygen, which was subsequently found to have the composition YBa2Cu3O7−δ. This new class of materials, dubbed the cuprate superconductors, reinvigorated the field and pointed to a new mechanism for superconductivity, where phonons no longer play a dominant role. Somewhat surprisingly, BCS theory, with slight modifications, can still provide meaningful predictions for these new high-Tc materials. Despite this, we know that modified BCS theory cannot be the fully correct microscopic picture as it relies on the existence of a well defined Fermi liquid, which is notably absent in the pseudogap phase of the cuprates. Many theories have been explored [17], such as the polaron-bipolaron model, and the resonating valence bond model, but a full theory of high-Tc superconductivity, starting from a microscopic Hamiltonian that also describes the normal state, is still beyond our grasp. With every newly-discovered superconductor comes a new opportunity to test the limits of our theories, identify which experimental features are consistent with them, and push the boundaries of our knowledge and associated technological applications. Two recent high-Tc discoveries, with Tc = 203 K in H3S [18], and Tc = 250 K in LaH10 [19], indicated that we were likely on the verge of a room temperature — albeit high-pressure — breakthrough.

1Later work showed that some of the assumptions were not valid, and in fact the transition temperature is not bounded for superconductors governed by electron–phonon interactions. Starting from Eliashberg theory, Tc can in fact be shown to be unbounded, growing like the square root of the coupling strength in the strong coupling regime.

3 In fact, during the late stages of writing this thesis, the elusive goal of achieving (cold) room temperature superconductivity was ﬁnally achieved in a photochemically synthesized ◦ carbonaceous sulfur hydride, with Tc = 288 K (approximately 15 C) at 267 GPa [20]! This discovery, if it withstands the close scrutiny and necessary repeatability tests, will surely renew the hope that a technologically viable room temperature superconductor might soon be found. A simpliﬁed timeline of some representative superconductors from several material families is shown in Figure 1.1.

1.2 Conventional and unconventional superconductivity

Up until about 1980, most superconductors that had been discovered were what we would now call “conventional” superconductors. They have s-wave gap symmetries, where the gap is isotropic, or at least does not change sign as one goes around the Fermi surface, and it is the phonons that provide the pairing glue that allows electrons to behave cooperatively. The transition temperatures of these materials are typically quite low, and non-magnetic impurities tend to have limited eﬀect on conventional superconductivity. The cuprate high-

Tc superconductors discovered in 1986 [15] are almost all “unconventional” superconductors with d-wave pairing symmetry, where phonon interactions do not provide the pairing glue; other fluctuations of the system, such as magnetic fluctuations, are thought to provide the necessary interaction to allow superconductivity to develop. Other unconventional superconductors may have pairing symmetries such as p-wave, d-wave, f-wave, etc. depending on the angular momentum of the Cooper pairs. In these unconventional materials the effects of disorder are much more pronounced and less well understood by the condensed matter community. A quick note on terminology is in order here. Unconventional superconductivity is a particularly vague term, as there are at least six different uses. Most authors understand it to mean that the superconductor is not mediated by the electron–phonon interaction. Another common use is to indicate whether or not the gap changes sign. A third interpretation, based on pairing symmetry, is more closely tied to the concept of crystallographic symmetries than to the mechanism of pairing. By this definition, a conventional superconductor is one which has the same symmetries as the crystal lattice, whereas an unconventional superconductor breaks one or more crystal lattice symmetries.2 Superconducting states that break time- reversal symmetry are also considered unconventional. Another possible interpretation is non-BCS-like superconductivity, e.g., superconductivity without pairing, although no clear examples of such a material are known. Finally, unconventional superconductivity can refer

2The sign-changing d-wave symmetry gap in the cuprate superconductors has blurred the lines as it changes sign. The s-wave case is thus sometimes referred to as conventional and the d-wave case referred to as unconventional, but one should be careful when the crystallographic symmetry is not the same as the pairing symmetry.

4 293 K LaH10 C-S-H HgBaCaCuO @ 30 GPa H3S HgBaCaCuO FeSe 1UC on SrTiO BiSrCaCuO 3 2 10 YBaCuO 77 K TlBaCuO MgB LaOFFeAs @ 4 GPa LaSrCuO 2 Cs3C60@7kPa LaBaCuO FeSe @ 8.9 GPz Nb3Ge RbCsC60 Nb3(Al,Ge) LaOFFeAs Nb3Sn PuCoGa5 NbN K3C60

) V3Si -(BEDT-TTF)2Cu(NCS)2 K Nb ZrN NbTi (

c 10

T Pb LaOFeP FeSe

-(BEDT-TTF)2AuI2 Hg 4.2 K

CeCoIn5 UPd2Al3

100 (TMTSF)2PF6 @ 1.2 MPa UBe13

CeCu2Si2 UPt3

1920 1940 1960 1980 2000 2020 Year Figure 1.1: A non-exhaustive list of transition temperaturesa of several classes of superconductor as a function of time, compiled from several sources. • are conventional superconductors [3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], are heavy fermion superconductors [31, 32, 33, 34, 35, 36], are the cuprates [15, 16, 37, 38, 39, 40, 41], H are the organics [42, 43, 44], I are the iron-based pnictides [45, 46, 47, 48, 49], are the fullerine superconductors [50, 51, 52], and N are the superconducting hydrides [18, 19].

aThe transition temperatures shown here are the currently accepted values, but the dates refer to the discovery of superconductivity in the material. Several of the early superconducting papers list transition temperatures that are signiﬁcantly lower than those shown in the ﬁgure, likely because of impurities, poor quality samples, or limitations of the measurement apparatus.

5 to some unusual aspect of the host material, such as the ultra-low charge carrier density in Nb-doped SrTiO3. There is unfortunately no single criterion that determines whether a material should be labelled as an unconventional superconductor.3 Unconventional superconductors can have s-wave symmetry, as we will see in Chapter 5, and there are even proposals for phonon-mediated d-wave superconductivity [53].

1.3 Impurity studies of superconductivity

The study of impurities has revealed many important clues to the mechanism underlying both conventional and unconventional superconductivity. Anderson’s theorem [54], formulated in 1959, describes how non-magnetic impurities homogenize the superconducting gap by pairing time-reversed states instead of states with opposite momentum k, as in the original BCS formulation. These new states are naturally spread over the Fermi surface as they are now the eigenstates of a disordered potential, with the loss of translational invariance meaning k is no longer a good quantum number. Based on this, Anderson pointed out that BCS theory applies more exactly to very dirty materials, since the disorder-homogenized gap is nearly isotropic, whereas in pure single crystals the gap can have substantial momentum variation. In superconductors with s-wave symmetry, Tc initially drops as increasing disorder progressively homogenizes the gap, then remains constant thereafter. This explains how some amorphous materials can superconduct despite the absence of long-range crystallographic order. Superconductors with higher angular momentum Cooper pairs, such as d-wave superconductors, have no such protection of Tc. Adding nonmagnetic impurities continuously decreases Tc until superconductivity is completely destroyed. On the experimental side, the dependence of Tc on impurities has been used to determine pairing symmetry in several materials. Some examples are Sr2RuO4, in which Al doping ruled out s-wave pairing [55, 56], and UPt3, where the symmetry is likely p-wave [57]. In addition, studies of impurities in the cuprate superconductors were essential in establishing the current consensus that the cuprates have predominantly d-wave pairing symmetry [58, 59, 60, 61]. Despite the long history of impurity studies, the somewhat counterintuitive effects of impurities in superconductors are often overlooked, misunderstood, or even willfully ignored when researchers believe their otherwise excellent samples are beyond such reproach. This thesis will start in Chapter 2 by exploring in detail the theory of dirty BCS superconductivity, with impurities treated in the self-consistent t-matrix approximation (SCTMA). The effects of impurities are calculated for the density of states, superfluid density, optical conductivity, and heat capacity, to help the reader appreciate the expected effects of impurities. Chapter 3 develops the electrodynamic response of a superconductor, and Chapter 4

3 The d-wave state in YBCO belongs to the trivial representation of the C2ν point group, so might be considered conventional by the symmetry argument, but YBCO is of course an unconventional superconductor by the ﬁrst and second criteria.

6 explains the experimental setup used to measure the electrodynamics of a superconductor using microwave spectroscopy. Finally, Chapters 5 and 6 explore three unconventional superconductors, expanding the theory in Chapter 2 to adapt to the unique features of each superconductor. In addition to the experimental results it presents, this thesis is an important step in pushing the boundaries of realistic simulations of superconducting materials, and provides a possible resolution to a recent controversy in cuprate materials.

7 Chapter 2

Theory

2.1 BCS theory

In 1957, Bardeen, Cooper and Schrieffer developed the first truly successful microscopic theory of superconductivity, now known as BCS theory [12, 13]. The key ingredient came from the realization by Cooper that the Fermi sea is unstable against the formation of a bound pair of electrons for an arbitrarily small attractive interaction [11]. The Cooper instability applies to the whole Fermi surface, and electron pairs will continue to condense until some equilibrium is reached. At first glance, it seems that the repulsive nature of the Coulomb interaction would preclude attraction between electrons. However, a dynamic polarization of the ionic lattice, in the form of a phonon, does in fact give just that. The real-space picture of this interaction, shown in Figure 2.1(a, b), is of an electron passing through the ionic lattice and polarizing it due to the Coulomb attraction. Another electron can then be attracted to this localized region of positive charge. The k-space picture, shown in Figure 2.1(c), is of two electrons with opposite momenta scattering from the ionic lattice via a phonon with momentum q. This elastic process must conserve momentum and so k → k + q and k0 → k0 − q. To maximize the probability of the electrons binding into a pair the momenta of the two electrons need to be opposite. The allowable k-states for this scattering process must also lie in a narrow region near the Fermi surface, and the Cooper pairs are therefore spatially extended. This large spatial extent allows the Cooper pairs to overlap strongly, and they therefore acquire a common centre-of-mass wavefunction throughout the superconductor, which can be identified as the superfluid wavefunction in the London theory of superconductivity. Inspired by the idea of pairing states with opposite momenta, the BCS Hamiltonian can be written as X † X † † H = kckσckσ + Vk,k0 ck↑c−k↓c−k0↓ck0↑ , (2.1) k,σ k,k0

† where ckσ is a quasiparticle creation operator, k is the quasimomentum (crystal momentum), σ represents the spin of the electron, and Vk,k0 is the attractive interaction. This

8 a) + + + + + + c)

+ + + + + + εF e- + + + + + +

+ + + + + + k' -k

-q Vkk' q b) + + + + + + -k' k + + + + + + e- + + + + + +

+ + + + + +

Figure 2.1: Pictorial representation of BCS pairing in real space (a, b), and k-space (c). (a) An electron first passes through the lattice and polarizes it. (b) A second electron with opposite momentum is attracted to the localized region of positive charge shown with the shaded grey area. (c) After [62]. States at or near the Fermi surface with opposite momentum pair via the attractive interaction Vkk0 . particular form of the Hamiltonial applies to the spin–singlet wavefunction, but can be gen- eralised to the spin-triplet wavefunctions with only slight modification. J. R. Schrieffer — who was a graduate student at the time — proposed the following ansatz for the ground state wavefunction: Y † † |BCSi = uk + vkck↑c−k↓ |0i , (2.2) k 2 where |0i is the vacuum state, uk represents the probability of not creating a Cooper pair, 2 and vk represents the probability of creating a Cooper pair. In the form given in Equation 2.1, the BCS Hamiltonian is an intractable problem, but progress can be made with a mean-field approximation, which is appropriate here since the number of particles involved in the BCS ground state is large, and Cooper pair number fluctuations should therefore be small. Because the BCS wavefunction is a phase-coherent superposition of pairs, we can define the expectation value of the pair operators

hbki = hBCS|c−k↓ck↑|BCSi , (2.3) ∗ † † hbki = hBCS|ck↑c−k↓|BCSi . (2.4)

9 Then in the mean-ﬁeld approximation we can write

c−k↓ck↑ = hbki + (c−k↓ck↑ − hbki) , (2.5) † † ∗ † † ∗ ck↑c−k↓ = hbki + (ck↑c−k↓ − hbki ) , (2.6) where the term in parentheses represents the ﬂuctuations around the expectation values. The BCS Hamiltonian can then be approximated with

X † X ∗ † † ∗ H = kckσckσ + Vk,k0 hbki + ck↑c−k↓ − hbki hbk0 i + c−k0↓ck0↑ − hbk0 i k,σ k,k0 X † X ∗ † † ∗ = kckσckσ + Vk,k0 hbki c−k0↓ck0↑ + hbk0 ick↑c−k↓ − hbki hbk0 i , (2.7) k,σ k,k0 where we have neglected the term that is quadratic in the ﬂuctuations. Now deﬁne a new parameter X ∆k = − Vk,k0 hbk0 i , (2.8) k0 so that X † X † † ∗ ∗ H = kckσckσ − ∆kck↑c−k↓ + ∆kc−k↓ck↑ − ∆khbki . (2.9) k,σ k The Hamiltonian can then be diagonalized by a canonical transformation, commonly referred to as a Bogoliubov transformation [63] (although also discovered independently by Valatin in the same year [64]):

∗ † ck↑ = ukγk0 + νkγk1 , † ∗ † c−k↓ = −νkγk0 + ukγk1 . (2.10)

2 2 With the condition that 1 = |uk| + |νk| , the coeﬃcients can be solved and are

2 2 1 k |νk| = 1 − |uk| = 1 − , (2.11) 2 Ek

∗ ∆k ukνk = q , (2.12) 2 2 2 k + |∆k| where q 2 2 Ek = k + |∆k| . (2.13) The mean-ﬁeld BCS Hamiltonian then becomes

X ∗ X † † H = (k − Ek + ∆khbki ) + Ek γk0γk0 + γk1γk1 , (2.14) k k

10 a) b) c)

2Δk

kF kF kF

Figure 2.2: After [65]. (a) Excitation spectrum for parabolic electron-like band. (b) Excita- tion spectrum for both particle and hole-like bands. Note the symmetry about the Fermi energy. (c) Opening of the superconducting gap ∆ caused by coupling between the electron- like and hole-like bands. The colors represent the degree of electron character (blue) and hole character (red). At the Fermi surface the quasiparticles are an equal admixture. where the first term describes the condensation energy of the quasiparticles and the second term describes the elementary excitations of the system, γk, known as Bogoliubov quasiparticles (or Bogoliubons). These elementary excitations have energy Ek and dispersion given by Equation 2.13. Figure 2.2 shows a schematic of the quasiparticle excitation spectrum of a superconductor, and how it develops from having electron-like and hole-like bands crossing at the Fermi level. From this picture it is clear that the minimum energy required to excite a Bogoliubov quasiparticle, i.e., break a Cooper pair, is 2∆k, which leads to the robustness of the superconducting state to small thermal or energetic fluctuations. The temperature dependence of the gap can be found by first converting Equation 2.8 with the Bogoliubov transformations into

X ∗ † † ∆k = − Vk,k0 uk0 νk0 h1 − γk0γk0 − γk1γk1i . (2.15) k0

Bogoliubons obey Fermi statistics with

† † h1 − γk0γk0 − γk1γk1i = 1 − 2f(Ek) , (2.16) so Equation 2.15 becomes

X ∗ ∆k = − Vk,k0 uk0 νk0 (1 − 2f(Ek)) (2.17) k0 X ∆k0 Ek0 = − Vk,k0 q tanh . (2.18) 2 2 2kBT k0 2 k0 + |∆k0 |

11 A more convenient computational form is found by writing the above equation in Matsubara form ωD X ∆k0,n ∆k = +2πT N(0) Vk,k0 . (2.19) q 2 2 ωn>0 ωn + ∆k0,n

2.2 Self-consistent t-matrix approximation

In many materials the effects of impurities play a dominant role in determining their physical properties. Here we describe the self-consistent t-matrix approximation (SCTMA), which is a simplified model allowing calculations of the properties of a material in the presence of disorder. In this section we follow the pedagogical approach of N. Hussey [66] and R. Joynt [67] by first introducing impurities in the context of a semiconductor, and then linking the formalism to the superconducting case through the Nambu form of the Hamiltonian for superconductivity. We start with the Hamiltonian for a single non-magnetic impurity in a semiconductor,

X † X † H = H0 + Himp = kckck + Uk,k0 ckck0 , (2.20) k k,k0

† where ck and ck are the creation and annhilation operators, Uk,k0 is the Fourier transform of the impurity potential, and the spin indices have been omitted to simplify the notation.

The unperturbed (or ‘bare’) Green’s function for H0 is

−1 G0k ≡ G0 (k, iω) = (iω − k) . (2.21)

The impurity-renormalized Green’s function can then be expressed as the perturbation series

0 X Gkk0 ≡ G k, k , iω = δkk0 G0k + G0kUkk0 G0k0 + G0kUkk00 G0k00 Uk00k0 G0k0 + ... (2.22) k00

This equation can then be converted into a Feynman digram

Ukk' Ukk" Uk"k'

= + + + ... k k' k k k' k k" k' where the thick orange arrow represents the renormalized Green’s function, the thin green arrow represents the unperturbed Green’s function, and the blue dot represents a scattering oﬀ the impurity potential. The Feynman diagram clearly shows that the perturbation series represents the process of a particle scattering multiple times from a single impurity. From the second term onwards, each diagram starts with a bare propagator (an alternative name

12 for a Green’s function in this context) with momentum k and ends with a propagator with momentum k0. We can thus factor out the common pieces to get

= + Tkk' k k' k k k'

U U U U Tkk' = kk' + Ukk'' Uk''k' + kk'' k''k''' k'''k' + ...

k'' k'' k''' where the yellow box is the t-matrix which describes the combined eﬀect of multiple scattering. Close inspection of the t-matrix reveals yet another opportunity to factorize the diagram by ﬁrst re-labelling the indices on the t-matrix

Converting back from diagram to equation gives

X Tkk0 = Ukk0 + Ukk00 G0k00 Tk00k0 , (2.23) k00

Gkk0 = G0k + G0kTkk0 G0k0 . (2.24)

This formula is exact for a single impurity. Note that the Green’s function depends on two momenta, which indicates that translational invariance is broken in the single scattering centre case. In a real material, there will be multiple scattering impurities, and the potential must therefore be a sum over all the impurity positions

N 0 X i(k−k )·Ri Vkk0 = Ukk0 e , (2.25) i where the Ri represent the positions of the impurities, and we have assumed that the impurity potential is identical for every impurity. As with the single impurity case, we start

13 with a perturbative expansion that includes the new disorder potential

0 X Gkk0 ≡ G k, k , iω = δkk0 G0k + G0kVkk0 G0k0 + G0kVkk00 G0k00 Vk00k0 G0k0 + ... (2.26) k00

One particular choice of the impurity positions is of no interest when discussing macroscopic properties of a material; we require an impurity-averaged Green’s function, which is given by the average of each term in the expansion. The ﬁrst term is trivial as there is no positional dependence. The average of the second term is

N N D 0 E D 0 E X i(k−k )·Ri X i(k−k )·Ri G0k Ukk0 e G0k0 = G0kG0k0 Ukk0 e (2.27) i i

= G0kG0k0 Ukk0 nimpδkk0 , (2.28) where nimp is the impurity concentration. This can be represented by the following diagram

Ukk'

k k' where the cross indicates an impurity-averaged interaction. The third term, written in full, is N N 00 00 0 X X i(k−k )·Ri X i(k −k )·Rj G0k Ukk00 e G0k00 Uk00k0 e G0k0 . (2.29) k00 i j There are clearly two diﬀerent parts to this equation; that of repeated scattering from the same impurity, where i = j, and that of scattering from two diﬀerent impurities, where i 6= j. For the former, the angle-averaged equation is

N D 00 00 0 E X X i(k−k )·Ri i(k −k )·Ri G0kG0k00 G0k0 Ukk00 Uk00k0 e e . (2.30) k00 i

This term reduces to

N D 0 E X X i(k−k )·Ri G0kG0k00 G0k0 Ukk00 Uk00k0 e , (2.31) k00 i

0 which has a single delta function that restricts the momentum to k = k , and nimp appears only once: X G0kG0k00 G0k0 Ukk00 Uk00k0 nimpδkk0 . (2.32) k00

14 Following a similar procedure, the scattering from two diﬀerent impurities is given by

X 2 G0kG0k00 G0k0 Ukk00 Uk00k0 nimpδkk00 δk00k0 , (2.33) k00

0 where there are two powers of nimp, and again the momentum is restricted to k = k . In fact, all terms after impurity averaging will have the same restriction on momentum; the impurity averaging has restored translational invariance. In terms of diagrams, the two- scattering terms are

k''

Note how in these diagrams we have omitted any terms where two impurity interaction lines cross; that is, we have neglected interference eﬀects between scattering events. The three-scattering event terms can then be enumerated as

The equivalent equations for these diagrams are simple to derive by inspection. Diagram- matically, the impurity averaged Green’s function can therefore be expressed as

15 Now we deﬁne the self-energy as the sum of all irreducible diagrams, that is, those which cannot be divided into subdiagrams by cutting a single bare propagator (green arrow).

where the delta functions required to preserve momentum are omitted for clarity. By inspection, we can see that, if we replace the bare propagator with the renormalized propagator, we would obtain the same series. For example, the fourth term in the diagram above is equivalent to the second term with the bare propagator replaced by the second term of the renormalized propagator. The self-energy thus takes the form

X Σk = nimpUkk0 δkk0 + nimp Ukk00 Gk00 Uk00k0 δkk00 δk00k0 + k00 X (2.34) nimp Ukk00 Gk00 Uk00k000 Gk000 Uk000k0 δkk00 δkk000 δk000k0 + ... k00k000 which can then be re-written in a self-consistent form: ! X Σk = nimpTkk0 δkk0 = nimp Ukk0 + Ukk00 Gk00 Tk00k0 δkk0 . (2.35) k00

16 That is X Tkk0 = Ukk0 + Ukk00 Gk00 Tk00k0 . (2.36) k00 The reducible terms in the renormalized propagator that do not appear in the self-energy can be obtained by multiplying the self-energy with the renormalized propagator. The renormalized propagator thus takes the form of a Dyson equation represented by

and in equation form by

Gk = G0k + G0kΣkGk . (2.37)

Rearranging this equation leads to

−1 −1 Gk = G0k − Σk , (2.38) which is the ﬁnal form for the SCTMA. Next we consider a short range s-wave scattering potential where the electron wavepacket is much larger than the impurity potential. The potential is constant in k-space and thus looks like a delta function (point defect) in real-space:

Ukk00 = U. (2.39)

Noting that both terms in Equation 2.36 no longer have any k dependence we get

! X Σ = nimpT = nimpU 1 + Gk00 T . (2.40) k00

The t-matrix has now lost its momentum dependence due to the form of the potential. Now multiply the t-matrix on both sizes by Gk and sum over momentum

X X X X GkT = U Gk + U Gk Gk00 T. (2.41) k k k k00

Deﬁning the left side as x,

X X X x = U Gk + U Gk Gk00 T, (2.42) k k k00

17 and then swapping summation indices gives

X X X x = U Gk + U Gk00 GkT (2.43) k k00 k X X = U Gk + U Gkx . (2.44) k k

This can then be rearranged into

P U Gk k x = P , (2.45) 1 − U Gk k and combining this with Equation 2.40 then gives

Σ = nimpU (1 + x) (2.46)  2 P  U Gk  k  = nimp U + P  (2.47) 1 − U Gk k 1 = nimp 1 P . (2.48) U − Gk k

Therefore, once the sum over Gk is carried out, and for the particular case of point-like scattering, the SCTMA self-energy no longer has any momentum dependence. The physics of a superconductor could not seem further from that of a semiconductor, but for the fact that, as hinted in Figure 2.2, the quasiparticles of a superconductor can be viewed as hole-like and electron-like excitations out of a band structure with a gap. This parallel means that the formalism that we have just presented can be applied to superconductors with only slight modiﬁcation. To see the parallel more clearly, the mean- ﬁeld Hamiltonian in Equation 2.9 can be re-cast in a particularly elegant form by introducing the Nambu spinor ! ck↑ ψk = † , (2.49) c−k↓ and the corresponding Hermitian conjugate

† † ψk = ck↑, c−k↓ . (2.50)

18 By expanding the sum over spins, recognizing that k = −k, and using the commutation relation for the fermionic operators, the kinetic energy term becomes

X † X † † kckσckσ = k ck↑ck↑ − c−k↓c−k↓ + 1 , (2.51) k,σ k " # ! X † k 0 ck↑ X = ck↑, c−k↓ † + k . (2.52) k 0 −k c−k↓ k

After dropping the constant term, a similar process can be performed for the pairing terms, which results in

" # ! ∆ c H = X c† , c k k↑ k↑ −k↓ ∗ † k ∆ −k c−k↓ " # † k ∆1 − i∆2 = ψk ψk ∆1 + i∆2 −k † = ψk [kτ3 + ∆1τ1 + ∆2τ2] ψk (2.53) where ∆ = ∆1 −i∆2 is a general form for the order parameter, and we have used the isospin matrices "1 0# "0 1# "0 −i# "1 0 #! (τ0, τ1, τ2, τ3) = , , , . (2.54) 0 1 1 0 i 0 0 −1 This Hamiltonian is quite analogous to the one for the semiconductor, but the quantities are all matrices now. Allowing for the possibility of a k dependent gap, the Green’s functions for a superconductor are then

−1 G0k ≡ G(k, iωn) = (iωnτ0 − kτ3 − ∆kτ1) , (2.55) and ˜ −1 Gk ≡ G(k, iωn) = iω˜nτ0 − ˜kτ3 − ∆k,nτ1 . (2.56)

The impurity Hamiltonian is

X † Himp = ψkUkk0 τ3ψk0 , (2.57) kk0 which takes this τ3 matrix form in order to preserve time-reversal symmetry. Due to the three matrix components of the Hamiltonian, there will be three self-energy terms instead of just one. Assuming again that the scatterers are point-like, each self-energy then reduces

19 to a momentum independent term

iω˜n = iωn − Σ0 , (2.58)

˜k = k + Σ3 , (2.59) ˜ ∆k,n = ∆k,n + Σ1 . (2.60)

In order to calculate the self-energy we must evaluate the sum over the renormalized Green’s function:

X X ˜ −1 Gk = iω˜nτ0 − ˜kτ3 − ∆k,nτ1 (2.61) k k iω˜ τ + ˜ τ + ∆˜ τ = − X n 0 k 3 k,n 1 . (2.62) 2 2 ˜ 2 k ω˜n + ˜k + ∆k,n

In order to do so we ﬁrst deﬁne the Fermi surface average

R Nk ... dSk h...iFS = R (2.63) NkdSk R N ... dS = k k , (2.64) N(0) where dSk is an element of Fermi surface, N(0) is total Fermi-surface density of states for a single spin, and V 1 Nk = 3 (2.65) (2π) ~vF,k is the k-dependent density of states. The momentum sum can thus be transformed into a Fermi surface average:

V Z V Z dS Z X G → d3k G = k d G () (2.66) k 3 k 3 v k k (2π) (2π) ~ F,k Z Z = Nk d Gk()dSk (2.67) Z = N(0) d Gk() . (2.68) FS

20 The energy integral of the Green’s function can then be performed in three parts and gives

* + 1 X iω˜ g0 ≡ Tr (τ0Gk) = q , (2.69) πN(0) 2 ˜ 2 k ω˜ + ∆ k,n FS * ˜ + 1 X ∆k,n g1 ≡ Tr (τ1Gk) = q , (2.70) πN(0) 2 ˜ 2 k ω˜ + ∆ k,n FS 1 g ≡ X Tr (τ G ) = 0 , (2.71) 3 πN(0) 3 k k where g0, g1, and g3 represent the momentum integrated parts of the Green’s function corresponding to the three τ matrices. The g3 term is only zero if there is strict particle- hole symmetry; this assumption breaks down near a van Hove singularity. Combining these equations with Equation 2.48 then gives

1 −1 Σ = n τ + πN(0)g τ + πN(0)g τ (2.72) imp U 3 0 0 1 1 nimp cτ3 − g0τ0 + g1τ1 = 2 2 2 , (2.73) πN(0) c − g0 + g1 where c = 1/ (2πN(0)U) is the cotangent of the scattering phase shift. The self-energy can be resolved into separate isospin components

nimp g0 Σ0 = 2 2 2 , (2.74) πN(0) c − g0 + g1 nimp g1 Σ1 = 2 2 2 , (2.75) πN(0) c − g0 + g1 nimp c Σ3 = 2 2 2 (2.76) πN(0) c − g0 + g1

The unitarity limit, when the cotangent of the scattering phase shift, c, is zero, represents strong or resonant scattering. The Born limit occurs when c → ∞, and represents weak scatterers. For the particle-hole symmetric case that is assumed here the Σ3k term corresponds to a shift in the chemical potential, and is therefore ignored. In the presence of disorder it can be shown that Equation 2.19 is renormalized in the following way ωD * ˜ + X ∆k0,n ∆k = +2πT N(0) Vk,k0 q . (2.77) 2 ˜ 2 ω >0 ω˜ + ∆ 0 n n k ,n FS Written out in full form we have the following self-consistent equations for the SCTMA, where the self-energy terms have been re-cast in the imaginary axis (or Matsubara) formalism hNk(˜ωn)iFS ω˜n = ωn + πΓ , (2.78) 2 2 2 c + hNk(˜ωn)iFS + hPk(˜ωn)iFS

21 hP (˜ω )i ∆˜ = ∆ + πΓ k n FS , (2.79) k,n k 2 2 2 c + hNk(˜ωn)iFS + hPk(˜ωn)iFS with ωñ Nk(˜ωn) = q , (2.80) 2 ˜ 2 ωñ + ∆k,n ˜ ∆k,n Pk(˜ωn) = q , (2.81) 2 ˜ 2 ωñ + ∆k,n where ωD is the Debye frequency and acts as a cutoff, Vk,k0 is the electron-electron interaction, and Γ = nimp/(πN(0)) is a scattering parameter that is proportional to the impurity concentration. We now have a set of equations that describe BCS superconductivity (in the mean-field approximation) that is valid for a superconductor with singlet pairing symmetry (s-wave or d-wave), arbitrary Fermi surface, any phase shift, but limited to low impurity concentrations. The rest of the theory section will evaluate some observable quantities for the case of an isotropic Fermi surface to provide the reader with some intuition for the effects of impurities in different scenarios. Chapter 5 and Chapter 6 will then show how these ideas can be applied to provide insight into the physics of three different unconventional superconductors.

2.2.1 Anderson’s theorem

In a material without any disorder, the Bloch states, labelled by k, are good eigenstates of the Hamiltonian. Introducing disorder breaks translational invariance, and k ceases to be a good eigenstate as the electron state spreads out around the Fermi surface. BCS theory was formulated by pairing k-states so this would seem to suggest that conventional superconductivity would be destroyed by impurities, but experimental evidence proves otherwise. When adding non-magnetic substitutional impurities, Tc initially drops in many conventional superconductors, but then remains constant even with the addition of enormous amounts of impurities. In 1959, Anderson realised that it is time-reversed states that provide the pairing, and that the (k ↑, −k ↓) pairing of BCS theory is one particular manifestation of this [54]. In pairing time-reversed states, Anderson showed that all the characteristics of BCS theory can be preserved. In fact, as Anderson pointed out, one would expect that the BCS theory of superconductivity would be more correct for dirty superconductors than for clean ones in which the energy gap can have strong momentum dependence around the Fermi surface. The addition of impurities connects distant parts of the Fermi surface, and homogenizes any anisotropy in the gap, but the time-reversed states persist. When this idea is applied to conventional superconductors with s-wave gap symmetry there typically is a slight decrease in Tc and ∆ as the gap homogenizes, and then they remain constant once the homogenization

22 is complete. This is not the case for unconventional superconductors where the gap changes sign; homogenization will result in a rapid decrease in both Tc and ∆, and superconductivity will eventually be destroyed at some critical impurity concentration. We now show how Anderson’s theorem is contained in the SCTMA equations derived in the previous section. In order to simplify the calculation, we assume an isotropic Fermi surface and an isotropic gap, which allows us to drop the Fermi surface average. The equations are then

2 ΓN c + 1 ωñ ωñ = ωn + q , (2.82) D 2 ˜ 2 ωñ + ∆n

2 ˜ ˜ ΓN c + 1 ∆n ∆n = ∆ + q , (2.83) D 2 ˜ 2 ω˜n + ∆n and ωD ˜ X ∆n ∆ = +2πT λq , (2.84) 2 ˜ 2 ωn>0 ω˜n + ∆n

2 2 2 2 where D = c +N(˜ωn) +P (˜ωn) , and ΓN = πΓ/(1+c ) is the normal-state scattering rate, which is related to the experimentally accessible transport scattering rate by Γtr = 2ΓN . Equation 2.82 and Equation 2.83 both have the same form, so dividing the unrenormalized quantity by the renormalized one, and rearranging gives

ω ∆ Γ c2 + 1 1 n = = 1 − N , (2.85) ˜ q ω˜n ∆n D 2 ˜ 2 ω˜n + ∆n and therefore ω˜ ω n = n . (2.86) ˜ ∆n ∆ Substituting this into Equation 2.84 gives

ωD ∆ ∆ = +2πT X V , (2.87) pω2 + ∆2 ωn>0 n which is the original BCS gap equation without impurities. This shows that both the energy gap and the transition temperature will be unaﬀected by nonmagnetic impurites. This is not to say that there will be no visible eﬀects of the impurities; Anderson’s theorem protects

∆ and Tc, but does not protect the superﬂuid density, as we will see in Section 2.3.2. Another point of interest that should be mentioned is that for the particular case of an isotropic gap and pointlike s-wave scatterers, no matter the Fermi surface shape, the scattering phase shift does not aﬀect the renormalized gap and frequency for a given normal-

23 state scattering rate. This can be seen from

2 2 2 D = c + N(˜ωn) + P (˜ωn) (2.88) ω˜2 ∆˜ 2 = c2 + n + n (2.89) 2 ˜ 2 2 ˜ 2 ω˜n + ∆n ω˜n + ∆n = c2 + 1 , (2.90)

2 which then converts the prefactor ΓN (c + 1)/D into ΓN .

2.2.2 Gap equation and Tc

In order to determine the temperature dependence of the gap we start with the general equations for BCS superconductivity in the SCTMA. For simplicity we assume a separable pairing interaction that has the same angular harmonics as the gap, Vk,k0 = V0ΩkΩk0 , with hΩkiFS = 1. Introducing the temperature dependent gap parameter ψ(T ), the gap equation then becomes

ωD * ˜ + X ψnΩk0 ∆k ≡ ψΩk = 2πT N(0) V0ΩkΩk0 q . (2.91) 2 ˜2 2 ω >0 ω˜ + ψ Ω 0 n n n k FS

In the absence of disorder, the quasiparticle energies are unrenormalized (ω˜n = ωn) and the 2 gap vanishes (ψ → 0) at the clean-limit transition temperature Tc0. Using Ωk FS = 1, we have at this temperature

ωD 1 X 1 2ωD = 2πTc0 ≈ ln , (2.92) V0 ωn(Tc0) 1.76Tc0 ωn>0 where the approximation is valid when ωD Tc0. This equation can be extended to arbitrary temperature by adding and subtracting

ω XD 2πT 1/ωn(T ) (2.93) ωn>0 and using the same logarithmic approximation:

1 ωD 1 T = 2πT X + ln . (2.94) V0 ωn(T ) Tc0 ωn>0

For the d-wave case, Equation 2.79 reduces to ψ˜n = ψ as the gap averages to zero around the Fermi surface. Applying this to Equation 2.91 and combining with Equation 2.94 allows us to eliminate V0, and rapid convergence allows the Matsubara sum to be taken to inﬁnity,

24 eliminating explicit dependence on ωD:   * + T ∞ 1 Ω2 c0 X  k  d-wave : ln = 2πT − q , (2.95) T ω 2 2 2  ω >0 n ω˜ + ψ Ω n n k FS

This equation is then solved self-consistently at each temperature to obtain ψ(T ) and ω˜n(T ) for any T < Tc. The equation for the s-wave case is similar, with an additional term due to the gap not averaging to zero over the Fermi surface.

  * + T ∞ 1 ψ˜ /ψ s-wave : ln c0 = 2πT X  − n  . (2.96) T ω q  ω >0 n ω˜2 + ψ˜2 n n n FS

Using Equation 2.86 allows all renormalized quantities to be eliminated from the equation leaving ∞ * + ! Tc0 X 1 1 s-wave : ln = 2πT − p . (2.97) T ωn ω2 + ψ2 ωn>0 n FS Solving these equations gives the curves in Figure 2.3. The s-wave gap does not change with increasing doping, but the d-wave gap does.

2.2.3 Abrikosov-Gor’kov formula

As we have just seen, in the presence of disorder the d-wave gap closes at a reduced transition temperature. For T ≥ Tc, Nk(˜ω) = 1, and the t-matrix equation describing impurity scattering reduces to πΓ ω˜(ω ) = ω + ≡ ω + Γ . (2.98) n n 1 + c2 n N

The imaginary part of the real-axis self-energy in the SCTMA is ΓN , which is the normal- state scattering rate due to impurities. To determine Tc we solve Equation 2.95 with T → Tc,

ψ → 0 and ω˜n → ωn + ΓN :

∞ Tc0 X 1 1 ln = 2πTc − (2.99) Tc ωn ωn + ΓN ωn>0 ! ∞ 1 1 = X − (2.100) n + 1 n + 1 + ΓN ωn>0 2 2 2πTc 1 ΓN 1 = ψ0 + − ψ0 , (2.101) 2 2πTc 2 where ψ0 is the digamma function. This is the famous Abrikosov-Gor’kov formula for a d-wave superconductor [68]. Figure 2.4 shows the expected drop in Tc relative to Tc0 as scattering increases.

25 unitarity

unitarity

Figure 2.3: (a) Gap for an s-wave superconductor as a function of temperature. (b) Gap for a d-wave superconductor as a function of temperature for both the Born and unitarity limits and several scattering rates: ΓN /Tc0 = 0.2, 0.4, 0.6, 0.8. (c) Zero temperature gap as a function of the scattering rate for a d-wave superconductor. The black line represents the clean d-wave result.

26 1.0

0.8

0.6 0 c T / c T 0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8

N/Tc0

Figure 2.4: Suppression of the transition temperature in a d-wave superconductor as a function of scattering rate based on the Abrikosov-Gor’kov formula.

2.2.4 Critical scattering rate

At a suﬃciently high impurity scattering rate, d-wave superconductivity is completely sup- crit pressed and Tc → 0. To calculate the critical scattering rate, ΓN we ﬁrst express the digamma function, ψ0, in its asymptotic expansion form

1 ∞ B ψ (z) = ln z − − X 2n , (2.102) 0 2z 2nz2n n=1 where Bk is the kth Bernoulli number. The argument of the ﬁrst digamma function in crit the AG equation is 0.5 + ΓN /(2πTc) which goes to inﬁnity in the limit of Tc → 0. We can therefore ignore all terms except log z in the expansion above. The AG formula then becomes crit ! Tc0 ΓN 1 ln = ln − ψ0 . (2.103) Tc 2πTc 2 Exponentiating both sides

crit ! ! Tc0 ΓN 1 = exp ln − ψ0 (2.104) Tc 2πTc 2 crit ΓN 1 = exp −ψ0 , (2.105) 2πTc 2

27 and then simplifying the expression gives

2πT Γcrit = c0 . (2.106) N 1 exp −ψ0 2

crit Evaluating this expression gives a critical scattering rate of ΓN ≈ 0.8819 Tc0 for d-wave superconductors.

2.3 Experimental observables

2.3.1 Density of states

The quasiparticle density of states (DOS) represents the number of states available to be occupied by the quasiparticles at a given energy. If there are no empty states at a particular energy then there can be no pair-breaking. The DOS therefore shows which energies are ‘gapped-out’ by the superconducting energy gap, and is therefore a valuable tool in determining gap symmetry and identifying the presence of gap nodes; if there are no states at the Fermi level, then there must be a ﬁnite energy gap around the whole Fermi surface. The general form for the quasiparticle density of states at a particular k point is

d|k| Ek Ns(Ek) = Nn(0) = Nn(0)q θ(Ek − |∆k|) , (2.107) dEk 2 2 Ek − |∆k| where θ (Ek − |∆k|) is the Heaviside step function. For an s-wave superconductor with isotropic Fermi surface and isotropic gap this reduces to

d|| E Ns(E) = Nn(0) = Nn(0) θ(E − ∆) . (2.108) dE pE2 − |∆|2

For a d-wave superconductor with isotropic Fermi surface, and gap function given by ∆(φ) = ∆ cos(2φ), the angle averaged density of states is

Z 2π Z 2π d|k| dφ E dφ Ns(E) = Nn(0) = Nn(0) (2.109) p 2 2 2 0 dEk 2π Ek=E 0 E − ∆ cos (2φ) 2π ! 2 ∆2 = EllipticK , (2.110) π E2 where EllipticK is the incomplete elliptic integral of the ﬁrst kind. See Appendix A for details on eﬃciently evaluating the elliptic functions. Calculations of the DOS for s-wave and d-wave superconductors are shown in Figure 2.5. For the s-wave case, the gap is clearly visible and introducing scattering via the SCTMA does not cause any change to the density of states. For the d-wave case, the linear density of states

28 a) b)

Figure 2.5: (a) Predictions of BCS s-wave density of states. Increasing elastic scattering does not aﬀect the s-wave DOS in any way. (b) From Ref. [69]. Variation of residual density of states with ΓN , proportional to impurity concentration, for the two scattering phase shifts in a d-wave superconductor. (c) From Ref. [69]. Predictions of dirty d-wave theory for the density of states N(ω)/N0 vs ω/∆00, where ∆00 is the clean-limit, zero-temperature gap magnitude.

29 at low energy, which gives the clean d-wave superconductor its characteristic properties, is clearly visible. As scattering is introduced into the system two distinct behaviours are observed. For the unitarity case, there is an immediate increase in the zero-energy density of states, which leads to the commonly accepted T 2 behaviour of the superﬂuid density for dirty d-wave. For the Born case, the density of states remains nearly identically zero and keeps an approximately linear energy dependence over a large range of scattering rates; a behaviour that is quite distinct from the usual understanding of impurities in a d-wave superconductor.

2.3.2 Superﬂuid density

The superfluid density or phase stiffness, ρs, can be thought of as the energy cost of twisting the superconducting wavefunction’s phase. A larger superfluid density will suppress phase fluctuations and make the superconducting state more robust. It is related to the penetration depth, λ, which is the depth to which magnetic fields penetrate into the bulk of a superconductor. A common way of calculating superfluid density comes from this formula:

2 ∞ ρs(T ) λ (0) Z df(E,T ) = 2 = 1 − 2 dE − Ns(E,T ) , (2.111) ρs(0) λ (T ) 0 dE where f(E,T ) is the Fermi function. The factor of 2 is simply a convenient way to deal with the particle/hole symmetry; it would be equivalent to integrate from negative infinity to infinity without the factor of 2. The derivative of the Fermi function acts as a thermal window. Therefore, the equation shows that the superfluid density is the product of the density of states and a thermal window. As temperature increases the number of states that can be thermally accessed will increase and lead to depairing of the superconducting condensate. An example of how this works is shown in Figure 2.6. The origin of the exponentially activated behaviour of the s-wave superconductor is clear from this picture. Figure 2.7 shows the density of states and superfluid densities for s-wave and d-wave superconductors. For the s-wave case the finite gap leads to exponential activation, and for the clean d-wave case the superfluid density is linear at low T since the DOS is linear at low energy. A more convenient formalism for dealing with the superfluid density when impurities are present is the Matsubara formalism. First we take the equation above and integrate by parts ρ (T ) Z ∞ ∆2 s = dEf(E,T ) . (2.112) 2 2 3/2 ρs(0) −∞ (E − ∆ ) iω The poles of the Fermi function are at T = iπ, so we can use the residue theorem to get

ρ (T ) ∞ ∆2 s = 2πT X , (2.113) 2 2 3/2 ρs(0) (ω + ∆ ) ωn>0 n

30 Increasing Temperature

Figure 2.6: (left) The density of states (blue) and derivative of the Fermi function (orange) for three different temperatures. The gap is calculated using the self-consistent formulae described in the text. (centre) Density of states, and product of Fermi function with DOS (solid orange) showing where the contribution from the DOS comes from in the integral. (right) Superfluid density calculated using Equation 2.111. Red dot indicates the superfluid density at the temperature the two other panels were calculated at.

31 Figure 2.7: Density of states (left) and superﬂuid density (right) for s-wave (top) and d-wave (bottom) superconductors without impurities.

32 1.0 N/Tc 0 0.25 0.8 0.5 1.0 2.0 0.6 5.0 0 s 10.0 / s

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

T/Tc

Figure 2.8: Normalized superﬂuid density ρs/ρs00 for an s-wave superconductor with circular Fermi surface. The degree of scattering is characterized by the normal-state scattering rate ΓN in units of the clean-limit transition temperature Tc0. The temperature dependence of the gap has been calculated self-consistently for each set of impurity parameters.

with ωn = 2πT (n + 1/2) being the fermionic Matsubara frequencies. To include the effects of impurities and allow for anisotropic gaps in the Matsubara formalism we simply replace the unrenormalized quantities by their renormalized versions and average over the Fermi surface * + ρ (T ) ∞ ∆˜ 2 s = 2πT X . (2.114) ρs(0) (˜ω2 + ∆˜ 2)3/2 ωn>0 n FS Evaluating this expression for an s-wave superconductor with varying amounts of impurity scattering gives Figure 2.8. The gap and transition temperature are protected by An- derson’s theorem, but the superfluid density is suppressed due to the impurities. Evaluating this expression for a d-wave superconductor with varying amounts of impurity scattering gives Figure 2.9. Based on the DOS plotted in Figure 2.5 (c) we can see how the unitarity limit scattering immediately leads to a large reduction in the superfluid densiy. Born scattering acts quite differently; for the same amount of Tc suppression there is much less superfluid density suppression, and the superfluid density remains almost linear to quite low temperatures even in the presence of significant disorder. This can again be seen from the density of states in Figure 2.5 (c) for the Born case, with the low energy DOS being almost linear up to large scattering rates. This will become very important in Chapter 6 where we evaluate the case of La2−xSrxCuO4.

33 Figure 2.9: From Ref. [70]. Normalized superﬂuid density ρs/ρs00 for a d-wave superconductor with circular Fermi surface. The degree of scattering is characterized by the normal-state scattering rate ΓN in units of the clean-limit transition temperature Tc0. The temperature dependence of the gap has been calcualted self-consistently for each set of impurity parameters.

2.3.3 Optical conductivity

The optical conductivity describes the linear response of a material exposed to an electromagnetic field. The electrical conductivity is primarily sensitive to the effects of light quasiparticles as they are more easily excited by electromagnetic fields than heavy quasiparticles. The typical framework for evaluating the optical conductivity starts with the Kubo formula ! i ne2 σ(ν) = Π(ν + iδ) + , (2.115) ν m where ν is the angular frequency of the electromagnetic field, ne2/m corresponds to the diamagnetic response, and Π(ν +iδ) is the paramagnetic response function. After significant reorganization the equation simplifies to this particularly compact form [71, 72, 73, 74, 75, 76] iΩ2 Z ∞ βω σ(ν) = p dω tanh [J(ω, ν) − J(−ω, ν)] , (2.116) 4πν 0 2

34 where

1 − N(ω)N(ω + ν) − P (ω)P (ω + ν) 1 + N ∗(ω)N(ω + ν) + P ∗(ω)P (ω + ν) 2J(ω, ν) = + , E(ω) + E(ω + ν) E∗(ω) − E(ω + ν) (2.117) q E(ω) = ω˜2 − ∆˜ 2 , (2.118) ω˜(ω + iδ) N(ω) = , (2.119) E(ω + iδ) ∆(˜ ω + iδ) P (ω) = . (2.120) E(ω + iδ) and ω˜ represents the analytic continuation of the Matsubara renormalized frequency to the real axis: hN (˜ω)i ω˜ = ω − iπΓ k FS , (2.121) 2 2 2 c + hNk(˜ω)iFS + hPk(˜ω)iFS Although quite the crude approximation, some intuition may be gained by thinking of the conductivity in terms of which states are available for scattering based on a shifted density of states. Imagine taking the density of states at positive frequency, shown in the left column of Figure 2.7, then taking the correlation with itself. For the s-wave case, at T = 0, we see that there will be no conductivity until the frequency reaches 2∆. In the d- wave case, there will be ﬁnite conductivity at all frequencies, and instead of the gap feature appearing at ω = 2∆ it will ﬁrst appear at ω = ∆. To put this on a more solid footing, Figure 2.10 shows the conductivity of s-wave and d-wave superconductors for several impurity scattering rates. Unitarity scattering results in a plateau in the conductivity. At low scattering rates a signature of the gap is clearly visible, but then disappears with increasing scattering rate. At very large scattering rates the conductivity begins to adopt a Drude-like shape. The Born case has a sharp peak at low frequency, which then turns into a somewhat Drude-like conductivity at a much lower scattering rate. The gap signature is washed out by Born scattering and only appears as a broad feature near the gap value.

2.3.4 Heat capacity

Heat capacity (speciﬁc heat) measurements probe the temperature dependence of the energy of a system. A simple way to access the heat capacity is to ﬁrst calculate the fermionic entropy from the density of states and the Fermi function

Z ∞ S(T ) = −kB N(E)(f(E) ln f(E) + (1 − f(E)) ln (1 − f(E))) dE . (2.122) 0

35 Figure 2.10: Dirty d-wave conductivity for the unitary and Born cases at T = 0.01Tc as a function of frequency in units of the clean, zero-temperature gap ∆00. The conductivity is measured in arbitrary units, but the relative values between each curve is preserved. Grey curves show the Drude conductivity in the normal state.

36 Figure 2.11: Heat capacity for an s-wave and d-wave superconductor with impurities. The normal state heat capacity is shown by a faint grey line.

37 The heat capacity can then be calculated using the usual formula

∂S CV (T ) = T (2.123) ∂T T,V

Evaluating this equation for the s-wave and d-wave cases gives Figure 2.11. Note that again the speciﬁc heat of an s-wave superconductor is not inﬂuenced by the presence of impurities. The two d-wave impurity cases are quite similar, but the heat capacity in the Born case does rise slightly faster at low temperature. The heat capacity is primarily sensitive to heavy quasiparticles, whereas the optical conductivity is primarily sensitive to light quasiparticles. This may go some way towards explaining the seemingly contradictory behaviour in some multiband systems, where the presence of nodes is observed in one quantity but not the other.

38 Chapter 3

Electrodynamics of superconductors

The superﬂuid density and optical conductivity of unconventional superconductors are of great interest as they reveal insights into the pairing state and the charge dynamics of quasiparticles. In order to measure these quantities we need a probe, such as microwave cavity perturbation, that is sensitive to the electrodynamics of a superconductor. This chapter will describe how electromagnetic ﬁelds interact with a superconductor, and how they can be used to measure the surface impedance and hence the complex conductivity.

3.1 London theory

Consider a perfect conductor that has been cooled through its critical temperature in an applied magnetic field. The current density in the material cannot be infinite, so based on Ohm’s law the electric field inside must be zero. Applying Faraday’s law then shows that −∂B/∂t = ∇ × E = 0, and Lenz’s law thus applies perfectly, with induced surface currents precisely opposing any change to the material’s internal magnetic field. As a result, fields will remain trapped even when the external field is removed. This conflicts with the observation of Meissner and Ochensfeld that a magnetic field is expelled from the bulk of a superconductor upon cooling below the superconducting transition temperature. A superconductor is therefore not simply a perfect conductor — there is something more to its electrodynamics. The first phenomenological model to simultaneously capture perfect conductivity and perfect diamagnetism was developed in 1935 by the brothers Fritz and Heinz London. They proposed that the superconducting state can be described by a macroscopic superfluid wavefunction extending throughout the superconductor. With the benefit of hindsight, we now know that the London superfluid wavefunction describes the centre-of-mass motion of Cooper pairs, which have charge 2e.

39 To derive the London equations, start by assuming that the macroscopic wavefunction takes the form √ iθ(r) ψ(r) = npe . (3.1)

1 This describes Cooper pairs with spatially varying phase, and uniform pair density np = 2 ns, where ns is the number density of superconducting carriers. In the case of plane waves the phase takes the form θ(r) = ks · r, and has associated momentum ~ks = 2mevs. The supercurrent density, js = −2ensvs, takes its usual gauge invariant quantum mechanical form 2 ie~ ∗ ∗ 2e ∗ js = (ψ ∇ψ − ψ∇ψ ) − ψ ψA , (3.2) 2me me where A is the vector potential. For the pair wavefunction (Equation 3.1) this becomes

2 e ns ~ js = − ∇θ(r) + A , (3.3) me 2e and reorganizing this equation gives

Λj = − ~ ∇θ(r) + A , (3.4) s 2e

2 where Λ = me/e ns is the London parameter. From Schrödinger’s equation we know that the rate of change of the pair wavefunction must be equal to the local pair energy, 2µ,

∂ψ i = 2µψ , (3.5) ~ ∂t ∂θ(r) = −2µ . (3.6) ~ ∂t

Taking the time derivative of Equation 3.4 then leads to the ﬁrst London equation:

∂Λj ∂∇θ(r) ∂A s = − ~ + (3.7) ∂t 2e ∂t ∂t ∇µ ∂A = − . (3.8) e ∂t

The term on the right side of Equation 3.8 is recognizable as an effective electric field so the first London equation reduces to

∂j Λ s = E . (3.9) ∂t eﬀ

This is an acceleration equation, with no dissipative term, which implies that the superfluid accelerates in response to an applied electric field, and does not decay once the field is removed. Equation 3.9 thus describes perfect conductivity. The second London equation

40 can be derived by taking the curl of Equation 3.4,

∇ × Λj = − ~ ∇ × ∇θ(r) − ∇ × A . (3.10) s 2e

The curl of a gradient is zero, and ∇ × A = B, so this reduces to

∇ × Λjs = −B . (3.11)

Using Ampère’s law, ∇ × B = µ0js, the previous equation then becomes

Λ ∇ × ∇ × B = −B , (3.12) µ0

µ ∇ × ∇ × B = − 0 B . (3.13) Λ Recalling that ∇ × ∇ × B = ∇ (∇ · B) − ∇2B and ∇ · B = 0, we then obtain

2 B ∇ B = 2 , (3.14) λL

1/2 2 1/2 where λL = (Λ/µ0) = (me/µ0nse ) is the London penetration depth. Solving this equation shows that the magnetic field decays exponentially with characteristic length λL. A schematic of the fields is shown in Figure 3.1. The London penetration depth is typically on the order of tens of nm in conventional superconductors, several hundred nm in high- temperature superconductors, and can be several micron for heavy-fermion superconductors. In all cases, as was observed by Meissner and Ochensfeld, the expulsion of magnetic flux from a bulk sample is almost complete, and thus a bulk superconductor is a good approximation of a perfect diamagnet.

3.2 Electrodynamics of superconductors

The response of a material to an applied electromagnetic ﬁeld is contained within the complex conductivity tensor, σij. For simplicity we will consider the material to be isotropic, meaning that we can use the scalar quantity, σ = σ1 − iσ2. Extending the equations to incorporate a tensor conductivity is straightforward. The simplest form of electrodynamics is local electrodynamics, where the response function directly relates the electric ﬁeld E at a point r to the current density j at the same point

j(r) = σE(r) . (3.15)

41 Figure 3.1: Electromagnetic fields in free space (z > 0) impinging on the surface of a superconductor. The Poynting vector (green) shows the direction of power flow. The magnetic field inside the superconductor (red) decays exponentially with characteristic length λ. The superfluid screening currents (green), which run perpendicular to the magnetic field, also decay exponentially with the same characteristic length.

Non-local effects arise when E(r) varies on length scales comparable to or less than the electron mean-free path or the superconducting coherence length. This occurs in some cases such as very clean superconductors, but they will not be treated here. To understand the electrodynamics of a superconductor we will proceed via a series of approximations. Consider first a superconductor at zero temperature, in the absence of disorder and neglecting any pair-breaking effect of the photons. The superfluid is the only contribution to the conductivity, as there are no quasiparticle excitations. Start with the London acceleration equation dj n e2 s = s E , (3.16) dt m where js is the superconducting current density, and ns is the number density of superconducting carriers. Taking the Fourier transform, the above equation becomes

n e2 iωj (ω) = s E(ω) . (3.17) s m

The imaginary part of the superﬂuid conductivity is then

2 nse 1 σs2 = = 2 , (3.18) mω ωµ0λL

q m where λL = 2 is the London penetration depth. µ0nse

42 To obtain the corresponding real part of the superﬂuid contribution we use the fact that the response function for the conductivity must be causal. There is therefore a relation between the real and imaginary parts of the response function in the form of the Kramers- Kronig relations:

Z ∞ 0 0 2 ω σ2(ω ) 0 σ1(ω) = P 02 2 dω (3.19) π 0 ω − ω Z ∞ 0 2ω σ1(ω ) 0 σ2(ω) = − P 02 2 dω . (3.20) π 0 ω − ω

It is easy to verify that a conductivity of the form σ1(ω) = Cδ(ω) satisﬁes the Kramers- Kronig relation for the superﬂuid contribution to the conductivity:

Z ∞ 0 2ω Cδ(ω ) 0 σ2(ω) = − P 02 2 dω (3.21) π 0 ω − ω C = . (3.22) πω

2 Comparing with Equation 3.18 we obtain C = π/µ0λL. The superﬂuid density, which indicates how much of the conductivity spectral weight has been shifted into the zero frequency 2 peak, is ρs = 1/λL, and so π π σ1(ω) = 2 δ(ω) = ρsδ(ω) . (3.23) µ0λL µ0

The next approximation to the conductivity of the material is to consider adding the effects of a small density of quasiparticles, either via thermal excitation or pair breaking. A simple way to incorporate this is to imagine that the quasiparticles act like electrons in a normal metal, with finite scattering rate Γ, but with an effective mass m and charge q that need not be that of an electron. In the absence of an external field we expect that scattering would return the quasiparticles to equilibrium within some characteristic time τ = 1/Γ. The probability of a scattering event occurring during a given time interval is dt/τ, so the average quasiparticle velocity will decay exponentially as

−t/τ v¯n(t) = v¯n(0)e . (3.24)

Taking the time derivative on both sides gives the equation of motion for quasiparticles that have been disturbed from equilibrium:

dv¯ (t) v¯ (0) v¯ (t) n = − n e−t/τ = − n . (3.25) dt τ τ

43 Finally, we add the response of the quasiparticles, with charge q, to an applied electric ﬁeld, and recover the full equation of motion for the quasiparticles

dv¯ (t) qE v¯ (t) n = − n . (3.26) dt m τ

Taking the Fourier transform allows us to write

1 qE iω + v¯ = . (3.27) τ n m

Recalling that j = nnqv¯n, we get

n q2τ 1 j = n E , (3.28) m 1 + iωτ from which we obtain the Drude conductivity

n q2τ 1 σ(ω) = n (3.29) m 1 + iωτ σ = 0 . (3.30) 1 + iωτ

The full complex conductivity, including contributions from the superﬂuid and quasiparticles is thus π 1 σ0 σ(ω) = 2 δ(ω) + 2 + . (3.31) µ0λ iωµ0λL 1 + iωτ In practice the Drude form of the conductivity is sometimes used to model the quasiparticle excitations, but wherever possible it should be replaced with a more detailed model of the quasiparticle conductivity.

3.3 Surface impedance

While conductivity can be accessed directly at low frequencies, by attaching leads to a sample and measuring E and j, this approach breaks down at high frequencies, requir- ing contactless methods to be used. In that limit, the accessible quantity is the surface impedance, Zs, which is defined as the ratio of the tangential components of the electric and magnetic fields at the surface of some material. For concreteness, consider a semi-infinite slab with the electric and magnetic fields applied in the xy plane. The surface impedance is then defined as Ex(z = 0) Ex(z = 0) Zs = R ∞ = = Rs + iXs , (3.32) 0 Jx(z)dz Hy(z = 0) where Ex(z = 0) and Hy(z = 0) are the electromagnetic fields at the surface of the slab, and Rs and Xs are the surface resistance and surface reactance respectively. To relate the surface impedance to the conductivity we start by considering a normal metal in the local

44 electrodynamics limit which follows Maxwell’s equations

∇ · E = 0 , (3.33) ∇ · B = 0 , (3.34) ∂B ∇ × E = − , (3.35) ∂t ∂E ∂E ∇ × B = µ j + = µσE + µ . (3.36) ∂t ∂t

Taking the curl of the third equation

∂B ∇ × ∇ × E = ∇ (∇ · E) − ∇2E = −∇ × , (3.37) ∂t and substituting Maxwell’s ﬁrst and fourth equations gives

∂ ∇2E = ∇ × B (3.38) ∂t ∂E ∂2E = µσ + µ . (3.39) ∂t ∂t2

Now assume that the solutions will be plane waves of the form

˜ E (z, t) = E (0) ei(kz−ωt) , (3.40) where k˜ = α + iβ is the complex wave number. Substituting this into Equation 3.39 and eliminating common factors gives an equation for the wave number

k˜2 = ω2µ + iωµσ . (3.41)

Solving for the real and imaginary parts then gives

s 1/2 rµ σ 2 α = ω  1 + + 1 , (3.42) 2 ω s 1/2 rµ σ 2 β = ω  1 + − 1 . (3.43) 2 ω

For a good conductor, σ ω, which means that the penetration depth can be written as

s 1 2 δ = = . (3.44) β ωµσ

45 The electric and magnetic ﬁelds can then be written as

E (z, t) = E (0) e−z(1+i)/δe−iωt , (3.45) B (z, t) = B (0) e−z(1+i)/δe−iωt . (3.46)

Applying Faraday’s law to the geometry described at the start of this section, with Ex and

By the only non-zero components, gives

∂B ∇ × E = − , (3.47) ∂t (1 + i) −E = −iωµH . (3.48) x δ y

Rearranging the equations ﬁnally gives a relation between the electric and magnetic ﬁelds

(1 − i) H (0) = E (0) . (3.49) y ωµδ x

Substituting this into Equation 3.32 then gives

1 + i Z = ωµδ . (3.50) s 2

Finally we can take the square of this to obtain an expression that relates the conductivity to the surface impedance.

1 + i2 iωµ Z2 = ω2µ2δ2 = . (3.51) s 2 σ

Now consider the case of a superconducting slab with its surface in the xy plane. The surface is exposed to a time-varying magnetic ﬁeld applied along the y direction,

iωt Hy(t) = Hy0e . (3.52)

From the Meissner eﬀect and the second London equation we know that the magnetic ﬁeld inside the sample is then −z/λ iωt By(z) = µ0Hy0e e , (3.53)

Applying Faraday’s law to this gives

∂B ∇ × E = − , (3.54) ∂t −z/λ iωt = −iωµ0Hy0e e , (3.55) which implies that

Ex = iωµ0λHy , (3.56)

46 The surface reactance is therefore proportional to the eﬀective penetration depth: Xs =

ωµ0λ.

To obtain Rs, start with the instantaneous local power absorption per unit volume, 2 dP/dV = σ1E . The time averaged power absorption per unit area is then given by

P Z = 1 σ E2dz . (3.57) A 2 1

Using the ﬁeld orientation described above and Equation 3.56 gives

P Z = 1 σ µ2ω2λ2H2dz (3.58) A 2 1 0 y Z 1 2 2 2 2 −2z/λ = 2 σ1µ0ω λ Hy0e dz (3.59) 1 2 2 3 2 = 4 σ1µ0ω λ Hy0 . (3.60)

The power ﬂowing through a surface can also be obtained by integrating the time average of the Poynting vector hSi,

Z Z n 1 ∗o P = − hSi · dA = − Re 2 E × H dA . (3.61) S S Substituting the phasor fields defined above, and using Equation 3.56 gives Z n 1 o P = Re 2 ExHy dA (3.62) S Z 1 2 = 2 RsHy0dA (3.63) S If the fields are uniform in the xy direction then the power per unit area is simply

P = 1 R H2 . (3.64) A 2 s y0

Comparing Equation 3.60 and Equation 3.64 shows that

1 2 2 3 Rs = 2 σ1µ0ω λ . (3.65)

From this we make the somewhat unintuitive observation that the surface resistance is directly proportional to the real part of the conductivity. The surface resistance has three powers of λ, two from the constraints imposed on E and H by Faraday’s law, and one from the exponential decay of the ﬁelds inside the material. Another consequence is that the 2 surface resistance is proportional to ω , making the determination of Rs at low frequencies very diﬃcult.

47 Chapter 4

Experimental methods

In this chapter we will build on the previous one by showing how we can measure Rs and Xs via cavity perturbation. We will then explore a novel helical resonator setup that I designed and built, which expands the range of frequencies over which we can perform experiments.

4.1 Cavity perturbation

Measurements of surface impedance provide direct access to the complex conductivity,

σ = σ1 − iσ2, which encodes the effects of thermally excited quasiparticles, the response of the quasiparticles to an applied electromagnetic field, and the response of the superfluid in a superconductor. Cavity perturbation techniques provide particularly accurate determi- nations of the surface impedance of a material, and overcome some of the difficulties that other techniques face. The contactless nature of cavity perturbation avoids the technical challenge of bonding wires to a small sample, and the issue of contact geometry. The cavity also acts as a natural impedance matching device, bypassing the problem that arises when the impedance of a sample is small compared to the impedance of the probe lines. Optical reflectivity measurements could in principle be used to measure surface impedance, but the sensitivity would be limited due to the reflectivity of metallic and superconducting samples being nearly indistinguishable from unity. Finally, the cavity perturbation technique works well in the low- to mid-GHz range, where many low-Tc, heavy fermion, or low scattering rate materials have interesting properties. For an intuitive picture of the technique, consider a sample placed within a resonant cavity. The sample acts as part of the inner surface of the cavity as shown in Figure 4.1(a).

Changes to the dissipation of the sample cause the resonant bandwidth, fB, to change, and changes to the skin depth — or penetration depth when in the superconducting state — modify the effective size of the cavity by allowing the fields to penetrate into the sample, leading to a shift in the resonant frequency, f0. Confinement of the electromagnetic fields within the cavity creates standing waves, enhancing the interaction with the sample by a factor Q, the quality factor of the resonator. Samples with very small surface resistance can

48 a) b)

Figure 4.1: (a) Cross-section of a cavity perturbation apparatus showing the sample per- turbing the H fields inside a cylindrical cavity. The sample is held at the end of a dielectric sample rod. (b) Schematic of the S21 parameter measured by a vector network analyzer. The blue and orange traces represent the sample at low and high temperature respectively. Typically as the sample temperature increases the dissipation increases and the fields penetrate deeper into the sample, effectively increasing the size of the resonator. The resonance then shifts to lower frequencies and broadens. thus be measured as the resonator provides efficient coupling between the sample impedance, which can be in the µΩ range, and the typical 50 Ω impedance of the coaxial cable that carries the microwave signal. Without this, the power transferred to the sample would be negligible, and the precision of the technique would suffer greatly. The signal quality from a microwave spectroscopy measurement is proportional to the filling factor, which measures the ratio of magnetic field intensity contained within the sample to the total magnetic field intensity within the resonator volume. This has practical consequences for performing measurements at low frequency as the cavity size typically grows as 1/ω, which is made worse by the surface resistance of a sample decreasing at low 2 frequency as Rs ∼ ω . Another important consideration is the distortion of the fields caused by sample geometry. A large sample placed perpendicular to the magnetic field will distort the fields, with increased field intensities near the edges of the sample. In this high demagnetization factor geometry, the current density can become nearly singular near the edges of the sample. It is therefore beneficial to work in the low demagnetization factor geometry where the sample provides negligible perturbations to the field, and the induced screening current density is uniform.

In our cavity perturbation experiments the measured quantity is the S21 (transmission) element of the S-matrix (or scattering matrix). A vector network analyzer is used to scan over frequency, ﬁnd the cavity resonances, and a Lorentzian lineshape is ﬁt to the peak

49 to extract the resonant frequency and resonant bandwidth. A reference temperature is then chosen, and the frequency and bandwidth at this temperature are subtracted from the values measured at other temperatures to give ∆fB(T ) and ∆f0(T ), which are the temperature-dependent changes associated with the electrodynamics of the sample. This is shown schematically in Figure 4.1 (b). From there the bandwidth shift and frequency shift are converted into surface impedance using the cavity perturbation formula. It has been derived elsewhere [77, 78] so the result is simply quoted here:

∆f ∆Z = ∆R + i∆X ≈ Γ B − i∆f . (4.1) s s s 2 0

4.2 Obtaining calibrated surface impedance from cavity perturbation

As shown in the previous chapter, in the local electrodynamic limit the complex conductivity is directly related to the surface impedance via

iωµ Z2 = 0 . (4.2) s σ

In order to obtain the conductivity we need a method to relate the cavity perturbation result in Equation 4.1 to the surface impedance in Equation 4.2. The ﬁrst step is usually to obtain the contribution to the resonant bandwidth that is strictly proportional to Rs (which we often refer to as ‘absolute’ fB, or simply fB). By measuring the resonant bandwidth of a resonator with and without the sample, while keeping everything else including the sample holder identical, we can, in principle, obtain the contribution of the sample. This works quite well at low frequencies where the resonant modes are well separated, but fails at higher frequencies where the resonant modes are more closely packed together. Small geometric perturbations can cause modes in close proximity to couple diﬀerently, which leads to geometry-dependent decay rates, and makes the subtraction procedure unreliable[79]. When this occurs, we use a bolometric technique, where the change in temperature of a sample is used in conjunction with a thermal model of the measurement apparatus to obtain the power absorbed by the sample, which can then be directly related to Rs [80, 79].

With absolute fB determined, the scale factor Γ in Equation 4.1 can be obtained by

first observing that in the normal state, where σ2 is small and Rs ≈ Xs the DC resistivity is given by 2 2Rs ρDC = . (4.3) ωµ0 Barring any finite-size effects, the scale factor is then

1 rωµ ρ Γ = 0 DC . (4.4) fB 2

50 Absolute f0 values are much more difficult to obtain as they are related to geometric effects. The presence of the sample in the cavity, when compared to the absence of the sample, is a significant perturbation to the field profile as the effective size of the cavity changes. A common method to overcome this difficulty is to use a known value of the absolute penetration depth — measured using a different technique such as µSR — to set

Xs(T → 0) ≈ ωµ0λL. This uniquely speciﬁes the oﬀset needed to convert ∆f0 into absolute f0, and then into Xs using the scale factor Γ.

An alternative method to determine absolute f0, useful when λL is not available, is to recognize that when the sample is in the normal state the conductivity is dominated by dissipative processes and is therefore predominantly real, σ1 σ2. The real and imaginary parts of the surface impedance are then approximately equal because

2 2 2 2 iωµ0 Zs = (Rs + iXs) = (Rs − Xs ) + 2iRsXs ≈ . (4.5) σ1

2 The real part of Zs must be small, so matching Rs and Xs over some range of temperatures in the normal state can be used to set the oﬀset required to obtain absolute f0.

4.3 Helical resonator

In order to obtain high-quality conductivity spectra of various materials we want to probe samples over a wide frequency range, typically from 2 GHz to 20 GHz. Our tool of choice is the cylindrical cavity resonator, which provides many useful resonant modes with high quality factor. Free-space resonators operating at low-GHz frequencies would be quite large, given that a 2 GHz wave has a wavelength of 0.15 m. The radius and height of a free-space cylindrical cavity resonator operating at this frequency are both approximately the same as the wavelength. Although it is possible to make such a large resonator, it would be of little practical use; the filling factor would be negligible and hence the resonator would have poor sensitivity, and the geometric constraints of our cryostat prohibit such a large apparatus. We instead employ dielectric resonators made of rutile which has a very large dielectric constant, confining the fields into a much smaller volume by reducing the group velocity. The small loss tangent of rutile allows the resonators to have very high quality factors, typically on the order of 106. Below 2 GHz, even dielectric resonators become too large, so another method is needed to create a small-volume and slow-wave resonator, while maintaining high Q. To this end, I built a novel self-resonant helical coil resonator, shown schematically in Figure 4.2. Superconducting Nb wire that has been annealed under vacuum to make it more malleable is wound on a cylindrical dielectric support and mechanically stabilized with either an adhesive, or an outer sheath wound around the coils. The helical coil acts as a slow-wave structure with charge running back and forth between the free-standing ends

51 Figure 4.2: (a) Schematic of the helical coil inside a Cu enclosure. Sapphire end-plates are used to hold the cylindrical former. The sample is inserted from above through a hole in the Cu enclosure and in the sapphire. (b) Cross-section of the inside of the enclosure showing the geometry used in the COMSOL simulation.

52 ID f1(MHz) : Q f2(MHz) : Q f3(MHz) : Q Description 1 153.5 : 2300 287.5 : 2620 409 : 2310 Glass former - 60 turns - GE outer 2 291 : 6500 511 : NA 893 : 4000 Glass former - 30 turns - Teflon outer 3 13.6 : 6600 55.85 : 8500 116.26 : 7600 Glass former - 60 turns in 2 layers - Epoxy outer 4 1177 : 87000 1649 : 87000 2111 : 168000 Teflon rod - 30 turns - Teflon outer 5 147.07 : 18500 285.9 : 41900 419.95 : 36000 Quartz tube - 30 turns - Teflon outer 6 238.4 : 8750 459.0 : 6400 656.9 : 5400 1266 Epoxy rod - 30 turns - Embedded 7 428.4 : 12500 1430.0 : 26200 2581.3 : 18300 Lead foil on quartz tube - Teflon outer 8 212.92 : 32500 435.06 : 56500 635.14 : 44500 Quartz tube - 30 turns - Dental floss outer 9 233.42 : 25500 448.00 : 41500 641.73 : 41000 High purity quartz tube - 40 turns - Teflon outer

Table 4.1: Several inner former materials, turn numbers, and outer cover materials were tested. Helix 9 was the ﬁnal successful version that balanced mechanical stability with the desired frequencies and high quality factor. Only the ﬁrst three resonant frequencies for each resonator are listed at 4.2 K. Some resonances were too weakly coupled to measure the quality factor and are listed as NA.

Figure 4.3: Picture of the first 8 helical resonators showing the different construction materials. Helix 5 is approximately 10 mm long and is similar to the final design. of the coil, allowing low-frequency operation within a small volume. If this were a coil in free-space the lowest frequency mode would have λ/2 = L, where L is the length of the wire. From this we can estimate that a 40 turn coil wound on a 4 mm diameter former would have a wavelength of ∼ 1 m, which is 300 MHz. Despite the low frequency, the helical resonator is quite compact, making the sample a significant portion of the resonator volume, and thereby achieving a reasonable filling factor. Nine coils were built and tested by measuring their resonant frequencies at room temperature and at 4.2 K in a Helium dip dewar. Several configurations were tested to find the right combination of material for the coil support, number of turns, and outer material. Ta- ble 4.1 provides an overview of the results, and Figure 4.3 shows the first 8 helical resonator attempts. The final design, Helix 9, was wound on a 4 mm diameter high-purity quartz tube, with length 10 mm. Teflon tape is wound around the coil to provide mechanical stability and prevent the bare Nb wires from touching. In the final design the ends of the coil are held close to the end walls of a cylindrical copper cavity coated in Pb:Sn. Capacitive coupling to the end walls shifts the resonant modes of resonator 9 down slightly from those shown in Table 4.1 to 202 MHz, 465 MHz, and 653 MHz. Although the resonator was originally designed solely to measure penetration depth at low frequency, the final coil has sufficient sensitivity to resolve both the superfluid and quasiparticle components of superconducting samples.

53 The magnetic fields in a helical resonator are expected to be similar to the TE01p modes in a cylindrical cavity, but the precise field profiles were not simulated at the time the helical resonators were built. Several years later we performed COMSOL simulations of helical resonator 9. All geometry and dielectric constants (including the anisotropic dielectric constant of sapphire) were faithfully reproduced in the simulations. The simulations were run in the full 3D environment with no axial symmetry applied in order to properly represent the helical structure. Figure 4.4 reveals in detail the field profiles for the first three resonant modes. Somewhat surprisingly, the frequencies from the simulation are all within 5 MHz of the physical resonator: 202 MHz, 462 MHz, and 658 MHz. The accuracy of the simulations shows that COMSOL can be a valuable tool to rapidly prototype complex resonant structures.

54 Figure 4.4: Results of COMSOL simulations of the helical resonator for the first three resonant modes. Black lines with arrows show the magnetic field lines. The colour profile represents the log of the magnetic field intensity.

55 Chapter 5

Multiband superconductivity

Multiband superconductivity is common, appearing in many materials such as MgB2;

SrTiO3; conventional elemental superconductors, in particular transition metals; many heavy fermion materials; and the pnictides. While most low energy properties of metals and superconductors can be expressed in terms of averages over the Fermi surface, it makes particular sense to treat the Fermi surface integral as a sum over distinct bands when the Fermi surface consists of several distinct pieces, in particular when these have different orbital character, such as the σ and π bands in MgB2. In many cases — distinct orbital character chief among them — it is natural for the strength of the pairing to be quite different in the different bands, with pairing in the subdominant bands induced mainly because of the proximity to the dominant seat of superconductivity. The pairing scale in each band is then expected to be associated with a distinct spectroscopic gap, and therefore identifiable using single particle spectroscopies (e.g., ARPES and STM), and two-particle spectroscopies (e.g., optical conductivity). In this chapter we will explore two examples of unconventional superconductivity. First in a very clean material, FeSe, where the multiband nature is readily apparent; and second in a dirty superconductor, Nb-doped SrTiO3, where the effect of disorder masks the multiband nature of the material.

5.1 FeSe

Due to the antagonistic relationship between magnetism and superconductivity, researchers had long avoided searching for superconductors containing magnetic elements, especially ferromagnetic ones such as iron. Although some rare cases of Fe-containing superconducting compounds had been reported (e.g., the rare-earth (RE) iron silicides, RE2Fe3Si5, with

Tc ∼ 2–6 K [82]; Th7Fe3 with Tc = 1.86 K [83]; and LaFe4P12 with Tc ∼ 4 K [84]), few would have anticipated the rapid evolution in our understanding of superconductivity that was to begin in 2006.

56 Figure 5.1: Adapted from [81]. The crystal structure of ﬁve common types of iron-based superconductors. The common iron-pnictogen/chalcogen layer, which is responsible for the superconductivity, is highlighted.

57 The ﬁrst iron-based superconductor to gain prominent attention, LaOFeP, was discovered in July 2006 by Hideo Hosono and co-workers [45]. LaOFeP was not a new material, having been synthesized and its crystal structure analyzed in 1995 by B. I. Zimmer et al. [85], but no electronic properties had been investigated at the time. While the transition temperature in LaOFeP is an unremarkable 4 K, increasing slightly with F-doping [45], its crystal structure and composition lend themselves favourably to systematic studies of superconductivity; substitution of the rare-earth, the transition metal, the pnictogen, or doping at the oxygen site, are all possible ways to tune the material. Shortly thereafter,

La[O1−xFx]FeAs was synthesized by Hideo Hosono and collaborators with a transition temperature of 26 K when x = 0.11 [46]. The transition temperature was then raised again in the same year to 43 K in La[O1−xFex]FeAs under hydrostatic pressure [47], ﬁrmly establishing the iron-based superconductors as high-Tc materials, and launching an intensive research eﬀort to understand this new class of superconductor. Many other iron-based superconductors (FeSCs) have been discovered since then, crys- tallizing in several distinct crystal structures such as the 11-type (e.g., FeAs), the 111-type

(e.g., LiFeAs), the 1111-type (e.g., LaOFeP), and the 122-type (e.g., SrFe2As2). Some of these crystal structures are shown in Figure 5.1, with the highlighted region drawing attention to the shared structural element, consisting of an Fe planar sheet tetrahedrally coordi- nated with pnictogen/chalcogen atoms above and below. The structure is highly reminiscent of the cuprate superconductors, with the superconductivity arising in the quasi-2D layers that form the common backbone of these materials. The 11-type materials can be viewed as a prototype for the other FeSCs. The compar- ative chemical simplicity of FeSe, and the lack of magnetic order, should facilitate the task of understanding how superconductivity arises in this class of materials, hopefully yielding clues about the general mechanism of high-Tc superconductivity. However, despite the apparent structural simplicity of FeSe, several eﬀects must be considered. At 90 K the crystal structure undergoes a structural transition from tetragonal to orthorhombic as it enters a nematic phase, lowering rotational symmetry without breaking translational symmetry

[86, 87]. Upon further cooling, FeSe becomes superconducting at Tc ≈ 9 K under ambient pressure [48], with Tc rising to 36.7 K under 8.9 GPa of quasihydrostatic pressure [49].

Single-layer FeSe grown on semi-insulating substrates such as SrTiO3 holds yet more sur- prises; the transition temperature increases further, with reported values of Tc up to 100 K [88, 89, 90, 91]. Angle-resolved photoemission spectroscopy (ARPES) [92, 93, 94, 95] and theoretical band-structure calculations such as those in Refs. [96, 97] have established the low-energy band structure of the FeSCs. The six valence electrons of Fe occupy 3d orbitals, with the states near the Fermi level having mainly dxy, dyz, and dxz character [96]. One or possibly two hole-like pockets are found at the Γ point, and at least one electron-like pocket crosses the Brillouin zone (BZ) boundary. As there are two inequivalent Fe positions in the lattice

58 Y

Figure 5.2: Adapted from Refs. [98, 99]. (a) Top-down view of the FeAs lattice. The green dashed line and solid blue line correspond to the 1-Fe and 2-Fe unit cells. (b) 1-Fe BZ representation of the 2D Fermi surfaces, with the arrow indicating the folding wave vector. (c) 2-Fe BZ representation of the Fermi surfaces. (d) Simpliﬁed picture of the band structure of FeSCs. The main characteristics are at least one hole pocket at the centre of the Brillouin zone and one or more electron pockets at the X and Y points of the 1-Fe BZ picture. the band structure should be viewed in the 2-Fe BZ picture. This however is somewhat harder to interpret, and some authors choose to work in the 1-Fe BZ picture [98], which allows for a more straightforward discussion of the physics; this convention will be adopted for the remainder of the discussion of FeSe. The 1-Fe and 2-Fe BZ pictures are shown schematically in Figure 5.2. Converting between the two representations involves folding the 1-Fe BZ along the line connecting the X and Y points.

5.1.1 Nodes or no nodes, that is the question

A key step in understanding how superconductivity arises in the Fe-based superconductors is the identification of the gap structure, which is intimately connected to the pairing glue. So far, most measurements agree on the multiband character of the superconductivity, but differ on the precise form of the gap, whether it has nodes, accidental or not, and whether it changes sign or not. A study of a high-quality FeSe sample with large residual resistivity ratio revealed substantial residual thermal conductivity, power-law penetration depth, and V-shaped tunneling spectra [100], all indications of the presence of nodes. Another scanning tunneling microscopy (STM) study again showed V-shaped tunneling spectra [101]. Although the shapes of the tunneling spectra seem to indicate a nodal gap it is puzzling that they show zero conductance at zero bias; the energy required to thermally excite a quasiparticle should be zero for nodal gaps, which would immediately lead to non-zero conductance at finite temperature. Other STM studies report features that resemble finite gaps [103, 102] with no presence of nodes. As shown in Figure 5.3, within experimental uncertainty, all these STM measurements show zero conductance at zero bias. This can only occur if there is a single-particle gap at least three times the measurement temperature, which on its own implies a non-zero gap minimum in the range ∆min/kB ≈ 1.2 − 4.5 K.

59 d

Figure 5.3: Tunneling conductance data from (a) Ref. [100], (b) Ref. [101], (c) Ref. [102], and (d) Ref. [103]. In all cases the conductance is zero within uncertainty at zero bias, indicating nodeless behaviour, despite the V-shaped spectra observed in several cases.

60 Many measurements on FeSe weigh in favour of non-zero gap minima, including heat capacity [104, 103], thermal conductivity [105], µSR [106], lower critical field [107], and penetration depth [108]. The heat capacity measurements of Bourgeois-Hope et al. [105] show almost no residual thermal conductivity, with κ/T = 6 ± 2 µW/K2cm, indicative of a non-zero gap, but are in direct contention with the measurements of Kasahara et al. [100] which showed a residual thermal conductivity close to 3 orders of magnitude larger: κ/T ≈ 4 mW/K2cm. It is unclear why the residual thermal conductivity would be so different between two high-quality samples. An explanation for the seemingly contradictory measurements, with some showing nodes, others not, may be that FeSe has accidental nodes that are lifted when the sample has slightly more impurities or when twin boundaries, which are known to exist below 90 K [101, 102], cause enough scattering to homogenize the gap. The differences would then simply be attributed to a particularly clean sample or one where the twin boundary density is not as high. Several pairing scenarios, shown schematically in Figure 5.4, have been proposed to account for the unusual properties of the FeSCs: simple s-wave (referred to as s++), non- nodal s+−, nodal s+−, d-wave both with and without nodes, and even more exotic pairing symmetries such as s + is, which would involve time reversal symmetry breaking. It is clear that determining whether or not the gap has nodes is an important step in identifying the correct symmetry, as this would heavily constrain which models should be considered. The rest of this section will present microwave measurements on high-quality FeSe crystals that help answer this question.

5.1.2 Experiment details

FeSe superconducts at the stoichiometric composition and can be grown as high quality single crystals using vapour transport methods [109]. The FeSe crystals used in this work were prepared at UBC by Ruixing Liang and Shun Chi. Meng Li cleaved a mm-sized platelet out of a thicker crystal, with the final sample having a thickness along the c-direction of 15 µm. The sample is dark and reflective, indicating a metallic character, and is malleable so great care had to be taken when mounting the sample. It is also air-sensitive, so it was mounted in a glove-box filled with argon, and left in argon atmosphere during transportation from UBC to SFU. I performed cavity perturbation measurements of surface impedance at two frequencies, 202 MHz and 658 MHz, using the self-resonant helical coil resonator described in Section 4.3 placed inside a Pb:Sn-coated enclosure that is mounted below the mixing chamber of an MX40 3He–4He dilution refrigerator [110]. The sample was positioned at a local maximum of the microwave magnetic field, Hrf , which was applied parallel to the FeSe layers. This geometry was specifically chosen so that the microwave response would be predominantly from in-plane screening currents, while minimizing geometric demagnetization factors and thus

61 Figure 5.4: From [98]. Schematic representation of possible gap symmetry scenarios, with colours representing the phase of the gap. The simplest scenario is the s++, which is just the conventional s-wave state in a multiband system. The s+− state has no nodes but the gaps have opposite signs. The nodal s+− state respects the full set of symmetries of the lattice, but has nodes on one or more of the bands — the nodes would be considered accidental in this case as they are not symmetry protected. The d-wave and nodeless d-wave states break C4 symmetry, but only the ﬁrst would show the signatures commonly associated with d-wave superconductivity such as linear penetration depth at low temperature. Finally the most exotic is the s + is state, which breaks time reversal symmetry.

62 ensuring the surface currents are fairly uniform. As a result, all electrodynamic quantities subsequently mentioned in this chapter are for in-plane currents. During the experiments the resonator temperature was maintained at 1.5 K, while the sample temperature was scanned between T = 0.1 and 20 K using a silicon hot-finger [111] thermally linked to the mixing chamber. In order to avoid nonlinearities the measurements were carried out under constant Hrf , and the microwave power was kept low enough to prevent self-heating of the sample. As explained in Section 4.1, measurements of absolute penetration depth using cavity perturbation are complicated due to frequency-shift systematic errors that are extremely difficult to properly account for. We therefore used an independently reported value for the zero-temperature penetration depth, in this case λ0 = 400 nm [107, 112, 100], to set the absolute zero-temperature surface reactance, Xs(0) = ωµ0λ0. At the low frequencies considered here background bandwidth subtraction is known to work well, so absolute bandwidth was obtained by subtracting the bandwidth of the empty eff resonator. The effective surface impedance is then given by ∆Zs (T ) = Γ(∆fB(T )/2 − i∆f0(T )), where ∆fB is the change in the resonator bandwidth with and without the sample. The resonator constant, Γ, is obtained by matching the DC resistivity, ρDC, of FeSe samples from the same batch [105]. At the low frequencies used in this experiment, finite size effects become important going into the normal state. The geometry chosen for the experiment allows us to use the 1D finite-size formula

eﬀ Zs = Γ(∆fB/2 − i∆f0) = Zs tanh(iωµ0t/2Zs) . (5.1)

eﬀ This formula can be numerically inverted given Zs to obtain the real surface impedance for a semi-inﬁnite sample Zs, given the constraints set by Xs(0), and ρDC.

5.1.3 Results

Surface resistance data for FeSe is shown in Figure 5.5. Below Tc = 9.1 K the surface resistance drops rapidly as Meissner screening currents take over from the normal-state skin eﬀect. The transition is sharp, indicative of a homogeneous sample. Rs initially drops sharply then forms a peak at ∼ Tc/2, with the peak moving to lower temperature with decreasing frequency. This is quite unusual; in most materials Rs would keep decreasing monotonically as the quasiparticles condense into the superﬂuid. In fact, at the time of publication of our results in Ref. [113], only one other material system — ultrahigh purity

YBa2Cu3O7−δ — has shown nonmonotonic behaviour. We have since found that very pure

CeCoIn5 also exhibits similar behaviour at frequencies below 1 GHz (unpublished results).

The nonmonotonic temperature dependence of Rs is immediately indicative of a system with rapidly collapsing scattering rate and extremely long-lived quasiparticles [114, 115, 116, 117].

63 Figure 5.5: From Ref. [113]. Surface resistance of FeSe with Rs plotted on a logarithmic scale. Inset shows a zoomed-in view of Rs in the superconducting state plotted on a linear scale with the surface resistance at the higher frequency scaled to remove the expected ω2 frequency dependence.

Further evidence for the very low scattering rate comes from the real part of the quasiparticle conductivity, shown in Figure 5.6. Passing through Tc, σ1(T ) starts to rise, but shows no frequency dependence, indicating that the quasiparticle relaxation rate is much larger than the measurement frequencies. σ1(T ) then rises rapidly as temperature decreases and develops substantial frequency dependence, which shows that the quasiparticle relaxation rate is becoming comparable to the measurement frequency. Together this points to a rapid collapse of quasiparticle scattering below Tc that outpaces the condensation of the quasiparticles into the superfluid, and to relaxation rates that are in the sub-GHz range. To quantify the quasiparticle relaxation rate Γ we fit a generalized two-fluid model [118, 119] to the complex conductivity.

ne2 f f 1 σ = s + n − iKK(ω/Γ0, y) , (5.2) m∗ iω Γ 1 + (ω/Γ0)y

0 y π where Γ = Γ 2 sin y is a convenient scaling that makes the integrated quasiparticle spectral weight independent of y, KK(ω/Γ0, y) is the Kramers-Kronig transform of the quasiparticle contribution to the conductivity, and fs + fn = 1 in order to conserve spectral weight. The frequency exponent, y, is between 1 and 2, with 2 being a pure Drude-like conductivity, and lower values indicating an energy-dependent scattering which gives the conductivity a cusp-like shape at low frequencies and a long high frequency tail [110, 120]. The best ﬁt value for y is 1.5 at low temperature, where the frequency dependence is appreciable. At

64 Figure 5.6: From Ref. [113]. Real part of the quasiparticle conductivity, σ1(T ). Inset shows the freeze-out of conductivity at low temperatures. higher temperature the frequency dependence is much smaller making the fits insensitive to the value of y. The frequency exponent was therefore fixed to 1.5 at all temperatures. The quasiparticle relaxation rate extracted from the conductivity fits is plotted in Fig- ure 5.7. Below T = 0.5 K there is insufficient quasiparticle conductivity to accurately fit the two-fluid model, so Γ is not plotted in this range. As expected from the qualitative behaviour of Rs(ω, T ) and σ1(ω, T ) there is a rapid collapse in Γ(T ) on cooling below Tc, which in the context of the cuprates, organics and CeCoIn5 is interpreted as a strong indication that the fluctuations responsible for inelastic scattering are electronic in origin and are gapped by the onset of superconductivity. At the lowest temperature the scattering rate is Γmin/2π ≈ 200 MHz, corresponding to ~Γmin/kB ≈ 10 mK. Assuming a Fermi ve- 4 locity ~vF = 440 meV Å (vF ≈ 7 × 10 m/s) [87] gives a quasiparticle mean free path of `0 = vF /Γmin ≈ 55 µm. This is the largest value we are aware of for a compound superconductor and is five times larger than that of the best YBa2Cu3O6.52 crystals, widely regarded as one of the cleanest superconducting materials, where the low temperature scattering rate reaches Γ/2π ≈ 3.3 GHz [120]. The surprisingly long quasiparticle mean free path is larger than almost any length scale in the sample, so nonlocal effects would normally be of concern. However, in this case, nonlocal effects are likely suppressed by four factors: the large in-plane penetration depth in FeSe (λ0 ≈ 400 − 500 nm) [100, 107, 112]; the quasi-2D electronic structure [87, 121, 122, 123], which, in the experimental geometry used here, forces quasiparticles to propagate at low angles relative to the sample surface; the vanishing quasiparticle group velocity near

65 Figure 5.7: From Ref. [113]. Quasiparticle relaxation rate, Γ(T ), obtained from the two- fluid model fits to the conductivity data. Inset shows the low temperature region, revealing a linear temperature dependence of Γ(T ) below 2.5 K. the gap minima; and the diffusive nature of small-angle scattering in a superconductor with anisotropic gap [124, 125]. The last two points work together to make the distance travelled by the quasiparticle wavepacket between large momentum-relaxing scattering events much smaller than the apparent mean free path. The main features of the superfluid density, shown in Figure 5.8 are a strong temperature dependence across most of the temperature range, indicating a broad distribution of energy scales in the gap; upwards curvature around Tc/3, hinting at the importance of multiband physics; and a plateau at low temperature, which is evidence of a non-zero gap minimum. In order to better understand these features we fit the normalized superfluid 2 2 density, ρs = λL(0)/λL(T ), to a two-band model based on the work of Kogan, Martin and Prozorov in Ref. [126], suitably modified to account for gap anisotropy.

5.1.4 Anisotropic two-band model

We start by allowing the gap to have the angle dependent form ∆ν(T, φ) = ψν(T )Ων(φ), where ν represents the Fermi surface (FS) index. To separate the eﬀects of anisotropy and

66 2 Figure 5.8: From Ref. [113]. Frequency dependent superfluid density, 1/δ ≡ ωµ0σ2. The 2 2 dashed line shows the London superfluid density, 1/λL ≡ 1/δ (ω → 0) obtained from fits to the two-fluid model described in the text.

magnitude, the function describing the angle dependence of the gap, Ων(φ) is normalized according to 2 hΩν(φ)iφ = 1, (5.3) where h...iφ denotes an angle average over the appropriate Fermi surface sheet. In terms of lowest-order cylindrical harmonics, common forms for the normalized Ων(φ) include:

Ων(φ) = 1, isotropic s-wave (5.4) √ Ων(φ) = 2 cos(2φ), d-wave (5.5) 1 √ Ω (φ) = 1 + 2α cos(4φ) , extended s-wave (5.6) ν p 2 ν 1 + αν which are shown in Figure 5.9. In the extended s-wave case that is of interest here, the parameter α controls the degree of gap anisotropy on each FS sheet, and is nodeless for ν √ the range 0 ≤ αν ≤ 1/ 2. Note that the correct form of the gap would have two-fold symmetry for FeSe due to the orthorhombic distortion and the C2ν symmetry of the Fermi surfaces. Here we used a four- fold symmetric gap as we performed the experiments on twinned crystals, with the screening currents picking up equal contributions from both a and b orientations. In de-twinned samples the superfluid density would be modified because the quasiparticle velocity would be different in different directions, leaving visible differences between the superfluid densities

67 2 2 s-wave d-wave

1 1

0 0

1 1

2 2 2 1 0 1 2 2 1 0 1 2

2 2 Extended s-wave Extended s-wave = 0 = 1 = 0.1 = 3 = 0.2 = 1 1 = 0.5 = 1/ 2

0 0

1 1

2 2 2 1 0 1 2 2 1 0 1 2

Figure 5.9: Polar plots of the normalized anisotropic gap function shown for the s-wave, d-wave, and extended√ s-wave cases. Note the presence of nodes in the extended s-wave case when α ≥ 1/ 2. In the extended s-wave ﬁgure on the right, the gaps are sign-changing.

68 measured on samples with screening currents flowing along different crystallographic directions. For FeSe we consider a simplified model of a two-band superconductor, with the FS broken into two circular (isotropic) pieces. For mathematical convenience we have assumed a separable form for the angular part of the pairing interaction:

0 N0Vk,k0 → nµVνµΩν(φ)Ωµ(φ ) . (5.7)

With these simpliﬁcations the self-consistent gap equation Equation 2.91 then becomes

ωD * 0 + X X 0 ψµΩµ(φ ) ψνΩν(φ) = 2πT nµVνµΩν(φ)Ωµ(φ ) . (5.8) q 2 2 0 2 2 ωn>0 µ=1,2 ψµΩµ(φ ) + ωn ~ φ0

The dimensionless interaction parameters Vνµ = Vµν form a symmetric coupling matrix, with Vνν representing intraband pairing and Vνµ representing interband pairing. nµ ≡ Nµ/N0 is the fraction of the total density of states associated with band µ, with n1 + n2 = 1. Can- celling common factors in Equation 5.8, rearranging, and dropping the prime, we have

ωD * 2 + X X 2πT Ωµ(φ) ψν = 2πT nµVνµψµ . (5.9) q 2 2 2 2 µ=1,2 ωn>0 ψµΩµ(φ) + ωn ~ φ

2 The gap parameters, ψν, are negligible near Tc so using hΩν(φ)iφ = 1 the inner sum can be linearized:

ωD X 2πTc 2 S = hΩν(φ)iφ , (5.10) ωn ωn>0 ~ ωD 2πT = X c , (5.11) ωn ωn>0 ~ 2 ω = ln ~ D (5.12) 1.76Tc

At Tc, Equation 5.9 then becomes

X ψν = nµVνµψµS, (5.13) µ=1,2 which is then recognizable as the eigenvalue problem

! ! ! ! n V n V ψ 1 ψ ψ 1 11 2 12 1 = 1 ≡ V˜ 1 . (5.14) n1V21 n2V22 ψ2 S ψ2 ψ2

69 Following the same procedure outlined in Section 2.2.2 we eliminate the Debye energy ~ωD and replace it with the measured Tc:

ωD 2πT 2 ω T T X ≈ ln ~ D − ln = S − ln . (5.15) ωn 1.76Tc Tc Tc ωn>0 ~

Adding and subtracting this from Equation 5.9 gives a rapidly convergent expression allowing the upper frequency cutoﬀ to be taken to inﬁnity. For any given temperature we then have     * + ∞ 2πT Ω2 (φ) 2πT T X  X  µ   ψν = nµVνµψµ   q −  + S − ln  . (5.16) 2 2 2 2 ωn Tc µ=1,2 ωn>0 ψµΩµ(φ) + ωn ~ ~ φ

It is convenient for numerical work to rewrite this equation with dimensionless gaps

ψν ψν 1 δν = = , (5.17) 2πT Tc 2πt with t = T/Tc. Equation 5.16 then becomes

T δ = X n V δ S − ln − A , (5.18) ν µ νµ µ T µ µ=1,2 c where   ∞ * 2 + X  1 Ωµ(φ)  Aµ =  −  . (5.19)  1 r 2  n=0 n + 2 2 2 1  δµΩµ(φ) + n + 2 φ

Aµ, which is now only implicitly dependent on T , can be tabulated once as a function of δµ and then re-used to determine δν at any temperature with a root ﬁnding algorithm.

Once the dimensionless energy gaps δν have been determined for a particular set of fit parameters, the normalized superfluid density can then be calculated. Each band contributes in parallel, with the total normalized superfluid density being a band-weighted average

ρs = γρs1 + (1 − γ)ρs2 , (5.20) where 0 ≤ γ ≤ 1 is a parameter that encodes the relative contribution each band makes to the total superﬂuid weight, and

* + ∞ δ2Ω2(φ) ρ = X ν ν . (5.21) sν 3/2 2 2 1 2 n=0 δνΩν(φ) + (n + ) 2 φ

70 Both the Fermi velocity and the density of states contribute to γ. In the absence of a direct measurement of these parameters it can be considered as a free parameter in the fit. However, since the expression governing the superfluid density, Equation 5.20, is a linear combination, a closed-form expression exists for the optimal choice of γ, which allows us to eliminate one parameter from the multidimensional search used in the fitting procedure. To obtain γopt, we represent the experimental superfluid density as a vector of temperature-dependent data, #– ρ s,expt. The model superfluid density is then also a vector of theoretical points evaluated #– #– #– at the same temperatures as the experimental data. By substituting ∆ ρ s = ρ s1 − ρ s2 into Equation 5.20 we get #– #– #– ρ s,model = γ∆ ρ s + ρ s2 . (5.22)

Then, deﬁning the merit function to be

2 #– #– 2 χ = | ρ s,expt − ρ s,model| (5.23) #– #– #– 2 = | ρ s,expt − ρ s2 − γ∆ ρ | (5.24) 2 #– 2 #– #– #– #– #– 2 = γ |∆ ρ | − 2γ∆ ρ · ( ρ s,expt − ρ s2) + | ρ s,expt − ρ s2| , (5.25) and minimizing with respect to γ, we obtain #– #– #– ∆ ρ · ( ρ s,expt − ρ s2) γopt = #– . (5.26) |∆ ρ |2

2 This procedure is carried out for each {n1,Vνµ, αµ} point and the χ value tabulated and minimized to ﬁnd the optimal ﬁt parameters. It is important to note that this formulation of multiband superconductivity does not require that there be a one-to-one mapping between Fermi surface sheets and bands. As a case in point, MgB2, which is known to have four distinct Fermi surface sheets, is well described by a two-band model [126, 127, 128, 129, 130].

5.1.5 Superﬂuid density ﬁts

Initial fitting attempts were made with isotropic gaps on both bands, but it quickly became apparent that the small gap energy scale implied by the flat part of the superfluid density could not be replicated. This prompted us to introduce gap anisotropy in the form of an √ extended s-wave-like gap, with ∆(φ) ∝ (1 + 2α cos(2φ)). The reason for this choice of gap is that the system undergoes a nematic phase transition, which should make the gap two-fold symmetric instead of four-fold. However, as the system enters the nematic phase, the grains orient randomly, which makes the experimentally accessible quantities equivalent in this case for four-fold and two-fold symmetric gaps. If the sample were de-twinned while cooling, a two-fold gap calculation that properly takes into consideration the orientation of the crystal axes relative to the screening currents would be necessary.

71 Figure 5.10: From Ref. [113]. (a) Fit to the superﬂuid density using a two-band extended s-wave model, with n1 = 0.35 and α2 = 0.42. Inset shows a polar plot of the C2ν symmetric version of the gaps at zero temperature used in the two-band ﬁt for multiple DOS parameter values 0.25 < n1 < 0.5. (b) Temperature dependence of the RMS gap amplitude on the two bands as well as the overall gap minimum, for the same range of n1.

The gap anisotropies were allowed to vary along with the other fitting parameters, giving a total of 6 parameters: α1 and α2, the gap anisotropies; V11, V12 = V21, and V22, the dimensionless effective interaction constants; and n1 = N1/Ntotal, the normalized density of states of the first band. γ was set according to the procedure outlined in the previous section. Further fit attempts indicated that the larger of the gaps is not particularly anisotropic, so α1 was fixed to 0. In order to improve convergence of the fitting procedure we reduced the number of fit parameters by performing a grid search over n1 in equal steps. Equally good fits to the superfluid density are obtained for 0.25 < n1 < 0.5. If we assume n1 = 0.3, the best-fit parameters are V11 = 0.212, V12 = 0.068, V22 = 0.022, α2 = 0.43 and γ = 0.35.

The ﬁt for n1 = 0.35 is shown in Figure 5.10(a). Although the gap maximum varies slightly between the ﬁts, the gap minimum is consistently ∆min/kB ≈ 0.25Tc = 2.3 K, as shown in

Figure 5.10(b), and the gap anisotropy is typically near α2 = 0.42. Within our multiband interpretation the large conductivity peaks seen in Figure 5.6 are the result of thermally excited quasiparticles near the gap minima in band 2. The deep minima on band 2 also have important consequences for the relaxation dynamics. First, there is a strong reduction in the phase space for recoil when low energy quasiparticles undergo elastic impurity scattering as the only available states are those near the gap minima; a situation reminiscent of the nodal cuprates [131, 124, 72]. Second, Umklapp scattering, which is necessary if inelastic processes are to relax the electrical current [132, 133], will be suppressed by the small Fermi

72 Figure 5.11: From Ref. [134]. (a) k-space structure of the gaps plotted on top of the Fermi surface from Bogoliubov quasiparticle interference measurements. The colour represents the sign of the gap. (b) Gap values as a function of angle measured from the centre of the individual Fermi surface pockets. surfaces of FeSe unless the scattering involves both bands. However, this requires the low energy quasiparticles near the gap minima to partner with quasiparticles on the large-gap band, and these excitations will therefore be strongly gapped. It is these effects that lead to the extremely long-lived quasiparticles inferred from the scattering rate in Figure 5.7. The model presented here is a minimal model that captures the properties of the superfluid density in FeSe. A full model which takes into the Fermi surface anisotropy will almost surely find differences in the details of the gap values and angular dependence. However, the importance of multiband effects and the presence of non-zero gap minima are robust conclusions. In fact, each of the gap values can be directly tied to qualitative features in the temperature dependence of the superfluid density: the gap minimum, ∆min, is fixed by the flat superfluid density region at T < 0.5 K; ∆2, the average value of the gap in the subdominant band, is linked to the upwards curvature in ρs(T ), which occurs in a range near Tc/3; and the dominant gap, ∆1, is set by Tc itself.

5.1.6 Outlook

The results of this study were published in 2016 in Ref. [113]. Since then there has been further progress with a simultaneous measurement of gap and Fermi surface using Bogoliubov quasiparticle interference imaging showing that the both gaps are highly anisotropic, both

73 having deep minima without nodes, and with opposite sign relative to each other [134], as shown in Figure 5.11. This would seem to indicate that the correct pairing symmetry is s+−. However, the nodal superconductivity seen in some measurements [100] remains a puzzle. A STM study pointed out that the FeSe gap near twin boundaries resembles a nodeless gap more than regions far from twin boundaries [102] raising the question of whether or not there are nodal and nodeless spatial regions coexisting within the same sample [135]. High precision measurements on carefully detwinned samples would therefore be extremely insightful.

5.1.7 Contributions

My contribution to this project was to first design and test the helical resonator that was an essential part of being able to resolve the extremely small Rs in FeSe, in the frequency range where the measurement frequency is of the same magnitude as the quasiparticle relaxation rate. I performed the microwave spectroscopy measurements in the dilution refrigerator in David Broun’s lab at SFU, and processed the data taken over the many temperature sweeps that were performed to reduce the uncertainty n the measurements. The code used to evaluate the two-band model was originally written in Mathematica, but was too slow to perform fits. I therefore rewrote it in Fortran and exported a Mathematica compatible library that was used to obtain the fits to the superfluid density.

5.2 Nb-SrTiO3

Having explored a two-band material with exceptionally high purity, FeSe, we now investigate the properties of a material where the eﬀects of impurities are unavoidable due to the chemical substitution process that induces superconductivity.

Stoichiometric SrTiO3 (STO) is an insulating perovskite, with a ∼ 3.2 eV direct band gap [136]. At room temperature it adapts a cubic crystal structure, shown schematically in Figure 5.12. Upon cooling below 106 K it undergoes a symmetry-lowering transition to a tetragonal structure [137]. STO has a large dielectric constant at room temperature, 4 r ≈ 300, which rises steadily upon cooling until 10 K where it plateaus at ≈ 2 × 10 . This large increase has been attributed to quantum critical phenomena, which leads to this material being designated a quantum paraelectric. STO becomes metallic with Fermi-liquid properties upon doping with Nb [139, 138, 140]. Quantum oscillation measurements [141, 142] and ﬁrst-principles calculations [138] have together shown that three electronic bands coming from the Ti 3d orbitals are present near the Fermi energy, and can be ﬁlled successively upon doping [142, 138]. The Fermi surfaces and calculated band structure of Nb-doped STO are shown in Figure 5.12(b, c). Early tunneling conductance measurements performed with In:Nb-STO tunnel junctions showed the existence of two superconducting gap features [143], and other measurements

74 Figure 5.12: (a) Perovskite crystal structure of SrTiO3. The Nb dopants replace Ti at the centre of the oxygen octahedron. (b) Fermi surface of Nb-doped SrTiO3 evaluated at a doping large enough that all three bands are populated, from Ref. [138]. (c) Band structure calculations of the Fermi surface of Nb-STO showing the three low-energy bands along the main crystallographic axes (left) and along two diﬀerent diagonal cuts (middle). The dashed lines indicate the minimum and maximum doping used in our study, and show that there are at least two bands at all dopings considered in this study. Modiﬁed from Ref. [138].

75 Figure 5.13: From Ref. [145]. Tunneling spectroscopy data for a sample with n = 2.5 × 1020 cm−3 revealing a single gap at all temperatures. The dashed black line is a ﬁt at 40 mK for a single-gap BCS superconductor. such as the ﬁeld dependence of thermal conductivity [144] have also been interpreted in terms of multigap superconductivity. More recent tunneling spectroscopy measurements have however put this picture into question as only one spectroscopic gap is resolved down to 40 mK [145], as shown in Figure 5.13. The superconductivity in STO is particularly strange due to the extremely low carrier density, which is surpassed only by superconducting Bi [146]. A high Debye temperature,

θD = 690 K (59.5 meV), and a Fermi energy on the order of tens of meV [142, 138] places STO well outside the range of validity of BCS theory which is formulated in the adiabatic limit where ωD/EF 1. The Fermi energy is too low for conventional phonon-mediated superconductivity [147, 145], thus leaving the mechanism for superconductivity in STO a matter of ongoing discussion. A wide variety of pairing mechanisms such as ferroelectric ﬂuctuations [148, 149, 150, 151], plasmons [152], longitudinal optical phonons [153], and localized phonons caused by substitution disorder [154] have so far been proposed.

Doping Nb into SrTiO3 induces superconductivity above a threshold concentration of 17 −3 n ≈ 0.5 × 10 cm [141]. Tc increases with additional doping before reaching a maximum 20 −3 21 −3 Tc ≈ 0.4 at n ≈ 10 cm , and then dropping back to zero at n ≈ 10 cm [155]. Nb- STO was the first material to show a dome of superconductivity in response to some tuning parameter; a now common feature across many different classes of materials such as the cuprates, organics, heavy fermions, and pnictides. In many of these materials the superfluid density also qualitatively tracks the rise and fall of Tc. However, the causal relationship between Tc and ρs remains a point of major controversy, in particular whether the superfluid

76 Nb doping Tc ρDC nnom nHall free carriers µ size supplier (wt. %) (K) (µΩ m) (1020 cm−3) (1020 cm−3) per unit cell (cm2/Vs) (mm3) 0.1 0.168 0.58 0.3 0.33 0.0020 3300 5 × 5 × 1 Crystec 0.2 0.254 0.57 0.7 0.59 0.0035 1800 5 × 5 × 0.5 Shinkosha 0.35 0.346 0.52 1.2 1.1 0.0065 1100 5 × 5 × 0.5 Shinkosha 0.5 0.278 0.42 1.7 2.0 0.012 800 5 × 5 × 1 Crystec 0.7 0.213 0.56 2.3 2.2 0.013 500 5 × 5 × 1 SurfaceNet

Table 5.1: Material properties and dimensions of the samples used in this study. nnom is the manufacturer-provided nominal doping level. The resistivity ρDC, charge carrier density nHall, and mobility µ were determined from Hall measurements at 2 K. The free carriers per unit cell were calculated from nHall assuming a unit cell volume of V = a3 = (0.3905 nm)3[162].

density places bounds on Tc or, instead, Tc controls the spectral weight available to form the superfluid [156, 1, 70, 157, 69, 158, 159]. From a somewhat naïve perspective, the absolute superfluid density should continuously increase with doping as the number of carriers available to condense into the superfluid increases. Tc would then be bounded by the stiffness of the superfluid condensate against phase fluctuations, ρs. This holds as long as the scattering rate is much smaller than 2∆, but breaks down when the scattering rate becomes comparable to the gap magnitude. When this happens the amount of particles that can participate in the condensate is limited by

∆ ∝ Tc and the superﬂuid density is suppressed from its maximum value. STO is an ideal model system to investigate the interplay between Tc and ρs as doping with Nb tunes the carrier concentration through the superconducting dome in a controlled manner, while also modifying the scattering rate. In the following sections we will show that the observation of a single gap is compatible with multiple superconducting bands, and also elucidate the causal relationship between Tc and ρs in Nb-STO. The low critical temperature in Nb-STO lends itself particularly well to this study as we can perform simultaneous measurements of superﬂuid density, optical conductivity, and superconducting gap, all on the same sample and in the same experimental apparatus.

5.2.1 Experiment details

The crystals used in this study were obtained from commercial suppliers (Crystec, Shinkosha, and SurfaceNet), who supplied nominal speciﬁcations of carrier concentration, nnom, by assuming that each Nb dopant contributes one free electron. The nominal charge carrier concentration of the samples that were provided were from 0.3 to 2.3 × 10−20 cm−3. Transport and microwave measurements were performed on the same samples, which were cut and polished from these crystals. Magnetotransport measurements were carried out in a Quantum Design Physical Properties Measurement System at magnetic ﬁelds up to 4 T.

The contacts were applied in a van der Pauw conﬁguration to obtain the resistivity, ρDC,

77 104 ) s V / 2 m c ( 103

This work Tufte et al. Frederikse et al. Lin et al. 102 1017 1018 1019 1020 3 nHall (cm )

Figure 5.14: Doping dependence of the electron mobility in Nb-doped SrTiO3. Hall mobility measured on the microwave samples used in this study (red circles) is compared with mobility data from Tufte et al. [160] at 2 K, Frederikse et al. [161] at 4.2 K, and Lin et al. [142].

and Hall coefficient, RHall. A single band analysis at 2 K was used to determine the carrier −1 concentration, nHall = (qRHall) , and the mobility, µ = RHall/ρDC. The results of this analysis are presented in Figure 5.14 and show that the samples used here are comparable to those used in previous studies. A summary of the sample properties is provided in Table 5.1. Comparing the measured carrier concentration to the calculated band structure shown in Figure 5.12(b) shows that there are at least two electronic bands crossing the Fermi level at all dopings studied here. Measurements of surface impedance were carried out at the University of Stuttgart by Markus Thiemann and Manfred Beutel. They used a microwave stripline resonator, shown in Figure 5.15 with centre conductor width 45 µm and thickness 1 µm, deposited on a sapphire plate of thickness 127 µm with dimensions 10 × 12 mm2. The sample acts as one of the ground planes in this resonator configuration. Coupling is provided by a small gap in the centre-conductor of width 50 µm, giving the resonator a quality factor of ∼ 4×104. The fundamental frequency of the stripline resonators is determined by the length of the central conductor section. Approximately evenly spaced overtones can also be used. In order to get good frequency coverage three resonators with different fundamental frequencies were used. The resonators were placed in a dilution refrigerator to allow measurements down to 20 mK.

78 Figure 5.15: From Ref. [163]. (a) Schematic of the microwave stripline resonator showing how the sample acts as a second ground plane. (b) Photograph of one of the resonators in its enclosure with the small coupling gap highlighted.

Raw bandwidth measurements were used to determine Tc, which is extracted from the 20 −3 sharp drop in fB in Figure 5.16. All of the samples, except nHall = 2.2 × 10 cm , show sharp superconducting transitions, indicating good sample homogeneity.

5.2.2 Results

Microwave surface impedance gives direct access to the complex conductivity, σ = σ1 − iσ2, which in turn encodes the electromagnetic response of both quasiparticles and superﬂuid.

The opening of the spectroscopic gap, 2∆(T ), is visible as a kink in σ1 in Figure 5.17(a) as some of the excitations become gapped and the missing spectral weight shifts into the zero- frequency superﬂuid peak. The solid lines in the left panels of Figure 5.17 are simultaneous

ﬁts of the Mattis–Bardeen (MB) equations [165, 166] to σ1/σN and σ2/σN :

σ 2 Z ∞ (f(E) − f(E + ω)) (E + ∆2 + ωE) 1 = ~ ~ dE 2 2 1/2 2 2 1/2 σN ~ω ∆ (E − ∆ ) ((E + ~ω) − ∆ ) 1 Z −∆ (1 − 2f(E + ω)) (E2 + ∆2 + ωE) + ~ ~ dE , (5.27) 2 2 1/2 2 2 1/2 ~ω ∆−~ω (E − ∆ ) ((E + ~ω) − ∆ )

σ 1 Z ∆ (1 − 2f(E + ω)) (E2 + ∆2 + ωE) 2 = ~ ~ dE , (5.28) 2 2 1/2 2 2 1/2 σN ~ω ∆−~ω,−∆ (∆ − E ) ((E + ~ω) − ∆ ) where f(E) is the Fermi function, and the low limit for the integral on σ2/σN changes from ∆ − ~ω to −∆ when ~ω > 2∆. These are single parameter fits, with ∆(T ) being the only free parameter in the theory. The good agreement of the fit with the data is the first indication that the electrodynamic response of Nb-STO is governed by a single gap. In addition, the MB equations were derived in the context of of materials that are in the dirty limit, where 2∆/~ < Γ [166], which provides the first clue that Nb-STO is a material where impurity scattering plays a dominant role in determining the physics.

79 Figure 5.16: From [164]. Temperature dependence of the resonant bandwidth, fB, at f0 ≈ 2 GHz for all ﬁve samples. Tc was determined from the sharp drop in fB and is indicated by arrows.

A more general formulation of the conductivity is given by the Zimmermann equations [167], which can be applied for any scattering rate. The right panels of Figure 5.17 show calculations with the Zimmermann equations, which demonstrate that the Mattis– Bardeen equations are indeed valid in the dirty limit of superconductivity, and that the conductivity of Nb-STO agrees well with the dirty limit. Note that the downturn above the gap energy in σ2 is a distinctive feature of the large scattering rate regime. The quality of the data is insuﬃcient to accurately estimate the scattering rate from ﬁts to the Zimmer- mann equations, but we can estimate a lower bound of Γ/2π = 10∆/~ from inspection of the calculated curves.

Kinks in the temperature dependence of σ1/σN , shown in Figure 5.18, are another method of extracting the spectroscopic gap value from the data without relying on a particular model for the conductivity. Deviation from linearity above a small threshold is taken to be the signature of the spectroscopic gap, and, indeed, good agreement between the methods is observed when plotted in Figure 5.19. The results from the gap analysis again evince only a single spectroscopic gap, and in fact the gap can be ﬁtted quite accurately with the

BCS s-wave gap equation, yielding a gap to Tc ratio that is in very good agreement with that expected for a weak-coupling BCS model. This leaves us with the question, why is only

80 Figure 5.17: From Ref. [164]. Left panels: Frequency dependence of the real (a) and imaginary (b) parts of the complex conductivity σ, normalized to the normal state conductivity 20 −3 σn, in the nHall = 2 × 10 cm sample. Symbols are the data and solid lines are ﬁts to the Mattis–Bardeen equations, with 2∆ being the only free parameter. Right panels: Fits to the 0.21 K data from the left panels but with the Zimmermann equations. The pronounced downturn in σ2 only appears at high scattering rates, and the Mattis–Bardeen equations match the very large scattering rate limit.

81 Figure 5.18: From Ref. [164]. Temperature dependence of σ1/σN measured at several frequencies for the same sample as shown in Figure 5.17. Kinks in σ1/σN mark the onset of dissipation caused by ω ∼ 2∆. 2∆ is measured from the point where σ1/σn departs signiﬁ- cantly from the linear region of the data. The grey line in the frequency-temperature plane shows the expected behaviour for a single-gap weak-coupling BCS superconductor. one energy gap observed when there is strong evidence for multiple superconducting bands in Nb-STO? A resolution of this puzzle emerges from Anderson’s theorem, which was discussed in Section 2.2.1. In short, conventional s-wave superconductivity (even multiband) is protected from nonmagnetic impurity scattering. Disorder does not completely suppress s-wave pairing, but instead homogenizes the energy gap over the Fermi surface. In a two-gap superconductor we would therefore expect that two gaps, distinct in the absence of disorder, would homogenize as scattering increases, leaving behind a single resolvable spectroscopic gap.

5.2.3 Anderson’s theorem in two-band superconductors

The model used in this section is a two-band model in the BCS weak-coupling limit. The formalism is similar to that outlined in Chapter 2, with slight modiﬁcations to take into account the two-band nature of the model. The energy gaps in the two bands, ∆1 and ∆2, are solutions to the coupled gap equations

ωD ˜ X X ∆µ,n ∆ν = 2πT nµVνµ q . (5.29) ˜ 2 2 ωn>0 µ=1,2 ∆µ,n +ω ˜n

1 Here ωn = 2πT n + 2 are the fermionic Matsubara frequencies and the Matsubara sum is cut oﬀ at the Debye frequency, ωD. The dimensionless interaction parameters, Vνµ = Vµν, for a symmetric coupling matrix, and nµ ≡ Nµ/N0 is the fraction of the total density

82 Figure 5.19: From Ref. [164]. (a) Temperature dependence of the spectroscopic gap 2∆. Open symbols were obtained from fits to the Mattis–Bardeen equations and closed symbols were obtained by inspection of σ1/σn as in Figure 5.18. Dashed lines are fits to a single- gap weak-coupling BCS temperature dependence with the zero-temperature gap, 2∆0, as the only adjustable parameter. (b) Gap to Tc ratio as a function of carrier density. Closed symbols are the data from this study and open symbols are from tunneling spectroscopy on thin films [145]. The dashed line represents the weak-coupling BCS value 2∆0/kBTc = 3.528.

83 of states associated with band µ, with n1 + n2 = 1. For simplicity we will use Born limit scattering and consider the case where the interband and intraband non-magnetic impurities according to   n1ωñ n2ωñ ωñ = ωn + πΓ q + q  , (5.30) 2 ˜ 2 2 ˜ 2 ωñ + ∆1,n ωñ + ∆2,n

 ˜ ˜  ˜ n1∆1,n n2∆2,n ∆µ,n = ∆µ + πΓ q + q  , (5.31) 2 ˜ 2 2 ˜ 2 ω˜n + ∆1,n ω˜n + ∆2,n where these equations are obtained by partitioning the average over the Fermi surface into two parts.

At each temperature, these equations are solved self-consistently to find ∆1(T ) and ˜ ˜ ∆2(T ). From inspection, we see that in the large Γ limit ∆1,n → ∆2,n. This is a manifestation of Anderson’s theorem, in which the effect of nonmagnetic impurities is to homogenize the energy gap over the Fermi surface. Once the energy gaps are obtained it is straightforward to calculate the normalized superfluid density as each band contributes in parallel. The total superfluid density is therefore a weighted average of the bands

ρs = γρ1 + (1 − γ)ρ2 , (5.32) ρs00 where 0 < γ < 1 is a parameter that encodes the relative contribution each band makes to the total superﬂuid weight. The superﬂuid density is calculated with

∞ ∆˜ 2 ρ = 2πT X µ,n . (5.33) µ 3/2 ω ˜ 2 2 n ∆µ,n +ω ˜n For illustrative purposes, calculations have been carried out with the following parameters: n1 = n2 = 0.5, V11 = 0.26, V22 = 0.25, V12 = 0, ωD = 500 K and relative superﬂuid weight γ = 0.5.

In Figure 5.20(a) we see that the unrenormalized gaps, ∆1 and ∆2, do not become equal even with large scattering rates, but this is unimportant as all physical observable quantities depend upon the renormalized gaps and frequencies. Figure 5.20(b) shows that the renormalized gaps homogenize as scattering rate is increased. The superﬂuid density is calculated for this case as well, and is shown in Figure 5.21. All signs of multiband superconductivity quickly disappear as scattering rate increases. The density of states and conductivity can similarly be calculated starting from given gap values. For small scattering rates the density of states shown in Figure 5.22(a) has two sharp peaks. As the scattering rate increases the peaks broaden as we would expect, and then merge. The density of states then becomes sharp again as scattering rate continues to increase. This last point seems counterintuitive, as scattering is normally associated with a broadening of the density of states, but this is only the case for inelastic scattering.

84 Figure 5.20: (a) The unrenormalized gaps, ∆1 and ∆2, are not completely homogenized by the eﬀects of disorder. Arrows denote the direction of increasing Γ. (b) In contrast, the ˜ ˜ renormalized gaps are homogenized in the large Γ limit. Here ∆1 and ∆2 are plotted at the ﬁrst Matsubara frequency.

Calculations of the conductivity in Figure 5.22(b) show that the two-gap nature of the material results in three distinct peaks at low scattering rate. Although not correct in detail, a simple way to view the conductivity is as the convolution of density of states with itself. The two peaks in the DOS therefore turn into three energy scales. As scattering rate increases the three features are quickly washed out, with a single spectroscopic gap signature remaining. It is clear from these examples that when the scattering rate is comparable to the superconducting gap value (Γ/∆ ≈ 1), then no signature of the multiband nature of the material will remain in conductivity, density of states, or superﬂuid density.

5.2.4 Is Nb-STO really a dirty superconductor?

As we have seen in the previous section, a scattering rate larger than the superconducting gaps homogenizes the gap values from diﬀerent Fermi surfaces, leaving behind a single spectroscopic gap, even if the two bands had well separated, distinct values in the absence of scattering. We must now establish that the scattering rate in Nb-STO is large enough to cause this homogenization. One way to quantify this is to extract the scattering rate from ∗ the normal-state Hall mobility, µ, via ΓHall ≡ e/(µm ) in a one-band interpretation. Based on quantum oscillation measurements [142] we assumed three possible values of m∗ = 1.3,

2, and 4me. Even in the lowest doped sample and with the highest assumed eﬀective mass the inferred scattering rate is Γ/(2π) > 20 GHz (0.96 K), which is already almost twice the spectroscopic gap, 2∆ ≈ 12 GHz (0.58 K), in that sample. The transport scattering rate

85 Figure 5.21: Superﬂuid density for a two-band superconductor, showing how two-gap eﬀects are washed out by disorder leading to the re-emergence of generic single-gap behaviour.

(a) (b)

Figure 5.22: (a) Density of states for a two-band superconductor with gaps separated by a ratio ∆2/∆1 = 1.3. At large scattering rate the two gaps merge into one sharp feature. Dashed lines represent the locations of the two original gaps and the ﬁnal merged gap in between. (b) Real part of the optical conductivity with increasing scattering rate. A single spectroscopic gap remains at high scattering rate, with the onset of dissipation occurring at the weighted average of the original gap values.

86 2 2 provides another avenue to estimate the scattering rate from Γρ = 0ωpρDC, where ωp is the plasma frequency estimated from optical studies of Nb-STO [168], and ρDC is the measured resistivity of our samples. Again, the inferred scattering rate in the lowest doped sample exceeds the spectroscopic gap in all samples. All scattering rates are plotted in Figure 5.23 for comparison. The roughly linear increase of Γ with nHall indicates that the dominant scattering mechanism is impurity scattering due to Nb dopants. Further confirmation of the single-band nature of Nb-STO comes from the temperature dependence of the superfluid density. The superfluid density does not show any of the common signs of multigap behaviour such as mid-range upwards curvature, as seen in V3Si [126] and FeSe (see Section 5.1 and Ref. [113]), or the presence of small activation energies, seen in both MgB2 [126] and FeSe [113].

I performed fits to the temperature dependence of ρs(T ) for all five samples, assuming a single-gap BCS weak-coupling model with nonmagnetic disorder. The plasma frequency, 2 ωp, from Ref. [168] was used to set the ‘clean’ superfluid density ρs00. Γsf , the scattering rate obtained from the fits, shown in Figure 5.23(c) falls between the scattering rates derived from other methods above, confirming the interpretation of single-gap superconductivity in terms of Anderson’s theorem. Although we clearly observe a single spectroscopic gap, field dependent measurements of thermal conductivity [144] as well as our own field dependent measurements (shown in Figure 5.24) clearly reveal that multiple bands contribute to superconductivity. The upper 2 critical field, Hc2 = Φ/2πξ , which is the point at which vortex spacing reaches the coherence length ξ and superconductivity is completely destroyed is visible in Figure 5.24(a) as the high-field kink. Slightly below this field there is a second kink, which indicates a longer coherence length, and which we call H∗. In a multiband system band-specific coherence lengths ξ ≡ ~vF,i naturally arise from band-to-band variation in either the energy gap i π∆i ∆i or the Fermi velocity vF,i. Even with the homogeneous spectroscopic gap reported here, quantum oscillation measurements indicate enough variation in vF,i that we expect multiple field scales to emerge [142]. With this in hand we can now comment on the evolution of superfluid density as a function of normal-state charge carrier density. A dome of superfluid density, quite similar in shape to the Tc and 2∆0 domes, can be seen in Figure 5.23(b). A similar superfluid dome has also recently been reported in Ref. [169]. ρs0 is proportional to the amount of spectral weight lost from σ1 in the superconducting state compared to the normal state. As the materials are in the dirty limit the transferred spectral weight is limited by 2∆, which is proportional to Tc. The Tc dome in STO therefore cannot be attributed to the superfluid stiffness, and instead it is the transition temperature that sets the limit on ρs in Nb-STO.

87 Figure 5.23: From Ref. [164]. Domes of Tc, 2∆0, and ρs0 as a function of charge carrier density are inferred from the fits. Scattering rate as a function of charge carrier density is also inferred from the fits.(a) Temperature dependence of the absolute superfluid density, ρs, for all five samples, measured at ω/2π ≈ 2 GHz. Solid lines are fits to the single-gap BCS superconductivity model in the presence of disorder, with plasma frequency set by infrared measurements of the Drude weight from Ref. [168]. In this model the scattering rate and Tc are the only adjustable fit parameters. (b) Domes of Tc, 2∆0, and ρs0 as a function of charge carrier density. The shaded area denotes the confidence band on ρs0, including a 10% scale factor uncertainty arising from resistivity measurements used to calibrate surface impedance. (c) Scattering rates inferred from infrared spectroscopy, ΓIR, resistivity, Γρ, and Hall effect with different choices of effective mass, ΓHall,m∗. Γsf is the fit parameter from the superfluid density fits with confidence interval shown by the shaded region. Γsf is between the lowest and highest inferred scattering rates for all dopings.

88 Figure 5.24: (a) Magnetic field dependence of Rs and Xs at 8.02 GHz for the 20 −3 nHall = 2.0 × 10 cm sample. Two kinks are clearly visible, with the lowest field kink ∗ corresponding to H and the higher field scale corresponding to Hc2. (b) Field-temperature phase diagram constructed from the data at four different measurement frequencies. There is clearly little to no frequency dependence.

89 5.2.5 Contributions

My main contribution to this project was the proposal of disorder and Anderson’s theorem to resolve the apparent discrepancy between the single spectroscopic gap that we observe in our measurements, and the clear evidence for multiple superconducting bands both from our own in-field measurements and from other research groups. In order to test this I wrote the code for the density of states, and conductivity calculations. I also postulated that the dirty s-wave model could provide a good understanding of the superfluid dome in Nb-STO and wrote the code to test this. Finally I assisted Markus Thiemann in writing the fitting routines for the Mattis–Bardeen and Zimmermann equations. The results of this study were published in Ref. [164].

90 Chapter 6

Disorder in La2−xSrxCuO4

The ﬁrst cuprate superconductor was discovered in 1986 by Bednorz and Müller when they found superconductivity at ∼ 30 K in BaxLa5−xCu5O5(3−y) [15]. This was followed shortly thereafter by the 1987 observation of superconductivity in Y1.2Ba0.8CuO4−δ at

Tc ∼ 90 K [16]; well above the boiling point of nitrogen. With such high transition temperatures, these new superconducting compounds do not fit within the original formalism of BCS theory that was developed for phonon-mediated electron-electron interactions. How- ever, putting aside the precise nature and origin of the pairing interaction, BCS theory can be suitably modified to use non-isotropic pairing mechanisms. As long as the pairing interaction overcomes the repulsive Coulomb interaction then the theory can still hold. Of course, one must be ever mindful when applying existing theories to new phenomena, lest some of the assumptions break down. In the case of BCS theory, the assumption of the existence of Cooper pairs is paramount. This issue was resolved early-on by measurements of the flux quantum that show the magnetic flux is quantized in units of ~/2e in Y1.2Ba0.8CuO4 [170], confirming that the superconducting charge carriers are electron pairs in at least some of the high-Tc materials. With the nature of the charge carriers understood, the next question is that of the pairing symmetry of the cuprates. The first clues came from Knight shift measurements which showed that the spin susceptibility falls rapidly to zero below Tc [171, 172], proving that the superconducting pairs form a spin-singlet state, such as s-wave or d-wave. Early measurements of the magnetic penetration depth were interpreted as showing conventional activated exponential behaviour with ∆/Tc ≈ 1.76 [173], in keeping with conventional s-wave BCS theory. Several other studies with various probes of penetration depth shared similar conclusions [174, 175]. A re-analysis of these data by J. Annett et al. [176] found that the penetration depth is actually proportional to T 2, which implies that the superconductivity is not conventional. They also concluded that the T 2 temperature dependence is likely not intrinsic as it changes from sample to sample, and suggested that it could be due to nonmagnetic impurity scattering. This was later confirmed when high-quality single-crystal

YBa2Cu3O7−δ samples made by Ruixing Liang and coworkers at UBC [177] became avail-

91 Figure 6.1: From Ref. [180]. (left) Schematic of the tetracrystal SrTiO3 substrate on which the Tl2201 was grown. The Tl2201 ﬁlm grows with the same orientation as the substrate. (right) Scanning SQUID microscopy reveals the presence of a spontaneously generated half- integer ﬂux quantum at the point where the four crystals meet. able. Microwave penetration depth [178, 115] and µSR measurements [179] on these high quality samples revealed a linear penetration depth at low temperature, in good agreement with what would be expected from a clean dx2−y2 order parameter. Deliberately introducing Zn impurities into YBCO recovers the T 2 temperature dependence of the penetration depth seen in other experiments and rapidly suppresses Tc [59]. Ni impurities also suppress

Tc, but have only limited eﬀect on the temperature dependence of the penetration depth at the impurity concentrations studied. One would expect that if the superconductivity in high-Tc materials is conventional then Zn, being non-magnetic, would have little eﬀect on

Tc, whereas Ni, being a magnetic dopant, would rapidly reduce Tc. Theoretical calculations of the superﬂuid density in d-wave superconductors with line nodes [61], show that the introduction of point-like unitary scatterers does indeed change the expected linear temperature dependence to a T 2 temperature dependence. Substantially more impurities (and correspondingly larger reduction in Tc) is required to obtain a similar curvature from Born scattering [60]. A natural interpretation is then that Zn acts in this case as a resonant scatterer, whereas Ni acts as a weak scatterer. Measurements on high-quality YBCO samples are thus consistent with d-wave pairing, where both magnetic and non-magnetic impurities cause a reduction in Tc. Although much of the evidence points to d-wave pairing symmetry, the techniques discussed so far have no phase sensitivity, which complicates the task of distinguishing between unconventional superconductivity and conventional s-wave superconductivity with accidental nodes or deep gap minima. A series of scanning SQUID measurements performed on

92 Figure 6.2: From Ref. [184]. (a) Crystal structures of several cuprate superconductors showing the common CuO2 layers in which superconductivity develops. (b) Orbital structure of the CuO2 planes showing the Cu dx2−y2 and O pσ orbitals.

YBCO rings patterned on a SrTiO3 substrate revealed a predominantly d-wave-like order parameter [181, 182]. However, for an orthorhombic system like YBCO the s-wave and d- wave pairings belong to the same irreducible representation of the C2ν symmetry, making it impossible to rule out a mixed d + s state [183]. Perhaps the clearest signature of pure d-wave pairing comes from the tetracrystal experiment of Tsuei et al. [180] in which they grew a Tl2Ba2CuO6−δ (Tl2201) ﬁlm on a tetracrystal substrate, made by fusing together two bicrystal substrates Figure 6.1. The choice of Tl2201 was motivated by its tetragonal

C4 symmetry, which forces the s-wave and d-wave symmetries to belong to distinct irreducible representations such that no admixture of the two is possible. The observation of a half-integer ﬂux quantum in this geometry is unambiguous proof that Tl2201 has a pure d-wave pairing symmetry. Despite the overall consensus on the dominant pairing symmetry in the cuprates, the nature of the pairing glue remains one of the biggest mysteries in the ﬁeld to date. A natural candidate can be found by looking at the parent compounds (i.e., the undoped cuprates), which all share the same CuO2 planar structure, as shown in Figure 6.2. Band structure calculations suggest that the orbitals originating from the CuO2 planes should produce

93 Figure 6.3: From Ref. [187]. Schematic representation of the phase diagram of the high-Tc cuprate superconductors. metallic behaviour as they are half-filled, with one electron per Cu atom. However, the consensus [185] is now that the parent compounds are in fact Mott insulators [186]. The physics governing Mott insulators is quite different from that of band insulators in which the Pauli exclusion principle prevents conduction due to fully filled electronic bands. The Mott insulator state is not limited by Pauli exclusion as the bands are only half-filled. Instead, it is the strong on-site Coulomb repulsion that prevents the creation of doubly occupied states, which are necessary for conduction to take place in a half-filled band. The resulting charge localization leads to the development of long-range antiferromagnetic order with an ordering wavevector Q = (π/a, π/a), which sets in at a Néel temperature on the order of 300 K. Adding charge carriers to the system destroys the long-range order, but short-range spin fluctuations persist, enhancing the spin susceptibility near the Q wavevector. If the wavevector of the antiferromagnetic spin fluctuations connects points on opposite sides of the Fermi surface then the BCS gap equation can be solved for a gap with d-wave form factor. Doping either electrons (not discussed further here) or holes into the parent compounds adds high-mobility carriers which screen the Coulomb interaction and quickly lead to the suppression of the AFM state by about 3–5% hole concentration [185]. As doping is increased further, superconductivity emerges and the transition temperature keeps rising with additional doping until it reaches a maximum at the so-called optimal doping value, as seen in Figure 6.3. The superconducting state on the underdoped side is complex, having to compete with various other orders such as charge density waves and spin density waves. Above

94 Figure 6.4: From Ref. [1]. (a) In-phase component of the mutual inductance pickup voltage. Inset shows a schematic of the coils and sample. (b) Out-of-phase component of the pickup voltage. Inset shows area around Tc revealing a relatively sharp transition. (c) Penetration depth and real part of the conductivity inferred from the mutual inductance measurements. (d) Schematic of one of the samples showing the LaSrAlO4 substrate, protective metallic La1.60Sr0.40CuO4 layers at the bottom and top of the stack, Tc suppression layers with Zn dopants, and the superconducting layer in dark blue at the centre.

Tc the underdoped cuprates enter a poorly understood state called the pseudogap regime, characterized by a spectroscopic gap, but lacking several features characteristic of a superconducting phase. Increasing temperature further causes the pseudogap to disappear, and a strange metallic state emerges with properties such as linear-in-T resistivity. Angle-Resolved Photoemission Spectroscopy (ARPES) measurements in the pseudogap regime reveal arcs instead of a well-formed Fermi surface. On the overdoped side of the phase diagram the picture is much simpler and the general consensus is that the d-wave superconducting state does not compete with other orders, and gives way at T > Tc to a normal metallic state with the usual T 2 power law for resistivity. In this region of the phase diagram the Fermi surface is also much simpler, with ARPES and quantum oscillation measurements revealing a large and somewhat cylindrical Fermi surface.

95 Figure 6.5: From Ref. [1]. (a) Temperature and doping dependence of the penetration depth of LSCO. (b, c) Superfluid density derived from the penetration depth. (d) Dependence of Tc on ρs0 ≡ ρs(T → 0). The red dashed line is proportional to the square root of the superfluid density, and the green line is a linear fit to the data with Tc > 10 K, showing the linear scaling of Tc with ρs0, with finite intercept.

96 A recent experiment by Božović et al. [1] has cast doubt on the simple picture of BCS d-wave superconductivity in the overdoped cuprates that has emerged over the last 30 years. In an experimental tour-de-force, they grew over 2500 LSCO films using atomic- layer-by-layer molecular beam epitaxy, and measured the penetration depth using a mutual inductance technique, as shown in Figure 6.4. By surrounding the superconducting layer with Zn poisoning layers they were able to limit the superconductivity to a region of known thickness, allowing them to extract absolute values for the penetration depth. The studied doping range covers the entire region from optimally doped to critically doped, where the material is no longer superconducting. The main results, shown in Figure 6.5, are a linear temperature dependence of the superfluid density over almost the full temperature range, and a scaling of Tc with ρs. The authors concluded that, due to the linearity of ρs(T ) and the sharp out-of-phase mutual inductance signal, the samples must be clean, but Tc is only expected to scale with ρs in the dirty limit where ∆ ∼ Tc acts as a bound on the amount of particles that can condense into the superfluid. To quote the authors directly

“Pair-breaking due to impurities and disorder can reduce ρs0, but we doubt that our data can be quantitatively explained using the standard ‘dirty’ d-wave BCS formalism, which includes the eﬀects of pair breaking on impurities and other defects, ...”. If they are correct then the current consensus of dirty d-wave BCS theory applying to the overdoped cuprates would need to be revised. It is therefore imperative to put these claims to the test using realistic and quantitative phenomenological models that try to capture as much of the relevant physics as possible while minimizing the amount of ﬁne-tuning required to explain the experimental results.

6.1 Dirty d-wave BCS theory on a realistic Fermi surface

The starting point for quantitative comparison with experiments is a realistic parametrization of the LSCO Fermi surface. A next-nearest-neighbour tight-binding parametrization of the dispersion relation in LSCO

0 00 k = 0 − 2t(cos kxa + cos kya) − 4t cos kxa cos kya − 2t (cos 2kxa + cos 2kya) , (6.1) obtained from fits to ARPES spectra as a function of hole doping [2] is used in the calculations here. The parametrization in the ARPES paper places the optimal doping point at x = 0.15, whereas the one used in Ref. [1] uses p = 0.16. A rigid shift of 0.01 is therefore used to convert between the two. In the ARPES study, t = 0.25eV and t00/t0 = −0.5 were 0 fixed. t and 0 are both doping-dependent parameters, which were interpolated using a cubic spline through the three highest doping points reported in Ref. [2], giving the smooth parameter evolution seen in Figure 6.6. This leads to the Fermi surfaces shown in Figure 6.7, defined by F ≡ k = 0 in Equation 6.1. As doping increases, the FS goes through a Lifshitz transition and associated van Hove singularity at p = 0.1914. At this doping, one might

97 Figure 6.6: From Ref. [70]. Interpolation of the energy oﬀset, 0, and next-nearest neighbour hopping integral, t0, through discrete points (circles) from Ref. [2].

Figure 6.7: From Ref. [70]. Constant energy contours in momentum space for optimally to overdoped La2−xSrxCuO4 based on a tight-binding parametrization to ARPES spectra [2]. Momentum is measured in units of inverse lattice spacing 1/a. Fermi surfaces are depicted 0 by solid black lines. Doping-dependent tight-binding parameters t and 0 are indicated in the bottom right corner of each panel, in units of nearest-neighbour hopping integral t. expect there to be significant changes in the electronic properties due to the abrupt modification of the Fermi surface topology and divergent density of states. This is however not the case as the x-axis velocity, vx, under-weights the regions where the dispersion becomes flat. Figure 6.8 shows the weighting factor — a combination of the Jacobian transformation 2 required to operate in cylindrical coordinates and vx, as we shall see shortly — that enter the calculations of electronic properties such as superfluid density and optical conductivity. At p = 0.1914 the weighting factor goes to zero at the anti-nodal points, precisely where the local density of states diverges.

98 p = 0.16 p = 0.185 p = 0.1914 p = 0.21 p = 0.26 0.25

| 0.20 2 ) (

x 0.15 F v )

( 0.10 J |

0.05

0.00 0 /2 3 /2 2 0 /2 3 /2 2 0 /2 3 /2 2 0 /2 3 /2 2 0 /2 3 /2 2

Figure 6.8: Weighting factor used in the calculation of electronic properties as a function of doping. Dashed lines indicate the directions along which the density of states diverges at the Lifshitz transition. Note that at p = 0.1914 the weighting factor goes to zero at these points.

6.1.1 Superﬂuid density

The zero-temperature, zero disorder penetration depth is closely related to the bare plasma frequency ωp, and the corresponding superﬂuid density is given by [188]:

1 2 ρs00 ≡ 2 = µ00ωp (6.2) λ00 Z + π Z 2 2 d dkz d k 2 = 2µ0e δ (F − k) vk,x . (6.3) π 2π (2π)2 − d

Here we specialize to a quasi-2D material with layer spacing d and in-plane energy dispersion 1 ∂ ∂ k. F is the Fermi energy, k = (kx, ky) is the in-plane momentum, and vk = , k ~ ∂kx ∂ky is the in-plane velocity. Equation 6.3 can be turned into an integral over the Fermi surface by transforming coordinates from (kx, ky) to (, φ), where φ is an angle in the plane, measured about the centre of the Fermi surface. The Jacobian for this transformation is

∂(k , k ) |k|2 J(φ) = x y = . (6.4) ∂(, φ) ~k · vk

When the energy and kz integrations are carried out we obtain

2 Z 2π 2 1 µ0e |kF | 2 2 = 2 vF,xdφ , (6.5) λ00 2π ~d 0 kF · vF where the Fermi wave-vector kF and Fermi velocity vF are functions of φ. The Fermi surface average from Chapter 2 must include the same Jacobian factor:

Z 2π Z 2π hA(φ)iFS ≡ J(φ)A(φ)dφ J(φ)dφ . (6.6) 0 0

99 When calculating the plasma frequency and superﬂuid density we need a velocity weighted version of the Fermi surface average, which we deﬁne as

Z 2π Z 2π 2 2 hhA(φ)iiFS ≡ J(φ)A(φ)vF,xdφ J(φ)vF,xdφ . (6.7) 0 0

For Fermi surfaces that are close to circular these distinctions are usually not important. However for the overdoped cuprates, the details of the Fermi surface averages turn out to be crucial to understanding the temperature dependence of superfluid density measured in experiments. The finite temperature superfluid density, in the presence of disorder, is most efficiently calculated using a Matsubara sum [189]. Normalized to ρs00 it is given by

** ++ ρ (T ) ∞ ∆2 s = 2πT X k , (6.8) 3/2 ρs00 2 2 ωn>0 ω˜n + ∆k FS where the eﬀects of disorder are built in via the renormalized Matsubara frequencies and gap.

6.1.2 Optical conductivity

The optical conductivity can be calculated within the same framework:

2 Z ∞ N0e D 2 E σ1(Ω) = − dω [f(ω) − f(ω + Ω)] vF,xRe [A++ − A+−] , (6.9) 2Ω −∞ FS where 2 0 0 ∆k +ω ˜+ω˜± + Q+Q± A+± = 0 0 , ω± = ω ± iη , (6.10) Q+Q± Q+ + Q±

0 ω± = ω + Ω ± iη , (6.11)

q 2 2 Q± = ∆k − ω˜± , (6.12)

q 0 2 02 Q± = ∆k − ω˜± . (6.13) The branch cut for the complex square root is the negative real axis, and the renormalized real axis frequencies are

ω˜±(ω) =ω ˜± (iωn → ω ± iη) . (6.14)

These equations, together with the renormalized frequencies fully deﬁne the temperature and frequency dependence of the real part of the optical conductivity. In the normal state

100 limit ∆k → 0, Equation 6.9 reduces to the Drude conductivity

2 ! 4ΓN σ1(Ω) = σN0 2 2 , (6.15) Ω + 4ΓN where the dc conductivity is 2 D 2 E e N0 v F,x FS σN0 = . (6.16) 2ΓN 6.2 Results

A subset of the experimental data from the supplementary information of Ref. [1] is shown in Figure 6.9(a). As mentioned previously, the main features are an almost linear temperature dependence over most of the temperature range, and a linear Tc vs ρs curve transitioning over to a square-root behaviour at low Tc. As the model we used in this study is based upon a mean-field theory, there is no way to capture the downward curvature near Tc in the samples near optimal doping. We therefore extrapolated from the linear part of the curve max to obtain Tc in a mean-field approximation, and then use a parabolic mapping of p to MF Tc as shown in Figure 6.10. A single scattering rate was used for all dopings to avoid fine-tuning the parameters and to explore how much of the physics is captured within a simple but realistic model. Calcu- lations of superfluid density were performed with varying amounts of Born and unitarity scattering and qualitatively compared to the data. The best agreement was found to be with Born unitarity ΓN = 17π K and ΓN = π K, shown in Figure 6.9(b), with the unitarity component mostly providing a slight downturn at low temperature, which is hinted at in some of the high-Tc data. At each value of the nominal doping, p, the underlying clean-limit transition temperature, Tc0, is set according to the Abrikosov-Gor’kov formula (Equation 2.101) so that Tc, the transition temperature in the presence of disorder matches the mean-field MF transition temperature, Tc shown in Figure 6.10. The superfluid densities calculated with Born unitarity ΓN = 17π K and ΓN = π K reproduce many of the qualitative features observed in Ref. [1]. In particular, ρs(T ) shows a strong, nearly linear temperature dependence over almost the full doping range, despite the strong suppression of superfluid density caused by disorder. Tc also correlates almost linearly with ρs, with finite intercept, crossing over to square-root-like behaviour at low Tc. We are not the first group to show this; very similar behaviour was observed for Born-limit scattering with calculations performed on a circular Fermi surface [190].

One aspect of the Tc-ρs0 relation which is not well captured by our model is that the linear to square-root transition does not occur as gradually in the experimental data as in the calculated curve. In the idealized form considered here, the dirty d-wave BCS theory assumes spatial homogeneity, whereas the real samples may in fact consist of an inho-

101 Figure 6.9: From Ref. [70]. (a) Main panel: Superfluid density for overdoped La2−xSrxCuO4 from Ref. [1]. Shading indicates the 2% error bands in superfluid density (1% in penetration depth) as stated in Ref. [1]. The dashed red line shows the method used to extract MF the mean-field transition temperature Tc . Inset: Correlation between measured Tc and zero temperature superfluid density re-plotted from Ref. [1]. (b) Main panel: Calculated dimensionless superfluid density based on a dirty d-wave BCS model that includes a realistic parametrization of the Fermi surface from ARPES data [2]. The scattering rates used in this U B plot are ΓN = π K and ΓN = 17π K. The values are normalized to the zero-temperature opt zero disorder superfluid density for the optimally doped material ρs00. Solid lines show the calculations over the same temperature ranges recorded in the experiments, and dashed lines are the extrapolation below the measured temperatures. Inset: Correlation between mean-field transition temperature and dimensionless superfluid density.

102 Figure 6.10: From Ref. [70]. Clean-limit transition temperature Tc0 and mean-field transition MF temperature Tc plotted along with the experimentally observed transition temperature Tc from Ref. [1] mogeneous mixture of superconducting and metallic regions, with the fraction of metallic regions increasing as doping increases [191, 192]. The doping mechanism in LSCO is based on random cation substitution. Inhomogeneity must therefore always be present despite the exquisite control over average composition in the experiments of Božović et al. [1]. In addition, the out-of-phase mutual inductance technique used in that study to character- ize the inhomogeneity is not a good probe of in-plane microscopic inhomogeneity as the superconducting coherence length diverges near Tc, which averages over short-length-scale inhomogeneity. Figure 6.10 shows the mean-field transition temperatures, measured transition temperatures from Ref. [1], and the clean-limit transition temperature Tc0. Although we should not read too much into the meaning of the clean-limit transition temperature, it is interesting to note that the Nernst effect appears at temperatures near Tc0 [193], and magnetoresis- tance measurements reveal precursor signatures of superconductivity at similar temperatures [194]. Temperature dependent gaps, calculated for the same dopings as in Figure 6.9 are shown in Figure 6.11. It is interesting to note that this choice of scattering rate satisfies the conventional definition of clean-limit scattering, Γtr = 2ΓN 2∆0, over approximately half of the doping range, with ΓN being comparable or below ∆0 only for samples with Tc < 20 K. Despite the designation of being a ‘clean’ superconductor, ∼ 60% of the quasiparticles are unable to condense into the superfluid in the highest Tc sample.

103 Figure 6.11: From Ref. [70]. Main panel: Temperature and doping dependence of the gap maximum on the La2−xSrxCuO4 tight-binding Fermi surface. Inset: Doping dependence of the zero-temperature gap ratio 2∆0/kBTc. Dashed line denotes the clean-limit d-wave BCS value 2∆0 = 4.28kBTc for a circular Fermi surface.

6.2.1 Finding the missing spectral weight

The most logical way to distinguish between the disorder scenario that we propose here and more exotic explanations is to measure the spectral weight of condensed and uncondensed carriers directly using optical probes. This is precisely what Mahmood et al. [157] set out to do with THz spectroscopy measurements of samples grown by ALL-MBE. They found that there is indeed a large uncondensed spectral weight but that the quasiparticle conductivity retains a Drude-like lineshape, with no sign of a superconducting gap feature; quite different from what comes to mind when thinking of d-wave superconductivity, which consists of a sharp peak at low frequency, and a small peak at ν = ∆. The authors express concern about our model and state that “the underlying calculations incorporate an unphysical infinite number of infinitely weak scatterers (Born limit) and assume a much higher Tc for overdoped

La2−xSrxCuO4 than what has been observed in both single crystals and thin films.” In this section we will show that our results are in good agreement with these new measurements, that in fact the absence of a gap feature is to be expected in d-wave superconductors with large scattering rates, and that the Born limit is a reasonable approximation to the scattering character of the out-of-plane defects in LSCO. Before diving into detailed calculations, we will first outline the expected behaviour of the conductivity in the clean limit for both Born and unitarity scatterers, as well as for the dirty limit. The first feature that is expected to be present in all cases is the existence of

104 Figure 6.12: Ref. [69]. Schematic of the conductivity spectrum in the superconducting (blue) and normal states (green). (a) Unitarity limit disorder scattering in the clean limit, with strong spin fluctuation scattering at higher frequencies. (b) Born limit disorder scattering in the clean limit, assuming weak spin fluctuation scattering. (c) Born limit disorder scattering in the dirty limit, again assuming weak spin fluctuation scattering. Note the complete absence of a spectroscopic gap feature.

a unique zero-frequency, zero-temperature limit of σ1(Ω), the so-called “universal” d-wave 2 2 1 conductivity σ00 = e N0vF ~/2π∆0, which is finite in the limit of vanishing disorder. The second feature that is common to the conductivity is the inelastic scattering of electrons by bosonic modes such as spin fluctuations. These excitations occur at frequencies much larger than the gap energy on the overdoped side of the phase diagram so they do not influence our model, and are in fact at frequencies above those measured in Ref. [157]. In general the presence of the pairing boson should result in a broad peak in the conductivity at 2∆0 + Ω0 where Ω0 is the frequency of the pairing boson, which typically occurs in the midinfrared. It should be noted that there are other proposals for the origin of the midinfrared peak, as discussed in Ref. [195]. For strong (unitarity) scatterers the residual density of states, γ = N(ω → 0) scales √ as γ ∼ ΓN ∆0, which creates a plateau in the density of states as shown in Figure 2.5. This leads to a relatively narrow peak at low frequency in the conductivity. Combining this with the two effects mentioned previously leads to the conventional behaviour of the d-wave conductivity pictured in Figure 6.12(a), which can be seen in Ref. [196] and is explored in further detail in Ref. [197]. For the weak scattering (Born) case, the shape of the conductivity spectra are far less well known. The conductivity must still tend to σ00 at low T , but this occurs at low frequencies in the clean Born limit as γ ∼ exp(−∆0/ΓN ). This is shown in Figure 6.12(b) where we have anticipated that spin-fluctuation scattering is weakened by the loss of low-frequency spin fluctuations in the overdoped cuprates. In

1 The expression for σ00 is not strictly independent of disorder as ∆0 varies slightly with varying impurity concentration.

105 Figure 6.13: Ref. [69]. Plasma frequency of La2−xSrxCuO4. Open circles are the experimental plasma frequencies inferred from Drude fits to the normal state conductivity in Ref. [157]. Solid curve shows the square of the plasma frequency calculated from the doping-dependent tight-binding dispersion of Ref. [2]. Inset: ratio of calculated and experimental plasma frequencies. We chose a renormalization factor of 0.3 based on this. the case of large scattering rate, shown in Figure 6.12(c), the picture is very different. The conductivity spectrum is completely dominated by disorder, and the states are spread out over a wide range of frequencies. The conductivity closely follows the normal state spectrum and therefore appears Drude-like even deep within the superconducting state. The gap feature, which was visible in the clean unitarity and Born limits as a small increase in the conductivity at Ω = ∆0 has completely vanished in the dirty Born limit. This is a consequence of the d-wave order parameter, for which disorder, especially in the dirty Born limit, rapidly smoothes out sharp gap features in both density of states and conductivity. In order to perform detailed calculations of the conductivity we need to take into account that the overdoped cuprates are not a simple noninteracting Fermi gas; interactions will renormalize properties such as mass and velocity. In interacting electron systems the plasma frequency (and hence the Drude weight) are reduced below the bare values calculated from band theory. The first effect that causes such a reduction is the local flattening of the dispersion near the Fermi surface, caused by many-body effects, which transfers spectral weight to higher frequencies. A common example of this is the so-called ‘phonon kink’ caused by electron-phonon coupling. The second is Fermi liquid effects, in which residual quasiparticle interactions induce a backflow that partially cancels the current response. As

106 Figure 6.14: Optical conductivity σ1(ν) of overdoped La2−xSrxCuO4. Each column shows normal state (T ≥ Tc, red) and low temperature (T = 1.6 K, blue) results for three different dopings. The shaded areas in between show the spectral weight that condenses at low temperature to form the superfluid. First column: experimental data from Mahmood et al. [157]. Columns 2 to 4: σ1(ν) calculated with Equation 6.9 from Ref. [69]. The calculated conductivities include the renormalization of the plasma frequency by a factor of 0.3. In all U cases the unitarity scattering parameter is set to ΓN = π K. Three different disorder models B B are presented: the second column has constant ΓN = 17π K; the third column adjusts ΓN B so that Tc0 is constant; and the fourth column has ΓN increasing linearly in doping by a factor of two between Tc = 27.5 K and Tc = 7 K. the theory we use in this chapter only considers states right at the Fermi level, the simplest way to account for these effects is to renormalize the Fermi velocities so that the calculated plasma frequency matches the experimentally measured one. The plasma frequencies measured from Drude fits to the THz conductivity are shown in Figure 6.13. Comparing the measured values to the ones calculated from the tight-binding 2 model of Ref. [2] reveals that the calculated ωp is larger than the experimental value by a factor of 3 to 4, consistent with previous work on LSCO which found that the effective mass is approximately 4me across the whole doping range [198]. Although there is some doping dependence of the renormalization factor (see inset of Figure 6.13), for simplicity a doping independent renormalization factor of 0.3 is chosen. This avoids fine-tuning the model, while revealing the effects that are intrinsic to the dirty d-wave model without competing effects.

Calculations of the optical conductivity of La2−xSrxCuO4 were performed using the exact same model as in the previous superﬂuid density section, and for the same three Tcs shown in the ﬁrst panel of Figure 6.14. The results in the second panel of Figure 6.14 show semiquantitatively similar features to the measured conductivity. The normal state curve is a pure Drude form with physical scattering rate Γtr = 2ΓN . The shaded area between

107 Figure 6.15: From Ref. [69]. Dimensionless optical spectral weights. Measured values (solid circles) from Ref. [157] determined by Drude ﬁts to the THz conductivity over the frequency range 0.3 − 1.7 THz. Lines are calculations performed at 1.6 K using the three disorder models described in the text. Su is uncondensed spectral weight from integrating σ1(Ω) and Sδ is the condensed spectral weight at 1.6 K. the normal state and superconducting state calculations represents the quasiparticles that condensed into the zero frequency δ function. Relaxing the constraint on having doping independent scattering rates, we also explored two other models: constant Tc0, and varying

ΓN by a factor of 2 between the critical doping and the p=0.22 point where Tc = 27.5 K, as shown in Figure 6.16. In each case Tc is set via the parabolic p–Tc relation. For the constant scattering rate and ×2 variable scattering rate models, Tc0 is calculated from the

Abrikosov-Gor’kov relations. For the case of constant Tc0 it is ΓN that is determined from the AG relations. The results of these calculations are shown respectively in the third and fourth panels of Figure 6.14, and are in quite good quantitative agreement with the measured conductivity. The doping dependence of the uncondensed dimensionless spectral weight can now be 2 R ∞ calculated by integrating the real part of the conductivity Su = 2 σ1(ω)dω. The πωp 0 condensed spectral weight can also be obtained from the superfluid density via Sδ = 2 ρs(T )/µ0ωp. The Ferrell-Glover-Tinkham sum rule implies that Su +Sδ = 1 at all temperatures, which we verified to ensure the calculations are correct. Figure 6.15 show the spectral weight as a function of Tc against the data from Ref. [157]. Good agreement between calculation and experiment was found for all three disorder models. As doping is increased towards the critical doping point, the condensed spectral weight vanishes despite the total carrier density remaining finite in the tight-binding model. The dimensionless quantities

108 Figure 6.16: From Ref. [69]. Upper panel: Scattering rate for the three disorder scenarios U outlined in the text with fixed ΓN = π K. Lower panel: Corresponding Tc and Tc0 for the three models. reported here are a more robust test of the models as they do not require any knowledge of the precise renormalization factor for the plasma frequency. The three models considered here capture most of the features of the optical conductivity and superfluid density. There are clearly some small discrepancies, especially near the more optimally doped samples, which reflect the simplicity of the models used in this study. We used as few free parameters as possible, and therefore do not capture the full doping dependence of the scattering rate, and we do not allow for a doping-dependent renormalization of the plasma frequency, despite the clearly varying renormalization in Figure 6.13. In order to perform more accurate calculations significant effort would need to be invested in understanding the actual sources of disorder and their scattering potentials. At the moment our assumption is that the unitarity scatterers are Cu vacancies, and the weak out-of-plane disorder is a combination of Sr dopants, whose number is reasonably well known, and O vacancies, whose number is poorly known. One region of the phase diagram where parameters of the weak-coupling model can be checked directly against experiment is near the overdoped critical point where Tc → 0. In the zero-temperature limit, inelastic scattering is irrelevant and pair-breaking can be attributed entirely to disorder. The transport scattering rate measured via THz spectroscopy can then be used to place a bound on ΓN . In the most overdoped sample of Ref. [157] the normal-

109 Figure 6.17: From Ref. [158]. Superfluid density calculated using the band structure of LSCO and a d-wave order parameter, for various impurity phase shifts. In each case the underlying scattering parameter has been adjusted to fix the zero-temperature superfluid fraction.

state spectrum has a Drude width of 1.75 THz, corresponding to Γtr = 84 K, placing a lower bound on the normal-state scattering rate such that ΓN ≥ 42 K. Equivalently, we can obtain a lower bound on the clean-limit transition temperature from the critical doping from AG theory. With Γc = 0.8819Tc0, we infer Tc0 ≥ 48 K. Against these experimental bounds, the values assumed in the weak-coupling model, ΓN = 18π = 56.5 K and Tc0(p → pc) = 64 K, are not unreasonable. We now address the concern of Mahmood et al. about the unphysical nature of the Born limit. First, Figure 6.17 shows the superfluid density calculated with different scattering phase shifts, but same scattering rate. Two constraints are imposed on the calculations: both Tc and ρs/ρs00 must remain the same, which is achieved by allowing Tc0 to vary. We see that essentially equivalent superfluid density curves can be obtained from scattering phase shifts as low as c = 2, far from the Born limit where c = ∞. The scattering phase shift can be expressed as c = 2/(πVimpN0), where Vimp is the impurity potential and N0 is the total density of states at the Fermi level. The density of states in the ARPES-derived −1 band structures is 7.5 eV for La2−xSrxCuO4, so impurities with potentials up to ∼ 0.1 eV (formula unit) are compatible with the Born limit results presented here. It is important to emphasize that the dirty d-wave explanation proposed here for the apparently surprising results of Božović and Mahmood does not imply that the overdoped side of the cuprate phase diagram can be described by a weakly interacting Fermi gas with

110 Figure 6.18: From Ref. [158]. Absolute superfluid density for La2−xSrxCuO4 calculated from the constant ΓN model outlined in the text. Open circles are the data from Ref. [1]. a d-wave pairing instability. In order to capture the quantitative behaviour of the superfluid density and conductivity we must take into account the large renormalization of the Fermi velocity. Born The calculations in this chapter are based on the scattering rates ΓN = 17π K and unitarity ΓN = π K, which were chosen solely on the basis of matching the qualitative shape of the superfluid density and linear Tc-ρs0 relation. Using this model, with only minor fine- tuning of the unitarity scattering rate, we later compared the absolute superfluid density in both LSCO and Tl2201 to the dirty d-wave BCS model, as shown in Figure 6.18. For Tl2201 we reduced the scattering rate by a factor of 3, to account for the fact that it is known to be a significantly cleaner material. The superfluid densities match quite well, and the difference in scattering rate naturally leads to different critical dopings for LSCO and Tl2201, which is observed experimentally [199, 200]. Overall, it is quite remarkable that such a consistent picture of the quasiparticle conductivity and absolute superfluid density can be obtained with virtually no fine-tuning, aside from the large plasma frequency renormalization, which is of similar magnitude to those reported in other cuprate systems.

6.3 Outlook

The dirty BCS model seems to explain many of the qualitative and quantitative features of the superﬂuid density and optical conductivity of LSCO. There are however some aspects of

111 the model which could be improved to provide further insight into the scattering mechanism in LSCO. The model presented in this chapter considers only point-like impurities, with s-wave scattering. A more realistic calculation would consider the eﬀects of small angle scattering, which we expect to be important in the cuprates as many of the scattering sites are away from the CuO2 superconducting layers. This is precisely the work that we are pursuing at the moment.

6.4 Contributions

My contribution to this project was to perform all calculations shown in this chapter. I started by focussing on the superfluid density, in response to the paper by Božović et al. [1]. When the follow-up paper by Mahmood et al. [157] appeared I built upon the code that I had created and extended it to do calculations of the conductivity of La2−xSrxCuO4. Following our second paper on this topic we decided to explore similar calculations in Tl2201, and also included the Sommerfeld coefficient, thermal conductivity, and field dependence of heat capacity in the calculation framework. These calculations were performed by Ulas Ozdemir and myself, and the resulting paper has been published recently [158].

112 Chapter 7

Conclusions

Finding a full theory of superconductivity that encapsulates all known phenomena, and explains high-Tc superconductivity, is one of the largest challenges in condensed matter physics today. A breakthrough in this search will likely come from the interplay between the discovery of new superconducting materials, the characterization and measurement of their physical properties, and the validation of current theories against new manifestations of superconductivity. To that end, we need to fully understand what phenomena are and are not contained within a given theory, which includes the essential but often overlooked and underappreciated eﬀects of impurities. Only then can we truly state that a new physical phenomenon has been observed that would provide clues to a new theory of superconductivity. As computing power increases it becomes incumbent on experimentalists to evaluate these more realistic models instead of toy models, and for theorists to explore experiment- driven parameter ranges. In this thesis, I have done precisely that, and have explored several aspects of unconventional superconductivity. I developed the tools needed to work with BCS theory in the presence of point-like impurities treated in the self-consistent t-matrix approximation. The theory was then applied to three materials that are of current interest:

FeSe, Nb-doped SrTiO3, and La2−xSrxCuO4. Using a novel helical resonator operating at frequencies below 1 GHz, I performed cavity perturbation measurements on the multiband iron-based superconductor FeSe. The electrodynamics reveal that FeSe is a particularly clean superconductor with an extremely long mean-free-path. Further, both superconducting bands have ﬁnite minima, without any sign of nodes. This strongly constrains which pairing symmetries should be considered in this material, and possibly all iron-based superconductors.

The electrodynamics of Nb-SrTiO3 were investigated using a stripline resonator between 2–20 GHz. Although there are at least two electronic bands present in all samples we measured, and one would expect two distinct gaps to be present, only a single spectroscopic gap could be observed. I proposed Anderson’s theorem as a resolution of this puzzle, and veriﬁed that the scattering rates in the sample are large enough to cause complete homogenization

113 of the two gaps. Single-band fits to the superfluid density then showed conclusively that the superfluid density is limited by the effects of impurities and gap magnitude. Tc therefore controls the superfluid density in Nb-SrTiO3. Our current understanding of the overdoped side of the cuprate phase diagram has been called into question through the measurements taken by of Božović et al. [1], and Mahmood et al. [157]. In response to their publications, I performed calculations of the superfluid density and conductivity for a dirty d-wave superconductor, starting from a tight-binding parametrization of the Fermi surface, and a many-body renormalization factor taken from the ratio of measured and calculated plasma frequencies. Most of the experimental features are replicated by the model using very few free parameters. As far as I am aware, this is the first time a quantitative approach that includes disorder has been taken to modelling superfluid density and conductivity in the cuprates. The results show that dirty d-wave BCS theory is a good description of the overdoped cuprates — at least on the hole-doped side of the phase diagram — as long as the strong many-body renormalization effects are properly accounted for.

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129 Appendix A

Evaluating elliptic integrals

If the Fermi surface is isotropic, many of the integrals used in dirty d-wave BCS theory can be reduced to elliptic integrals with complex arguments. It is therefore important to develop efficient ways to perform these integrals. Many computational packages have definitions of the elliptic integrals for real values of the function argument, but as far as I know, at the time of writing, only Mathematica provides elliptic functions for complex arguments. The complete elliptic integral of the first kind, usually referred to as the elliptic K function, is defined as

Z π/2 dθ K(m) = q . (A.1) 0 1 − m sin2(θ)

The complete elliptic integral of the second kind, usually referred to as the elliptic E function, is similarly deﬁned as

Z π/2 p E(m) = 1 − m sin2 θdθ . (A.2) 0

A brute-force approach to numerically evaluating these functions would be to perform the integrals as-is. There is however a much better way to approach the problem through the identity π 1 K(m) = √ , (A.3) 2 AGM 1, 1 − m where AGM is the arithmetic geometric mean of the two quantities in brackets. Similarly, the elliptic E function can be evaluated with

∞ ! X n−1 2 E(m) = K(m) 1 − 2 cn , (A.4) n=0

130 p 2 2 where cn = |an − gn|, or via the following equation, which is obtained by taking the derivative of K(m) and rearranging terms,

dK(m) E(m) = (1 − m) 2m + K(m) . (A.5) dm

A.1 Arithmetic geometric mean

The arithmetic geometric mean, AGM(x, y), is obtained from the convergence of the two sequences deﬁned by

a0 = x (A.6)

g0 = y (A.7) 1 an+1 = (an + gn) (A.8) √2 gn+1 = angn . (A.9)

These sequences converge to within a computer’s numeric precision very rapidly, often within less than ten iterations. When computing the AGM for complex arguments, care must be taken to ensure the proper branch cut is chosen, so that both the amplitude and the phase of the arithmetic and geometric means converge. Start from the phasor representation

iθn an = |an| e , (A.10)

iφn gn = |gn| e . (A.11)

A.2 Reference implementation

Mathematica is one of the few programming languages or mathematical libraries that are capable of calculating the elliptic integrals for complex valued input. We therefore provide a reference implementation in Python of the complex AGM, and the complete elliptic integrals of the ﬁrst and second kind.

131 import numpy as np def geometric_mean(a, b): return np.sqrt(np.abs(a)*np.abs(b))*np.exp(0.5j*(np.angle(a)+ np.angle(b))) def agm(a, b): prec= 1e-12 maxit= 10000

gn=a an=b

n=0 while np.abs(angn)> prec and n< maxit: tmp= geometric_mean(an, gn) an= (an+ gn)/2.0 gn= tmp n +=1

return gn def EllipticK(m): return np.pi/2.0/agm(1.0, np.sqrt(1.0- m)) def EllipticE(m): # Equation A.5 # Use a central difference to take the derivative dm= 1e-6 return (1- m)*(2*m*(EllipticK(m+dm)- EllipticK(m-dm))/(2*dm)+ EllipticK(m))

Figure A.1 shows the results of evaluating these functions over a small region of the complex plane centred around the origin. The values have been compared to the Mathematica implementation, and agree to better than 10−6.

132 2 2 EllipticK((kx + iky) ) EllipticE((kx + iky) ) Abs Abs 2.0 2.0 3.00 2.94

1.5 2.76 1.5 2.70

2.52 1.0 1.0 2.46

2.28 0.5 0.5 2.22

2.04 1.98

ky 0.0 ky 0.0

1.80 1.74 0.5 0.5 1.56 1.50

1.0 1.0 1.32 1.26

1.5 1.5 1.08 1.02

2.0 0.84 2.0 0.78 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 kx kx

Re Re 2.0 2.0 2.94 2.58

1.5 2.70 1.5 2.34

1.0 2.46 1.0 2.10

2.22 1.86 0.5 0.5

1.98 1.62

ky 0.0 ky 0.0

1.74 1.38 0.5 0.5 1.50 1.14

1.0 1.0 1.26 0.90

1.5 1.5 1.02 0.66

2.0 0.78 2.0 0.42 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 kx kx

Im Im 2.0 2.0 1.28 1.6

1.5 1.5 0.96 1.2

1.0 1.0 0.64 0.8

0.5 0.32 0.5 0.4

ky 0.0 0.00 ky 0.0 0.0

0.5 0.32 0.5 0.4

0.64 0.8 1.0 1.0

0.96 1.2 1.5 1.5

1.28 1.6 2.0 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 kx kx

Figure A.1: Contour plots of the magniture, real component, and imaginary component of the EllipticK and EllipticE functions.

133 Appendix B

Scanning tunnelling spectroscopy

Given the right conditions, a current can flow between two electrodes separated by several angstrom of vacuum, due to the process — familiar to all who have taken a quantum mechanics course — of quantum mechanical tunnelling. The amount of current that is able to tunnel across the vacuum barrier is related exponentially to the distance between the electrodes. By forming one electrode into an atomically sharp tip, and scanning it across the surface of the other electrode (the sample), one can obtain stunning subatomic resolution pictures. In 1982, Gerd Binnig and Heinrich Rohrer built the first scanning tunnelling microscope (STM) based on these principles [201]. They shared the 1986 Nobel Prize in Physics for the development of this important new tool with Ernst Ruska who developed the first electron microscope. The number of states available for electrons to tunnel from and into also influences the tunnelling current. When a bias voltage is applied between tip and sample the effective Fermi levels of the sample and tip are shifted relative to each-other and electrons will tend to flow from filled states to empty states. A measurement of the tunnelling current at constant distance from tip to sample, and with varying bias voltage, therefore provides an atomically resolved picture of the local density of states. Bardeen developed a time-dependent perturbation approach to calculating the tunnelling current. Consider the sample and tip to be isolated and unperturbed systems described by the Schrödinger equations:

Eµψµ = (T + US) ψµ , (B.1)

Eνχν = (T + UT ) χν , (B.2) where the unperturbed wavefunctions are non-zero only within their respective materials. The probability per unit time of an electron tunnelling from the sample to the tip is then given by Fermi’s golden rule

2π 2 wµν = |Mµν| δ (Eν − Eµ) , (B.3) ~ where Mµν is a matrix element that describes the coupling between tip and sample states. This is of course not the total current, as it only describes tunnelling between levels of

134 equal energy. Summing wµν over all possible tip and sample states, and multiplying by the electron charge gives 2πe X 2 I = |Mµν| δ (Eν − Eµ) . (B.4) ~ µν The sum can then be converted into an integral by the usual transformation involving the density of states weighted by the Fermi function, f, to account for thermal broadening. Tunnelling from the sample to the tip is then given by Z ∞ 4πe 2 Isample→tip = |Mµν| f()ρS() (1 − f( + eV )) ρT ( + eV ) , (B.5) ~ −∞ and from tip to sample, Z ∞ 4πe 2 Itip→sample = |Mµν| (1 − f()) ρS()f( + eV )ρT ( + eV ) . (B.6) ~ −∞ The two can be added to give the total tunnelling current Z ∞ 4πe 2 I = |Mµν| ρS()ρT ( + eV )(f( + eV ) − f()) . (B.7) ~ −∞ If we take the matrix element and density of states of the tip to be constants, the equation is further simplified to Z ∞ 4πe 2 I = |M| ρT (0) ρS()(f( + eV ) − f()) d . (B.8) ~ −∞ Taking the derivative of the tunnelling current with respect to the applied voltage eV then gives Z ∞ dI 4πe 2 df( + eV ) = |M| ρT (0) ρS() d . (B.9) dV ~ −∞ dV The derivative of the Fermi function is a window of width ∼ kBT at finite temperature, and is a delta function at zero temperature. The derivative of the tunnelling current is therefore either a convolution of the local density of states with a finite width distribution in the former case, or the density of states itself in the latter.

B.1 Temperature dependence of tunneling

As temperature is increased the Fermi function broadens which allows tunneling to take place over a broad range of energies. Figure B.1 shows the tunneling conductance for clean BCS s-wave and d-wave superconductors. The gap value is ﬁxed to the zero-temperature gap value for each superconducting pairing symmetry. The temperature at which the zero bias tunneling conductance deviates from zero should therefore be considered an upper bound. The absence of quasiparticle states over an energy range (−∆, ∆) in the s-wave case ensures that the tunnelling conductance at zero bias is eﬀectively zero for temperatures up to approximately 0.3∆. Due to the presence of quasiparticle states at all energies in the d-wave case, the tunneling conductance

135 at zero bias immediately lifts from zero at ﬁnite temperature. These observations lead to the conclusion that the tunneling spectra in Figure 5.3, which all have zero conductance at zero bias within experimental uncertainty, are nodeless spectra. For these nodeless gaps, the gap value must be at least 3 times the measurement temperature in order to ensure that the conductance is zero, which leads us to the conclusion that the gap is at least 1.2 K, based on the 0.4 K measurement temperature.

136 s-wave

T/Tc 0.01 0.05 0.1 0.5 1 ) b r A (

V d / I d

4 2 0 2 4 T (K)

d-wave

T/Tc 0.01 0.05 0.1 0.5 1 ) b r A (

V d / I d

4 2 0 2 4 T (K)

Figure B.1: Tunneling density of states as a function of temperature for s-wave and d-wave superconductors.

137